{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"Aggregated Source Repository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Yang, Dongyang","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"Date Available","value":"2024-10-01T18:37:18Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"Date Issued","value":"2024","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree (Theses)","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"Degree Grantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Atomically thin Transition metal dichalcogenide (TMD) with different stacking orders can exhibit distinct quantum phenomena when interacting strongly with external fields. Rhombohedral (R)-stacked TMDs, where neighboring layers are oriented in the same direction, can be created via chemical synthesis or artificial stacking with a small twist. Understanding how atomic registry determines the properties of TMD homo-bilayer is crucial for revealing exotic physics in 2D semiconductors. This thesis uses advanced optical spectroscopy to explore excitonic and correlated phenomena in both homogeneous and twisted R-stacked TMD homo-bilayers.\r\nIn R-stacked MoS\u2082 bilayers, we observe spontaneous electrical polarization induced by asymmetric interlayer coupling and Berry phase effects. Excitonic effects help reveal the electronic band structure, while a novel photovoltaic effect, driven by the depolarization field (DEP) in Gr\/R-MoS\u2082\/Gr heterostructures, shows potential for scalable optoelectronic applications. Using non-degenerate pump-probe photocurrent spectroscopy, we extract an intrinsic photocurrent speed of \u223c2 ps, contributing to the understanding of ultrafast carrier dynamics.\r\nThe out-of-plane electrical polarization in R-stacked MoS\u2082 can be switched under an external electric field through in-plane sliding motion, known as sliding ferroelectricity. We develop an optical method to probe domain wall motion in R-stacked MoS\u2082 homo-bilayers and trilayers, demonstrating that the pinning and de-pinning of domain walls drive the polarization switching. This leads to unconventional two-dimensional ferroelectric behavior, which could open new avenues for device engineering.\r\nFinally, we investigate the strongly correlated physics in R-stacked MoSe\u2082 bilayers with a small twist, revealing a series of correlated insulating states at both integer and fractional fillings in \u0393-valley flat bands. Contrary to continuum-band calculation, we observe a Mott-insulator state instead of a semi-metal on a half-filled honeycomb lattice, offering insights into the semi-metal-to-insulator transition. This thesis provides the latest understanding of optical and electronic properties in rhombohedral-stacked TMDs, making them a promising platform for exploring both fundamental condensed matter physics and future applications.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"Digital Resource Original Record","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/89310?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"Full Text","value":"Emergent optical and electronic properties in atomicallythin rhombohedral-stacked transition metaldichalcogenidesbyDongyang YangB.Sc., Nanjing University, China, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)September 2024\u00a9 Dongyang Yang, 2024The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Emergent optical and electronic properties in atomically thin rhombohedral-stacked transition metal dichalcogenidessubmitted by Dongyang Yang in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Physics.Examining Committee:Ziliang Ye, Associate Professor, Physics, University of British ColumbiaSupervisorAndrew Potter, Associate Professor, Physics, University of British ColumbiaSupervisory Committee MemberSabrina Leslie, Associate Professor, Physics, University of British ColumbiaSupervisory Committee MemberJoshua Folk, Professor, Physics, University of British ColumbiaUniversity ExaminerAlireza Nojeh, Professor, Electrical and Computer Engineering, University of BritishColumbiaUniversity ExaminerAdina Luican-Mayer, Associate Professor, Department of Physics, University ofOttawaExternal ExaminerAdditional Supervisory Committee Members:Andrea Damascelli, Professor, Department of Physics and Astronomy, Universityof British ColumbiaSupervisory Committee MemberiiAbstractAtomically thin Transition metal dichalcogenide (TMD) with different stacking or-ders can exhibit distinct quantum phenomena when interacting strongly with exter-nal fields. Rhombohedral (R)-stacked TMDs, where neighboring layers are orientedin the same direction, can be created via chemical synthesis or artificial stackingwith a small twist. Understanding how atomic registry determines the propertiesof TMD homo-bilayer is crucial for revealing exotic physics in 2D semiconductors.This thesis uses advanced optical spectroscopy to explore excitonic and corre-lated phenomena in both homogeneous and twisted R-stacked TMD homo-bilayers.In R-stacked MoS2 bilayers, we observe spontaneous electrical polarization in-duced by asymmetric interlayer coupling and Berry phase effects. Excitonic effectshelp reveal the electronic band structure, while a novel photovoltaic effect, drivenby the depolarization field (DEP) in Gr\/R-MoS2\/Gr heterostructures, shows poten-tial for scalable optoelectronic applications. Using non-degenerate pump-probephotocurrent spectroscopy, we extract an intrinsic photocurrent speed of \u223c2 ps,contributing to the understanding of ultrafast carrier dynamics.The out-of-plane electrical polarization in R-stacked MoS2 can be switchedunder an external electric field through in-plane sliding motion, known as slidingferroelectricity. We develop an optical method to probe domain wall motion inR-stacked MoS2 homo-bilayers and trilayers, demonstrating that the pinning andde-pinning of domain walls drive the polarization switching. This leads to uncon-ventional two-dimensional ferroelectric behavior, which could open new avenuesfor device engineering.Finally, we investigate the strongly correlated physics in R-stacked MoSe2 bi-layers with a small twist, revealing a series of correlated insulating states at bothiiiinteger and fractional fillings in \u0393-valley flat bands. Contrary to continuum-bandcalculation, we observe a Mott-insulator state instead of a semi-metal on a half-filled honeycomb lattice, offering insights into the semi-metal-to-insulator transi-tion. This thesis provides the latest understanding of optical and electronic prop-erties in rhombohedral-stacked TMDs, making them a promising platform for ex-ploring both fundamental condensed matter physics and future applications.ivLay SummaryUnderstanding the mechanisms underlying quantum phenomena contributes to thedesign of new quantum materials and the development of functional devices. Theway how atomically-thin materials are assembled determines the interactions be-tween atomic layers and therefore is crucial to the material\u2019s properties. This thesisinvestigates a series of novel optical and electronic phenomena observed in atom-ically thin rhombohedral-stacked transition metal dichalcogenides. By measuringthe color change of light reflection or emission, we can probe how electrons inthe material respond to external-field excitations at the microscopic level. Throughthese optical measurements, we reveal the intertwined relationship between exci-tons, emergent ferroelectricity, and correlated effects, all of which arise from theunique crystal structure of the material. The works presented in this thesis high-light the versatility of transition metal dichalcogenides in rhombohedral stackingas a platform for both fundamental scientific research and practical applications.vPrefaceThe work included in this thesis represents my scientific activity during my Ph.D.study at the University of British Columbia. For all the works presented here, Iam either a primary investigator or a key contributor. However, the research ontwo-dimensional materials is highly collaborative in nature and demands expertisein both experiments and theory. I have established a fruitful collaboration withresearchers from Ye, Franz, and Jones groups. In this section, I will list the detailedcontributions I and others have made for each scientific project.Chapter 3 This chapter details our basic understanding of the excitonic effectsin 3R-MoS2 homobilayer. The results are based on the publication \"Optically Prob-ing the Asymmetric Interlayer Coupling in Rhombohedral-Stacked MoS2\" by JingLiang\u2020, Dongyang Yang\u2020, and Jingda Wu\u2020, et.al.Physical Review X, 2022, 12(4):041005. I am listed as a co-first author. This work was conceived by Ziliang Ye,Jing Liang, and Dongyang Yang. The experiments were carried out by Jing Liang,Dongyang Yang, and Jingda Wu. Jerry I. Dadap assisted with all the measure-ments. The boron nitride crystal used in all experiments in this thesis was providedby Kenji Watanabe and Takashi Taniguchi from the National Institute for MaterialsScience, Tsukuba, Japan. The electrostatic model included in the publication wasbuilt by Dongyang Yang with assistance from Jing Liang. Jing Liang, DongyangYang, and Ziliang Ye analyzed the data. Ziliang Ye supervised the whole project.Chapter 4 This chapter contains two sections. The first section reports thediscovery of an efficient photovoltaic effect in 3R-MoS2 and reveals the micro-scopic mechanism behind it. The results of section 4.1 are based on the publi-cation \"Spontaneous-polarization-induced photovoltaic effect in rhombohedrallystacked MoS2\" by Dongyang Yang\u2020, Jingda Wu\u2020, et.al.Nature Photonics, 2022,vi16(6): 469-474. I am the leader and co-first author of this project. Ziliang Ye andDongyang Yang conceived this work. Most experiments including sample fabrica-tion, set-up building, and measurements were performed by Dongyang Yang andJingda Wu. Jing Liang performed the KPFM measurement with the assistance ofDongyang Yang. Kashif Masud Awan helped with the nanofabrication and TeriSiu contributed to the sample fabrication. Toshiya Ideue and Yoshihiro Iwasa pro-vided 3R-MoS2 bulk crystals from the University of Tokyo, Japan. Benjamin T.Zhou built the tight-binding model included in the publication under the supervi-sion of Marcel Franz. Dongyang Yang analyzed and interpreted the data under thesupervision of Ziliang Ye.The second section of Chapter 4 details the time-resolved measurements onintrinsic photocurrent speed. We developed a non-degenerated pump-probe pho-tocurrent spectroscopy to dis-tangle the competition between thermal and elec-tronic effects. The results of section 4.2 are from the publication \"Ultrafast re-sponse of spontaneous photovoltaic effect in 3R-MoS2\u2013based heterostructures\" byJingda Wu\u2020, Dongyang Yang\u2020, and Jing Liang\u2020, et.al. Science Advances, 2022,8(50): eade3759. This work was conceived by Ziliang Ye, Jingda Wu, and Dong-yang Yang. Jingda Wu built the experimental set-up with the assistance of Dong-yang Yang. Dongyang Yang and Jing Liang fabricated the samples. Jingda Wuand Dongyang Yang performed the measurements and collected the data with thehelp of Jing Liang, Yunhuan Xiao, and Jerry I Dadap. Jingda Wu and ZiliangYe analyzed the data. Jingda Wu built the two-temperature model to explain theobservations. Dongyang Yang built the tunneling junction model to explain thetemperature dependence of photocurrent. Max Werner, Evgeny Ostroumov, andDavid Jones developed an optical parametric oscillator as the light source. ZiliangYe supervised the project for both the experimental design and theoretical analysis.Chapter 5 This chapter also contains two sections. Section 5.1 presents an op-tical method to probe the sliding ferroelectricity in a natural 3R-MoS2 homobilayer.The results come from the publication \"Non-volatile electrical polarization switch-ing via domain wall release in 3R-MoS2 bilayer\" by Dongyang Yang\u2020, Jing Liang\u2020,and Jingda Wu\u2020, et.al. Nature Communications, 2024, 15(1): 1389. This workwas conceived and supervised by Ziliang Ye. Dongyang Yang and Jing Liang andfabricated the device. The experiments were carried out by Dongyang Yang, JingviiLiang, and Jingda Wu with the assistance of Yunhuan Xiao and Jerry I. Dadap. Zil-iang Ye and Dongyang Yang analyzed and interpreted the data. Section 5.2 revealsthe pathway of ferroelectric domain switching in 3R-MoS2 trilayers. This workentitled \"Resolving polarization switching pathways of sliding ferroelectricity intrilayer 3R-MoS2\" by Jing Liang\u2020, Dongyang Yang\u2020 is now under peer review.Ziliang Ye and Jing Liang conceived this work. Jing Liang and Dongyang Yangfabricated all the devices and carried out the measurements with the help of JingdaWu, Yunhuan Xiao, and Jerry I.Dadap. Ziliang Ye, Jing Liang and Dongyang Yanganalyzed the data.Chapter 6 This chapter details our discovery on the strongly correlated statesin twisted MoSe2 homobilayer. The results are latest and unpublished. I amthe leader of this project. Ziliang Ye and Dongyang Yang conceived this work.Dongyang Yang and Jing Liang fabricated the samples, built the experimental set-up, and collected the data with the assistance of Jingda Wu, Yunhuan Xiao, JerryI.Dadap, and Haodong Hu. Dongyang Yang analyzed the data and performed theHartree-Fock calculation of the Hubbard model on a half-filled honeycomb lat-tice under the supervision of Ziliang Ye. Nitin Kaushal performed the unrestrictedHartree-Fock calculation of twisted MoSe2 homobilayer under the supervision ofMarcel Franz.viiiContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . 11.1 Stacking order of two-dimensional materials . . . . . . . . . . . . 11.2 Atomically thin transition metal dichalcogenides . . . . . . . . . 31.2.1 TMD monolayer . . . . . . . . . . . . . . . . . . . . . . 31.2.2 TMD homo-bilayer with different stacking orders . . . . . 51.2.3 Asymmetric interlayer coupling and spontaneous electricalpolarization in 3R-MoS2 bilayer . . . . . . . . . . . . . . 82 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Optical spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 13ix2.1.1 Reflection Contrast Spectroscopy . . . . . . . . . . . . . 132.1.2 Magnetic Circular Dichroism (MCD) . . . . . . . . . . . 162.1.3 Photoluminescence Spectroscopy . . . . . . . . . . . . . 162.1.4 Non-degenerate pump-probe photocurrent spectroscopy . 172.2 Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 183 Excitonic effects in 3R-MoS2 homobilayer . . . . . . . . . . . . . . . 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Optically probing the asymmetric interlayer coupling . . . . . . . 253.3 Effective four band model . . . . . . . . . . . . . . . . . . . . . 363.4 Electrostatic modelling of a dual-gated device . . . . . . . . . . . 384 Spontaneous-polarization-induced photovoltaic effect in 3R-MoS2 bi-layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1 Probing the photovoltaic effect in 3R-MoS2 bilayer . . . . . . . . 434.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.2 Revealing the mechanism of photovoltaic effect in 3R-MoS2bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.3 Electrostatic modelling on the Gr\/3R-MoS2\/Gr device . . 564.1.4 Multiple photocurrent components in the Gr\/3R-MoS2\/Grdevice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.5 The mechanism of exciton dissociation . . . . . . . . . . 664.2 Ultrafast response of spontaneous photovoltaic effect in 3R-MoS2\u2013basedheterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.2 Ultrafast photocurrent response . . . . . . . . . . . . . . 694.2.3 Non-degenerate pump-probe photocurrent measurement . 794.2.4 Modelling the photocurrent temperature dependence . . . 824.2.5 Two-temperature model . . . . . . . . . . . . . . . . . . 865 Sliding ferroelectricity in 3R-MoS2 . . . . . . . . . . . . . . . . . . . 885.1 Non-volatile polarization switching in 3R-MoS2 homobilayer . . . 885.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 885.1.2 Optically probing the interlayer sliding motion . . . . . . 91x5.1.3 Intrinsic domain wall motion vs extrinsic interfacial chargetrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1.4 Joule heating effect . . . . . . . . . . . . . . . . . . . . . 1055.2 Resolving polarization switching pathways of sliding ferroelectric-ity in 3R-MoS2 trilayer . . . . . . . . . . . . . . . . . . . . . . . 1075.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.2 Polarization switching in 3R-MoS2 trilayer . . . . . . . . 1105.2.3 Stacking configuration of 3R-MoS2 trilayer . . . . . . . . 1125.2.4 Polarization switching pathway in 3R-MoS2 trilayer . . . 1185.2.5 Electrostatic simulation on dual-gated 3R-MoS2 trilayerwith ABC stacking . . . . . . . . . . . . . . . . . . . . . 1296 Strongly correlated insulating states in twisted MoSe2 homobilayer 1346.1 Introduction to moir\u00e9 superlattice and strongly correlated physics . 1346.2 Moir\u00e9 flat bands in \u0393 valley . . . . . . . . . . . . . . . . . . . . . 1386.3 Rydberg exciton sensing technique . . . . . . . . . . . . . . . . . 1406.4 Magnetic property for the \u03bd state . . . . . . . . . . . . . . . . . . 1506.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190A.1 A tight-binding model for 3R-MoS2 bilayer . . . . . . . . . . . . 190A.2 Mean field model for \u03bd =\u22122 on a half-filled honeycomb lattice . 194A.3 Theoretical model of the twisted MoSe2 bilayer . . . . . . . . . . 199A.3.1 Continuum model for MoSe2 homobilayer . . . . . . . . 199A.3.2 Estimation on the dielectric permittivity of device D1 ofChapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 200A.3.3 Unrestricted Hartree-Fock Calculation . . . . . . . . . . . 200A.3.4 Spin Model for \u03bd =\u22121 state . . . . . . . . . . . . . . . . 202B List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 204xiList of TablesTable 1.1 Values of C3-angular momentum quantum number mz for statesat K with AB-stacking. . . . . . . . . . . . . . . . . . . . . . 10xiiList of FiguresFigure 1.1 Stacking order and material\u2019s property . . . . . . . . . . . 2Figure 1.2 Basic introduction to TMD and exciton . . . . . . . . . . . 4Figure 1.3 TMD homobilayer of H-stacking and interlayer exciton . . 6Figure 1.4 TMD homobilayer of R-stacking and interlayer exciton . . 7Figure 1.5 Schematic of band structure in a 3R-MoS2 homobilayer . . 11Figure 2.1 Reflection contrast spectroscopy and Photoluminescence spec-troscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.2 Experimental set up . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.3 Ultrafast technique . . . . . . . . . . . . . . . . . . . . . . 19Figure 3.1 Schematic of asymmetric interlayer coupling in the 3R-MoS2 bilayer . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.2 Structure and optical image of a dual-gated 3R-MoS2 bi-layer device . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.3 Doping-dependent reflectance contrast spectrum of intralayerexcitons in the 3R-MoS2 bilayer . . . . . . . . . . . . . . . 27Figure 3.4 Electric-field-dependent reflectance contrast spectrum ofintralayer excitons in 3R-MoS2 bilayer at a fixed electrondoping density . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 3.5 Doping-dependent and electric-field-dependent photolumi-nescence spectrum of \u0393-K interlayer excitons in the 3R-MoS2 bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . 33xiiiFigure 3.6 Doping-dependent Reflection contrast and photolumines-cence spectrum of K-K intralayer excitons in the 3R-MoS2bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.7 Band structure and energy derivative of the reflectance con-trast spectrum of the 3R-MoS2 bilayer . . . . . . . . . . . 37Figure 3.8 Schematic of the dual-gated 3R-MoS2 bilayer device . . . 38Figure 4.1 Crystal structure and electronic band structure of 3R-MoS2bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.2 Photocurrent generation in Graphene (GR)\/3R-MoS2\/GRheterostructure . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.3 Mechanism of photocurrent . . . . . . . . . . . . . . . . . 51Figure 4.4 I-V curve of GR\/2H-MoS2 bilayer\/GR device . . . . . . . . 53Figure 4.5 Scalability of the photovoltaic effect in 3R-MoS2 . . . . . . 54Figure 4.6 Additional devices . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.7 Electrostatic simulation . . . . . . . . . . . . . . . . . . . . 58Figure 4.8 Fitted extrinsic contribution of photocurrent . . . . . . . . 61Figure 4.9 I-V curve of monolayer region in device C2 . . . . . . . . . 62Figure 4.10 Photo-thermal electric effect (PTE) Effect . . . . . . . . . . 63Figure 4.11 Exciton dissociation . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.12 Photocurrent generation in a GR\/3R-MoS2\/GR device . . . 70Figure 4.13 Optical image of the device . . . . . . . . . . . . . . . . . . 71Figure 4.14 Temperature-dependent measurements on BL region . . . 73Figure 4.15 Additional autocorrelation signals . . . . . . . . . . . . . . 74Figure 4.16 Pump-probe photocurrent measurement . . . . . . . . . . 77Figure 4.17 Pump-probe photocurrent measurements from the 4-layerregion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.18 Photocurrent power dependence under IR-pulse excitation 82Figure 4.19 Circuit model of the GR\/3R-MoS2\/GR photodetector . . . . 83Figure 4.20 Band alignment of GR\/3R-MoS2\/GR junction . . . . . . . 85Figure 4.21 Temperature dependence of photocurrent and resistance . 85Figure 4.22 Graphene electronic heat capacity . . . . . . . . . . . . . 87xivFigure 5.1 Coupling interlayer excitons with stacking orders . . . . . 89Figure 5.2 Observing polarization switching through quantum Starkshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 5.3 Polarization switching of sample 2 . . . . . . . . . . . . . . 94Figure 5.4 Non-volatile polarization switching enabled by domain wallrelease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 5.5 Polarization switching of sample 1 at room temperature . . 97Figure 5.6 Pre-existing domain walls and polarization switching in sam-ple 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 5.7 Observing polarization switching through intralayer exci-tons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 5.8 Hysteresis of intralayer excitons . . . . . . . . . . . . . . . 102Figure 5.9 Domain switching vs Interface charge trapping . . . . . . 105Figure 5.10 Schematic of an ideal model for the domain switching . . 106Figure 5.11 Polarization switching in a 3R-MoS2 trilayer involving ABAstacking as the intermediate state . . . . . . . . . . . . . . 109Figure 5.12 Determining the stacking configuration of the initial andfinal states during the polarization switch . . . . . . . . . 111Figure 5.13 RC spectrum with finite doping without field . . . . . . . 114Figure 5.14 Determining the stacking configuration of the intermediatestate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.15 Screening effects on excitonic peaks . . . . . . . . . . . . . 117Figure 5.16 Resolving polarization switching pathways with ABA andBAB stacking as intermediate states. . . . . . . . . . . . . 119Figure 5.17 Mapping of stacking orders under various electric fields. . 120Figure 5.18 Schematics of the polarization switching pathway shown inFig.5.11(b). . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.19 Direct transition transition from ABC to CBA stacking . . 122Figure 5.20 Half-way transition transition from ABA to CBA stacking 123Figure 5.21 Half-way transition transition from ABC to ABA stacking 124Figure 5.22 Multiple-cycle electric field scanning from the same deviceshown in Fig.5.11. . . . . . . . . . . . . . . . . . . . . . . . 125xvFigure 5.23 Multiple-cycle electric field scanning from the other newlyfabricated device. . . . . . . . . . . . . . . . . . . . . . . . 126Figure 5.24 Single-gate-dependent polarization switching pathways. . . 128Figure 5.25 Simulation on an ABC-stacked 3R-MoS2 trilayer . . . . . 133Figure 6.1 Moir\u00e9 superlattice and Mott-Hubbard physics . . . . . . . 136Figure 6.2 Rydberg exciton sensing technique . . . . . . . . . . . . . 140Figure 6.3 Honeycomb lattice of a twisted MoSe2, device structure,electronic band structure and reflection contrast spectrumof sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure 6.4 \u0393 valley flat bands and correlated insulating states of tMoSe2bilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure 6.5 Reflection contrast and its first derivative spectra of 2s ex-citon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Figure 6.6 Experimental evidence of \u0393 valley. . . . . . . . . . . . . . . 146Figure 6.7 Additional data for another position P2 of device D1. . . . 147Figure 6.8 Temperature dependence of correlated insulating states. . 148Figure 6.9 Additional temperature dependence data of \u03bd =\u22121 state. . 149Figure 6.10 Additional temperature dependence data of correlated in-sulating states. . . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 6.11 Generalized Wigner crystals . . . . . . . . . . . . . . . . . 151Figure 6.12 Magnetic property of the charge transfer insulator at \u03bd=-1. 153Figure 6.13 Theoretical simulation on the spin-spin interaction at \u03bd =\u22121.154Figure A.1 Calculated band structure . . . . . . . . . . . . . . . . . . 192Figure A.2 Momentum distribution of Pk(k) . . . . . . . . . . . . . . . 192Figure A.3 Mean field model . . . . . . . . . . . . . . . . . . . . . . . 195Figure A.4 Simulation parameters . . . . . . . . . . . . . . . . . . . . 203xviList of AbbreviationsAQN Azimuthal quantum numberBOL Bolometric effectBPVE Bulk photovoltaic effectDEP depolarization fieldDW Domain wallEQE External quantum efficiencyGR GrapheneMCD Magnetic circularly dichroismMBZ mini-Brillouin ZonePL PhotoluminescencePTE Photo-thermal electric effectPV Photovoltaic effectQAM Quasi-angular momentumRC Reflection contrastSHG Second Harmonic GenerationTMD Transition metal dichalcogenide2D Two-dimensionalxviiAcknowledgmentsThe journey to obtaining my Ph.D. has been one of the most exciting and challeng-ing periods of my life, and it would not have been possible without the support andencouragement of those around me.To my family: This thesis is dedicated to my parents, Zhongmei Liu and LiangYang. Thank you for raising me and supporting me every step of the way. I alsoextend my gratitude to my grandfathers, Runcai Yang and Xiangdong Yang, forintroducing me to physics and opening the door to the world of science. I wouldalso like to thank my girlfriend, Xinyao Fan, Ph.D. in statistics, for your invaluablecompanionship and support.To my supervisor, Ziliang Ye: Ziliang, you are an exceptional scientist and atrue role model. Over the past five years, you have taught me how to think andwork like a physicist, always offering insightful and helpful suggestions no mattermy challenges. Thank you for allowing me to pursue my Ph.D. in such a wonderfulenvironment, for giving me the freedom to explore my interests while keeping meon the right track, and for always believing in me and pushing me to succeed. Youmake me feel safe during my Ph.D. research. It has been an honor to work withyou.To Jingda Wu: Jingda, as the first postdoctoral fellow I worked with at QMI,your hands-on approach, and excellence in experimental skills have taught me agreat deal. You make me realize how to become a good researcher. It has been apleasure to collaborate with you.To Jing Liang: Jing, you are not only an excellent scientist but also a talentedartist. This thesis would not have been possible without your help. Thank you forteaching me to work in an organized way. It is really a pleasure to work with youxviiiand understand many interesting phenomena together in the lab.To the senior member, Jerry I. Dadap: Jerry, you are a kind, experienced, andknowledgeable scientist. Thank you for always encouraging me, sharing your valu-able experiences, and assisting with paper writing. Your time and effort have beengreatly appreciated.To my collaborators: I extend my gratitude to my theoretical collaboratorsBenjamin T. Zhou, Nitin Kaushal, Marcel Franz, and Joerg Rottler. Thank you allfor your efforts and continuous support.To my friends and colleagues: Yunhuan Xiao, Teri Siu, Manabendra Kuiri,Jiabin Yu, Yuan Xie, Haodong Hu, Tianxiao Niu, Melisa Ozen, and Xin Xin, HeMa, it has been a pleasure working alongside you and learning from each of you.xixChapter 1Introduction and motivation1.1 Stacking order of two-dimensional materialsFinding a new solution to further increase transistor density on chips is crucial forthe next information revolution, especially in the post-Moore era. The emergenceof two-dimensional van der Waals (vdW) materials offers such an opportunity dueto their reduced dimensionality, various atomic structures, and different electronicand optical properties. To achieve real applications of these vdW materials, thecapability of engineering their properties flexibly is highly desirable in both thescientific community and industry. One promising approach is to control thesematerials\u2019 properties by engineering the stacking orders between their atomicallythin layers. Due to the weak vdW forces in the out-of-plane direction, the stackingorder can be easily manipulated through lateral displacement or interlayer rotation,both of which typically require minimal energy.Adjusting the stacking order between vdW layers can significantly modify in-terlayer coupling1, spin-orbit interactions2, and electronic correlations3, poten-tially giving rise to exotic phases of matter. For instance, when the stacking or-der of few-layer graphene changes from Bernal to Rhombohedral stacking, thedispersive electronic bands become flattened, enhancing electron-electron correla-tions4;5. This stacking-order-induced change in the band structure leads to a varietyof novel phases, including unconventional superconductivity6, correlated insula-tors7, and fractional quantum Hall states8 (Fig.1.1(a)-(c)). The ability to precisely1ABABABACBAa b cde fFigure 1.1: a. Bernal (ABA) and rhombohedral (ABC) stacking in graphenefew-layer5. b. Emergent superconductivity in rhombohedral-stackedtrilayer graphene6 (Reproduced from Ref.6). c. Fractional quan-tum Hall state in rhombohedral-stacked graphene8 (Reproduced fromRef.8). d. Monoclinic and rhombohedral stacking in bilayer CrI3. e,f.Magnetic field response for bilayer and five-layer regions before (e) andafter (f) applying pressure10. d-f are reproduced from Ref.10.control stacking order in graphene systems offers valuable insights for designingnext-generation quantum materials.Changes in stacking order can also modify long-range magnetic orders in two-dimensional magnetic materials, such as Chromium triiodide (CrI3). In atomicallythin CrI3, there are two common stacking orders: monoclinic and rhombohedral9.These two stacking configurations can be inter-converted under hydrostatic pres-sure. When the stacking order transitions from rhombohedral to monoclinic, theinterlayer magnetic coupling shifts from a ferromagnetic to an antiferromagneticstate10 (Fig.1.1(d)-(f)). The coupling between stacking order and magnetic stateoffers promising potential for the development of high-performance spintronic de-vices.Controlling the stacking orders in a single transistor device based on Two-2dimensional (2D) materials can expand the family of vdW materials and potentiallylead to the development of multi-functional devices. In this thesis, we will explorehow stacking orders influence the optical and electronic properties of atomicallythin Transition metal dichalcogenide (TMD). A deeper understanding of the mi-croscopic behavior of electrons in transition metal dichalcogenides with differentstacking orders can drive advances in both fundamental physics and optoelectronicapplications.1.2 Atomically thin transition metal dichalcogenides1.2.1 TMD monolayerTransition metal dichalcogenides (TMDs) are a class of materials that have gainedsignificant attention in the condensed matter physics community since their atom-ically thin counterparts were first examined in the lab11;12. A monolayer of TMDsconsists of transition metal atoms (M) sandwiched between layers of chalcogenatoms (X) (Fig.1.2(a)), forming a honeycomb lattice with broken inversion sym-metry. This symmetry breaking opens a band gap at the corners of the BrillouinZone (K and K\u2032 points). Unlike monolayer graphene, the first two-dimensionalmaterial isolated experimentally13, TMDs are semiconducting, exhibiting a directband gap near the K (K\u2032) valley (Fig.1.2(b)) when thinned to the monolayer limit14.The direct band gap of monolayer TMDs leads to strong light-matter interac-tions, making them a promising platform for both fundamental studies of two-dimensional (2D) systems13;15 and potential optoelectronic applications16;17. Oneof the most important features that make TMDs appealing in the atomically thinlimit is the reduced dielectric screening in 2D systems, which results in the for-mation of tightly bound excitons (electron-hole pairs) with binding energies in therange of hundreds of meV18. Because both direct and exchange Coulomb interac-tions are enhanced in the 2D limit19, monolayer TMDs provide an ideal platformfor studying many-body physics. In recent years, a variety of complex excitonicspecies have been observed, including charged excitons (trions)20, bi-excitons21,and charged bi-excitons22. These emergent quasi-particles underscore the signifi-cance of interactions in monolayer TMD systems.3a bc dFigure 1.2: a. Top and side view of the crystal structure of monolayer TMD.M is the transition metal atom while X represents the chalcogen atom. b.Band structure of a TMD monolayer calculated by DFT23(Reproducedfrom ref.23). c. Rydberg exciton states probed by reflection contrastspectroscopy in monolayer WS2 24 (Reproduced from ref.24). d. Exci-ton photoluminescence from a monolayer MoSe2 25 (Reproduced fromref.25) .On the other hand, excitonic properties can also provide insights into the elec-tronic properties of the material. The crystal symmetry and orbital character ofthe bands in monolayer TMDs lead to unique chiral optical selection rules14;26. Inmonolayer TMDs, the orbitals in the center of the hexagonal Brillouin zone (the \u0393valley) feature strong hybridization between the pz orbitals of the chalcogen atomsand the dz2 orbitals of the transition metal atoms27;28. In contrast, the conductionand valence band states at the K and K\u2032 valleys are primarily localized in the tran-sition metal atoms. At the corners of the Brillouin zone, the valence band states aremainly composed of dx2\u2212y2\u00b1 idxy orbitals, while the conduction band states are pri-marily dz2 29. The Bloch wave functions of the valence band states remain invariant,whereas the conduction band states transform with an angular momentum of \u00b11under the C3 operator. As a result, left-handed (right-handed) circularly polarized4light couples exclusively to transitions at the K (K\u2032) valley30. Additionally, thestrong spin-orbit coupling induced by the transition metal atoms produces a spin-valley locking effect30;31, where electrons at the Brillouin zone corners becomeboth valley- and spin-polarized when excited by circularly polarized light.Probing the excitonic states in monolayer TMDs can be achieved through var-ious optical methods. Transitions from the vacuum to exciton energy states canbe observed using absorption, reflectance, or photoluminescence (Photolumines-cence (PL)) spectroscopy32. For example, Fig.1.2(c) displays a series of Rydbergexciton states in the reflection contrast spectrum of a WS2 monolayer24. The ex-citon binding energies in monolayer TMDs can be approximated using the hydro-genic series33. Photoluminescence is another common technique used to probeexciton states25 (Fig.1.2(d)). Due to the direct band gap at the K and K\u2032 points,photon emission at cryogenic temperatures is highly pronounced. In the followingchapters (2-5) of this thesis, we will demonstrate that combining these fundamen-tal tools with other experimental techniques not only enables the measurement ofexciton behavior in atomically thin systems but also reveals the rich electronicproperties of atomically thin TMDs.1.2.2 TMD homo-bilayer with different stacking ordersMany efforts have been made to uncover the electronic and excitonic propertiesof monolayer TMDs. However, a detailed understanding of monolayer TMDs doesnot directly translate to a straightforward understanding of the physics when twomonolayers are stacked to form a homo-bilayer. In this bilayer system, a newdegree of freedom, the layer index, is introduced, which can give rise to novel phe-nomena. The exploration of the distinct physics unique to the homo-bilayer systemis a key motivation behind this work. This approach aligns with the fundamentalphilosophy of condensed matter physics: \"More is different34\".Two different stacking orders naturally exist in TMD homo-bilayers. The mostcommon one is hexagonal stacking (H-stacking), where the two monolayers arestacked in an anti-parallel configuration (Fig.1.3(a)). H-stacked TMD homo-bilayerscan be readily obtained via mechanical exfoliation from a bulk crystal in the 2Hphase, which has led to extensive investigation11;37 shortly after the discovery of5a bc dFigure 1.3: a. Top and side view of crystal structure of a TMD homobilayerin H-stacking. M is the transition metal atom while X represents thechalcogen atom. b. Temperature dependence of reflection contrast inH-stacked MoS2 homobilayer35 (Reproduced from ref.35) c. Schematicof Interlayer transitions H-stacked MoS2 homobilayer, according to thetheoretical analysis36 d. Quantum stark effect of interlayer exciton inH-stacked MoS2 homobilayer36.c and d are reproduced from ref.36monolayer TMDs.Compared to the reflection contrast spectrum of a monolayer (Fig.1.2(c)), thespectrum changes significantly with the addition of an extra layer. A new excitonfeature emerges between the well-known A and B exciton peaks (Fig.1.2(b)) andpersists even at room temperature35. This new feature is attributed to an interlayerexciton, where the electron and hole are spatially separated2.The band structure of a 2H-MoS2 homo-bilayer consists of two copies of themonolayer band structure with opposite spin and valley indices, resulting from a180\u25e6 rotation (Fig.1.3(c)). Due to interlayer coupling, the interlayer transitionsstrongly hybridize with the intralayer B exciton35;38, acquiring a large oscillatorstrength. The interlayer nature of this exciton was confirmed by further studies,6a bc2H 3RdFigure 1.4: a. Top and side view of crystal structure of a TMD homobilayerin R-stacking. M is the transition metal atom while X represents thechalcogen atom. b. Optical microscopic image of the surface of 2H and3R MoS2 crystal40(Reproduced from ref.39). c. Second Harmonic Gen-eration (SHG) mapping of an exfoliated 3R-MoS2 terrace sample41d.Layer dependence of SHG in both 2H- and 3R-MoS2 41. c and d are re-produced from ref.40which have demonstrated that the exciton transition energy can be tuned by an out-of-plane electric field, known as the quantum Stark effect (Fig.1.3(d))36. Thesefindings highlight a fundamental fact: interlayer interactions must be considered tounderstand the excitonic and electronic properties of TMD homo-bilayers fully.The second stacking order in TMD homo-bilayers is rhombohedral stacking (R-stacking), where the monolayers are stacked in a parallel arrangement. One layeris laterally shifted by one-third of a unit cell relative to the other, completing a unitcell after three shifts. This stacking order is often referred to as 3R (Fig.1.4(a)).Rhombohedral stacking in TMDs was first synthesized by Dr.Wildervanck in 197039,and later re-discovered by Dr. Iwasa\u2019s group in 201440. Fig.1.4(b) shows opticalimages of the surface of MoS2 crystals in both 2H- and 3R-phases, where distinctscrew dislocations reflect their differing crystal symmetries40.7Unlike the H-stacked homo-bilayer, rhombohedral-stacked TMD homo-bilayerslack inversion symmetry. This unique crystal structure has drawn the attention ofthe non-linear optics community. Enhanced second harmonic generation (SHG)has been observed in 3R-MoS2 multi-layer terraces (Fig.1.4(c))41. The SHG signalsgenerated by the two layers interfere constructively, leading to a four-fold enhance-ment in SHG in the 3R-MoS2 bilayer. Moreover, the SHG efficiency scales quadrat-ically with the number of layers, suggesting that rhombohedral-stacked TMDs holdpotential for scalable, practical applications (Fig.1.4(d)).After identifying these effects tied to the crystal structure, a key question arises:How does a 3R-stacked TMD homo-bilayer differ from a monolayer or an H-stacked homo-bilayer?To address this question, we have conducted systematic optical measurementsof the excitonic effects in the 3R-MoS2 bilayer, as presented in Chapter 3 of thisthesis. Through these measurements, we reveal a unique asymmetric interlayercoupling between the conduction band of one layer and the valence band of theother. This asymmetric coupling induces a redistribution of charge between thetwo layers, resulting in spontaneous electrical polarization along the out-of-planedirection. Before delving into the experimental results, we will first provide a brieftheoretical overview of the electronic band structure of the 3R-MoS2 homo-bilayer.1.2.3 Asymmetric interlayer coupling and spontaneous electricalpolarization in 3R-MoS2 bilayerThe conduction and valence band states at the three-fold (C3)-invariant K-pointsin monolayer MoS2 exhibit different C3-angular momentum quantum numbers,also known as quasi-angular momentum (Quasi-angular momentum (QAM)), withdiscrete values of mz = 0,\u00b11. For a monolayer transition metal dichalcogenide(TMD), the conduction band basis |dz2\u27e9 and valence band basis |dx2\u2212y2 \u00b1 idxy\u27e9 areeigenvectors of the C3 operator, with eigenvalues of 1 and e\u00b1i2\u03c03 . Depending onwhether the valley index is K or K\u2032, the mz of the valence-band Wannier func-tion is \u22121 or +1, respectively, while mz is zero for the conduction band basis, aspreviously reported1.Using the basis of the conduction band and valence band, we can construct theBloch wave function at K valley. N is the number of unit cells and R j denotes the8position of transition atom (Mo) in the jth unit cell.\u27e8r|c,K\u27e9= 1\u221aN\u2211je\u2212iK(r\u2212R j) \u27e8r\u2212R j|dz2\u27e9\u27e8r|v,K\u27e9= 1\u221aN\u2211je\u2212iK(r\u2212R j) \u27e8r\u2212R j|dx2\u2212y2 + idxy\u27e9 (1.1)In addition to the orbital wave functions \u27e8r\u2212R j|dz2\u27e9 and \u27e8r\u2212R j|dx2\u2212y2 + idxy\u27e9,the plane wave e\u2212iK(r\u2212R j) acquires an additional Berry phase of e\u2212iK(C3R j\u2212R j) underthe C3 rotation operation. This Berry phase depends on the choice of the rotationcenter, which could be the Mo atom, the S atom, or the center of the honeycomblattice, resulting in phase factors of 1, e\u2212i2\u03c03 , and e+i2\u03c03 , respectively. Therefore,the total angular momentum must account for both the orbital contribution and theBerry phase of the plane wave.In the rhombohedral stacking (R-stacking) configuration, the relative lateralshift between the two layers modifies these angular momentum quantum numbersat the K valleys. The relative interlayer displacement along the armchair directionresults in two inequivalent stacking configurations: AB or BA stacking, where theMo atom in the top layer aligns directly over the S atom in the bottom layer forAB stacking40;41. Since these two stacking domains are inversion symmetric witheach other, we will focus on the AB stacking domain.In the AB-stacking configuration (Fig.1.5(a)), if we choose the Mo atom in thetop layer as the rotation center, the top-layer wave function remains unchangedunder the C3 operation. Consequently, the rotation center for the bottom layerbecomes the center of the honeycomb lattice. The bottom-layer wavefunction ac-quires a 2\u03c03 Berry phase under the same operation1. Therefore, from the perspectiveof the top layer, the states at K in the bottom layer gain an additional angular mo-mentum of +1. This results in the conduction band state |c,K, t\u27e9 in the top layerand the valence band state |v,K,b\u27e9 in the bottom layer sharing the same mz = 0.Conversely, the conduction band state |c,K,b\u27e9 in the bottom layer has mz = +1,and the valence band state |v,K, t\u27e9 in the top layer has mz = \u22121 (see Table 1.1).Consequently, the interlayer coupling at +K is non-zero only between |c,K, t\u27e9 and|v,K,b\u27e9 (Fig.1.4(b)), while |c,K,b\u27e9 and |v,K, t\u27e9 remain decoupled. Specifically, inthe basis9|c,K, t\u27e9 , |v,K,b\u27e9 , |c,K,b\u27e9 , |v,K,b\u27e9, the effective Hamiltonian at K is:Heff,K =\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0Ect ,K 0 0 t\u22a50 Evt ,K 0 00 0 Ecb,K 0t\u22a5 0 0 Evb,K\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb . (1.2)Here, Ect ,K = Ecb,K = Ec and Evt ,K = Evb,K = Ev denote the energies at theconduction and valence band edges in each decoupled monolayer, and t\u22a5 representsthe effective interlayer coupling strength. From equation (1.2), it is evident that theasymmetric interlayer coupling causes level repulsion only between |c,K, t\u27e9 and|v,K,b\u27e9, leading to a splitting of the layer degeneracy as schematically illustratedin Fig.1.5(b).Table 1.1: Values of C3-angular momentum quantum number mz for states atK with AB-stacking.State at K mz|c,K, t\u27e9 0|v,K, t\u27e9 \u22121|c,K,b\u27e9 +1|v,K,b\u27e9 0The asymmetric interlayer coupling leads to a type-II band alignment at the Kand K\u2032 points, arising from the Berry phase of the wavefunction winding. ThisBerry phase can be understood more intuitively by considering the bulk limit,where a large number of MoS2 monolayers are rhombohedral stacked along thez-direction with periodic boundary conditions. With momentum kz as a good quan-tum number, the corresponding effective bulk Hamiltonian at K is:He f f ,+K(kz) =[Ec t\u22a5e\u2212ikzdzt\u22a5eikzdz Ev](1.3)10bv,+K,ttop bottomav,+K,bt\u2534AB stackingc,+K,t c,+K,b c,tc,bv,tv,byx\u03b41\u03b42\u03b43Figure 1.5: a, Schematic of AB-stacking configuration in 3R-MoS2 bilayer.b,Schematic band diagram at K in two decoupled monolayers (left panel)and bilayer with rhombohedral-stacking (right panel). The asymmetricinter-layer coupling causes the energy level repulsion between |c,K, t\u27e9and |v,K,b\u27e9 and splits the layer degeneracy.Equivalently, the effective bulk Hamiltonian can also be expressed asHe f f ,+K =Ec+Ev2\u03c40+Ec\u2212Ev2\u03c4z+ t\u22a5 cos(kzdz)\u03c4x+ t\u22a5 sin(kzdz)\u03c4y, (1.4)where the Pauli matrices \u03c4i=0,x,y,z are defined in the conduction-valence band space.This Hamiltonian resembles the classic Su-Schrieffer-Heeger (SSH) model42,with an added sublattice asymmetry (Ec \u0338= Ev) in each unit cell, resulting in anontrivial Berry phase. Analytic calculations show that the Berry phase (\u03a60) ofHe f f ,K(kz) is given by \u03a60 = \u03c0(1\u2212 Ec\u2212Ev2\u221at2\u22a5+(Ec\u2212Ev)2\/4), reflecting the origin of theasymmetric interlayer coupling43.Due to the asymmetric interlayer coupling, states in the lower valence bandare a superposition of states from different layers, although the wavefunction re-mains highly localized in the bottom layer due to the significant energy differencebetween the original unperturbed states |c,K, t\u27e9 and |v,K,b\u27e9. This wavefunctionmixing causes the Wannier center of the lower valence band to shift slightly to-ward the top layer. In contrast, the higher valence band at K is formed by states inthe top layer only. Consequently, the overall Wannier center of all occupied valenceband states at K is displaced from the midpoint between the two layers (closer tothe top layer), indicating a spontaneous electrical polarization in the out-of-planedirection, as described by modern Berry phase theory of polarization44.This spontaneous electrical polarization induces a built-in electric field, which11is anti-parallel to the depolarization field direction, further increasing the conduc-tion and valence band offset at the K and K\u2032 valleys. Utilizing this spontaneouspolarization, we will demonstrate a strong photovoltaic effect in a heterostructureof Graphene\/3R-MoS2\/Graphene (Gr\/3R-MoS2\/Gr) in the first section of Chapter4. This effect scales with thickness, suggesting the potential of rhombohedral-stacked TMDs for fundamental optoelectronic applications. In the second sectionof Chapter 4, we use ultrafast non-degenerate pump-probe spectroscopy to resolvethe real-time response of the photocurrent in the Gr\/3R-MoS2\/Gr device, revealingan intrinsic photocurrent speed of 2 ps and highlighting the competition betweenthermal and electronic effects.Chapter 5 will explore the switching of out-of-plane polarization by applyinga vertical electric field in both rhombohedral-stacked homo-bilayers and trilayers.This results in emergent 2D ferroelectricity, termed sliding ferroelectricity, wherepolarization switching relies on domain wall motion rather than domain nucleation,differing from traditional ferroelectric materials.Chapters 3-5 offer the latest insights into the rhombohedral-stacked MoS2homo-bilayer, demonstrating distinct excitonic and electronic behaviors comparedto monolayers and 2H bilayers. The emergence of unconventional interfacial fer-roelectricity represents a new frontier in 2D materials research. Our contributionsto the field underscore the versatility of rhombohedral-stacked TMDs as a platformfor both fundamental research and practical applications.The TMD homobilayer, whether in hexagonal or rhombohedral stacking, is anintrinsic example where the stacking order remains spatially homogeneous on amacroscopic scale (typically with domain sizes larger than 1 \u00b5m). This is becausethe homo-bilayers discussed in Chapters 3-5 are primarily obtained through me-chanical exfoliation from chemically synthesized crystals. However, when twomonolayers of TMD are artificially stacked with a small twist angle, the stackingorder can vary continuously from point to point. This is now known as moir\u00e9TMD homobilayer. Chapter 6 will explore our discovery of strongly correlatedinsulating states in rhombohedral-stacked MoSe2 homo-bilayer with a small twistangle. The physics of this system is markedly different from that of homogeneousstacking, opening new avenues for research and application.12Chapter 2Experimental techniques2.1 Optical spectroscopyWe employ various optical spectroscopy techniques to probe the electronic prop-erties of 2D materials, including reflection contrast spectroscopy, photolumines-cence spectroscopy, and non-degenerate pump-probe photocurrent spectroscopy.Additionally, magnetic circular dichroism (Magnetic circularly dichroism (MCD))is used to measure magnetization in 2D materials. This section introduces theseexperimental techniques.2.1.1 Reflection Contrast SpectroscopyReflection contrast (Reflection contrast (RC)) spectroscopy is commonly used toprobe excitonic features in transition metal dichalcogenides (TMDs). RC is definedas the difference in reflected light intensity between the sample (R) and the sub-strate (Rs), normalized by the reflection of the substrate:RC =R\u2212RsRs(2.1)The measured reflection contrast is proportional to the sample\u2019s absorption if thesample is atomically thin, placed on a transparent substrate, and illuminated withweak light. A simplified model (Fig.2.1(a)) shows the sample on a transparent,semi-infinite substrate, dividing space along the z-axis into three regions: vacuum13(1), sample (2), and substrate (3), each with different refractive indices. The refrac-tive index of vacuum is n0 = 1, the sample\u2019s complex refractive index is n= nr+ i\u03ba ,and the substrate\u2019s refractive index is ns, assumed to have a negligible imaginarypart.The incident and reflected light at the interface between materials 1 and 2 mustsatisfy electromagnetic wave boundary conditions:E1i\u2225+E1r\u2225 = E2i\u2225+E2r\u2225B1i\u22a5+B1r\u22a5 = B2i\u22a5+B2r\u22a5 (2.2)Here, the subscripts i and r denote incident and reflected electric and magneticfields in materials 1 and 2, while \u2225 and \u22a5 indicate the parallel and perpendicularfield projections. Based on the transfer matrix method, the reflection coefficient r(ratio between reflected and incident electric fields) is given by45:r =E1rE1i=r12+ r23e\u22122i\u03c61+ r12r23e\u22122i\u03c6e\u22122i\u03c60 (2.3)=1\u2212ns\u2212 i(ns\u2212n2)\u03c60 sin\u03c60\u03c601\u2212ns\u2212 i(ns+n2)\u03c60 sin\u03c60\u03c60e\u22122i\u03c60 (2.4)where the reflection coefficients at the vacuum-sample and sample-substrateinterfaces are denoted as r12 and r23.r12 =n0\u2212nn0+nr23 =n\u2212nsn+ns(2.5)The phase shift \u03c6 = 2\u03c0nd\u03bb is the distance light travels through the sample andvacuum, where d is the sample thickness and \u03bb is the light wavelength in the vac-uum. For 2D materials, d is typically much smaller than \u03bb . Under this condition,the equation can be approximated:14r =1\u2212ns+ i(n2\u22121)\u03c601\u2212ns\u2212 i(n2\u22121)\u03c60 (2.6)For weak light intensities, this can be linearized:r = 1\u2212 2i(n2\u22121)\u03c601\u2212n2s(2.7)The reflection contrast RC can be derived from the reflection coefficient r.Keeping only the leading orders in Equation (2.8) and dropping all terms withn2\u03c60 under the weak light excitation, we get:RC =R\u2212RsRs= |r|2\u22121 (2.8)\u2248 4\u00d72nr\u03ba\u03c60n2s \u22121=4\u03c60n2s \u22121Im(\u03b5)=4nrn2s \u221214\u03c0\u03bad\u03bb=4nrAn2s \u22121This shows that reflection contrast is proportional to the imaginary part of thedielectric permittivity, Im(\u03b5) = 2nr\u03ba , and the absorption A= 4\u03c0\u03bad\u03bb of the dielectricmaterial. For a 2D semiconductor, the dielectric permittivity can be modeled bymultiple Lorentz oscillators:\u03b5 = 1+\u2211ifi\u03c92i \u2212\u03c92\u2212 i\u0393i\u03c9i(2.9)where fi is the oscillator strength, \u03c9i is the resonance frequency of the i-th os-cillator, and \u0393i represents the damping constant. Excitonic features are detectablethrough the Lorentz peaks in the RC spectrum, which correspond to resonance ab-sorption at specific photon energy. If the substrate is not transparent, the reflectioncontrast becomes a combination of the real and imaginary parts of the dielectric15Incident ReflectionEBza bFigure 2.1: a. Schematic of a semi-infinite substrate model to illustrate thebasic idea of reflection contrast. b. Schematic of hot carrier cooling andphoton emission process46. This figure is reproduced from ref.46permittivity, arising from the multi-layer reflection.2.1.2 Magnetic Circular Dichroism (MCD)Reflection spectroscopy not only probes exciton states but also measures the mag-netization of 2D materials. Due to the spin-valley-locked band structure and valley-dependent selection rules, left-handed and right-handed circularly polarized lightcan couple exclusively with transitions at the K and K\u2032 valleys, where spins arepolarized30. The difference in reflection between left- and right-handed circularlypolarized light is proportional to the sample\u2019s net magnetization (M), which is de-fined as magnetic circular dichroism (MCD):MCD =R\u03c3+\u2212R\u03c3\u2212R\u03c3+ +R\u03c3\u2212\u221d M (2.10)2.1.3 Photoluminescence SpectroscopyAnother conventional method to probe exciton states is photoluminescence (PL)spectroscopy. Under the illumination of an above-gap laser, electrons are excited16from the valence band to the conduction band through vertical transitions (see Fig.2.1(b)). These hot electrons and holes quickly cool down via phonon emission,forming quasi-Fermi levels at both conduction and valence band edges, a processthat typically takes around 100 fs. Subsequently, electrons and holes combine toform electron-hole pairs or excitons, which emit photons through radiative recom-bination. Probing the photoluminescence mainly detects the exciton ground state.Both reflection contrast and photoluminescence spectroscopy are measured us-ing our home-built confocal microscopy system (see Fig. 2.2). For reflection con-trast, a white light source is focused onto a 10 \u00b5m pinhole, which acts as a pointsource. The light is collimated by a lens after the pinhole and then focused ontothe sample by a 60x cryogenic objective lens with a numerical aperture of 0.65.The estimated spot size at focus is around 1 \u00b5m with an optical power of 50 nW .The reflected signal is guided into a spectrometer (Princeton Instruments) equippedwith a Blaze camera. A combination of a linear polarizer and a quarter-wave plateis used to generate circularly polarized light for MCD measurements.For photoluminescence measurements, a 532 nm laser serves as the excitationsource, normally incident on the sample. A long pass filter (600 nm) is placedbefore the entrance of the spectrometer to remove the reflected excitation laser. Theoptical power at the focus is around 50-100 \u00b5W. By combining reflection contrastand photoluminescence spectroscopy with a dual-gated sample geometry, we cansystematically study how the optical spectrum evolves with external carrier dopingand electric fields.2.1.4 Non-degenerate pump-probe photocurrent spectroscopyTo measure the dynamics of photo-excited carriers, we have developed a non-degenerate pump-probe photocurrent spectroscopy setup. A 1030 nm ultrafast lasersource with a pulse duration of 60 fs and a repetition rate of 70 MHz is split intotwo paths, as shown in Fig.2.3. In the first path, the 1030 nm light serves as thepump beam, which passes through a translation stage used to control the time de-lay. Two gratings are employed to compress the ultrafast pulse. In the second path,the laser enters a home-built optical parametric oscillator to generate visible lightas the probe beam. After this, the pump and probe beams are combined into a sin-17TMDs Sample\/SubstrateObjective lens45o MirrorLampCollimated LenspinholeFocal lensCubic Beam splitterLaser\/White light Beam splitter 2Beam splitter 1MirrorMirrorMirrorMirrorSpectrometerMirrorMirrorlenslenslenslens45o Mirrorpolarizerpolarizerquarter wave plateFigure 2.2: Schematic of experimental set up for reflection contrast spec-troscopy (RC), magnetic circular dichroism (MCD), and photolumines-cence spectroscopy (PL). The red color represents the light source forexcitation while the blue color is the illumination path. The reflectedlight in orange will be collected by a grating-based spectrometer. Thecombination of a quarter wave plate and a linear polarizer is used forMCD measurement.gle path and focused onto the sample surface. By measuring the time-dependentphotocurrent response, we can resolve the photocarrier dynamics.2.2 Sample fabricationThe samples investigated in this thesis are fabricated using a dry-transfer method47on a custom-built transfer stage. First, monolayer or bilayer TMDs, few-layergraphene, and hexagonal boron nitride (hBN) are exfoliated onto a silicon sub-strate with a 285 nm SiO2 layer, chosen to enhance the optical contrast of the tar-get flakes. The thickness of each flake is confirmed using atomic force microscopy(AFM) in contact or tapping mode.To encapsulate the sample, we first fabricate a polymer stamp as a handle,composed of a thin layer of polycarbonate (PC) on a polypropylene-carbonate-18Ultrafast laser1030 nmdelay stagebeam splitterOptical parametric oscillatorGratings beam splitterPumpProbelensMirror MirrorAsampleMirrorMirrorFigure 2.3: Schematic of experimental setup for non-degenerated pump-probe photocurrent measurement. The system is essentially a pump-probe set up with the sample itself working as a detector.coated polydimethylsiloxane (PDMS) block. The flakes are then picked up layerby layer and assembled in the following sequence: cover hBN, top graphite gate,top hBN, graphene contact, TMD bilayer, bottom hBN, and bottom graphite gate.The cover hBN serves as an adhesion layer and protects the entire sample fromdegradation or contamination. To improve the success rate of sample fabrication,every flake is fully covered by the hBN. The graphite gate and contact are shapedusing a 1030 nm ultrafast laser.Once all flakes are assembled, the entire stack is transferred onto a clean sub-strate with pre-patterned gold electrodes by melting the PC film at 180 \u25e6C. Thesegold electrodes are fabricated using mask-less photo-lithography. The gates andcontacts are aligned with the electrodes to ensure good electrical contact. Finally,chloroform is used to dissolve the polymer, followed by an isopropyl alcohol (IPA)cleaning step.19Chapter 3Excitonic effects in 3R-MoS2homobilayer3.1 IntroductionInterlayer coupling, a ubiquitous ingredient in van der Waals (vdW) materials, of-fers unprecedented freedom to tailor the electronic band structure of 2D materi-als. Through precise control of the layer orientation in an artificial 2D assembly,one can create a periodically modulated interlayer coupling in moir\u00e9 superlattice,which leads to many intriguing phenomena different from their monolayer sub-stituents48\u201353. In semiconducting transition metal dichalcogenides (TMDs), suchmoir\u00e9 potentials can quench the kinetic energy of electrons and localize them at theenergy local minimum of the moir\u00e9 superlattice, giving rise to a new platform forcreating quantum emitter arrays and simulating various correlated physics3;54\u201363.In order to have a full grasp over these emerging phenomena, it is therefore impor-tant to understand the underlying interlayer coupling at different localization siteswith different layer orientations.Recently, parallel stacked TMD layers have attracted significant attention, asthey have a broken mirror symmetry with an out-of-plane spontaneous electricalpolarization64\u201371. In an artificially stacked TMD bilayer with a marginal twistangle, the crystal structure may spontaneously relax into multiple rhombohedraldomains with alternating polarization72, which can be electrically switched via in-20plane sliding motion. As a result, the rhombohedrally (R) stacked TMD becomesa new ferroelectric semiconductor with a promising range of electronic and opto-electronic applications70;71;73. Here, we optically probe the spontaneous polariza-tion and interlayer coupling in an R-stacked TMD bilayer. Unlike in the commonhexagonal polytype, the interlayer coupling in the rhombohedral polytype is inter-estingly asymmetric: The conduction band in one layer couples only to the valenceband in the other, but not vice versa, due to the layer-dependent Berry phase effect.Such an asymmetric coupling is also the electronic origin of spontaneous polariza-tion and interlayer potential in TMDs with natural rhombohedral crystal structureor artificial parallel stacking with a marginal twist. Here we perform both electric-field and doping-dependent optical spectroscopy in a dual-gated device made of ahomogeneous bilayer MoS2 exfoliated from a chemically synthesized 3R crystal.In a 3R-MoS2 bilayer, the adjacent layers are stacked in the same direction(Fig.3.1(a)). A relative interlayer displacement along the armchair direction givestwo inequivalent stacking configurations: BA or AB stacking order, where the Satom in the top layer lies directly on the top of the Mo atom in the bottom layer forBA stacking order40;41. The parallel stacking direction maintains the broken inver-sion symmetry of the monolayer, while the lateral shift breaks the mirror symmetrybetween the top and bottom layer, resulting in the C3v point group of the 3R-MoS2bilayer. Under C3 rotational operation, the geometric phase of the Bloch states at Kpoints, the high symmetry points at the Brillion zone edge, has two contributions:C3\u03a8K = e\u2212(m+m\u2032)2\u03c0\/3\u03a8K , where m is the magnetic quantum number associatedwith the rotation of the atomic orbital and m\u2032 is associated with the Berry phasewhen hopping occurs from one site to the next, which is dependent on the rotationcenter (Fig.3.1(b))1;26;74\u201376. The sum m+m\u2032 is the total azimuthal quantum number(Azimuthal quantum number (AQN)).The dependence of the Berry phase on the rotation center has a profound im-pact on interlayer coupling in 3R-MoS2. In the monolayer, the AQNs between theconduction and valence bands are different by +1 or -1 , depending on the valleyindex, which gives rise to the well-known valley selection rule26. In the 3R-MoS2bilayer, the lateral shift leads to a distinct rotational center of two layers, whichcauses a different Berry phase in different layers (Fig.3.1(b)). Take the BA-stacked3R-MoS2 bilayer as an example.21The states at K points in the bottom layer obtain an extra quantum number fromthe Berry phase of m\u2032, resulting in the overall AQN of 1 and 0 for the conductionand valence bands (Fig.3.1(c)). In contrast, the AQN of the conduction and valencebands in the top layer remains 0 and -1. Since electrons can tunnel only betweenbands of the same AQN, the interlayer coupling t exists only between the conduc-tion band of the top layer and the valence band of the bottom layer at K points.Neither the conduction nor the valence bands at K points are directly coupled be-tween the two layers, so the band edges can be distinctively defined for each layerin the 3R-MoS2 bilayer at K points, in contrast to the strongly hybridized valenceband at the \u0393 point. Such an interesting asymmetric interlayer coupling induces alevel repulsion between the two coupled bands and splits the layer degeneracy, re-sulting in a staggered gap at K points where the valence band edge at the Brillouinzone corner is localized in the top layer while the conduction band edge is localizedin the bottom layer, which we refer to as an effective type-II band alignment at Kpoints in a 3R-MoS2 homobilayer(Fig.3.1(c))1;73;76;77.22Figure 3.1: a. Side view of atomic structures of 3R-MoS2 bilayer with AA(left) and BA (right) stacking configurations, where the yellow andgreen spheres denote the S and Mo atoms, respectively. b. Schematicof the phase winding at K points in monolayer MoS2 when the C3 ro-tational center is located at a Mo site (left) and hollow center of thehexagon formed by Mo and S (right). The middle is the top view of theatomic structure of the 3R-MoS2 bilayer with BA stacking configura-tion. The small, dashed orange circle denotes the C3 rotational center.c. Electronic band structure at K points of a 3R-MoS2 bilayer with BAstacking configuration. Yellow (green) dashed lines denote the origi-nal uncoupled bands. Yellow (green) solid lines denote the bands ofthe top (bottom) MoS2 layer. Numbers denote the overall azimuthalquantum number (AQN) of the conduction or valence band edge at Kpoints. Interlayer coupling t exists only between the conduction bandof the top layer and the valence band of the bottom layer, resulting in alevel repulsion t2Edgbetween these two bands, where Edg is the direct bandgap of MoS2. d. Schematic of spontaneous out-of-plane polarization Ppointing from the top layer (yellow) to the bottom layer (green) in a 3R-MoS2 bilayer with BA stacking configuration. e. Schematic of directband gap at the K point of the top (yellow) and bottom (green) layer in a3R-MoS2 bilayer with BA stacking configuration. The atomic structureshows the nonequivalent environments of the Mo atom in the top andbottom MoS2 layers.23On the other hand, the asymmetric interlayer coupling also mixes the Blochwave functions from different layers, which causes the Wannier center of the va-lence band from the bottom layer to shift toward the top layer while the valenceband of the top layer is unchanged. Consequently, the overall Wannier center of alloccupied valence band states shifts to the top layer, resulting in a downward out-of-plane polarization (P) according to modern Berry phase theory (Fig.3.1(d))1;76;78\u201380The interlayer electrostatic potential associated with the polarization, \u03c60 =Pd0\/\u03b50\u03b5m,generates an additional band offset in the 3R-MoS2 bilayer, where \u03b50 and \u03b5m denotethe vacuum permittivity and the out-of-plane dielectric constant of MoS2, respec-tively, and d0 is the interlayer distance.The local chemical environments of the two layers are nonequivalent due tothe lateral shift. In the BA stacking, the Mo atom in the bottom layer lies directlybeneath the S atom in the top layer, but the Mo atom in the top layer does notcoincide with the S atom in the bottom layer. Such an asymmetric atomic con-figuration is known for causing some high-order electronic coupling, which leadsto a larger band gap in the bottom layer than in the top1;77. The band-gap energydifference \u03b4 is observed in both artificial and natural TMD bilayers through opticalspectroscopy (Fig.3.1(e)). As a result, the total band offset at the K points in a 3R-MoS2 bilayer has three contributions: the asymmetric interlayer coupling strengtht, the polarization-induced intrinsic interlayer potential \u03c60, and the band-gap differ-ence \u03b4 , which, according to our effective four-band model, can be approximatelyexpressed as:\u2206c = e\u03c60+t2Edg\u2212 \u03b42(3.1)\u2206v = \u2206c+\u03b4 = e\u03c60+t2Edg+\u03b42(3.2)Here, \u2206c and \u2206v are the conduction and valence band offsets, respectively, and Edgdenotes the direct band gap of the top layer MoS2. Clearly, if we measure the totalband offsets and unravel the contributions from each term, the coupling constant tcan be determined.24Figure 3.2: a. Schematic of a dual-gated 3R-MoS2 bilayer device using hBNand few-layer (FL) graphene as top and bottom gates. b. Optical imageof a representative dual-gated 3R-MoS2 bilayer device. Scale bar, 5\u00b5m.The yellow dashed line marks the boundary of a bilayer sample.3.2 Optically probing the asymmetric interlayer couplingIn this section, we determine each contribution of the band offset in 3R-MoS2bilayer using doping and field-dependent optical spectroscopy.In contrast to artificial stacks, the sample is uniformly stacked with a BA stack-ing order, which is assigned by the interlayer exciton dipole direction. A dual-gated device is fabricated from exfoliated vdW materials using a layer-by-layerdry-transfer method47, which allows independent control of the doping density(n) and the vertical electric field (Ez) between two layers Figure 3.2 illustrates theschematic device structure and the optical image of the dual-gated device. Accord-ing to the thickness ratio of the bottom hexagonal boron nitride (h-BN) gate andthe top h-BN gate, one can introduce the external doping or electric field by apply-ing a top gate (Vt) in proportion to the bottom gate (Vb) voltage with the same oropposite polarity.We first investigate the optical response of the 3R-MoS2 bilayer as a functionof carrier density by reflectance contrast (RC) spectroscopy. All measurements areperformed at 8 K. We also take the energy derivative of the RC to highlight smallfeatures Fig.3.3(b). As shown in Fig.3.3(a), the doping-dependent RC spectrumcan be divided into three regions. In region II, the Fermi level is inside the bandgap of both layers. As in the case of its monolayer counterpart, the 3R-MoS2 bi-layer exhibits two prominent exciton peaks, termed A and B excitons, associated25with the direct optical transition at the K points20. In contrast to the monolayer,the intralayer A exciton is split into two peaks: the low-energy A exciton at 1.918eV (XAt ) and the high-energy A exciton at 1.929 eV (XAb). Such a peak splitting(\u03b4 =11.0\u00b10.5 meV) is observed in the artificial and natural TMD bilayers and isattributed to the intralayer exciton emission from two layers, which is a powerfultool for probing the optical response from different layers. Here, the photon en-ergy difference of the intralayer excitons is directly read from the peak separationin the RC spectra, since the gate-dependent RC spectra are complex due to the h-BN encapsulation and the presence of top and bottom gates. The uncertainty ofthe band-gap difference is determined by the spectral resolution of 0.5 meV. Fur-thermore, the full width at half maximum of the intralayer exciton in the 3R-MoS2bilayer is about 12 meV, broader than that of the monolayer MoS2 81, which couldarise from the phonon scattering-induced exciton lifetime reduction, as the MoS2bilayer is an indirect-gap semiconductor with a valence band maximum at the \u0393point.In the electron doping region I, the electron doping dependence of the twoA excitons XAt and XAb are clearly different(Fig.3.3(a) and Fig.3.3(b)). With in-creasing electron doping density, the high-energy XAb blueshifts and transfers itsoscillator strength to another emergent peak on the low-energy side of XAt . Theobservation can be explained by the interaction between the exciton and degen-erate Fermi sea(FS) of excess charge carriers. As shown in Fig.3.3(c), the XAbexcitons are dressed by excitations (electron-hole pairs) of the FS and split intotwo branches: a low-energy attractive exciton polaron (XP\u2212\u2032Ab) and a high-energyrepulsive exciton polaron (XP\u2212Ab)82\u201386. Around the transition area of region I andregion II, we estimate the binding energy of XP\u2212\u2032Ab to be 25 meV. With increasingelectrostatic gating, the exciton-FS interaction is enhanced due to the expandingFS, which enlarges the repulsion between the XP\u2212\u2032Ab and XP\u2212Ab . Consequently, theXP\u2212\u2032Ab redshifts whereas XP\u2212Ab blueshifts with increasing electron density, and thehigh-energy repulsive exciton-polaron branch XP\u2212Ab transfers its oscillator strengthto the low-energy attractive exciton-polaron branch XP\u2212\u2032Ab , which is consistent withprevious studies of monolayer MoS2 20. On the other hand, the intralayer A ex-citon from the top layer XAt remains largely unchanged except for a slight initialredshift, potentially due to the weak screening or a Pauli-blocking-free exciton po-26Figure 3.3: a. Contour plot of the doping-dependent reflectance contrastspectra of the 3R-MoS2 bilayer. The gate voltage is applied propor-tionally on top and bottom gates with relation Vb = 0.85Vt . The blackdashed lines divide the spectrum into three regions when the Fermi levelis in the (I) conduction band, (II) bandgap, and (III) valence band. b.First energy derivative of the reflectance contrast spectrum at differenttop gate voltage. Dashed gray lines denote different exciton-polaronin region-I and region-III. c. The electronic band structure of the 3R-MoS2 bilayer with BA stacking configuration at the region-I (top panel)or region-III (bottom panel) in a. The yellow (green) color denotes theelectronic states localized in the top layer (bottom layer). The dashedgray line denotes the Fermi level. In region I, electrons are doped intothe conduction band at the K point of the bottom layer. XP\u2212\u2032Ab andXP\u2212Ab represent attractive and repulsive exciton-polarons, respectivelyfrom the bottom layer when the intralayer excitons XAb are dressed withelectron-hole pairs in the Fermi sea (FS). In region III, holes are dopedinto \u0393 point. XP+\u2032Ai and XP+Ai represent attractive and repulsive exciton-polarons, respectively. The subscript i is t or b, which denotes the opticaltransitions at the K points from the top or bottom layers.27laron effect with the charges in the other layer (Fig.3.3(c)). The different dopingdependence of the two A excitons suggests that the electrons are doped into thelayer with the larger band gap, indicating a type-II band alignment at the K pointsof the 3R-MoS2 bilayer87In contrast to the distinctive response in electron doping, the two MoS2 layersrespond similarly to hole doping. The two prominent exciton peaks XAt and XAbare quickly replaced by two emergent peaks with lower energy, which redshifttogether with the increasing negative voltage (Fig.3.3(a) and Fig.3.3(b)). Such adoping dependence arises from the valence band maximum at the \u0393 point in the3R-MoS2 bilayer, where the two layers hybridize strongly. When holes are dopedinto the bilayer, they become nearly equally distributed between two layers. As aresult, both XAt and XAb excitons are dressed by electron-hole pairs of the FS atthe \u0393 point and become split into low-energy attractive exciton polarons (XP+\u2032At andXP+\u2032Ab) and high energy repulsive exciton polarons (XP+At and XP+Ab), as shown inthe first energy derivative of the RC spectra (Fig.3.3(b) and (c)). The more abruptoscillator strength evolution than that in the electron-doping region might originatefrom the higher density of states at the \u0393 point associated with the larger effectivemass. Since X+At is expected to have similar energy as X+\u2032Ab at low doping andshould rapidly diminish with increasing doping, we cannot distinguish them in thefirst derivative spectra, but their opposite doping dependence can be found in thesecond derivative spectra (Fig.3.6(a)). As hole doping density further increases,the FS expands, and, consequently, the X+\u2032At and X+\u2032Ab redshift at the same rate. Thebinding energy of X+\u2032At (X+\u2032Ab) at a low doping density limit is estimated to be 18meV (11 meV). The binding energy difference could be caused by the finite layerpolarization of the hole at the \u0393 point due to the interlayer potential77. Our dopingassignment agrees with the photoluminescence(PL) spectra in Fig.3.6(b)-(c). Thedoping dependence of optical response could also be explained by the trion modelwithout affecting the main conclusion of this work20;87\u201389.Next, we utilize the field dependence of the RC spectra in the electron-dopingregion I to determine the intrinsic interlayer potential \u03c60. Since the exciton ismainly dressed with the electron-doped in the same layer to form exciton-polarons,as shown in Fig.3.3, the oscillator strengths of the excitons and exciton polaronsbecome efficient probes of the doping imbalance between two layers, which is de-28termined by the conduction band offset \u2206c. As a part of \u2206c arises from the interlayerpotential associated with the electric field between two layers, one can tune \u2206c byapplying an external electric field with an anti-symmetric top and bottom gate,\u2206V =Vt \u2212Vb. When \u2206c becomes zero, the doped electrons are equally distributedin two layers, and XAt and XAb should have the same oscillator strengths. The spe-cific relationship between \u2206c and \u2206V depends on the ratio between the averagedgate capacitance C\u02c6 and the geometric and quantum capacitance of MoS2 (Cm andCq, respectively), which can be calculated from the film thickness and dielectricconstants (The derivation is included in the section 3.1.3.). In particular, when theFermi level is in the conduction band of both layers,\u2206c(\u2206V )\u2248 2eCm2Cm+Cq+2C\u02c6\u03c60\u2212 2eC\u02c62Cm+Cq+2C\u02c6\u2206V (3.3)where e is the electron charge. We note that \u2206c has a different \u03c60 dependence whenthe Fermi level is inside the gap of one layer or two. Only when the Fermi level isin the conduction bands of both layers can the intrinsic interlayer potential \u03c60 beobtained by measuring \u2206V at \u2206c = 0 condition (Section 3.1.3). We now considerthe electric field dependence of the RC spectra at a fixed electron doping density(n = 6\u00d71011 cm\u22122. As Vt is swept between 5.5 and 9.5 V, Vb is always 0.85 timessmaller than Vt with an opposite polarity to compensate for the h-BN thicknessdifference (Fig.3.4(a)). When the external field is small (Vt < 6.5 V), the electronsare located in the bottom layer. So the exciton in the top layer (XAt ) is observed to-gether with the exciton polaron in the bottom layer (XP\u2212\u2032Ab and XP\u2212Ab), similar to theregion I in Fig.3.3(a). With increasing electric field, the offset between two con-duction bands decreases, until the Fermi level reaches the conduction band edge ofthe top layer (Vt = 6.5 V). From this point on, the electron begins to migrate fromthe bottom layer to the top layer (Fig.3.4(d)). The interaction between XAt and FSin the top layer forms an attractive and repulsive exciton-polaron, XP\u2212\u2032At and XP\u2212At .As XP\u2212At blueshifts and diminishes with the expanding FS, the XP\u2212Ab redshifts andgains oscillator strength. Such an oscillator strength transfer terminates when theFermi level is sufficiently away from the conduction band edge of the bottom layer(Vt =8.2 V). As discussed above, we can quantify the intrinsic interlayer poten-tial \u03c60 by measuring when the electrons become equally distributed in two layers29(middle panel in Fig.3.4(d)). Since the oscillator strength of the intralayer excitonis a sensitive indicator of the electron doping in that layer, we extract the strengthof their peaks from the RC spectrum at fixed photon energy (1.917 and 1.928 eV).Because the two excitons can have different intrinsic oscillator strengths and theirobserved peak strength can be convoluted with the local field factor and the exciton-polaron formation in the neighboring layer, we normalize the peak strength changewith respect to the maximum value within our gate tuning range (Fig.3.4(e)). Un-surprisingly, we observe a slope in both excitons where the peak strength variesrapidly with the gate voltage. The middle point of the slope region coincides withthe crossing point of the two curves (Vt=7.35\u00b10.15 V) indicating the \u2206c = 0 condi-tion. The uncertainty is estimated from the difference between two middle points,and the crossing point is found to be largely insensitive to the total doping den-sity n. According to Eq. (33), the intrinsic interlayer potential \u03c60 is 58\u00b11.5 mV,corresponding to an out-of-plane polarization P= 0.55\u00b10.02 \u00b5C \u00b7cm\u22122. Our mea-sured interlayer potential is about 20% larger than the previous report in artificialstacks70;71, which is expected since our optical measurement is performed in a ho-mogeneous sample without mixed domains and does not require domain flipping.Additionally, the downward polarization also confirms the BA stacking-order as-signment. Multiple sets of field-dependent measurements are taken at differentlocations, and they all show quantitatively similar results as Fig.3.4(a), confirm-ing the homogeneous BA-stacking order of our sample and the intrinsic nature ofour observation. The determination of charge distribution via exciton contrast isreliable, since the low exciton density excited by the broadband white light sourceduring the RC measurement does not affect the carrier distribution and the finitedoped electrons do not cause significant screening of excitons.To quantify the asymmetric interlayer coupling strength t, we also need to knowthe total intrinsic band offset. Here, we measure this offset optically by investigat-ing the photoluminescence (PL) spectra of the interlayer excitons. It is reportedthat, in the bilayer MoSe2, the interlayer exciton comprises a hole residing in the\u0393 point and an electron at the K point (Fig.3.5(b))77. Since the electrons locatedin the layers with different band edges emit photons of different wavelengths, theintrinsic conduction band offset \u2206c can be measured by resolving the interlayerexciton PL peaks. In our dual-gated 3R-MoS2 bilayer device, the intrinsic inter-30Figure 3.4: a. Contour plot of the electric-field-dependent reflectance con-trast spectrum of 3R-MoS2 bilayer at a fixed electron doping density.The yellow and green arrows denote the intralayer A-exciton, XAt andXAb , from the top and bottom layers respectively. b. Reflectance con-trast spectrum at different top gate voltages in steps of 0.5 V. The bot-tom gate voltage is Vb =\u22120.85Vt +1.65. The yellow and green dashedlines denote XAt and XAb from the top and bottom layers, respectively.c. From low-energy to high-energy, electric-field dependence of XP\u2212\u2032At(attractive exciton-polaron in the top layer), XP\u2212\u2032Ab (attractive exciton-polaron in the bottom layer), XAt or X\u2212At (intralayer exciton or repulsiveexciton-polaron in the top layer) and XAb or X\u2212Ab (intralayer exciton orrepulsive exciton-polaron in the bottom layer) energies extracted froma. d Schematics of band alignment and existing optical transitions ofa BA-stacked 3R-MoS2 bilayer at a fixed electron doping density when(I) the Fermi level reaches the conduction band edge of the top layerwith \u2206c = nCq (II) the Fermi level is in the conduction band of both lay-ers and electrons are equally doped into two layers with \u2206c = 0, (III)the Fermi level is at the conduction band edge of the bottom layer with\u2206c = nCq . e. Normalized intensity changes of XAt and XAb extracted froma along the fixed photon energy indicated by yellow and green arrows.The crossing point denotes the same oscillator strength of X\u2212At and X\u2212Abwhen the conduction band offset is zero as shown in the middle panelof d.31layer PL spectrum has six peaks around 1.45 eV. These six peaks can be groupedinto three pairs with an energy separation of about 20 meV in each pair. Oneprior explanation for such an energy separation is that each interlayer exciton hasa phonon replica, since phonons are needed for momentum-indirect optical tran-sitions90. Available phonons for scattering the electron from the K to \u0393 pointhave energies of approximately 26 meV for the acoustic branch and approximately47 meV for the optical branch90. The energy splitting in each pair of interlayerexcitons is approximately 20 meV, comparable to the energy difference betweenacoustic and optical phonons, suggesting the two exciton peaks can be assistedby different phonon branches. More studies are required to determine the specificphonon modes responsible for the optical transition. In the following, we focus onthe low-energy branch of each pair.The origin of the three interlayer exciton pairs is clarified by measuring theirdoping and electric field dependence. Since the two high-energy pairs disappearin the doped regimes (Fig.3.5(a)), we attribute them to the intrinsic interlayer exci-ton, XIb and XIt . The highest-energy pair is emitted from the top layer, as the toplayer has a higher conduction band than the bottom one at zero field. The broadpeaks of the lowest energy are likely to result from trions emitted from the bottomlayer (TIt ). They are observable in region II, since the bilayer is always doped bybond charges induced by the spontaneous polarization. Such an assignment is con-firmed by the electric field dependence of the emission energy (Fig.3.5(c)). WithEz varying from negative to positive, XIt redshifts and XIb blueshifts, respectively,and they cross each other at Vt =6.5 V. Similar to the picture in Fig.3.4, such aStark shift is due to the tuning of the conduction band offset at the K point, whilethe valence band maximum at the \u0393 point remains largely unchanged, since thetwo layers hybridize strongly. When the conduction band offset is zero (\u2206c =0V), XIt and XIb have the same photon energy corresponding to the crossing pointin Fig.3.5(c). The dipole moment of the interlayer exciton can be extracted fromthe slope of the Stark shift to be 0.31 e\u00b7nm, which agrees with the picture that theelectron is localized in one layer while the hole is shared between the two, so theirout-of-plane distance is about half of the interlayer distance (d \u2248 0.65nm). Themeasured dipole moment agrees with the calculated value of the \u0393\u2212K interlayerexciton, which is much larger than that of the \u0393\u2212Q interlayer exciton, thus further32Figure 3.5: a. Contour plot (on a logarithmic scale) of the doping-dependentPL spectrum of momentum-indirect interlayer excitons in the 3R-MoS2bilayer. The gate voltage is applied proportionally on top and bottomgates with relation Vb = 0.85Vt . At zero doping, from low-energy tohigh-energy, three pairs of peaks indicate TIb(interlayer trion), XIb (in-terlayer \u0393-K exciton), and XIt (interlayer \u0393-K exciton), respectively.The XIt is more visible in c indicated by a yellow arrow. b. Elec-tronic band structure and momentum-indirect \u0393-K transitions in intrin-sic BA-stacked MoS2 bilayer. c. Contour plot (on a logarithmic scale)of the electric-field dependent PL spectrum in the 3R-MoS2 bilayer. Thegreen (yellow) arrow denotes the interlayer exciton XIb (XIt ). The gatevoltage is applied proportionally on top and bottom gates with relationVb = \u22120.85Vt . d. Electric-field dependence of interlayer excitons ex-tracted from (c). Solid lines denote the linear fitting results. The dashedgray line denotes the same energy of XIb and XIt when \u2206c = 0.33confirming our assignment77. Interestingly, we find that the interlayer trion peakshave a similar field-dependent Stark shift as the interlayer exciton.When Ez =0, the energy difference between XIt and XIb of 58.0\u00b10.5 meV isequal to the intrinsic conduction band offset \u2206c. The uncertainty of \u2206c is also de-termined by the spectral resolution of 0.5 meV as that of \u03b4 . The spin configurationof conduction bands should not affect the determination as long as the two setsof spin bands have the same band offset. Considering the optically determined\u2206c, \u03b4 , and \u03c60, we extract the asymmetric interlayer coupling of the 3R-MoS2 bi-layer to be t=100 \u00b1 25 meV. This conclusion demonstrates that the asymmetricinterlayer coupling is not negligible, in contrast to some estimation in the litera-ture80. Instead, our measured asymmetric coupling strength in the 3R bilayer isof the same order as the predicted interlayer coupling strength between valencebands in the 2H bilayer80, in agreement with first principles calculations76. Ourfour-band Hamiltonian should be regarded as an effective model which describesthe essential physics involving states near the conduction and valence band edgesof the 3R-MoS2 bilayer. As such, the interlayer coupling parameter t should alsobe regarded as an effective parameter which includes, in principle, contributionsfrom all interlayer tunneling processes between the conduction band in one layerand the valence band in the other. Besides the band-gap difference \u03b4 , the higher-order corrections of remote bands to the band offset are qualitatively discussed inthe section 3.1.4.As a summary, the polarization-induced intrinsic interlayer potential \u03c60, theinterlayer coupling t, and the direct band-gap difference \u03b4 correspondingly con-tribute 58 \u00b1 1.5, 6 \u00b1 2.5, and -5.5 \u00b1 0.25 meV to the intrinsic conduction bandoffset \u2206C of 58 meV at K points in 3R-MoS2. The experimental results agree withprevious self-consistent calculations which relate the intrinsic interlayer couplingand interlayer potential73. Since \u03c60 is determined only by the ratio of the averagedgate capacitance C\u02c6 and the geometric capacitance of bilayer MoS2 (Cm), neitherthe size of the spin-orbit coupling-induced conduction band splitting nor the spinconfiguration of the lowest band affect our measurement conclusions91. The con-tribution from the Q points is not expected to have a qualitative impact on ourconclusions either for the same reason.In conclusion, we leverage the distinctive sensitivity of the Fermi polaron to34ab cFigure 3.6: a. Doping-dependent maps of the second energy derivative ofthe reflectance contrast d2(\u2206R\/R)dE2 in the 3R-MoS2 bilayer. b. Contourplot of the doping-dependent photoluminescence spectra of momentum-direct intralayer excitons in the 3R-MoS2 bilayer. c. Photoluminescencespectra of momentum-direct intralayer excitons in 3R-MoS2 bilayer atdifferent top gate voltages Vt in steps of 2 V. The bottom gate voltage ina-c is Vb = 0.85Vt .charges doped in different layers to probe the intrinsic interlayer potential and, thus,the spontaneous polarization in a homogeneous 3R-MoS2 bilayer. Compared to thegraphene electrical sensing approach to determining spontaneous polarization, ouroptical technique does not rely on polarization switching via interlayer sliding,rendering it suitable for quantifying spontaneous polarization in a wide range ofmaterials. In combination with the optically measured band gaps and band offsets,we quantitatively determine the strength of an asymmetric interlayer coupling atK points, which is fundamental to the ferroelectric and optoelectronic applicationsof semiconducting TMDs where sliding ferroelectricity is observed70. Last but notleast, our results lay an important foundation for understanding the moir\u00e9 super-lattice formed by hetero- or twisted homo-structures. Besides the exotic physicspredicted from the position dependent interlayer coupling92, the hopping betweenlayers is known for determining the moir\u00e9 potential depth as well as the bandwidthof the moir\u00e9 bands, whose competition with the Coulomb interaction has led to arange of correlated insulating states observed in various experiments93. Therefore,35the full understanding of interlayer coupling in different stacking orders is crucialfor exploring these new semiconducting moir\u00e9 materials.3.3 Effective four band modelThe effective four-band Hamiltonian for BA-stacked 3R-MoS2 bilayer at K pointsbased on k\u00b7p calculation is given by80:Heff,+K =\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0Etc+e\u03c62 \u03b33q+ \u03b3ccq\u2212 t\u03b33q\u2212 Etv+e\u03c62 \u03b3vcq+ \u03b3vvq\u2212\u03b3ccq+ \u03b3vcq\u2212 Ebc \u2212 e\u03c62 \u03b33q+t \u03b3vvq+ \u03b33q\u2212 Ebv \u2212 e\u03c62\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (3.4)where Etc =Edg2 , Etv = \u2212(Edg )\/2, Ebc = (Edg + \u03b4 )\/2, and Ebv = \u2212(Edg + \u03b4 )\/2 arethe corresponding electron energies of the conduction and valence bands in thetop and bottom MoS2 layers. Here the direct optical band gap of the top layer isEdg =1.918 eV while the direct bandgap in the bottom layer is larger by \u03b4=11 meV.\u03c6 is the interlayer potential induced by the external field and internal polarization.q = qx\u00b1 iqy denotes the wave-number measured from the K point of the Brillouinzone (BZ). \u03b33 denotes the intra-layer coupling of the conduction band and valenceband. \u03b3cc and \u03b3vv is the interlayer coupling between the two monolayers\u2019 conduc-tion bands (valence bands). \u03b3vc denotes the interlayer coupling between the valenceband from the top layer and the conduction band from the bottom layer. t is the in-terlayer coupling between the conduction band from the top layer and the valenceband from the bottom layer. At K points, q\u00b1 is zero, and the contribution to theHamiltonian from the intralayer coupling \u03b33, interlayer coupling \u03b3vc, \u03b3cc, and \u03b3vvall becomes zero. Therefore, only the interlayer coupling t remains. Consequently,the effective four-band Hamiltonian at K points becomesHeff,+K =\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0Etc+e\u03c62 0 0 t0 Etv+e\u03c62 0 00 0 Ebc \u2212 e\u03c62 0t 0 0 Ebv \u2212 e\u03c62\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (3.5)36d(\u2206R\/R)\/Figure 3.7: a. Type-II band alignment of BA-stacked MoS2 bilayer at Kpoints of the Brillouin zone. Spin-up (spin-down) bands are denotedby solid (dashed) lines. XAt , XBt (XAb , XBb) denote the momentum-directintralayer A,B excitons in the top MoS2 layer (bottom MoS2 layer). b.First energy derivative of the reflectance contrast spectrum of the 3R-MoS2 bilayer.The conduction band splitting could be extracted from the equation (3.) afterdiagonalization.\u2206c = e\u03c6 +t2Edg\u2212 \u03b42(3.6)\u2206v = \u2206c+\u03b4 = e\u03c6 +t2Edg+\u03b42(3.7)The equation has been linearized under the condition e\u03c6 ,\u03b4 << Edg . \u2206c repre-sents the conduction band offset. The first term denotes the spontaneous polariza-tion induced band splitting. The second term arises from the interlayer couplingbetween the conduction band of the top layer and the valence band of the bottomlayer. The third term comes from the intrinsic bandgap difference between thetop and bottom layer MoS2. The high-order tunnelling terms between the remotebands and band edges have been dropped. The effective four band model capturesthe type-II band alignment at K point in 3R-MoS2 bilayer, as shown in Fig.3.7.37Figure 3.8: P and Ei represent spontaneous electric polarization and the elec-tric field between the top and bottom layer, respectively. Downward istaken to be the positive direction.3.4 Electrostatic modelling of a dual-gated deviceWe consider a parallel plate capacitor model to derive the interlayer electrostaticpotential in the 3R-MoS2 bilayer with an out-of-plane spontaneous polarization P.The electrostatic interlayer potential \u03c6 is determined by the top gate voltage Vt ,the bottom gate voltage Vb and the spontaneous polarization P. Figure 3.8 showsthe schematic of our dual-gate 3R-MoS2 bilayer device. The middle two layersdenote 3R-MoS2 bilayer with interlayer distance d0 (0.65 nm) and out-of-planedielectric constant \u03b5m (\u223c7.0). The top (bottom) gate is composed of a few-layer(FL) graphene flakes and top (bottom) hBN dielectric with thickness dt \u223c28.0 nm(db \u223c23.8 nm) and dielectric constant \u03b5bn (\u223c2.7).The quantities nt and nb are the sheet carrier densities at the top and bottomlayer of the 3R-MoS2 bilayer. The electric fields from the top gate, bottom gate,and 3R-MoS2 bilayer are denoted as Et , Eb, and Ei, respectively. The positivedirection is taken to be from top to bottom. For the 3R-MoS2 bilayer with a BA-stacking configuration, the polarization is downward. The electric field and the38sheet carrier density are related by the Gauss\u2019s Law:nte = \u03b50\u03b5bnEt \u2212 \u03b50\u03b5mEt \u2212P (3.8)nbe =\u2212\u03b50\u03b5bnEb+ \u03b50\u03b5mEt +P (3.9)Here e is the electron charge and \u03b50 is the vacuum electric permittivity. Theelectric fields Et , Eb, and Ei should satisfy the self-consistent equations related tothe total chemical potentials of each layer. Here the total chemical potential \u00b5 isdefined as the sum of the electrochemical potential \u00b5e and chemical potential \u00b5c,\u00b5 = \u00b5e + \u00b5c. The top and bottom gate electrochemical potentials are eVt and eVb,respectively. The chemical potential of the top and bottom electrodes is denoted as\u2212W1, where W1 is the work function for few-layer graphene. The fields Et , Eb, andEi are given byEt =\u2212W1+ eVt \u2212\u00b5tedt(3.10)Ei =\u00b5t \u2212\u00b5bed0=\u2212 \u03c6d0(3.11)Eb =\u00b5b\u2212 (\u2212W1+ eVb)edb(3.12)Here \u00b5t and \u00b5b are the chemical potentials measured from the middle of the bandgapof the top and bottom MoS2 layer while their electrochemical potential is zero sinceboth top and bottom layers are grounded. \u03c6 is the interlayer electrostatic potentialdifference in the 3R-MoS2 bilayer.We express the Fermi energy of each MoS2 layer in terms of the carrier density.We only consider the electron doping case. We assume doping densities n0t and n0bare required to fill all the in-gap states of the top and bottom MoS2 layers, respec-tively, and are related to the direct bandgap energy Edg and the in-gap quantumcapacitance Cq0 as Edg =n0t e2Cq0and Edg +\u03b42 =n0be2Cq0. When the Fermi level aligns with39the conduction band edge, the quantum capacitance becomes Cq = 2m\u2217e2\u03c0\u210f2 >>Cq0,where m\u2217 and \u210f are the effective mass and Planck\u2019s constant, respectively. Then,the Fermi energy in terms of the top and bottom layer parameters has the followingexpression.\u00b5t =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\u2212W2+ nt e2Cq0 nt <n0t2\u2212W2+ Edg2 +t2Edg+(nt\u2212 n0t2 e2)Cqnt \u2265 n0t2(3.13)\u00b5b =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\u2212W2+ nbe2Cq0 nt <n0b2\u2212W2+ Edg2 +\u03b42 +(nb\u2212 n0b2 e2)Cqnb \u2265 n0b2(3.14)The work function of few-layer graphene W1 and MoS2 bilayer W2 is \u223c4.5 eVand \u223c5.1 eV, respectively94;95. The terms t2Edg and\u03b42 arise from the asymmetricinterlayer coupling and the direct-bandgap difference, respectively. Equation 3.8-3.12 can be combined to yield(nt \u2212nb)e =\u22122P+(CtgVt \u2212CbgVb)\u22122\u03b50\u03b5mEi\u2212Ctg\u00b5t \u2212Cbg\u00b5be (3.15)(nt \u2212nb)e\u2248\u22122Cm\u03c60+2C\u02c6(Vt \u2212Vb)+2Cm\u03c6 \u2212 2C\u02c6e\u03c6 (3.16)Here Cm = \u03b50\u03b5md0 is the geometric capacitance of the bilayer MoS2, \u03c60 =PCmis thepolarization-induced intrinsic interlayer potential, C\u02c6 = Ctg\u2212CbgCtg+Cbg , where Ctg =\u03b50\u03b5bndtand Cbg = \u03b50\u03b5bndb are geometric capacitance of the top and bottom gates. We cannow combine the Equations 3.6, 3.13-3.16 to express the interlayer potential \u03c6 andconduction band offset \u2206c for three different cases.40The Fermi level is inside the gap of both layers.\u03c6 =2Cm2Cm+Cq0+2C\u02c6\u03c60\u2212 2C\u02c62Cm+Cq0+2C\u02c6(Vt \u2212Vb) (3.17)\u2248 \u03c60\u2212 C\u02c6Cm+C\u02c6(Vt \u2212Vb) (3.18)\u2206c = e\u03c60\u2212 eC\u02c6Cm+C\u02c6(Vt \u2212Vb)+ t2Edg\u2212 \u03b42(3.19)The Fermi level is in the gap of the top layer and the conduction band of thebottom layer.\u03c6 =CmCm+C\u02c6\u03c60\u2212 C\u02c6Cm+C\u02c6(Vt \u2212Vb)\u2212 (nb\u2212nt)e2Cm+2C\u02c6(3.20)\u2248 \u03c60\u2212 C\u02c6Cm+C\u02c6(Vt \u2212Vb)\u2212 ne2Cm+2C\u02c6(3.21)\u2206c = e\u03c60\u2212 eC\u02c6Cm+C\u02c6(Vt \u2212Vb)\u2212 ne22Cm+2C\u02c6+t2Edg\u2212 \u03b42(3.22)Since electrons are mainly doped into the bottom layer, we assume nb\u2212nt \u2248 nb\u2248 n.Here n= e(Ctg(Vt\u2212Vt0)+Cbg(Vb\u2212Vb0)) is the total electron doping density, Vt0 =0.4V , Vb0 = 0.85Vt0 is the top and bottom gate voltage extracted from Fig.3.3(a)when Fermi level starts to align with the conduction band edge of the bottom layer.The Fermi level is in the conduction band of both layers.\u03c6 =2Cm2Cm+Cq+C\u02c6\u03c60\u2212 2C\u02c62Cm+Cq+C\u02c6(Vt \u2212Vb)+ Cqe(2Cm+Cq+C\u02c6)(\u2212 t2Edg+\u03b42)(3.23)\u2248 2Cm2Cm+Cq+C\u02c6\u03c60\u2212 2C\u02c62Cm+Cq+C\u02c6(Vt \u2212Vb)+\u2212 t2Edg +\u03b42e(3.24)41\u2206c =2eCm2Cm+Cq+C\u02c6\u03c60\u2212 2eC\u02c62Cm+Cq+C\u02c6(Vt \u2212Vb) (3.25)Since Cq0,Ctg,Cbg << Cm << Cq , high order terms have been dropped for sim-plicity.42Chapter 4Spontaneous-polarization-induced photovoltaic effect in3R-MoS2 bilayer4.1 Probing the photovoltaic effect in 3R-MoS2 bilayer4.1.1 IntroductionThree intrinsic photovoltaic mechanisms exist in polar materials96. Two of thesemechanisms are second-order effects of the optical field, arising from the shift cur-rent and injection current97. They have recently attracted much attention becauseof the potential for overcoming the Shockley\u2013Queisser limit in energy harvestingapplications98\u2013101. The third mechanism is related to the depolarization field (de-polarization field (DEP)). The DEP in a polar material is an electric field generatedby the bond charge at the surface or interface where the polarization is terminated.If a polar thin film is sandwiched between two metal electrodes, the DEP is nonzerowhen the induced image charges in the electrode do not fully compensate the polar-ization charges102. Upon excitation, the photocarriers drift under the DEP, forminga Photovoltaic effect (PV) current. In contrast to the first two, the third mecha-nism is linearly dependent on the depolarization field and has been observed in43oxide-based ferroelectric films103\u2013105. Since the DEP\u2019s strength depends on thefilm thickness and screening in electrodes103;106;107, much effort has been dedi-cated to growing ultra-thin ferroelectric oxide film108 and optimizing the contactmaterial109. Here we push the DEP effect to the atomically thin limit with few-layer rhombohedral MoS2 and monolayer Graphene (GR) as the electrode, whichwe find can preserve as much as 95% of the DEP due to its reduced screening intwo dimensions.Stacking order in van der Waals (vdW) materials determines the coupling be-tween atomic layers and is therefore key to the materials\u2019 properties. By explor-ing different stacking orders, many novel physical phenomena have been real-ized in artificial vdW stacks51;52;110;111. Ferroelectricity, a phenomenon exhibitingreversible spontaneous electrical polarization, has been observed in zero-degreealigned hexagonal boron nitride (hBN)67;69;112 and graphene-hBN113 heterostruc-tures, holding promise in a range of electronic applications114. The rhombohe-dral stacking order in carbon systems has led to some exotic transport phenom-ena6;115;116. In MoS2, as the two sublattice sites are not equivalent, the rhombohe-dral (R) stacking means the neighboring layers are oriented in the same direction,in contrast with the hexagonal (H) stacking (Fig.4.1(a)). In R-stacking, adjacentlayers shift laterally by a third of the unit cell and three shifts complete a unit cell,for which the phase is named 3R40;41. The relative lateral shift modifies the effec-tive angular momentum at K and K\u2032 valley and leads to an asymmetric interlayercoupling between two monolayer MoS2. As a result, an emergent spontaneouspolarization arises along the out of plane direction according to our Berry phasecalculation based on an realistic tight-binding model (Details in A.1).A schematic band structure of a bilayer 3R-MoS2 is shown in Fig.4.1(b). Pro-tected by three-fold rotational symmetry, the Bloch state at the K point in each layerremains almost localized with a weak interlayer coupling1;61. The interlayer po-tential induced by the spontaneous polarization and the asymmetric interlayer cou-pling117 cause an energy shift in both the conduction and valence bands, forming atype-II band alignment at K point. Since the band offsets are slightly different, theinter-band transitions in the top and bottom layers can be distinguished in energy(Fig.4.1(c)). The extracted A exciton splitting is about 13 meV, in agreement withthe measured results in artificial stacks61;77 and theoretical calculations1;80. One44important difference of the bilayer MoS2 from the monolayer is that its bandgap isindirect11, which has a large impact on the PV effect as discussed later.4.1.2 Revealing the mechanism of photovoltaic effect in 3R-MoS2bilayerThe schematic of our device structure and band alignment is shown in Fig.4.2(a). Apiece of atomically thin 3R-MoS2 is sandwiched between two graphene electrodes.Similar structures have been used to build 2H-MoS2 based photodetectors94;118.Unlike the 2H stacking, the spontaneous polarization from the 3R stacking inducesthe same amount of charge with opposite signs in top and bottom graphene elec-trodes, thus modifying graphene\u2019s chemical potential. (The graphene electrodes arecut from the same original piece to keep initial doping levels the same.) At zerobias, a DEP is established in proportion to the induced potential difference. Dueto the small density of state near the Dirac point, the DEP in a graphene-contacteddevice can be 95% of that in a completely unscreened film (Section 4.1.3). Sincethe MoS2 layer is atomically thin (0.7 nm for a monolayer), the Schottky junctioneffect in our device is negligible, as confirmed by the symmetric current-voltagecharacteristic (I-V) curve. The DEP drives photocarriers into a photovoltaic currentwhich is measured by a sourcemeter. A graphite back gate is added outside thehBN encapsulation to tune the doping level. All photocurrent measurements arecarried out at room temperature.Our first device (D1) contains a series of terraces including one, two, and threelayers of 3R-MoS2 (Fig.4.2(b)), which is directly exfoliated from a 3R bulk crystalgrown by chemical vapor transport method and characterized by both linear andnonlinear optical probes. First, we obtain a spatial map of the short-circuit currentby scanning the sample under laser focus (\u03bb = 532 nm) (Fig.4.2(c)). On average,we observe a responsivity of 1.0 mAW\u22121 in the bilayer, while the photoresponsein the monolayer is about an order of magnitude weaker and close to the noisefloor. Moreover, we find an even stronger responsivity with the same polarity in thetrilayer (1.5 mAW\u22121). The photocurrent is independent of the laser polarization,indicating a likely different origin from the shift current or injection current. Thesharp contrast between the monolayer and few layers MoS2 at zero bias suggestsan intrinsic PV effect in the rhombohedral device.45SMo H-stacking R-stacking1.8 1.9 2 2.1 2.2 2.3Photon energy (eV)00.511.52 R\/R  ab cP\u0413 K\u0394 VMonolayer3R Bilayer\u0394CQtopbottomA BP = 0Figure 4.1: a, Schematics of H-stacking (2H) and R-stacking (3R) of bilayerMoS2. The yellow and magenta colored balls correspond to the S andMo atoms, respectively. The black arrow represents a spontaneous po-larization between layers. b, Schematic of the band structure of a 3R-MoS2 bilayer. Brown and green bands represent the top and bottom lay-ers\u2019 contribution respectively. The smallest band gap is an indirect typewith the highest valence band edge at the \u0393 point. At the K point,theconduction and valence bands in the two layers are decoupled. Finitesplittings of \u2206c and \u2206v are a manifestation of the spontaneous polariza-tion. The bands at \u0393 and Q points are layer coupled but the Q point ishighly layer polarized as a result of the depolarization field. c, Reflec-tion contrast spectra of monolayer (grey) and bilayer 3R-MoS2 (black).The bilayer spectrum is shifted upward by 0.6 for clarity. The mono-layer only shows two prominent peaks of A and B excitons. In contrast,in the bilayer case, the A exciton resonance splits into two peaks, cor-responding to the inter-band transitions in the top (brown arrow) andbottom (green arrow) layer due to the asymmetric splitting in conduc-tion and valence bands (\u2206c \u0338= \u2206v).46To further examine the nature of the PV effect, we study the I-V and the pho-toresponse spectrum. Under dark condition, we observe a linear I-V curve thatcrosses the origin, absent of any Schottky behavior from -0.1 mV to 0.1 mV(Fig.4.2(d)). When the laser is focused on the bilayer area, the I-V curve is shiftedupward, indicating a negligible photoconductivity effect. As the device resistanceis almost unchanged, the phenomenological open-circuit voltage (Voc) is linearlydependent on the short-circuit current. We attribute this tunable Voc to a small tun-neling resistance times a nearly constant photocurrent generation, which can bemodeled as an ideal current source in this limited bias range (Section 4.2.4). Thephotocurrent will vary over a broader bias range as discussed later. In Fig. 4.2(e),we report the photocurrent spectra measured in the bilayer and trilayer regions.Two peaks corresponding to the A and B exciton absorption are observed. Theelectron and hole will be separated into different layers under the DEP and becomedissociated as free carriers via the interfacial charge transfer119 (section 4.1.7).The resemblance between photocurrent spectra and MoS2 absorption confirms thephotocurrent origin.Similar to the Bernal stacked bilayer graphene120, there are two possible stack-ing domains in a 3R bilayer MoS2 (Fig.4.3(a)): In the AB domain, the molybde-num atom in the top layer is above the sulfur atom in the bottom layer, while inthe BA domain, the same amount of shift happens along the opposite direction,rendering an in-plane mirror image. The spontaneous polarization direction andphotocurrent polarity are opposite between these two domains. Among the ten de-vices we fabricated, most devices show homogeneous photoresponse (more devicemappings are in Fig.4.6), except one sample (D2) shows both positive and negativephotoresponse, which we attribute to the coexistence of AB and BA domains. Thephotocurrent mapping of D2 is shown in Fig.4.3(a), where the AB-BA domainsshow nearly symmetric responses. The monolayer region in D2 has a close-to-zerophotoresponse, in agreement with D1.Made of atomically thin materials, the doping level of our device can be elec-trostatically tuned. Since the Dirac point of graphene is in the middle of MoS2bandgap, most carriers are doped into graphene rather than MoS2 (Section 4.1.3).The back-gate voltage dependence of the zero-bias photocurrent in D2 is shownin Fig.4.3(b). In both AB and BA domains, the photoresponse drops quickly with47b cad e01L2L3Lh    +   +   +3R-MoS2Gr hBNAV bias V gateBilayerTrilayer1.6 1.8 2 2.2 2.4Photon Energy (eV)01Photoresponsivity (a.u.)-0.1 0 0.1Vbias (mV)-80-4004080I (nA)DarkLaser on-0.05 0.05Top GrBottom Gr1.6-  -  -   +   +    +   + MoS2-  -  -  -  PEDEPI (mA\/W)e\u0127\u03c9ABFigure 4.2: a, The top panel is a schematic of the tunneling junction in ourdevice. The spontaneous polarization (white arrow) leads to polariza-tion charges at the interface (black), which induces image charges inthe Graphene electrodes (white). Since the image charges do not fullycompensate the polarization charges, a depolarization field exists (blackarrow). The bottom panel illustrates the band alignment in the device.The top and bottom graphene are electron and hole doped by the sameamount of image charges. Photocarriers generated in the MoS2 aredriven to the interface and collected by graphene electrodes. b, Op-tical image of the D1 device, consisting of one, two, and three layerswithin the white dashed box. Graphene and 3R-MoS2 are outlined byblack dashed line and brown solid line. The scale bar is 5 \u00b5m. c, Scan-ning photocurrent map of D1, corresponding to the white dashed box ofb. Monolayer, bilayer, and trilayer regions are indicated. The scale baris 2 \u00b5m. d, Current-voltage (I-V) curves of the bilayer region as mea-sured in dark (black) and illuminated conditions. (\u03bb = 532 nm, brown).e, Photoresponsivity spectra of the bilayer and trilayer region. The twopeaks near 1.8 and 2 eV correspond to the A and B excitons, respec-tively.48increasing positive voltage when the top and bottom graphene are electron doped.Due to the weak screening in graphene, the top electrode can be doped to have asmaller but same order-of-magnitude amount of charge compared with the bottomelectrodes94;121. According to our electrostatic model, the Fermi level of the bot-tom (AB domain) and top (BA domain) graphene can be raised to 230 and 150 meVabove the Dirac point respectively, when a Vg = 8 V is applied (Section 4.1.3). Onthe other hand, the decrease is much slower in the negative voltage range when thegraphene is hole doped (In a small negative range, the AB domain has an increasingphotocurrent with the increasing voltage, which we attribute to the decrease in thephoto-thermal electric effect as discussed in detail in the section 4.1.4).The simi-lar back-gate dependence in AB and BA domains suggests that the photoresponsechange is not induced by an electric field in MoS2 94;118 but by a doping effect ingraphene - the extra charges doped into the graphene electrodes shift the chemicalpotential and occupy the states to which the photocarriers can be transferred to.This trend is reproducible in other single-domain devices we studied.The asymmetric response of D2 to the electron and hole doping suggests ourphotocurrent is dominated by the electron transfer process. In bilayer MoS2, thehighest valence band edge is at the \u0393 point while the K and Q points share a similarenergy which is lowest in the conduction band. More importantly, the couplingbetween the top and bottom layer at the K and Q point is either strictly zero orcomparable to the DEP induced interlayer potential, leading to a complete or sig-nificant layer polarization in the conduction band edge77. In each valley, the lowerconduction band has more contribution from the layer with a lower potential, andthe higher conduction band has a larger weight in the other. Initially, electronsare excited in both layers through inter-band transitions. After quickly relaxing tothe lowest conduction band edge, they acquire a corresponding layer polarization,a process we call interlayer relaxation. Since the tunneling probability decreasesexponentially with distance, the electrons in the top MoS2 layer mostly tunnel tothe top graphene, while those in the bottom layer mostly tunnel to the oppositeelectrode. Consequently, the layer polarization in the conduction band leads to anet imbalance between the two counter charge flows, and ultimately, to a PV cur-rent. An efficient PV effect based on this mechanism requires that the interlayerrelaxation happens at a similar or faster rate than the charge tunneling to graphene.49On the other hand, at the \u0393 point, where most photo-excited holes relax to, theinterlayer coupling is too large, about 400 meV, which is much larger than theDEP potential and leads to a negligible layer polarization77. Therefore, the holescontribute much less to the photocurrent. Together with the band filling effect ingraphene, the asymmetric band structure in bilayer MoS2 can explain the asym-metric doping dependence in the experiment.Using this electron transfer picture, we can also understand the device\u2019s per-formance over a broad bias range. Since the tunneling current becomes significantunder large bias, we define the photocurrent (PC) as the difference of current mea-sured between laser on and off conditions. As shown in Fig.4.3(c) and (d), the PCis highly non-monotonically dependent on the bias. Across the full measurementrange, the PC changes sign multiple times, showing negative slopes at the largebias ends. The response of the AB and BA domains are not symmetric either. Weattribute this complicated bias dependence to the PV effect, plus the photo-thermalelectric (Photo-thermal electric effect (PTE)) (IPT E) and bolometric (Bolometric ef-fect (BOL)) (IBOL) effects in device122\u2013124. Here we leave the detailed discussionof thermal effects in the section 4.1.4 and focus on the PV effect.The impact of the external bias on the PV effect can be modeled by a two-band Hamiltonian (section 4.1.4). When the bias field is parallel to the DEP, theelectrostatic potential difference between the top and bottom layer increases andthe layer polarization is enhanced. If the bias is anti-parallel to the DEP, the layerpolarization is reduced and can even be reversed. After fitting the experimentaldata using IPC = IPV + IBOL+ IPT E , the pure PV contribution (IPV ) is extracted andshown in Fig.4.3. (The thermal contributions of IBOL and IPT E are shown in section4.1.4.)The PV effect accounts for most of the PC at zero bias and the saturation be-havior at high bias. A compensation voltage, defined as the bias level when thePV current diminishes, is observed at 0.30(\u00b10.01) V and independent of the lightintensity, a hallmark of the DEP-induced PV effect103. The extracted compensationvoltage and short-circuit current are symmetric between the AB and BA domains.The experimental compensation voltage is a few times larger than the spontaneouspolarization induced interlayer potential from first-principles calculations77;78, po-tentially due to only a fraction of the applied bias drops across the tunneling junc-50a cbI  ( A)-8 -4 0 4 8Vg (V)-0.2-0.100.10.2PCABBAABBA1LI-3 3 (mA\/W)BAABhhee-0.2-0.100.10.2I PC-1 0 1Vbias (V)-0.4-0.200.20.4I PV  ( A)I PC (A)I PVbias-0.2-0.100.10.2-1 0 1V  (V)-0.4-0.200.20.4dPlaser (   W)10 70 10 70Plaser (   W) (A) (A)AB BAFigure 4.3: a, Left panel is a schematic of two possible stacking domains(AB and BA) of a 3R bilayer MoS2. Top layer (solid) shifts towards left(AB) and right (BA) respectively, relative to bottom layer (translucent).Right panel is the scanning photocurrent map of device D2. Positive andnegative photoresponse areas correspond to AB and BA domains withalmost symmetric responsivity (\u00b1 3 mA\/W). A monolayer region is ad-jacent to the bilayer domains, exhibiting negligible photoresponse. Thescale bar is 1 \u00b5m. b, Back-gate voltage dependence of the photocurrentin the AB (Blue dots) and BA (red dots) domains. The inset of b, showsa schematic of charge transfer between 3R-MoS2 and graphene (black).Electrons (green) are partially layer polarized while holes (brown) areequally distributed between two layers. c and d, top panels are the biasdependence of the photocurrent (PC) in the AB (blue series) and BA(red series) domains. The dots represent the measured PC at each biasvoltage at different laser powers between 10 and 70 \u00b5W. The solid linesare fits to the data based on the model of IPC = IPV + IBOL + IPT E . IPVis the intrinsic photovoltaic effect. IBOL and IPT E are bolometric andphoto-thermal electric effect, both of which are thermal contributionsfrom graphene electrodes. Bottom panels are the extracted photovoltaicI-V dependence. The compensation voltage where photocurrent stops is\u00b10.3 V in AB and BA domains.51tion in the experiment. The spontaneous layer polarization at zero bias, determinedby the ratio between the experimentally measured interlayer potential and inter-layer coupling, is 66(\u00b13)% top and 34(\u00b13)% bottom for the AB domain, which isclose to the theoretical value predicted for the Q valley (59.1% top and 40.9% bot-tom)77. This agreement suggests the major contribution to the photocurrent arisesfrom the photocarrier in the Q valley, likely due to its extra valley degeneracy com-pared to the K valley. As the bias voltage becomes much larger than the interlayercoupling, the band edge becomes completely layer polarized and the AB-BA do-mains show identical saturation behavior. Overall, our model captures the mainfeatures of the bias dependence and confirms a dominant PV effect arising from thephoto-excited electrons.In contrast with 3R, the mirror symmetry is restored in the 2H-MoS2. Here westudy a 2H bilayer sample with the same configuration as a control device (C1). InC1, we observe a negligible PC at zero bias and the External quantum efficiency(EQE) is about two orders of magnitude smaller than D2. The bias dependence isalso totally different. The PC increases monotonically with external bias, which isknown to be caused by the bias-induced tunneling barrier difference between thetop and bottom electrodes125 (Fig.4.4). Without any spontaneous polarization, thetop and bottom graphene are undoped with a symmetric chemical potential, andthus are free of the PTE and BOL effects.As a commensurate crystal structure, the 3R stacking and its spontaneous po-larization induced PV effect can be scaled up beyond bilayer. By assuming a con-stant polarization, we calculate the DEP in a graphene-contacted device with differ-ent MoS2 thickness (Fig.4.5(a)). In agreement with the conventional ferroelectrictunneling junction, the DEP decreases with the thickness102, but the decrease isslower than 1\/(L-1), since the graphene doping increases with thickness. As aresult, the DEP-induced potential difference between the top and the bottom inter-facial MoS2 layers increases with thickness, leading to an increasing layer polar-ization. This enhanced layer polarization in combination with the higher opticalabsorption in the thicker device should improve the PV efficiency. In the thicklimit, other factors such as the optical skin depth, Schottky junction effects, andimperfect stacking order need to be taken into account to estimate the ultimate PVefficiency. In in-plane devices, the DEP and its related PV effect become negligible52-1 0 1Vbias (V)-0.03-0.02-0.0100.010.020.03I PC (A)2H bilayerFigure 4.4: Bias dependence of the photocurrent in the 2H bilayer device,C1. With a similar laser illumination condition (P = 20 \u00b5W ), C1 showsa near zero photocurrent under zero bias. The photocurrent linearlyincreases with the bias, with no thermal contribution observed.as the polar material becomes a few \u00b5m long99;101.To verify the layer dependence prediction, we fabricated another device withabout ten layers of 3R-MoS2 (D3). Compared with D2, the photoresponse of D3is significantly stronger, reaching a responsivity of 70 mAW\u22121 or an EQE of 16%.The high efficiency suggests the interlayer relaxation within MoS2 is no slowerthan the ultrafast interfacial charge transfer between MoS2 and graphene2. In con-trast with previous bulk photovoltaic devices based on WSe2 101, our photocurrentincreases linearly with the laser power up to \u223c mW level and does not becomesquare-root power dependent in the saturation regime, confirming again a differentorigin. The estimated shift current response in 3R-MoS2 bilayer along the out-of-plane direction is one to two orders smaller than what we observed in the experi-ments (D1, D2, and D4) (Fig.4.6), indicating the shift current is not the dominantmechanism in our device.Finally, we benchmark our device against other unconventional photovoltaicdevices. Compared with the most efficient devices based on the bulk photovoltaiceffect99;101, ours is over ten times more efficient, polarization-independent, and53ab2 4 6 8 101234567891011Depolarization Field (V\/m)1071\/(L-1)L (Number of Layers) 10-6 10-5 10-4 10-3 10-2Power (W)10-410-310-210-1100101Photocurrent (A)0Power (mW)00.40.81.21.63 6I (A)pcFigure 4.5: a, Simulated depolarization field (DEP) strength versus the MoS2thickness (L) assuming a thickness-independent polarization (blackrectangular). The blue dash line decreases as 1\/(L-1), which is fasterthan the calculated DEP-layer dependence, indicating a stronger re-sponse in the thicker device. b, Laser power dependence of the pho-tocurrent at zero bias for device D2 (black) and D3 (orange). The dotsare experimental data and the solid lines correspond to a linear depen-dence fit. The external quantum efficiency of D3 is one order of magni-tude larger than that of D2. The inset shows the power dependence ofD2 in the linear scale.54abcd012345Responsivity (mA\/W)10 \u00b5m 5 \u00b5m -80-60-40-200Responsivity (mA\/W)10 \u00b5m 5 \u00b5m Figure 4.6: a and b, Optical image of D3 and D4. D3 is composed of a\u223c 10-layer MoS2 device. D4 is a 3R bilayer device. Brown curvesoutline the 3R MoS2 area and black dashed lines represent the top andbottom graphene. c and d, Scanning photocurrent map of D3 and D4.In D4, there is a small bubble area where the responsivity is lowest. Theresponsivity of D3 (60-70 mA\/W) is one order of magnitude larger thanthat of D4 (3-5 mA\/W).gives a much larger photocurrent before saturation. Despite not being ideal forenergy-harvesting applications due to its negligible Voc, our device has a large zero-bias EQE, which is comparable with the other TMD-based photo-detectors relyingon the extrinsic electric fields, either induced by the gate voltage94;118 or in thep-n and Schottky junctions118;126. Compared with these extrinsic effects, our de-vice benefits from requiring zero external voltage and involving less interfaces.Thescalability, fast carrier extraction, and potentially switchable spontaneous polar-55ization make it possible to build memory-integrated photodetectors based on 3R-MoS2 67;69. Although similar DEP induced PV effect has been observed in polar ox-ides103;105, our device explores the atomically thin limit of this effect. The exciton-enhanced light-matter interaction and smaller band gaps in TMDs also make themmore appealing for many optoelectronic applications32;127.4.1.3 Electrostatic modelling on the Gr\/3R-MoS2\/Gr device1. Depolarization FieldThe mechanism accounting for the photocurrent generation inside the device isattributed to the depolarization field induced by spontaneous out-of-plane polariza-tion in 3R-MoS2. In traditional metal-polar insulator-metal junctions, most of thedepolarization field is compensated by the induced charges in the metal. However,if the metal is atomically thin like graphene, most of the field will be preserved.To model the depolarization field (DEP) inside our GR\/3R-MoS2\/GR device,we adopt the polarization of 3R MoS2 bilayer from our Berry phase calculationP \u2248 0.60 \u00b5C\/cm2, which is close to the reported value from density functionaltheory (DFT)78.This polarization will induce the same amount of charges but with oppositesigns in the top and bottom graphene layers. The total field inside the junction isdenoted as EDEP. In this case, EDEP should satisfy a self-consistency equation.\u00b5t \u2212\u00b5b = eEDEPd1\u2212 e(Et +Eb)d2 (4.1)where \u00b5t and \u00b5b are the chemical potential of top and bottom graphene, d1 =0.70 nm is the interlayer distance, d2 is the van der Waals gap between 3R MoS2and graphene interface. Et and Eb are field within the gap. For monolayer graphene,the chemical potential is related to the carrier density n and it is given by\u00b5 =\u00b1\u210fv f\u221a\u03c0 |n+n0| (4.2)where v f is the Fermi velocity and n0 is the initial doping of graphene. Here weassume n0 is much smaller than polarization induced doping and therefore ignorethe initial doping in the following calculation. n and \u00b5 are positive when graphene56is electron doped and become negative when graphene is hole doped.The displacement field is continuous inside the junction when the 3R-MoS2 ischarge neutral. Thus,P\u2212 \u03b5MoS2\u03b50EDEP = \u03b52\u03b50Eb = \u03b52\u03b50Et (4.3)where \u03b52 is the relative permittivity in the gap, which we take as \u03b52 =\u03b5MoS2+\u03b5Gr2 .Here we take \u03b5MoS2 = 7.0 and \u03b5Gr = 3.0.Based on equations (4.1)-(4.3), we can calculate the depolarization field (DEP)within the tunnel junction. Comparing with the value without any screening E0 =P\u03b5MoS2\u03b50d1, we can conclude thatEDEPE0\u2248 0.95 (4.4)Thus based on our calculation, about 95% of the depolarization field is preservedin the polar material when graphene is used as a contact.2. Gate tunabilityTo simulate the doping density in the device when a backgate voltage Vg isapplied, we adopt a modified electrostatic model based on the one previously usedin 2H devices128 by taking an out-of-plane polarization into consideration. In theundoped case, the Dirac points of graphene are near the middle of the MoS2 bandgap because of their similar work functions94. As a result, most carriers will dopeinto graphene rather than MoS2 and the displacement field in the junction willremain continuous. Our device D2 is encapsulated with hBN with thickness ofabout 30 nm and an out-of-plane relative permittivity \u03b5hBN = 2.7.Based on the capacitance model and electrostatic boundary condition, the chargedensities in the bottom and top graphene layers are given by\u2212enb =\u2212\u03b5hBN\u03b50 VgdhBN + \u03b52\u03b50Eb (4.5)and\u2212ent =\u2212\u03b52\u03b50Et (4.6)57a bc d-5 0 5Vg (V)-4-2024Doping density (cm-2)1012GrbGrtg-5 0 5V  (V)-200-1000100200EF (meV)GrbGrt-5 0 5Vg (V)-200-1000100200EF (meV)GrbGrt-5 0 5 (V)-4-2024Doping density (cm-2)1012GrbGrtVgFigure 4.7: a,c Simulated top and bottom graphene doping levels versus gatevoltage for AB and BA stacking, respectively. b,d, Simulated top andbottom graphene Fermi energy versus gate voltage for AB and BAstacking, respectively. Black represent the bottom graphene while bluedenotes the top.Using the self-consistent equation (4.1), we can simulate the top and bottomgraphene Fermi-level change with gate voltage. The simulation results are shownin Fig.4.7.For AB stacking, the photo-excited electrons move towards the bottom graphenebecause the polarization is upward. The charge transfer at the interface is domi-nated by the inelastic tunneling due to the momentum mismatch, so the chargetransfer rate should be proportional to the total number of empty states in graphenebetween the Dirac point and MoS2 band edge. When a large positive gate (8V)is applied, significant amount of electrons are doped into the bottom graphene(Fig.4.7(a)), where the Fermi energy becomes 230 meV above the Dirac point(Fig.4.7(b)) (Here we take P \u2248 1.0 \u00b5C\/cm2 for simplicity). As a result, the inelas-tic tunneling of electron between graphene and MoS2 becomes suppressed, whichreduces the photocurrent.As for BA stacking, the photo-excited electrons move towards the top graphene,58opposite to the AB case. The gate dependence of the doping density and Fermi en-ergy in both graphene electrodes are shown in Fig.4.7(c) and (d). The Fermi energyrising of the top graphene in BA domain is about 150 meV under Vg = 8 V, whichis smaller but has the same order of magnitude compared to the bottom graphene inAB domain. The calculation indicates that the electron transfer rate at the GR\/TMDinterface as well as the photocurrent strength in BA domain can also be suppressedby the doping induced by a positive back gate.4.1.4 Multiple photocurrent components in the Gr\/3R-MoS2\/Grdevice1. Bolometric EffectThe negative differential photoconductivity at large bias in D2 can be explainedby the bolometric effect in the device, whose resistance increases upon laser heat-ing. Under the bias, the device resistance increase leads to a total current decrease,equivalent to a negative photoconductivity. The bolometric effect is reported to bestronger in doped graphene123;129, which is the case in our junction area. Based onOhm\u2019s law, this photocurrent from resistance change can be written asIBOL =VR+\u2206R\u2212 VR=\u2212 VR2\u2206R (4.7)where\u2206R = (dRdT)\u2223\u2223\u2223\u2223T=300K\u2206T (4.8)where R is the device resistance under dark condition. And \u2206R is the laser-induced resistance change. Here we assume the device is always at room temper-ature in the dark condition by neglecting the Joule heating induced by the bias.Equation (4.8) suggests IBOL is a straight line with negative slope and crosses theorigin. Since the temperature change is proportional to the incident laser power,we use IBOL = \u2212\u03b2 (P) VR2 to fit the bolometric contribution in Fig.4.3. The fittingresults are shown in Fig.4.8.The bolometric contribution is clearly seen in the bias dependence of the pho-tocurrent in the monolayer region in device D2 (Fig.4.9). Since the monolayer59doesn\u2019t have any spontaneous polarization, the bias dependence is dominated bythe bolometric effect in the device, with the negative slope at two bias ends ofabout -0.03 \u00b5A\/V, corresponding to a photo-responsivity of 0.75 mA\/W\/V whichis close to the reported value123. At zero bias, only close-to-zero photoresponse isobserved. A small range near zero bias shows a positive slope, which is a manifes-tation of the bias-induced tunneling barrier difference between the top and bottomelectrodes, similar to the 2H bilayer result. This positive-slope contribution is sat-urated and covered by the graphene bolometric effect at large bias.Next, we estimate the maximal light-induced temperature increase. When thelaser is shone on the junction area, the sample temperature increases compar-ing with the surrounding area, which can be modeled by the radical equation122:2\u03c0\u03badGr\u2206T = P\u03b1 where \u03ba = 5\u00d7 103 Wm\u22121K\u22121 is the in-plane thermal conduc-tivity of graphene, dGr = 0.3 nm is the graphene thickness, \u03b1 is the absorptionof the junction including bilayer MoS2, estimated to be 0.20, and P is the laserpower. Since the maximum laser power used in Fig.4.3 is 72 \u00b5W , the largest tem-perature increase is estimated to be 1.5 K. We expect the temperature increase inreality is smaller than this since the thermal conduction in hBN and substrate arenot considered in this model.2. Photothermal electric effectThe photothermal electric effect (PTE) accounts for the non-zero thermal con-tribution to the photocurrent at zero bias in Fig.4.8. PTE exists because the sponta-neous polarization induces an asymmetric doping in the graphene electrodes. Tak-ing AB stacking as an example, the top and bottom layer are electron and holedoped in the overlapping area with different Seebeck coefficients, St and Sb. Asa result, the photothermal voltage becomes VPT E = (St \u2212 Sb)\u2206T . The direction ofsuch a voltage is opposite to the DEP. By using a room-temperature Seebeck co-efficient in the slightly doped graphene (S \u2248 \u00b1100 \u00b5V\/K)130;131, we estimate themagnitude of the PTE-based photocurrent and compare it with the experimentalvalue (Fig.4.10). The calculation is based on the temperature estimation discussedin the subsection of bolometric contribution without hBN encapsulation involved,so the estimation is, unsurprisingly, larger than the experimental values.The PTE effect is sensitive to doping. At certain back gate voltages (negative forAB and positive for BA as shown in Fig.4.7), the PTE current is expected to vanish60-1 0 1Vbias (V)-1-0.500.51-1-0.500.51I thermal (A)Plaser (   W)10 7010 70Plaser (   W)Figure 4.8: Extracted thermal contributions of the AB (top panel) and BA(bottom panel) domains in device D2. The straight lines with negativeslopes are due to bolometric effect IBOL, which crosses zero at zero bias.The non-zero intercept is caused by the photo-thermal electric effectIPT E , which is always opposite in direction to IPV . The IPT E is approxi-mately independent of the bias.61-1 0 1Vbias (V)-0.1-0.0500.050.1I pc (A)MonolayerFigure 4.9: Bias dependence of the photocurrent in the monolayer area ofD2. The laser power is 20 \u00b5W . Photocurrent at zero bias is negligiblecompared with the bilayer area. Negative differential photoconductivityis observed at large bias.when the graphene doping becomes identical132;133. Since the IPT E and IPV haveopposite directions, the decrease of IPT E leads to an enhancement in the total pho-tocurrent. In the negative back-gate range of the AB domain where both grapheneelectrodes become equally hole-doped, such enhancement is visible since the IPVis mostly unchanged by hole doping. The similar enhancement in the electrondoping side of the BA domain is covered by the fast decrease of the photovoltaiccontribution.3. Photovoltaic effectAs discussed in the section 4.1.2, the intrinsic photovoltaic effect is dominatedby electrons in the conduction band edge. Depolarization field (DEP) induced in-terlayer potential difference leads to a complete or significant layer polarization inconduction band electrons at K or Q point. Thus, a net imbalance of interfacialcharge transfer between MoS2 and graphene due to the finite layer polarizationleads to a photovoltaic current IPV , which is proportional to the layer polarization.Here we assume electrons in top layer all tunnel into the top graphene while thosein bottom layer all tunnel into the bottom electrode. The external bias can either620 20 40 60 80Plaser ( W)00.020.040.060.080.10.120.14I PTE (A)TheoryExperiment0 20 40 60 80Plaser ( W)-0.14-0.12-0.1-0.08-0.06-0.04-0.020I PTE (A)TheoryExperimenta bFigure 4.10: Photocurrent contribution induced by the photothermal electriceffect. a, Black dots are experimental values from the bias dependenceseries in the BA domain. Red line is from the theoretical calculationbased on the reported Seebeck coefficient130;131. b, Black dots are theexperimental values and the blue line is from the theoretical calculationof AB stacking. The overestimation could be caused by a much lowersample temperature due to the thermal conduction in hBN.enhance or reduce the layer polarization depending on whether external field isparallel or anti-parallel with the DEP. The impact of the external bias on the layerpolarization and PV effect can be analysed using an effective coupled two-bandmodel, as shown below. The effective Hamiltonian is given byHeff =[\u2212(\u2206eff2 + eVbias2 ) \u03b3\u03b3 (\u2206eff2 +eVbias2 )]where \u2206eff is the DEP induced interlayer potential and \u03b3 is the interlayer coupling.In the lower conduction band edge, an electron wavefunction is in a superpositionstate of the top and bottom layer, |\u03a8\u27e9= c1 |b\u27e9+ c2 |t\u27e9 and the photovoltaic currentis proportional to |c1|2\u2212|c2|2. As a result, we getIPV = I0\u221a1+( 2\u03b3\u2206eff+Vbias )2\u22121( 2\u03b3\u2206eff+Vbias )2\u2212\u221a1+( 2\u03b3\u2206eff+Vbias )2+1(4.9)63where I0 is the saturation current when the bias is sufficiently large and the twobands become completely layer polarized.The bias dependence of the photocurrent can be quantitatively understood bycombining the photovoltaic, bolometric and photothermal electric effects.IPC = IPV + IBOL+ IPT E (4.10)Applying nonlinear global fitting of this model to the two series of bias depen-dence curves, we can separate the intrinsic photovoltaic effect and thermal effects.The extracted interlayer potential \u2206eff is 0.30(\u00b10.01) V, whose sign depends on thestacking order, positive and negative for AB and BA, respectively, and the inter-layer coupling \u03b3 is 0.43(\u00b10.02) eV. Uncertainty is from the global fitting. The fitted\u2206eff is a few times larger than the DFT calculation results134, possibly because onlya fraction of applied bias drops across the tunneling junction. Nevertheless, the ra-tio between \u2206eff and \u0393 is independent of the voltage drop reduction. Based on thisresult, we calculate the layer polarization under zero bias is 66(\u00b13)% of top and34(\u00b13)% bottom (AB stacking), which is close to the DFT calculation result at Qpoint (59.1% and 40.9%)77. The deviation could be due to the contribution fromK and \u0393 points or the uncertainty in the van der Waals gap in the DFT calculation.4. BPVE responseThere could be another photocurrent component in our device, which is theshift current response from the bulk photovoltaic effect(Bulk photovoltaic effect(BPVE)). Next, we estimate the shift current response in our bilayer 3R-MoS2.We consider the case of shift current Jz along the z-direction generated by lightwith linear x-polarization (characterized by electric field Ex), which is directly rel-evant to our photo-current measurement. The shift current response \u03c3xxZ is definedvia the relation: Jz = \u03c3xxZE2x , where we use the capital letter Z to denote the currentdirection in the response function.According to the general formula introduced in the Reference97, the shift cur-64rent response for a bulk crystal can be expressed as:\u03c3xxZ = 2\u03c0e(e\u210fm\u03c9)2\u2211cv\u222b d3k(2\u03c0)3\u27e8v,k|P\u02c6x|c,k\u27e9\u27e8c,k|P\u02c6x|v,k\u27e9Rz(k)\u00d7\u03b4 (\u03c9c\u2212\u03c9v\u2212\u03c9),(4.11)where c,v denotes conduction and valence band indices, P\u02c6x denotes the momentumoperator along x-direction, Rz(k) is known as the \"shift vector\" along the z-directionwhich is essentially given by the inter-band Berry connection135.To estimate \u03c3xxZ for atomically thin 3R MoS2, we first observe that the integra-tion of Rz(k) over kz essentially amounts to the relative spatial shift in Wanniercenters within a unit cell along the z-axis which is spanned by inter-layer dis-tance dz, thus in the atomically thin limit we replace\u222b dkz2\u03c0 Rz(k) by the constant\u03bbz = Rz\/dz, where Rz denotes the spatial difference between Wannier centers ofconduction and valence band states. Based on the Wannier centers of conductionand valence band states extracted from our tight-binding model, we found a typicalvalue of \u03bbz \u2248 5\u00d7 10\u22123. Next, we note that the value of the inter-band velocityterm Ve f f = \u27e8c,k|P\u02c6x|v,k\u27e9\/m can be extracted from the well-known k \u00b7 p model forMoS2 14 , with Ve f f \u2248 ate f f\u210f , where a\u2248 0.32 nm is the in-plane lattice constant andte f f \u2248 1.1 eV is the effective hopping energy. These observations allow us to obtainan approximate analytic expression for the shift current response:\u03c3xxZ =\u03bbze3a2t2e f f m\u2217(\u210f\u03c9)2\u210f3. (4.12)Here, m\u2217 \u2248 0.4 me 136 is the effective mass of electron band in MoS2, \u210f\u03c9 = 2.33eV denotes the photon energy of the 532 nm laser used in the experiment. The shiftcurrent response is estimated to be \u03c3xxZ \u223c 1\u00d710\u22127A \u00b7V\u22122.In our experiment, the typical responsivity at 532 nm of bilayer our devices(D1, D2, D4) is around 1-10 mA\/W. The size of the laser spot is around 0.5 \u00b5m.According to these experimental conditions, total photo-current response \u03c3tot. \u22482\u00d710\u22126\u22122\u00d710\u22125A \u00b7V\u22122. It is one to two orders larger than the estimated shiftcurrent response \u03c3xxZ \u223c 1\u00d710\u22127A \u00b7V\u22122.Based on these estimations, we expect that the shift current from BPVE only65contributes to a small percentage of the total photo-current observed in our bilayerdevice and it is not the dominant mechanism in our devices.4.1.5 The mechanism of exciton dissociationIn this section, we discuss the exciton dissociation mechanism in GR\/3R-MoS2\/GRjunction. The strong photocurrent response in our experiments suggests an efficientexciton dissociation and carrier extraction. Generally, we expect there are multipleexciton dissociation mechanisms in our devices, depending on the device thickness.For 3R-MoS2 bilayer, the DEP induced interlayer potential is about 50 meV(P \u2248 0.6 \u00b5C\/cm2) according to our tight-binding calculation. Meanwhile, thebinding energy of the K \u2212K interlayer exciton is \u223c 100 meV smaller than thebinding energy of intralayer exciton77;136. Therefore, the binding energy differ-ence between intra- and interlayer excitons is similar as interlayer potential, indi-cating that the DEP is sufficient to separate the electron and hole into two layers viainterlayer tunneling, as shown in Fig.4.11(a). When the electron and hole reachesthe graphene and MoS2 interface, the electronic band alignment plays an importantrole on the exciton dissociation. Because of the large energy difference betweenthe band edge of 3R-MoS2 and Graphene Dirac point (larger than \u223c 0.1 eV)137,the exciton binding energy can be compensated when the carriers tunnel into thestates near the Fermi-level of graphene, similar to what happens at the interfaceof the organic solar cell and metal contacts119, shown in Fig.4.11(b). Due to theultrafast charge transfer within TMD layers and at the GR\/TMD interface137\u2013139, theexciton dissociation and carrier extraction are efficient enough to generate a strongphotoresponse in our devices.In the trilayer or thicker layer 3R-MoS2 , the electron of the interlayer excitoncan continue to tunnel into the other neighbouring layer, since the interlayer exci-ton binding energy further decreases as the Bohr radius increases36 (Fig.4.11(a)).In the bulk limit, the electron and hole can be separated and can transport to the bot-tom and top interfaces through a cascade of tunneling under the DEP field, whicheventually frees the carriers from their bound state. In other words, the exciton getsdissociated via tunneling ionization. The tunneling ionization rate can be quantita-66tively estimated by the asymptotic solution at the weak-field limit140:\u0393\u2248 12\u03c90(2k2FDEP)2Z\/k\u22121exp(2kDz\u2212 2k33FDEP) (4.13)The dimensionless parameters in equation (4.13) are k =\u221a2 m\u00b5 (ab)2\u039einterb\u210f2 , Dz =dzab, Z = m\u00b5e2ab4\u03c0\u03b5\u210f2 and FDEP =m\u00b5 (ab)3eEDEP\u210f2 , respectively. Other parameters are inatomic units: the frequency \u03c90 = 13.6eV\u210f ; m\u00b5 denotes the reduced effective mass ofthe interlayer exciton; ab is the Bohr radius of the hydrogen atom. The \u039einterb \u2248 0.3eV is the interlayer exciton ground state (1s) binding energy estimated from thereferences77;121. The interlayer exciton has an out-of-plane dipole dz \u2248 0.6 nm.Since eEDEPdz << \u039einterb , the weak field assumption is valid in our case140. Weapproximate EDEP to be 0.1 V\/nm and take the effective mass of electron and holeme = 0.38 m0, mh = 0.46 m0 136. We get the interlayer exciton dissociation rate \u0393,which is on the order of 1012\u22121013 Hz. Such a fast rate corresponds to a dissoci-ation time of less than 1 ps, which is shorter than charge transfer time at GR\/TMDinterface137.4.2 Ultrafast response of spontaneous photovoltaic effectin 3R-MoS2\u2013based heterostructures4.2.1 IntroductionWhen two layers of transition metal dichalcogenides (TMDs) are stacked in paral-lel, both the inversion symmetry and mirror symmetry are spontaneously broken,leading to an electric polarization along the out-of-plane direction77. Under anelectric field larger than the coercive field, the polarization direction can be flippedas one layer slides relative to the neighboring layer, which has been termed slid-ing ferroelectricity68;70;71;141. While the artificially stacked bilayers have limiteddomain sizes due to twist misalignment, homogeneous polarization in a scale aslarge as the entire flake has been observed in bilayers directly exfoliated from a3R-MoS2 bulk crystal117, making it ideal for certain optoelectronic applications.In particular, when the 3R-MoS2 is sandwiched between graphene73, the sponta-67Edep=0Edep>0a b clayer 1layer 2layer 3P\u2206E Figure 4.11: a, Schematic of electron and hole separation under the uniformDEP. The cascade electron tunneling happens through the layers in afew layer 3R-MoS2. b, Illustration of interlayer charge transfer at theGr\/MoS2 interface. The large energy difference between the the bandedge of MoS2 and Dirac point compensate the exciton binding energyand facilitate the exciton dissociation. c, illustration of tunneling ion-ization process. The black lines represents the Coulomb potential bar-rier in the absence (dash) and presence (solid) of depolarization field,respectively. Solid blue curve denotes the interlayer exciton (1s) enve-lope wave function.neous polarization gives rise to a photovoltaic effect, known as spontaneous photo-voltaic effect101, where photoexcited carriers in MoS2 transfer asymmetrically tothe graphene under a largely unscreened depolarization field. An external quantumefficiency (EQE) of 16% has been observed in a device with ten-layer 3R-MoS2.The high EQE in the graphene\/3R-MoS2\/graphene heterostructures suggestsa potentially ultrafast photocurrent dynamics comparable to the charge-transferprocess at the MoS2-graphene interface, which lasts approximately one picosec-ond138;139;142\u2013144. Such an ultrafast photocurrent, if confirmed, can potentially beleveraged for high-speed optical communications. Although the MoS2 photocarrierdynamics generally can be measured by ultrafast techniques such as pump-probespectroscopy2, it remains a challenge to probe the intrinsic dynamics of the afore-mentioned photovoltaic effect, due to the existence of multiple photocurrent con-tributions. Because the spontaneous polarization naturally induces image chargesof opposite polarities at the two electrodes, both the bolometric and the photo-thermoelectric (PTE) effects exist in the device123;130;145\u2013148. In addition, the asym-68metric tunnelling barrier is also known for generating a photocurrent without exter-nal bias125;149. Here, we disentangle these electronic and thermal contributions andmeasure their corresponding response times by performing ultrafast photocurrentautocorrelation and non-degenerate pump-probe photocurrent measurements. Weconclude that the photocurrent has approximately 2-ps fast dynamics for devicesof various thicknesses and a slow dynamics that lasts for 25 ps or more.Ultrafast optical techniques are powerful tools for probing the photocarrier dy-namics2;139;150. Photocurrent autocorrelation technique has been previously usedto study the photocurrent dynamics128;133;151;152 in 2D-material based photode-tectors. In this approach, two equal-power laser pulses are split from a singlefemtosecond pulse and illuminate the heterostructure to generate a photocurrent(Fig.4.12(a)). When these two pulses coincide on the device, a saturation in thephotocurrent can be observed due to either electronic or thermal mechanisms. Forexample, electronic saturation may arise from absorption saturation and increasedinteractions between carriers, whereas thermal saturation may arise from the non-linear temperature increase under illumination153\u2013156. By scanning the delay be-tween these two pulses, a characteristic time associated with the saturation mecha-nism can be measured. When multiple mechanisms contribute to the photocurrent,the device response speed is usually limited by the slowest process. The intrinsicphotocurrent time response can be obscured.4.2.2 Ultrafast photocurrent responseWe first perform the photocurrent autocorrelation measurement on a heterostruc-ture composed of both bilayer (BL) and four-layer (4L) 3R-MoS2. The opticalimage of the heterostructure is shown in Fig.4.13. From the current-voltage char-acteristics shown in Fig.4.12(b), a spontaneous photovoltaic behavior is clearlyobserved. The zero-bias photocurrent mapping shows a finite photoresponse inBL and 4-layer regions and the photocurrent distribution agrees with the MoS2thickness in the graphene overlapped area (Fig.4.12(c)). Such photocurrent growssublinearly with optical power (Fig.4.5), giving rise to a negative autocorrelationsignal. A representative autocorrelation result is shown in Fig.4.12(d). A dip of thephotocurrent strength is observed at zero delay, with symmetric recovery dynam-69Figure 4.12: (A) A schematic of the device and ultrafast time-resolved pho-tocurrent measurement. The heterostructure consists of a bilayer orfew-layer 3R-MoS2 flake that is sandwiched between two grapheneelectrodes. The whole device is encapsulated in hBN. The photocur-rent is measured with a lock-in amplifier through graphene electrodes.The two ultrafast pulses are delayed by a time delay \u2206\u03c4 , and focusedcollinearly on the device. (B) Photocurrent-voltage characteristics ona bilayer device at room temperature. The device shows photovoltaiceffect with nonzero photocurrent without bias. (C) Photocurrent mapof a device consists of both bilayer and 4-layer regions at zero bias.The resolution of the scan is 400 nm in both directions. The darkerblue regions are also bilayer but with a different ferroelectric polarity.Scale bar: 2 \u00b5m. Both (B) and (C) are measured under 20 uW 532 nmCW laser illumination. (D) Autocorrelation signal from the BL and 4Lregion with 770 nm femtosecond pulse excitation, which is above theindirect bandgap of the material.70Figure 4.13: The device region where graphene electrodes overlaps coversboth bilayer and 4-layer MoS2. Scale bar: 8 \u00b5m.ics at both positive and negative delays, as expected for conventional photocurrentautocorrelation signals128;133;151;152.More quantitative analysis of the autocorrelation signal in the bilayer area ispresented in Fig.4.14. With a fluence of 140\u00b5J\/cm2 for a spot size of approxi-mately 1\u00b5m in diameter at 770 nm (50 fs pulse width) for each pulse, the saturationdip at zero time delay is approximately 15% of the steady state signal. The recov-ery time is fitted to be 17 ps, more than one order of magnitude slower than thecharge-transfer time at the MoS2\/graphene interface138;143, suggesting the satura-tion mechanism is likely not electronic. If such a slowing is caused by the chargetransfer at the MoS2\/MoS2 interface, a previous study suggests an external elec-tric field can speed up the process when the TMD is not monolayer128. However,when we apply a bias voltage from -0.5 V to 0.5 V across the device, no significantchange is observed in the recovery time (Fig.4.14(a)).Instead, we suggest that this photocurrent saturation originates from photo-thermal effects. As shown inFig.4.14(c), the photoresponse of our device has a surprisingly large temperaturedependence. When the heterostructure is heated by 10 K above room temperature71(RT), the photocurrent drops by more than 25% (Fig.4.14(c) inset). More strik-ingly, the photoresponsivity increases by more than one order of magnitude aftercooling the heterostructure from RT to 3 K (Fig.4.14(c), full temperature range inFig.4.21(a)). At the A-exciton resonance, the EQE approaches 10 %, which is re-markable for an atomically thin device. Besides that, we also observe a significanttemperature dependence of the total resistance of the device (Fig.4.21(b)), whichmotivates us to develop a shunt resistance model. In this model, within a few tensof Kelvin above room temperature, which is relevant to our pump-probe photocur-rent measurement, only the shunt resistance changes due to its tunnelling currentorigin157. By modelling the measured photocurrent to be proportional to the shuntresistance, which exponentially decreases with increasing temperature, we are ableto fit the temperature dependence of the photocurrent (Fig.4.14(c) inset, detailsin the section 4.2.4). Other factors that can contribute to this strong temperaturedependence include the changes in spontaneous polarization69;158, photocarrier re-combination time, and exciton linewidth and wavelength, which affects the absorp-tion of the heterostructure. As a result, when an ultrafast laser pulse is absorbed bythe device, the transient temperature of the illuminated region quickly rises, whichdecreases the shunt resistance and subsequently decreases the photocurrent gener-ated by the following pulse. In this case, the saturation recovery time correspondsto the device heat dissipation time. At low temperatures, the photocurrent becomesless temperature dependent (Fig.4.21(a)), which is consistent with the smaller au-tocorrelation signal observed at 3K, possibly due to the internal quantum efficiencylimit.We confirm the photo-thermal origin of the saturation by measuring the pho-tocurrent autocorrelation at cryogenic temperature (Fig.4.14(b)). After most phononmodes are frozen at low temperature, the thermal conductivity of the 2D materialssignificantly decreases159;160, which should lead to a longer heat dissipation time.Experimentally, we find a recovery time that is approximately three times longer(55 ps) than that at RT, when the heterostructure is cooled to 3 K. All these evi-dences suggest that the dynamics observed in the autocorrelation measurements islimited by the device heat dissipation.In addition to this long recovery time, we also observe a transient middle peakin the autocorrelation measurement at 3 K. Exponential fitting of that small peak72Figure 4.14: Autocorrelation signals of the bilayer region measured at (A)room temperature and (B) 3K. The dotted lines are from experimen-tal measurements, and black lines are exponential fittings to extract thetime constants. Inset of (A): photocurrent time constant extracted frombias-voltage-dependent autocorrelation measurements. The dashedline indicates a photocurrent time constant, \u03c4 , of 17 ps. (C) Energy-dependent photoresponsivity of the bilayer region measured at roomtemperature and 3K. The A and B peaks correspond to the A-excitonand B-exciton absorptions in bilayer 3R-MoS2. The average power iskept at around 0.5 \u00b5W at all wavelengths to avoid saturation. Inset:photocurrent of the BL region at temperatures above RT with 532 nmCW laser excitation. The dots are experimental data and the solid lineis fitted based on the circuit model.73Figure 4.15: (A) Autocorrelation signal from the 4L region on the device pre-sented in this section under 770 nm femtosecond pulse excitation. (B)Autocorrelation signal from a pure BL device under 800 nm femtosec-ond pulse excitation. The peaks at zero delay are from the PTE pho-tocurrent saturation as discussed in this section.suggests a time constant of approximately 4 ps. Similar peaks have been observedin the 4L (Fig.4.15(a)) region and on other bilayer devices of varying amplitudesup to RT (Fig.4.15(b)). We interpret this feature as a result of the electronictemperature saturation in the graphene electrodes. As discussed in our previouswork73, the top and bottom graphene electrodes are naturally doped with chargesof opposite polarities in these devices, which consequently acquire opposite See-beck coefficients. Upon laser heating, a PTE current in a direction opposite to thephotovoltaic-current is generated, which reduces the net photocurrent. In the au-tocorrelation experiment, the two subsequent laser pulses with a small delay timesaturate the graphene electronic temperature, which decreases the PTE effect andtherefore increases the overall photocurrent. The characteristic time of the middlepeak also agrees with graphene electronic temperature relaxation through super-collision151;152;161.Since the photo-thermal effect of the laser pulse obscures the electronic re-sponse speed in autocorrelation measurements, an alternative method is neededto separate the thermal and electronic contributions. Here we develop a non-degenerate pump-probe photocurrent spectroscopy technique that permits the de-termination of the intrinsic speed of our devices (Fig.4.16(a)). First, we use astrong sub-bandgap infrared pulse (1030 nm, 50 fs, 350\u00b5J\/cm2, IR-pulse) to heat74the graphene electrodes, which subsequently heats the MoS2. Second, we employa weak visible pulse (670 nm, 50 fs, 2\u00b5J\/cm2, VIS-pulse) that is resonant with theA-exciton of MoS2 to generate a photocurrent pulse but without generating muchheat.At negative time delays when the VIS-pulse arrives earlier than the IR-pulse,even though the IR-pulse generates a finite photocurrent, the photo-thermal satu-ration effect should be negligible because little heat is deposited by the VIS-pulse(details in Section 4.2.3). The photocurrent should decrease significantly imme-diately after the zero delay since the substantial heating induced by the IR pulsewould reduce the photocurrent generated by the VIS-pulse, in accordance withthe results from our autocorrelation experiments. At a large positive delay, thephotocurrent should be restored to a steady state as the heat is fully dissipated.Therefore, as we scan the time delay, we expect to see an asymmetric pump-probephotocurrent signal similar to photocarrier dynamics from an optical pump-probemeasurement, which exhibits a quick change followed by an exponential decreasein signal. Since the pump-probe photocurrent dynamics is approximately a cross-correlation between the photocurrent and device temperature dynamics, the decayof the signal is likely limited by the heat dissipation, while the rise of the signalshould be determined by the photocurrent decay and lattice heating.Experimentally, a highly asymmetric temporal response is indeed observed inthe pump-probe photocurrent measurement (Fig.4.16(b)). The pump-probe pho-tocurrent signal captures the photocurrent change due to thermal saturation. Asexpected, we find a very sharp drop of the photocurrent around the zero delay. Thedrop lasts for approximately 4 ps (Fig.4.16(b) inset), which gives an upper boundfor the characteristic time of the transient photocurrent. It is followed by a 20-psslow recovery dynamics similar to that observed in the autocorrelation experiment,corresponding to the cooling of the device. Such pump-probe results confirm theexistence of a picosecond photocurrent response. In addition, a small photocurrentsaturation is observed ahead of the sharp drop before zero delay, suggesting multi-ple dynamics could be associated with the photocurrent. The delayed IR-pulse canaffect the photocurrent generated earlier by the VIS-pulse if the photocurrent lastslong enough to overlap with the heating pulse. With the fast photocurrent com-ponent determined from the quick drop, the saturation long before the zero delay75indicates the existence of a slow photocurrent component. We further symmetrizethe pump-probe photocurrent signal by adding itself to its time-reversed copy, andthe result shows very much the same dynamics as the photocurrent autocorrela-tion signal (details in section 4.2.3). This observation confirms the validity of thispump-probe photocurrent technique to distinguish various photocurrent generationand saturation mechanisms.To have a quantitative understanding of the photocurrent dynamics, a phe-nomenological model is developed to model the joint response of the device tem-perature evolution (thermal pulse) and transient photocurrent (photocurrent pulse).In this model, each VIS-pulse generates two concurrent photocurrent pulses in thedevice with independent decay constants I(t) = I1e\u2212t\/t1 + I2e\u2212t\/t2 . Upon overlap-ping with the thermal pulse, the instantaneous photocurrent is reduced to \u03b1I, where\u03b1 is a dimensionless saturation factor and its temperature dependence is defined insection 4.2.4. As the device temperature is a function of time, \u03b1 is consequentlytime-dependent. After calculating the evolution of the transient lattice temperatureof the device excited by an IR-pulse at a delay of \u2206\u03c4 through a two-temperaturemodel (Section 4.2.5), we can obtain the average photocurrent through the cross-correlation relation I\u00af(\u2206\u03c4) = 1\u03c4rep\u222b \u03c4rep0 \u03b1(t)I(t +\u2206\u03c4)dt, where I\u00af is the average pho-tocurrent and \u03c4rep is the repetition time of the laser, which is much longer thanthe photocurrent response time. The calculated I\u00af fits our experimental results well(Fig.4.16(b)) with the transient photocurrent profile I(t) presented in Fig.4.16(c).The fast decay in the early stage has a time constant t1 of approximately 2 ps,which is similar to the graphene\/TMD charge-transfer time. The measured timeconstant is nearly independent of the power of the IR or VIS pulses. This char-acteristic time of the transient photocurrent approaches that of the charge transferat the graphene-TMD interface138;139;142\u2013144, suggesting a likely faster interlayercharge-transfer process at the MoS2 interface, similar as those observed in TMDheterostructures139. On the other hand, the slow component has a time constant t2of approximately 25 ps. It was previously found that the two-component photocur-rent can originate from the defect-related processes154;156;162, which agrees withthe dynamics we observe.To further confirm the electronic and thermal contributions in the photocurrentdynamics, a comparison is made between the BL and 4L regions with the pump-76Figure 4.16: (A) Illustration of the heating and cooling dynamics of the de-vice by a sub-bandgap pulse (IR-pulse). Graphene is firstly heated bythe IR-pulse and then heats MoS2. The photocurrent is generated bythe VIS-pulse, and it gets smaller when the device is thermally sat-urated upon heating. (B) Pump-probe photocurrent measurement ona bilayer device. The red dotted line represents experimental resultsand the solid black curve is the fitting from the numerical model. Themeasurements are carried out with an IR-pulse(VIS-pulse) fluence of350(2) \u00b5Jcm\u22122 . The pump-probe photocurrent signal is normalizedfor comparison with the numerical model. The 20 ps scan around zerodelay is performed at a finer temporal resolution than the rest. Inseton the left: A zoom-in semilog plot of the pump-probe photocurrentsignal for the 40 ps in the middle. The black straight lines are for guid-ance. (C) Transient photocurrent responses for different processes.The total profile consists of two photocurrent generation processes,with the fast and slow processes having characteristic times of 2 psand 25 ps, respectively.77probe photocurrent measurements. Similar to the BL region, an asymmetric timedependence before and after zero-delay as well as a sharp drop in photocurrentaround zero-delay are observed in the 4L (Fig.4.17(a)), indicating a similarly fastdynamics at the graphene\/MoS2 interface. Nevertheless, the pump-probe photocur-rent signal before zero-delay is much more prominent in the 4L than BL, suggest-ing a much larger slow component contribution in the thick area. This contrastmight result from a larger defect density in 2D or the extra MoS2-MoS2 interfacesin the 4L. Interestingly, an external bias can be applied to tune the slow componentof the photocurrent, which is observed before zero delay (Fig.4.17(a)). As fittedby the aforementioned model, the slow component becomes faster when the elec-tric field is along the photocurrent direction, and vice versa (Fig.4.17(b)). On theother hand, the recovery time at positive delay is dominated by the device coolingdynamics, and is therefore largely bias independent. These results further confirmthe thermal saturation nature and the existence of a two-component photocurrentresponse in the 3R-MoS2 photovoltaic device.In conclusion, we have observed a large temperature dependence in the sponta-neous photovoltaic effect in heterostructures comprising atomically thin 3R-MoS2and graphene. At low temperatures, the EQE can reach 10 % even in the thinnestarea. Upon laser heating, such a temperature dependence leads to a strong thermalsaturation of the photocurrent, which dominates in the autocorrelation measure-ment. Alternatively, we have developed a non-degenerate pump-probe photocur-rent spectroscopy technique, which can distinguish the electronic and thermal dy-namics. In particular, we find that the transient photocurrent has two contributionswith distinct temporal responses. We quantify the fast photocurrent response tobe on the picosecond level, which is similar to the charge-transfer process at thegraphene-TMD interface, suggesting an intrinsic device bandwidth of hundreds ofgigahertz. At room temperature, the photoresponsivity and speed of our homo-bilayer device are comparable to that of TMD heterobilayer devices with type-IIband alignment126;139;163. Our results may stimulate future uses of ferroelectricvan der Waals devices in optoelectronic applications that require high performancewith low power consumption and built-in memory function, such as in high-speedoptical communications and optical computing. In the future, the non-degeneratepump-probe photocurrent spectroscopy technique can also be applied to study the78Figure 4.17: (A) Bias-dependent pump-probe photocurrent measurements onthe 4L region with bias voltages from -0.25 V to 0.5 V. The measure-ments are carried out with an IR-pulse(VIS-pulse) fluence of approxi-mately 350(2) \u00b5Jcm\u22122. (B) Numerical results with the slow photocur-rent components having different response speeds from 35 ps to 50 psto model the bias-dependent measurement.interplay between various photo-excitations in other photocurrent devices.4.2.3 Non-degenerate pump-probe photocurrent measurementAs mentioned in the section 4.2.2, the signal measured by the non-degeneratepump-probe photocurrent spectroscopy can be expressed as a cross-correlation be-tween the photocurrent generated by the visible beam and heat generated by the IRbeam, which isI\u00af(\u2206\u03c4) =1\u03c4rep\u222b \u03c4rep0\u03b1IR(t)IV IS(t+\u2206\u03c4)dt, (4.14)79or in a compact formI\u00af(\u2206\u03c4) \u221d \u03b1IR \u22c6 IV IS, (4.15)where subscripts denote the pulses with which the physical processes are associ-ated.On the other hand, since the two beams are symmetric in the autocorrelationexperiment, each beam can cause a photo-thermal saturation effect on the pho-tocurrent generated by the other beam. As a result, the autocorrelation signal canbe expressed as the sum of two contributions.I\u00af(\u2206\u03c4) \u221d \u03b11 \u22c6 I2+\u03b12 \u22c6 I1, (4.16)where subscripts refer to the beam index. With the two beams being identical butare symmetric in time, we can rewrite the expression toI\u00af(\u2206\u03c4) \u221d (\u03b11 \u22c6 I2)(\u2206\u03c4)+(\u03b11 \u22c6 I2)(\u2212\u2206\u03c4). (4.17)The second term is the time-reversed copy of the first term, suggesting the autocor-relation signal can be reproduced by symmetrizing the pump-probe signal if onecan distinguish the effect of individual beams. In the following, we apply such asymmetrization procedure to the non-degenerate pump-probe photocurrent signal:I\u00af(\u2206\u03c4) \u221d (\u03b1IR \u22c6 IV IS)(\u2206\u03c4)+(\u03b1IR \u22c6 IV IS)(\u2212\u2206\u03c4). (4.18)As shown above, the symmetrized pump-probe photocurrent signal shares sim-ilar dynamics with the autocorrelation data collected in the same device at the sametemperature, which justifies our simple model. There exists a mismatch betweenthe two at zero delay, where the symmetrized signal has a sharper dip relatedto the ultrafast photocurrent response. We attribute the difference to the photo-thermoelectric (PTE) effect in the graphene electrodes, as discussed in the section4.2.2. When an ultrafast pulse heats up the electrons in graphene, the PTE effect in-duced by the other pulse is reduced, and thus causing a sharp feature with a positivesign. The magnitude of the PTE peak is highly device and temperature dependent,80and usually dominates the signal near zero delay in the autocorrelation experiment(Fig.4.12(b)). In contrast, the probe beam in the pump-probe experiment is veryweak and causes neglibile heating, thus avoiding the interference from the PTEeffect. This comparison highlights the benefit of the non-degenerate pump-probephotocurrent technique we developed.Another important detail we need to clarify is that the IR-pulse also generatesa finite photocurrent (Fig.4.18) in the non-degenerate pump-probe photocurrentmeasurement. Such a photocurrent could have multiple extrinsic origins such asa) hot-carrier injection from graphene164, b) two-photon excitation of photocarri-ers, c) excitation of MoS2 mid-gap defect states154;156;162. For case a) and b), thephotocurrent should have a linear or super-linear temperature\/power dependence,while for case c), the photocurrent should have a similar saturation behavior asabove-bandgap excitation. Our measured IR-photocurrent power dependence fa-vors the latter mechanism.With the assumption of defect states excitation, the VIS- and IR-pulse-inducedphotocurrents are modulated based on the same circuit. As a result, we can applythe same correlation formula to both photocurrents, with the temperature increasegenerated by the opposite pulse, namelyI\u00afIR(\u2206\u03c4) =1\u03c4rep\u222b \u03c4rep0\u03b1V IS(t)IIR(t+\u2206\u03c4)dt, (4.19)andI\u00afV IS(\u2206\u03c4) =1\u03c4rep\u222b \u03c4rep0\u03b1IR(t)IV IS(t+\u2206\u03c4)dt. (4.20)In our experimental configuration, the IR-pulse (350 \u00b5J cm\u22122) generates two or-ders of magnitude more heating and 9 times more photocurrent than the VIS-pulse(2 \u00b5J cm\u22122). Thus, the maximum change in I\u00afIR is more than one order of mag-nitude smaller than that in I\u00afV IS. As a result, we conclude the IR-pulse inducedphotocurrent does not affect our main result and is neglected in the data analysis.81Figure 4.18: The photocurrent excited by 1030 nm femtosecond pulse illu-mination at different power levels.4.2.4 Modelling the photocurrent temperature dependenceFig.4.19 shows an equivalent circuit model of the GR\/3R-MoS2\/GR device. In thephotovoltaic mode, the device is modelled as an ideal current source in parallel (se-ries) with a shunt (series) resistor.156;165 Itotal is the total photocurrent generated inthe device and Iph is the measured output current. The shunt resistance RSH mainlyarises from the tunnelling or thermionic transport, with the latter dominating atabove room temperature.157 Rs mainly consists of graphene contact resistance Rc.As a result, the measurable output photocurrent is Iph = Itotal RSHRSH+Rc . At room tem-perature, Rc is much larger than RSH .73 Thus, we approximate the output currentasIph \u2248 Itotal RSHRc . (4.21)Above room temperature, the shunt resistance of the device is dominated byover-the-barrier thermionic transport.157 The thermionic current from top graphene82Figure 4.19: Itotal is the total photocurrent generated in the device, Iph is themeasured output current, RSH is the shunt resistance of the device, Rsis the series resistance.to bottom graphene can be written asItherm = Aeh\u222b +\u221e\u2206Dt(E +\u00b5t)Db(E +\u00b5b)[ f0(E\u2212E f t)\u2212 f0(E\u2212E f b)]dE. (4.22)where A is a normalization factor, e is the charge of an electron and h is the Planckconstant. \u2206 is the energy difference between the graphene Dirac point and theMoS2 conduction band edge (Fig.4.20). Dt and Db are the density of states (DOS,2|E|\u03c0(\u210fv f )2) in the top and bottom graphene, respectively. f0(E) = 1exp( EkBT )+1is theFermi-Dirac distribution, while E f t and E f b are the Fermi levels in the top and bot-tom graphene electrodes, respectively. At zero bias, E f t = E f b = 0. The chemicalpotentials relative to the Dirac points for the bottom and top layer graphene aredenoted as \u00b5t and \u00b5b (Fig.4.19), respectively. To derive the conductivity at zerobias, we assume there is a small bias \u03b4 between two electrodes, leading to a finiteFermi level difference E f t \u2212E f b = e\u03b4 . The zero bias conductivity is the limit ofI\/\u03b4 at \u03b4 \u2192 0.lim\u03b4\u21920Itherm\u03b4= Ae2h\u222b +\u221e\u2206Dt(E +\u00b5t)Db(E +\u00b5b)(\u2212\u2202 f0\u2202E )dE, (4.23)For the case where \u2206>> kBT , we can approximate the Fermi-Dirac distribution as83the Boltzmann distribution, and \u2212 \u2202 f0\u2202E = 1kBT exp( \u2212EkBT ). Thus we obtain\u03c3 = 4Ae2h1\u03c02(hv f )4\u222b +\u221e\u2206(E +\u00b5t)(E +\u00b5b)1kBTexp(\u2212 EkBT)dE, (4.24)\u2248 4Ae2h1\u03c02(hv f )4\u222b +\u221e\u2206E21kBTexp(\u2212 EkBT)dE, (4.25)= 4Ae2hk2BT2\u03c02(hv f )4((\u2206kBT)2+2\u2206kBT+2)exp(\u2212 \u2206kBT), (4.26)The chemical potentials in the top and bottom graphene are negligible comparedto \u2206. Since \u2206>> kBT , the inequality ( \u2206kBT )2 >> \u2206kBT >> 1 also holds. Hence, wecan further simplify equation (4.38) as\u03c3 \u2248 \u03c30 exp(\u2212 \u2206kBT ), (4.27)where \u03c30 = 4A\u22062\u03c02(hv f )4e2h is a constant. The shunt resistance is the inverse of conduc-tance, yieldingRSH \u221d 1\/\u03c3 \u221d exp(\u2206kBT). (4.28)Combining equation (4.33) and (4.40), we haveIph \u221d exp(\u2206kBT). (4.29)By fitting the photocurrent temperature dependence shown in the inset of Fig.4.14(c)in the section 4.2.2 with equation (4.41), we extract \u2206 = 0.14 eV.From equation (4.41), we can also derive the photocurrent I\u2032ph at a device tem-perature Td with respect to the photocurrent I\u2032 at the ambient temperature T0.I\u2032ph = I\u2032 exp(\u2206kB(1Td\u2212 1T0)). (4.30)We can then assign \u03b1 = exp( \u2206kB (1Td\u2212 1T0 )) as the dimensionless photocurrentsaturation factor mentioned in the section 4.2.2.Fig.4.21 shows two set of data points on the full temperature dependence of84Figure 4.20: The Fermi-levels in the top and bottom graphene are aligned.The chemical potentials are denoted as \u00b5t and \u00b5b in top and bottomgraphene, respectively.Figure 4.21: A. The photocurrent under 20 \u00b5W 532 nm CW laser illumina-tion from 3 K to room temperature. B. Total resistance of the devicefrom 3K to room temperature.85the photocurrent and resistance, whose trend qualitatively agrees with our simplemodel. However, quantitative understanding of the full range temperature depen-dence requires a more sophisticated model, which may need more theoretical ef-forts in the future study.4.2.5 Two-temperature modelThe temperature evolution is modelled through a two-temperature model consist-ing of the graphene electronic temperature Te and device temperature Td , whichrepresents graphene and MoS2 lattice temperature. After graphene is excited byintense pulses, the electronic temperature rapidly cools through supercollision withacoustic phonons151;152;161, with the interaction term \u0393e\u2212ph = A(T 3e \u2212T 3l ), whereA= 0.75Wm\u22122K3 is the supercollision coupling constant152 and Tl is the graphenelattice temperature. With only four atomically thin layers in a symmetric stackingorder, graphene quickly reaches thermal equilibrium with MoS2, and both con-tribute to the change in the shunt resistance. Therefore, we can use the device tem-perature Td to capture the photocurrent dynamics. The whole device cools throughthe thick hBN encapsulation, which is approximately at room temperature T0. Wemay then write the rate equations asCe\u2202Te\u2202 t=\u2212A(T 3e \u2212T 3d )+QIR(t), (4.31)Cd\u2202Td\u2202 t=A(T 3e \u2212T 3d )\u2212GG\u2212hBN(Td \u2212T0). (4.32)Ce is the graphene electronic heat capacity, which can be calculated from the tem-perature derivative of the electrons\u2019 total energy using the tight-binding model.166The calculated temperature dependence of Ce is plotted in Fig.4.22. Cd =Cl\u2212Gr +Cl\u2212MoS2 is a combination of graphene and MoS2 lattice heat capacity. Since thedevice is symmetric along out-of-plane direction, the model is based on top halfof the stack, i.e., Cl\u2212Gr(Cl\u2212MoS2) is the lattice heat capacity of monolayer graphene(MoS2). Cl\u2212Gr has linear temperature dependence around room temperature and isapproximately three thousand times larger than Ce at around room temperature.167Cl\u2212MoS2 is obtained from reference168. QIR is the optical energy absorbed by amonolayer graphene from the IR-pulse, which is approximately 2% of the pulse86Figure 4.22: The values are calculated from the tight-binding model.energy. GG\u2212hBN = 55MWm\u22122K3 is the thermal conductivity between grapheneand hBN. Since the VIS-pulse is much weaker than the IR-pulse, the pulse heat-ing effect is neglected. As a result, we can quantitatively track the evolution ofthe transient temperature of the device. By incorporating the experimental con-dition of the pump-probe photocurrent measurement in Fig.4.16(b), we obtain aTd maximum of approximately 330 K after IR-pulse excitation. The photocurrentmodelling is carried out in the time domain with the cross-correlation relation anda time-dependent transient photocurrent profile as presented in the section 4.2.2.87Chapter 5Sliding ferroelectricity in3R-MoS25.1 Non-volatile polarization switching in 3R-MoS2homobilayer5.1.1 IntroductionThe rise of two-dimensional (2D) ferroelectric materials169 has offered many newopportunities for developing novel nano-electronic and optoelectronic applications,such as non-volatile memories170;171, high-performance photodetectors73;172;173,and integrated ferroelectric transistors174\u2013176. Recently, an interfacial phenomenonhas been discovered in artificial stacks of two layers of non-polar 2D materials withmarginal twists67\u201369, where a network of domains with alternating stacking con-figurations arises out of the atomic reconstruction. These domains exhibit spon-taneous electrical polarization due to an asymmetric inter-layer coupling at theinterface and the net polarization can accumulate over multiple layers117;177\u2013180.Under an external electric field, atoms at the domain wall (Domain wall (DW))slide perpendicular to an external field181\u2013183, in contrast to conventional ferro-electric materials. As a result, the domains of favorable polarization expand whilethe others contract, leading to a switch of the total polarization. This phenomenon,88referred to as sliding ferroelectricity64;141, has been observed in a wide range of vander Waals materials70;71;184, greatly broadening the family of 2D ferroelectrics.acVtgVbg3R-MoS2hBNGrAB BA 1.3 1.4 1.5 1.6Photon Energy (eV)00.10.20.30.40.5 PL Intensity (a.u.)IT2IX1IX2IT1Fex=0 V\/nmFex=-0.04 V\/nmIX1IX2IT1IT2DWbK \u0413 K-+-+AB BA +-\u03c9\u210ftop bottom- -Figure 5.1: a. Left panel: schematic of a dual-gated device with a 3R-MoS2bilayer composed of both AB and BA domains with a domain wall lo-cated at the interface. Right panel: side and top views of atomic struc-tures in AB (MotopSbottom) and BA (StopMobottom) stacking configura-tions. The positive polarization direction is defined as vertical upward.b. Left panel: the electronic band structure of an AB-stacked 3R-MoS2bilayer. The red and blue lines at the K point represent the conduc-tion and valence bands in the top and bottom layers, respectively. Themomentum-indirect \u0393-K excitons are composed of localized electrons(orange) at the K point and layer-shared holes (green) at the \u0393 point.The wavy line represents the photoluminescence from the \u0393-K exci-tons. Right panel: the real-space charge distribution of the \u0393-K exci-tons. c. Normalized PL spectrum of AB-stacked 3R-MoS2 bilayer withzero (black) and -0.04 V\/nm (blue) external electric field. The former isshifted upward by 0.1 for clarity. Two phonon replicas of \u0393-K excitonsare labeled as IX1 and IX2 while their trion counterparts are denoted asIT1 and IT2, respectively.89Rhombohedral stacking also exists naturally in chemically synthesized singlecrystals, such as rhombohedral molybdenum disulfide (3R-MoS2), which exhibitsa large polarization-induced spontaneous photovoltaic effect73. Switchable as it is,the polarization in these crystals proves to be more challenging compared to arti-ficial stacks70;117;185. Here we show the challenge lies in the underlying switchingmechanism, that is, the release of pre-existing domain walls in single crystallineMoS2 flakes. By optically mapping the polarization distribution and its variationduring the switching, we reveal that a freely propagating DW can switch the po-larization of a large single domain. The polarization switch is non-volatile, asDWs become localized by pinning centers. In contrast to DW networks in artifi-cial stacks, the DWs in 3R-MoS2 can be released from pinning centers and sweepacross nearly the entire flake under a sufficiently strong external electrical field.Our findings highlight the crucial role of DWs in sliding ferroelectricity and sug-gest a promising pathway for 3R-MoS2 to serve as fundamental building blocks forprogrammable optoelectronic devices186.The polarization switching in our experiments is achieved by applying an out-of-plane electric field and monitored by leveraging the strong light-matter interac-tion of MoS2 11;32. As illustrated in Fig.5.1(a), a bilayer 3R-MoS2 is encapsulatedin a dual-gated device, which allows us to apply an electric field without doping.The bilayer 3R-MoS2 has two stacking configurations: if the Mo atom in the toplayer sits on top of the sulfide atom in the bottom layer, we define it as AB stacking,and the opposite structure is termed BA stacking. Previously, we have shown thatthe interlayer potential arising from the polarization gives rise to a finite energyoffset in both conduction and valence bands at K point, leading to an effectivelytype-II band alignment (Fig.5.1(b))117. In such a band structure, optically excitedelectrons rapidly relax to the conduction band edge at the K point, which is local-ized in one of the two layers, while holes transfer to the valence band edge at the\u0393 point, which is delocalized between two layers, forming a momentum-indirectinterlayer exciton. Depending on the stacking configuration, the interlayer excitonexhibits an out-of-plane electric dipole moment along either the upward (AB) ordownward (BA) direction61;77.The dipole moment of interlayer excitons can be measured through the quan-tum Stark shift of their photoluminescence (PL) peaks excited by a continuous-90wave laser (2.33 eV). At zero electric field and a temperature of 4 K, four distinctPL peaks are observed in the 1.4-1.6 eV range. Two of them have been attributedto phonon sidebands of the \u0393-K transition187, while the rest likely originated fromtheir trion counterparts117. Under a finite electric field, all four peaks shift to-ward the same direction, as a result of their out-of-plane dipole moments. Whena downward (negative) electric field is applied to an AB-stacked domain, the fieldis parallel to the dipole moment and the peaks undergo a redshift (Fig.5.1(c)). Thepeaks will blueshift if the electric field is anti-parallel to the dipole moment. Con-sequently, by measuring the slope of the Stark shift, we can determine the dipolemoment direction and infer its corresponding stacking order, thus identifying whenthe switch happens.5.1.2 Optically probing the interlayer sliding motionWe first study the quantum Stark shift of an AB-stacked bilayer (Fig.5.2(a)). Withina small field range (-0.068 V\/nm<F<0.085 V\/nm), all four peaks shift positivelywith the field, confirming a positive dipole moment. The slope of the Stark shiftcorresponds to a dipole moment of about 0.3 e\u00b7nm, which varies slightly betweenexcitons and trions (\u00b51-\u00b54 in Table 1). Such a variation can result from the differ-ence between interlayer excitons and interlayer trions117. When the field exceeds0.085 V\/nm, the differential slope of the Stark shift changes from positive to neg-ative. This change occurs because the large positive external field (Fex) overcomesthe built-in depolarization field (Fdep) and reverses the energy order of conductionbands between the two layers. As a result, optically excited electrons relax to theother layer, reversing the direction of the dipole moment of the interlayer exciton.Such a dipole moment switch has been observed in artificially stacked bilayers77 -it happens when the electric field is parallel to the polarization but does not repre-sent a change in the stacking order.The signature of stacking order switching is clearly observed at a large nega-tive external field (Fc=-0.068 V\/nm). As the field is continuously scanned towardsthe negative range, an abrupt change is observed in the interlayer trion peak, char-acterized by a much stronger amplitude and a blueshift of tens of meV. We assignthe major peak as an interlayer trion since it has a similar Stark shift slope as IT291prior to the switch (Fig.5.2(b)). A comparable shift can be found for the two in-terlayer excitons, although their emission intensity becomes very weak after theswitch. This sudden change in the Stark shift, involving the same dipole momentand external field, implies a change in the depolarization field, indicating a switchof stacking order from AB to BA. The sudden change in the trion emission ampli-tude may also be related to the stacking order change, as the free carrier distributesdifferently in different structures. With a fitted blueshift of 38(3) meV for the IT2peak, we conclude a \u2206Fdep of 0.17(2) V\/nm, corresponding to a Fdep \u22480.09(1)V\/nm and an interlayer potential of 58(7) meV, consistent with the previous obser-vations70;71;117.The switch in stacking order is non-volatile and shows hysteretic behavior. Fol-lowing the stacking order being switched from AB to BA, we observe that both theStark shift slope near zero field and the depolarization compensation field becomenegative, consistent with the BA stacking order (Fig.5.2(c)). This switch remainspersistent up to room temperature, as we will discuss later. When we apply a largepositive field Fex (0.073 V\/nm), the stacking order and associated polarization re-verse back to AB, as indicated by another abrupt change in the peak position. Thehysteresis becomes more apparent if we plot the dipole moment versus the trainingelectric field Ftr (Fig.5.2(d)). To train the polarization, we initially apply a train-ing field for 0.5 s. Due to the linear-stark shift under an external field less than0.07 V\/nm, the exciton dipole moment is determined by normalizing the PL peakenergy shift to the small field applied. The measured hysteresis loop has a rect-angular shape, exhibiting two opposite dipole moments along with two coercivefields matching the abrupt features in the spectra.92-0.2 -0.1 0 0.11.41.451.51.55Peak Energy (eV)Fex (V\/nm)IX1IX2IT1IT22\u03bc4Fdep-0.15 -0.1 -0.05 0 0.05 0.1Fex (V\/nm)1.41.451.51.551.6Photon Energy (eV)0 0.4PL (a.u.)Fc-0.15 -0.1 -0.05 0 0.05 0.1Fex (V\/nm)1.41.451.51.551.6Photon Energy (eV)a bdc Fc0 0.4PL (a.u.)Dipole (e  nm)\u25cfAB-0.2 0 0.2-0.400.4Ftr (V\/nm)BAFigure 5.2: a. Photoluminescence spectra of \u0393\u2212K transitions as a functionof the external electric field (Fex). Fex is swept towards the negativedirection. IX1, IX2, IT1, and IT2 are labeled by green, yellow, purple,and red arrows. At the coercive field Fc=0.068 V\/nm, the stacking orderswitches from AB to BA at the focus spot. b. The fitted peak energyis plotted as a function of Fex. The peak shift of T2 before and afterthe switch corresponds to the change of the depolarization field timesthe dipole moment (\u00b54). c. PL spectra of \u0393\u2212K transitions versus Fexwhen the stacking order switches back to AB. The domain switchinghappens at Fc=0.073 V\/nm. d. The dipole moment of IT2 as a functionof training electric field (Ftr) in the forward (blue) and backward (red)scan directions, showing a hysteresis behavior.The coercive field is an important parameter for ferroelectric materials and canreveal the mechanism of the polarization switching. The coercive field reportedhere is asymmetric in the negative and positive field directions. Since our opticalsensing approach provides a diffraction-limited spatial resolution, we can mea-sure the coercive field at different locations, where we also find different coercive93fields. Moreover, in another device, we observe an AB-to-BA transition at Fc =-0.21 V\/nm (Fig.5.3), which is nearly two times larger. Such significant variationin Fc suggests that the switching field is not an intrinsic property of 3R-MoS2 crys-tal, e.g., for nucleating new domains. The ferroelectric polarization switching in asingle-domain 3R-MoS2 bilayer is challenging likely due to the domain nucleationenergy141. The three-fold symmetry in the structure leads to three equivalent di-rections for the sliding to occur, which makes the switching nondeterministic188.Therefore, we attribute the observed switching behavior to the propagation of apre-existing domain wall (DW), with the coercive field corresponding to the pin-ning potential of a pinning center that localizes the DW.0.05 0.1 0.15Fex(V\/nm)1.41.451.51.55Photon Energy(eV)-0.25 -0.2 -0.15 -0.1 -0.05Fex(V\/nm)1.41.451.51.55Photon Energy(eV)Fc Fca bPL0 1.0PL0 1.0Figure 5.3: PL spectra of \u0393-K excitons as a function of external electric field(Fex) in sample 2.a. Fex is swept towards the negative direction. At thecoercive field of Fc=-0.21 V\/nm, the domain switches from AB to BAstacking. b. Fex scans towards the positive direction. The stacking orderswitches from BA back to AB at Fc=+0.10 V\/nm.The domain wall in 3R-MoS2 is an \u223c10 nm wide region connecting two po-larization domains, where the stacking order smoothly transitions from one to theother189;190. DWs are rare in chemically synthesized single crystals but can befound in mechanically exfoliated flakes, where shear strain can induce avalanchesof interlayer sliding191. These DWs are usually trapped by pinning centers like de-fects, bubbles, or edges of the sample. When an external electric field is applied,one stacking order becomes more energetically favorable than the other, providing94a driving force for the DW to propagate. In our device, as Fex becomes larger thanFc, the free energy difference is expected to overcome the local pinning poten-tial and release a DW. The sharp transition in Fig.5.2(d) suggests that the DW canpropagate throughout the focus spot without getting pinned. The variation in Fc istherefore a result of the random distribution of pinning potentials. The asymmetriccoercive fields in different switching directions arise from the difference in pinningcenters between the initial and final states of a switch.Such domain-wall-propagation-based polarization switching picture is furthersupported by polarization mapping at room temperature and ambient conditions.As shown in Fig.5.4(a), although the fine peak splitting is no longer resolvable,the overall Stark shift is still distinguishable at room temperature under Fex=-0.04V\/nm. The PL peaks in AB and BA domains shift in opposite directions. Herewe use the Stark shift as a qualitative representation of the dipole moment, whichyields a hysteresis loop similar to the one observed at low-temperature (Fig.5.4(b)).No obvious temperature dependence of the coercive fields is observed up to theroom temperature, indicating the pinning potential is larger than the thermal energy(Fig.5.5). More importantly, the long-term stability at ambient conditions allowsus to map the real-space distribution of the polarization, through which we observedifferent intermediate DW positions during the switching.951.3 1.4 1.5 1.6Photon Energy (eV)00.20.40.60.81PL Intensity (a.u.)ABBAFex=-0.04 V\/nm1.7abcPeak shift (meV)-25 25Ftr=+0.10 V\/nmFtr=-0.10 V\/nmFtr=+0.05 V\/nmFtr=-0.08 V\/nm-0.2 0 0.21.4651.471.475Peak Energy (eV)Ftr (V\/nm)ABBAFigure 5.4: a. Interlayer PL spectra of AB and BA stacked domains at roomtemperature under an external electric field of Fex=-0.04 V\/nm. b. Roomtemperature hysteresis loop. Blue and red arrows denote the forwardand backward scan direction of the training field. c. Real-space map-ping of the Stark shift after four different training fields (Ftr) are ap-plied. The negative shift corresponds to an AB-stacked domain whilethe positive shift indicates a BA-stacked domain. The dashed circle inthe upper-left panel outlines a bubble area where the initial domain walls(DWs) are likely pinned. One DW in the lower-left panel is labeled bya magenta dashed line and its local propagation directions are indicatedby the arrows. Scale bar: 5 \u00b5mThe polarization mapping is performed in a way similar to the hysteresis loopmeasurement. We first apply a large training electric field (Ftr) and then map theStark shift across the entire flake with a diffraction-limited focus spot at a smallfield (Fex=-0.04 V\/nm). In the upper-left panel of Fig.5.4(c), we first apply a pos-itive training field (Fex=0.1 V\/nm) to prepare the sample in a single AB-stackeddomain. As a result, the entire sample exhibits a redshift, supporting our stack-ing order assignment. Subsequently, we apply a negative training field (Fex=-0.0896V\/nm), leading to a polarization switching in over half of the device, including thedevice center, which corresponds to the spot probed in Fig.5.2. Finally, an evenlarger negative field is applied (Fex=-0.1 V\/nm), resulting in further expansion ofthe BA domain. By comparing the two states, we can conclude that the polariza-tion switching is achieved through the propagation of domain walls, one of whichis highlighted in magenta in the lower-left panel of Fig.5.4(c). When the electricfield exceeds the local pinning field, the DW is released, then moves westward, andfinally becomes pinned again by stronger pinning potentials near the edge of theflake. The intermediate and final positions of DWs are non-volatile, as the observa-tion is independent of the Stark shift field.-0.088 -0.084 -0.081.31.41.51.60 0.05 0.11.31.41.51.6Photon Energy (eV)Fex (V\/nm)Photon Energy (eV)Fex (V\/nm)a b0 1.0PL (a.u.)0 1.0PL (a.u.)Figure 5.5: PL spectra of \u0393-K excitons as a function of external electric field(Fex) at room temperature in sample 1. a. Fex is swept towards thepositive direction. At the coercive field of Fc=+0.063 V\/nm, the domainswitches from BA to AB stacking. b. Fex scans towards the negativedirection. The stacking order switches from AB back to BA at Fc=-0.084 V\/nm.To study the hysteresis effect in DW, we further apply a positive training fieldto reduce the BA domain. As shown in the upper-right panel of Fig.5.4(c), the DWmoves to a new location when we apply a training field Ftr=0.05 V\/nm. The newDW position is different from the previous intermediate state, indicating that the97depinning and pinning process strongly depends on the history. When we apply aneven larger positive field (Ftr=0.1 V\/nm), the BA domain shrinks to a size smallerthan the detection limit, and the DWs likely move into a bubble area in the uppermiddle area of the flake, which cannot be accessed by our optical probe. Hencewe complete the demonstration of a reversal polarization switching in a natural3R-MoS2 bilayer over a \u223c100 \u00b5m2 area. Such a series of spatial distributions ofpolarization domains also agree with the hysteresis loop measured at the centerspot. We note there are potentially more intermediate states between our appliedtraining fields - the efficiency of the PL probe focus us on the few conditions aspresented. Overall, we find that 4 out of 9 fabricated devices are switchable. Wethink the repeatability is limited by the probability of having pre-existing domainwalls191 and large domain nucleation energy in 3R-MoS2 bilayers. The existenceof domain walls in another switchable device (sample 4) is confirmed by Electro-static force microscopy (EFM) imaging prior to the device encapsulation (Fig.5.6).Compared with DWs in artificially stacked homo-bilayer, it is clear that our DWsare not always attached to certain pinning centers and can propagate through theentire flake under a sufficiently large field.98b155 uV 648 uVa-0.1 0 0.1Fex(V\/nm)1.41.451.51.551.6Photon energy (eV)c d-0.1 0 0.1Fex(V\/nm)1.41.451.51.551.6Photon energy (eV)Fc Fc0 1.0PL (a.u.)0 1.0PL (a.u.)Figure 5.6: a. Optical image of the exfoliated 3R-MoS2 bilayer in sample 4.Scale bar: 10 \u00b5m. b. EFM of the flake in (a) prior to its encapsulation.The black lines of a. and b. indicate the boundary between the regionsof monolayer and bilayer. c. Photoluminescence spectra of \u0393\u2212K tran-sitions as a function of the external electric field (Fex). Fex is swepttowards the negative direction. At the coercive field Fc=-0.057 V\/nm,the stacking order switches from AB to BA at the focus spot. d. PLspectra of \u0393\u2212K transitions versus Fex when the stacking order switchesback to AB. The domain switching happens at Fc=0.046 V\/nm.In addition, we report that the polarization switching in 3R-MoS2 bilayers canbe optically observed not only in the Stark shift of interlayer exciton but also in theintensity of intralayer exciton. Despite being an indirect band gap semiconductor,the bilayer 3R-MoS2 exhibits hot PL from direct band gaps in K valleys within eachindividual layer (Fig.5.7). As shown in Fig.5.7(a), two distinctive PL peaks with anenergy separation of 11 meV near 1.9 eV are observed because the nonequivalentlocal environment of Mo atom induces a small band gap difference between the top99and bottom layers.73;117. Based on the optically determined BA stacking order, thehigher energy peak (Xh) originates from the bottom layer while the lower energyspecies (Xl) emits from the top layer. Under zero external field, the PL intensity ofXh is much stronger than Xl , because the asymmetrical coupling in 3R-MoS2 leadsto a type-II band alignment between the two layers - the photoexcited electrons inthe top layer quickly relax to the bottom layer, which quenches Xl (Fig.5.7(b)).Since the holes in both layers relax to the \u0393 point at a similar rate73;172, the valenceband offsets at the K points do not contribute to the intensity imbalance.Such an intralayer PL intensity contrast can be reversed by a large external elec-trical field applied in either direction. In the negative direction, the external fieldis anti-parallel to the built-in depolarization field. When this field is sufficientlystrong to compensate the depolarization field, the band alignment changes to type-I as the conduction band minimum of the top layer becomes lower than that of thebottom layer, leading to a quenching of the bottom layer PL (Xh) (Fig.5.7(b)). Thisreversal in PL intensity shares the same origin as the dipole moment change in theinterlayer exciton (Fig.5.2(c)).100ab--++--++\u03c9\u210f--++\u03c9\u210ftop bottom\u03c9\u210f-0.15 -0.1 -0.05 0 0.05 0.1Fex (V\/nm)1.851.91.952Photon Energy (eV)XhXl1.00PL (a.u.)FcFdepFex > FcFex= 0Fex< FdepFigure 5.7: a. PL spectra of intralayer excitons as a function of external elec-tric field (Fex). Xh and Xl are attributed to the excitons in the bottomand top layers, respectively. The white dashed lines indicate when theintensity ratio between the two peaks changes. b. Schematics illustrat-ing the interlayer charge transfer process that accounts for the intensityratio variation between Xh and Xl . The red and blue lines represent theband gaps in the top and bottom layers, respectively. Left panel: at alarge negative field, the interlayer potential is compensated, causing theconduction band in the bottom layer to be higher than that in the toplayer. The photoluminescence of Xh is therefore suppressed. Middlepanel: with zero external field, the PL of Xl is quenched because theelectrons migrate from the higher conduction band in the top layer tothe lower one in the bottom layer. Right panel: when Fex exceeds thecoercive field, the domain switches from BA to AB and the band gap inthe bottom layer becomes smaller. The large external field causes theband alignment to change from type-II to type-I, leading to a quench ofthe PL from Xh.101Another sudden PL intensity change is observed when a large positive electricalfield is applied (F=Fc). In this case, the external field is anti-parallel to the electricalpolarization and the abrupt intensity change is caused by a switch in the stackingorder. When the stacking order changes from BA to AB, the band gap in the toplayer becomes larger than that in the bottom layer, and the conduction band offsetshould be opposite to that in the middle panel of Fig.5.7(b). Nevertheless, similar asthe situation in the left panel of Fig.5.7(b), a large positive field can change the bandalignment from type-II to type-I, thus quenching the PL from the top layer withhigher energy. Importantly, the switching field Fc corresponds to the coercive fieldobserved in Fig.5.2(c), confirming the picture that the stacking order is switchedfrom BA to AB. Hysteresis is also observed in the intralayer PL intensity whenthe stacking order reverts back to BA (Fig.5.8). The comparable intensity ratiobetween Xh and Xl when there is no external field in Fig.5.7 and Fig.5.8 indicatesthat the observed hysteresis behavior originates from intrinsic domain switchingrather than the interfacial charge trapping effect (The detail discussion is includedin section 4.1.3).-0.15 -0.1 -0.05 0 0.05 0.1Fex (V\/nm)1.851.91.952Photon Energy (eV)XhXl01.0PL (a.u.)Figure 5.8: PL spectra of intralayer excitons as a function of external electricfield (Fex). Xh and Xl are attributed to the excitons in the top and bottomlayers for AB stacking, respectively. Fex is swept towards the negativedirection.102In conclusion, we have optically observed a non-volatile switch in the electricalpolarization in a natural 3R-MoS2 bilayer. By probing the Stark shift of interlayerexciton with a diffraction-limited focus spot, we map the spatial distribution ofpolarization domains and their variations during the switch. Most importantly, weidentify that this switch is enabled by the propagation of pre-existing domain wallsthat are released by the external electric field. The polarization switch can alsobe optically read through the relative photoluminescence intensity of intralayerexcitons between the two layers.Our findings demonstrate the interplay between rich excitonic effects and slid-ing ferroelectricity, which enables a non-volatile control of the optical propertiesof 2D semiconductors. The polarization-dependent optical response of 3R-MoS2provides a promising foundation for optical data storage, optical communication,and optical computing applications. Currently, the formation of domain walls inour flakes is not controlled, which agrees with the observation that only a fractionof our bilayer devices are switchable. In the future, it will be important to explorehow to systematically generate domain walls in homogeneous rhombohedral tran-sition metal dichalcogenide films, such as by exerting shear strain near the criticalpoint191 or applying strong THz field79;192, in order to improve the repeatabilityand scalability of the switching behavior for the future applications of sliding fer-roelectricity.5.1.3 Intrinsic domain wall motion vs extrinsic interfacial chargetrappingThe interface charge trapping can be a potential extrinsic contribution to the 2Dferroelectricity phenomena. In this section, we will discuss how to distinguish be-tween the intrinsic domain wall motion and the extrinsic interfacial charge trappingeffect.Experimentally, we observed the sign of the dipole moment (\u00b5) of \u0393-K ex-citons can be switched by a large training field (Fig.5.2), indicating the lowestconduction band in the bilayer can be changed from one layer to the other in a non-volatile manner. One possible mechanism behind it is the sliding ferroelectricityeffect we describe in section 5.1.2: the polarization associated with the stackingorder is switched by the training field, and the band alignment between the two103layers is reversed by the change in the depolarization field. Nevertheless, there isanother possible explanation: the defects in hBN or MoS2 may become ionizedunder a large training field, and the ionized charges could get trapped at the inter-faces, thus causing a built-in electric field that can also change the band alignment.As discussed below, we can distinguish between these two scenarios by closelyexamining the intralayer photoluminescence (PL) spectra in Fig.5.7.One way to distinguish the trapped charge effect from sliding ferroelectricity isby comparing the intralayer exciton\u2019s PL spectra at zero external electric fields afterthe interlayer exciton\u2019s dipole moment is trained towards different orientations. Inour previous work, we have reported that a bilayer 3R-MoS2 has an effective type-II band alignment at the K point, where the direct band gaps also have differentenergies between two layers. In the BA stacking case, where the Mo atom in thetop layer is positioned on top of the Sulfide atom in the bottom layer, the bandgap in the bottom layer is slightly larger than that in the top layer, correspondingto two intralayer exciton emission peaks near 1.9 eV (Xl and Xh). Moreover, dueto the type-II band alignment, the photoexcited electron in the top layer relaxesquickly to the bottom one, causing the low-energy PL peak, Xl , to be weaker thanXh (Fig.5.9(a)).After a large field training, the interlayer exciton\u2019s dipole moment is switched.If such a switch is caused by a built-in field from the trapped charge rather thana stacking-order change, we expect that the band gaps in both layers remain thesame, and only the conduction band alignment is switched by the built-in field(Fig.5.9(b)). As a result, the low-energy peak Xl should become stronger than Xhat zero external field. However, this is not observed in the experiment. As shownin Fig.5.9(d), the relative strength of Xl and Xh at zero external fields is almostunchanged after training using the largest fields. The high-energy peak is alwaysstronger than the low-energy one. As discussed in section 5.1.2, such a switchin the interlayer exciton dipole moment with no change in the intralayer excitonstrength can only be explained by a change in the stacking order from BA to AB.In the AB stacking, the top layer has a larger band gap and a lower conductionband (Fig.5.9(c)). As a result, the Xh peak remains stronger than Xl , although theyare emitted from different layers after the switch.104AB stacking--++ trapped charge effect-+-+\u03c9\u210f\u03c9\u210f--++\u03c9\u210f1.8 1.85 1.9 1.95 2Photon Energy (eV)00.20.40.60.8PL Intensity (a.u.)Postive field trainedNegative field trainedtop bottomBA stackinga b c dXhXlFigure 5.9: a. Schematic of the band alignment for a BA-stacked 3R-MoS2bilayer. b. Schematic of the conduction band inversion under the built-in field induced by trapped charges at the interface. c. Schematic ofthe band structure after the stacking order is changed from BA to AB.d. PL spectra of intralayer excitons in sample 1 measured at zero exter-nal electric field. Red: The sample is trained by a large positive field,indicating the AB stacking domain. Blue: The sample is trained by alarge negative field and the domain, corresponding to the BA stackingdomain.5.1.4 Joule heating effectIn this section, we provide an estimation on the temperature increase of the 3R-MoS2 bilayer induced by delocalized electrons during the propagation of the do-main wall.105vBA ABDWLDW+ + + ++ + + + +- - - -- - - - -Figure 5.10: Top and side view of the polarization switching in a 3R-MoS2bilayer. The simplified flake is a 10x10 \u00b5m square. The domain wall(Grey) is moving towards the right edge.We first assume a 10x10 \u00b5m square 3R-MoS2 bilayer (Fig.5.10) with a domainwall initially pinned at the left edge. The length of the domain wall LDW is there-fore 10 \u00b5m and its width is negligible compared to the flake size. Under an externalelectric field, the domain wall is released from the pinning center and moves to-wards the right. We estimate the domain wall propagation speed v to be in the sameorder as the sound speed in MoS2, about 104 m\/s193. The time interval \u2206t for thedomain switch to finish is therefore about 1 ns. Within such a short time, thermalconduction has not finished and the process is approximately adiabatic. Thus, theinstantaneous temperature increase \u2206T can be estimated using the equation:C\u2206T = IU\u2206t (5.1)Here C \u2248 10\u221213 J\/K is the heat capacity of this 10x10 \u00b5m 3R-MoS2 bilayer, es-timated according to the literature168. The right side of Equation 5.1 is the Jouleheating induced by the delocalized charges. I is the current when electrons transferfrom one layer to another layer (Equation 5.2).I =\u2206q\u2206t=2Pv\u2206tLDW\u2206t= 2PvLDW (5.2)Here P is the polarization, which is around 0.55 \u00b5C\/cm2. U is the total interlayer106potential when the external electric field is close to the coercive field Fc (Equa-tion.5.3)U = \u03c60+Fc\u00d7d0 (5.3)The depolarization field-induced interlayer potential \u03c60 is 58 mV. The coercivefield Fc is around 0.07 V\/nm in sample 1. d0 \u2248 0.70 nm is the interlayer distance.Thus, the total interlayer potential is about 0.1 V at the switching point.Based on Equations 5.1-5.3, the estimated temperature increase due to Jouleheating is about 1 K. Since we do not observe any obvious temperature dependencefrom 4 K to 300 K, we do not expect that such Joule heating caused by the interlayertransfer of delocalized electrons can significantly affect the domain wall motion.5.2 Resolving polarization switching pathways of slidingferroelectricity in 3R-MoS2 trilayer5.2.1 IntroductionThe stacking order in 2D materials has introduced a new dimension for explor-ing novel phenomena and functional devices. Notably, sliding ferroelectricity hasrecently been observed in rhombohedral-stacked transition metal dichalcogenides(R-TMDs), both artificially stacked and chemically synthesized, where a sponta-neous electrical polarization arises from an asymmetric interlayer-coupling-inducedBerry phase64;70;71;117;141;177;180;185;188;191;194;195. Unlike conventional ferroelectricmaterials, polarization switching in sliding ferroelectricity involves the collectivemotion of atomic layers along the in-plane direction, perpendicular to the drivingelectric field178;181;183.When the stacking is coherent among layers, the polarization-induced inter-layer potential can accumulate in multilayer 3R-TMDs, which are semiconductorswith sub-eV bandgaps and strong excitonic effects, leading to novel applicationsin sensing and computing73;172;173. Conversely, incoherent stacking allows an n-layer material to host 2n\u22121 polarization configurations, significantly expanding thephase space for applications related to information storage180;185;191. However,107electrically distinguishing various intermediate stacking configurations from a co-existence of polarization domains has proven challenging. Here we show that theconfiguration of an incoherently stacked 3R-TMD trilayer can be resolved by lever-aging the unique excitonic response of each substituent layer. In particular, we usereflection contrast spectroscopy with a diffraction-limited resolution to fully iden-tify the stacking configuration of a trilayer 3R-MoS2 in its initial, intermediate, andfinal states during multiple polarization switching cycles.The polarization switching process in bilayer 3R-TMDs has been extensivelystudied. When subjected to an out-of-plane electric field, the domain of the polar-ization along the same direction experiences lower free energy and tends to expand,while the domain of the opposite polarization tends to contract, thus creating a driv-ing force for the movement of the domain wall (DW)70;72;177;194. Upon surpassingthe free energy difference relative to the trapping potential of the pinning centerthat initially localizes the DW, the DW becomes depinned, thus triggering polariza-tion switching. The switch can be non-volatile if the DW finally becomes pinnedagain195.A similar process is expected in trilayer 3R-TMDs, introducing two interfacialpolarizations and four potential polarization configurations. Specifically, we des-ignate states with aligned interfacial polarizations as ABC and CBA stacking, andthose anti-aligned interfacial polarizations as ABA and BAB stacking. Theoreti-cal predictions suggest that one of the anti-aligned states (ABA) may be energet-ically favored during the transition from ABC to CBA stacking, with the otheranti-aligned state (BAB) energetically favorable in the reverse process. However,such contrast has not been discerned by the transport measurements due to thedegeneracy in the net polarization185.108Figure 5.11: a.Schematic of reflectance contrast (RC) spectrum measurementin a dual-gated trilayer device. The arrows indicate the polarization di-rection at each interface. Three domains of distinct net polarization areseparated by one domain wall located at the top interface (DWt) andthe other at the bottom interface (DWb). b. Electric field-dependentfirst energy derivative of the RC (dRC) spectra in the backward (leftpanel, from positive to negative field) and forward scan (right panel,from negative to positive field). The blue and red arrows highlight thepolarization-switching events. c. The net polarization as a function ofan electric field in the forward (red) and backward (blue) scan direc-tions, shows a hysteretic behavior.In contrast to electrical measurement, optical spectroscopy offers the capabilityto resolve the stacking configuration of different polarization states based on theirdistinctive excitonic responses. In bilayer 3R-MoS2, the broken symmetry betweenthe top and bottom layer results in different band edges, equivalent to a type-IIalignment at the K point77;117. Upon optical excitation, photocarriers migrate tothe layer with a lower potential, forming an interlayer exciton whose dipole orien-tation can indicate the stacking order. Moreover, the intralayer excitons in the twolayers exhibit approximately a 10-meV difference in peak energy, and a compar-109ison of their oscillator strengths with finite doping can further reveal the stackingconfiguration177. Since the interlayer exciton yields identical responses betweenABA and BAB stackings, here we employ reflection contrast (RC) spectroscopyto detect polarization switching events and identify stacking configurations in atrilayer 3R-MoS2. In analogy to the bilayer, we expect a trilayer with aligned po-larizations will exhibit an RC spectrum having three excitonic peaks, reflectingthe diverse chemical environments of the three layers11;20;32;36;117;196. Conversely,if the polarizations become anti-aligned, the two outer layers become symmetric,resulting in a spectrum similar to that of the bilayer.5.2.2 Polarization switching in 3R-MoS2 trilayerWe investigate the RC spectra of a 3R-MoS2 trilayer with double-side encapsula-tion and with both top and bottom graphene gates that provide control over dopinglevel and electric field (Fig.5.11(a)). All measurements are performed at 1.6 K.In Fig.5.11(b), the field dependence of the first energy derivative of the RC spec-tra (dRC) is presented as the electric field is scanned from the positive to negativelimit. Three intralayer excitonic peaks at approximately 1.9 eV are observed atthe positive limit, suggesting an initial polarization state to either ABC or CBAstacking, consistent with the 3R bilayer analogy.At E1=-0.064 V\/nm, a sharp transition in the dRC spectra occurs, with an in-creased separation between excitonic peaks and a reduction in peak number totwo, indicating a change in stacking configuration to either ABA or BAB. Subse-quently, at E2=-0.087 V\/nm, a second pronounced change in the exciton energysignifies another switch to a third state. Similar to the initial state, a three-peakdRC spectrum is observed in the final state. Besides the backward scan, we per-form a forward electric field scan, where we observe two analogous transitions attwo positive fields (E3=0.084 V\/nm, E4=0.100 V\/nm). Importantly, the immediatestate exhibits similar spectroscopic features as in the backward scan: two excitonicpeaks with comparable peak splitting and oscillator strength ratio. Consequently,we conclude that the polarization switching pathways in the forward and back di-rections involve the same intermediate state in this device, contrary to the previousprediction of one intermediate state being more favorable than the other.110Figure 5.12: a. Electric field-dependent dRC spectra of a sample with CBAstacking. The arrows indicate interlayer excitons (IX\u2212) emerging at anegative field. Xi represents the intrinsic intralayer exciton. The sub-script i represents t, m, or b, denoting the optical transitions in the top,middle, or bottom layers. b. Schematics of interlayer transitions inCBA stacking under a negative electric field. c. RC spectrum of CBAstacking at largest positive (red) and negative fields (blue), which aresimilar to Fig.5.11(b). d-f are similar to a-c but from a sample withABC stacking, whose polarization is opposite to CBA. g. Electricfield-dependent dRC spectra of CBA stacking at a fixed electron dop-ing density. The dashed lines divide the spectrum into three regionswhere the electrons are doped into: bottom layer (region-I), middlelayer (region-II), and top layer (region-III). h. Schematics of the bandalignments and optical transitions in CBA stacking with a fixed elec-tron doping residing in different layers.The hysteresis loop depicting the two-stage switching process observed in de-vice 1 is illustrated in Fig.5.11(c). In the positive and negative field limits, the111stacking configurations are identified as CBA and ABC, respectively. The com-mon intermediate state in this process is an ABA state, where the two anti-alignedinterfacial polarizations give rise to a zero net polarization. The single hysteresisloop suggests that such an intermediate state is metastable, indicating the absenceof an anti-ferroelectric phase in our system. However, the absence of BAB states inthis scan cycle does not preclude their possibility. Fig.5.16 demonstrates oppositeintermediate states in different cycles of electric field training, and we attribute thisdiversity in the switching pathway to the competition in trapping potential amongpinning centers in different layers.5.2.3 Stacking configuration of 3R-MoS2 trilayerThe stacking configuration of different polarization states can be identified by an-alyzing their optical response as a function of electric field and doping. In thebackward scan of Fig.5.11(b), the middle peak diminishes as the electric field isswept from positive to zero, and the low energy peak intersects with a rapidly red-shifting peak with the increasing negative field, just before the stacking switchesat (1). These spectral features agree well with our observations in an unswitch-able CBA-stacked trilayer (Fig.5.12(a)). Similar to the bilayer device, not everytrilayer device exhibits switchability, which we attribute to the absence of pre-existing DWs.As depicted in Fig.5.12(a), two avoided-crossing features emerge in the lower-energy peak, occurring at -0.05 V\/nm and -0.12 V\/nm. These features are identifiedas resonances between the intralayer exciton and the interlayer exciton (IX\u2212) tran-sition, involving the valence band in one layer and the conduction band in the other(Fig.5.12(b)). When the external electric field is anti-aligned with the spontaneouspolarization direction, the interlayer potential increases, and the band offsets be-tween layers become larger, thus causing a redshift in the interlayer exciton. Theout-of-plane electric dipole moments, determined from the Stark shift slopes areapproximately 0.62 e\u00b7nm. Interlayer excitons, characterized by a smaller bindingenergy and weaker oscillator strength compared to intralayer excitons, cannot beobserved in the RC spectra until they hybridize with the intralayer species. Sincethe avoided crossings are exclusively observed in the negative field range, we con-112clude the initial stacking order is coherent among the three layers, defined as theCBA stacking. Between the two interlayer excitons, a peak splitting of approxi-mately 46 meV is observed, similar to that observed in 3R-MoS2 bilayers. Thissplitting significantly exceeds the spin-orbit-induced spin splitting in the conduc-tion band. Furthermore, the low-energy interlayer exciton can only hybridize withthe low-energy intralayer exciton. These features suggest that phonon-assisted in-tervalley interlayer transitions may play a crucial role, warranting further investi-gations36;91;197\u2013199.Similar to the initial state, the stacking configuration of the final state in Fig.5.11(b) is identified as ABC by comparing the spectra with another unswitchable device(Fig.5.12(d)). Both devices show avoid-crossing features between the intralayerexciton and interlayer exciton at approximately 0.05 V\/nm. The interlayer exci-ton redshifts with a positive field, indicating the band alignment is opposite to theinitial state. The missing inter-intra exciton crossing in Figure 1b also confirmsthe switching in the stacking configuration. In contrast to CBA, the ABC stackingdisplays three peaks at the negative field limit (Fig.5.12(c) and (f)) and the middlepeak gradually diminishes as the external field is swept positively.113Figure 5.13: a.Doping-dependent dRC spectrum of CBA stacking. Thedashed lines divide the spectrum into five regions when the Fermilevel is in: the conduction band of the bottom layer (region-I(1)),the conduction band of the middle layer (region-I(2)), the conduc-tion band of the top layer (region-I(3)), bandgap(region-II), and va-lence band (region-III). Xi represents the intrinsic intralayer A-exciton.XP\u2212i , XP\u2212\u2032i represent repulsive exciton-polarons and attractive exciton-polarons with electrons doped into the conduction band at K points.XP+i , XP+\u2032i represent repulsive exciton-polarons and attractive exciton-polarons with holes doped into the \u0393 point. The subscript i representst, m, or b, denoting the optical transitions at K points from the top,middle, or bottom layers. b. dRC spectra at different doping densities.The doping level is labeled by a number near the curve, with a unit of1012 cm\u22122. c. Band alignment corresponding to the electron dopingregions (I(1), I(2), and I(3)) in (a). d.Band alignment corresponding tothe hole doping region (III) in (a).To explore the layer origin of three intralayer excitons and to confirm the stack-ing assignment, we analyze the electric field dependence of a CBA-stacked deviceat a fixed electron doping density of 1.4\u00d71012 cm\u22122 (Fig.5.12(g)). When the exter-nal field is small (E<0.077 V\/nm), the electrons reside in the bottom layer, resultingin the observation of excitons in the top and middle layers (Xt , Xm), and the attrac-114tive and repulsive exciton polarons (XP\u2212\u2032b , XP\u2212b ) in the bottom layer (left panel inFig.5.12(g)), similar to the RC spectrum with finite doping and without electricfield (Fig.5.13).Figure 5.14: a. Electric field-dependent dRC spectra of a sample with ABAstacking. Unlike ABC or CBA stacking, two pairs of interlayer exci-tons are observed and labeled as IX+ and IX\u2212. b,c Band alignmentsat K points and interlayer transitions in ABA stacking under a positiveb and negative c. electric field. d. Doping-dependent dRC spectrumof the same sample. Two lines divide the spectrum into three regions,which are intrinsic or electron- or hole-doped. e,f. Band alignmentscorresponding to regions I and III in d. The gray line denotes the Fermilevel position.As the electric field increases, the conduction band offset \u2206tc (between the topand middle layers) and \u2206bc (between the middle and bottom layers) decrease untilthe Fermi level reaches the conduction band edge of the middle layer (E=0.077V\/nm). From this point on, electrons begin to migrate from the bottom layer to themiddle layer (middle panel in Fig.5.12(h)), which leads to the formation of attrac-tive and repulsive exciton polarons, XP\u2212\u2032m and XP\u2212m , in the middle layer. Due tothe similar energy of XP\u2212mand Xt , XP\u2212m only enhances the oscillator strength of Xt(Fig.5.13). Simultaneously, XP\u2212b experiences a redshift and transfers its oscillator115strength to the intrinsic Xb. As the electric field continues to increase, \u2206tc decreasesfurther while \u2206bc is reversed. At E=0.152 V\/nm, the Fermi level reaches the conduc-tion band edge of the top layer, causing electron migration from the middle layerto the top layer (right panel in Fig.5.12(h)). This change causes XP\u2212\u2032m to blueshiftand transfers its oscillator strength to intrinsic Xm. The interaction between Xt andthe Fermi surface also results in the formation of attractive and repulsive excitonpolaron in the top layer, enhancing the oscillator strength in Xm and Xb due to theircomparable energies (Fig.5.13).Overall, the field-dependent spectra support our stacking assignment and sug-gests the three exciton peaks, from low to high energy, are originated from themiddle, top, and bottom layers respectively in the CBA stacking. Interestingly, thetwo-staged transition suggests that the intrinsic conduction band offset \u2206tc is largerthan \u2206bc . Additionally, the oscillator strength of the highest-energy Xb peak de-creases as the positive field increases (Fig.5.12(a)). This change can be attributedto field-assisted exciton dissociation: as the band offset \u2206bc decreases with increas-ing positive field, it facilitates exciton dissociation through electron transfer fromthe bottom layer to the middle layer, leading to a decrease in the oscillator strengthof Xb. Opposite changes can be observed in the middle Xt peak.In contrast to the initial and final state, the immediate state exhibits remark-able differences in the RC spectra, where two excitonic peaks are observed witha larger separation of approximately 34 meV. Fig.5.14 is based on a stable inter-mediate domain, where switching to either ABC or CBA requires a large coercivefield, which allows us to study the doping and field dependence within the pre-sented range. Field dependence measurements in this device reveal two pairs ofavoided crossing features, with one pair occurring at positive fields and the other atnegative fields. These avoided crossing features resemble the spectra in both CBAand ABC stacked devices, suggesting they originate from two types of interlayerexcitons with opposite dipole moments. This corresponds to the band alignment inan incoherent stacking of either ABA or BAB.In either ABA or BAB stacking configurations, there exist two types of in-tralayer excitons, which we label as XA from the A layer and XB from the Blayer. Previous 3R-bilayer study has established that XA exhibits higher energythan XB 117. In trilayer configurations, the A and B layers are in different dielectric116environments (Fig.5.15). In ABA stacking, the B layer is sandwiched by two MoS2layers, while the A layer is adjacent to one MoS2 layer and one BN layer. Sincethe dielectric constant of MoS2 is larger than BN, XB experiences more screen-ing than XA, resulting in a relative redshift of XB and an increase in the excitonpeak splitting. Conversely, the A layer undergoes stronger screening in the BABconfiguration, causing a redshift in the high-energy exciton peak and reducing thepeak splitting. If this peak splitting is smaller than the linewidth, the spectral re-sponse effectively merges into a single peak that comprises all oscillator strengths.Therefore, we attribute the observed intermediate state in Fig.5.11, with signifi-cant splitting, to ABA, and the intermediate state in Fig.5.16(d), characterized bya single strong excitonic peak, to BAB. The large oscillator strength of the singleexcitonic peak in the BAB configuration and the ratio of the oscillator strengthbetween the two peaks in the ABA configuration all corroborate our assignment.Figure 5.15: a. Schematic representation of the screening effect on excitonicpeak positions in ABA and BAB polarization states. b. RC spectrafrom ABA and BAB polarization states.The band alignment at the K point among three layers in ABA stacking is illus-trated in Fig.5.14(b), where the conduction band in the middle layer is higher thanthe two outer layers. Under an external electric field, the conduction band offsetbetween the middle layer and one of the outer layers decreases, leading to a red-shift in one pair of interlayer excitons, as observed in Fig.5.14(b) and Fig.5.14(c).This band alignment is further confirmed by the doping dependence in Fig.5.14(d),where the responses can be classified into three regions, electron-doped (I), intrin-sic (II), and hole-doped (III) regions.In region II, where the Fermi level lies inside the bandgap of all layers, thespectrum exhibits two intrinsic excitonic peaks. In region I, the trilayer is doped117by electrons, and the doped electrons should reside in the top and bottom layersaccording to the band alignment (Fig.5.14(e)). Experimentally, the high-energypeak transforms into a blueshifted branch and a redshifted branch, known as repul-sive and attractive exciton polaron (XP\u2212t,b and XP\u2212\u2032t,b), respectively. The low-energypeak remains largely unchanged, except for some redshift at large doping, likelydue to the screening effect. In region III, the trilayer is hole-doped at the \u0393 point(Fig.5.14(e)), where the three layers strongly hybridize, converting all excitons intoexciton-polaron peaks with varying redshifts in different layers. The binding en-ergy differences (4 meV of XP+\u2032t and XP+\u2032b , and 16 meV of XP+\u2032m , respectively)may arise from finite layer polarization-induced unequal hole distribution betweenthree layers at the \u0393 point.5.2.4 Polarization switching pathway in 3R-MoS2 trilayerTo investigate the polarization switching mechanism, we conduct the electric fieldon the same device as in Figure 1 for multiple cycles, where we observe the switch-ing pathway and coercive fields can vary. For example, in the backward scan ofa different cycle (Fig.5.16(a)), the spectrum is similar as in Fig.5.14(a), indicatingthe same ABA stacking in the intermediate state, except that the switching fieldsbecome different. On the other hand, in the intermediate state of the forward scan(Fig.5.16(c)), the RC spectrum exhibits a large peak at 1.918 eV, which we attributeto the intermediate state with BAB stacking. The hysteresis loop illustrating thetwo-stage switch in Fig.5.16 is summarized in Fig.5.16(e). In the positive and neg-ative field limits, the stacking configuration is CBA and ABC, respectively, whilethe intermediate state is BAB in the forward scan and ABA in the backward scan.The two-stage switch occurs as the external electric field provides a driving forceto depinning the domain wall (DW) at different interfaces sequentially, as depictedin Fig.5.16(b).118Figure 5.16: a,c Electric field-dependent dRC spectrum of the Fig.5.11 sam-ple in a different scan cycle. The intermediate states, which are be-tween the coercive field (1) and (2), and between (3) and (4), clearlyexhibit distinctive optical responses. b,d RC spectra of the two in-termediate states. e. The net polarization as a function of an electricfield in the forward (red) and backward (blue) scan directions, showsa different hysteretic behavior compared to Fig.5.11. f. Schematics ofthe polarization switching pathways in this cycle. The green, orange,yellow, and blue regions correspond to CBA, ABC, ABA, and BABstacking, respectively.Initially, the sample has two pre-existing DWs located at the top and bottominterfaces. These DWs are trapped by pinning centers such as defects and bubblesoutside the optical probing area, resulting in a homogenous CBA response. Whenthe external field reaches E1=-0.043 V\/nm, the top domain wall (DWt) is releasedfrom the pinning center with a weaker pinning potential pt , causing a local switch119from CBA to ABA. Subsequently, at E2=-0.087 V\/nm, the bottom domain wall(DWb) is released from the pinning centers with a stronger pinning potential pb,causing ABA to switch to ABC. The difference between pt and pb determinesthe electric field window where the intermediate state ABA can be observed. Thesequential release of DWt and DWb, which form the intermediate ABA state, issupported by the electric-field-dependent reflection contrast map that indicates thedistribution of domains and domain walls (Fig.5.17).Figure 5.17: Scale bar, 2 \u00b5m. Yellow and green solid lines represent domainwalls at the top and bottom interfaces, respectively. Small black arrowson each map show the direction of domain wall movement. Whitecircles with black dashed lines indicate bubble areas in the dual-gatedevice.After being released from the initial pinning centers on one side of the opti-cal probe, the DW can quickly sweep through the focus spot and finally becomestrapped again by the pinning centers on the other side of the probe. Consequently,the sequence of the DW release and the intermediate state\u2019s stacking in the reversescan depends on the strength of the final pinning centers (p\u2032t and p\u2032b). In the scancycle in Fig.5.16, p\u2032t is smaller than p\u2032b, and therefore DWt is released earlier thanDWb, causing ABC to switch back to CBA through a BAB intermediate state. How-ever, in the scan cycle in Fig.5.11, p\u2032t is larger than p\u2032b, and the intermediate statebecomes ABA (Fig.5.18).120Figure 5.18: The green, orange, and yellow regions correspond to CBA,ABC, and ABA stacking, respectively. Brown and orange circles rep-resent the Mo and S atoms. The switching pathway is enabled by alarger pinning potential at the top interface than the bottom interfacein the initial state and the opposite condition in the final state.Experimentally, our devices exhibit additional switching pathways, and we canattribute these processes to different combinations of pinning potentials (pt andpb). In Fig.5.19, we illustrate a single sharp transition between CBA and ABCstacking without an observable intermediate state. This behavior is consistent withthe scenario where the pinning potentials in the upper and lower interfaces are com-parable, i.e., pt \u223c pb and p\u2032t \u223c p\u2032b. In another trilayer device, we observed differentspots where the switch occurs exclusively between CBA and ABA (Fig.5.20) oronly between ABA and ABC (Fig.5.21). These observations suggest that somepinning centers in one layer can be exceptionally strong, thereby allowing only theDW in the other interface to sweep through the probe spot.121Figure 5.19: a,c Electric field-dependent dRC spectra of a sample thatswitches between CBA and ABC stacking. The blue and red arrowsindicate the coercive fields in the backward and forward scans. b.Schematics of the polarization switching pathways are shown in (a)and (c). Such a switching pathway is due to the similar pinning poten-tials at the top and bottom interface.122Figure 5.20: a,c Electric field-dependent dRC spectra of a sample thatswitches between ABA and CBA stacking. The blue and red arrowsindicate the coercive fields in the backward and forward scans. b.Schematics of the polarization switching pathways shown in (a) and(c). Such a switching pathway is caused by a much larger pinningpotential at the top interface than at the bottom interface.123Figure 5.21: a,c Electric field-dependent dRC spectra of a sample thatswitches between ABA and ABC stacking. The blue and red arrowsindicate the coercive fields in the backward and forward scans. b.Schematics of the polarization switching pathways shown in (a) and(c). We associate such a switching pathway with a much smaller pin-ning potential at the top interface than at the bottom interface.To understand the variation in the polarization switching pathway, we per-formed statistical analysis on the coercive fields, which are summarized in Fig.5.22and Fig.5.23. Clearly, the coercive field varies up to 30% between cycles and dif-ferent devices can exhibit very different switching pathway statistics. We attributesuch a stochastic behavior to the presence of a network of pinning centers with dif-ferent potentials \u2013 despite having the same electric field loop applied in each cycle,domain walls are not deterministically pinned in the same configuration195.124Figure 5.22: a,c Coercive field for different cycles in the backward (a) andforward (c) scanning directions. Open circles represent the coercivefield for either the second switch from the intermediate state to the finalstate or the direct switch from the initial state to the final state. Greenspots indicate the ABA intermediate state, and orange spots indicatethe BAB intermediate state. b,d Representative field-dependent dRCspectra in the backward (b) and forward (d) scanning directions.125Figure 5.23: a,c Coercive field for different cycles in the backward (a) andforward (c) scanning directions. Open circles represent the coercivefield for either the second switch from the intermediate state to the finalstate or the direct switch from the initial state to the final state. Greenspots indicate the ABA intermediate state. b,d Representative field-dependent dRC spectra in the backward (b) and forward (d) scanningdirections.In addition, statistical analysis shows that the ABA stacking appears muchmore frequently than the BAB stacking. (BAB stacking is only observed occa-sionally in the forward scan of one device, out of five switchable devices). Ratherthan to the energy difference between two intermediate stackings, here we attributethe prevalence of ABA stacking to an asymmetric screening due to n-type initialdoping. When a thin layer of semiconductor is placed in a uniform electric field,the electronic band structure near both surfaces will be bent within the space chargelayer. Similar to the Debye length, the space charge layer thickness is determinedby the local density of states (DOS) at the Fermi level. If the semiconductor ischemically n-type doped with the Fermi level close to the conduction band edge,an electric field out of the film that induces hole doping can cause a much largerband bending than an electric field into the film that causes an additional electrondoping, because the DOS increases significantly if the Fermi level enters the con-duction band. As a result, one surface can experience much more band bending126and admit more electric field than the other surface.Experimentally, we observed a large contrast in the screening of two gate fields(Fig.5.24): when the trilayer is in CBA stacking, only the top gate that generatesan electric field out of the film can achieve the switching; the bottom gate appearsmostly screened by the A layer. The Fermi level in this device is clearly close tothe conduction band edge, as reflected in the early appearance of the polaron peak(Fig.5.24(b)), and such n-type doping is common among MoS2 crystals. Mostlycomprising the electric field from the top gate, the total displacement field in theCBA stacked MoS2 is, therefore, stronger at the upper interface than at the lowerinterface (Fig.5.24(a)). If the pinning centers at both interfaces have similar trap-ping potentials, the domain wall at the upper interface will then experience largerstress and become unpinned first. As a result, the CBA stacking will switch toABA first and then switch to ABC.In the opposite switching direction from ABC to CBA, both the external fielddirection and screening asymmetry are reversed (Fig.5.24(c) and Fig.5.24(d)). Thetotal displacement field thus becomes stronger at the lower interface than at theupper interface and moves the bottom domain wall first. Consequently, ABA is al-ways the favored intermediate state whichever the switching direction is. Overall,such an asymmetric screening effect should play a significant role in any multilay-ered devices with n-type doping, and the switching pathway in a 3R-MoS2 deviceis jointly determined by many extrinsic factors including the pinning center anddomain wall distribution, as well as the doping that can screen the external field.127Figure 5.24: a,c Illustration of the external electric field contributions fromthe top (blue arrows) and bottom (red arrows) gates to the 3R-MoS2trilayer, with the initial state being CBA (a) and ABC (c) polariza-tion states, respectively. The middle panel depicts the first switchingpathway, while the right panel illustrates the band structure of the ini-tial polarization state. b,d Opposite switching behaviors due to theasymmetric screening effect for initial states of CBA (b) and ABC (d)stacking, respectively.In conclusion, our investigation into the excitonic response of 3R-MoS2 tri-layers under varying electric fields and doping carrier densities has allowed us todistinguish different stacking configurations in a trilayer, even when net polariza-tions are identical. This resolution enables the identification of possible pathwaysduring polarization switches, revealing that intermediate states are primarily de-termined by the sequence of DW releases and, therefore, the relative strength ofpinning centers at distinct interfaces that initially localize the DWs. Variability inswitching behavior across multiple scan cycles indicates a potential impact of DWmotion on the distribution of pinning centers. Moreover, we found one interme-diate state occurs more frequently than the other, which can be attributed to anasymmetric screening effect arising from the band alignment with an n-type dop-ing. Although this study is focused on the 3R-MoS2 trilayer as a prototype, ourinsights should apply to other sliding ferroelectric material systems with multipleinterfaces.1285.2.5 Electrostatic simulation on dual-gated 3R-MoS2 trilayer withABC stackingIn this section, we develop an electrostatic model for ABC-stacked trilayer MoS2to quantitatively explain why the top and bottom interfaces experience differentexternal electric fields in a dual-gated device. We assume the trilayer MoS2 is en-capsulated by top and bottom hBN layers of equal thickness, dt = db = d = 10nm, as shown in Fig.5.25(a). The band structure of the ABC-stacked trilayer ispresented in Fig.5.25(b). Assuming a flat vacuum level, there is an intrinsic workfunction difference between the top, middle, and bottom layers. The work func-tion difference between adjacent layers is equal to the intrinsic interlayer potential,Wb\u2212Wm =Wm\u2212Wt = e\u03c60.The charge density \u03c1 of each layer (t, m, and b) can be calculated by the parallelcapacitor model (Equation 5.4-5.6).\u03c1t +P =Cg(\u03c6t \u2212\u03c6tg)+Cm(\u03c6t \u2212\u03c6m) (5.4)\u03c1m =Cm(\u03c6m\u2212\u03c6t)+Cm(\u03c6m\u2212\u03c6b) (5.5)\u03c1b\u2212P =Cg(\u03c6b\u2212\u03c6bg)+Cm(\u03c6b\u2212\u03c6m) (5.6)Here P is the spontaneous polarization. \u03c6i (i = t,m,b, tg,bg) denotes the electro-static potential of each layer in the dual-gated device. Cg = \u03b5hBN\u03b50d and Cm =\u03b5MoS2\u03b50d0are the geometrical capacitance of the gate and adjacent bilayer, respectively.We also have boundary conditions where the electrostatic potentials of the topand bottom gate are set by the gate voltages (Vt ,Vb) and trilayer MoS2 is grounded.The electrochemical potential of a semiconductor is defined as the summation ofelectrostatic potential and chemical potential \u2212e\u03a8=\u2212e\u03c6 +\u00b5 .\u03c6tg =Vt (5.7)\u03c6bg =Vb (5.8)\u2212e\u03a8t =\u2212e\u03c6t +\u00b5t = 0 (5.9)\u2212e\u03a8m =\u2212e\u03c6m+\u00b5m = 0 (5.10)\u2212e\u03a8b =\u2212e\u03c6b+\u00b5b = 0 (5.11)129The chemical potential of each layer (t,m,b) varies as the gate voltage changes.Thus, we can write the chemical potential of \u00b5t , \u00b5m, and \u00b5b as the summation ofintrinsic work function and their change \u2206\u00b5 .\u00b5t =\u2212Wm\u2212 e\u03c60+\u2206\u00b5t (5.12)\u00b5m =\u2212Wm+\u2206\u00b5m (5.13)\u00b5b =\u2212Wm+ e\u03c60+\u2206\u00b5b (5.14)Based on the equation (5.4)-(5.15), we can express the charge density \u03c1t\/m\/b ofeach layer as a function of chemical potential change \u2206\u00b5t\/m\/b.\u03c1t =\u2212Cg(Vt \u2212V\u00aft)+Cg\u2206\u00b5te +Cm\u2206\u00b5t \u2212\u2206\u00b5me(5.15)\u03c1m =Cm(\u2206\u00b5m\u2212\u2206\u00b5t)+Cm(\u2206\u00b5m\u2212\u2206\u00b5b) (5.16)\u03c1b =\u2212Cg(Vb\u2212V\u00afb)+Cg\u2206\u00b5be +Cm\u2206\u00b5b\u2212\u2206\u00b5me(5.17)The V\u00aft =Wt\u2212Wtge and V\u00afb =Wb\u2212Wbge are the work function difference between graphiteelectrodes and MoS2 layer. Since the reference point of the real experiment is theground of our source meter rather than the real vacuum, we need to re-define thezero point by offsetting a voltage constant of V\u00aft and V\u00afb. We manually set V\u00aft = V\u00afb = 0for simplicity in the following discussion.Next, we calculate the charge density of each layer as a function of chemicalpotential change \u2206\u00b5t\/m\/b. For a 2D semiconductor with a given chemical potential,the total in-band carrier density can be computed based on Fermi-Dirac distribu-tion.N =\u222b \u221e0dE1+ exp(E\u2212\u00b5kBT )= kBT ln(1+ exp(\u00b5kBT)) (5.18)Thus, the in-band electron density of each layer nt\/m\/b can be calculated from130Equation 5.18.n j = gcN(\u00b5 j\u2212Ec j,kBT )\u2212 13 \u2211i=t,m,bgvN(Ev\u2212\u00b5i,kBT ) (5.19)j = t,m,bHere Ec j ( j = t,m,b) are the energy level of the conduction band edge of the top,middle, and bottom layers, which will also be changed under an external electricfield. gc =2m\u2217ce\u03c0\u210f2 and gv =2m\u2217ve\u03c0\u210f2 are the density of states (unit: m\u22122eV\u22121) at theconduction and valence band edges, respectively, where we take the effective massof m\u2217c = 0.50m0 and m\u2217v = m0. Since the highest valence band of trilayer MoS2is at \u0393 valley, the in-band holes are equally distributed within three layers. Weshould note that the chemical potential of a specific layer is not well defined whenthe Fermi-level is at \u0393 valley. Because the MoS2 is usually n-doped, the chemicalpotential is away from the valence band. The classical model of equation 5.19 doesnot affect the simulation results.We assume the energy separation between the conduction band edge of themiddle layer and its intrinsic chemical potential to be \u2206Ec = 160 meV, suggestingthe MoS2 is n-doped due to the chemical doping. Thus, we can calculate the energyof band edges as a function of the chemical potential change.Ect =\u2212Wm+\u2206Ec+ e(\u03c6m\u2212\u03c6t) =\u2212Wm+\u2206Ec\u2212 e\u03c60+ e(\u2206\u00b5m\u2212\u2206\u00b5t) (5.20)Ecm =\u2212Wm+\u2206Ec+ e(\u03c6t +\u03c6b\u22122\u03c6m) (5.21)=\u2212Wm+\u2206Ec+ e(\u2206\u00b5t +\u2206\u00b5b\u22122\u2206\u00b5m)Ecb =\u2212Wm+\u2206Ec+ e(\u03c6m\u2212\u03c6b) =\u2212Wm+\u2206Ec+ e\u03c60+(\u2206\u00b5m\u2212\u2206\u00b5b) (5.22)Ev =\u2212Wm+\u2206Ec\u2212Eg (5.23)Due to the strong interlayer hybridization, the energy of the valence band edgeat \u0393 valley remains unchanged under a weak external field (Eg = 1.51 eV is thesemiconductor band gap of trilayer MoS2, measured from the middle layer). Ifwe set \u2206\u00b5t\/b\/m = 0, we can calculate the residue in-band carrier density n\u00aft\/m\/bdue to chemical doping using equation 5.19. Here we set the work function ofthe middle layer to be Wm = 4.2 eV and the intrinsic interlayer potential between131adjacent layers to be \u03c60 = 60 meV. The total free carrier density in each layer underthe external gate can be computed by the following equation, considering both thein-band and in-gap density of states.\u03c1t =\u2212e(nt \u2212 n\u00aft)+Cq0\u2206\u00b5te (5.24)\u03c1m =\u2212e(nm\u2212 n\u00afm)+Cq0\u2206\u00b5me (5.25)\u03c1m =\u2212e(nb\u2212 n\u00afb)+Cq0\u2206\u00b5be (5.26)For an intrinsic semiconductor, the in-gap quantum capacitance is close to zeroCq0 \u2192 0. However, we are studying a n-doped semiconductor, which is usually thecase for MoS2. The Fermi-level of trilayer MoS2 is close to the conduction bandedge, making it highly susceptible to band tailing effects and defect energy levels,which leads to a finite Cq0. In our simulation, we take a constant Cq0, which is 5times the gate capacitance Cg.Numerically solving the equation of 5.15-5.26, we can get the chemical poten-tial change of each layer as a function of top gate voltage at a temperature of T = 4K, when the device is anti-symmetrically gated Vt =\u2212Vb, as shown in Fig.5.25(c).The conduction band edge, the chemical potential, and the carrier density of eachlayer as a function of gate voltage are presented in Fig.5.25(d) and Fig.5.25(e).When the chemical potential of the top layer is tuned into the top-layer conductionband, electrons are doped in the top layer while holes are doped into the middleand bottom layers. The free carriers in the middle layer cause the discontinuity ofthe displacement field within the trilayer. As shown in Fig.5.25(f), the interlayerpotentials at the top and bottom interfaces are different under positive gate volt-age where the external electric field is anti-parallel to the spontaneous polarization.The bottom layer always experiences a stronger field than the top layer, leadingto the DW of the bottom interface moving first. So the intermediate state will beABA when the stacking order changes from ABC to CBA. These simulation resultsexplain the experimental fact that the ABA state appears more frequently than theBAB state.132dd0PVtVbVacuumtop middle bottom \u00b5t \u00b5m \u00b5b\u0393a b ctmbbgtg\u2206Ec-5 0 5Vt (V)-300-200-1000100200  (meV) t m b-5 0 5Vt (V)-4.6-4.4-4.2-4-3.8Energy (meV)tmbEctEcmEcb-5 0 5Vt (V)-1000100200interlayer potential () (meV)interface topinterface bottom-5 0 5Vt (V)-4-20246 ( cm-2)1012tmbd e fFigure 5.25: a. Schematic of a dual-gated 3R-MoS2 trilayer device with ABCstacking order. b. Electronic band structure of ABC-stacked 3R-MoS2trilayer. c-f Simulation results. c. Chemical potential change of top,middle, and bottom layers as a function of top gate voltage Vt . d. Con-duction band edge and chemical potential of top, middle, and bottomlayers as a function of top gate voltage Vt . f. The carrier density ofeach layer. f. The interlayer potential of the top and bottom interfacesas a function of top gate voltage when the device is anti-symmetricallygated.133Chapter 6Strongly correlated insulatingstates in twisted MoSe2homobilayer6.1 Introduction to moir\u00e9 superlattice and stronglycorrelated physicsUtilizing a controllable or accessible quantum system to simulate the quantumphysics in a far more complicated system has been regarded as one of the mostpromising methods to address the computational challenges inherent in many-bodyproblems200\u2013202. The diverse platforms for quantum simulation have been signifi-cantly developed, such as photons203, ultra-cold atoms204\u2013206 and superconductingcircuits207. These artificial quantum systems provide valuable tools for investigat-ing complex phenomena in condensed matter physics, including quantum phasetransitions208\u2013210, magnetism208;211;212, and high-Tc superconductivity213;214.In parallel, quantum materials based on moir\u00e9 superlattice have introduced anew pathway for studying emergent phenomena driven by Coulomb interactionsbetween electrons50;51;215\u2013217. Transition metal dichalcogenides (TMDs) homo-and hetero-bilayers, in particular, have been utilized in realizing the extended Hub-bard model on a triangular lattice57;93;110, where various quantum phenomena have134been revealed, including a plethora of correlated insulating states3;111;218\u2013223, Mott-Hubbard phase transition62;224\u2013226, as well as quantum anomalous Hall states227.The ability to control the energy competition between the onsite Coulomb interac-tion(U) and kinetic energy(t) of electrons by external field or twist angle makes themoir\u00e9 TMD bilayer an ideal platform to study the strongly correlated physics93;216.The formation of moir\u00e9 materials requires stacking of two layers of van derWaals materials, either with a small twist (\u03b8 ) or a lattice mismatch (\u03b4 ). The in-terference of atomic lattice gives rise to a new set of periodic structures known asmoir\u00e9 superlattice. Since the twist angle or lattice mismatch is usually quite small(Equation 6.1), the superlattice constant aM is typically many times larger than theatomic lattice constant a. Consequently, the Brillouin zone in the momentum spacebecomes folded into a smaller one referred to as the mini-Brillouin zone (mBZ)49(Fig.6.1(a)). As a result, the low energy physics in these materials are effectivelydetermined by electrons moving in a smooth periodic potential at the large scaleaM.aM =a\u221a\u03b8 2+\u03b4 2>> a (6.1)The moir\u00e9 superlattice of TMD hetero-bilayers, which arises from stacking twomonolayers with different lattice constants, is first investigated in the experiment.The superlattice in TMD hetero-bilayers typically possesses a period of approx-imately 10 nm93, approximately one order of magnitude larger than the atomiclattice constant of 0.3 nm. Consequently, the moir\u00e9 density (\u223c a\u22122M ) in these sys-tems is around 1012 cm\u22122, which is two to three orders of magnitude smaller thanthe real atomic density a\u22122. In typical field-effect devices, the maximum dopingdensity achievable before dielectric breakdown is approximately 1% of the atomicdensity a\u22122. This means that electrostatic gating can fill a significant number ofelectrons within each artificial atom, which has a size on the order of aM. Thisaspect is of particular significance because each integer filling of electrons withinthe moir\u00e9 superlattice can be viewed as analogous to a different chemical elementin real materials.There are also two different stacking configurations in an angle-aligned TMDheterobilayer (Fig.6.1(b)): 0\u25e6 stacking and 180\u25e6 stacking, which is also referred135d e fmini BZa b cR-type H-typeFigure 6.1: a. A schematic of mini-Brillouin zone for twisted bilayergraphene49.b.A schematic of R-stacking and H-stacking as well as theirhigh symmetric points93. c. Schematic illustration of an array of moir\u00e9atoms that trap electrons, which can tunnel between neighboring siteswith amplitude t and experience on-site Coulomb repulsion U93.b andc are reproduced from ref.93. d. STM image of a Wigner crystalstate in angle-aligned WS2\/WSe2 heterostructure221 (Reproduced fromref.225). e.The magnetic susceptibility \u03c7 of moir\u00e9 bilayer is measuredby magnetic circular dichroism (MCD). The negative Curie-Weiss tem-perature indicates an anti-ferromagnetic interaction of the correspond-ing correlated state110. f. WS2\/WSe2 heterostructure simulates the Mottphysics at half-filling110. e-f are reproduced from ref.110to as R-type and H-type93;228, similar to the TMD homobilayer. As shown inFig.6.1(b), for R-type stacking, three high-symmetry points exist in the superlatticeof the hetero-bilayer, which are MM, XM, and MX , respectively. The XM sites de-note the transition metal atom (M=Mo, W) directly on top of the chalcogen atom X(X=S, Se) while the MX sites are the inversion of XM. The metal atom lays on topof the metal atom in another layer, which is called MM site. These high-symmetrysites are different for H-type stacking, in which they are defined as MM, XX , andXM sites.The superlattice exhibits periodic variations in the local interlayer registry,136which refers to the in-plane displacement from one layer\u2019s metal atom to a nearbymetal atom in the other layer229. As a result, quantities such as the interlayer dis-tance (d) and the local bandgap (Eg) experience lateral modulation within a singlemoir\u00e9 unit cell, leading to a varying potential energy landscape (VM) for electronsmoving within the heterobilayer (Fig.6.1(c))93.By applying Bloch\u2019s theorem to the moir\u00e9 superlattice, it is possible to de-scribe the low-energy physics quantitatively using a continuum Hamiltonian HM =P2M2m\u2217 +VM for electrons within the moir\u00e9 superlattice, where PM represents the quasi-momentum operator, and m\u2217 denotes the effective mass of the electrons. In thelimit of large moir\u00e9 superlattice aM >> a, where a is the atomic lattice constant,the largest wave vector in the first mini-Brillouin zone of the moir\u00e9 superlattice isgiven by KM = 4\u03c03aM . Based on the wave vector KM, the bandwidth W of the firstmoir\u00e9 mini-band can be estimated to be approximately 10 meV, taking the effectivemass of band electrons m\u2217 \u2248 0.5m0 (m0 is the mass of free electrons)230.The narrow bandwidth suggests that the kinetic energy of electrons in the su-perlattice is largely suppressed, leading to the formation of a set of flat bands. Onthe other hand, the on-site Coulomb repulsion U \u2248 e24\u03c0\u03b5\u03b50aw of electrons is about100 meV, taking the dielectric constant of substrates \u03b5 \u2248 4 and the characteristicsize of Wannier orbital aw \u2248 2 nm. The repulsion between electrons of the nearest-neighbor moir\u00e9 sites is estimated as V = e24\u03c0\u03b5\u03b50aM \u2248 50 meV. The estimation ofU > V >>W indicates that the Coulomb interaction between carriers dominatesover the kinetic energy and the single particle picture does not hold for moir\u00e9 het-erobilayers of TMDs. Therefore, the moir\u00e9 hetero-bilayer provides a new platformto simulate the strong-correlation physics which is beyond the band picture.The lowest moir\u00e9 potential localizes XM and MX sites for R-type and H-typestacking configurations respectively, leading to a formation of a triangular latticewithin the moir\u00e9 hetero-bilayer of TMDs216. Various model Hamiltonians can bemapped into the triangular lattice. Quantum simulations of the Hubbard model110and the Kane-Mele-Hubbard model227 have been achieved in different TMDs heter-obilayers, such as WSe2\/WS2 and MoTe2\/WSe2. By continuously tuning the band-filling factor in the hole-doped regime, a wide range of the Hubbard phase diagramcan be explored57. At half-filling, corresponding to one hole per moir\u00e9 unit cell, aninsulating state is observed (Fig.5.1(d)), contradicting the single-particle picture. In137this regime, the on-site interaction dominates, and hopping between different moir\u00e9sites is significantly suppressed, giving rise to a Mott insulating state110;111. Theemergence of the Mott insulator suggests the localization of charge carriers andthe formation of local magnetic moments. The interaction between these magneticmoments is confirmed to be antiferromagnetic based on the negative Curie-Weisstemperature extracted from temperature-dependent measurements of the magneticsusceptibility (\u03c7) (Fig.6.1(e)). The sample magnetization can be measured usingmagnetic circular dichroism (MCD), taking advantage of the spin-valley lockingand optical selection rules in monolayer TMDs30.Due to the C3 symmetry of the triangular lattice, the moir\u00e9 heterobilayers canalso simulate other exotic phases of matter when the doping density deviates fromhalf-filling222. In the strong correlation limit U >V >W , the minimization of theextended Coulomb repulsion between electrons drives the spontaneous formationof incompressible charge-ordered states at commensurate fractional filling factors,which is also referred as to generalized Wigner crystals(Fig.6.1(f))221 220. In ad-dition to charge-ordered states that preserve the rotation symmetry of the system,the stripe phases223, which breaks the rotational symmetry, can also form due tothe competition between short-range and long-range Coulomb interaction. Theformation of different charge-ordered patterns aims to minimize the energy asso-ciated with extended Coulomb repulsion, thus dramatically enriching the strong-correlation phase diagram.6.2 Moir\u00e9 flat bands in \u0393 valleyAnother way to achieve flat bands in TMD-based materials is to stack two mono-layers of the same material with a small twist, as what has been done in thegraphene system50;51. The ground state of the Hubbard model not only depends onthe ratio of U\/t, but also depends on the symmetry of underlying moir\u00e9 superlat-tice57;58;231. A honeycomb lattice is formed in rhombohedral-stacked TMD homo-bilayers with a small twist angle, providing a different lattice structure to explorevarious many-body states with distinct symmetries92;232;233. Recently, quantummagnetism234, integer and fractional Chern insulator235\u2013237, as well as quantumspin hall states238;239 have been observed in rhombohedral-stacked moir\u00e9 MoTe2138and WSe2 homobilayer with a direct band-gap at K point of momentum space.In contrast, rhombohedral-stacked homo-bilayers of other TMD families such asMoSe2, MoS2, and WS2 exhibit an indirect band gap, with the highest valenceband localized at \u0393 valley, the center of the Brillouin Zone73;117. To date, themoir\u00e9 effect of a honeycomb lattice in \u0393 valley remains unexplored, partially be-cause of the indirect band-gap nature of these twisted TMD homo-bilayers, whichimpedes direct optical measurements as achieved in twisted MoTe2.The moir\u00e9 bands localized in the \u0393 valley are qualitatively different from theflat bands in the K valley63;240;241. In the absence of spin-orbit coupling, spin de-generacy occurs at the \u0393 valley, leading to the emergence of a new quantum simula-tor characterized by spin SU(2) symmetry242. Analogous to monolayer graphene,the effective low-energy Hamiltonian of the first \u0393-valley moi\u00e9 band on a honey-comb lattice has a C6 symmetry242, providing a platform for investigating DiracFermions at the strong-correlation limit243;244. It has been theoretically predictedthat a semi-metal to insulator phase transition occurs when the ratio of U\/t ex-ceeds a critical threshold, opening a charge-gap at Dirac point245;246. The chiralsymmetry is spontaneously broken along with the semi-metal to insulator phasetransition, which serves as the mechanism for Dirac Fermions to acquire mass246.However, this picture has not been fully experimentally verified due to the weakelectron-electron interaction in monolayer graphene247. The moir\u00e9 bands in the\u0393 Valley of twisted TMD homobilayer may offer such an opportunity, as the largeeffective mass of the \u0393 Valley can further suppress kinetic energy and highlight theimportance of interaction248.The topmost orbital in the \u0393 valley is a mixture of p- and d-wave orbitals63, giv-ing rise to a different size of the Wannier function57. The theoretically predictedmoir\u00e9 potential in twisted homo-bilayer is around five times larger than the hetero-bilayer, significantly enhancing the ratio of U\/t. On the other hand, the honeycomblattice is bipartite. The geometrical frustration is removed, which can potentiallylead to different magnetic ground states, such as different generalized Wigner crys-tal220, quantum spin-liquid242, and unconventional superconductivity249.139a bphoton energy (eV)Reflection contrastconductive Insulating 2s excitontop gatetop hBNsensorsamplebottom hBNcontactbottom gatecontact gateFigure 6.2: a. Schematic of exciton sensing in a dual-gated device. The 2sexciton is sensitive to the dielectric environment change in the twistedMoSe2 bilayer. b. Schematic of 2s exciton spectra when the twistedMoSe2 bilayer is conductive or insulating.6.3 Rydberg exciton sensing techniqueDue to the indirect band-gap nature, there are no direct optical transitions involvingthe \u0393 valley. Thus, it is challenging to probe the possible correlated insulating statesarising from the \u0393-valley moir\u00e9 flat bands. To overcome this challenge, we employa novel experimental technique called 2s-exciton sensing, proposed by Dr. Kin FaiMak and Jie Shan\u2019s group at Cornell University219;220;250.As we have discussed in section 1.1, the strong light-matter interaction in themonolayer TMD leads to the formation of a series of exciton states, similar to theRydberg states of a 2D hydrogen atom. These exciton states, such as 2s and 3sstates, have a significantly larger Bohr radius compared to the monolayer thickness.For example, the Bohr radius of a 2s exciton in the WS2 monolayer is estimated tobe 4-5 nm. As a result, these excitonic states are highly sensitive to the dielectricenvironment in proximity to the monolayer. By placing a monolayer TMD close toa moir\u00e9 superlattice (Fig.6.2(a)), insulating states at different filling factors can beprobed by optically measuring the resonance and oscillator strength of the excitedexciton states in the monolayer. If there is a charge-gap opening at a certain fillingdensity, the dielectric environment becomes more insulating, leading to a reductionof screening in 2s resonance. Thus, the 2s exciton will blueshift, along with anenhancement of oscillator strength (Fig.6.2(b)).140In our experiment, we designed a dual-gated device with an embedded WS2monolayer sensor (Fig.6.2(a)). The twisted MoSe2 homo-bilayer and MoS2 sensorare separated by a 1 nm thick hBN spacer. This spacer is chosen to suppress elec-tronic coupling between the sample and the sensor while maintaining proximitybetween them. The dual-gated geometry of the device allows independent controlof carrier density and electric field within the sample. The sum of the top and bot-tom gate voltages determines the total carrier density, while the voltage differencebetween them determines the total electric field.Correlated insulating states in tMoSe2In this section, we report the realization of \u0393-valley moir\u00e9 flat bands on a honey-comb lattice in a twisted MoSe2 homobilayer (t-MoSe2), where the high symmetrypoints with the lowest moir\u00e9 potential are at MX and XM sites63 (Fig.6.3(a) andFig.6.3(b)). Within the \u0393 valley of the t-MoSe2 homo-bilayer, we observe a sub-lattice polarized Wigner state at the filling of one hole per moir\u00e9 unit cell as wellas a series of correlated insulating states at various fractional fillings. Addition-ally, we identify a Mott-insulator phase on the half-filled honeycomb lattice of the\u0393-valley, contradictory to the semi-metal prediction proposed by the continuousmodel251. Our experimental findings establish a new system capable of simulat-ing the Heisenberg model on a honeycomb lattice and exploring the semi-metal toinsulator transition by controlling the twist angle and dielectric screening252.To investigate the correlated states at the \u0393-valley, we use the 2s exciton sens-ing technique as discussed in section 6.3, placing a monolayer WS2 as a sensornear the twisted MoSe2 homo-bilayer (Fig.6.3(c)), separated by a \u223c 1 nm hBN(3-4 layers). The Fermi-level of the moir\u00e9 MoSe2 bilayer can be tuned to reach the\u0393-valley using top and bottom gates (Fig.6.3(c)). Due to the type-II band alignmentbetween t-MoSe2 homobilayer and WS2 monolayer, carriers are mainly doped intothe MoSe2 bilayer, while maintaining the Fermi-level within the gap of the sensorlayer (Fig.6.3(d)). This method has been previously employed to detect the gen-eralized Wigner crystal states220as well as excitonic insulator states250;253 in TMDhetero-bilayers. Since the 2s exciton state is sensitive to the surrounding dielectricenvironment220, it can serve as an optical probe to access the moir\u00e9 flat bands in141\u0393-valley. The full spectrum of WS2\/hBN\/t-MoSe2 at T=7K without any doping isshown in Fig.6.3(e). The moir\u00e9 A-exciton of t-MoSe2 homobilayer is at 1.617 eVwhile the peak of 2s exciton of WS2 is around 2.190 eV. The correlated states areidentified by monitoring the change in the peak position and oscillator strength ofthe 2s exciton of the sensor when the Fermi-level is tuned within the moir\u00e9 bandsof the sample. If there is a correlated insulating state, an energy blueshift alongwith an enhancement in the oscillator strength of the 2s exciton is expected.We first perform a band structure calculation on a 3.6\u25e6 tMoSe2 bilayer (ourtarget angle). The topmost two valence bands arising from the \u0393 valley are wellisolated from the remote bands, indicating that the effective low-energy Hamil-tonian can be mapped onto a two-orbital Hubbard model of a honeycomb lattice(Fig.6.4(a)). The continuum model confirms the existence of a Dirac point at thek point of the mini-Brillouin zone, corresponding to the filling of two holes permoir\u00e9 unit cell.The Fermi-level of the tMoSe2 bilayer is tuned to reach the \u0393 valley by apply-ing a negative voltage to the top and bottom gates. The gate dependence of the 2sexciton spectra in the WS2 sensor is shown in Fig.6.4(b). We use the first deriva-tive of the reflection spectrum to enhance the contrast. Between the gate voltageVtg =\u22124 V and Vtg =\u22121.2 V, we observe a series of resonance blueshifts of the 2sexciton accompanied by an enhancement of oscillator strength, which signaturesthe emergence of insulating states. We attribute the two most prominent featuresobserved at Vtg = \u22122.36 V and Vtg = \u22123.48 V to the integer fillings, correspond-ing to the filling numbers \u03bd =\u22121 and \u03bd =\u22122, respectively. Using these two statesas landmarks, we can calibrate the moir\u00e9 density to nM \u2248 4.6\u00d7 1012cm\u22122, corre-sponding to a twist angle of 3.77\u25e6 (close to the target angle) and a 5.0 nm moir\u00e9period aM.The correspondence between the enhancement of the 2s exciton and insulatingstates becomes clearer when we plot the oscillator strength R2s as a function ofthe filling number \u03bd (Fig.6.4(c)). The filling numbers are assigned by assuming alinear dependence between gate voltage and doped hole density. We use the peak-to-peak amplitude of the 2s exciton to represent the oscillator strength, as defined inthe Fig.6.5. Between \u03bd =\u22121 and \u03bd = 0, we observe a cascade of insulating states,all appearing at commensurate fractional fillings such as \u03bd =\u221216 ,\u221214 ,\u221213 ,\u221212 ,\u221223 ,142a bXMMMMX\u0398 -1 0 1X (a M)-101Y (aM)-1000100200300moire potential (meV)MMXMMXXMMXMXXMdc eVtVbGraphitehBNWS2  spacert-MoSe2  KWS2  \u0413 Kt-MoSe2  1.5 2 2.5Photon energy (eV)-3.5-3-2.5-2-1.5-1-0.5R\/R1s2smoire excitonMo Se Figure 6.3: a. Schematic of a small angle twisted MoSe2 in R-stacking. Col-ored circles highlight high symmetry points. The yellow circle repre-sents the MX sites where the Mo atom is on top of the Se atom whilethe green and purple circles correspond to XM and MM sites, respec-tively. b. Simulated real-space distribution of moir\u00e9 potential for holes.c. Schematic of a dual-gated device structure. d. Type-II band align-ment between t-MoSe2 and WS2. The spin is degenerated in \u0393 valley.e. Reflection Contrast of WS2\/hBN\/t-MoSe2143140120100806040200\u03b3 m \u03b3Energy (meV)k 2.15 2.2 2.25V tg(V)-4-3.5-3-2.5-2-1.5-1-0.50Photon energy (eV)derivative of R\/R-4 9a b cR2sFilling =-2=-1Figure 6.4: a. Calculated band structure of tMoSe2 bilayer along the di-rection of \u03b3-k-m-\u03b3 of the mini-Brillouin zone. Grey and black dashlines denote the Fermi-level position when the filling is at \u03bd = \u22121 and\u03bd = \u22122. b. First derivative of Reflection contrast (\u2206R\/R) of the de-vice near the 2s exciton resonance versus top gate voltage (Vtg). Theback-gate voltage is applied with a fixed ratio of Vbg = 1.15Vtg. Theexperimental temperature is 7 K. c. Filling number (\u03bd) versus oscillatorstrength of the 2s exciton (R2s). Grey and black dash lines label the twointeger fillings.and \u221256 , suggesting the reliability of our assignments. Additionally, we observea continuous band of insulating states within the filling range of \u221232 < \u03bd < \u22121.All of these insulating states remain insensitive to the external electric field dueto the strong interlayer hybridization of \u0393 valley (Fig.6.6(a)). The observation ofmomentum-indirect \u0393-K exciton also confirms the \u0393-valley origin of moir\u00e9 flatbands (Fig.6.6(b)). Following the same procedure, another point (P2) of the deviceD1 is also examined. Similar insulating states at both integer and fractional fillingscan be resolved at P2, despite a slightly smaller calibrated angle of 3.55\u25e6 due to thetwist inhomogeneity (Fig.6.7).1442.15 2.2 2.25Photon Energy(eV)-0.100.10.20.30.40.50.60.7RC2.15 2.2 2.25Photon Energy(eV)-20020406080100120140160180dRC\/dE1.6 K3 K5 K7 K9 K13 K15 K18 K20 K25 K30 K35 K40 K45 K50 K60 K70 Ka bR2sFigure 6.5: a. Reflection contrast spectra of the 2s exciton at different tem-peratures. The doping is fixed at the charge neutral point of \u03bd = 0. Weuse the peak-to-peak amplitude at the 2s resonance to represent the os-cillator strength, denoted as R2s. b. The first derivative of the spectra in(a) with respect to photon energy.The insulating state observed at \u03bd =\u22122 is unexpected since it corresponds to ahalf-filled honeycomb lattice with the Fermi level at the Dirac point. Different fromthe twisted MoTe2 and WSe2, where spin-orbit coupling opens a single-particlegap at the k point of the mini-Brillouin zone, no such single-particle gap exists inthe small-angle twisted MoSe2 homobilayer251. The only possible explanation forour results is that strong correlation effects open a charge gap at the Dirac point,making the ground state of the half-filled honeycomb lattice become a correlatedinsulator rather than a semi-metal.The correlated nature of these insulating states is confirmed by measuring thetemperature dependence of the 2s exciton spectra (Fig.6.8(a)). These insulatingstates gradually disappear, and the enhancement of 2s exciton resonance becomesinvisible as the temperature rises. To characterize this melting behavior, we track145a b-4 -3.5 -3 -2.5 -2 -1.5 -1Vtg (V)-60-40-200204060E (mV\/nm)R2s0.0450.0151.3 1.4 1.5 1.6 1.7 1.8 1.9Photon energy (eV)00.511.52Counts104K-K excitons\u0393-K excitonFigure 6.6: a. The phase diagram of R2s as a function of doping (Vtg ) andelectric field (E). The insulating states with corresponding filling num-bers are labeled by yellow and black dashed lines. The region labeledby a magenta dashed line indicates the region where the doped carriersstart to leak into the sensor layer. b. The photoluminescence spectrumof tMoSe2 homobilayer at T=1.6 K. The low-energy emission between1.4-1.6 eV is the momentum indirect \u0393-K excitons.the changes in the 2s spectral weight across different temperatures (Fig.6.9(a)). Wedefine a critical temperature Tc, where the value of 2s spectral weight decreases toabout 20% of its maximum value. As shown in the upper panel of Fig.6.8(b), the2s spectral weight of \u03bd = \u22122 state remains nearly flat at low temperatures anddecreases at high temperatures, indicating that the insulating state at half-filling in-deed arises from the electronic correlation. In contrast, the temperature dependenceof 2s spectral weight for the quarter filling case (\u03bd =\u22121) exhibits a Pomeranchukeffect224;254. The tMoSe2 system at \u03bd =\u22121 becomes more insulating at an elevatedtemperature T \u2217 since the 2s spectral weight nearly doubles at T \u2248 15 K. A Similarphenomenon for \u03bd =\u22121 can be observed in another device D2 (Fig.6.9(b)).The critical temperature Tc, at which band crossover occurs due to thermalbroadening, can be used to estimate the size of the charge gap (\u2206c) for differentinsulating states. The results are summarized in Fig.6.8(c), with additional tem-perature dependence data provided in Fig.6.10. Generally, the charge gap \u2206c isgiven by \u03b1kBTc, where the coefficient \u03b1 varies with different filling numbers. For1462.15 2.2 2.25V tg(V)-4-3.5-3-2.5-2-1.5-1-0.50Photon energy (eV)  Photon energy (eV) R2sFilling Filling a b cderivative of R\/R-4 9derivative of R\/R-4 9Figure 6.7: a. First derivative of Reflection contrast (\u2206R\/R) with respect tophoton energy as a function of top gate voltage. The experimental tem-perature is 7 K. b. The same data as a with filling number as y-axis.Filling number (\u03bd) versus oscillator strength of the 2s exciton (R2s) atthe position P2.fractional fillings, the critical temperature ranges from 13 K to 25 K. These frac-tional insulating states are essentially generalized Wigner crystals with a chargegap of approximately 2-4 meV, estimated as twice the melting temperature219. The\u03bd =\u22121 state is identified as a sublattice-polarized Wigner crystal state in the pres-ence of two orbitals on a honeycomb lattice, with a charge gap of about 10 meV(\u223c 2kBTc). However, no ferroelectricity is observed in this state. The proposedcharge configuration of these generalized Wigner crystal states and their tempera-ture dependence are summarized in Fig.6.11. For the \u03bd =\u22122 state, the gap-closuretemperature is around 60 K. Using a value of \u03b1 \u2248 5 from numerical simulations fora honeycomb lattice in the strong correlation limit255, we can estimate a U \u2248 25meV Mott gap for the half-filled case, where U is the onsite Coulomb repulsion.From the temperature dependence measurement, the ratio between the Coulomb1472.15 2.2 2.25-4-3-2-10V tg(V)2.15 2.2 2.25 2.15 2.2 2.25 2.15 2.2 2.25 2.15 2.2 2.25Photon energy(eV)derivative of R\/R-4 9aT= 9 K T= 13 K T= 15 K T= 25 K T= 60 K\u03b80 30 60 90charge gap (meV)2.5 3.0 3.5 4.0 4.5 5.020406080100correlated insulatorsemi-metaldD1\u03b5 Tc (K)c0 20 40 60T (K)01=-1b12 10-4=-22s spectral weigthtTCTCT*Figure 6.8: a. First derivative of Reflection contrast (\u2206R\/R) as a function ofVtg measured at different Temperatures (9 K, 13 K, 15 K, 25 K, and 60K). b. Temperature dependence of 2s exciton spectral weight for theinsulating states at \u03bd =\u22121 (red) and \u03bd =\u22122 (black). The dashed linesare guides to the eye. Tc denotes the critical temperature. T \u2217 \u2248 15 Kdenotes the temperature where the 2s spectral weight exhibits a peak. c.Critical Temperature (Tc) of all observed insulating states at both integerand fractional fillings. d. Calculated charge gap of \u03bd = \u22122 state as afunction of effective dielectric permittivity (\u03b5) and twist angle (\u03b8 ). Thedashed line in orange denotes the boundary between correlated insulatorand semi-metal. The yellow dot denotes the parameter vicinity of deviceD1 in the experiment.1480 20 40 60 80T (K)0122s spectral weigtht10-4=-1T*TC2.15 2.2 2.25Photon energy (eV)-0.4-0.3-0.2-0.1R\/Ra bFigure 6.9: a. Reflection contrast spectrum of 2s exciton at \u03bd = \u22121. Thered area near 2s exciton resonance is integrated to obtain the 2s spectralweight.b. Temperature dependence of 2s spectral weight for \u03bd = \u22121state of another device D2. The peak of the 2s spectral weight is around25 K in this sample.interaction (U) and the kinetic energy (t) of our device D1 can also be estimated.The kinetic energy t is calculated from the energy bandwidth w, based on the twistangle and moir\u00e9 period: t = 13 w =13\u210f22m\u2217(4\u03c03aM)2\u2248 4.5 meV, with an effective massm\u2217 = 1.17m0 63. The factor 13 arises from the honeycomb lattice structure. Theestimated ratio of U\/t is 5.6, significantly larger than the critical value of 2.2,where a semi-metal phase transitions into a correlated Mott insulator245;246 (Ap-pendix A.2). We also perform an unrestricted Hartree-Fock calculation to mapthe phase diagram of tMoSe2 at \u03bd = \u22122, as shown in Fig.6.8(d) (Details are inAppendix A.3). The parameter \u03b5 is the effective dielectric permittivity, which rep-resents the screening of the three-dimensional dielectric environment, includingthe metallic graphite gate. The effective dielectric permittivity of our device isaround 30 based on the measured U , which is consistent with other TMD moir\u00e9materials57;110;226;240;242. The rough parameter space of device D1 is labeled bythe yellow dot, confirming that a charge gap is opened at the Dirac point due to thecorrelation effect (Fig.6.8(d)).1492.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=1.6K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=3K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=5K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=18K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=20K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=30K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=35K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=40K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=45K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=50K2.15 2.2 2.25Photon energy (eV)-4-3-2-10V tg (V)T=70Kderivative of R\/R-4 9Figure 6.10: Contour plot of gate dependence data at T=1.6 K, 3 K, 5 K, 18K, 20 K, 30 K, 35 K, 40 K, 45 K, 50 K, and 70 K.6.4 Magnetic property for the \u03bd stateWe next investigate the magnetic properties of the insulating state at \u03bd =\u22121. Mea-suring the magnetization of t-MoSe2 under a magnetic field is challenging becausethere are no direct optical transitions at the \u0393 valley. In the moir\u00e9 TMD hetero-bilayers or homo-bilayers with a direct band gap, magnetic circular dichroism(MCD) of excitons is commonly utilized to probe the spin configuration of car-riers at K valley110;219;234. This method relies on the chiral optical selection rule1500 20 40 60T (K)0122s spectral weigtht 10-4=-1\/40 20 40 60T (K)0122s spectral weigtht 10-4=-1\/30 20 40 60T (K)0122s spectral weigtht 10-4=-1\/20 20 40 60T (K)0122s spectral weigtht 10-4=-1\/60 20 40 60T (K)0122s spectral weigtht 10-4=-2\/3abcdeFigure 6.11: a-e. Left: Proposed charge configuration for the Wigner crys-tal state at fractional fillings. Right: Temperature dependence of 2sspectral weight.151and the spin-valley locking in monolayer TMDs, where the imbalance of reflectedlight intensity between left-handed(\u03c3+) and right-handed (\u03c3\u2212) circularly polarizedlight is proportional to the sample magnetization (M)93. However, this method cannot be directly applied to tMoSe2 due to its indirect band gap nature.Instead, we use the MCD of \u0393-K exciton Fermi-polarons to reflect the magne-tization of \u0393-valley carriers. When the Fermi level is within the moir\u00e9 band, moir\u00e9excitons in the K and K\u2032 valleys can bind with carriers at the \u0393 valley, forming\u0393-K Fermi-polarons, as shown in the lower panel of Fig.6.12(a). Multiple speciesof these exciton Fermi-polarons (M1-M3) have been observed. Left-handed light(\u03c3+) excites a spin-singlet exciton at the K valley of both layers in the tMoSe2 ho-mobilayer (upper panel of Fig.6.12(a)). Due to exchange interactions, the K-valleyexciton tends to bind with a spin-up hole in the \u0393 valley, forming a Fermi-polaronsince it contains a spin-down hole256. Similarly, K\u2032-valley excitons excited byright-handed light (\u03c3\u2212) are more likely to bind with spin-down carriers. Thus,the magnetization of \u0393-valley carriers is proportional to the MCD of \u0393-K excitonFermi-polarons.Near the resonance of \u0393-K exciton Fermi-polaron, the MCD signal is enhanced,as shown in Fig.6.12(b). Different species, such as M1 and M3 exhibit oppositeresponses to the magnetic field, indicating an opposite optical selection rule forM1 and M3. Focusing on a spectral window of 5 meV near the resonance ofM1, we measure the magnetic field dependence of MCD for the \u03bd = \u22121 stateat various temperatures (Fig.6.12(c)). The MCD signal follows a paramagneticresponse and saturates at |Bs|= 3.8\u00b10.2 T for T=1.6 K, suggesting the existenceof local moments. The saturation field Bs increases, and the magnetic susceptibility(\u03c7 = limB\u21920 \u2202M\u2202B ) decreases with rising temperature, a trend qualitatively similar tothe MCD of excitons measured in TMD heterobilayers110;220.These observations suggest that the measured MCD of \u0393-K exciton Fermi-polarons can effectively probe the spin configuration of \u0393-valley carriers. Themagnetic susceptibility at a low magnetic field follows a linear Curie-Weiss (CW)law with a small positive CW temperature (Fig.6.12(d)). The measured CW tem-perature for position P1 is \u0398CW = +0.27(\u00b10.20) K, indicating weak ferromag-netic\/paramagnetic interactions between local moments. For position P2, the CWpoint is \u0398CW = +1.10(\u00b10.40) K. We attribute the variations in the CW tempera-152-4 -3 -2 -1 0Vtg(V)1.561.581.61.621.64Photon energy (eV)0.890.90.910.920.93R\/R1.54 1.56 1.58 1.6 1.62 1.64Photon energy (eV)-0.03-0.02-0.0100.010.020.03MCDB=3 T-6 -4 -2 0 2 4 6B (T)-0.03-0.02-0.0100.010.020.03MCD T=1.6 KT=4 KT=7 Ka bdc\u03c3+ \u03c3-M1M2M3M1 =-1-2 0 2 4 6 8 10 T (K)0204060801\/ (a.u.)=-1P2P1Figure 6.12: a. Upper panel: K(K\u2019)-valley excitons pumped by circularly po-larized light selectively couple with carriers in \u0393 valley, forming \u0393-Kexciton Fermi-polarons. Lower panel: Reflection contrast spectrumof moir\u00e9-A excitons in tMoSe2 homobilayer as a function of top gatevoltage Vtg . M1-M3 are three species of \u0393-K exciton Fermi-polarons.b. MCD versus photon energy for \u03bd =\u22121 at B=+3 T. c. Magnetic fielddependence of MCD near M1 \u0393-K exciton Fermi-polarons at differenttemperatures. d. Curie-Weiss fit for the magnetic susceptibility (\u03c7) atzero magnetic field as a function of temperature at \u03bd =\u22121. P1 and P2are two positions measured in device D1. The fitted Curie-Weiss tem-perature is \u0398CW = +0.27(\u00b10.20) K and \u0398CW = +1.1(\u00b10.40) K forP1 and P2 respectively. The uncertainty arises from the linear fitting.1530-0.1-0.2-0.3-0.4 0.1\u0398=3.6o\u03b55 10 15 20 25 30 40J (meV)a b1234Figure 6.13: a. A schematic of the hopping processes contributing to theleading terms in the ferromagnetic correlation via spin exchange.Black arrows represent the local moments. Green, yellow, and redboxes outline the second, third, and fourth-order processes, respec-tively. The color of the arrows corresponds to the step number in thehopping process. b. Spin-spin coupling constant J as a function of ef-fective dielectric permittivity \u03b5 , for the twist angle of 3.6\u25e6. The yellowstar denotes the parameter region of the device D1.ture across different positions to the twist inhomogeneity within our device.Such a ferromagnetic correlation between local moments at \u03bd =\u22121 state is in-duced by the direct exchange interaction, which overcomes the anti-ferromagneticsuper-exchange interaction. The explanation is supported by theoretical simula-tions of spin-spin coupling J for the \u03bd = \u22121 state in a 3.6\u25e6 twisted MoSe2 ho-mobilayer (calculation details are included in Appendix A.3). The effective spin-spin interaction of \u0393-valley carriers is modeled using the Heisenberg model on ahoneycomb lattice. We incorporate spin-exchange processes contributing to spin-spin correlation by performing t\/U perturbation up to the fourth order, includingboth direct and super-exchange interactions (Fig.6.13(a)). We find that the ferro-magnetic spin-spin coupling J remains robust even at large dielectric permittivity(Fig.6.13(b)). For our device, where \u03b5 is around 30, J \u2248 \u22120.1 meV is consistent154with the energy scale of the measured CW temperature. Our results highlight theimportance of non-local spin-spin interactions, which are crucial for determiningthe magnetic ground state of correlated insulating states on a honeycomb lattice257.6.5 DiscussionIn conclusion, we have observed a range of correlated insulating states at both in-teger and fractional fillings in the twisted MoSe2 homobilayer, arising from the\u0393-valley moir\u00e9 flat bands. Our experiments confirm that, at a large U\/t ratio, theground state of a half-filled honeycomb lattice is a Mott insulator rather than asemi-metal. The charge gap opening at the Dirac point implies chiral or AB sublat-tice symmetry breaking246 in the twisted MoSe2 homobilayer, often accompaniedby the emergence of long-range magnetic order. Through MCD measurementsof \u0393-K exciton Fermi-polarons, we have revealed the magnetic interactions of \u0393-valley carriers at \u03bd =\u22121. However, the high doping density (\u223c 1013 cm\u22122) at the\u03bd = \u22122 state limits further exploration of its magnetic ground state. Future stud-ies should focus on samples with smaller twist angles and improved homogeneity,examined at lower temperatures. Additionally, exploring experimental methods totune the ratio between U\/t in such systems could offer new opportunities to studythe semi-metal to insulator phase transition. The removal of geometrical frustrationin the honeycomb lattice may also lead to the discovery of exotic magnetic orderssuch as quantum spin liquids258, distinct correlated insulating states223, and uncon-ventional superconductivity249. Similar phenomena are anticipated in other TMDfamilies with the highest valence band located at the \u0393 valley, such as small-angletwisted MoS2 and WS2 homo-bilayers, largely expanding the family of moir\u00e9 TMDhomo-bilayers.These results discussed above only scratch the surface of the correlated phe-nomena arising from the \u0393-valley moir\u00e9 flat bands. Many open questions requiremore experimental effort. First of all, the magnetic ground state of \u03bd = \u22122 statehas not been resolved in our experiment yet. We do not get any significant MCDsignal even at a strong magnetic field of 5T. For a 3.6\u25e6-twisted bilayer, the carrierdensity of \u03bd = \u22122 is around 9.2\u00d71012 cm\u22122, leading to a large broadening of the\u0393-K exciton Fermi-polaron. The broad linewidth reduces the sensitivity of MCD155measurement and makes the signal unresolvable.Another possible reason for the negligible MCD at \u03bd = \u22122 is that there couldbe a strong in-plane anti-ferromagnetic order along with the semi-metal to insulatortransition. The magnetic field we apply is not sufficient to tilt the spins from theireasy plane. The future experiment on the magnetic ground state study of \u03bd =\u22122 state requires a twisted bilayer with a smaller twist angle. A smaller angleleads to lower moir\u00e9 density, which can increase the MCD sensitivity at half-filling.Meanwhile, reducing the twist angle can also further enhance the U\/t ratio, givingrise to a weaker anti-ferromagnetic coupling which is proportional to t2U .Secondly, the semi-metal-to-insulator phase transition on a honeycomb latticeis different from the continuous metal-to-insulator transition on a triangular latticein a TMD hetero-bilayer. Whether the chiral symmetry is broken in the twisted ho-mobilayer at half-filling remains unclear. Revealing the nature of the semi-metal toinsulator phase transition will be a tough problem, which requires transport mea-surements with good contact between the TMD bilayer and metal electrodes. Fi-nally, understanding the nature of those generalized Wigner-crystal at fractionalfillings is also challenging. At certain fractional filling, there could be some spe-cial stripe phases as predicted by some numerical simulations242. To resolve thereal charge configurations for the fractional fillings, a new technique with highspatial resolution is highly demanding.The \u0393-valley moir\u00e9 flat bands provide a novel system to simulate the correlatedphysics. With the improvement of the twist homogeneity and optimized twist an-gle, more striking phenomena such as unconventional superconductivity249, super-solids259, and quantum spin liquid260 can be possibly achieved in this system.156Chapter 7ConclusionIn this thesis, we have revealed the emergence of spontaneous polarization and itsmicroscopic Berry phase origin in atomically thin, rhombohedral stacked transi-tion metal dichalcogenides. The depolarization field generated by this spontaneouspolarization induces a finite band offset (\u2206\u2248 60 meV) in both the conduction andvalence bands, giving rise to a type II band alignment at the K valley. Such aunique band structure of the 3R-MoS2 homo-bilayer is inferred from the excitoniceffects as observed through optical methods, which we detail in Chapter 3. Theintrinsic band gap difference between the two layers allows us to distinguish thelayer index of charges and excitons, providing a comprehensive understanding ofthe electronic properties of the 3R-MoS2 homo-bilayer.The picture of band shifting can be extended into the trilayer and multi-layersystem with coherent rhombohedral stacking. As the layer number N increases,the band gap continuously shrinks until the conduction band and valence bandedges converge at N \u2248 Eg\/(2\u2206)\u223c 16, where Eg is the semiconductor band gap of amonolayer. Once the band gap closes, further increases in the layer number lead toself-doping, preventing the conduction and valence bands from overlapping. Thesethick rhombohedral stacked transition metal dichalcogenides may offer a prototypefor studying the physics of excitonic insulator.In the two sections of Chapter 4, we explore the potential of constructing anultrafast photodetector using the 3R-MoS2 bilayer and few-layer structures. Weobserve a significant optical response from the Gr\/3R-MoS2\/Gr heterostructure,157although the intrinsic photovoltaic effect is somewhat masked by unavoidable ther-mal effects. To differentiate between the thermal and electronic effects, we de-veloped a non-degenerate pump-probe spectroscopic technique to measure the in-trinsic photocurrent speed in the Gr\/3R-MoS2\/Gr heterostructure. The measuredintrinsic photocurrent speed (2 ps) establishes an upper limit for the charge trans-fer time at the MoS2\u2013MoS2 and Gr\u2013MoS2 interfaces. However, the actual charge-transfer time within the 3R-MoS2 homo-bilayer remains unknown. A direct pump-probe measurement, similar to the approach used for the WS2\/MoS2 hetero-bilayerby Dr. Feng Wang\u2019s group139, does not apply to the 3R-MoS2 homo-bilayer. Thisis because the energy splitting between the A exciton resonances of the two layersis only 10 meV. Filtering such a narrow bandwidth from an ultrafast laser wouldcause significant pulse broadening, ultimately reducing measurement sensitivity.One possible experimental proposal is that the interlayer charge transfer time couldbe deduced from the many-body correlation between these two A excitons mea-sured by the Four-wave mixing spectroscopy, which is currently under develop-ment in our group.The thermal effects in Gr\/3R-MoS2\/Gr photodetectors play a significant rolebecause the conduction band edge of the 3R-MoS2 bilayer is very close to the Diracpoint of graphene\u2014approximately 100-200 meV, as extracted from our temperature-dependent measurements. This relatively small energy separation results in a lowtunneling resistance, which in turn leads to a reduced open-circuit voltage. Thesmall tunneling resistance causes the device very sensitive to the photo-thermaleffects, induced by the finite temperature gradient. To suppress the thermal effectand improve the device\u2019s performance, we need to increase the device resistance byeither changing to other materials, such as 3R-WS2 or 3R-WSe2 where the Diracpoint is deeply in the gap of the semiconductor or etching the device to reduce theoverlapping area. Growing these new crystals of different families of rhombohe-dral stacked transition metal dichalcogenides requires additional efforts from thematerial science community.Chapter 5 describes how we use optical techniques to investigate sliding fer-roelectricity in both the 3R-MoS2 bilayer and trilayer. Our findings indicate thatthe polarization direction can be switched as long as pre-existing domain walls arepresent in the natural flakes exfoliated from chemically synthesized crystals. These158domain walls are formed during the exfoliation process due to shear-strain-inducedslip avalanches191. An external electric field is applied to release the pre-existingdomain wall from its original pinning center, allowing it to propagate across thesample until it encounters stronger pinning centers. The pinning and depinning ofdomain walls are tracked through various spectroscopy techniques. In multi-layersystems, such as trilayer 3R-MoS2, the dynamics of polarization switching are de-termined by the competition between the pinning potential strengths at differentinterfaces, giving rise to diverse polarization transition pathways. The discoveryof unconventional interfacial ferroelectricity significantly expands the family offunctional 2D materials. A similar strategy can be applied to 2D ferromagneticmaterials, such as CrI3 bilayers, potentially leading to the discovery of novel typesof multi-ferroic materials.On the other hand, 2D materials with a twist, known as moir\u00e9 TMD bilayers,offer a fascinating playground for scientists to study new phenomena arising fromelectronic correlations. These materials have widely tunable optical and electronicproperties and may host many exotic phases of matter, such as various correlatedinsulators, quantum magnetism, and topological insulators. The final part of thisthesis delves into the strongly correlated physics of a honeycomb lattice using flatbands from the \u0393 valley. We have revealed a series of generalized Wigner crystalstates, as well as charge-transfer and Mott insulators appearing at both fractionaland integer fillings. However, whether superconductivity can exist near the Mottinsulator phase in such a bipartite lattice geometry remains an open question. Ourexperimental results have the potential to stimulate a range of correlated phenom-ena in a moir\u00e9 superlattice with a distinct symmetry.In general, scientific research on 2D materials involves two main directions.Since the discovery of the first monolayer graphene, significant efforts have beenmade to achieve real-world applications, such as field-effect transistors and opti-cal modulators, using these new materials due to their reduced dimensionality andextraordinary carrier mobility. However, we must acknowledge that there is still along way to go before 2D materials can be truly used in industry. The challengeslie in both fundamental science and material synthesis. In reality, it is really hard tofind a material system with all kinds of good properties for industrial applications.Therefore, flexibly changing the material\u2019s property by stacking order engineer-159ing is highly desirable in this field. This thesis presents the current progress inunderstanding the emergent phenomena in transition metal dichalcogenides whentheir stacking order changes from hexagonal to rhombohedral. Our work demon-strates the existence of an unconventional 2D ferroelectricity and makes positiveexploration in the practical applications, although highly repeatable and large-scaledevice fabrication method requires further study. The importance of the work in-cluded in the thesis lies in these experimental results, which indicate that stackingorder engineering is a powerful tool to unlock the full potential of 2D materials.160Bibliography[1] Yong Wang, Zhan Wang, Wang Yao, Gui-Bin Liu, and Hongyi Yu. Inter-layer coupling in commensurate and incommensurate bilayer structures oftransition-metal dichalcogenides. Physical Review B, 95(11):115429, 2017.\u2192 pages 1, 8, 9, 21, 22, 24, 44, 190[2] Chenhao Jin, Eric Yue Ma, Ouri Karni, Emma C Regan, Feng Wang, andTony F Heinz. 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Physical Review B, 47(3):1651, 1993. \u2192 page 193189Appendix AAppendixA.1 A tight-binding model for 3R-MoS2 bilayerThe TB model is constructed based on a third-nearest-neighbor (TNN) tight-bindingmodel for monolayer MoS2 261, and includes inter-layer coupling terms derivedfrom the C3v point group symmetries of 3R-MoS2 1.In the basis of {\u03a8(l)\u03c3\u03b1k} ({|\u03a8(1)\u2191,dz2 ,k\u27e9 , |\u03a8(1)\u2191,dxy,k\u27e9 , |\u03a8(1)\u2191,dx2\u2212y2 ,k\u27e9 , |\u03a8(1)\u2193,dz2 ,k\u27e9 , |\u03a8(1)\u2193,dxy,k\u27e9 ,|\u03a8(1)\u2193,dx2\u2212y2 ,k\u27e9, |\u03a8(2)\u2191,dz2 ,k\u27e9 , |\u03a8(2)\u2191,dxy,k\u27e9 , |\u03a8(2)\u2191,dx2\u2212y2 ,k\u27e9 , |\u03a8(2)\u2193,dz2 ,k\u27e9 , |\u03a8(2)\u2193,dxy,k\u27e9 , |\u03a8(2)\u2193,dx2\u2212y2 ,k\u27e9}),where \u03c3 =\u2191,\u2193 is the spin index, \u03b1 = dz2 ,dxy,dx2\u2212y2 is the orbital index, and l = t,bis the layer index, the 12-band TB Hamiltonian for bilayer 3R MoS2 is written as:HT B(k) =[H0(k)+\u03d52 \u00d7 I6\u00d76 TR(k)T \u2020R (k) H0(k)\u2212 \u03d52 \u00d7 I6\u00d76](A.1)Here, \u03d5 = \u2212ePd1\/(\u03b5MoS2\u03b50) accounts for the electrostatic potential differenceacross the two layers induced by nonzero polarization P and d1 is the interlayerdistance. H0(k) denotes the third-nearest-neighbor(TNN) tight-binding Hamilto-nian for monolayer MoS2 261, and TR(k) denotes the inter-layer coupling terms,where the subscript R refers to rhombohedral stacking. The form of H0(k) is ex-plicitly given by the Reference1 while the interlayer coupling term can be obtainedaccording to Reference14 by considering spin-preserving nearest-neighbor inter-layer tunneling processes: TR(k) = \u03c30\u2297 tR(k). The C3v point group symmetriesenforce tR(k) to take the form of:190tR(k) =\uf8ee\uf8ef\uf8f0 w0 f0(k) i\u03b30\u221a2( f+(k)\u2212 f\u2212(k)) \u03b30\u221a2( f+(k)+ f\u2212(k))i \u03b31\u221a2( f+(k)\u2212 f\u2212(k)) w1 f0(k) 0\u03b31\u221a2( f+(k)+ f\u2212(k)) 0 w1 f0(k)\uf8f9\uf8fa\uf8fb ,(A.2)where the special functions in tR(k) are defined as:\uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3f0(k) = (eik\u00b7\u03b4 1 + eik\u00b7\u03b4 2 + eik\u00b7\u03b4 3)\/3,f+(k) = (eik\u00b7\u03b4 1 +\u03c9+eik\u00b7\u03b4 2 +\u03c9\u2212eik\u00b7\u03b4 3)\/3,f\u2212(k) = (eik\u00b7\u03b4 1 +\u03c9\u2212eik\u00b7\u03b4 2 +\u03c9+eik\u00b7\u03b4 3)\/3.Here, \u03b4 1 =\u2212 a\u221a3 y\u02c6, and \u03b4 n =C(n\u22121)3z \u03b4 1, shown in Fig.1.4(a). w0, w1, \u03b30, and \u03b31 arecoupling parameters. The special functions have the following properties: f0(0) =1, f\u00b1(0) = 0,f0(\u00b1K) = 0, f+(+K) = 0, f\u2212(+K) = 1, f+(\u2212K) = 1, f\u2212(\u2212K) = 0.The basis Bloch wave functions have the general form of:\u03a8(l)\u03c3\u03b1k(r,z) =1\u221aN \u2211R(l)eik\u00b7R(l)\u03c6\u03c3\u03b1(r\u2212R(l),z\u2212R(l)z ), (A.3)where r = (x,y) labels the 2D spatial coordinate, z labels the coordinate alongvertical c-axis, N refers to the total number of sites in the 2D triangular lattice.\u03c6\u03c3\u03b1(r\u2212R(l),z\u2212R(l)z ) denotes a localized orbital with spin \u03c3 and orbital character\u03b1 located at site (R(l),R(l)z ), where R(l) is the 2D triangular lattice sites for layer l.R(l)z = (\u22121)(l\u22121) d12 labels the sites along the c-axis.Fig.A.1 shows a calculated band structure based on a realistic 12-band tight-binding (TB) model which allows us to take all k-points into account. Each bandshown in Fig.1.5 is a superposition of the 12 basis and its color represents thewavefunction weight from the top of the layer. We find that the wavefunction atK point is almost layer polarized while it is equally distributed in both layers at \u0393point.As we have explained in subsection 1.2.3, the origin of the electric polariza-tion in 3R-MoS2 bilayer arises from the asymmetric inter-layer coupling, which iscaptured by the TB model in equation (1.4). On the other hand, the nonzero spon-taneous polarization P obtained from the TB Hamiltonian will, in turn, establish anon-zero interlayer potential \u03d5 = \u2212ePd1\/(\u03b5MoS2\u03b50) as discussed before and mod-ify the on-site energy in the TB Hamiltonian as what we suggested in the equation191Energy (eV)0-2012K\u0413 \u0413M|\u03a8t|2Figure A.1: Calculated band structure along \u0393-K-M-\u0393 direction based onthe TB Hamiltonian in the equation (A.2). The colorbar indicates theweight of the total wave function from the top layer denoted as |\u03a8t |20-8P0 x 10-3\u0413 +K-KFigure A.2: Momentum-resolved Pk(k) in terms of polarization quantumP0 \u2261 e\/\u21260 \u2248 180\u00b5C \u00b7 cm\u22122. The major contribution is concentratednear the two K-points where the band offsets are most prominent.(A.4). Thus P and \u03d5 must be determined self-consistently via the equation:\u03d5 =\u2212 ed1\u03b5MoS2\u03b50P[HT B(\u03d5)], (A.4)The value of P can be calculated from the TB Hamiltonian HT B(\u03d5) using the192Wannier function approach44. The polarization P in terms of Wannier functions ofall occupied valence bands is given by262:P =e\u2126 \u2211n\u2208 f illed\u222bdz\u222bd2rz|Wn(r,z)|2 (A.5)where \u2126 is the volume of the 3D unit cell, and the Wannier function of band n ofHT B(g,\u03d5) is defined as:Wn(r,z) =1\u221aN\u2211keik\u00b7runk(r,z), (A.6)where unk(r,z) denotes the periodic part of the Bloch state at k for band n, whichis explicitly given by:unk(r,z) =\u2211l\u03c3\u03b1c(l)n,\u03c3\u03b1(k)u(l)\u03c3\u03b1k(r,z), (A.7)u(l)\u03c3\u03b1k(r,z) =1\u221aN \u2211R(l)eik\u00b7(R(l)\u2212r)\u03c6\u03c3\u03b1(r\u2212R(l),z\u2212R(l)z ).Here, the coefficients c(l)n,\u03c3\u03b1(k) for band n are obtained by exact numerical diago-nalization of HT B(k). Substituting equation (A.7) into equation (A.5),and making use of the identity\u27e8\u03c6\u03c3 \u2032\u03b1 \u2032(R(l\u2032),R(l\u2032)z )|z|\u03c6\u03c3\u03b1(R(l),R(l)z )\u27e9=R(l)z \u03b4ll\u2032\u03b4\u03c3\u03c3 \u2032\u03b4\u03b1\u03b1 \u2032derived from the symmetry properties of the d-orbitals, we find equation (1.8) canbe simplified as:P =1N \u2211k\u2208BZPk(k) (A.8)with Pk(k) =e\u2126 \u2211n\u2208 f illed \u2211l\u03c3\u03b1|c(l)n,\u03c3\u03b1(k)|2R(l)z= P0{12\u2211\u03c3\u03b1|c(1)n,\u03c3\u03b1(k)|2\u2212\u2211\u03c3\u03b1|c(2)n,\u03c3\u03b1(k)|2}. (A.9)Here, P0 \u2261 e\/\u21260 \u2248 180\u00b5C \u00b7 cm\u22122 is the polarization quantum for 3R-MoS2 with\u21260 =\u221a32 a2 denoting the area of the 2D unit cell (note: \u21260d1 = \u2126). Pk is themomentum-dependent electric polarization at each wave vector k. The polarizationin the real space is an average of all Pk in the whole Brillouin Zone. Fig.A.2shows the calculated momentum resolved Pk in terms of polarization quantum P0.It shows that the major contribution of the polarization is from\u00b1K points while thepolarization arising from \u0393 point is close to zero. This is consistent with the band193structure calculation in Fig.A.1.The sum of coefficients \u2211\u03c3\u03b1 |c(l)n,\u03c3\u03b1(k)|2 in equation (A.9) essentially amountsto the weight |\u03a8(l)n (k)|2 from layer l in the eigenstate wave function at momentumk and band n as shown in Fig.A.1. The equation (A.9) allows us to obtain the valueof total P from HT B(\u03d5) by solving the self-consistent equation (A.4). We obtaina self-consistent value of P = \u22120.6 \u00b5C\/cm2 for the AB-stacking configuration(P = +0.6 \u00b5C\/cm2 for BA-stacking configuration), which is consistent with theLDA calculation in the previous report78.A.2 Mean field model for \u03bd =\u22122 on a half-filledhoneycomb latticeIn this section, we calculate the ground state at half-filling using Hartree-Fock(HF) approximation. This is essentially a mean-field calculation to capture thesemi-metal to insulator phase transition at a large U\/t limit. We start with the tight-binding expression for a Hubbard model on a honeycomb lattice (Equation A.10).c\u2020ia\/b\u03c3 and c ja\/b\u03c3 are the creation and annihilation operators for holes at a or b sub-lattice, where \u03c3 labels the spin index. The index i = (x,y) denotes the coordinatesof a unit cell in the two-dimensional honeycomb lattice. < i, j > represents thenearest hopping between a and b sublattice. nia\/b\u03c3 is the particle number operatorat two sub-lattices. For holes, the hopping integral is positive.H =\u2212 \u2211<i, j>,\u03c3t(c\u2020ia\u03c3c jb\u03c3 +h.c.)+U\u2211i(nia\u2191nia\u2193+nib\u2191nib\u2193) (A.10)We first assume a Hartree-Fock Hamiltonian using momentum representation inEquation A.11.HHF =\u2211k\u03c3Ek\u03c3a\u2020k\u03c3ak\u03c3 (A.11)The new creation operator is a linear combination of c\u2020ia\u03c3 and c\u2020ib\u03c3 .a\u2020k\u03c3 = \u2211i=(x,y)(\u00b5k\u03c3 (i)c\u2020ia\u03c3 +\u03bdk\u03c3 (i)c\u2020ib\u03c3 ) (A.12)ak\u03c3 = \u2211i=(x,y)(\u00b5\u2217k\u03c3 (i)cia\u03c3 +\u03bd\u2217k\u03c3 (i)cib\u03c3 ) (A.13)Next, we use the method of equation of motion (EOM) to find the solution of theHartree-Fock state (\u00b5k\u03c3 and\u03bdk\u03c3 ). We have independent equations for \u00b5k\u03c3 (i) and\u03bdk\u03c3 (i), respectively. Here the notation {} and [] denote the anti-commutator and1940 1 2 3 4 50.00.10.20.30.40.5baa bxy(x,y)(x-1,y) (x,y+1)x1y1\u03b41\u03b42\u03b43Figure A.3: a. A schematic of a honeycomb lattice. b. Calculated staggeredmagnetization ms versus the ratio of U\/t.commutator operation. The state |HF\u27e9 denotes the Hartree-Fock ground state.Ek\u03c3\u00b5k\u03c3 (i) = \u2211j=(x\u2032,y\u2032)(\u00b5k\u03c3 ( j)\u27e8HF |{cia\u03c3 , [H,c\u2020ja\u03c3 ]}|HF\u27e9+\u03bdk\u03c3 ( j)\u27e8HF |{cia\u03c3 , [H,c\u2020jb\u03c3 ]}|HF\u27e9) (A.14)Ek\u03c3\u03bdk\u03c3 (i) = \u2211j=(x\u2032,y\u2032)(\u00b5k\u03c3 ( j)\u27e8HF |{cib\u03c3 , [H,c\u2020ja\u03c3 ]}|HF\u27e9+\u03bdk\u03c3 ( j)\u27e8HF |{cib\u03c3 , [H,c\u2020jb\u03c3 ]}|HF\u27e9) (A.15)To solve the EOM in Equation (A.14) and (A.15), we first calculate four commuta-tors in the following equations. The coordinates (x\u2032,y\u2032),(x\u2032\u00b11,y\u2032), and (x\u2032,y\u2032\u00b11)are the three unit cells where nearest hopping between a and b sublattice happens.[ \u2211(x,y)\u03c3c\u2020(x,y)a\u03c3c(x,y)b\u03c3 + c\u2020(x,y)a\u03c3c(x\u22121,y)b\u03c3 + c\u2020(x,y\u22121)a\u03c3c(x,y)b\u03c3 +h.c.,c\u2020j=(x\u2032,y\u2032)a\u03c3 ]= c\u2020j=(x\u2032,y\u2032)b\u03c3 + c\u2020j=(x\u2032\u22121,y\u2032)b\u03c3 + c\u2020j=(x\u2032,y\u2032\u22121)b\u03c3 (A.16)195[ \u2211(x,y)\u03c3c\u2020(x,y)a\u03c3c(x,y)b\u03c3 + c\u2020(x,y)a\u03c3c(x\u22121,y)b\u03c3 + c\u2020(x,y\u22121)a\u03c3c(x,y)b\u03c3 +h.c.,c\u2020j=(x\u2032,y\u2032)b\u03c3 ]= c\u2020j=(x\u2032,y\u2032)a\u03c3 + c\u2020j=(x\u2032+1,y\u2032)a\u03c3 + c\u2020j=(x\u2032,y\u2032+1)a\u03c3 (A.17)[\u2211inia\u2191nia\u2193+nib\u2191nib\u2193,c\u2020ja\u03c3 ] = c\u2020ja\u03c3n ja\u2212\u03c3 (A.18)[\u2211inia\u2191nia\u2193+nib\u2191nib\u2193,c\u2020jb\u03c3 ] = c\u2020jb\u03c3n jb\u2212\u03c3 (A.19)Using above algebra, the equation of motion in Equation (A.14) and Equation(A.15) can be simplified:Ek\u03c3\u00b5k\u03c3 (x,y) =U \u2211x\u2032,y\u2032\u00b5k\u03c3 (x\u2032,y\u2032)\u27e8HF |{c(x,y)a\u03c3 ,c\u2020(x\u2032,y\u2032)a\u03c3n(x\u2032,y\u2032)a\u2212\u03c3}|HF\u27e9\u2212 t\u2211x\u2032,y\u2032\u03bdk\u03c3 (x\u2032,y\u2032)\u27e8HF |{c(x,y)a\u03c3 ,c\u2020(x\u2032,y\u2032)a\u03c3 + c\u2020(x\u2032\u22121,y\u2032)a\u03c3 + c\u2020(x\u2032,y\u2032\u22121)a\u03c3}|HF\u27e9=\u2212t(\u03bdk\u03c3 (x,y)+\u03bdk\u03c3 (x\u22121,y)+\u03bdk\u03c3 (x,y\u22121))+U < n(x,y)a\u2212\u03c3 >HF \u00b5k\u03c3 (x,y)(A.20)Ek\u03c3\u03bdk\u03c3 (x,y) =U \u2211x\u2032,y\u2032\u03bdk\u03c3 (x\u2032,y\u2032)\u27e8HF |{c(x,y)b\u03c3 ,c\u2020(x\u2032,y\u2032)b\u03c3n(x\u2032,y\u2032)b\u2212\u03c3}|HF\u27e9\u2212 t\u2211x\u2032,y\u2032\u00b5k\u03c3 (x\u2032,y\u2032)\u27e8HF |{c(x,y)b\u03c3 ,c\u2020(x\u2032,y\u2032)b\u03c3 + c\u2020(x\u2032+1,y\u2032)b\u03c3 + c\u2020(x\u2032,y\u2032+1)b\u03c3}|HF\u27e9=\u2212t(\u00b5k\u03c3 (x,y)+\u00b5k\u03c3 (x+1,y)+\u00b5k\u03c3 (x,y+1))+U < n(x,y)b\u2212\u03c3 >HF \u03bdk\u03c3 (x,y)(A.21)< n(x,y)a\u2212\u03c3 >HF and < n(x,y)b\u2212\u03c3 >HF denote the expected occupation value of spin\u2212\u03c3 at HF state. For the half-filled honeycomb lattice, the total expectation of196occupation at each lattice should be 1. ms is defined as the staggered magnetization.< n(x,y)a\u2191 >HF +< n(x,y)a\u2193 >HF= 112(< n(x,y)a\u2191 >HF \u2212< n(x,y)a\u2193 >HF) = ms< n(x,y)b\u2191 >HF +< n(x,y)b\u2193 >HF= 112(< n(x,y)b\u2191 >HF \u2212< n(x,y)b\u2193 >HF) =\u2212ms(A.22)We assume the spin index in (A.20) and (A.21) is \u2191 (The conclusion is the samefor \u2193).Ek\u2191\u00b5k\u2191(x,y) =\u2212t(\u03bdk\u2191(x,y)+\u03bdk\u2191(x\u22121,y)+\u03bdk\u2191(x,y\u22121))+U(12 \u2212ms)\u00b5k\u2191(x,y)(A.23)Ek\u2191\u03bdk\u2191(x,y) =\u2212t(\u00b5k\u2191(x,y)+\u00b5k\u2191(x+1,y)+\u00b5k\u2191(x,y+1))+U(12 +ms)\u03bdk\u2191(x,y)(A.24)We choose the Bloch function as the trial function.\u00b5k\u03c3 = \u03b1k\u03c3ei\u20d7kR\u20d7xya\u221aN\u03bdk\u03c3 = \u03b2k\u03c3ei\u20d7kR\u20d7xyb\u221aN(A.25)Substituting Equation (A.25) into Equation (A.23) and (A.24), we can get:Ek\u2191\u03b1k\u2191ei\u20d7kR\u20d7xya\u221aN=\u2212t(\u03b2k\u2191 ei\u20d7kR\u20d7xyb\u221aN+\u03b2k\u2191ei\u20d7kR\u20d7x\u22121yb\u221aN+\u03b2k\u2191ei\u20d7kR\u20d7xy\u22121b\u221aN)+U(12\u2212ms)\u03b1k\u2191 ei\u20d7kR\u20d7xya\u221aNEk\u2191\u03b2k\u2191ei\u20d7kR\u20d7xyb\u221aN=\u2212t(\u03b1k\u2191 ei\u20d7kR\u20d7xya\u221aN+\u03b1k\u2191ei\u20d7kR\u20d7x+1ya\u221aN+\u03b1k\u2191ei\u20d7kR\u20d7xy+1a\u221aN)+U(12+ms)\u03b2k\u2191ei\u20d7kR\u20d7xyb\u221aN(A.26)197The three nearest vectors \u03b4\u20d71-\u20d7\u03b43 from b to a are shown in Fig.A.3(a).\u03b4\u20d71 = R\u20d7xyb\u2212 R\u20d7xya\u03b4\u20d72 = R\u20d7x\u22121yb\u2212 R\u20d7xya =\u2212R\u20d7x+1ya+ R\u20d7xyb\u03b4\u20d73 = R\u20d7xy\u22121b\u2212 R\u20d7xya =\u2212R\u20d7xy+1a+ R\u20d7xyb (A.27)In the end, we are able to get the HF solutions from the following equation.(Ek\u2191\u2212U2 )[\u03b1k\u2191\u03b2k\u2191]=[\u2212Ums \u03b5k\u03b5\u2217k Ums][\u03b1k\u2191\u03b2k\u2191](A.28)Here \u03b5k = \u2212t\u2211 j=1,2,3 ei\u20d7k\u03b4\u20d7 j . This term is the same as the tight-binding model ofmonolayer graphene.Ek\u03c3 =U2\u00b1\u221a\u03b5k\u03b5\u2217k +(Ums)2=U2\u00b1\u221at2(1+4cos(32kx1aM)cos(\u221a32ky1aM))+4cos2(\u221a32ky1aM))+(Ums)2(A.29)Without the correlation effect, the dispersion is the same as monolayer graphene.The Dirac points are at the corner of the mini-BZ. If the staggered magnetizationms is non-zero, then there will be a charge-gap opening at the Dirac point, whichturns the system into an anti-ferromagnetic insulator.\u2206\u22122 = 2Ums (A.30)The value of ms is determined by the following self-consistent equation:ms =12(< n(x,y)a\u2191 >HF \u2212< n(x,y)a\u2193 >HF)=12\u221a32 a2M(2\u03c0)2\u222b \u222bmini\u2212BZdkx1dky1Ums\u221a\u03b5k\u03b5\u2217k +(Ums)2(A.31)ms can be numerically solved for a given ratio of U\/t. ms = 0 is one possiblesolution for Equation (A.31), which means the ground state of a half-filled honey-comb lattice is a semi-metal, the same as the monolayer graphene. At the limit ofU\/t \u2192 \u221e, the upper bond of ms is 1\/2. Since the ms can not exceed 1\/2, the mea-sured charge gap is smaller than the onsite Coulomb interaction U. The numericalresults are shown in Fig.A.3(b). The critical value of U\/t for a semi-metal-to-198insulator transition is around 2.2.A.3 Theoretical model of the twisted MoSe2 bilayerA.3.1 Continuum model for MoSe2 homobilayerIn this section, we will discuss the procedure to solve the continuum model in de-tail. We start from a continuum model for twisted MoSe2 homo-bilayer, followingthe Reference63, as shown below:H\u03c4(r) =\u03a8\u2020\u03c4(r)(\u2212\u210f2k\u02c622m\u2217+\u2206(r))\u03a8\u03c4(r) (A.32)where \u03a8\u2020\u03c4(r) is the field operator for the spin \u03c4 . The \u2206(r) =\u2211s\u22116j=1Vsei(gsj\u00b7r+\u03c6s) isthe moir\u00e9 potential as shown in Figure.1b. We fixed the parameters of the contin-uum model corresponding to the twisted MoSe2 bilayer system, i.e. {V1,V2,V3}={36.8,8.4,10.2} meV, \u03c6s = \u03c0 , m\u2217 = 1.17me, and monolayer lattice constant a0 =0.33 nm.We use the plane wave expansion \u03a8\u03c4(r) = 1\u221aA \u2211k f\u2020k\u03c4eik\u00b7r, with inverse trans-formation f \u2020k\u03c4 =1\u221aA\u222b\u03a8\u03c4(r)e\u2212ik\u00b7rd2r, where f \u2020k\u03c4 is the hole creation operator withmomentum k and spin \u03c4 . A is the area of the sample. The non-interacting Hamil-tonian i.e. H\u03c4 =\u222bH\u03c4(r) d2r, after performing the plane wave expansion, can bewritten as H\u03c4 = \u2211k\u2208mbz H\u03c4(k), whereH\u03c4(k) =\u2212\u2211G[\u03b5(k+G) f \u2020k+G\u03c4 fk+G\u03c4 + \u2211j=1,3,5sVs(ei\u03c6s f\u2020k+g j+G\u03c4 fk+G\u03c4 +h.c.)](A.33)where, \u03b5\u03c4(k) = \u2212\u210f2k22m\u2217 . The sum on G spans over the grid of moir\u00e9 reciprocallattice vectors around the momentum k which lies in the moir\u00e9 Brillouin zone(mini-Brillouin Zone (MBZ)). We noticed finite grids of G of size L\u02dc\u00d7 L\u02dc quicklygive convergence in the low energy eigenspectrum of H\u03c4(k), we use 18\u00d7 18 gridof G . The single particle Hamiltonian is diagonal in moir\u00e9 Bloch states c\u2020kn\u03c4 =\u2211G unG\u03c4(k) f\u2020k+G\u03c4 i.e. H\u03c4(k) = \u2211n \u03b5\u02dcn\u03c4(k)c\u2020kn\u03c4ckn\u03c4 , where n is moir\u00e9 band index.The total number of moir\u00e9 bands we get in the continuum model is L\u02dc2, out ofwhich mainly the few lowest energy moir\u00e9 bands (in hole representation, as usedin this work) are generally relevant.199A.3.2 Estimation on the dielectric permittivity of device D1 ofChapter 6In this section, we will estimate the effective dielectric permittivity \u03b5 of the de-vice D1 examined in the main text of Chapter 6, which characterizes the three-dimensional screening including the dielectric hBN and metallic graphite gate. Themoir\u00e9 potential VM at the XM and MX sites is approximately 200 meV, as deter-mined by simulation results in Figure.1b. The moir\u00e9 lattice constant of device D1is around 5 nm. Then we can estimate the size of the Wannier orbitals (aw) underharmonic approximation217.aw = 2(\u03c0)\u221212\u221aaM(\u210f2m\u2217VM)\u2248 2 nm (A.34)The onsite Coulomb repulsion U = 14\u03c0\u03b5\u03b50e2aw\u2248 25 meV according to the temperaturedependence measurement in Figure.3c. Thus, the effective dielectric permittivity\u03b5 can be estimated to be around 30 based on the U . This parameter is crucial forbenchmarking our device in the large parameter space of the following calculations.A.3.3 Unrestricted Hartree-Fock CalculationWe start with interaction Hamiltonian written for the holes, as shown belowHint =12A \u2211k1k2q\u03c4\u03c4 \u2032V (q) f \u2020(k1\u2212q)\u03c4 f\u2020(k2+q)\u03c4\u2032 fk2\u03c4 \u2032 fk1\u03c4 (A.35)where, V (q) = 2\u03c0e2tanh(|q|d)\u03b5|q| , \u03b5 is the dielectric constant, and d \u2248 10 nm is distancebetween the gate and the sample. The area of the sample is A = (\u221a3\/2)l1l2a2M,with l1(2) moir\u00e9 unit cells along the Bravais lattice vector a1(2). In equation (A.35),the sum over all possible momentum\u2019s namely \u2211k can be replaced by \u2211k+G wherenow k \u2208 mbz (mini-Brillouin zone). Moreover, using fk+Gl\u03c4 = \u2211n ckn\u03c4unGl\u03c4(k), wewrite the interaction term in the moir\u00e9 band basis as followingHint =12A \u2211k1k2q\u03c4\u03c4 \u2032n1n2n3n4V\u02dc \u03c4\u03c4\u2032n1n2n3n4(k1,k2,q)c\u2020(k1\u2212q)n1\u03c4c\u2020(k2+q)n2\u03c4 \u2032ck2n3\u03c4 \u2032ck1n4\u03c4 (A.36)200where,V\u02dc \u03c4\u03c4\u2032n1n2n3n4(k1,k2,q) =\u2211GV (q+G)\u03bbn1n2\u03c4(k1\u2212q,k1+G)\u03bb \u2217n3n2\u03c4 \u2032(k2,k2+q+G)(A.37)and the form factors \u03bbn1n2\u03c4(k1,k2) = \u2211G(un1G\u03c4(k1))\u2217un2G\u03c4(k2). In equation (A.35),the total number of terms is 4(l1l2)3L\u02dc6, whereas if we project the equation (A.36)to only few Nb relevant moir\u00e9 bands then the total number of terms in Equation(A.36) is 4N4b (l1l2)3. One may look at the total single particle Fermionic states re-quired for both equations, for equation (A.35) 2l1l2L\u02dc2 are required whereas for theequation (A.36) 2Nbl1l2 states are required, which suggests a huge gain if we usethe interaction Hamiltonian projected to few moir\u00e9 bands of interest. In our calcu-lations, we project the interactions only to the first two moir\u00e9 bands by restrictingthe sum over band indices in the equation (A.36).Then we perform the mean field decomposition of equation (A.36) to get thefollowing Hartree and Fock termsHHartree =12A \u2211k1k2q\u03c4\u03c4 \u2032n1n2n3n4(V\u02dc \u03c4\u03c4\u2032n1n2n3n4(k1,k2,q)+V\u02dc\u03c4 \u2032\u03c4n2n1n4n3(k2,k1,\u2212q))\u27e8c\u2020(k1\u2212q)n1\u03c4ck1n4\u03c4\u27e9c\u2020(k2+q)n2\u03c4 \u2032ck2n3\u03c4 \u2032 (A.38)HFock =\u2212 12A \u2211k1k2q\u03c4\u03c4 \u2032n1n2n3n4(V\u02dc \u03c4\u03c4\u2032n1n2n3n4(k1,k2,q)+V\u02dc\u03c4 \u2032\u03c4n2n1n4n3(k2,k1,\u2212q))\u27e8c\u2020(k1\u2212q)n1\u03c4ck2n3\u03c4 \u2032\u27e9c\u2020(k2+q)n2\u03c4 \u2032ck1n4\u03c4 (A.39)The total mean field Hamiltonian can be written as follows,HHF =HHartree+HFock\u2212 12(\u27e8HHartree\u27e9+ \u27e8HFock\u27e9)\u2212 12A \u2211k2q\u03c4n1n2n3V\u02dc \u03c4\u03c4n1n2n3n2(k2+q,k2,q))c\u2020k2n1\u03c4ck2n3\u03c4+12A \u2211k1k2\u03c4\u03c4 \u2032n1n2n4V\u02dc \u03c4\u03c4\u2032n1n2n2n4(k1,k2,0))c\u2020k1n1\u03c4ck1n4\u03c4 (A.40)The last two quadratic Fermionic terms come from the anti-commutation of Fermionswhile reordering Fermionic operators for the Hartree and Fock terms. The HHF is201solved for the given filling of holes maintaining self-consistency in the order pa-rameters namely the single-particle density matrix elements \u27e8c\u2020k1n1\u03c4ck2n2\u03c4 \u2032\u27e9. Wewill be solving the above mean field Hamiltonian using unrestricted Hartree-Focknamely starting from random initial conditions for the order parameters and thengaining the self-consistency for the order parameters in an iterative way.After gaining convergence, we also calculate the charge gap \u2206c to deduce ifthe system is metal or insulator, as shown in the phase diagram of Fig.6.3(d). Inaddition, we calculate the real space resolved density of doped holes \u27e8n(r)\u27e9 =\u2211\u03c4\u27e8\u03a8\u03c4(r)\u03a8\u2020\u03c4(r)\u27e9 to visualize the Wigner Crystals using the equation (26). Thetheoretical proposed charge configuration of the generalized Wigner crystals us-ing above algorithm and their critical temperatures observed in experiments arepresented in Fig.6.11.\u27e8\u03a8\u03c4(r)\u03a8\u2020\u03c4 \u2032(r)\u27e9=1A \u2211G1G2k1k2n1n2eir\u00b7(k1\u2212k2)eir\u00b7(G1\u2212G2)(un1G1\u03c4(k1))\u2217un2G2\u03c4(k2)\u27e8c\u2020k1n1\u03c4ck2n2\u03c4 \u2032\u27e9 (A.41)A.3.4 Spin Model for \u03bd =\u22121 stateIn this section, we perform a theoretical simulation on the spin-spin coupling Jfor the \u03bd = \u22121 state. We derived the effective low-energy spin 1\/2 model H =J\u2211\u27e8i j\u27e9Si \u00b7S j by performing the t\/U perturbation theory up to the fourth order. Themost dominant exchange processes contributing to the spin correlation J are shownin Fig.6.13. They are all included in the following equation,J \u2248 JFM2 +4t22U+t21 t2U21+8t21 t2UU1+t41UU21+2t41(U +2U1)U21(A.42)We also include the second nearest neighbor (in terms of underlying Honey-comb lattice) direct ferromagnetic exchange JFM2 as it is in direct competition to an-tiferromagnetic super-exchange terms coming from second and fourth-order terms.t1 and t2 are the nearest and second nearest hopping energy on a honeycomb lat-tice while U and U1 denote the on-site and nearest neighbor Coulomb repulsion.It should be noted that the negative sign of t2 (see Fig.A4(c)) also favors ferro-magnetism via the third-order terms present in the equation A.42. In deriving theabove exchange term using perturbation theory, for simplicity, we have only useddensity-density repulsion terms up to the nearest neighbor and also ignored themuch weaker longer-range hopping. We believe including the long 1\/r tail of thedensity-density repulsion and other weaker interaction terms should not change the202\u03b8t 1t 2t 32.0 3.0 4.0 5.00.04.08.012.0t (meV)\u03b82.0 3.0 4.0 5.01101001000energy (meV)\u03b5U\u03b5U1\u03b5J1FM\u03b5J2FMc d-0.03 +0.03-0.03 +0.03a bFigure A.4: a and b. Wannier orbitals attained using the first two moir\u00e9 bandsin Fig.6.4(a) for A and B sublattices. c. various interaction param-eters as a function of twist angle, namely onsite Coulomb repulsion\u03b5U , nearest neighbor density-density repulsion \u03b5U1, nearest neighbor(JFM1 ) and next nearest neighbor direct (JFM2 ) ferromagnetic exchanges.d. Hopping parameters, (nearest t1, second nearest t2 and third nearestneighbor t3) as a function of the twist anglequalitative picture attained by this analysis.To estimate J, we calculated the hopping and interaction parameters using theWannier functions \u03a8A(B)(r) (shown in Fig.A4(a) and Fig.A4(b))240. The depen-dence of these real-space model parameters on twist angle is shown in Fig.A4(c)and Fig.A4(d). Fig.A4(b) depicts the evolution of the spin exchange as a functionof \u03b5 , for the twist angle of 3.6\u25e6. The divergence for very small \u03b5 is expected be-cause all Coulomb interaction terms JFM1 ,U , and U1 diverges as \u03b5\u22121. We foundfor the twist angle \u03b8 = 3.6\u25e6 the system favors ferromagnetism up to large \u03b5 ofnearly 42 while the \u03b5 of device D1 is around 30 (Section.A.3.2). The above studyexplains our experimental finding of the weak ferromagnetic spin-spin correlationfor \u03bd =\u22121 and suggests that the direct exchange and the second nearest neighborhopping overcomes the anti-ferromagnetic super-exchange interaction.203Appendix BList of Publications\u2022 Dongyang Yang\u2020, Jingda Wu\u2020, Benjamin T. Zhou, Jing Liang, Toshiya Ideue,Teri Siu, Kashif Masud Awan, Kenji Watanabe, Takashi Taniguchi, YoshihiroIwasa, Marcel Franz, Ziliang Ye. \u201cSpontaneous-polarization-induced photo-voltaic effect in rhombohedrally stacked MoS2\". Nature Photonics, 16(6), 469-474 (2022).\u2022 Dongyang Yang\u2020, Jing Liang\u2020, Jingda Wu\u2020, Yunhuan Xiao, Jerry I. Dadap,Kenji Watanabe, Takashi Taniguchi, Ziliang Ye. \u201cNon-volatile electrical polar-ization switching via domain wall release in 3R-MoS2 bilayer\". Nature Commu-nications, 15(1), 1389 (2024).\u2022 Dongyang Yang\u2020, Jing Liang\u2020, Nitin Kaushal, Haodong Hu, Jerry I. Dadap,Kenji Watanabe, Takashi Taniguchi, Marcel Franz, Ziliang Ye. \u201cCorrelated insu-lating states arising from the \u0393 valley flat bands in moir\u00e9 MoSe2 homobilayer\".in submission (2024)\u2022 Jing Liang\u2020, Dongyang Yang\u2020, Jingda Wu, Yunhuan Xiao, Kenji Watanabe,Takashi Taniguchi, Jerry I. Dadap, Ziliang Ye. \u201cResolving polarization switchingpathways of sliding ferroelectricity in trilayer 3R-MoS2\". under review (2024)\u2022 Jing Liang\u2020, Dongyang Yang\u2020, Jingda Wu\u2020, Jerry I Dadap, Kenji Watanabe,Takashi Taniguchi, Ziliang Ye. \u201cOptically Probing the asymmetric interlayercoupling in Rhombohedral-Stacked MoS2 bilayer.\" Physical Review X, 12(4),041005 (2022)\u2022 Jingda Wu\u2020, Dongyang Yang\u2020, Jing Liang\u2020, Max Werner, Evgeny Ostroumov,Yunhuan Xiao, Kenji Watanabe, Takashi Taniguchi, Jerry I Dadap, David Jones,Ziliang Ye. \u201cUltrafast response of spontaneous photovoltaic effect in 3R-MoS2\u2013based heterostructures.\" Science Advances, 8(50), eade3759 (2022)204\u2022 Jing Liang, Dongyang Yang, Yunhuan Xiao, Sean Chen, Jerry I. Dadap, JoergRottler, and Ziliang Ye. \u201cShear Strain-Induced Two-Dimensional Slip Avalanchesin Rhombohedral MoS2.\" Nano Letters (2023)\u2022 Takatoshi Akamatsu, Toshiya Ideue, Ling Zhou, Yu Dong, Sota Kitamura, MaoYoshii, Dongyang Yang, Masaru Onga, Yuji Nakagawa, Kenji Watanabe, TakashiTaniguchi, Joseph Laurienzo, Junwei Huang, Ziliang Ye, Takahiro Morimoto,Hongtao Yuan, Yoshihiro Iwasa. \u201cA van der Waals interface that creates in-planepolarization and a spontaneous photovoltaic effect.\" Science, 372, 68 (2021)\u2022 Yunhuan Xiao, Jingda Wu, Jerry I Dadap, Kashif Masud Awan, DongyangYang, Jing Liang, Kenji Watanabe, Takashi Taniguchi, Marta Zonno, MartinBluschke, Hiroshi Eisaki, Martin Greven, Andrea Damascelli, Ziliang Ye. \u201cOp-tically Probing Unconventional Superconductivity in Atomically ThinBi2Sr2Ca0.92Y0.08Cu2O8+\u03b4 \" Nano Letters (2024)\u2022 Dongyang Yang, Ziliang Ye. \u201cQuantum simulator comes in pair.\" Nature Nan-otechnology, 17(9), 902-903 (2022)\u2022 Jing Liang\u2020, Yuan Xie\u2020, Dongyang Yang\u2020, Shangyi Guo, Kenji Watanabe, TakashiTaniguchi, David Jones, Ziliang Ye. \u201cFerroelectric Switching of 2D Excitons foran Ultrafast Nonvolatile Optical Memory\" under review, (2024)205","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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