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Study of two dimensional materials graphene and FeSe Yang, Rui 2018

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Study of two dimensional materialsgraphene and FeSebyRui YangB.Sc., Zhejiang University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2018c© Rui Yang 2018AbstractThe properties of several two-dimensional (2d) materials are studied. Themain content is the study of magnetic impurities in graphene through phasecoherent transport phenomena, including weak localization (WL) and uni-versal conductance fluctuations (UCF). Magnetic impurities manifest them-selves in the in-plane magnetic field (B‖) and temperature dependence ofthe dephasing (phase breaking) rate. Our experiments unambiguously revealthe existence of magnetic impurities through the B‖ dependence of WL andUCF. The properties of the magnetic impurities are further studied throughthe dephasing rate as a function of magnetic field and temperature. TheWL dephasing rate as a function of B‖ shows a non-monotonic behaviour,which is rooted in the existence of magnetic impurities with a Lande´ g factordifferent from that of the free electron. The collapse of the dephasing rate asa function of temperature is a sign of the quenching of magnetic impurities,which could come from Kondo coupling between magnetic impurities andconduction electrons or Ruderman-Kittel-Kasuya-Yosida (RKKY) couplingbetween magnetic impurities. The implications of our graphene experimentsare two-fold: on the one hand, they provide new knowledge about the in-terplay among magnetic moments and electrons in graphene; on the otherhand, they also established an effective tool to reveal the existence of mag-netic impurities in 2d systems.In addition to graphene, we also studied the exfoliation and stabil-ity of thin sheets of FeSe — a member of the Fe-based superconductorsfamily. Our Raman Spectroscopy, Atomic Force Microscopy, Optical Mi-croscopy and Time-of-Flight-Secondary-Ion-Mass-Spectroscopy experimentsshow that FeSe nanosheets decay in air, precipitation of Se and oxidationlikely occurring during the decay process. Our transport measurements showiiAbstractthat FeSe nanosheets exposed briefly to air can still retain superconductivity.iiiLay SummaryIn this thesis, properties of several 2d materials were studied. The main con-tent is about graphene, which has potential microelectronics applications.One potential application is spintronics, which employs the spin to carryinformation. We revealed the existence of magnetic impurities in differentkinds of graphene, which helps us understand the limiting factors for spin-tronics. We also revealed signs of quantum coupling between the magneticimpurities and the electrons, which has its own significance for the under-standing of the rich physics related to the magnetic impurities.The layered superconductor FeSe was also studied. 2d superconductorsserve as a good platform for studying phenomena such as quantum phasetransitions, thus improving our understanding of superconductivity. Oneway to realize 2d superconductors is through the exfoliation of layered su-perconductor crystals. We studied the stability and superconductivity ofexfoliated FeSe flakes. This lays the foundation for high quality thin filmFeSe devices.ivPrefaceThis thesis describes work associated with the following publication:• Lundeberg, M. B., Yang, R., Renard, J., Folk, J. A., Defect-mediatedspin relaxation and dephasing in graphene. Phys. Rev. Lett. 110,156601 (2013). [Chapter 5]However, most of the research described here happened after, and wasinspired by, that initial publication. The SiC graphene experiments de-scribed in Chapter 6 are ready for publication, and in fact I have writtena manuscript describing this work. Before publication, however, Prof. Folkwishes to connect these results to an analogous experiment (led by othergroup members) on CVD graphene. Similarly, the results I found on FeSeflake degradation (Chapter 8) are ready for publication. However, the deci-sion has been made to build on these results to make a new generation ofFeSe devices, rather than publishing the degradation results independently.The following is a statement of my contributions to the experiments de-scribed in this thesis:Chapter 5 is about the magnetic impurities in exfoliated graphene. Pro-fessor Joshua Folk and Dr. Mark Lundeburg conceived the experiment inChapter 5. Mark Lundeburg made the device and obtained the first half ofthe data. I took the second half of the data and participated in the dataanalysis. Two bilayer graphene devices made by me were also measuredbriefly to verify the observed in-plane magnetic field dependence of the weaklocalization dephasing rate. The experiment in Chapter 5 established theresearch direction for the graphene part of this thesis.vPrefaceChapter 6 is about the magnetic impurities in SiC graphene. It is in-spired by the 2015 paper of Dr. Samuel Lara-Avila[1]. That paper wasmainly focused on phase coherent transport in the high temperature region.We explored the lower temperature region experimentally. For the experi-ment in Chapter 6, I made the samples and performed the measurements.After the measurements, I did the analysis with Dr. Silvia Folk and Pro-fessor Joshua Folk. In this chapter, we also compared the phase coherentphenomena of SiC graphene to that of annealed CVD graphene; this com-parison provides information about the source of the magnetic impurities.Chapter 7 is about searching for signatures of quantum coupling ofmagnetic impurities (such as the collapse of the dephasing rate as a func-tion of temperature) in an exfoliated graphene/hexagonal boron nitride(graphene/h-BN) heterostructure. For the experiment in this chapter, Imade the sample, performed the measurements and did the data analysis.Chapter 8 is about the stability of exfoliated FeSe nanosheets. For theexperiments in this chapter, Dr. Shun Chi from the superconductivity group(led by Professor Douglas Bonn) grew the bulk crystal and launched the col-laboration with me. I did almost all of the experiments and the data analy-sis. The Tc measurement was done in the Physical Properties MeasurementSystems (PPMS) of the superconductivity group, with help from PinderDosanjh and Shun Chi. For the Raman measurement, I received trainingand help from Weijun Luo and Professor Guangrui Xia of the MaterialsEngineering Department of UBC. The Time-of-Flight-Secondary-Ion-Mass-Spectroscopy (ToF-SIMS) measurement was outsourced to the InterfacialAnalysis and Reactivity Laboratory (IARL, led by Professor Reinhard Jet-ter) located in the Advanced Materials and Process Engineering Laboratory(AMPEL) building.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Bandstructure of graphene . . . . . . . . . . . . . . . . . . . . 72.1 Crystal structure and chemical bonds of graphene . . . . . . 72.2 The tight-binding-approximation Hamiltonian of graphene . 93 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Transport properties . . . . . . . . . . . . . . . . . . . . . . . 173.3 Drude conductivity . . . . . . . . . . . . . . . . . . . . . . . 183.4 Phase coherent transport . . . . . . . . . . . . . . . . . . . . 203.4.1 Weak localization . . . . . . . . . . . . . . . . . . . . 213.4.2 Universal conductance fluctuations . . . . . . . . . . 25viiTable of Contents3.4.3 Influence of magnetic impurities on the temperatureand B‖ dependences of the dephasing rate . . . . . . 293.4.4 Manifestation of the magnetic impurities beyond itsinfluence on the dephasing rate . . . . . . . . . . . . 374 Experimental techniques . . . . . . . . . . . . . . . . . . . . . 394.1 Graphene materials . . . . . . . . . . . . . . . . . . . . . . . 394.1.1 Exfoliated graphene . . . . . . . . . . . . . . . . . . . 394.1.2 SiC graphene . . . . . . . . . . . . . . . . . . . . . . . 424.1.3 CVD graphene . . . . . . . . . . . . . . . . . . . . . . 434.2 Nanofabrication . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Electrical measurement . . . . . . . . . . . . . . . . . . . . . 495 Experiment 1: phase coherence in exfoliated graphene . 515.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Dephasing rates extracted from UCF and WL . . . . . . . . 545.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 Experiment 2: magnetic impurities in SiC graphene . . . 626.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Magnetic impurities in SiC graphene and signs of freezingmagnetic impurities . . . . . . . . . . . . . . . . . . . . . . . 656.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Experiment 3: graphene on h-BN heterostructure . . . . . 737.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . 747.2 Phase coherent transport in the exfoliated graphene/h-BNheterostructure . . . . . . . . . . . . . . . . . . . . . . . . . . 758 Other two dimensional materials . . . . . . . . . . . . . . . . 808.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.2 FeSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2.1 Experiment setup . . . . . . . . . . . . . . . . . . . . 84viiiTable of Contents8.2.2 Study of the stability of the FeSe nanosheets . . . . . 878.2.3 Study of the superconductivity . . . . . . . . . . . . . 928.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 939 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . 959.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A More details behind phase coherent transport . . . . . . . 117A.1 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . 117A.2 Universal conductance fluctuations . . . . . . . . . . . . . . . 120B Sensor network for cryogenics operation . . . . . . . . . . . 123ixList of Tables8.1 Tc and thicknesses . . . . . . . . . . . . . . . . . . . . . . . . 93xList of Figures2.1 Graphene lattice . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Hybridization of the orbitals of a carbon atom in graphene . 92.3 The band structure of graphene . . . . . . . . . . . . . . . . . 122.4 Valleys in the graphene band structure . . . . . . . . . . . . . 143.1 Boltzmann transport picture . . . . . . . . . . . . . . . . . . 193.2 The phase coherent corrections . . . . . . . . . . . . . . . . . 203.3 Details of extracting the dephasing rate from UCF . . . . . . 283.4 Theoretical UCF dephasing rate as a function of in-planemagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Influence of magnetic impurities on the temperature depen-dence of the dephasing rate . . . . . . . . . . . . . . . . . . . 354.1 Exfoliated graphene on SiO2 . . . . . . . . . . . . . . . . . . . 424.2 Structure of epitaxial SiC graphene . . . . . . . . . . . . . . . 434.3 Growth of CVD graphene . . . . . . . . . . . . . . . . . . . . 444.4 Photos of a SEM and a graphene device . . . . . . . . . . . . 464.5 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 4-terminal measurement setup . . . . . . . . . . . . . . . . . . 505.1 A photomicrograph of the sample studied in this chapter . . . 535.2 The measurement setup and UCF signal . . . . . . . . . . . . 555.3 UCF dephasing rate vs in-plane magnetic field . . . . . . . . 565.4 The WL signal after averaging . . . . . . . . . . . . . . . . . 565.5 WL dephasing rate vs in-plane magnetic field of an exfoliatedgraphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58xiList of Figures5.6 Gate-dependence of the UCF dephasing rate as a function ofin-plane field . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.1 SiC graphene sample . . . . . . . . . . . . . . . . . . . . . . . 646.2 Perpendicular magnetoconductivity and parallel magnetore-sistance in SiC graphene . . . . . . . . . . . . . . . . . . . . . 666.3 In-plane magnetic field dependence of the dephasing rate inSiC graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4 Temperature dependence of the dephasing rate in SiC graphene 706.5 Temperature dependence of the dephasing rate before andafter a background subtraction . . . . . . . . . . . . . . . . . 717.1 Device wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Dirac peak and quantum Hall effect . . . . . . . . . . . . . . 767.3 Magnetoresistance trace of the graphene/h-BN stack . . . . . 767.4 Steps to get the UCF dephasing rate . . . . . . . . . . . . . . 787.5 Steps to get the averaged WL trace . . . . . . . . . . . . . . . 787.6 Temperature and gate voltage dependence of the dephasingrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.1 Crystal structure of FeSe . . . . . . . . . . . . . . . . . . . . 838.2 Band structure of superconducting FeSe . . . . . . . . . . . . 838.3 Optical and AFM characterizations of the FeSe nanosheets . 858.4 The electrodes fabrication methods . . . . . . . . . . . . . . . 878.5 Shadow mask made with a focused ion beam microscope . . . 888.6 Resistance vs temperature for the bulk FeSe . . . . . . . . . . 888.7 Raman spectrum for FeSe flakes with different thicknesses . . 898.8 Raman spectrum of FeSe before and after long time exposurein air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.9 Optical and AFM images of FeSe nanosheets before and afterlong time exposure in air . . . . . . . . . . . . . . . . . . . . . 918.10 A ToF-SIMS image for a FeSe flake exposed in air . . . . . . 928.11 Superconducting transition for FeSe flakes with different thick-nesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93xiiList of Figures9.1 Saturating dephasing rate in the black phosphorus . . . . . . 98B.1 Wiring for the Arduino-Zigbee module . . . . . . . . . . . . . 123B.2 Photos of the sensor reading and receiving system . . . . . . 124B.3 Sensor on a liquefier and the webpage monitor . . . . . . . . 124xiiiAcknowledgementsThanks to my supervisor — Joshua Folk. His vision and support in the pastyears made this thesis possible. His vision led to the topic of this thesis. Hissupport made the experiments possible. Thank you for letting me use theprecious fridge time and scanning electron microscope time. Thanks to Dr.Mark Lundeburg and Dr. Silvia Folk’s help during my research in UBC,I learnt lots of experimental skills from you. It is a great fun to discussphysics with Professor Folk, Dr. Silvia Folk and Dr. Mark Lundeburg.Thanks to Ebrahim and other members in the lab, for the help on themaintenance of the cryogenics facilities in our lab. Thanks to ProfessorDouglas Bonn and Dr. Shun Chi, your vision and help made the FeSeresearch possible. Thanks to Weijun Luo, Professor Guangrui Xia, yourhelp made the Raman measurements possible! Thank you to everyone whohelped me in my research!Thanks to my parents, their support and encouragement help me passmy hard times!xivChapter 1IntroductionGraphene is an isolated single layer of graphite. It was first isolated bymechanical exfoliation of graphite crystals and electrically characterizedby Geim’s group in 2004. Since then, there has been heightened inter-est in graphene research. The band structure of graphene’s charge car-riers is unique. Its effective Hamiltonian is that of massless relativisticDirac fermions, in contrast to non-relativistic massive particles obeying theSchro¨dinger equation seen in common semiconductors. Graphene opened abrand new field of mimicking particle physics with quasi-particles in con-densed matter systems. Besides theoretical implication, graphene also hasthe potential for widespread industrial applications, and graphene forms(e.g. SiC graphene) suitable for industrial-scale production have been in-vented. Its high mobility (10 times that of silicon) and transparency makegraphene useful for conventional electronics such as microwave transistors,touch screens, light detectors, etc. Graphene is sometimes referred to aspotentially the “next silicon” for modern electronics. This belief is furtherstrengthened by the recent discovery of a special kind of SiC graphene witha bandgap[2].Furthermore, graphene is also a promising candidate for a potentiallydisruptive next-generation technology — spintronics. Graphene is expectedto be an ideal material for spin transport because of its low spin-orbit cou-pling and small hyperfine interactions[3–6]. However, the spin relaxationmeasured from spin-valve and quantum transport experiments is unexpect-edly fast, making the actual spin relaxation length lower than the predictedvalue[1, 7, 8]. This short spin relaxation length was long believed to becaused by magnetic impurities or spin-orbit interaction[9].Though magnetic impurities are problematic for spin transport, mag-1Chapter 1. Introductionnetic impurities together with their quantum nature in graphene form afascinating topic. Magnetic moments are usually associated with f or d elec-trons. Since graphene does not have f or d electrons, the magnetic momentformation would be non-trivial, for example, related to defects such as thevacancies[9]. A further area of interest is the interaction of the magnetic im-purities with the conduction electrons, which gives rise to Kondo couplingand RKKY coupling. Kondo coupling is the quantum screening of magneticmoments in the Fermi sea. RKKY coupling is the indirect coupling betweenthe magnetic moments mediated by the conduction electrons. On the theoryside, the Kondo effect and RKKY coupling in graphene have been studiedfor several years[10–13]. Several papers argued about the strength of Kondoand RKKY couplings in graphene. On the experiment side, however, thereis very little input. The primary evidence for the Kondo effect comes froma 2011 paper by J.H. Chen, et al.[14], which observed the characteristicresistance–vs–temperature curve caused by the Kondo effect. However, dueto the possible interference from the electron-electron interaction and inho-mogeneity, the 2011 paper is still not conclusive for the existence of Kondocoupling in graphene. The primary evidence for the RKKY effect comes froma 2009 paper by J. Cervenka, et al.[15], which indirectly supports the exis-tence of RKKY coupling by the observation of weak ferromagnetic orderingin graphite. Therefore, whether the Kondo coupling and RKKY couplingreally exist in graphene is still a puzzle, and more investigation is needed forclarifying this puzzle. The study of the manifestation of magnetic impuritiesand their quantum nature is the major topic for the graphene part of thisthesis.A single flake of graphene is too small for bulk magnetometry to studymagnetic impurities. Instead, we use phase coherent transport phenomena(weak localization and universal conductance fluctuations) as our tool forstudying magnetic impurities in graphene. This is a proven method fordetecting magnetic impurities with low Kondo temperatures (as well as weakRKKY couplings)[16–18]. Phase coherent transport is so sensitive that itcan detect Kondo coupling even if there is no obvious manifestation of Kondoeffect in the resistivity–vs–temperature plot[17, 18].2Chapter 1. IntroductionIn phase coherent transport, a basic metric is the dephasing rate, whichreflects the phase relaxation of quantum mechanical waves. The dephas-ing rate contains contributions from various phase breaking processes, suchas inelastic electron-electron interactions, magnetic scattering, etc. Mag-netic impurities manifest themselves in the parallel field and temperaturedependence of the dephasing rate. In the first two experiment chapters ofmy thesis, we use the effect of in-plane magnetic field on the dephasing rate(the dephasing rate is extracted from universal conductance fluctuations andweak localization), to unambiguously demonstrate the existence of magneticimpurities. Furthermore, interactions between magnetic impurities as wellas between the magnetic impurities and the conduction electrons are alsostudied. Below 1K, possible signals from Kondo coupling or RKKY cou-pling were observed, such as the collapse of the dephasing rate as a functionof temperature, as well as an unusual magnetoresistance.The graphene experiments described above benefited from the fact thatgraphene is extremely inert in air relative to most other 2d materials. Graphenerepresents the beginning of 2d material research, but recently, more andmore other members of the 2d material family have emerged, though not asinert as graphene in air. This thesis also explored another member of thegrowing 2d material family — FeSe. FeSe is a transition metal chalcogenides(TMX) that is also a superconductor. It is also the simplest Fe-based super-conductor, thus is regarded as a key for the understanding of this family. Wesuccessfully exfoliated FeSe crystal into nanosheets, examined their chemicalstability with various instruments, and also studied their superconductivityby transport method. Our study of FeSe nanosheets shows that FeSe de-cays in air, and the decay process is accompanied by oxidation as well asSe precipitation. The transport measurement also shows that supercon-ductivity can be retained for flakes exposed briefly to air. When the FeSeexperiment was first conceived, the target was to realize gate-tunable FeSethin flake devices, which provides a great playground for gate-tunable su-perconductivity and quantum phase transitions[19–24], because of the lowerdimensionality and tunable areal carrier density (2d layers have much lowerareal carrier density when comparing with bulk material). We spent a year3Chapter 1. Introductionsolving various experimental challenges. In the end, we found that a conven-tional SiO2 back gate has negligible effect on the superconductivity, sincethe carrier density of thin FeSe flakes is still too large to be affected by thegate. Moreover, the lifespan of exfoliated thin flakes is extremely short, thusencapsulation with h-BN or other forms of protection would be necessary.Therefore, this quest ended with the study of FeSe flake stability. However,the experiments on FeSe nanosheets lay a solid foundation for future studyof superconductivity in FeSe nanosheets.Outline of this thesisChapters 2 to 7 are dedicated to the graphene research. Chapter 2 re-views the crystal structure and electronic structure of graphene. Chapter 3reviews the electron gas in crystals and its transport properties. This is thefoundation for calculating transport properties in graphene. Several usefulconcepts widely used in condensed matter research are reviewed, includ-ing Fermi gas, Fermi level, and effective mass approximation. The Drudeconductivity from semiclassical Boltzmann transport theory, as well as cor-rections from coherent transport, are also reviewed. Two phase coherentphenomena—weak localization and universal conductance fluctuations—aredescribed in detail. These two phenomena serve as fundamental tools forthe experimental findings discussed in this thesis.The next chapter (Chapter 4) is dedicated to experimental techniquessuch as nanofabrication and cryogenics, and to the weak signal measure-ments for nanodevices. We use these techniques to make our graphene nan-odevices and observe the phase coherent transport phenomena in graphene.The experiment described in Chapter 5 is a study of the phase coher-ence time of electrons in exfoliated graphene, using both weak localizationand conductance fluctuations as probes. At this time I was a junior PhDstudent working with a senior student Mark Lundeburg. This experimentdemonstrates the existence of magnetic impurities in graphene by lookinginto the in-plane magnetic field dependence of dephasing rate extracted withUCF and WL. This could explain, at least partially, the shortened dephasing4Chapter 1. Introductiontime observed in graphene. This experiment shows that magnetic impuri-ties are important in graphene and phase coherent transport is a good wayto investigate them. The anomalies observed in the in-plane magnetic fielddependence of the dephasing rate extracted with WL was a starting pointfor the experiments that followed.The anomalies observed in the in-plane magnetic field dependence of thedephasing rate in Chapter 5 were obscured by the ripples and the small sizeof the exfoliated graphene. To get a better understanding of the anomaly,we did more experiments on other kinds of graphene that did not sufferfrom the same issues. The experiment described in Chapter 6 is a studyof magnetic impurities in SiC graphene. In this experiment, the anomaliesof the dephasing rate extracted with WL as a function of in-plane mag-netic field were studied in detail. Because the SiC graphene is much flatterthan the exfoliated graphene, it gives us a clearer look at the anomalousbehaviour. This anomaly can be explained with the dynamics of the mag-netic impurities. The Lande´ g factor of the magnetic impurities can be ex-tracted. At ultra-low temperatures, signatures of Kondo or RKKY physicsare also observed, such as the collapse of the dephasing rate as a functionof temperature[16, 25], as well as an unusual magnetoresistance[26]. Wealso compare the phase coherent phenomena from annealed CVD graphene(the measurement was done by other members of this lab) with the SiCgraphene data, which indicates that intrinsic defects could be the source ofthe magnetic impurities.Comparing Chapter 6 and Chapter 5, we can see that the signaturesof magnetic impurities are quite different in exfoliated graphene and SiCgraphene. In particular, there are at most weak signatures of quantumcoupling (Kondo or RKKY coupling of the magnetic impurities) in the de-phasing rate of exfoliated graphene studied in Chapter 5. The final chapterrelated to graphene (Chapter 7) describes an effort to induce quantum cou-pling in exfoliated graphene. In this chapter, the phase coherent transportresults of a graphene/hexagonal boron nitride (h-BN) device annealed informing gas (5% hydrogen balanced argon) are included. Inspired by thecollapse of the dephasing rate as a function of temperature of CVD graphene5Chapter 1. Introductionenhanced by annealing (there is very weak collapse of the dephasing rate inraw CVD graphene, annealing probably can introduce quantum magneticimpurities), we conjectured that annealing probably can induce the signa-tures of quantum coupling in exfoliated graphene. In this experiment, weprocessed the exfoliated graphene on h-BN heterostructure with a similarannealing process used for the annealing of CVD graphene, but we did notsee any clear signatures of quantum coupling in the dephasing rate.In recent years, other 2d crystals beyond graphene have been explored.The next chapter (Chapter 8) examines the experiments on a layered crystalbeyond graphene – FeSe. As described previously, this part of my thesis isabout the material science of exfoliated FeSe nanosheets. In this chapter,the chemical stability of exfoliated FeSe nanosheets is studied. We findthat FeSe is unstable in air and unstable under electron beam lithographyprocessing; in addition, we find that oxidation and Se precipitation probablytake place during the decay process. Although FeSe nanosheets decay in air,short exposure allows for the retention of superconductivity.The final chapter (Chapter 9) summarizes the experiments and pointsout future research directions. Our techniques developed in these years canhelp us clarify whether magnetic impurities also exist in other 2d materials,such as black phosphorus and fluorinated/nitrogenated-graphene, becausehints of magnetic impurities seem to be found in these materials.6Chapter 2Bandstructure of graphene2.1 Crystal structure and chemical bonds ofgrapheneGraphene is a two-dimensional allotrope of carbon. It is an isolated singlelayer of graphite. Carbon atoms in graphene are arranged in the honeycomblattice, see Fig. 2.1.Figure 2.1: Graphene lattice, red (site A) and blue (site B) are two inequiv-alent atom sites (from https://wiki.physics.udel.edu/phys824/)To represent graphene lattice, we need to represent the position vectorof any atom in graphene. The base of these vectors are called the primitivelattice vectors:72.1. Crystal structure and chemical bonds of graphenea1 =√32axˆ+32ayˆ (2.1)a2 =−√32axˆ+32ayˆ (2.2)where a = 0.142nm is the distance between two neighboring carbon atoms.The linear combinations of a1 and a2 can represent the position of anycarbon atom in the lattice.The smallest repeating unit in the honeycomb lattice is called a unit cell.Figure 2.1 is an illustration of graphene lattice structure; carbon atoms arelabeled as circles, and nearest-neighbor bonds are labeled as thick lines. Theyellow rhombus is the boundary of the unit cell. We can see that there aretwo inequivalent atoms in one unit cell, one sitting at R = n1a1 +n2a2, theother one sitting at R + τ , with τ = −(a1 + a2)/2.The atomic orbitals and chemical bonds in graphene determine its elec-tronic structure. An isolated single carbon atom has 4 valence electrons,with 2 occupying 2s orbitals and 2 occupying 2p3 orbitals. In the graphenecrystal, the original isolated atomic orbitals of a carbon atom are changed,because of the perturbations from neighboring atoms. The 2s orbital andtwo of the three 2p3 orbitals (2px and 2py) are hybridized to form 3 sp2 or-bitals, pointing to three symmetric directions. The pz orbital is unchanged[27].After this hybridization, the occupation of the electrons also changes: 3electrons occupy the 3 sp2 orbitals and 1 electron occupies the pz orbital.The overlap of the two sp2 orbitals from two neighboring carbon atomsin graphene is called a σ bond. The overlap between the two pz orbitals iscalled a pi or pi∗ bond. The σ bond provides the crystal integrity of graphenebut doesn’t conduct current (as the σ orbital is fully-filled). The unfilled piand pi∗ bonds carry current and are responsible for the electrical propertiesof graphene.82.2. The tight-binding-approximation Hamiltonian of graphene(a) (b)Figure 2.2: (a)Hybridization of the orbitals of an isolated carbon atom(from http://www.adichemistry.com); (b)Bonds in graphene (from DOI:10.1063/1.4915611)2.2 The tight-binding-approximationHamiltonian of grapheneThe Schro¨dinger equation for graphene’s infinite-dimensional Hamiltonian inreal-space representation is impossible to solve directly. Methods have beendeveloped to simplify the Hamiltonian into a finite-dimensional problem,using representations with linear combinations of atomic orbitals as basis.Huckel’s tight-binding-approximation method is suitable to give an ap-proximate Hamiltonian based on localized atomic orbitals, especially effec-tive for carbon systems with pi electrons. It only considers interactions be-tween pi orbitals from neighboring carbon atoms. Applying Huckel’s methodto graphene, the tight-binding-approximation Hamiltonian represented inthe localized atomic orbitals reads:H = −t∑R(|R〉〈R + τ |+ |R〉〈R + a1 + τ |+ |R〉〈R + a2 + τ |+ h.c.)(2.3)|R〉 is a pz state localized at type A atom at site R, |R + τ〉 is a pz statelocalized at type B atom at site R+τ , t is the hopping integral, representinghopping energy between neighboring pi orbitals.92.2. The tight-binding-approximation Hamiltonian of grapheneDirectly solving the eigenvalue problem of this infinite-dimensional ma-trix is not easy in general. For a finite system, we can find the coefficients bydirectly solving the linear equations system. The symmetry of the graphenelattice can help us find the new representation to get a finite-dimensionalmatrix of the Hamiltonian.The translational symmetry in a crystal lattice gives a strong restrictionon the form of the wavefunction (the Bloch theorem). It requires that thewavefunction (probability amplitudes) on one unit cell only differs from thewavefunction on another unit cell by a modulation factor[28]. Thus thewavefunction of conduction electrons in graphene can be written as:|ψ〉 = 1√N∑RCR|R˜〉 (2.4)with|R˜〉 =[|R〉|R + τ〉](2.5)where CR = [C1R, C2R] represents the probability amplitudes at the twoatomic orbitals in a single unit cell. |R˜〉 represents the basis set (pz orbitalsat the two inequivalent carbon atoms) in a single unit cell.According to Bloch’s theorem, CR = exp(ik ·R)C0, and C0 = [C1, C2]is a constant. Hence, |ψ〉 = 1√N∑R CR|R˜〉 = 1√N∑R exp(ik ·R)C0|R˜〉.Therefore we have:|ψ〉 = 1√N∑Rexp(ik ·R)(C1|R〉+ C2|R + τ〉) (2.6)= C1|k, A〉+ C2|k, B〉 (2.7)102.2. The tight-binding-approximation Hamiltonian of graphenewith|k, A〉 = 1√N∑Rexp(ik ·R)|R〉, (2.8)|k, B〉 = 1√N∑Rexp(ik ·R)|R + τ〉. (2.9)This is a new basis set that can cast the single electron Hamiltonian intoblock-diagonalized finite dimensional form:HAB(k) = 〈k,A|H|k,B〉 = 1√N∑R,R′exp(ik · (R−R′))〈R|H|R + τ〉= −t(1 + exp(ik · a1) + exp(ik · a2)) = e(k), (2.10)HBA(k) = −t(1 + exp(−ik · a1) + exp(−ik · a2)) = e(k)† (2.11)H =[E0 HABHBA E0]=[E0 e(k)e(k)† E0](2.12)The single particle eigenenergy is therefore (k) = ±|e(k)|, with |e(k)| =t√1 + 4 cos(√3kxa2 ) cos(kya2 ) + 4 cos2(kya2 ). Write e(k) in the form e(k) =|e(k)|eiφ, accordingly, the corresponding eigenvectors read:|k,−〉 = 1√2[eiφ/2−e−iφ/2](2.13)|k,+〉 = 1√2[eiφ/2e−iφ/2](2.14)In the reciprocal space, the points where two bands touch are calledvalleys. In the first Brillouin zone of graphene, there are two inequivalent112.2. The tight-binding-approximation Hamiltonian of grapheneFigure 2.3: The band structure of graphenevalleys. In the original Hamiltonian, we deal with a general case in whichall points in the reciprocal space are accessible by electrons. However, inmost cases, only states around the two valleys contribute to the electricalconduction, because the Fermi level is never too far from the valleys. In otherwords, only k vectors around the corners of certain valleys are accessible. Wecan get a simpler effective Hamiltonian for states near the two valleys. ThisHamiltonian is much easier to handle, compared to the original Hamiltonianwe discussed before.Taylor expansion is used to simplify matrix (2.12) to get an effectiveHamiltonian for states near the valley centers. The Taylor expansion ofe(k) reads1:1https://wiki.physics.udel.edu/phys824.122.2. The tight-binding-approximation Hamiltonian of graphenee(k) = −t(1 + ei(K+δk)·a1 + ei(K+δk)·a2)≈ −t(1 + (−12− i√32)(1 + iδk · a2) + (−12+ i√32)(1 + iδk · a2))= −t(−12iδk · (a1 + a2) +√32iδk · (a1 − a2)) = −t(√32|δk|ae−ipi/2e−iθ)= −√32at|δk|(−i)(cos(θ)− i sin(θ)) = −√32at(−iδkx − δky) (2.15)It is convenient to extract an overall constant factor and thus write theabove equation as:e(k) ≈ −√32at(δkx − iδky) (2.16)e(k)† ≈ −√32at(δkx + iδky) (2.17)Therefore, the Hamiltonian (2.12) can be simplified into:H0 =[0 −√32 at(δkx − iδky)−√32 at(δkx + iδky) 0]= −√32at1~[0 ~(δkx − iδky)~(δkx + iδky) 0]= −vFσ · p (2.18)where vF =3ta2~ ≈ 106m/s, σ = (σx, σy) is defined in the state space of thetwo inequivalent carbon atoms (known as isospin) in a unit cell.There are two copies of this effective Hamiltonian, one for each valley.These two Hamiltonians are connected with each other with the transfor-mation HK′ (k) = −HTK(k).Combining the two copies of the Hamiltonian together, we can get the132.2. The tight-binding-approximation Hamiltonian of grapheneeffective Hamiltonian with valley as a degree of freedom:H = vF[−σ∗ · p 00 σ · p](2.19)where p is the momentum defined with respect to the center of the valley.K and K′are the reciprocal space vectors of the centers of the two valleys,respectively. The valley degree of freedom is referred to as a pseudospin.Figure 2.4: Valleys (K and K′) in the graphene band structure (fromhttp://phz389.ust.hk/lab2)This effective Hamiltonian around valley centers is similar to the Hamil-tonian for massless Dirac Fermions, possessing a linear dispersion relation.This is different from the Shro¨dinger-like effective Hamiltonian for valley-centers of normal semiconductors. In equation (2.15), the phase of e(k) is−θ − pi/2. Since the wavefunction’s phase is one half of the phase of e(k),therefore, for a path circling the valley center, θ changes by 2pi, the corre-sponding phase change of the wavefunction is thus pi. This phase change iscalled Berry phase, which leads to the novel Landau energy levels in strongmagnetic fields and weak antilocalization (see Chapter 3) in the absence ofintervalley scattering.14Chapter 3Transport3.1 Fermi gasIn the last chapter, we modeled a single electron in the graphene crystal.There are billions of electrons in a piece of crystal. How do we understandthe properties of these electrons as a whole? In most cases, to the lowestorder, we can regard the electron gas in the crystal as composed of freeindependent particles, thus we can use statistical mechanics to model itsproperties. The basic picture of identical particles living in a crystal latticeis: the particles fill up single-particle quantum states, and these particlesfollow collective statistical distributions as a whole. This is the require-ment imposed by the exchange symmetry of identical quantum particles,even though there are no interactions among them. Electrons in crystalsare fermions; they obey Fermi-Dirac statistics. The Fermi-Dirac functiondescribes the probability f(E) of a single-particle state at energy E beingoccupied:f(E) =11 + exp(E−µkBT )(3.1)T is the absolute temperature, kB is the Boltzmann’s constant. Theparameter µ is called the chemical potential. At 0K, the Fermi-Dirac dis-tribution becomes a step function, so all states below certain energy level(called the Fermi level EF ) are occupied and all states above that are empty.For T > 0K, some particles get thermally excited into states above theFermi level, so the occupation at T > 0K deviates from the step function153.1. Fermi gas(thus the µ at T > 0 differs from that of 0K), and the boundary betweenoccupied states and unoccupied states becomes blurred. For finite temper-atures we meet in the lab, the distribution only deviates from the T = 0Kcase slightly, thus only states very close to the Fermi level contribute to thetransport properties.In a 2d system, the chemical potential is related to density of charge carri-ers per unit area. The integration (with respect to energy) of the density-of-states multiplied by the Fermi distribution function gives the areal densityof particles[28]. For graphene, the density of states near the Dirac pointreads:ν(E) = gsgv|E − E0|2pi~2v2F(3.2)where E0 is the energy of the Dirac point, gs and gv are spin and pseudospindegeneracy, respectively. vF is the Fermi velocity.The extra carrier density above the carrier density of charge–neutralgraphene reads:ns =∫ ∞−∞dEf(E)ν(E)− n0. (3.3)n0 =∫ E0−∞ dEf(E)ν(E) is chosen to be the electron density in charge–neutralgraphene, with the chemical potential sitting at the points K and K ′ of thereciprocal space. ns > 0 means the graphene is electron-doped. ns = 0means the graphene is undoped.At T ≈ 0K, we can approximate the Fermi distribution as a step functionf(E) ≈ Θ(EF − E), thus:|ns| ≈∫ EFE0dEν(E) =µ2pi~2v2F. (3.4)µ = EF − E0 is the chemical potential with respect to E0 for T ≈ 0K(it only depends on |ns| when T ≈ 0K). In experiments, we can tunethe areal charge carrier density easily with a gate voltage, hence changing163.2. Transport propertiesthe chemical potential and Fermi level accordingly. The Fermi wavevectorcan also be expressed as a function of the carrier density. In graphene,pF = ~kF = µ/vF , thus |ns| = k2Fpi , and kF =√|ns|pi. As a comparison,kF =√2|ns|pi for the parabolic bands in usual semiconductors[28].3.2 Transport propertiesTransport phenomena in (semi-)conductors are related to currents of elec-trons driven by an electric field and other external fields. How can we modelthese phenomena? How does the electron gas respond to electric field?In an ideal crystal without relaxation mechanisms, DC bias will giveoscillating conductivity (Bloch Oscillation)[29], because of the periodicity ofthe band structure. However, the Bloch Oscillation is basically never seenin realistic crystals. This is because scattering leads to relaxation, and thisrelaxation prevents electrons from accelerating too much. In most cases, thestudy of transport is the study of scattering. Finite conductivity comes fromscattering of electrons from defects, impurities and other electrons. Twoscattering times are important for modeling the transport process: one isthe momentum relaxation time, which describes the relaxation of momentumdue to scattering from defects/impurities/other electrons; another one isphase-relaxation time, which describes how long a quantum particle canmaintain the phase of its wavefunction.To model conductivity, we need to model how electrons respond to anexternal electric field. In transport phenomena of a degenerate Fermi gas,only states near the Fermi level matter. For most semiconductors, mo-bile electrons/holes live in high-symmetry valleys of the energy bands. Ingraphene, the Fermi level is always near the K and K ′ points, thereforeonly electronic states near these points contribute to transport. Dynam-ics of a wavepacket composed of Bloch eigenstates around high-symmetryvalley points can tell us how electrons/holes respond to external field. Inthe effective mass approximation, quantum states can be approximated byenvelope functions, and the dynamics of a wavepacket follow Shro¨dinger’sequation[30], with mass replaced by effective mass m∗ (all information about173.3. Drude conductivitythe bandstructure and crystal potential is stuffed into the effective mass).To handle scattering from impurities, we have to start with the scatter-ing of the quantum mechanical wavefunction. With quantum mechanics, wecan calculate how the wavefunction scatters through the disorder potentialwithin the phase-coherent volume, which gives several contributions to theconductivity. The lowest order contribution is from single scatterings, andit coincides with Drude conductivity from the semiclassical Boltzman trans-port model. The next-order correction is from the interference of scatteredwaves — multiple scattering effect. There are two such corrections: weaklocalization (WL) and universal conductance fluctuation (UCF). For a largecrystal composed of many phase-coherent units, the UCF correction getsaveraged out, but WL still exists.3.3 Drude conductivityWith the dynamics from the effective mass approximation[30], we can usea semiclassical approximation (Boltzmann transport theory) to calculateconductivity without phase coherent effect, which is a good starting pointfor the further study.For T ≈ 0K, the conductivity can be written as[31]:σ0 = e2τm∫dk4pi31~∂vk∂kf0(k) = e2τm∫occupieddk4pi31m∗k(3.5)where f0(k) is the Fermi distribution function, and τm is the momentumrelaxation time.For parabolic dispersion, σ0 =nee2τmm∗ (where τm is the momentum re-laxation time or transport time), thus |j| = σ0|E| = nee2τmm∗ |E|. In theDrude picture, |j| = nee|v|, so v = eτmm∗ |E|. The proportionality constant,µ = eτmm∗ , is called mobility. Thus the conductivity can be written in the fa-miliar Drude form: σ = neµ. Therefore, the mobility can be extracted aftermeasuring conductivity and carrier density (via Hall effect). Another char-acteristic number is mean free path lm = vF τm, where τm can be calculated183.3. Drude conductivityfrom mobility and effective mass. kF is another characteristic parameter,which is related to carrier density: kF =√2pine[28].For graphene, the peculiar linear dispersion around band extrema givesdifferent formulas[27, 32]. The effective mass for graphene is momentum-dependent: m∗ = pFvF =~kFvF= EFv2F[32]. Thus, σ = e2τm · 2 · 2∫kd2k(2pi)21m =e2τm · 2 · 2∫kd2k(2pi)2vF~kF = e2τm · 4pik2F4pi2vF~kF = e2τmvF~pi kF [31], where kF =√nepi(see section 3.1). vF ≈ 106m/s is a constant determined by the band struc-ture. The mean free path in graphene thus can be written as lm = vF τm =σh2e2√pine, and the diffusion constant can also be calculated employing therelation D = 12vF lm. These are the equations for getting basic Boltzmanntransport parameters of graphene.In general, conductivity σ is a tensor, but for weak magnetic field, re-sistivity and conductivity can be connected simply by σxx ≈ 1/ρxx. In ex-periments, conductivity and carrier density can be extracted by measuringresistivity in a magnetic field. Two voltage drops Vxx and Vxy are measuredin experiments: Vxx is the voltage drop along the current path, and Vxy isthe voltage drop perpendicular to the current path. Resistivity and carrierdensity can be calculated easily with ρxx = Vxx/I and ne = I/(eVxy/B).Figure 3.1: Boltzmann transport picture193.4. Phase coherent transport3.4 Phase coherent transportThis section is about the correction to the Drude conductivity caused bycoherent multiple scatterings of Bloch waves[33]. In the previous treatment,we assume scattered waves from different impurities don’t interfere witheach other. This treatment gives the results of classical Boltzmann trans-port. This picture holds when wavelength is much smaller than the distancebetween scattering events. However, in general, we have to take into consid-eration the interference among scattered waves. This is especially importantfor understanding transport at low temperatures. Coherent correction hastwo kinds: one is weak localization (WL), and another one is universalconductance fluctuation (UCF). UCF is only influenced by the dephasingprocesses. WL can be influenced by the dephasing processes as well as thebroken time-reversal-symmetry. These two phase-coherent corrections canprovide information about the quantum scatterings and the magnetic impu-rities in samples.(a) (b)Figure 3.2: (a)Quantum corrections; (b)WL and UCF in mesoscopic samples(the asymmetry in the magnetic filed is caused by the mixture of Hall voltagesignal (Vxy) in the irregular-shaped graphene flake)203.4. Phase coherent transport3.4.1 Weak localizationConductivity can be calculated from the quantum scattering of Bloch waveswith the help of the Green’s function. Neglecting multiple interference ef-fects from scattered waves, classical Drude conductivity is recovered in thecalculation. Weak localization is the higher order correction to the Drudeconductivity caused by interference of scattered Bloch waves in a phase co-herent domain. The size of the domain is set by the phase-coherent time τφ(the details of the calculation can be found in appendix A). τφ is introducedas a phenomenological parameter, and results from various phase-breakingprocesses. The WL correction is related to the returning probability of thequantum diffusion process. It is the Laplace transform of the returningprobability with respect to time, which can be calculated with a quantumBrownian motion picture[33].For the weak localization correction at zero magnetic field, the correctionreads[34]:∆σ = − e2pihlnτφτm. (3.6)It contributes to the temperature dependence of resistivity through thetemperature dependence of the phase-coherent time τφ (as a comparison,in normal metal, the Drude contribution to resistivity flattens out at lowtemperature[31]). τφ is related to the linear size of the phase coherent path— Lφ =√Dτφ.The weak localization effect also shows up in the magnetoconductance,∆σ = − e2pih[ψ(12+BtrB)− ψ(12+BφB)], (3.7)where Btr =~4|e|Dτm , Bφ =~4|e|Dτφ , the meaning of Bφ is the magnetic fieldwhich makes a quarter of flux quantum penetrate through the area definedby phase coherent length (Lφ =√Dτφ =√~4|e|Bφ , thus L2φBφ =~4|e| =Φ04pi ,213.4. Phase coherent transportΦ0 is the flux quantum).For intermediate and low fields (B ≤ ~eDτ )[33]:∆σ(B) ≈ − e2pih[ln(BφB)− ψ(12+BφB)] (3.8)This determines the central region of the magnetoconductivity trace.The dephasing rate is manifested in the curvature of the ‘dip zone’ of themagnetoconductivity trace.For intermediate and high fields B  Bφ (for Lφ ≈ 1µm, Bφ is about1mT )[33]:∆σ(B) ≈ − e2pih[ψ(12+~4eDBτm)− ψ(12)] (3.9)WL in this region is basically removed, and a background value that hasno dependence on the phase coherence time is left. In this region, magne-toconductivity from other effects such as the electron-electron interactionshows up. If there is no electron-electron interaction, the only thing left willbe a constant conductivity (no dependence on the magnetic field).At fields higher than ~eDτm , the cyclotron radius of an electron is so smallthat the wavefunction maintains coherence for one cyclotron motion, whichis the regime where the Shubnikov-de Hass oscillations and the quantumHall effect show up.These are the formulas for the simplest WL case. The only relaxationrates are the dephasing rate and momentum relaxation rate. The effect ofweak localization is always suppressing conductivity. When there are otherdegrees of freedom and relaxation rates in the system, the WL formula willbe more complicated, with new relaxation rates included. In general, theweak localization formula has this form[35]:223.4. Phase coherent transport∆σ(B) = −e2Dpi~∫d[−n′F ()]×∫ddq(2pi)d4∑i=1αiCi(, ω = 0,q) (3.10)where −n′F () = 1/[4kBT cosh2( − µ)/(2kBT )] is the derivative of Fermidistribution function. Ci(, ω = 0,q) =1Dq2+Γi()is the Cooperon (seeappendix A).Performing integration over momentum, the equation above becomes:∆σ = − e2pi~4∑i=1αiFd(B,Γi) (3.11)Fd(B,Γ) has different forms depending on the conductor geometry (d = 2for films and d = 1 for wires). αi and the phase relaxation rate Γi aredetermined by the details of the scattering processes (Γi can be brokendown into several dephasing rates). For 2d, F2(B,Γi) ∝ [ψ(12 + Γi4DeB/~) +ln 4eBDτm~ ].Whether the effect of weak localization is to enhance or suppress conduc-tivity depends on the interplay among various relaxation rates. It is possiblefor the weak localization correction to enhance conductivity, in which casethis reversed-sign weak localization is called weak antilocalization.WL with valley scattering and chiralityIn graphene, due to the valley-scattering and the peculiar chiral nature of theeffective Hamiltonian, the weak localization correction has a new form[36]:∆ρ(0)−ρ2 = −e2pih[ln(1 + 2τφτi)− 2 ln τφ/τtr1 +τφτ∗] (3.12)∆ρ(B) = −e2ρ2pih[F (BBφ)− F ( BBφ + 2Bi)− 2F ( BBφ +B∗)] (3.13)where F (z) = ln z + ψ(12 +1z ), Bφ,i,∗ =~4eDτφ,i,∗ .233.4. Phase coherent transportIn addition to the phase-breaking rate, there are some new relaxationrates that come into play. 1τi and1τ∗ are the invtervalley scattering rateand intravalley scattering rate, respectively. 1τφ (1τmag, 1τeei , ...) is the com-bined dephasing rate as a result of various phase-breaking processes (suchas magnetic scattering and electron-electron inelastic scattering). It can bebroken down into different kinds of phase-breaking processes, with some ofthe processes being influenced by external fields, leading to external-fielddependence of the combined dephasing rate.Weak localization in graphene is determined by the chirality (of Diracfermions) and scattering. The interplay among dephasing rate and (elastic)inter/intra-valley scattering rates determines the sign of the weak localiza-tion correction. Depending on the weights of these relaxation rates, theweak localization correction in graphene can suppress (weak localization) orenhance (weak antilocalization) the conductivity.Ideally, near the vicinity of valley centers, the effective Hamiltonian isthat of chiral massless Dirac fermions. The Berry phase factor accumulatedfor backscattering is −1, which corresponds to suppressed backscatteringand a weak antilocalization effect. In actual graphene samples, the chiral-ity is approximate, and the dispersion relation deviates from linear not farfrom the Dirac point (trigonal warping). This breaks conservation of chi-rality and restores conventional weak localization[36, 37]. Besides deviationfrom chiral conservation caused by trigonal warping, various (elastic) val-ley scattering also break chirality and lead to WL, this can take the formof intravalley scattering caused by smooth scattering potential from remoteCoulomb scatterers and ripples, or intervalley scattering caused by point-like sharp scattering potential from atomic defects[36, 37]. Specifically, theweak localization phenomenon is observed when τϕ > τi, τ∗.Therefore, graphene usually exhibits weak localization. The curvatureof the tip zone of the WL magnetoconductance trace is controlled by thedephasing rate 1τφ , which is not sensitive to valley scattering rates. Thevalley scattering rates control the morphology of the magnetoconductance243.4. Phase coherent transporttrace at higher magnetic fields. The dephasing rate extracted from fittingthe whole WL magnetoconductivity is the same as that extracted from fit-ting the tip zone of the magnetoconductivity (as long as the dephasing rateis much smaller than the inter/intravalley scattering rate, or equivalently,Bi,∗ is much larger than Bφ). We can extract dephasing rate informationefficiently by analyzing the curvature of the tip zone of the magnetocon-ductivity, τ−1WL =eD~ (3pi4he2κ)−12 ∝ 1√κ(κ is the curvature of the tip)[7].Moreover, another advantage of the metric around the center is that thedephasing rate can be more reliably characterized when the magnetic scat-tering causes deviation from the standard weak localization formula. Thestandard graphene weak localization formula only considers non-magneticdephasing mechanism, it could be distorted when suffering from substantialmagnetic scattering[7].This is the classical spinless WL in graphene. There is no intrinsic spindegree of freedom involved. The vanishingly small spin-orbit interaction ingraphene makes it an ideal system for studying extrinsic spin-related effectsfrom magnetic impurities.3.4.2 Universal conductance fluctuationsIn the derivation of the weak localization formula, it is assumed that thereare lots of different realizations of disorder potential landscapes to averageover. What will happen if the sample is small enough to contain only afew phase coherent units? In this case, there is no ensemble average, andthe measured conductivity reflects the peculiar interference pattern of wavestraveling through a specific realization of the disorder potential landscape.If we change the disorder potential landscape by changing the configurationof the scattering sources, a different interference pattern will emerge. Ex-perimentally, we can adjust the scattering potential landscape continuouslyby changing the gate voltage or magnetic field applied to the sample, anda continuous conductance change will be observed accordingly. The result-ing conductance as a function of magnetic field or gate voltage looks like arandom but repeatable trace, like the pseudo-random numbers generated by253.4. Phase coherent transportalgorithms. In other words, for such a small sample, we can use magnetic orgate voltage knobs to iterate through different disorder potential landscapesof an ensemble. This fluctuating conductance is called universal conduc-tance fluctuation or mesoscopic conductance fluctuation. The variance ormagnitude of UCF is proportional to (Lφ/L)4−d (where Lφ =√Dτφ is thephase coherent length, d is dimension). For the regime where Lφ is com-parable with L, the fluctuations are universal, meaning they always havemagnitude ( e2h )2. That’s why it is called ‘universal’ conductance fluctu-ation. In the opposite case, where Lφ  L, the fluctuations decrease asδG2(L,Lφ) ∝ (LφL )4−d. UCF quickly vanishes as the sample becomes large.That’s why we can only see UCF in mesoscopic samples[33].The UCF coexists with WL. UCF appears to be some noise which smearsthe WL signal. For small samples, WL traces are buried in UCF ‘noise’. Inorder to see a clear WL magnetoresistance trace from the fluctuating traces,we need ensemble averaging to remove UCF. The ensemble average canbe done by averaging over magnetoconductance traces from different gatevoltages.The statistical properties of this UCF itself also contains informationabout phase coherent transport. In the multiple-scattering regime, the co-herent effects associated with the Cooperon modify the average values ofthe conductivity, leading to the weak localization correction. Similarly, thevariance and correlation functions of the fluctuating part of the magnetore-sistance are also affected by the interference of scattered waves. In the caseof UCF, they are determined not only by Cooperon, but also by Diffusons(the constructive interference of electrons and holes traveling along the samepath[38]). A basic measure for stochastic processes is the autocorrelationfunction. It reflects to what extent the trace repeats itself. Each disor-der configuration has a unique correlation function. The ensemble-averagedcorrelation function provides information about the phase coherence. Wecan get dephasing rates by comparing (fitting) the theoretical correlationfunction with the experimental correlation function.263.4. Phase coherent transportThe theoretical ensemble-averaged correlation function of the fluctuatingpart of the conductivity is, in general, a two dimensional function of energyand magnetic field.At T = 0K, consider a phase coherent rectangle (Lx by Ly) in 2 dimen-sional space, in which the formula for the autocorrelation as a function ofFermi energy and magnetic field reads[39]:F0(δE, δB) = δG0(E,B)δG0(E + δE,B + δB)= Ce4h2LyDτφL3xK0(, β). (3.14)withK0(, β) =1piIm[ψ(12+1 + iβ)]+12piβRe[ψ′(12+1 + iβ)], (3.15)where ≡ δE · τφ/~, β ≡ |δB| · 2eDτφ/~.For T > 0K, the conductance and correlation function are smeared bytemperature (more details can be found in appendix A):F (δµ, δB) = Ce4h2LyDτφL3xKT(′, β). (3.16)Note that here, ′ = δµτφ/~. The kernel reads:KT(′, β) =∫ ∞−∞dκ(/T1)T1K0(− ′, β), (3.17)where T1 ≡ kBT ·τφ/~, κ(x) = 12(x2 coth x2−1)/ sinh2 x2 , and C is a constant.The cross-section at δµ = 0 gives the autocorrelation function with re-273.4. Phase coherent transportspect to B⊥. Comparing the experimental autocorrelation function of themagnetoconductance with the theoretical model, we can quantitatively getthe dephasing rate.In practice, we need to occupy the dilution refrigerator for a very longtime to get a good enough ensemble-averaged correlation function to com-pare with theory. Because we need a tremendously large magnetic field rangeand gate voltage range, it will take a very long time to finish. In addition,the systematic error introduced during background conductance subtractionalso makes it difficult to get a good experimental autocorrelation function.Since we can not occupy the dilution refrigerator for too long, we need to getan experimentally viable metric to measure the dephasing rate in a fast andreliable manner. A theory paper[39] compared some different easy-to-getmetrics such as half-maximum field B1/2 and inflection field Binflect. It wasfound that only Binflect gives a robust metric with reasonable error. ThusBinflect is a practical metric for getting phase coherent information fromUCF traces, and τ−1UCF ≈ 2eDBinflect3~ , where d2fdδB2|δB=Binflect = 0[39].Figure 3.3: UCF metric: (a) Raw dataset (b) Autocorrelation function foreach gate voltage (c)Averaged autocorrelation functionFig. 3.3 shows an example of how to extract the dephasing rate fromUCF. Fig. 3.3(a) show the raw UCF data. It is conductance as a func-tion of gate voltage Vg and perpendicular magnetic field B⊥, denoted asG(B⊥, V g). For each gate voltage Vg, there is a unique UCF magneto-283.4. Phase coherent transportconductance trace. The fluctuating part of the UCF dataset is obtainedfrom the raw G(B⊥, V g) by subtracting off a background function. Thebackground function is obtained by fitting a polynomial of the form a0 +a1B⊥+a2B2⊥+a3V g+a4V g2 +a5B⊥V g to the raw data G(B⊥, V g), wherea0, a1, a2, a3, a4, a5 are the free fitting parameters. After getting the fluctu-ating part of the UCF dataset, we calculate the autocorrelation function foreach Vg: FV g(δB) =1B2−δB−B1∫ B2−δBB1δG(B⊥, V g)δG(B⊥ + δB, V g)dB⊥(see Fig. 3.3(b)). Then, we average over Vg to get the final ensemble-averaged autocorrelation function: f(δB) = 1[∑V g]∑V gFV g(δB), where [∑V g] de-notes the number of different gate voltages (see Fig. 3.3(c)).3.4.3 Influence of magnetic impurities on the temperatureand B‖ dependences of the dephasing rateThe phenomenological dephasing rate introduced above actually can bebroken down into various basic phase-breaking processes, such as inelas-tic scattering caused by electron-electron interaction, or inelastic scatteringcaused by scattering off magnetic impurities. In particular, magnetic scat-tering depends on the polarization of the magnetic impurities and temper-ature. Therefore, magnetic impurities manifest themselves in the B‖ andtemperature dependences of the dephasing rate. The specific dependencedepends on the dimension of the system and the type of phase coherentphenomenon[35, 38].Influence of magnetic impurities on B‖-dependence of the WLdephasing rateNow let’s bring the spin degree-of-freedom into the picture. Spinful im-purities exert a non-scalar scattering potential on electrons Jσ · Sδ(r− ri)(where S is a vector representing the orientation of the impurity spin, J isthe coupling constant, ri is the location of the magnetic impurity). Whenmagnetic scattering is involved, a new relaxation rate — magnetic scatteringrate — will play a role in the weak localization correction. We can use weaklocalization as a tool to probe the properties of magnetic scattering sources.293.4. Phase coherent transportDue to the lack of spin-orbit interaction, graphene is an ideal clean systemto probe the properties of magnetic impurities.Dephasing happens when an inelastic interaction occurs between anelectron and the environment2, because the interaction leaks informationabout the quantum state of the electron. The previously-introduced phe-nomenological dephasing rate can be broken down into basic dephasing pro-cesses. Many dynamical variables in the environment can dephase electrons.Electron-electron interactions can dephase electrons by means of thermalcharge fluctuations. Spins also fluctuate and serve as a dynamic environmen-tal variable to dephase electrons. After introducing magnetic impurities, thebasic picture is that electrons travel through fluctuating spinful configura-tions and get dephased. According to the theory, only fluctuating magneticimpurities can dephase electrons (by inelastic scattering), thus anythingsuppressing the thermal fluctuation of magnetic impurities can suppress themagnetic dephasing[40]. Strong in-plane magnetic fields polarize the spins,thus enhancing the coherence time of the electrons traveling in the forestof fluctuating magnetic impurities 3. Recent studies point out that, besidesthe polarization effect, a subtle new effect will occur for weak localization.When the Lande´ g factor of the magnetic impurities is different from thatof the electron, a small in-plane field actually adds extra dephasing onto theweak localization effect[1, 42]. The polarization effect of a small in-planefield is negligible. However, if the Lande´ factors of the magnetic impuritiesand the electron are different, there will be a phase difference between thetwo time-reversal-symmetric paths an electron can take, due to the differentLarmor precession frequencies. Therefore, the dephasing rate is increasedby a small in-plane field. The extracted dephasing rate from weak localiza-tion contains contributions from all these polarization and precession effects.The dephasing rate thus shows a dependence on the in-plane magnetic field.2we only talk about dephasing caused by inelastic scattering in this thesis, equivalent‘dephasing’ caused by the broken time reversal symmetry is not in the scope of this thesis3the crossover B‖ value in this case is temperature dependent, proportional to kBT [41].As a comparison, though weak spin-orbit coupling of a special kind can also lead tosuppressed dephasing rate in B‖, but the crossover B‖ is set by ~τ−1spin−orbit, does notdepend on temperature[41]303.4. Phase coherent transportThe B‖ dependence of the dephasing rate depends on the dimension ofthe system and the type of phase coherent phenomena[35, 38]. Recently, amodel was developed to study the B‖ dependence of the WL dephasing ratein 2d systems caused by precessing classical magnetic impurities[1, 42]. Thismodel quantitatively gives the suppression of the dephasing rate caused bythe in-plane magnetic field, as well as the precession effect. In this theory themagnetic scattering rate couples with the dephasing rate from the electron-electron interaction in a nonlinear way, contributing to the extracted WLdephasing rate.The weak localization correction in the model developed in [42] reads:σ(B⊥, B‖)− σ(0, B‖) = −e22pih∫d∑α=±n′F (α)× (F ( B⊥B1,α) + (F (B⊥B0,−)− F ( B⊥B0,+))Aα) (3.18)where F (z) = ln z + ψ(12 +1z ), Bβ,α =~/(4e)Dτβ,α, and the other terms read:A± = (τ−11,±1 − τ−1T )/(τ−10,+ − τ−10,−)τ−10,± = [1 +M2 −M1(nF (+)− nF (−))]τ−1s ±√τ−11,+1τ−11,−1 − Ω2B + τ−1Tτ−11,±1 = [1−M2 ∓M1(1− 2nF (∓))]τ−1s + τ−1TMn = 〈Snz 〉/S(S + 1)−n′F () =14kBT cosh2[ σ2kBT ]σ = − σgiµBB‖, σ = ±ΩB =(ge − gi)µBB‖~ψ is the digamma function. τ−1s and τ−1T are the magnetic scattering rateand the inelastic electron-electron scattering rate, respectively. D is thediffusion constant. gi and ge are Lande´ factors for the magnetic impurity313.4. Phase coherent transportand the electron, respectively. Mn is related to the magnetization of themagnetic impurities, S = 0 or 1.This formula can be regarded as a modification of the usual weak lo-calization formula, taking into consideration the difference in precession fre-quencies of electrons and magnetic impurities as well as the induced impuritymagnetization caused by external magnetic field[1, 42].The S in Kashuba’s formula refers to the spin of two-electron propa-gators (Cooperons). In the case where electrons and magnetic impuritiesprecess with the same frequencies in an external magnetic field, the WLcorrection is a simple summation of 4 Cooperons CS,M (3 from the S=1triplets (S = 1,M = 0,±1), 1 from the S=0 singlet (S = 0,M = 0)), andall these modes decay with the same rate. In the case of electrons and mag-netic impurities having different Lande´ factors and precessing with differentfrequencies (ΩB 6= 0), the Cooperons mix into new modes with differentrelaxation rates. τ−11,±1 represents decay rates of two of the S = 1 Cooper-ons. On the other hand, S = 0,M = 0 and S = 1,M = 0 Cooperons mixtogether due to the difference in precession frequencies, leading to combinedmodes with relaxation rates τ−10,±1.Precession physics is reflected in τ−10,±1 via the ΩB =(ge−gi)µBB‖~ term.The polarization effect of the external magnetic field is reflected in τ−11,±1 andτ−10,±1 via the 〈S2z 〉 and 〈Sz〉 terms. They are related to induced magnetizationand are non-zero only for S = 1. 〈S2z 〉 and 〈Sz〉 can be obtained by theclassical partition function method from statistical mechanics. For S = 1:〈Sz〉 = 2 sinh(a)2 cosh(a) + 1(3.19)〈S2z 〉 =2 cosh(a)2 cosh(a) + 1(3.20)a = giµBB‖/(kBT ) (3.21)The curvature around the central zone of the WL magnetoconductanceis equal to:323.4. Phase coherent transportκ =d2σdB2⊥|B⊥=0 =2e23pih∫d∑α=±(−n′F (α))× ((Dτ1,α~/e)2 + (τ−11,α − τ−1T )(τ0,+ + τ0,−)Dτ0,+~/eDτ0,−~/e) (3.22)= κ(B‖, τs, gi, ge = 2, S = 1). (3.23)This curvature is related to the (effective) dephasing rate extracted withthe conventional weak localization formula[7]:τ−1WL =eD~(3pi4he2κ)−12 ∝ 1√κ. (3.24)As long as ΩB 6= 0 (gi 6= ge), all gi values can give non-monotonicbehaviour for the dephasing rate as a function of magnetic field. For gi = 0,it can give high-field saturation of the dephasing rate[42].This is the only theoretical model we know of to explain the non-monotonicityin the effective dephasing rate (extracted from the curvature of the cen-tral zone of WL magnetoresistance) as a function of parallel magnetic field,though some deviations still exist. If correct, this model can also help usextract information on the Lande´ factors of magnetic impurities.Influence of magnetic impurities on B‖-dependence of UCFdephasing rateSimilar to the WL dephasing rate, the UCF rate is also a function of basicphase-breaking processes: the UCF rate in general can be expressed as afunction of various basic dephasing rates, i.e. as τ−1UCF (1τmag, 1τeei , ...). Thereis also in-plane magnetic field dependence of the UCF dephasing rate. Thein-plane magnetic field influences τ−1UCF through its influence on τ−1mag, whichgoes into the calculation of the autocorrelation function.In order to quantitatively study the influence of in-plane magnetic fieldon τ−1UCF , we need to consider the 2d autocorrelation function. In a magnetic333.4. Phase coherent transportfield, the 2d autocorrelation function of UCF reads[7]:F (δB, δE) = 2F0[τ−1↑↑ , T ](δB, δE) + F0[τ−1↑↓ , T ](δB, δE + Ez)+ F0[τ−1↑↓ , T ](δB, δE − Ez), (3.25)where F0 is the aforementioned spinless correlation function given by UCFtheory in 2d (see equations 3.14–3.15). We can expand the terms as: τ−1↑↑ (B) =τ−10 + [1 − P (B)2]τ−1mag, τ−1↑↓ (B) = τ−10 + [1 + P (B)2]τ−1mag [7, 35]. τ−10 rep-resents the inelastic scattering rate with magnetic scattering suppressed bythe in-plane magnetic field, i.e. the dephasing rate when the in-plane fieldis large. τ−1mag represents the magnetic scattering rate, i.e. the differencebetween high and low in-plane field dephasing rates.P (B) is the magnetization function for magnetic impurities. For the sim-plest case, P (B) = tanh(gµB/(2kBT )) for classical free magnetic impurities.Magnetization functions for quantum impurities are more complicated.Given P (B), τ−1mag, and τ−10 , convolving F (δB, δE) with a thermal smear-ing factor (see equation 3.16), we can get the final 2d autocorrelation func-tion as a function of in-plane magnetic field. The cross-section at δE = 0gives the autocorrelation function on which the inflection point is defined.Therefore, we can use numerical computations to calculate τ−1UCF as a func-tion of in-plane magnetic field.The resulting in-plane dependence of the UCF rate shows a crossoverbehaviour (see Fig. 3.4). Unlike the WL dephasing rate, there is no non-monotonic behaviour or peak in the τ−1UCF (B‖). Comparing with the modeldeveloped by Kashuba, et al. for the in-plane magnetic field dependence ofthe WL dephasing rate, we can see that, in the UCF case, only polarizationphysics exists, and there is no precession physics of the magnetic impurities.343.4. Phase coherent transportFigure 3.4: The red curve is an example theoretical UCF dephasing rate vsin-plane magnetic field curve of free classical magnetic impurities, with T =1K, g = 1.5, τ0 = 22.137ns−1, τmag = 7.126ns−1 (these values are typicalvalues for fitting the theoretical dephasing rate vs in-plane field curve to thedata of exfoliated graphene, see Chapter 5). The reduction of dephasingrate at higher field is clearFigure 3.5: Influence of magnetic impurities on the temperature dependenceof the dephasing rate, the black dash-dotted curves are fits to Nagaoka-Suhlformula, solid lines are fits to numerical renormalization group calculation.These two figures are from Phys. Rev. Lett. 95, 266805[16].Influence of magnetic impurities on the temperature dependenceof the dephasing rateMagnetic impurities not only manifest themselves in the B‖ dependence ofthe dephasing rate, but also in the temperature dependence of the dephasingrate due to Kondo or RKKY coupling. Fig. 3.5 shows the influence of Kondo353.4. Phase coherent transportcoupling on the temperature dependence of the dephasing rate of Au withFe impurities. This effect comes about because the dephasing rate has acontribution from the magnetic scattering rate[7, 42], and Kondo or RKKYcoupling can influence the magnetic scattering rate[16, 25]. Kondo or RKKYcoupling can both cause a collapse in the magnetic scattering rate at lowtemperature[16, 25].The Kondo effect is a strong coupling between a magnetic moment andthe conduction electrons. In Kondo effect, the conduction electrons screenmagnetic impurities and quench their spins, hence changing the magneticscattering rate and the dephasing rate. The exact solution can be obtainedwith numerical renormalization group (NRG) calculation. It shows that thecollapse of the magnetic scattering rate occurs below the Kondo temperatureTK , following a universal line shape[16, 40]. An approximate analyticalform such as the Nagaoka-Suhl formula can also be used to model of thetemperature dependence of the magnetic scattering rate above TK [16]. TheNagaoka-Suhl formula reads[16, 38]:τ−1mag =nm2piν34pi234pi2 + ln2( TTK ), (3.26)where nm is the volume density of the magnetic impurities, ν is the density ofstates, TK is the Kondo temperature. This formula applies to 3-dimensionalsystems with a parabolic band structure and impurities with a single Kondotemperature.The Nagaoka-Suhl formula only applies for temperatures above TK , andan approximate analytical form that matches with the NRG result bothabove and below TK was also proposed[43]:τ−1mag =pinmS(S + 1)ν(ln2(TTK) + pi2S(S + 1)[(TKT)2 + 4])−1 (3.27)where S is the spin quantum number of the magnetic moment, all the otherparameters are the same as that in the Nagaoka-Suhl formula.The RKKY coupling is an indirect coupling between magnetic impurities,363.4. Phase coherent transportmediated by conduction electrons. In an ordered lattice, RKKY couplingwill lead to ferromagnetism or antiferromagnetism. In a disordered lattice,RKKY coupling frustrates magnetic moments (forming spin glass or otherspin-correlated state) and reduces their fluctuations, leading to a reducedmagnetic scattering rate[25, 44]. Unlike the case of Kondo effect, we know ofno formula describes the temperature dependence of the magnetic scatteringrate caused by RKKY coupling.To make matters even more complicated, Kondo physics and RKKYphysics can coexist and interfere with each other[43, 44]. In general, whenthe density of magnetic impurities is very low (diluted case), we can ignoreRKKY coupling between the magnetic impurities. Only Kondo screeningremains and the localized magnetic moments disappear in the T = 0Klimit. When the density of magnetic impurities is high, RKKY couplingis not negligible, and can lead to the formation of a spin-glass or othermagnetically ordered state, thus quenching the Kondo screening[43]. Evenif one coupling dominates over the other, it is still difficult to distinguishbetween the Kondo-dominated and RKKY-dominated regimes based on thedephasing rate, because they will give rise to a similar trend in the dephasingrate as a function of the temperature[16, 25, 45] (or magnetic field[7, 44]).3.4.4 Manifestation of the magnetic impurities beyond itsinfluence on the dephasing rateBesides leaving marks on the dephasing rate, in some cases magnetic im-purities can also manifest themselves in magnetoresistance (or magnetocon-ductance) directly. When magnetic impurities are dilute, WL and UCFmagnetoresistance dominate and magnetic impurities only leave marks onthe dephasing rate; when magnetic impurities are dense enough, contri-butions from magnetic scattering can be substantial to the magnetoresis-tance. If RKKY coupling dominates the interaction between the magneticmoments, hysteretic behaviour might occur. If Kondo coupling dominatesthe interaction between the magnetic moments, anomalous magnetoresis-tance will show up. Previous experiments show that magnetic impurities373.4. Phase coherent transportcan cause anomalously large negative magnetoresistance beyond the weaklocalization contribution, which can be explained with a Kondo-like spinscattering model[46–48]. There are also cases where strong negative Kondomagnetoresistance can show up in the parallel magnetoresistance[26, 49].Of course, besides the manifestation in the dephasing rate and magne-toresistance, magnetic impurities can also manifest themselves in the tem-perature dependence of the resistivity. For the RKKY-dominated case, var-ious behaviours are possible, depending on the details of magnetically or-dered state[50–53]. The most classical manifestation of Kondo coupling isthe logarithmic resistivity vs temperature dependence at low temperature.However, though this is the classical manifestation of Kondo coupling inAu/Ag doped with magnetic ions, it is not necessarily a clear evidence in 2dsystems, as other effects (weak localization and electron-electron interaction)in 2d can also lead to logarithmic resistivity vs temperature dependence atlow temperature[54–56].38Chapter 4Experimental techniquesThe experiments studied in this thesis are based on a variety of techniques.First of all, there are techniques for the sample synthesis. This thesis ex-plores mostly phase coherent measurement with very similar measurementphilosophies but carried out in very different material systems. Particu-larly, Chapter 5 describes exfoliated graphene on SiO2. Chapter 6 describesSiC graphene (and a little CVD graphene). Even though each one of thosegraphene materials contains a similar lattice of carbon atoms, each of themhas its own synthetic method, and even some peculiar features. The de-tails of each graphene type will be described in section 4.1. Given a pieceof graphene material, we employ nanofabrication techniques to add micro-electrodes onto it and shape it into a desired geometry for electrical mea-surement. The details of device fabrication are described in section 4.2.Given the graphene devices suitable for electrical measurement, the phasecoherent phenomena in graphene can only be observed at ultra-low temper-atures with weak signal detection methods. The details of the cryogenics toreach ultra-low temperature are described in section 4.3. The details of theelectrical measurement setup we use for studying phase coherent phenomenain graphene are described in section Graphene materials4.1.1 Exfoliated grapheneThe mechanical exfoliation method or ‘Scotch Tape method’ was first de-veloped by Geim’s group in Manchester. It was the first method that suc-cessfully isolated monolayer graphene. Since 2004, this method has become394.1. Graphene materialsa standard bench top method for making graphene flakes.The basic principle is simple: just put a small piece of graphite crystal onsome tape, peeling off the tape repeatedly to make small exfoliated graphiteparticles. Then, press the tape onto a SiO2 substrate and finally peel it offto leave monolayer graphene on the substrate. However, there are actuallymany tricks to making exfoliated graphene more efficiently. The yield isvery low if we just throw together some random graphite, tapes and siliconwafers (capped with a SiO2 layer). The graphite crystal, the tape, substratecleanliness, and peeling-off method—all these details matter.Previously, the average graphene size in our lab was very small (< 10micrometer). The yield of large-size graphene was extremely low. After look-ing through relevant papers and testing various parameters in the exfoliationprocedure, some tricks were introduced, and the new method successfullyimproved the size of the exfoliated graphene. The first trick is graphiteselection, which greatly affects the yield of large exfoliated graphene. Thegraphite crystal from NGS company in Germany facilitates exfoliation oflarge graphene flakes. This flaky graphite is branded as the best graphitefor exfoliation. Another trick is the choice of tape. Both blue Nitto tapeor heat release tape will work fine. They have much less residue, comparedto Scotch tape. The peeling operation also matters. In our experiment,we found that a single piece of tape should not be peeled more than 5times, otherwise the graphene flakes will be shattered into pieces, reduc-ing the probability of getting large flakes. Another trick is oxygen plasmatreatment of the substrate. A 20min oxygen plasma treatment right beforepressing tape on top of the SiO2 surface usually will greatly improve adhe-sion between the graphene flakes and substrate. Another trick is to heatthe tape/SiO2 stack on a hotplate before peeling it off. The heating willdeplete any air trapped between the graphene and the SiO2, leaving partialvacuum between the graphene and the substrate surface. The air pressureduring cooling will improve contact between the graphene and the SiO2,thus facilitating the peeling-off of few-layer graphene[57].The whole optimized exfoliation procedure is as follows. The exfoliationis started with large NGS graphite particles and pieces of blue Nitto tape.404.1. Graphene materialsMeanwhile, a piece of silicon wafer with a 285nm SiO2 layer (bathed inacetone for 5 min and IPA for another 5 min, then blow-dried with nitrogenflow) is going through oxygen plasma cleaning in a plasma etcher for 20min (oxygen flow is 15 SCCM, the power is 100W, and the pressure is200 mTorr). When the oxygen plasma cleaning is nearly finished, the tapecarrying graphite particles is peeled about 5 times. Then, the peeled tapeis pressed onto the cleaned wafer and the tape-covered wafer is put on ahotplate set at 100◦C for 2 min. Then, the wafer covered with tape is takenoff and allowed to cool in air for 5 min. After that, the tape is peeled offfrom the wafer surface. During the peeling, the tape is kept as parallel tothe surface as possible. After that, the wafer is placed under a microscopeto hunt for graphene. These steps usually can give a graphene flake largerthan 100 microns in length for one quarter of a 4 inch wafer. By the way,this method is good for graphene but not for FeSe and black phosphorus.The tricks of the exfoliation method for them can be found in Chapter 8.In addition to the yield of large-size graphene, doping from the sub-strate is another factor to consider. Lower doping is usually favored to seethe charge neutrality point in graphene during gate voltage scanning. Pre-viously, the SiO2 wafers we used were wafers from Nova corporation (drychlorinated oxide), which tend to cause high doping in exfoliated grapheneand the charge neutrality point is usually not observable for graphene ona bare Nova wafer. The dangerous RCA cleaning process (invented by the‘Radio Corporation of America’, a cleaning procedure including the usageof hydrochloric acid and hydrogen peroxide) is needed to reduce doping inSiO2, thus we can see the charge neutrality point (the gate voltage for thecharge neutrality point is usually still above 30V). The new wafers (fromSilicon Quest International, part number: 20020227, dry thermal oxide) westarted to use in recent years give much lower doping, and we can see thecharge neutrality point (the gate voltage for the charge neutrality point isbelow 10V) even on fresh SQI wafers.414.1. Graphene materialsFigure 4.1: Exfoliated graphene on SiO24.1.2 SiC grapheneThough exfoliated graphene began the era of graphene research, it is nota practical method for industrial application. The microelectronics indus-try requires materials to be produced on the scale of a whole wafer. Theinvention of SiC graphene and CVD (chemical vapor deposition) graphenemade the production of large-area graphene wafers possible. There are somepeculiarities about SiC graphene. Its structure is shown in Fig. 4.2. Com-paring with the exfoliated graphene, there is a new structure between theSiC graphene and the substrate — the buffer layer. The existence of thebuffer layer can influence the electronic properties of SiC graphene sittingon top of it[2, 11, 58].Epitaxial graphene grown on SiC (SiC/G) draws much attention. Itis more or less compatible with modern Si-based microelectronics. TheSiC graphene wafer has the standard size used in Si-based industrial pro-cessing. Many prototype devices have been developed for microwave elec-tronics, spintronics, etc. Our SiC graphene is epitaxial graphene grownon the Si face of a semi-insulating 4H-SiC crystal by the thermal sublima-tion method. The manufacturer is graphensic.com, and its SiC grapheneproduct is shipped with product documentation (with information such asRaman/AFM characterizations). Its high quality has been demonstrated in424.1. Graphene materialsa series of papers[1].Figure 4.2: Structure of epitaxial SiC graphene grown by thermal sublima-tion (from https://physics.aps.org/articles/v8/91)4.1.3 CVD grapheneAnother way to make graphene on an industrial scale is CVD (chemicalvapor deposition) synthesis of graphene. It is by far the easiest and cheapestway to make graphene in an industrial scale.The CVD process includes two steps. The first step is the decompositionof the precursor gas (compounds contain carbon atoms) to form carbon. Thesecond step is the creation of the graphene structure out of the dissociatedcarbon atoms. A catalyst is usually used to lower the reaction temperature.A standard method is to use methane gas to react with copper (copperserves as a catalyst) at about 1000◦C so that graphene layers form on thecopper foil. Many commercial CVD graphene samples grown on copperare also available these days. Since the substrate used for the synthesis ofCVD graphene is copper, the CVD graphene has to be transferred onto atarget substrate before use (usually one uses a polymer to transfer graphenefrom copper to other substrates). Our CVD graphene was purchased fromGraphenea.com. It was already transferred onto the SiO2 substrate by thecompany.434.2. NanofabricationFigure 4.3: Growth of CVD graphene (from: https://www.intechopen.com)4.2 NanofabricationIn our lab, the main tool for patterning graphene and other nano-materialsis electron beam lithography (EBL). Lithography is a major tool for pattern-ing micro/nanodevices in research as well as industry. The basic operationprinciple of lithography technology is similar to photography (as in camera):some beam-sensitive chemicals are exposed to a beam to sketch the desiredpattern; then, the exposed chemicals are developed and the patterned ge-ometry shows up. In electron beam lithography, a polymer (we usually usepolymethyl methacrylate (PMMA)) sensitive to an electron beam is spin-coated onto a sample. Then, an electron beam from a Scanning ElectronMicroscope (SEM) is used to sketch patterns on the polymer. The polymer,when exposed with a high enough electron beam dose, will change its sol-ubility, as the chains in the polymer have been cut by the electron beam.Then, the sample covered with exposed polymer will be put in developer—achemical that can selectively dissolve the exposed part of the polymer. Inmy experiments, I usually use a MIBK (methyl isobutyl ketone):IPA (iso-propyl alcohol) (1:3 volume ratio) mixture. After developing and rinsingin IPA to clean the surface, the exposed polymer will be removed, leavingthe designed trenches. Thus the polymer layer with designed trenches canserve as a stencil mask for further processing. For making electrodes, thenext step is to deposit metal into these trenches, then, to do liftoff in hotacetone to remove the remaining polymer layer and the metals on top of it,leaving only the metal in the trenches to form the electrodes. For etching444.2. Nanofabricationoperations, the sample will be put into a plasma etcher for plasma etching,then the exposed nano-materials in the trenches will be moved away. Thuswe shape the sample into the designed geometry.In the fabrication of our graphene devices, a two-layer polymer (PMMAA2 and PMMA C4, MicroChem Corp.) is spin-coated onto the sample, withspin-speed 5000 rpm, the spinning time is 50s. After spin-coating with A2,baking at 180◦C for 10 min is carried out to dry the polymer, then, afterspin-coating with C4, baking at 180◦C for 10 min is carried out to hardenthe polymer. This combination will give the polymer layer an undercut, thusfacilitating lift-off of metals after depositing thick metals into the patternedtrenches.The pattern is designed in KLayout software and transferred into suit-able format for NPGS (Nano Patterning Graphic System) with the help ofdesignCAD software. NPGS takes the designed patterns and generates anexecutable run file for driving the electron beam of the scanning electron mi-croscope (ZEISS SmartSEM) to expose the patterns. For precise patterning,some alignment marks will be made first with electron beam lithography.After that, the patterning of the device can proceed with the help of thealignment marks. NPGS will acquire imaging of the alignment marks andmatching it with the designed geometry to calculate calibration parametersfor beam control. Then, the desired pattern will be written with the electronbeam. The exposure dose is 330 µC/cm2. This is the appropriate dose, asdetermined by an exposure test.For the beam intensity, for patterning small devices, a standard 30 µmaperture is used. For patterning big devices, a 60 µm or a 120 µm apertureis used. Large apertures provide a much stronger beam and reduce the timefor patterning. The working distance is also adjusted accordingly.After patterning, the exposed bilayer polymer stack is put in MIBK:IPAdeveloper to develop for 3.5 min, and rinsed in IPA for 1.5 min. TheMIBK:IPA developer produces fine features and a relatively clean surface.The exposed PMMA stack is removed and we have the stencil mask carryingthe desired patterns for further processing.Then, for making electrodes, the sample, covered with the PMMA sten-454.3. Cryogenicscil mask, is loaded into either a thermal evaporator or an ebeam sputter-ing evaporator. Carefully-chosen metal combinations are deposited to formelectrodes. For electrodes on graphene devices, we use 4nm Ti/91nm Au or40nm Pd.For shaping Hall bars, the sample covered with the PMMA stack stencilmask is loaded into a plasma etcher. Oxygen plasma is generated by the100W RF generator (Model: Plasma-Etch PE-50). The pressure is kept at250 mTorr, the flow is 25 SCCM, and the processing time is ∼3min. Afterthe plasma etching process, the graphene in the trenches of the PMMA stackstencil mask is removed, leaving a Hall bar-shaped graphene piece.The standard EBL process mentioned above is suitable for making elec-trodes on samples that go well with PMMA and the heating process. Forsamples sensitive to chemicals and heat (such as FeSe), new ways must bedeveloped for making electrodes. I developed two other ways to make elec-trodes: transfer onto pre-defined electrodes, and shadow mask evaporation(with a metal shadow mask made by a Focused Ion Beam microscope).These methods will be described in detail in the FeSe chapter.(a) (b)Figure 4.4: (a)ZEISS SmartSEM; (b)One of our exfoliated graphene Hallbars464.3. CryogenicsFigure 4.5: Experiment setup4.3 CryogenicsThe low temperature and magnetic field is provided by an Oxford Instru-ments Kelvinox MX250 He-3:He-4 dilution refrigerator and superconductingmagnet system. The sample is mounted on the socket at the end of thefridge. After mounting the sample, vacuum cans are loaded, and the fridgeis pumped. Then, the whole assembly is loaded into a dewar filled withliquid Helium (∼4.2 K). The dewar has a large superconducting magnet toprovide magnetic field parallel to the sample plane, with magnetic field upto 12 Tesla. On the fridge itself, a smaller superconducting magnet is alsomounted. This small magnet can provide a magnetic field perpendicular to474.3. Cryogenicsthe sample plane, with magnetic field up to ∼200 mT. This small magnetis the one we use for measuring weak localization magnetoresistance. Thissmall magnet can also correct the alignment of the big parallel magneticfield by cancelling out its perpendicular component.After activating circulation of the He-3:He-4 mixture, ultra-low tem-peratures down to ∼20 mK can be realized. To prevent heating from theenvironment, layers of radiation shields are used. Since electrical measure-ments must be done through wires connected to the samples, preventingheat from sneaking down the wires to heat up the sample is an importantconsideration. The heating effect of electrical noise is especially strong atsuch low temperatures, thus a few carefully-designed filters are introducedto block heating effects from various electrical noise sources.The thermometry is realized with a vendor-supplied RuO2 thermometer.The temperature control is realized with the heater mounted on the mixingchamber.Maintaining the fridge’s operation is not an easy job. Since helium isprecious, we have to recover helium boil-off. The recovery involves a com-plicated recovery system. Helium boil-off is first collected by a bag, then,a compressor compresses the helium gas into high-pressure bottles. Then,the helium in pressured bottles is sent to a purifier to remove moisture fromthe helium. After that, the helium is sent to a liquefier to be convertedinto liquid helium. Due to the capacity of the purifier, it saturates everyfew days and has to be regenerated. Many transducers are used to monitorthe condition of the system. The condition of the purifier is monitored byan oxygen meter. The pressure of the helium gas is monitored by pressuretransducers. The helium level in the liquefier and our dewars are monitoredby level meters. To free us of the heavy duties of monitoring the health ofthe recovery system and keep log of the amount of helium, I developed thefirst Arduino-Zigbee system used in our lab. The details of this system canbe found in the appendix B.484.4. Electrical measurement4.4 Electrical measurementFor the electrical measurement, in general, there are two kinds of measure-ment modes: current bias mode and voltage bias mode. In current biasmode, a big resistor (bias resistor) is connected in series with the sample formaintaining a constant bias current. Resistance can be read out by divid-ing the measured voltage across the sample by the current. This methodis suitable for the case where the sample’s resistance is much smaller thanthe bias resistor. In voltage bias mode, a voltage divider is used to applya constant voltage across the sample, the current passing through the sam-ple is measured directly, and thus the resistance can be obtained. Voltagebias mode is suitable when the sample resistance is much larger than thevoltage source segment of the voltage divider. For the current bias mode, afour-terminal measurement is usually used to remove the contact resistance,where the two voltage probes directly measure the voltage drops across thesample, thus actual sample resistance can be measured directly.The experiments in this thesis always use current bias mode. We usefour-terminal measurement on Hall bar-shaped samples to get resistivity.SRS (Stanford Research Systems) lock-in amplifiers are used to performlow-frequency lock-in measurements to measure the voltage drop across oursample (a preamplifier is usually used to further improve signal-noise-ratio).A big resistor with resistance R is connected in series with the sample. Thusthe current passing through the sample is constant, set by V/R, where Vis the excitation voltage from the lock-in amplifiers. Low bias current isusually used to avoid heating the samples.The carrier density for graphene devices with back gates can be con-trolled with back gate voltages. The graphene sample, SiO2 dielectric, andthe conductive Si form a capacitor. A voltage difference is applied betweenthe graphene sample and the conductive Si base to charge up the graphene.The SiO2 dielectric in our wafer is about 285nm thick, and can withstandvoltage up to ∼90V. We use a K2400 source-meter to provide the back gatevoltage. For SiC graphene, there is no back gate to tune the carrier density.The carrier density is measured through Hall voltage probes on our Hall494.4. Electrical measurementFigure 4.6: 4-terminal measurement in current bias modebar samples. The basic principle is the Hall effect: the magnetic field perpen-dicular to the sample exerts a Lorentz force on the charge carriers, leadingto the build-up of charges on the two sides of the Hall bar, inducing a volt-age drop between the opposite voltage probes on the Hall bar. The carrierdensity can be extracted from the slope of the Hall-voltage plotted againstthe perpendicular magnetic field.We use the Igor Pro software to automate the measurements. Igor scriptsare used to control back gate voltages, magnetic field, fridge temperatures,as well as collecting data from lock-in amplifiers.50Chapter 5Experiment 1: phasecoherence in exfoliatedgrapheneWhen I started my PhD study in 2012, several aspects of the decoherenceand spin relaxation in graphene were not yet understood. Weak localizationexperiments had been showing a saturation in the electron phase coherencetime at low temperature[59], and spin current experiments were detectingsignificant spin relaxation[8, 9]. The spin relaxation has significant effects onweak localization, thus it was suspected that there was likely one commoncause for both decoherence and spin relaxation: either spin-orbit interac-tions, or magnetic defects.One possible cause of the saturated coherence is spin-orbit interaction.Spin-orbit interaction usually will cause weak antilocalization if it is strongenough, but spin-induced antilocalization was never observed in graphene.There are other manifestations of spin-orbit interactions; spin-orbit inter-actions (though elastic) can mimic decoherence in weak localization if theyonly couple to the out-of-plane spin component[41]. The intrinsic spin-orbitterm in graphene is of this type, but it is expected to be extremely weak[6].Another possible cause of the coherence saturation in graphene is mag-netic impurities. Over a decade ago, the electron coherence time was ob-served to saturate in pure metals such as Cu, Ag and Au[18]. It was even-tually found that very dilute magnetic impurities (e.g., Fe or Mn) wereresponsible for the coherence saturation in these metals. A similar mech-anism could also exist in graphene, as graphene could possess a more gen-51Chapter 5. Experiment 1: phase coherence in exfoliated grapheneeral kind of magnetic defect such as an unpaired spin, or some other formof odd-electron localized state. The quantum nature (such as Kondo orRKKY coupling of the magnetic impurities) of these magnetic impuritieshas been studied intensively in theory[10, 12, 13], but few experiments arerelevant[14].We decided to solve this issue of spin-orbit vs magnetic defects by reex-amining UCF, for which the dephasing rate is not influenced by the ripples(WL dephasing rate is influenced by the ripples[60]) and the spin-orbit in-teractions cannot mimic decoherence[33] (the spin-orbit interaction only re-duces the magnitude of UCF but does not distort the autocorrelation func-tion). The experiment described in this chapter provides the first phasecoherent transport evidence that magnetic impurities exist in exfoliatedgraphene. In the experiment, both weak localization and universal con-ductance fluctuation were employed to extract phase coherent informationin graphene. The temperature dependence of dephasing rates extracted withUCF and WL were consistent with each other. A large in-plane field wasused to polarize possible magnetic impurities in the sample. The influenceof in-plane magnetic field on the dephasing rate revealed the existence ofmagnetic impurities. The crossover behaviour in the in-plane magnetic fielddependence of the UCF rate provided clear evidence of the existence of mag-netic impurities, thus we solved this issue of spin-orbit vs magnetic defects.Moreover, the in-plane magnetic field dependence of the WL rate, thoughobscured by the dephasing background from the ripples, also showed hintsof non-monotonic behaviour, which was a mystery during the time this ex-periment was performed. In the following years, our experiments on SiC andCVD graphene as well as the development of a recent model by theorists[42]make us understand that this non-monotonic behaviour is further evidenceof the existence of magnetic impurities (see Chapter 6).Though we revealed the existence of magnetic impurities in exfoliatedgraphene. The quantum nature of these magnetic impurities was not mani-fested clearly in this experiment — the data can be explained with classicalspins, no Kondo or RKKY coupling (which will occur on quantum magneticimpurities) was clearly observed. Later on, in our experiments on SiC and525.1. Experiment setupCVD graphene, we found clearer evidence of the quantum nature of themagnetic impurities (see Chapter 6).Figure 5.1: A photomicrograph of the sample studied in this chapter. Theyellow lines are the electrodes. The purple flake surrounded by the electrodesis the graphene flake.5.1 Experiment setupThe graphene device (see Fig. 5.1) used here was made by Dr. Mark Lun-deberg. Conventional electron beam lithography was used to shape thegraphene into a 4-terminal device and to define the electrodes. The aspectratio is about 1.04. The substrate is a doped silicon wafer with a layer of275nm thick SiO2. The carrier density is about 4 × 1012cm−2 and the dif-fusion constant D ∼ 0.03m2/s. The basic experiment setup is described inChapter 4. The measurement was performed in 4 terminal geometry (seeFig. 5.2(a)). The measurement was performed with SRS830 lockin ampli-fiers, the measurement frequency is 870Hz. A relatively large bias currentwas used to improve the signal-noise-ratio in UCF measurement. In the mea-surements below 300mK, the bias current is ≤ 20nA; for measurements athigher temperature, bias current up to 45nA was used. Taking the bias heat-ing into consideration (based on Wiedemann-Franz law), the actual devicetemperature is approximately T =√T 2cryostat + T2bias, where Tbias =√3eVxx2pikB,and Vxx is the voltage across the sample[7]. In the measurement, resistanceas a function of magnetic field was measured in a small gate voltage window.535.2. Dephasing rates extracted from UCF and WLThus the data structure is a series of 2d scan matrices with different tem-perature or in-plane magnetic field for each 2d scan. Each 2d scan has twoaxes: a gate voltage axis and a magnetic field axis. At each gate voltage, themagnetoresistance trace looks like a random trace. Calculating autocorre-lation for each magnetoresistance trace (after subtracting the slow-varyingbackground conductance from the 2d scan) and averaging over gate voltages,we can get the UCF rate from the inflection point of the averaged autocor-relation function (as explained in Chapter 3, see Fig. 3.3). Averaging themagnetoresistance traces over the gate voltages, we can get the WL trace.The WL dephasing rate was obtained from the curvature around the peakof the WL magnetoresistance trace.5.2 Dephasing rates extracted from UCF andWLAs described in Chapter 3, UCF in magnetoresistance traces contains in-formation of the dephasing rate. In this experiment, we measured conduc-tance fluctuations from ∼ 50mT to ∼ 150mT (see Fig. 5.2(c)), for severalback-gate voltages in a narrow gate voltage window. After subtracting thebackground conductance from the 2d scan, the ensemble-averaged autocor-relation function was obtained by calculating autocorrelation functions ofthe conductance fluctuation traces at each gate voltage and averaging overall these autocorrelation functions (see Fig. 5.2(d)). The inflection pointof the ensemble-averaged autocorrelation function was used to extract thedephasing rate (see Fig. 5.2(d)): τ−1φ ≈ 2eDBIP3~ , where d2fdδB2|δB=BIP = 0,and D = 0.03m2/s is the diffusion constant.After turning on the in-plane magnetic field, the extracted UCF dephas-ing rate decreases (see Fig. 5.3). The in-plane magnetic field dependenceof the dephasing rate reveals an interesting crossover from low field to highfield dephasing rates. And the crossover region shifts to higher fields athigher temperatures. This is clear evidence of the existence of magneticimpurities (see Chapter 3). Quantitatively, the decreased amount of the545.2. Dephasing rates extracted from UCF and WLFigure 5.2: (a) Illustration of the 4-terminal measurement performed on thedevice studied in this chapter. (b)Conductance as a function of back gatevoltage. (c)An example UCF trace (d)Averaged autocorrelation functionof the UCF trace and the inflection point (indicated by BIP ). From Phys.Rev. Lett. 110, 156601 (2013)dephasing rate is about 5ns−1 at 0V gate voltage, giving an estimate ofthe scattering rate from polarizable magnetic impurities. Fig. 5.3 showsthe experimental and theoretical in-plane magnetic field dependence of thedephasing rates extracted with UCF analysis. The crossover behaviour isconsistent with the thermodynamics of free magnetic moments (see Chapter3). The theoretical curves calculated from classical spin 1/2 defects ap-proximately match the observed crossover. The dashed and solid lines aretheoretical curves calculated based on the magnetization function (Curie’slaw, P (B) = tanh(gµB/2kBT )) of classical spinful impurities. However, wecan see that below 200mK, the theoretical lines from the magnetization func-tion of classical spinful impurities are not good fit to the data anymore. Thedeviation from classical spin polarization below 200mK might come from thequantum nature of the magnetic impurities. For quantum magnetic impuri-ties, either Kondo coupling or RKKY coupling can lead to a magnetizationfunction different from that of classical magnetic impurities[26, 49].555.2. Dephasing rates extracted from UCF and WLFigure 5.3: UCF dephasing rate vs in-plane magnetic field of the exfoliatedgraphene sample studied in this chapter: solid symbols are experimentaldata, solid lines are simulations assuming classical magnetic impurities withg=2, dashed lines are for g=1. From Phys. Rev. Lett. 110, 156601 (2013)Figure 5.4: Averaging to get WL signal, the example shows averaging over agate voltage window spanning from -2.5V to 2.5V, the temperature is about70mKAnalogous measurements were performed for WL. The WL trace was ob-tained by averaging over magnetoresistance traces at several gate voltages ina narrow gate voltage window (see Fig. 5.4). The extracted Bi and B∗ aremuch larger than Bφ (at 1K, Bi=0.94mT, B∗=37mT and Bφ=0.17mT, and565.2. Dephasing rates extracted from UCF and WLLφ ∼ 1µm), as expected from the weak localization regime of graphene (seeChapter 3). The dephasing rate was extracted from the curvature of the av-eraged WL magnetoconductance trace — τ−1WL =eD~ (3pi4he2LWd2GdB2⊥|B⊥=0)−12 ;it provides a reliable metric of the dephasing even when there are magneticscatterers (see Chapter 3). The temperature dependence of the WL dephas-ing rate (as well as the UCF rate) is linear (see Fig. 5.5(b)), consistent withthe dephasing caused by electron-electron-interactions. After turning on thein-plane magnetic field, a decrease of the WL dephasing rate caused by thepolarization of magnetic impurities (similar to the decreasing in UCF ratevs B‖) was also observed in a range of the magnetic field (see Fig. 5.5(c)).However, comparing with the monotonic B‖ dependence of the UCF de-phasing rate (UCF rate always decreases asB‖ increases), new behaviour wasobserved at very low magnetic field and very high magnetic field for the WLrate (see Fig. 5.5(c)). There are three stages as the in-plane magnetic fieldincreases: in the first stage (for 0mT ≤ B‖ ≤ 50mT ), the WL dephasing rateincreases; in the second stage (for 50mT ≤ B‖ ≤ 400mT ), the WL dephas-ing rate decreases; in the third stage (for B‖ ≥ 400mT ), the dephasing rateincreases again (see Fig. 5.5(c)). The third stage (B‖ ≥ 400mT ) is causedby the additional dephasing rate caused by the ripples in graphene[60]. The2nd stage (50mT ≤ B‖ ≤ 400mT ) of the B‖-dependence of the WL dephas-ing rate was immediately understood as being caused by the polarizationof the magnetic impurities. The 2nd stage was predicted (but deemed noteasy to observe) by a theoretical paper published by Glazman[35]. However,the initial increase of the dephasing rate (1st stage, 0mT ≤ B‖ ≤ 50mT )was a mystery during the time this experiment was finished and it was alsoconfirmed in our follow-up experiments in exfoliated bilayer graphene (un-published). It was later understood much better after detailed experimentsperformed on SiC and CVD graphene samples (see Chapter 6). The initialstage is related to precession physics of the magnetic impurities[42], not po-larization physics (which I will explain in detail in Chapter 6). It is clearevidence of the existence of magnetic impurities with the Lande´ g factordifferent from that of electron. In an ideal graphene sample with much lessrippling, we expect to see a clear trend caused by the precession and polar-575.2. Dephasing rates extracted from UCF and WLization of the magnetic impurities, without additional background causedby the ripple effect.Figure 5.5: WL dephasing rate vs in-plane magnetic field of the exfoliatedgraphene sample studied in this chapter, from Phys. Rev. Lett. 110, 156601(2013)Let us consider why it would be that UCF and WL dephasing rates havesuch different behaviours. We can see that the dephasing rates extractedwith UCF and WL have different behaviours in the in-plane magnetic field(comparing Fig. 5.3 and Fig. 5.5(c)). The difference in the behaviours ofUCF and WL dephasing rates is probably rooted in the measurements. Inthe WL measurement, only the large B‖ is used as a polarization field, thepolarization effect of the small B⊥ is negligible. Under the influence of thislarge B‖, the subtle 1st stage increase of the WL dephasing rate will besuppressed at B‖ ≤∼ 50mT (see the inset of Fig. 5.5(c)). However, in theUCF case, the measurement was always performed at large field (B⊥ scanwas from ∼ 50mT to ∼ 150mT , to avoid interference from the WL signalnear B⊥ = 0mT ). Thus the subtle initial increase of the dephasing rate hadalready been suppressed due to the large magnetic field.Although it is not a primary focus of this experiment, we also checked the585.3. DiscussionFigure 5.6: Gate-dependence of the crossover in the UCF dephasing rateof the exfoliated graphene sample studied in this chapter, from Phys. Rev.Lett. 110, 156601 (2013)back-gate voltage or carrier-density dependence of magnetic scattering rate.Fig. 5.6 shows the crossover curves in the in-plane magnetic field dependenceof the extracted UCF rates at different gate voltages. We can see that thedifferences between the low field and high field rates can be tuned by thecarrier density. The magnetic scattering rate is the difference between thelow field and high field rates. The inset shows the magnetic scattering rateas a function of the carrier density. It seems to be proportional to the carrierdensity, higher carrier density leads to higher magnetic scattering rates. Thisis consistent with the theoretical expectation that magnetic scattering rateis proportional to the product of the density of states (which is proportionalto the carrier density) and the magnetic coupling constant[1].5.3 DiscussionIn this experiment, we saw the existence of magnetic impurities in exfoli-ated graphene, by inspecting the in-plane magnetic field dependence of theUCF and WL dephasing rates. The importance of this experiment can besummarized in the following three points:First of all, this experiment was the first experiment clearly demonstrat-ing that magnetic impurities instead of spin-orbit interaction are the cause595.3. Discussionof the fast spin relaxation in graphene. Though now this is a well-acceptedfact, when this paper came out, it was an unresolved question. The keyto this observation came down to the use of UCF (which is not affected byspin-orbit interaction or the ripples, but is affected by magnetic impurities);the suppression of UCF rate at strong parallel magnetic field is a directmanifestation of the magnetic impurities.Secondly, this experiment observed for the first time the non-monotonicdependence of the WL dephasing rate as a function of a parallel magneticfield, which was not a focus for this experiment (because we did not un-derstand it at the time) but turned into a topic for the following experi-ments. The ripple-caused enhancement of the dephasing rate in strong B‖ ingraphene was revealed many years ago[60], but nobody inspected the low B‖region carefully enough to observe the interesting non-monotonic behaviour!As far as we know, we are the first group to reveal this non-monotonic be-haviour in the dephasing rate. The decreasing-before-increasing part of theB‖-dependence of the dephasing rate (2nd stage , 50mT ≤ B‖ ≤ 400mT )was immediately understood with the interplay between polarization physicsand ripple physics. It was predicted by Glazman[35], though it was regardedas not easy to observe in experiments. However, the initial increasing de-phasing rate (1st stage, 0mT ≤ B‖ ≤ 50mT ) was a total mystery duringthe time we finished this experiment; nobody expected such behvaiour at all(this behaviour was also confirmed in our follow-up experiments in exfoliatedbilayer graphene (unpublished)). The rippling effect also obscured us fromstudying it in details. Further understanding of this phenomenon was ac-complished in the following years. In the following chapters, this behaviouris investigated in detail with SiC and CVD graphene and is explained withthe model developed in reference [42].Thirdly, it is interesting to note that this experiment did not see Kondoor RKKY couplings of the magnetic impurities, though spins are supposedto be quantum objects. The crossover behaviour of UCF rate as a functionof the in-plane magnetic field can be explained with classical spins (see Fig.5.3). Considering the quantum nature of the magnetic impurities, it is quitea surprise that simple classical magnetic impurities model can give a not-so-605.3. Discussionbad explanation. There might be some elusive signs pointing to quantummagnetic impurities: the deviation of our model based on the classical spinsfrom the B‖-dependence of the UCF rate below 200mK (see Fig. 5.3)may be a sign of the quantum nature of the magnetic impurities, sinceeither Kondo or RKKY coupling can lead to polarization curve differentfrom that of classical spinful impurities. Clearer evidence of the quantumnature of magnetic impurities was later found in other graphene samples,see the chapters below.61Chapter 6Experiment 2: magneticimpurities in SiC grapheneExfoliated graphene is just a beginning of the graphene research. Among themany graphene materials that have been invented, SiC graphene is suitablefor industrial-scale wafer production, and has the potential to become thenext “Silicon”. Many prototype devices such as the microwave transistor andspintronic devices[9, 61], have already been developed on the basis of SiCgraphene. The range of behaviours, from the usual semi-metallic grapheneto the semiconducting graphene, demonstrates the polymorphism of SiCgraphene. Especially, the discovery of semiconducting SiC graphene showsthat the SiC graphene could be useful for the fabrication of conventionaltransistors[2].The early phase coherent transport in SiC graphene showed that there isadditional decoherence, which is regarded as indirect evidence of scatteringfrom magnetic impurities[62]. The dephasing time in SiC graphene is alsoorders of magnitude shorter than that of exfoliated graphene[62]. Clarifyingwhether such a limit on the coherence time is the result of the magneticimpurities is a direct motif for our study of the phase coherent transportin SiC graphene. In addition, magnetic impurities are quantum objects.In many cases, instead of behaving like classical free magnetic moments,they tend to couple with electrons through Kondo coupling or RKKY cou-pling. On the theoretical side, many papers predicted Kondo[10] and RKKYcouplings[11–13] for magnetic impurities in graphene. On the experimentalside, recent experiments already showed that there are magnetic impuritiesin graphene[1, 7, 58, 63], which can explain the unexpectedly fast spin relax-62Chapter 6. Experiment 2: magnetic impurities in SiC grapheneation observed in graphene[1, 7, 8]. A question arises naturally: could anyKondo or RKKY coupling be seen in the magnetic impurities of graphene?So far, only a few indirect pieces of experimental evidence exist[14]; moreexperiments are still needed. This question can be addressed by checkingthe influence of the magnetic impurities on the magnetoresistance (or mag-netoconductance) and the dephasing rate.This chapter is inspired by a 2015 paper of Dr. Samuel Lara-Avila[1].That paper was mainly focused on the phase coherent transport in the hightemperature region and proved the existence of magnetic impurities by thein-plane magnetic field dependence of the dephasing rate. We explored thelower temperature region and found more interesting phenomena relatedto the quantum coupling of the magnetic impurities. The experiment intro-duced in this chapter is a detailed investigation into the magnetic impuritiesand their quantum nature in SiC graphene using magnetotransport. The ex-istence of magnetic impurities in SiC graphene can be again unambiguouslyrevealed from the B‖ dependence of the dephasing rate extracted from theweak localization effect. In addition, an interesting collapse temperature de-pendence of the dephasing rate was also observed. This is a sign of freezingor quenching of the magnetic impurities, because only dynamic magneticimpurities can dephase electrons[40]. The freezing or quenching mechanismmay be related to Kondo or RKKY coupling.Moreover, our measurements also show that magnetic impurities mani-fest themselves not only in the dephasing rate, but also in the magnetoresis-tance (or magnetoconductance) itself. The perpendicular magnetoconduc-tance in 2d systems has contributions from the weak localization as well asthe magnetic scatterers. When magnetic impurities are dilute, the weak lo-calization magnetoconductance dominates and the magnetic impurities onlymanifest themselves in the dephasing rate; when the magnetic impuritiesare dense enough, the magnetoconductance contributions from the magneticscattering can be substantial. Previous experiments from other groups showcases where magnetic impurities cause an anomalously large magnetocon-ductance beyond the weak localization contribution, which can be explainedwith a spin scattering effect such as the Khosla-Fisher model[46–48]. Sim-636.1. Experiment setupilar anomalous magnetoconductance was also observed in our experiments.The B⊥ magnetoconductance above 300mK can be well captured by theweak localization model and there is no B‖ magnetoresistance. However, attemperatures below 300mK, the B⊥ magnetoconductance is too strong tobe completely accounted for by the conventional weak localization (but itcan be killed by a strong in-plane magnetic field, thus restoring the usualweak localization in a strong in-plane magnetic field), which may indicatedirect effects of the spin scattering[26, 46, 47, 49]. A prominent negativein-plane magnetoresistance was also observed below ∼200mK, which couldalso come from the spin scattering effect[26, 47, 64].6.1 Experiment setupFigure 6.1: SiC graphene sample, the red part represents a graphene Hallbar, yellow parts are electrodesSamples in this experiment are large area (600µm long, 50µm wide) SiCgraphene Hall bars fabricated using standard electron beam lithography.The epitaxial 4H-SiC graphene wafers were purchased from Graphenesic.They are the same kind of SiC graphene used for the measurements in aprevious paper by Dr. Lara-Avila, et al.[1]. The graphene layer was grownby sublimation on 4H-SiC. In the samples used in Dr. Lara-Avila’s exper-iment, an additional polymer layer was spin-coated onto the graphene for646.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impuritiesthe photochemical doping, but no such layer was deposited onto the sam-ples for this experiment (pure SiC graphene layers in our samples). Electronbeam lithograph and plasma etching were used to shape the Hall bars. Af-ter shaping the graphene into Hall bars, another electron beam lithographystep was used to define the pattern of electrodes. Then, 4nm Ti and 90nmAu were deposited using an ebeam evaporator to form the electrodes. TwoHall bars (hb1 and hb2) have been measured in detail. The measurementswere performed in a dilution refrigerator with a two-axis magnet for inde-pendent control of the out-of-plane and in-plane magnetic fields B⊥ andB‖ (see Chapters 4 and 5). Resistance vs B⊥ traces were measured in afour-terminal configuration at different in-plane magnetic fields and tem-peratures. SRS830 lockin amplifiers were used to perform the measurement,the bias current is set at 0.35nA to avoid heating. The carrier density isabout 7.0 × 1016/m2, the diffusion constant is about 0.013m2/s, and Lφ isabout 330nm at 500mK for B‖ = 0mT . The data from Hall bars hb1 andhb2 are similar, in this chapter, we only present the data from hb2 as arepresentative.There are some advantages of the SiC graphene samples over our previousexfoliated graphene samples. The large size eliminates the interference fromUCF, and gives a clear weak localization correction. The greater flatness ofthe SiC graphene also eliminates the dephasing caused by the ripples. Thesetwo advantages make SiC graphene an ideal system for studying in detail thenon-monotonic in-plane magnetic field dependence of the dephasing rate.6.2 Magnetic impurities in SiC graphene andsigns of freezing magnetic impuritiesFig. 6.2 shows the perpendicular magnetoconductance and parallel magne-toresistance traces at different temperatures. At temperatures higher than300mK, the magnetoconductivity traces shown in Fig. 6.2a can be well cap-tured by the WL model: ∆σ(B)WL =e2pih [F (BBφ)−F ( BBφ+2Bi )−2F ( BBφ+B∗ )],where F (z) = ln z + ψ(12 +1z ), Bφ,i,∗ =~4eDτφ,i,∗ (where τφ,i,∗ represent de-656.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impurities(a) (b)Figure 6.2: (a)Perpendicular magnetoconductivity. Blue lines and circles arethe data with a big magnetic field range (∼ −120mT to∼ 120mT ). The dataat 16mK only has a few points, at each point, we read for a long time periodto check the temporal stability. No strong temporal change was found.Yellow lines are fits to the weak localization magnetoconductivity plus thespin-scattering magnetoconductivity from the Khosla-Fisher model[46]. Redlines are fits to the WL formula only. The black, green and cyan lines arefor B‖ = 9mT and T=100mK, but with a smaller magnetic field range. Theblack line is the data for B‖ = 9mT and T=100mK. The green line is thefit to the weak localization magnetoconductivity plus the spin-scatteringmagnetoconductivity from the Khosla-Fisher model[46]. The cyan line isthe fit to the WL formula only;(b)Parallel magnetoresistance at differenttemperaturesphasing time, inter-valley scattering time and intra-valley scattering time,respectively). However, at temperatures lower than 300mK, an extra magne-toconductance background beyond WL emerged, which could come from theKondo magnetoconductance[26, 49], could also come from a spin scatteringmodel developed by Khosla et al.[46, 47]: ∆σspin−scattering = a2 ln(1+b2B2)(a2 and b2 are the fitting parameters[46, 47]). In Fig. 6.2a, the yel-low and green lines below 300mK are fits with both WL and the Khosla-Fisher contributions to the magnetoconductivity taken into consideration:666.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impurities∆σ(B⊥) = ∆σWL + ∆σspin−scattering. The valley around B⊥ = 0 is dom-inated by the WL[1, 65] and the dephasing rates were extracted by fittingthe WL model to this region (red and cyan lines). Fig. 6.2b focuses onthe in-plane magnetic field dependence of the resistivity. In 6.2b, for eachin-plane field, the resistivity is the value at the center (B⊥ = 0) of theweak localization magnetoresistance curve. Above ∼200mK, the in-planemagnetoresistance is quite flat. However, a prominent negative in-planemagnetoresistance also showed up below ∼200mK both in the low and highparallel field (see Fig. 6.2b). This prominent negative magnetoresistanceis not caused by the weak localization. Because weak localization can onlylead to positive in-plane magnetoresistance; when the in-plane magneticfield increases, it will lead to enhanced phase coherence and thus a strongerweak localization correction; therefore the resistivity will be larger at higherin-plane field. The origin of this negative in-plane magnetoresistance couldbe spin scattering, as negative in-plane magnetoresistance in 2d systems isusually caused by the suppression of spin-scattering in a parallel magneticfield[26, 47, 64].The dephasing rate for the weak localization effect can be captured fromthe central part of the WL magnetoconductivity (see Chapter 3); we ex-tracted all the dephasing rates in this experiment from the central regionof the WL magnetoconductivity. The extracted dephasing rate is a combi-nation of different phase relaxation processes in the system[1, 7, 42], suchas the electron-electron inelastic scattering, magnetic scattering, etc. Themagnetic scattering rate depends on the polarization and precession of themagnetic impurities, thus in-plane magnetic field can influence the dephas-ing rate. The theoretical calculation of the influence of the magnetic field onthe dephasing rate depends on the dimension of the system and the type ofphase coherent phenomenon[35, 38]. A model (by Kashuba, et al.) has beendeveloped to quantitatively describe theB‖ dependence of the WL dephasingrate in 2d systems caused by precessing classical magnetic impurities[1, 42],see section 3.4.3 of Chapter 3.Fig. 6.3 shows the B‖ dependence of the dephasing rate. The open circlesare the experimental data. A clear non-monotonic dependence is observed.676.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impuritiesAs the in-plane magnetic field increases, the dephasing rate first increasesuntil reaching a peak value, then, the dephasing rate decreases. According tothe model developed in references [1, 42], the non-monotonic B‖ dependenceof the dephasing rate proves the existence of the magnetic impurities andthe non-monotonic behaviour at low magnetic fields can be explained withthe magnetic moments precessing with Larmor frequencies different fromthat of the electrons (due to different Lande´ g factors). The red curvesare the fits to this model (section 3.4.3 of Chapter 3). There are threefitting parameters in the model described in section 3.4.3 of Chapter 3: themagnetic scattering rate τs, the Lande´ factor gi of the magnetic impuritiesand a residual dephasing rate. We also added a common linear backgroundproportional to B‖ (∼ αB‖, α denotes the slope) into the model. In thefits (red lines) shown in Fig. 6.3, the slope of the linear background is acommon parameter and the other three fitting parameters (the magneticscattering rate τs, the Lande´ factor gi and a residual dephasing rate) gofree. The extracted slope α is 4.16 ns−1mT−1; the magnetic scattering rateτs and Lande´ factor gi are indicated in the inset of Fig. 6.3. The extractedLande´ factor (∼ −1.5 at 500mK) is quite different from that of the freeelectron; it is possible that the magnetic moments located in the bufferlayer of SiC graphene could be the source. The buffer layer was revealed bya few previous studies on SiC graphene to be highly paramagnetic at hightemperature, containing lots of localized magnetic moments[58, 63] and theg factor of these localized magnetic moments is believed to differ from thatof the free electron (though the g factor obtained previously is not the sameas ours)[58, 63].Besides manifesting themselves in the non-monotonic B‖ dependence ofthe dephasing rate, magnetic impurities could also manifest themselves inthe temperature dependence of the dephasing rate via Kondo or RKKYcoupling. The famous Kondo effect is a strong coupling between a magneticmoment and the conduction electrons. In the Kondo effect, the conductionelectrons screen the magnetic impurities and ‘dissolve’ their spins, hencechanging the magnetic scattering rate and the dephasing rate. The exactsolution can be obtained with the numerical renormalization group (NRG)686.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impuritiesFigure 6.3: In-plane magnetic field dependence of the dephasing rates, opencircles are data, red curves are fits. Inset shows the temperature dependenceof the magnetic scattering rate and the Lande´ g factor extracted from thefitscalculation. It shows that, the collapse temperature dependence of the de-phasing rate can occur around the Kondo temperature TK and a saturatingdephasing rate will occur if TK is very low[16]. The RKKY coupling is anindirect coupling between the magnetic impurities, mediated by the conduc-tion electrons. It frustrates the magnetic moments and makes them static,leading to the collapse dephasing rate. Kondo physics and RKKY physicscan coexist and interfere with each other[43, 44]. In general, when the den-sity of the magnetic impurities is very low (diluted case), we can ignore thecouplings between the magnetic impurities, and only Kondo coupling exists.When the density of magnetic impurities is high, the RKKY coupling isnot negligible, leading to the formation of spin-glass or other magneticallyordered state.In our experiment, the temperature dependence of the dephasing rateshows the sign of freezing or quenching of the magnetic impurities. Fig. 6.4shows the temperature dependence of the dephasing rates at different in-696.2. Magnetic impurities in SiC graphene and signs of freezing magnetic impuritiesFigure 6.4: The anomalous temperature dependence of the dephasing rateis shown clearly in the pink circles. The red triangles represent the dephas-ing rate (below 200mK) in B‖ = 1T , it clearly shows the reduction of thedephasing rate caused by the polarization of the magnetic impurities in aparallel field. The bottom black line is the (theoretical) contribution fromthe electron-electron-interaction (eei dephasing) simply calculated from theconductivityplane magnetic fields. The decrease of the dephasing rate at higher B‖indicates that a large portion of the dephasing rate is from the scattering offthe magnetic impurities, as the polarization of the magnetic impurities inan in-plane field will reduce the magnetic scattering rate and the dephasingrate[7]. At ultra-low temperatures, an intriguing collapse of the dephasingrate showed up (Fig. 6.5 shows the comparison of the collapse dephasingrate before and after removing a background). Similar anomalous tempera-ture dependence of the dephasing rate was also observed in mesoscopic metalwires doped with dilute magnetic impurities[16, 25]. It was ascribed to thefreezing or quenching of the quantum magnetic impurities caused by theKondo coupling or RKKY coupling[16, 25], as only dynamic spinful impu-706.3. Discussionrities can dephase the electrons[40]. As far as we know, it was the first timethat collapse temperature dependence of the dephasing rate was observedin graphene.(a) (b)Figure 6.5: (a), (b) show the temperature dependence of the dephasing ratebefore and after the background being subtracted, respectively. We can seethe clear collapse of the dephasing rate6.3 DiscussionComparing with the phase coherent experiment on the exfoliated graphene,this experiment provides the evidence of the quantum magnetic impuritiesin graphene. The collapse temperature dependence of the dephasing rateis a direct manifestation of the quantum nature of the magnetic impurities.The difference between the temperature dependence of the dephasing ratein SiC graphene and exfoliated graphene may be rooted in the differentconcentrations of magnetic impurities.Similar anomalous temperature dependence of the dephasing rate andthe negative in-plane magnetoresistance have also been observed in our an-nealed CVD graphene samples (unpublished, the measurement was done byother members of this lab). Before annealing, the dephasing rate of CVDgraphene is very low, the in-plane field dependence of the dephasing rateand the resistance, as well as the collapse temperature dependence of thedephasing rate are very weak. After annealing the CVD graphene, some-thing drastic happens: the dephasing rate is one magnitude higher, and all716.3. Discussionof the three phenomena are enhanced greatly. Therefore, the existence ofquantum magnetic impurities in graphene is a very universal phenomenon.Though we revealed the quantum freezing or quenching of the magneticimpurities, we still do not know the precise source of these magnetic mo-ments. They are highly likely to be intrinsic in the graphene systems, be-cause previous studies of the B‖ dependence of the dephasing rate in GaAsquantumwells didn’t see any anomalies[66]. The drastic effects of anneal-ing in CVD graphene also indicate that defects (which can be generatedduring annealing process) could be the source of the magnetic impurities.The magnetic impurities could be located in graphene itself[10, 14, 15, 67],also could be located in the buffer layer of SiC graphene, since the bufferlayer has been revealed to be highly paramagnetic at high temperature andlocalized magnetic moments with a Lande´ factor different from that of thefree electron is believed to exist in the buffer layer[1, 11, 58, 63, 68–70]. Toclarify the source of the observed magnetic moments, local probes such asSTM, are needed.72Chapter 7Experiment 3: graphene onh-BN heterostructureIn the previous experiment chapters, indications of the magnetic impuritieswere found in the exfoliated graphene as well as in SiC graphene. In bothcases, the non-monotonic in-plane magnetic field dependence of the WL de-phasing rate was observed. However, the collapse of the dephasing rate asa function of temperature (as a manifestation of the quantum coupling ofthe magnetic impurities) was only observed in SiC graphene (and also in theexperiments on the annealed CVD graphene). A question arises naturally,could the exfoliated graphene also show a collapse of the dephasing rate as afunction of temperature? Inspired by the collapse of the dephasing rate as afunction of temperature of CVD graphene enhanced by annealing (there isvery weak collapse of the dephasing rate in un-annealed CVD graphene, an-nealing probably can introduce quantum magnetic impurities in graphene),we subsequently conjectured that probably the exfoliated graphene afterannealing may also show a collapse of the dephasing rate as a function oftemperature. The experiment introduced in this chapter aims to investigatewhether annealing in exfoliated graphene can induce a similar collapse ofthe dephasing rate as a function of temperature. In this experiment, an ex-foliated graphene/h-BN heterostructure was fabricated and measured. TheBN substrate greatly improved the quality of the graphene, as there are nodangling bonds on the surface of the h-BN. The graphene/h-BN device wasannealed in a hydrogen/argon mixture, in a manner similar to the anneal-ing process performed on CVD graphene. The phase coherence signal wasprominent, both universal conductance fluctuations and weak localization737.1. Experiment setupwere observed in the magnetoresistance. However, no obvious hints of acollapse of the dephasing rate were observed.7.1 Experiment setupThe exfoliated graphene (the size is about 20µm by 10µm) was transferredonto an exfoliated flake of h-BN crystal at the University of Washington (byDr. Zaiyao Fei). The graphene flake and the h-BN flake were exfoliatedseparately. A piece of polycarbonate (PC) film on top of a small piece ofpolydimethylsiloxane (PDMS) cushion was used to pick up the grapheneflake[71–74]. The PC/PDMS stack was mounted on a glass-slide. A mi-crostage under an optical microscope was used to locate the PC film on topof the graphene flake. The PC film was then lowered to touch the grapheneflake. After heating the silicon wafer to around 90◦C, the graphene flakewas picked up by the PC film, since the adhesion between the graphene andthe PC film is stronger than the adhesion between the graphene and SiO2at 90◦C. After that, the graphene/PC film stack was positioned on top of apiece of flat h-BN flake prepared before-hand. Then, the graphene flake waslowered onto the BN surface. After that, the silicon wafer will be graduallyheated up to about 190◦C to melt the PC film and release the graphene flakeonto the BN flake. Next, the whole graphene/BN stack will be washed inchloroform for a few hours to remove the PC residue. Then, the stack wasannealed in the 5% hydrogen balanced argon gas for about 1hr, at 400◦C.The transfer method is now our standard procedure for transferring 2d ma-terials, we make heterostructures such as the graphene/h-BN stacks, blackphosphorus stacks with this method. The electrodes were made in UBC,with the conventional electron beam lithography technique (see Fig. 7.1).The electron beam lithography recipe is similar to that described in Chapter4. A single layer PMMA A4 was spin-coated on the graphene/h-BN stackfor lithography (4000 rounds/min for 53s, baking at 200◦C for 20min). TwoEBL writings were performed, the alignment marks were defined after thefirst writing and developing step, then, the developed alignment marks weredirectly used for the second writing of the electrodes pattern. 4nm Cr and747.2. Phase coherent transport in the exfoliated graphene/h-BN heterostructure95nm Au were deposited to form the electrodes. Four-terminal measure-ment was employed. But the geometry is not a well-defined Hall bar, thuswe do not know the exact aspect ratio of the sample. The measurementdetails were described in Chapter 4. The measurement was performed inour new bluefors dry dil-fridge, with the base temperature as low as 9mK.The magnetic field was applied perpendicular to the sample surface.Figure 7.1: Device wiring7.2 Phase coherent transport in the exfoliatedgraphene/h-BN heterostructureThe graphene/h-BN heterostructure showed very high quality. The dopinglevel is very low, the charge neutrality point is around 4V. The charge neu-trality peak as shown in the resistance vs gate voltage plot is very sharp andsymmetric, see Fig. 7.2(a). The mobility is larger than 30000cm2/(V ·S) at0V gate voltage, one order of magnitude higher than the mobilities we havein the exfoliated graphene on SiO2. The high mobility of graphene on h-BNis expected based on the previous research[75]. Quantum Hall effect (QHE)can be observed at a magnetic field as low as 1 Tesla. Fig. 7.2(b) shows theQHE (inchoate QHE, not fully-developed ideal QHE plateaus/oscillations)at 3 Tesla. It is one of the best graphene devices ever made in our lab.757.2. Phase coherent transport in the exfoliated graphene/h-BN heterostructure(a) (b)Figure 7.2: (a)Dirac peak (b)Quantum Hall effect (inchoate) at 3TFigure 7.3: Magnetoresistance trace (symmetrized), the fluctuating part isUCF, the peak around zero field is from WLThe measurement of the phase coherent phenomena was performed onthis device. The phase coherent phenomena were prominent: both weaklocalization and universal conductance fluctuations were observed. Fig. 7.3shows the magnetoresistance. The red line is the data, the inset shows767.2. Phase coherent transport in the exfoliated graphene/h-BN heterostructurethe optical image of the graphene/h-BN device. In the magnetoresistancetrace, we can see both weak localization and universal conductance fluctu-ations. The peak near the center comes from the weak localization effect,it reflects the enhanced return probability of the quantum diffusion pro-cess. The strongly fluctuating part of the magnetoresistance is the universalconductance fluctuations, each point represents a specific disorder potentiallandscape controlled by the magnetic field. Due to the strong interferencefrom the UCF, the WL peak was obscured. A better visibility can be ob-tained after averaging away UCF in a small gate voltage window. Tempera-ture and gate voltage dependences of the extracted UCF and WL dephasingrates were obtained. We use the inflection point of the averaged autocorre-lation function of the UCF traces to get the UCF dephasing rate; and usingthe curvature of the WL trace averaged over a few gate voltages to get theWL dephasing rate (see Chapter 3). Fig. 7.4 shows the steps to get theUCF dephasing rate. Fig. 7.4a shows the UCF magnetoconductance tracesat different gate voltages; Fig. 7.4b shows the autocorrelation function cal-culated for each gate voltage; Fig. 7.4c shows the averaged autocorrelationfunction, whose inflection point can be converted into the UCF dephasingrate. Fig. 7.5 shows the weak localization magnetoconductance. Fig. 7.5ashows the magnetoconductance measured in a small gate voltage window,from which we can get an averaged weak localization magnetoconductanceto suppress the interference from UCF. Fig. 7.5b shows the averaged weaklocalization magnetoconductance. Fig. 7.5c shows the vertically-shifted andzoomed-in magnetoconductance traces at different gate voltages. The cur-vature of the center of the averaged weak localization magnetoconductancecan be used for the calculation of the WL dephasing rate. The mathematicaldetails about how to extract the dephasing rate from UCF and WL can befound in Chapter 3.Fig. 7.6 shows the temperature and gate voltage dependences of the UCFand WL dephasing rates. The magnitude of the rates here are closer to thatin the SiC and annealed CVD graphene samples (see Chapter 6), higher thanthe dephasing rate in the usual exfoliated graphene samples (see Chapter5). From the temperature dependence, we did not see clear hints of collapse777.2. Phase coherent transport in the exfoliated graphene/h-BN heterostructureFigure 7.4: Steps to get the UCF dephasing rate: (a) The raw dataset isthe magnetocondutances at different gate voltages (b) Calculated autocor-relation function for each gate voltage (c) Averaging over the gate voltagesto get the final averaged autocorrelation function, its inflection point can beconverted into the UCF dephasing rateFigure 7.5: (a) Magnetoconductances measured in a small gate voltage win-dow, a zoom-in of the traces can be found in (c); (b) Averaged magnetocon-ductance shows the weak localization magnetoconductance lineshape; (c)Vertically shifted and zoomed-in plot, to show the center part of the weaklocalization magnetoconductance tracesdephasing rate (shown in the Fig. 7.6a). The temperature dependences ofthe extracted UCF and WL rates have similar slopes in a big temperaturerange, reflecting that both rates were influenced by the electron-electroninelastic scattering[7, 76]. The gate voltage dependence of the dephasingrate (shown in the Fig. 7.6b) is similar to that observed in the previousstudies[77], usually is ascribed to the gate voltage tuning of the inelastic787.2. Phase coherent transport in the exfoliated graphene/h-BN heterostructure(a) (b)Figure 7.6: (a)Temperature dependence of the dephasing rates (b)Gate volt-age dependence of the dephasing rates, the error bars in (a) and (b) aremainly from the uncertainty in the aspect ratioelectron-electron interaction[77].In summary, we tried to make the collapse of the dephasing rate as afunction of temperature occur in the exfoliated graphene. Though the exfo-liated graphene went through an annealing process, there was still no clearevidence of the collapse of the dephasing rate as a function of temperaturein the exfoliated graphene.79Chapter 8Other two dimensionalmaterials8.1 IntroductionThe study of two-dimensional or layered crystals can be traced back tothe 1960s. Many excellent works about the crystal structure and electronicproperties of the layered crystals have been published between the 1960s and1980s. For example, the book edited by Grasso[78] summarizes the struc-ture, bonding, and electronic properties of transition metal chalcogenides(even black phosphorus and h-BN). A book edited by Le´vy summarizedthe crystallography and crystal chemistry of layered crystals[79]. A bookedited by Lee summarized the optical and transport properties of layeredcrystals[80]. The research of Klemm, et al. studied the properties of layeredsuperconductors[81]. These works provided the condensed matter commu-nity a solid understanding of the basic properties of the layered crystals.However, these early studies were mainly focused on the bulk form. Thediscovery of graphene provides an easy way to make nanosheets of two-dimensional materials, thus reviving the field. Graphene is not an end, buta beginning. Since the discovery of graphene, various 2d crystals nanosheetshave been explored. A recent study claims that there are more than 500structurally stable 2d materials awaiting to be explored in the nanosheetsform[82].Transition metal chalcogenides is a star 2d crystal family drawing muchattention. There are hundreds of members in this family, ranging fromsemiconductors to superconductors. Various physics can be found in this808.2. FeSefamily[83, 84]. For example, the charge density wave is found in TiSe2[85]and TaS2[86]. WTe2 serves as a possible candidate for a natural 2d quantumspin hall insulator. Prior to WTe2, 2d quantum spin hall insulator statewas only observed in man-made quantum-wells. Superconductivity is foundin NbSe2 and FeSe. Exfoliated superconducting 2d crystals are regardedas good systems to study the superconductivity. Because comparing withbulk material, exfoliated 2d layers of these superconductors have reduceddimensionality and tunable carrier density (a few 2d layers have much lowerareal carrier density, which are tunable with the ion-gate or the usual SiO2gate technique), providing a great playground for the study of physics suchas the quantum phase transitions[19–24].In this chapter, a member of the 2d material family — FeSe was explored.FeSe is the simplest member of the Fe-based superconductors, thus FeSe isregarded as a key to understand the unconventional superconductivity in Fe-based superconductors (the pairing mechanism might come from the spinfluctuations). We were among the first labs to realize superconducting FeSenanosheets. The stability and degradation of the FeSe nanosheets were in-vestigated. This research lays the foundation for realizing high quality FeSenanosheets with the tunable areal carrier density to study the quantumphase transition and the pairing mechanism of the Fe-based superconduc-tors.8.2 FeSeWithin the last several years, another 2d material family emerged as a sig-nificant focus of research — layered transition metal chalcogenides com-pounds. Among these compounds, a spectrum of different behaviour wasobserved. FeSe is a superconducting member in this family. The reduceddimension and carrier density (2d layer has much lower areal carrier den-sity, which should be tunable with the ion-gate or usual SiO2 gate tech-nique) makes exfoliated superconductor nanosheets an ideal playground forthe tunable superconductivity and quantum phase transitions[19–24]. FeSealso belongs to another mysterious and intriguing family — the Fe-based818.2. FeSesuperconductors[87–92]. This superconductor family is believed to host theunconventional spin-fluctuation pairing mechanism[93, 94]. As its simplestmember in this family, FeSe is regarded as a key to understanding the Fe-based superconductor family[89].Figure 8.1 is an illustration of the crystal structure of FeSe. At ambi-ent pressure, there is a structural transition as the temperature goes down.The lattice of FeSe is tetragonal above 90K but the lattice will be distortedinto orthorhombic structure below 90K. Superconducting FeSe has an or-thorhombic lattice, with band structure and Fermi surface illustrated inFig. 8.2. The Fermi energy is 3.6 meV[95]. The Debye temperature oforthorhombic FeSe is 210K and the superconducting transition temperatureof bulk FeSe is 8K[96]. However, monolayer FeSe grown on the SrTiO3 sub-strate (by Molecular Beam Epitaxy (MBE)) reached transition temperatureTc above 100K[97]! This discovery made FeSe immediately a star in thecondensed matter community.Questions concerning the high-Tc of monolayer FeSe arise naturally:Could FeSe thin films made in a top-down fashion also reach such hightransition temperature? Can we tune the Tc of the 2d FeSe film to observethe quantum phase transition in 2d? What is the stability of FeSe? We thusconceived and carried out this experiment.We first studied the exfoliation of FeSe nanosheets and found a fewtricks to improve the yield of nanosheets. We also studied the air stabilityof the exfoliated FeSe nanosheets with the Raman spectroscopy, AFM char-acterization, as well as the ToF-SIMS. It is concluded that exfoliated FeSenanosheets decay in a few hours in air, and amorphous Se nanoparticlesare probably formed on the surface of the decayed FeSe nanosheets. Wefound that FeSe also decays during the conventional EBL process. We thusdeveloped two new methods (shadow mask evaporation and transfer-onto-electrodes) to add electrodes onto FeSe. We also found that exfoliated FeSenanosheets exposed in air briefly can still maintain superconductivity. Werealized superconducting FeSe nanosheets in October of 2014. Around thattime, no superconducting FeSe nanosheets were reported. We were probablyamong the first few researchers who observed the superconductivity in FeSe828.2. FeSeFigure 8.1: Illustration of the crystal structure of FeSe, from DOI:10.1039/C2CS35090DFigure 8.2: (a)Calculated band structure of superconducting FeSe (in or-thorhombic lattice) (b)Calculated Fermi surface in the first Brillouin zone.The points Γ, Y, T, S, and R are high symmetry points in the first Brillouinzone. Γ point is in the center of the first Brillouin zone, Z point is above Γpoint; (a) and (b) are from Phys. Rev. B 90, 144517 (2014)838.2. FeSenanosheets. We also tried to tune Tc with a back gate voltage (FeSe wastransferred onto a silicon wafer with a SiO2 dielectric layer, the conductivesilicon below the SiO2 layer is used as a back gate), no obvious tuning effectwas observed.We spent a year solving various experimental challenges, though the ulti-mate goal of realizing exfoliated FeSe flake devices with gate-tunable super-conductivity was not reached in this lab in the end. During our experimenton FeSe, a group in China (led by Professor Xianhui Chen of the Universityof Science and Technology of China) reported the tuning of Tc in the FeSethin flakes by ion gating[93]. It claimed that Tc in FeSe can be increaseddrastically, from 5.2K to over 40K, by doping alone. If only doping is in-volved in the enhancement of Tc, this result would offer a strong evidencefor the doping mechanism (the increased Fermi level caused by doping isexpected to induce a spin-fluctuation paring mechanism in Fe-based super-conductors, which gives rise to the high Tc of the Fe-based superconductors)behind the observed high Tc in the FeSe-derived superconductors. However,whether only doping is involved in the ion-gating experiment is not clear.It’s possible that chemical changes are also involved[98, 99]. The knowledgeof the stability of FeSe we obtained in this experiment could deepen our un-derstanding of the results of the ion-gating experiment. Our study also laysthe foundation for building the exfoliated FeSe devices with higher quality.8.2.1 Experiment setupWe use a polydimethylsiloxane (PDMS)-based exfoliation method (similarto the scotch tape method developed in the graphene research) for produc-ing nanosheets and transferring them onto different substrates[100]. In ourmethod, we use a piece of low-residue Nitto tape to do the exfoliation of theFeSe crystal, and peel off the tape covered with FeSe crystal from a pieceof PDMS polymer. Some nanosheets are left on the PDMS and ready foruse. This is a simple way to obtain ∼10nm thick flakes. We use AtomicForce Microscope (AFM, model type: Asylum Research MFP3D) to char-acterize the thicknesses. The color of the fresh nanosheets are yellow for848.2. FeSesheets larger than 60nm thick, purple for sheets less than 20nm thick. Bothoptical and AFM images show that the surface of the nanosheet is flat andsmooth (Fig. 8.3).Figure 8.3: Optical and AFM characterizations of the FeSe nanosheets, thepurple ones in the left image are nanosheets less than 20nm thick, the yellowones are more than 60nm thick.The typical lateral size of the exfoliated nanosheets is 10 to 20 microns.For the stability test, we employ the optical microscopy to monitor thecolor change and the micro-Raman spectroscopy (Horiba LabRAM HR) tomonitor the change of the phonon modes. The Time-of-Flight SecondaryIon Mass Spectroscopy (ToF-SIMS, model: PHI Trift V nanoToF-SIMS) isused to characterize the chemical change. For the Tc measurement, sincethe volume of a flake is too small for the bulk magnetometry method tomeasure the Tc, thus transport measurements are used to check for thesuperconductivity. In order to do the electrical characterization, at leasttwo electrodes must be added to a nanosheet as the source and drain. Atthe beginning, we tried the conventional electron beam lithography (EBL)method to pattern these contacts. However, EBL caused serious degradationof the FeSe thin flakes and they did not conduct. As a comparison, the blackphosphorus (another novel 2d material, see Chapter 9) thin flake devices wemade with EBL conduct well. Thus FeSe serves as an example of the 2dmaterials incompatible with the widely-used EBL process. We postulatethat some chemical reactions happening during the EBL process causedthe degradation of the FeSe nanosheets. The degraded FeSe flakes havethe same color as the heavily-oxidized ones. To solve this problem, we858.2. FeSedeveloped two methods for the micro-electrodes fabrication that avoid EBL,and also avoid chemicals and heating. Both of the two methods give goodelectrical contacts on FeSe. These two methods are summarized in Fig. 8.4and described below:1. Method 1, transfer of the thin flake onto pre-fabricated electrodes,includes two steps:(a) Exfoliate FeSe with the Nitto tape and press it onto a piece ofPDMS.(b) Position the PDMS piece covered with flakes with a micromanip-ulator and transfer flakes onto the pre-fabricated pads by simplytouching.2. Method 2, shadow mask evaporation[101], includes three steps:(a) Make nanosheets by exfoliation.(b) Use a mask aligner to align a stencil (shadow mask, see Fig. 8.5)on top of a flake. I developed the techniques for making micron-sized features on the shadow mask with a Focused Ion BeamMicroscope.(c) Put the whole setup into a metal evaporator and evaporate throughthe patterned shadow mask to make the contacts.868.2. FeSeFigure 8.4: Illustration of the electrodes fabrication: (a) lithography (b)shadow mask evaporation (c) direct transfer8.2.2 Study of the stability of the FeSe nanosheetsOur study is based on the FeSe crystal grown in the UBC superconductivitygroup. Experiments performed by the superconductivity group on the bulkcrystal, including the magnetometry and resistivity experiments (Fig. 8.6),confirmed the high quality of the crystal.We successfully exfoliated FeSe into nanoflakes. The thickness can be assmall as 5nm. Fig. 8.7a shows several flakes with different thicknesses, thethinnest flake is about 5nm thick, according to the AFM measurement.After the exfoliation, we used the AFM and micro-Raman spectroscopyto characterize the flakes as soon as possible. Fig. 8.7b shows the Ramansignal of relatively fresh FeSe flakes, with air exposure of about 1hr. TheRaman spectra of flakes with different thicknesses were plotted. The twopeaks around 200/cm are the characteristic peaks of the FeSe lattice[102].They are related to the phonon-modes of P4/nmm lattice. The left one isthe Ag(Se) peak from the vibration of the Se atoms, the right one is the878.2. FeSeFigure 8.5: Shadow mask made with a focused ion beam microscopeFigure 8.6: Resistance vs temperature for the bulk FeSe crystalBg(Fe) peak from the vibration of the Fe atoms. Looking into the Ramanspectrum of flakes with different thickness carefully, we can see that, as thethickness decreases and the flake becomes more and more transparent, apeak from the SiO2 (300/cm peak) shows up. Besides this peak, a new peakaround 250/cm also shows up. Meanwhile, the characteristic peaks of the888.2. FeSe(a) (b)Figure 8.7: (a)Exfoliated thin FeSe flakes, the inset is an AFM image;(b)Raman spectrum for FeSe flakes with different thicknessesFeSe lattice diminish. There are no signs of the characteristic peaks of FeSelattice in the thinnest flakes.What caused the diminishing of the characteristic peaks of the FeSelattice, and the emergence of the new peak around 250/cm ? One plausiblescenario is the degradation during the air exposure. If this were the case,long time air exposure could make thick flakes also show this new Ramanpeak. We performed an aging test in the open air and found evidencesupporting this scenario. Fig. 8.8 shows the change of the Raman spectrumof the FeSe nanosheets during the prolonged exposure in air (Fig. 8.8a isfor a piece of bulk crystal, Fig. 8.8b is for a ∼ 10nm-thick nanosheet). Wecan see that (see Fig. 8.8a), the new peak at 250/cm shows up in the bulkcrystal after a long-time exposure to air. The degradation of the FeSe flakesin air can also be judged from the changes in color. An estimate of the speedof the degradation process can be made. There is no obvious change withinthe first hour of air exposure, small changes can be seen after 3 hours of airexposure, obvious change can be observed after 24 hours exposure to air.Therefore, the typical life time of a thick exfoliated FeSe nanosheet is about24 hours.What is the microscopic origin of this peak around 250/cm ? Afterlooking through the Raman peaks of different materials. I found that898.2. FeSe(a) (b)Figure 8.8: Raman spectrum of FeSe before and after long time exposure inair: (a) a piece of bulk crystal; (b) a ∼ 10nm-thick nanosheetthe peak position is consistent with the Raman signal of amorphous Senanoparticles[103]. Further evidence can also be found from the AFM char-acterization. Fig. 8.9 shows the AFM images taken at different exposuretime durations for a FeSe flake. The Fig. 8.9(a) shows the optical images offlakes before lengthy air exposure (up) and after 24 hours exposure to air(bottom). The change in the color is obvious. The left part of Fig. 8.9(b)shows an AFM image of a fresh FeSe nanoflake, in which the surface is quiteclean. The right part of Fig. 8.9(b) shows the AFM image after 24 hoursof air exposure, in which nanoparticles and bumps show up on the surface.These particles are likely the precipitation of Se (nanoparticles). The ori-gin of the new peak around 250/cm may therefore be the Se precipitationinduced by air exposure.A question arises after knowing the degradation of FeSe in air: whatchemical processes are involved during the air exposure? Raman and opticalmicroscopy can not tell us the details of the chemical change, especially whathappened during the first hour of the air exposure (no obvious change canbe seen with Raman or optical microscopy). The most obvious conjecturewould be the oxidation. In order to find out whether this is the case, we needto perform a precise chemical composition analysis of the FeSe nanosheets.Our choice of instrument is the Time-of-Flight Secondary Ion Mass Spec-908.2. FeSeFigure 8.9: Optical and AFM images of FeSe nanosheets before and afterlong time exposure in airtrometry (ToF-SIMS). ToF-SIMS can do nanoimaging and depth profiling ofthe chemical compositions[104]. The nanoimaging of chemical compositionscan tell us the lateral distribution of different chemical compositions. Thedepth profiling provides information about the chemical compounds that ex-ist at different depths. It is only within the last few years, researchers havestarted to use ToF-SIMS to characterize the nanosheets of layered crys-tals. For example, in a study of the stacks of nanosheets, researchers usedthe ToF-SIMS to obtain the chemical details of the transferred nanosheets.They found that the chemicals used in the transferring process are trappedin the stacks[104]. In our own ToF-SIMS measurements (see Fig. 8.10),we characterized the chemical compositions for FeSe nanosheets before andafter lengthy exposure to air. For nanosheets exposed to air for less than 20minutes, we found that the nanosheets remain intact, no obvious oxidationwas found. For nanosheets exposed to air for about 2 hours, we found thatonly a thin surface layer (around 1nm thick) is degraded after exposure.The sample is intact deep into the flake. For nanosheets exposed to air formore than 24 hours, we found that oxidation penetrated about 5nm into thenanosheets. The composition of the oxidized nanosheets is also determined,several iron oxides (such as FeO) were detected. This oxidation processmight be the reason for the precipitation of Se.918.2. FeSeFigure 8.10: A ToF-SIMS image for a FeSe flake exposed in air for 24 hours8.2.3 Study of the superconductivityIn the last section, we checked the stability of the FeSe nanosheets exposedto air. A question arises naturally, whether these nanosheets are still su-perconducting? The exfoliated nanosheets are too small for the bulk mag-netometry to check the superconductivity. We thus employed the transportmethod to measure Tc. The electrodes fabrication details were describedin the experiment setup section. Our transport measurement showed that,although FeSe flakes completely degrade in air in 24 hours, for thick FeSenanosheets with less than 2 hours air exposure, the superconductivity canstill be retained. So far, we have performed transport measurements on 6flakes with thicknesses ranging from 10nm to 60nm; 4 of these flakes werefound to be superconducting. We found that the superconducting transitionis not as sharp as that in the bulk for any of these nanosheets, and Tc wasreduced compared to that in the bulk, see Table 8.1 and Fig. 8.11. Thelower Tc and broadened superconducting transition is also consistent withoxidized superconductor thin films. Gating by SiO2 was also attempted onour 10nm-thick flake. No gating effect was found, thus the carrier densityof the FeSe nanosheet is way beyond the manageable range of a SiO2 gate.928.2. FeSeTable 8.1: Tc and thicknessesTc Thicknessbulk 9 K -flake 1 7 K ∼ 60 nmflake 2 6 K ∼ 15 nmflake 3 5 K ∼ 10 nmflake 4 6 K ∼ 15 nmflake 5 semiconducting ∼ 10 nmflake 6 semiconducting ∼ 10 nmFigure 8.11: (a): Data for a ∼60nm thick FeSe flake; (b): Data for a ∼15nmthick FeSe flake; (c): Data for a ∼10nm thick FeSe flake; (d): Zoom-in of(a), (b), (c) for T<10 K (contact resistance subtracted)8.2.4 SummaryIn summary, the experiments in this chapter show that: FeSe can be ex-foliated into nanosheets; FeSe flakes degrade in air and thin flakes degradequickly. The life-time for flakes thicker than 10nm is about 24 hours andis much shorter for thinner flakes; The new peak (around 250/cm) whichoccurred during degradation matches the Raman signal of the Se nano-particles. AFM monitoring also found nanoparticles occurred during air ex-posure, consistent with the Se nanoparticle scenario; Oxidation takes placein air, as the ToF-SIMS shows that oxygen penetrates the FeSe crystal.Furthermore, FeSe also degrades in the conventional EBL process for mak-938.2. FeSeing electrodes, and two alternative methods have been developed for makingelectrodes. Transport measurements showed reduced Tc and broadened tran-sition in the exfoliated thin flakes. Our study provides a clearer picture ofthe stability of FeSe nanosheets and the chemical change happening in theexfoliated FeSe nanosheets exposed to air. The knowledge obtained pavesthe way for building FeSe nanodevices with higher quality. It could also beuseful for a thorough understanding of the experimental results based on theFeSe nanosheets exposed in air, such as the recent ion-gating experiment[93].94Chapter 9Summary and outlook9.1 SummaryThe bulk of this thesis is about the quantum magnetic impurities in graphene.We developed an effective method to reveal the magnetic impurities in two-dimensional materials, such as graphene — the first and by far the most fa-mous member in the 2d material family. In this method, we use phase coher-ent transport such as weak localization and universal conductance fluctua-tions to extract the dephasing rate. The existence of magnetic impurities willaffect the in-plane magnetic field dependence of the dephasing rate. In thein-plane magnetic field dependence of the UCF dephasing rate, a crossoverbehavior was observed. It can be ascribed to the polarization of magneticimpurities. The complex phenomena in the in-plane magnetic field depen-dence of the WL dephasing rate are captured by a recent model (developedby Kashuba, et al.)[42], which takes into consideration both the polarizationand the precession of the magnetic impurities: in strong field, suppressedWL dephasing rate will emerge; in weak field, a non-monotonic dependenceexists when electron and magnetic impurities have different Lande´ g factors.Our experiments on the exfoliated graphene, SiC graphene and CVDgraphene demonstrated the existence of the magnetic impurities in variousgraphene systems. In the exfoliated graphene, both UCF and WL were usedto study the influence of the magnetic impurities on the dephasing rates. Inthe experiments on the SiC graphene, detailed studies have been performedon the influence of the magnetic impurities on the dephasing rate extractedwith weak localization. In the SiC graphene, we also observed signaturessuch as collapse dephasing rate, which are probably related to the freezingor quenching of the quantum magnetic impurities.959.2. OutlookThough similar in-plane field dependence of the WL dephasing rate existsin both exfoliated and SiC graphene. The collapse dephasing rate was onlyobserved in SiC graphene (and also CVD graphene), whose dephasing rate isone order of magnitude higher than that of the exfoliated graphene. To in-vestigate whether the collapse rate can also exist in the exfoliated graphene,phase coherent transport on an annealed exfoliated graphene/h-BN stackwas also studied. Though the dephasing rate of the exfoliated graphene/h-BN stack was brought to the same magnitude of that in SiC graphene byannealing, no hints of the collapse temperature dependence of the dephasingrate was observed in the exfoliated graphene.To sum up the graphene part of this thesis, our contribution to thegraphene research can be highlighted into three points: First, we developedan effective phase coherent transport method to reveal the magnetic impuri-ties in 2d system; Secondly, we demonstrated the existence of the magneticimpurities in various graphene systems, which could explain the fast spin re-laxation observed in graphene; Thirdly, we also demonstrated the signaturesof Kondo or RKKY coupling in the magnetic impurities of graphene.Other members of the 2d material family were also studied. This partmainly focuses on the exfoliated FeSe nanosheets. This study proved thatatomically thin FeSe nanosheets can be obtained by the mechanical exfoli-ation method. We found that FeSe flakes decay in air. We also found thatFeSe will be damaged during the conventional EBL process. The charac-terization with optical microscopy, Raman microscopy, AFM, ToF-SIMS, aswell as transport experiments revealed the chemical instability of the FeSenanosheets. The decay product of FeSe in air is consistent with the Senanoparticles. This result strengthened the complex chemical process oc-curring on the FeSe nanosheets and laid the foundation for the future studyon the exfoliated FeSe nanosheets.9.2 OutlookAs an outlook, more studies about magnetic impurities in the 2d materi-als can be done in the future. Besides the original SiC graphene, another969.2. Outlooknew member of the SiC graphene family was introduced — hydrogenatedgraphene. It is believed to be a quasi-free-standing graphene, with hydrogenatoms sitting between the graphene layer and the buffer layer. Bulk mag-netometry revealed the existence of magnetic order and localized magneticmoments[58] in the hydrogenated SiC graphene. Whether the localized mag-netic moments living in the hydrogenated graphene system can leave markson the quantum transport is also something worth investigating. Besidesthe SiC graphene, CVD graphene family also has interesting new members.For example, in the commercially-available nitrogen-doped CVD graphene,recent research found the nitrogen atom will distort the graphene latticelocally and form localized states[105]. Whether magnetic moments relatedto the nitrogen dopants exist and whether they couple with the electrons ina Kondo manner is also an interesting topic.Moreover, recently, in the exfoliated black phosphorus thin flakes, peo-ple also observed weak localization and universal conductance fluctuations.The dephasing rate extracted with the weak localization magnetoresistanceis saturating at low temperature[106, 107] (see Fig. 9.1). This saturation issimilar to the saturation (residue dephasing rate) caused by the dilute mag-netic impurities in graphene and other metal wires[18, 59, 108]. Therefore, itis possible that there are also magnetic impurities in the black phosphorus.Because of the importance of black phosphorus in the next generation tran-sistors and spintronic devices, it is worth investigating whether the sourceof the saturation is from the magnetic impurities.So far, we only considered magnetic impurities in systems without spin-orbit interactions. The effect of the magnetic impurities on the dephas-ing rate in systems with spin-orbit coupling is an unsolved question. Re-cent phase coherent transport on graphene on transition metal dichalco-genides indicates that spin-orbit coupling can be introduced into graphenesuccessfully[109–111]. In such system with spin-orbit coupling, whether sim-ilar non-monotonic in-plane magnetic field dependence[41, 42] exists is alsoan interesting topic.979.2. 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In a phase coherent unit, the scattering of wave through aspecific disorder potential can be expressed by Green’s function.The Hamiltonian with disorder potential reads[34]:Hc = − ~22m∗∇2 + U(r) (A.1)where U(r) is a specific realization of the disorder potential.The Green’s function in this phase coherent unit reads:GR = [(E + iηϕ)I −Hc]−1 (A.2)ηϕ represents the loss of coherence by phase-breaking processes.Plug it into the Kubo formula, we can get the conductivity:σxx =2e2h~2Ld∑k,k′vx(k)vx(k′)〈|GR(k,k′)|2〉 (A.3)where ” <> ” represents the ensemble average over different realizations117A.1. Weak localizationof disorder potential. GR(k,k′) is the Fourier transform of the Green’sfunction, it is related with the Fourier transform of the Hamiltonian anddisorder potential:Hc(k′,k) =~2k22mδk′,k + U(k′,k) (A.4)U(k′,k) = 〈k′|U(r)|k〉 =∫U(r) exp[−i(k′ − k) · r] drL2(A.5)If the scattering potential were absent:U = 0 (A.6)G0(k′,k) = δk′,k1E − ~2k22m∗ + iηϕ(A.7)In general, we can write the Fourier transform of the Green’s functionas:GR(k′,k) = δk′,kG0(k) +G0(k′)UqG0(k) +G0(k′)Uq2G0(k + q1)Uq1G0(k),(A.8)where Uq = U(k′,k), q = k′ − k, as U(k′,k) depends only on the differencebetween the wavevectors (k′ − k).The Green’s function is different for each phase-coherent unit becausethe scattering potential is different. The ensemble-average is calculated byaveraging over all the phase-coherent units assuming appropriate statisticalproperties for the set of scattering potentials.The Drude result corresponds to the approximation that 〈|GR(k′,k)|2〉 ≈|〈GR(k′,k)〉|2, this represents the case without coherent multiple scatterings.WL correction is the difference between 〈|GR(k′,k)|2〉 and |〈GR(k′,k)〉|2,which reads:118A.1. Weak localization∆σ2e2/h=~2L2∑k,k′vx(k)vx(k′)[〈|GR(k′,k)|2〉 − |〈GR(k′,k)〉|2]=~2L2∑k,k′vx(k)vx(k′)|G(k)|2|G(k′)|2Γ(k′,k)≈ −2DτmL2∑βC(β) (A.9)where Γ(k,k′) = (U2/L2)C(β), β = k + k′, C(β) ≈ 1/Dτm(1/Dτφ)+β2is knownas the Cooperon, τφ is introduced as a phenomenological parameter rep-resenting phase coherent time, which is a result of various phase-breakingprocesses.Thus,∆σ = − 12pi∫ (Dτm)−10d(β2)β2 + (1/Dτφ)= − e2pihlnτφτm(A.10)Above is for weak localization correction at zero magnetic field. It showsup in the temperature dependence of resistivity through the temperature de-pendence of the phase coherent time τφ. Weak localization effect also showsup in the magnetoresistance. In a perpendicular magnetic field, electronsform Landau levels and β takes a new form:β2n = (n+12)4|e|B~Thus:∆σ =2e2h(− 12pi4|e|B~)a∑n=01β2n + (1/Dτφ)= − e2pih[ψ(12+~4|e|BDτ )− ψ(12+~4|e|BDτφ )]= − e2pih[ψ(12+BtrB)− ψ(12+BφB)] (A.11)119A.2. Universal conductance fluctuationswhere Btr =~4|e|Dτm , Bφ =~4|e|DτφTransforming the Cooperon equation in momentum space back into thereal space, we can appreciate the meaning of WL correction in real space.[Dβ2 +1τφ]C(β) ≈ 1τ, β = k + k′ (A.12)[−D + 1τφ]C(ρ) ≈ 1τδ(ρ), (ρ = r− r′) (A.13)This is the diffusion equation, C(r, r′) represents the probability of find-ing an electron at r after it is introduced at r′.∆σ2e2/h≈ −Dτ [C(ρ)]|ρ=0 = −Dτ [C(r, r)] (A.14)We can see that the WL correction is related to the probability of aparticle returning to its point of origin. The returning probability is theLaplace transform of the returning probability as a function of time, whichcan be calculated with the quantum Brownian motion picture[33].The physical meaning of the Cooperon introduced here is the probabil-ity of going from one point to another by a coherent process of multiplescattering. Another similar mathematical object in the calculation of dif-fusion probability is the Diffuson, which will be used in the calculating ofuniversal conductance fluctuation. The Diffuson is the probability of goingfrom one point to another point by a classical process of multiple scattering(sequential scattering)[33].A.2 Universal conductance fluctuationsHow to calculate the theoretical ensemble averaged correlation function ofthe fluctuating part of the conductivity? Similar to the weak localization,it can also be modeled with the Green’s function and Kubo’s formula. Thecorrelation function eventually can be expressed as contributions from Dif-120A.2. Universal conductance fluctuationsfusons and Cooperons[33] and it is in general a 2-dimensional function ofenergy and magnetic field.At T = 0K, consider a phase coherent rectangle (Lx by Ly) in the 2dimensional space, the formula for autocorrelation function as a function ofFermi energy and magnetic field reads[39]:F0(δE, δB) = δG0(E,B)δG0(E + δE,B + δB)= Ce4h24D2L4x∑n[1|λn|2 +12Re1λ2n], (A.15)λn =2De|δB|~(n+12) + τ−1φ − iδE~, (A.16)G0(E,B) is conductance at zero temperature. C is a constant dependingon the symmetries in the system.This autocorrelation function can be further written into the followingform:F0(δE, δB) = Ce4h2LyDτφL3xK0(, β). (A.17)K0(, β) =1piIm[ψ(12+1 + iβ)]+12piβRe[ψ′(12+1 + iβ)]. (A.18)where ≡ δE · τφ/~, β ≡ |δB| · 2eDτφ/~.This expression can not be calculated directly when either β or  are zero;taking limits, one obtains the variance K0(0, 0) =32pi and the one-parametercorrelations:K0(, 0) =tan−1 pi+12pi(1 + 2)−1, (A.19)K0(0, β) =32piβψ′(12+1β). (A.20)121A.2. Universal conductance fluctuationsFor T > 0 the conductance and correlation function are smeared bytemperature.The smeared conductance reads:G(µ,B) =∫ ∞−∞dE f ′F (E − µ)G0(E,B), (A.21)where f ′F (δE) =14kBTsech2( 12kBT δE) is the derivative of the Fermi-Diracdistribution function.The smeared correlation function, F (δµ, δB), is then obtained by aconvolution of F0(δE, δB) with the function1kBTκ( δEkBT ), where κ(x) =12(x2 cothx2 − 1)/ sinh2 x2 .Equivalently, the core of the correlation function gets smeared: K0 →KT by the convolution:KT(′, β) =∫ ∞−∞dκ(/T1)T1K0(− ′, β). (A.22)where T1 ≡ kBT · τφ/~.The thermal smeared correlation function is then written asF (δµ, δB) = Ce4h2LyDτφL3xKT(′, β). (A.23)Note that here, ′ = δµτφ/~ (not δE).122Appendix BSensor network forcryogenics operationFigure B.1: Wiring for the Arduino-Zigbee moduleMaintaining the fridge’s operation is not an easy job. Since helium isprecious, we have to recover helium boil-off. The recovery job involves acomplicated recovery system. Helium boil-off is first collected by a bag, then,a compressor compresses the helium gas into high-pressure bottles. Then,the helium in pressured bottles is sent to a purifier to remove moisture fromthe helium. After that, the helium is sent to a liquefier to be converted intoliquid helium. Due to the capacity of purifier, it saturates every few daysand has to be regenerated. Many transducers are used to monitor the healthof the system. The health of the purifier is monitored by an oxygen meter.The pressure of the helium gas is monitored by pressure transducers. Thehelium level in the liquefier and our dewars are monitored by level meters.To free us of the heavy duties of monitoring the health of the recovery systemand keep log of the amount of helium, I developed the first Arduino-Zigbee123Appendix B. Sensor network for cryogenics operationsystem used in our lab. The Arduino board is used to read the signals fromthe pressure gauge and oxygen meter. Some hacking on the transducer isnecessary to make the analog signal readable with the Arduino. Then, theArduino communicates with a Zigbee wifi module to transmit readings to aserver. At the beginning, a Raspberry-Pi based server was used, the drivercode on the Arduino is written in Arduino script, and Python code was usedto communicate with the Zigbee modules. Later, other group members (suchas Mohammad Samani) upgraded it into a desktop computer server.(a) (b)Figure B.2: (a) Hacked pressure sensor being read with an Arduino board(b)The Raspberry Pi computer is the server end(a) (b)Figure B.3: (a) Sensor on a liquefier; (b) Webpage monitorcode on the Arduino-Zigbee module:124Appendix B. Sensor network for cryogenics operation#include <SoftwareSerial.h>SoftwareSerial xbee(8, 9); // RX, TXint pin = A0;//int switchPin = 5;int awake=4;//unsigned long duration;//unsigned long now;//unsigned long lastmillis=0;//unsigned long sum;//unsigned long tem1;//int count=0;void setup(){Serial.begin(115200);//pinMode(pin, INPUT);//pinMode(switchPin, OUTPUT);//digitalWrite(switchPin,HIGH);pinMode(awake,INPUT);// set the data rate for the SoftwareSerial portxbee.begin(19200);}void loop(){//delay(10000);// pinMode(switchPin, INPUT);// pinMode(switchPin, OUTPUT);// digitalWrite(switchPin, LOW);125Appendix B. Sensor network for cryogenics operation// pinMode(switchPin, INPUT);//// delay(1000);//generate a square wave to turn on pressure gaugeint button = digitalRead(awake);//Serial.println(button);if (button == HIGH) {//digitalWrite(switchPin,LOW);//delay(100);int sensorValue = analogRead(pin);float voltage = sensorValue * (5.0 / 1023.0);Serial.println(voltage);Serial.println(" ");xbee.print(voltage);//digitalWrite(switchPin, HIGH);}delay(2000);}126Appendix B. Sensor network for cryogenics operationcode on server side, it periodically logs sensor values:from xbee import ZigBeeimport timeimport structimport sqlite3import osimport globser=serial.Serial(’/dev/ttyUSB0’,19200)xbee=ZigBee(ser)myRouter=’\x00\x13\xa2\x00\x40\x79\xa8\x08’dbname=’/var/www/templogmuc.db’xbee.remote_at(dest_addr_long=myRouter, command=’IR’,parameter=’\X00\X00’)xbee.remote_at(dest_addr_long=myRouter, command=’D4’,parameter=’\X05’)time.sleep(10)xbee.remote_at(dest_addr_long=myRouter, command=’IR’,parameter=’\X00\X00’)reply=xbee.wait_read_frame()if reply[’source_addr_long’]==’\x00\x13\xa2\x00@y\xa8\x08’:127Appendix B. Sensor network for cryogenics operationprint ’%f, %f’%(time.time(),float(reply[’rf_data’]) )conn=sqlite3.connect(dbname)curs=conn.cursor()curs.execute("INSERT INTO temps values(datetime(’now’),(?))",(float(reply[’rf_data’]),))conn.commit()conn.close()xbee.remote_at(dest_addr_long=myRouter, command=’D4’,parameter=’\X04’)time.sleep(1)ser.flushInput()128


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