Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Vishnu Radhan, Aswin
Master of Science - MSc
University of British Columbia
Two dimensional (2D) van der Waals materials with a lattice mismatch or a relative twist angle, stacked atop each other forms a moiré superlattice. The electronic properties of such a system can be modified by controlling the relative twist angle between the layers, the most famous example being magic-angle twisted bilayer graphene. Since the observation of correlated insulator states and superconductivity in this flat band moiré system, investigations on other 2D moiré systems has gathered pace. Two Bernal stacked bilayer graphene sheets twisted relative to each other, i.e., twisted double bilayer graphene, gives the additional opportunity of tuning the electronic structure by a displacement electric field. This thesis presents the fabrication and electrical transport measurements of twisted double bilayer graphene devices. In order to compare the efficacy of various fabrication techniques, we fabricated the devices using two different modified dry transfer techniques: “tear-and-stack” and “cut-and-stack”. By calculating the twist angles from the electrical transport data, we find the “cut-and-stack” technique to give a better control of the twist angle of the fabricated devices. Subsequently, we studied the transport properties of a device with a twist angle of 1.39 degree, in order to replicate the electronic phase diagram of twisted double bilayer graphene devices. Tuning the displacement electric field and carrier density independently in a double gated geometry, we were able to observe correlated insulating states at quarter, half and three-quarter filling of the moiré band for a range of electric field values. The evolution of these insulating states in an in-plane magnetic field suggests spin/valley polarization. Additionally, the metallic states surrounding the correlated insulating state at half-filling displays a strong temperature dependence and nonlinear current-voltage characteristics, the nature of which remains ambiguous in our measurements.
Electronic transport in twisted doublebilayer graphenebyAswin Vishnu RadhanB. Tech., Mahatma Gandhi University, Kerala, India, 2011M.Tech., AcSIR, India, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2020c© Aswin Vishnu Radhan 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Electronic Transport in Twisted Double Bilayer Graphenesubmitted by Aswin Vishnu Radhan in partial fulfillment of the require-ments for the degree of Master of Science in Physics.Examining Committee:Joshua Alexander Folk, Associate Professor, Physics and Astronomy, UBCSupervisorZiliang Ye, Assistant Professor, Physics and Astronomy, UBCAdditional ExamineriiAbstractTwo dimensional (2D) van der Waals materials with a lattice mismatch ora relative twist angle, stacked atop each other forms a moire´ superlattice.The electronic properties of such a system can be modified by controllingthe relative twist angle between the layers, the most famous example beingmagic-angle twisted bilayer graphene. Since the observation of correlatedinsulator states and superconductivity in this flat band moire´ system, inves-tigations on other 2D moire´ systems has gathered pace. Two Bernal stackedbilayer graphene sheets twisted relative to each other, i.e., twisted doublebilayer graphene, gives the additional opportunity of tuning the electronicstructure by a displacement electric field. This thesis presents the fabricationand electrical transport measurements of twisted double bilayer graphene de-vices. In order to compare the efficacy of various fabrication techniques, wefabricated the devices using two different modified dry transfer techniques:“tear-and-stack” and “cut-and-stack”. By calculating the twist angles fromthe electrical transport data, we find the “cut-and-stack” technique to givea better control of the twist angle of the fabricated devices. Subsequently,we studied the transport properties of a device with a twist angle of 1.39degree, in order to replicate the electronic phase diagram of twisted dou-ble bilayer graphene devices. Tuning the displacement electric field andcarrier density independently in a double gated geometry, we were able toobserve correlated insulating states at quarter, half and three-quarter fillingof the moire´ band for a range of electric field values. The evolution of theseinsulating states in an in-plane magnetic field suggests spin/valley polariza-tion. Additionally, the metallic states surrounding the correlated insulatingstate at half-filling displays a strong temperature dependence and nonlinearcurrent-voltage characteristics, the nature of which remains ambiguous inour measurements.iiiLay SummaryWhen two layers of graphene, an atom-thick sheet of carbon, are stacked ontop of each other with an exact twist angle of 1.1◦ (magic angle) betweenthem, it displays an array of exciting properties not seen in the originalgraphene, such as superconductivity. However, the precise requirement forthis misalignment angle makes it very challenging to fabricate such devicessparking a search for alternate systems. Bilayer graphene, i.e., a two-atomthick sheet of carbon, is one such candidate where in addition to twist anglethe properties can be modified by fabricating a dual gated field-effect tran-sistor out of it. In this thesis, we fabricate a two-gated device where twobilayer graphene sheets are twisted at a small angle with respect to eachother and show that its properties can be controlled by applying voltages tothese dual gates.ivPrefaceChapter 4 is based on the work done at Quantum Devices group at thedepartment of Physics and Astronomy, University of British Columbia by theauthor Aswin Vishnu Radhan, postdoctoral fellow Dr. Manabendra Kuiri,undergraduate research assistant Ms. Teri Siu, research associate Dr. SilviaLu¨scher Folk and Prof. Joshua Folk. The exfoliation and assembly of vander Waals heterostructures were done by the author and Ms. Teri Siu. Theelectron beam lithography processes and the low temperature measurementswere done by Author and Dr. Manabendra Kuiri. The analysis presentedin this thesis was done by the author under the supervision of Prof. JoshuaFolk. The work presented is unpublished as of the writing of this thesis.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theory and Background . . . . . . . . . . . . . . . . . . . . . 32.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Bilayer graphene . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Moire´ Superlattices . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Twisted graphene layers . . . . . . . . . . . . . . . . 113 Experimental Techniques. . . . . . . . . . . . . . . . . . . . . . 163.1 Exfoliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1 Dry transfer . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Fabrication of Twisted Bilayer Graphene devices . . . 203.3 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Electrical Transport Measurements . . . . . . . . . . . . . . 28viTable of Contents4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 304.1 Twist angle measurements . . . . . . . . . . . . . . . . . . . 304.2 Correlated Insulator States for device CS2 . . . . . . . . . . 365 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47AppendixA Resistance maps of all the fabricated devices . . . . . . . . 54viiList of Tables4.1 The target angle (θt) and measured angle (θm) of all the de-vices fabricated during this work. T1 and T2 corresponds tothe temperatures at which the stamp touches the substrateand the stamp progresses until the edge of hBN/cutting linerespectively. Devices CS3 and TS4 indicated by (SL) aretwisted bilayer graphene devices included for the sake of com-pleteness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32viiiList of Figures2.1 Crystal structure and electronic dispersion of graphene. . . . 42.2 Field effect in monolayer graphene. . . . . . . . . . . . . . . . 62.3 Crystal structure and low energy electronic dispersion of bi-layer graphene. . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Moire´ pattern, mini Brillouin zone and electrical characteris-tics of graphene-hBN heterostructure. . . . . . . . . . . . . . 102.5 Moire´ pattern, superlattice Brillouin zone and flat band for-mation in twisted bilayer graphene. . . . . . . . . . . . . . . . 122.6 Calculated band structure and device characteristics of magicangle twisted bilayer graphene. . . . . . . . . . . . . . . . . . 132.7 Calculated band structure with and without a perpendicu-lar electric field and device characteristics of twisted doublebilayer graphene. . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Optical micrograph of graphene and hBN flakes exfoliatedonto SiO2 (285nm)/Si substrates. . . . . . . . . . . . . . . . . 173.2 Schematic of the process flow for hBN pick-up. . . . . . . . . 193.3 Schematic of the “tear-and-stack” process of fabrication oftwisted devices. . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Optical micrographs of “tear-and-stack” fabrication processflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Optical micrographs of failed tearing attempts. . . . . . . . . 243.6 Optical micrographs of “cut-and-stack” fabrication processflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 Optical and AFM image of a completed stack. . . . . . . . . . 263.8 Electron-beam lithography process for fabrication of doublegated devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Schematic of two-probe and four-probe measurement tech-niques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29ixList of Figures4.1 Map of Rxx vs V tg and V bg for device TS2 and line cut cor-responding to D = 0 plotted as a function of carrier density(n) at T = 4.2K. . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Map of Rxx vs V tg and V bg and their corresponding line cutsat D = 0 plotted as a function of carrier density (n) fordevices TS1 ((θm = 0.68◦), CS2(θm = 1.39◦) and TS5(θm =1.87◦) at 4.2K. . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Map of Rxx vs V tg and V bg for device CS2, θm = 1.39◦ atT = 16mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Band structure evolution in a perpendicular electric field of atwisted double bilayer graphene device. . . . . . . . . . . . . 374.5 Zoomed-in map of Rxx around thens2 correlated insulatingstate and it’s dependence on an in-plane magnetic field fordevice CS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6 Rxx map over the small region region indicated in Fig. 4.5(a)at 16 mK and 100 mK for device CS2. . . . . . . . . . . . . . 404.7 dV/dI measurements as a function of Idc on the low resistanceregions of device CS2. . . . . . . . . . . . . . . . . . . . . . . 414.8 The device structure, optical image and two probe resistancemap across the voltage leads for the device CS2. . . . . . . . 43A.1 V tg vs V bg resistance maps of all the “cut-and-stack” twisteddouble bilayer devices at 4.2K. . . . . . . . . . . . . . . . . . 54A.2 V tg vs V bg resistance maps of the “tear-and-stack” twisteddouble bilayer devices fabricated at T 1, T 2 < 100◦at 4.2K. . . 55A.3 V tg vs V bg resistance maps of the “tear-and-stack” twisteddouble bilayer devices fabricated at T 1, T 2 > 100◦at 4.2K. . . 56xAcknowledgementsFirst and foremost, I would like to thank my supervisor Prof. Joshua Folkfor providing me the opportunity to work in this exciting field of 2D van derWaals materials and moire´ superlattices. I would like to thank him for hisconstant support and guidance during the two years I spent here.I would also like to thank Dr. Manabendra Kuiri for teaching me vari-ous experimental techniques and providing day to day guidance on experi-ments. I am lucky benefit from his expertise and knowledge of 2D materials.I would also like to thank the other members of my group Teri Siu, Dr.Ebrahim Sajai, Christian Olsen, Tim Child, Owen Sheekey and Dr. SilviaFolk for their help at various points of this project.I would also like to thank Prof. Ziliang Ye for agreeing to be the secondreader of this thesis and for providing his invaluable comments.I am thankful to my friends in UBC, Max Werner, Martin Cross, ShadabAhmed, and Sebastian Gitt for their company and invigorating discussions.Finally, I would like to thank my parents and my sister for everything theyhave done for me.xiDedicationI would like to dedicate this thesis to my grandmother who left us a few yearsback and whom I dearly miss.xiiChapter 1IntroductionThe ability to isolate[1, 2] and assemble[3, 4]one or more layers of a largenumber of two-dimensional (2D) crystals have opened new frontiers for ex-ploring low dimensional physics and engineering novel devices. Graphene, amonolayer of graphite, was one of the first 2D materials to be extracted. Theextraordinary properties of graphene and the simplicity of the mechanicalcleavage technique employed for its extraction led to an explosion of re-search activities in the field of 2D materials. Today a variety of 2D crystalsencompassing a wide spectrum of properties have been identified from insu-lators (hBN[5]) to semiconductors (MoS2[6]) to superconductors (NbSe2[7])and ferromagnets (Cr2Ge2Te6[8]). Furthermore, we now have developedtechniques to form heterostructures by combining these diverse materialsutilizing van der Waals interactions to hold them together without beingconstrained by the rules of traditional epitaxy. This presents numerous pos-sibilities for modifying the properties of fabricated devices.One such parameter that modifies the properties of van der Waals het-erostructures is the the relative alignment between the neighbouring crystals.An earliest example of this is the appearance of moire´ patterns on graphene-hBN heterostructures[9–12]. Owing to their lattice mismatch, when stackedatop each other it leads to the formation of a geometric interference patternwhose periodicity can be controlled by the relative alignment between them.Acting as a superlattice potential with a periodicity different form the un-derlying lattice parameters, moire´ pattern modifies the electronic structureof the heterostructure. This has lead to observation of several interestingphenomena such as symmetric secondary Dirac peaks and a fractal specturmin a magnetic field known as Hofstader butterfly[10–12].The most famous example yet of moire´ potential affecting the electronicstructure of van der Waals heterostructures is the emergence of a flat mini-band at low energies when two graphene layers are twisted by a certainangle θmagic(= 1.1◦) with respect to each other[13–15]. The large densityof states of these flat bands makes them an ideal playground for corre-lation physics where the Coulomb interactions exceed the electron kineticenergy set by the bandwidth. Twisted bilayer graphene has been experi-1Chapter 1. Introductionmentally observed to display correlated insulating[14] behaviour at partialfilling of the moire´ superlattice band with the appearance of superconduc-tivity in the regions between these insulating states[15]. Additionally, itdisplays ferromagnetism[16] and quantum anomalous hall effect[17] makingit an ideal system to study the effect of electron correlations and band topol-ogy. But the strict requirements on the twist angle to be within a narrowrange of θmagic makes it challenging to fabricate twisted bilayer graphenedevices.It is here the twisted double bilayer graphene system gains relevance.Unlike monolayer graphene, Bernal stacked bilayer graphene has a parabolicdispersion that can be modulated by a perpendicular electric field[18]. Hence,two Bernal stacked graphene bilayers twisted by an angle θ, presents theadditional opportunity of tuning the band widths and band gaps by dis-placement electric field[19–22]. Previous studies have found the presence ofcorrelated insulating states in twisted double bilayer graphene over a finiterange of displacement fields, which becomes prominent under the applica-tion of an in-plane magnetic field[19, 21, 22]. It was also seen that themetallic region surrounding the correlated insulator states shows non-linearI −V characteristics and a sharp drop in resistivity with temperature remi-niscent of superconductivity[19, 23]. These characteristics and their similar-ity/difference from the former twisted bilayer system makes it an interestingsystem to investigate.In this thesis, we fabricate twisted double bilayer graphene devices usingtwo different fabrication techniques: “tear-and-stack”[24–26] and “cut-and-stack”[27]. By measuring the twist angles of the fabricated devices using lowtemperature electrical transport we determine the efficacy of our fabricationtechniques. Afterwards, we studied the transport properties of a device witha twist angle, θm = 1.39◦, in order to replicate the electronic phase diagramof twisted double bilayer graphene devices.This thesis is organized as follows. Chapter 2 gives the theoreticalframework for monolayer/bilayer graphene and moire´ superlattices. It alsopresents the relevant experimental background of twisted bilayer and twisteddouble bilayer graphene devices. Chapter 3 details the experimental tech-niques employed for the fabrication of twisted double bilayer graphene de-vices. Chapter 4 presents the results of the electrical transport measurementon fabricated devices. Finally, chapter 5 summarizes the key observationsthat emerged out of this thesis.2Chapter 2Theory and BackgroundThis chapter gives a brief review of the relevant properties of monolayer/bilayergraphene and moire´ superlattices.2.1 GrapheneGraphene is a monolayer of sp2 hybridized carbon atoms arranged on ahoneycomb lattice. This can be thought of as two interpenetrating triangularsublattices. Hence, a unit cell of graphene has two inequivalent carbon atomsdenoted as A and B in Fig. 2.1. If a ≈ 1.42 A˚ is the distance between thenearest neighbours, the primitive lattice vectors are given bya1 =a2(3,√3),a2 =a2(3,−√3). (2.1)As each atom in the sublattice A is surrounded by three atoms from sub-lattice B and viceversa, the nearest-neighbour vectors can be representedbyδ1 =a2(1,√3), δ2 =a2(1,−√3), δ3 = −a(1, 0) (2.2)and the second-nearest neighbour vectors byδ′1 = ±a1, δ′2 = ±a2, δ′3 = ±(a2 − a2). (2.3)Also, the reciprocal lattice is spanned byb1 =2pi3a(1,√3),b2 =2pi3a(1,−√3). (2.4)The first Brillouin zone has the following high symmetry points which aregiven by the wave vectorsΓ = (0, 0),M =2pi3a(1, 0),K =2pi3a(1,1√3),K ′ =2pi3a(1,− 1√3). (2.5)The two nonequivalent corners K and K ′ of the first BZ are known as theDirac points and play an important role in defining the physics of graphene.32.1. GrapheneA Ba1a2321(a) (b)(c)Figure 2.1: (a) Honeycomb lattice. The unit cell is represented by dashedlines and the two sublattices composed of atoms A and B are indicated bythe yellow and blue dotted triangles. (b) the first BZ of graphene with thehigh symmetry points. (c) electronic dispersion of graphene. The zoomedpart shows the linear band crossing near the K(K ′) point. Adapted from[28].42.1. GrapheneThe electronic structure of graphene can be derived using the tight bind-ing approximation [28, 29] taking into account of the nearest and next-nearest neighbour hopping. The Hamiltonian can be written asH = −t∑〈i,j〉,σ(a†σ,ibσ,j + H.c.)− t′∑〈i,j〉,σ(a†σ,iaσ,j + b†σ,ibσ,j + H.c.) (2.6)where ai,σ(a†i,σ) is the annihilation (creation) operator for an electron onsublattice A with spin σ and bi,σ(b†i,σ) for sublattice B. t = 2.8 eV and t′ =0.1 eV are the nearest and next-nearest neighbour hopping energies. Theabove Hamiltonian can be diagonalized to derive the electronic dispersionof the formE±(k) = t√3 + f(k)− t′f(k) (2.7)wheref(k) = 2 cos(√3kya) + 4 cos(√32kya) cos(32kya). (2.8)We can expand the above equation by substituting for k=K+q to get thedispersion close to the Dirac points. For |q| |K|, the energy bands aregiven byE±(k) ≈ h¯vF|q|+O[(q/k)2] (2.9)with q being the momentum measured with respect to the Dirac pointsand vF =3at2h¯ is the Fermi velocity. The key feature of the above equationis that the energy bands linearly cross at the K(K ′) points. The aboveapproximation is valid for small carrier densities and hence the grapheneacts as a zero band gap material with a linear dispersion. From equation(2.9), the Hamiltonian near the K(K ′) points can be written asHk =[0 ∆k∆∗k 0]= h¯vFσ · q, (2.10)where σ is the corresponding Pauli matrices. This is the equation for mass-less chiral Dirac fermions in 2D where the speed of light has been replacedby vF and with a pseudospin spinor structure related to the graphene sub-lattices. Many of the interesting properties of graphene can be mapped backto this unique dispersion.Figure 2.2(a) and 2.2(b) gives the device configuration and the variationof graphene sheet resistivity, ρ, under the application of an electric fieldrespectively [31]. The inset of the Fig. 2.2(b) gives the energy dispersionnear the Dirac points with the position of the Fermi energy, EF. As the52.2. Bilayer grapheneFigure 2.2: (a) The schematic of the graphene field effect device. (b) Thevariation of graphene sheet resistivity with the application of gate voltage.The inset shows the location of Fermi energy, EF within in the Dirac cone.Panel (a) is adapted from [30] and panel (b) is adapted from [31].gate voltage is tuned from negative to positive values, ρ increases sharplypeaking at V g ≈ 0 and falling off rapidly towards higher positive values.The application of a negative (positive) V g shifts the position of EF down(up) inducing holes (electrons) on graphene in concentrations, n = −CV g/e,where C is the capacitance of the gate dielectric. The peak in ρ versus V gsignifies the charge neutrality point with EF located at the Dirac point.The Hall coefficient, RH, (not shown in figure) further exhibits a sharpreversal of sign at the charge neutrality point marking a transition fromhole to electron transport [2]. The presence of unintentional doping due toimpurities/disorder or a work function difference between the graphene andgate electrode often tend to shift the charge neutrality point from V g = 0[32].2.2 Bilayer grapheneTwo layers of graphene can be stacked atop each other to form bilayergraphene. In AB stacking or Bernal stacking arrangement (Fig. 2.3 (a)),the two layers are shifted by one atomic spacing i.e, half the atoms of thetop layer are aligned to the atoms in the bottom layer and the other halfis above the center of the hexagon of the bottom layer. The energy disper-sion of bilayer graphene can be derived by constructing the corresponding62.2. Bilayer graphenetight-binding Hamiltonian [28, 29, 33] allowing for both in-plane nearestneighbour hopping, γ0 = t ≈ 2.8 eV, and interlayer hopping, γ1 = t⊥ ≈ 0.4eV, between atoms that directly lie on top of each other. In the simplest ap-proximation, we ignore the weaker hopping terms (γ3 and γ4) in fig. 2.3(a).The resulting Hamiltonian is given byH = −γ0∑〈i,j〉,m,σ(a†i,m,σbj,m,σ + H.c.)− γ1∑j,σ(a†j,1,σbj,2,σ + H.c.) (2.11)where ai,m,σ(bi,m,σ) annihilates an electron on sublattice A (B) with spin σon plane m = 1, 2. When an electrochemical potential ∆ is added betweenthe layers by raising the energy on plane 2 by ∆/2 and lowering the energyon plane 1 by −∆/2 the Hamiltonian in the k space near the K(K ′) pointscan be represented asHk =−∆2 vFk 0 0vFk−∆2 γ1 00 γ1∆2 vFk0 0 vFk∆2 (2.12)and the resulting energy spectrum is2±k =∆24+ v2Fk2 + t2⊥/2±√∆2v2Fk2 + t2⊥v2Fk2 + t4⊥/4. (2.13)This equation gives rise to the four bands shown in Fig. 2.2(c) near theK(K ′) points for ∆ = 0 (left) and ∆ 6= 0 (right). For ∆ = 0 and vF t⊥,the lowest two bands can given by±k ≈ v2Fk2t⊥. (2.14)Thus both pristine single and bilayer graphene do not feature a band gap;however, unlike single layer, bilayer has a parabolic dispersion with an ef-fective mass, m∗ = t⊥2v2F. When ∆ 6= 0 and for V t, the lowest two bandswill be±k ≈ V − 2V v2Fk2/t⊥ + v4Fk4/2t2⊥V. (2.15)Thus bilayer graphene provides the additional opportunity to tune the bandgap by the application of a perpendicular external electric field [18, 34].72.3. Moire´ Superlattices(b)(c)Δg(a)BAγ1γ0γ3γ4Figure 2.3: (a) The AB stacking arrangement of bilayer graphene andthe corresponding (b) Brillouin zone. γ0, γ1, γ3 and γ4 are the respectivehopping energies (c) The bilayer graphene energy dispersion near the K(K ′)point in the absence of an interlayer potential, ∆ = 0 (left) and ∆ 6= 0(right). Panel (a) is adapted from [35], panel (b) is adapted from [28], andpanel (c) is adapted from [18].82.3. Moire´ Superlattices2.3 Moire´ Superlattices2D crystals with a lattice mismatch or a relative twist angle between themwhen stacked on top of each other give rise to a geometric interferencepattern, a moire´ pattern. For a two layer scenario involving graphene [36],the wavelength of the moire´ pattern, λ, is given byλ =(1 + δ)a√2(1 + δ)(1− cos θ) + δ2 (2.16)where a is the lattice constant of graphene, δ is the lattice mismatch be-tween the two different 2D crystals and θ is the twist angle between thetwo lattices. This wavelength becomes much larger than the underlying lat-tice constants for small θ. Moire´ patterns acts as an effective superlatticepotential with a periodicity different from the original 2D lattice hence mod-ifying the electronic structure significantly when λ becomes comparable tothe Fermi wavelength, λF = 2√pin .One instance where the electronic structure of graphene gets modified asa result of the moire´ superlattcie potential is the appearance of secondaryDirac points in aligned graphene-hBN heterostructures [9–12]. The presenceof a large superlattice period λ, results in smaller size of the superlattice BZ.The resulting zone folding of the graphene band structure into the smallersuperlattice BZ leads to the formation of minibands and replication of Diracpoints at higher energies given byE =h¯vF|G|2=2pih¯vF√3λ(2.17)where G is the reciprocal superlattice vector. These secondary Dirac pointsappear as dips in the local density of states symmetrically placed aroundthe original Dirac point in scanning tunnelling spectroscopy [9]. They canalso be detected in transport measurements as satellite peaks appearingsymmetrically on either side of the main charge neutrality point [10–12]as shown in Fig.2.4. The large moire´ wavelength of aligned graphene-hBNheterostructures, λ, also gets comparable to the magnetic length, lB =√h¯eBat moderately high magnetic fields (≈ 40 T) resulting in the appearanceof a fractal energy diagram know as Hofstader butterfly [10–12]. Moire´superlattices thus provide an exciting opportunity to alter the electronicstructure and engineer novel device properties. Incidentally, the very firststudies on moire´ potential affecting the electronic structure were done onCVD grown few layer graphene films where the alternate layers are naturally92.3. Moire´ Superlatticestwisted relative to each other [37–39]. In that case, the expression for themoire´ wavelength 2.16 reduces toλ =a2 sin(θ/2). (2.18)The next section will talk about the properties of twisted graphene layers.(a) (b)(c)Figure 2.4: (a) Moire´ pattern for graphene (gray) on hBN (red and blue)for θ = 0 and an exaggerated lattice mismatch of 10%. (b) Brillouin zonefolding in the moire´ superlattice. (c) The resistance of aligned graphene-hBN device as a function of the gate voltage. The inset shows the imagesof the heterostructure before and after thermal annealing and the schematicof the modified band structure. Panel (a) is adapted from [12], panel (b) isadapted from [40], and panel (c) is adapted from [36].102.3. Moire´ Superlattices2.3.1 Twisted graphene layersTwisted bilayer graphene (TBG)Similar to graphene-hBN heterostructure, two graphene layers twisted at anangle θ with respect to each other results in the formation of a mini BZ [13].This happens as the twist angle θ displaces the individual monolayer Diracpoints by a wave-vector |kθ| = 2|K| sin θ/2. There exists a finite interlayercoupling between the two monolayer graphene sheets causing hybridizationof the moire´ bands leading to gap openings at the intersection of the Diraccones [13, 14, 41]. Hybridization also renormalizes the Fermi velocity atthe Dirac points [38, 39]. Bistritzer and MacDonald [13] uncovered a seriesof magic angles, θ(i)magic = 1.05◦, 0.5◦.., at which the Fermi velocity at theDirac points vanishes resulting in a flat moire´ band with a large Dirac pointdensity-of-states. These flat bands near charge neutrality points are a con-sequence of the competition between the interlayer hybridization energy andthe kinetic energy. When the hybridization energy, 2w, gets comparable toor larger than h¯vFkθ (Fig.2.5(c)), the lower hybridized states are pushed tozero energy resulting in bands with extremely narrow band width.The presence of flat bands with large DOS near the charge neutralitypoint causes break down of the single particle picture as the Coulomb in-teractions far exceeds the kinetic energy. When the Fermi energy is tunedwithin the flat bands, twisted bilayer graphene is driven into various stronglycorrelated and topological states. Experimentally it has been observed tohost correlated insulating (CS) states [14], superconductivity[15], ferromag-netism and quantum anomalous Hall effect [16, 17] . Each moire´ superlatticeband is four fold degenerate at low twist angles and the carrier denisty re-quired to fill one superlattice band is given byns =4A=8θ2√3a2(2.19)where A is the area of the moire´ unit cell and a is the lattice constant ofgraphene. As seen in Fig.2.6(c), the resistance values peak at full filling(ν = 4) and also at quarter (ν = 1), half (ν = 2) and three-quarter filling(ν = 3). The filling fraction ν = 4n/ns defines the number of electronsper moire´ unit cell. While the presence of the insulating states at full fill-ing arise on account of the superlattice gap (band insulator), the states atν = 1, 2 and 3 occurs as a result of electronic correlations. Additionally,in proximity to the insulating states the resistances goes to zero signallinga superconducting state. The superconducting states appear in the regions112.3. Moire´ Superlattices(a) (b)(c)Figure 2.5: (a) Moire´ pattern in TBG. (b) The moire´ superlattice Brillouinzone. (c) Schematic of the effect of interlayer hybridization, w, on the moire´bands. Adapted from [14].122.3. Moire´ Superlatticessurrounding the correlated insulating states. The remarkable similarity ofthis superconducting state to the cuprates suggests a plausible unconven-tional all electronic origin to it.(a) (c)(b) (d)Figure 2.6: (a) Calculated band structure at θ = 1.08◦ for TBG. (b) AFMimage of the magic angle TBG device. Scale bar is 2µm. (c) The longitudinalresistance as a function of the carrier denisty. (d) Plot of longitudinal resis-tance vs carrier density and temperature indicating various band insulator(BI), correlated insulator (CS) and superconducting (SC) states. Transitionfrom normal metal state to SC is defined at 50% resistance value. Panel (a)adapted from [14], (b)-(d) from [42].Twisted double bilayer graphene (TDBG)As mentioned in the section 2.2 the band gap of the AB stacked bilayergraphene can be modified with the application of an electric field. Hence,two AB stacked graphene bilayers twisted at an angle θ with respect to eachother provides the extra freedom to tune the bandstructure using a trans-132.3. Moire´ Superlatticesverse electric field [19–22]. Fig. 2.7 (a) and (b) shows the calculated bandstructure of twisted double bilayer graphene for θ = 1.23◦ in zero and finiteexternal displacement fields respectively. It can be seen that the displace-ment field widens the charge neutrality gap analogous to bilayer graphene,but reduces the superlattice gaps above and below the superlattice bands.Hence for a range of D values the moire´ superlattice band becomes flat andisolated and results in strong correlations. This has been experimentallyobserved using a double gated device [19–22] as shown in Fig.2.7(c). In sucha scenario, both the displacement electric field (D) and carrier density (n)can be independently controlled and are related to the V tg and V bg byn = [Ctg(V tg − V tg,0) + Cbg(V bg − V bg,0)]/e (2.20)D = [−Ctg(V tg − V tg,0) + Cbg(V bg − V bg,0)]/2 (2.21)where Ctg (Cbg) and V tg,0 (V bg,0) are the top (bottom) gate capacitanceand offset respectively. Like TBG the moire´ bands in TDBG are also fourfold symmetric. The plot of resistance of θ = 1.09◦ vs both V tg and V bgreveals the effect of displacement field on the band structure. For D = 0,the resistance peaks at n = ns. As D is increased, new insulating states areinduced at ns/2 for a range of |D| values. This appearance of a gap in thehalf filled moire´ band exhibits the electric field tunability of correlations inTDBG heterostructures.142.3. Moire´ Superlattices(a)(c)(b)(d)Figure 2.7: Calculated band structure and denisty of states (DOS) at θ =1.23◦ for TDBG for (a) D = 0 and (b) D 6= 0. (c) Schematic of the TDBGdevice. (d) Plot of longitudinal resistance vs V tg and V bg. Red (blue) arrowindicates the direction of increasing carrier density (displacement field) nsindicates the carrier density at which the moire´ band is fully filled. Adaptedfrom [22].15Chapter 3Experimental TechniquesThis chapter presents the techniques employed in the fabrication and mea-surement of twisted graphene devices. At first, we describe the process ofexfoliation of graphene and hBN onto SiO2 (285nm)/Si substrates. Next, inorder to assemble the exfoliated flakes to make the twisted devices we usedtwo different modified dry transfer techniques using a transfer stamp made ofa hemispherical handle substrate made of two polymers polydimethylsilox-ane (PDMS) and Poly(Bisphenol A) carbonate (PC). The prepared stacksare then patterned using electron-beam lithography and plasma etching ontoa double gated hall bar device for electrical transport measurements.3.1 ExfoliationGraphene flakes were mechanically exfoliated onto SiO2 (285nm)/Si sub-strates ( ∼ 1× 1 cm2) using the Scotch Ruben Magic tape [1, 2]. A portionof natural graphite was peeled on and subsequently cleaved to the freshregions of the tape. Care is taken to exfoliate the initial graphite pieceuniformly so that it does not get split into tiny parts. This process is con-tinued carefully until we get a region on the tape that is completely coveredwith thin and translucent graphite flakes. This region is then transferred tothe substrate and pressed using a cotton swab to remove the blisters andimprove the adhesion. Finally, the tape is slowly peeled off from the sub-strate and examined under an optical microscope for monolayer and bilayerflakes distinguished by optical contrast. Figure 3.1(a) shows the opticalmicorgraph of a graphene flake exfoliated onto SiO2 (285nm)/Si substratesidentifying the mono/bilayer regions. As the whole process relies on the vander Waals attraction between the graphene flakes and the substrate, remov-ing the organic adsorbates from the substrate surface prior to exfoliationhelps improve the adhesion and thereby increase the yield [3, 43]. Typically,we achieve this in our lab by cleaning the substrate in oxygen plasma (PE-50plasma etcher) for about 10-15 minutes. The plasma power is usually setto 100 W at an oxygen flow rate of 15 cubic centimeters per minute whichgives a pressure of 250 mTorr.163.1. ExfoliationHexagonal Boron Nitride (hBN) flakes are used to sandwich the graphenebilayers for improved carrier inhomogeneity [44], which also doubles as agate dielectric. As the bulk hBN crystals are smaller in size compared tographite crystals, we exfoliate them using a similar but slightly modifiedtwo-step process. A thin and flat hBN crystal is stuck and cleaved to a bluetape (Nitto Denko). Once the whole of the tape is covered with thin andsemi-transparent flakes, it is transferred to a scotch tape. Finally, the scotchtape is transferred to the substrate and annealed at 100◦C for 20s. Aftercooling the substrate back to room temperature the tape is slowly peeled offfrom the substrate. Annealing pushes the trapped air out of the interfacebetween the tape and the substrate thereby facilitating a uniform contactover a large area. We use hBN flakes of thickness 10-40 nm to encapsulategraphene bilayers which appears purple to bluish-green colour under opticalmicroscope. 3.1(b) shows the optical micrograph of an exfoliated hBN flakeof thickess ∼ 25µm.monolayerbilayer(a) (b)Figure 3.1: Optical micrographs of flakes exfoliated onto SiO2 (285nm)/Sisubstrates.(a) graphene Mono/bilayer regions are identified as indicated inthe image. The scale bar is 10 µm. (b) hBN of thickess ∼ 25nm. The scalebar is 20µm173.2. Transfer3.2 Transfer3.2.1 Dry transferIn order to assemble the exfoliated flakes into van der Waals heterostructureswe use a dry transfer technique [4, 24] involving a transfer stamp made of ahemispherical handle substrate made of polydimethylsiloxane (PDMS) witha thin film of adhesive polymer Poly(Bisphenol A) carbonate (PC) on topof it. The hemispherical handle is used to selectively pickup flakes fromSiO2/Si one after the other and finally drop them onto a clean wafer.Preparation of Transfer StampThe hemispherical handle of PDMS is made using the same process as ex-plained in [24]. Sylgard 184 prepolymer, and curing agent were mixed in10:1 molar ratio. The bubbles formed during the mixing were removed by in-termittent pumping (for about 2 minutes) and venting of the mixture insidea desiccator using a scroll pump. Whilst pumping the bubbles expand andcoalesce which bursts upon venting the desiccator. This process is repeateduntil the mixture appears clear and bubble-free. Care should be taken so asto not keep the mixture pumping for more than 30 minutes. Next, a glassslide is prepared by laying a planar PDMS (Gel-Pak) (∼ 3×3 mm2 ) on andannealing it at 180◦C for 5 minutes on a hot plate. Lastly, the mixture ispicked up using a tooth pick and dropped on top of the planar PDMS andcured at 180◦C for 5 minutes.PC films were made inside a wetbench from a 6% (by weight) solutionof Poly(Bisphenol A) carbonate in chloroform (it is left overnight for PC todissolve). Once dissolved a few drops of the solution is released on to a glassslide using a pipette and spread to form a thin film by pressing and slidinganother glass slide on top of it. The film is left to dry on the glass slide for20-30 minutes.Finally, the transfer stamp is made by peeling off the PC film from theglass slide and laying it down on top of the hemispherical PDMS with thehelp of a Kapton tape. We stick half portion of the tape to one corner of theglass slide with PC film on it and use the other side to gently peel off thePC film from the slide. We then carefully put it on top of the hemisphericalPDMS so as to cover it completely. After cutting off the excess film with asharp blade, we anneal the stamp at 215◦C for a few minutes to remove thewrinkles on the PC film. With each PC slide we are able to make at least 3stamps.183.2. TransferGeneral Pick-up ProcessGlassPDMSSiO2/SiPChBN(a) (b)(d)(c)(e) (f)Figure 3.2: Process flow illustrating the pick-up of hBN (a) the stampis lowered so as to touch the substrate close to the hBN at 90◦C. (b) thetemperature is increased to 110-120◦C so that the contact area expands andcovers the whole hBN flake (c) after waiting for 1-2 minutes the temperatureis lowered back to 30◦C when the stamp traverses back while picking up hBNalong with it. Optical micrograph showing (d) the stamp touching substrateat 90◦C (e) the stamp covering the whole hBN flake (f) hBN flake pickedup on the stamp. Scale bar is 50 µm.The prepared transfer stamp can be used to pick-up exfoliated flakes asillustrated in Fig. 3.2. The general pick up process can be explained asfollows. For picking up the first flake, which is usually hBN in our case,a region of the transfer stamp close to the center is chosen and alignedcarefully to be over the flake using the micro-manipulator. For hBN, thesubstrate is first heated to a temperature of 90◦C. The transfer stamp isthen lowered down until it touches substrate very close to the hBN. Then,the temperature of the substrate is slowly increased to 110-120 ◦C (at therate of 5 degree in 30s) so that it expands and the whole hBN is contactedby the stamp. After waiting for 1-2 minutes at this temperature, we set thetemperature back to 30 ◦C so that the stamp traverses back as the substratecools naturally and while doing so picks up the hBN flake along with it. In193.2. Transferorder to expedite the cooling process, a handheld, battery operated fan isused. The subsequent layers can be picked up by utilizing the van der Waalsattraction between the different layers by carefully aligning them on top ofeach other. After the entire heterostructure has been stacked we transferit from the stamp to a clean SiO2/Si wafer with pre-patterned alignmentmarks for nano-fabrication. This is done at a temperature of 190◦C whenthe PC gets transferred to the wafer. Finally, the wafer is cleaned for 30minutes in chlorofom to dissolve the PC followed by rinsing it in iso-propylalcohol (IPA) for 5 minutes.3.2.2 Fabrication of Twisted Bilayer Graphene devicesThe twisted bilayer graphene devices were stacked by modifying the drytransfer process. In order to make sure that the crystallographic orientationsof the two graphene flakes are the same, they were made out of a singlelarge graphene flake. The twisted bilayer graphene devices in this thesiswere fabricated using two similar but different techniques.“Tear-and-Stack” TechniqueIn the first“tear-and-stack” process [24–26] we use the van der Waals attrac-tion between hBN and graphene to tear a single large graphene flake by theedge of an hBN flake.The schematic of the process flow is shown in Fig. 3.3. The hBN onstamp is used to tear the first half of graphene (Fig. 3.3 (a)-(c)). For this,we choose an hBN flake that is considerably wider than the graphene itself.The hBN on stamp is then aligned such that during contact it covers onlyhalf of the graphene flake. Then, we lower the stamp to touch the SiO2/Sisubstrate at a temperature T1 and then increase the temperature slowly toT2 ( at the rate of 5◦C in 30s) so that the film progresses up to the edge ofthe hBN flake. At this point one half of the graphene is bonded to the hBNflake by van der Waals force and the other half is bonded to the substrate.So when the stamp traverses back, the stronger van der Waals bondingbetween the hBN and graphene forces the part of graphene contacted byit to be lifted up along with it. After tearing the first half, the stage isrotated by the desired angle and then the same process is repeated to pickup the second half of the graphene (Fig. 3.3 (d)-(f)). We intentionally rotatethe stage by an angle 0.2-0.3◦higher than the target angle to account for therelaxation of the graphene bilayers [45]. We fabricated the devices at variousT1 and T2 temperatures in order to optimize the fabrication process.203.2. TransferθGlassPDMSSiO2/Si(a) (b) (c)(d)(e)(f)PC hBNgrapheneFigure 3.3: Schematic of the “tear-and-stack” process of fabrication oftwisted devices. (a) Align hBN on stamp over the first half of the graphene.(b) laminate the first half of graphene with the hBN on stamp. (c) pick-upthe first half of graphene. (d) rotate the second half of graphene by thedesired angle. (e) laminate the second half of graphene with graphene/hBNon stamp (f) pick-up the second half of graphene213.2. Transfer(a) (b)(c)(d)Figure 3.4: Optical micrographs of (a) bilayer graphene. The scale bar is20 µm.(b) Aligning hBN for tearing the first half of graphene. (c) Grapheneon substrate after tearing and (d) picking up the second half of grapheneafter rotating the desired angle. Scale bar is 50 µm.223.2. TransferFig. 3.4 shows the optical micrographs of the “tear-and-stack” process.Fig. 3.4 (a) shows the bilayer graphene flake to be torn. Fig. 3.4(b) showsthe tearing process using the top hBN as explained in the previous para-graph and Fig. 3.4(c) shows the second half of graphene remaining on thesubstrate after tearing. After rotating the stage by the desired angle thesecond graphene is also picked up Fig. 3.4(d). The stack is then completedby picking up a bottom hBN to sandwich the graphene and also at times afew layer graphite as back gate electrode.Though the “tear-and-stack” modification gives a method to control thetwist angle, we have encountered several problems during the fabricationprocess. When the tearing is done at low temperatures (T1 and T2< 100),a greater bonding force at the graphene-substrate interface compared to thegraphene-hBN interface meant one of the three things in Fig. 3.5 happenedoften. Fig. 3.5(a) and (b) shows partial pick up of the graphene regioncovered hBN. Fig. 3.5(c) and (d) shows graphene not being torn at theedge of the hBN and getting stretched and folded while the stamp traversesback. Finally, in the extreme scenario the hBN drops to the graphene onthe substrate as seen Fig. 3.5(e) and (f). Hence it is always preferableto use high temperatures for tearing graphene. The effect of the tearingtemperatures on the measured twist angles of the fabricated devices will bepresented in section 4.1.“Cut-and-Stack” TechniqueThe complications that arose during the “tear-and-stack” process promptedit’s modification to the “cut-and-stack” technique [27]. The significant dif-ference here is that the graphene flakes were cut prior to stacking usinga surgical needle of 1 µm diameter tip attached to a micro-manipulatorsetup. This would help drastically reduce the strain compared to tearingthe graphene using hBN [27] and also allowed pick-up of graphene at muchlower temperatures (50-65◦C). The cutting process is described briefly asfollows. After bringing the graphene flake to be cut under the field of viewof the optical microscope, we also bring the needle close to the flake usingthe micro-manipulator. We then lower the needle such that it touches thesubstrate on one side of the flake. We then move the needle gently over theflake along the appropriate cutting line.The process flow of “cut-and-stack” is very similar to the one shownin Fig. 3.3, with the main difference being the graphene is pre-cut priorto staking. Fig. 3.6 shows the optical micrographs of the twisted bilayerpickup process. The graphene flake in Fig. 3.6(a) is cut using the needle into233.2. Transfer(a)(b)(c)(d)(e)(f)Figure 3.5: Optical micrographs of failed tearing attempts at temperatures<100◦C. (a) and (b) only a part of graphene gets picked up. (c) and (d) Thegraphene doesn’t tear at the edge and gets folded. (e) and (f) hBN dropson to the graphene during tearing. Scale bar is 50 µm.243.2. Transfer(a) (b)(c) (d)Figure 3.6: Optical micrographs of graphene (a) before and (b) after cut-ting. The scale bar is 20 µm. Pick up process for (c) first half of graphene(d) second half of graphene. Scale bar is 25 µm.253.2. Transferthree parts as shown in Fig. 3.6(b). Fig. 3.6(c) shows the pick-up processof the first half of graphene using a top hBN. After aligning the edge of thehBN flake with the cutting line, the stamp is allowed to touch the SiO2/Sisubstrate at 50◦C (T1) and then the temperature is slowly increased to65◦C (T2) ( at the rate of 5◦C in 30s) so that the film progresses up tothe cutting line. After picking up the first half, the stage is rotated by thedesired angle and then the same process is repeated to pick up the secondpart of the graphene Fig. 3.6(d), here also a bottom hBN sandwiches thegraphene flakes and also at times a few layer graphite is picked up as backgate electrode.(a) (b)Figure 3.7: (a) Optical image of a completed stack. The scale bar is 20 µm.(b) The AFM image of the region enclosed in the red boundary in (a).Thescale bar is 8 µm.After stacking we perform the Atomic Force Microscopy (AFM) in tap-ping mode to find the regions of the stack free of bubbles. Figure 3.7 showsthe optical and atomic force micrographs of a completed stack. The clean re-gion in the middle of the two graphene flakes, as seen in the AFM, would bepatterned into a device using electron-beam lithography and plasma etching.263.3. Device Fabrication3.3 Device FabricationElectron-beam lithography on the completed stacks were done using the100kV JEOL JBX-8100FS system at the Advanced Nanofabrication Facility(ANF), UBC. Since the twisted graphene part is completely encapsulatedby hBN, in order to make metallic contacts we follow the procedure in [4],whereby only the edge of the graphene is exposed for metalization throughplasma etching.(a)(c)(b)(d)Figure 3.8: Optical micrographs (a) of the completed stack. The scale baris 20 µm. (b) after top gate evaporation (c) after contact metalization (d)final device.The scale bar for (b),(c) and (d)is 10 µm.We use PMMA 495A4/ PMMA 950A4 double layer resists spin coatedat 4000 rpm for all the lithography steps. After spin coating each layerof the resist, the stack is annealed at 120◦C for 3 minutes. For pattern-ing we set the dose to be 1000µC/cm2 and develop in a 7:3 solution ofIPA:deionized water for 50s. Optical micrographs of the entire fabrication273.4. Electrical Transport Measurementsprocess is demonstrated in Fig. 3.8. First, we pattern PMMA to define theshape of the top gate and then evaporate 5/80 nm of Ti/Au as top gateelectrode(3.8(b)) by electron-beam evaporation (AJA UHV hybrid evapo-rator) and lift off in hot Acetone. Next, for contact metalization, patternfor the leads are written and etched using a PLASMAQUEST: ECR plasmaetcher. We use CHF3/O2 at a flow rate of 40 standard cubic centimeters perminute (sccm)/ 4 sccm and a pressure of 40 mtorr to generate the plasma.For a microwave power of 200 W and the RF power of 30 W, the etch rateis around 45 nm/min. After etching, the contact leads are evaporated rightaway to this exposed edges of the graphene( 3.8(c)). The contacts are alsomade to the graphite back gate electrode (not shown in the figure for thisdevice) during this same step. In the final step, the remaining regions areetched out to get a hall bar shape ( 3.8(d)). Once the device fabricationis completed, they are pasted on to a chip carrier using silver paste andannealed at 110◦C for 10 minutes. After this the device is wire bonded tothe chip carrier using gold wires (West.Bond Wedge-Wedge). Care is takenso as to prevent the SiO2 dielectric (sometimes used as a back gate in theabsence of graphite back gate electrode) from cracking under the force ofbonding. This is typically done by finding the lowest possible force to bondby testing on a blank bondpad.3.4 Electrical Transport MeasurementsThe initial characterization and testing of the fabricated devices are doneat liquid helium temperatures. The chip carrier is placed inside a dunker,which in turn gets inserted into a liquid helium dewar. After the initialcharacterization of the twist angle and contact resistances, the device wasmeasured in a dilution refrigerator (Bluefors-XLD). The measurements ofthe device resistances were done in four-probe and two-probe configurationsusing low frequency lock-in techniques (Fig. 3.9).283.4. Electrical Transport MeasurementsVoltage dividerCurrent pre-amplifiero/pi/pLock-in amplifierRLo/pi/pLock-in amplifierVoltage pre-amplifier(a)(b)13 42SSDD1 23 4IFigure 3.9: Schematic of measurement configurations (a) Two-probe; Abias voltage of 10-100µV is applied using a lock-in/voltage divider to oneof the terminals of the hall bar and the signal is measured across anotherterminal after amplifying it through a current amplifier using the lock-in.(b) Four-probe; a current of 0.5-5nA is applied across S and D terminalsusing the lock-in/RL and the voltage across terminals 1 and 2 is measuredwith the lock-in after amplifying it with a voltage pre-amplifier.29Chapter 4Results and DiscussionIn this chapter I will summarize the main results of electrical transportmeasurements done on twisted double bilayer graphene devices. First, inorder to compare the efficacy of our fabrication techniques, we have calcu-lated the twist angles of the devices from the electrical transport data (θm)and compared it with the target angle (θt) for both the “tear-and-stack”(TS1-TS8) and “cut-and-stack”(CS1-CS3) methods. We observe that the“cut-and-stack” technique gives a better control of the twist angle of thefabricated devices. Subsequently, we present the low temperature transportdata for one of the devices with measured angle, θm = 1.39◦, which displayedcorrelated insulating peaks at 1/4, 1/2 and 3/4 filling fractions.4.1 Twist angle measurementsIn order to extract the twist angles, we performed electrical transport mea-surements on the fabricated devices at 4.2K. Fig. 4.1(a) gives the map ofthe four probe longitudinal resistance, Rxx, versus the top gate voltage, V tg,and the back gate voltage, V bg, for a device, TS2, measured at 4.2K. Usingequations 2.20 and 2.21,n = [Ctg(V tg − V tg,0) + Cbg(V bg − V bg,0)]/eD = [−Ctg(V tg − V tg,0) + Cbg(V bg − V bg,0)]/2we can convert the gate voltages to carrier density (n) and electric field (D),where Ctg and Cbg are the capacitance values of the top and back gatesrespectively, estimated from the dielectric thickness (hBN and SiO2). Thediagonal axes indicate the direction of increasing D (red) and n (white). Weobserve insulating states at carrier densities denoted by n = ±ns over a widerange of D values. These insulating states corresponds to the single particlegaps when each moire´ superlattice band is completely filled. The bottompanel of Fig. 4.1(b) gives the line cut corresponding to D = 0 (n axis)plotted as a function of carrier density where the insulating states appear304.1. Twist angle measurements !" #""#"!" $%&'() *" + " + *" ,%&'()!"!-!.!- !.!-"#" *+ *" + "#//&'01)#"*+*"+"#//&'01) # " #!&'*"*#23 #) *4" "4+ "4" "4+ *4"!5!.'6) '$)$-!4#7Figure 4.1: (a) Map of Rxx vs V tg and V bg for device TS2 at 4.2K. Thediagonal axes indicate the direction of increasing D (red) and n (white).The peaks corresponding to the full filling of the moire´ superlattice bandsare denoted by n = ns and n = −ns. n = 0 corresponds ot the chargeneutrality peak. (b) Line cut of Rxx plotted as a function of carrier density(n) at D = 0 (n axis). The value of θm calculated is 1.12◦. The mirror axisdenotes the filling fraction nns .symmetrically at superlattice densities, |ns| ≈ 2.94× 1012 cm-2. We can usethis value of ns to extract the twist angle using equation 2.19 consideringeach moire´ superlattice band to be four fold symmetric (which is the casefor both twisted bilayer and twisted double bilayer devices). Substitutingthis value of ns inθm2 =√3a2ns8we find the value ofθm = 1.12◦. (4.1)This value of θm is lower than the target angle of the device (θt = 1.4◦),which is defined as the angle to which the stage was rotated before pickingup the second graphene bilayer at the time of stacking. However, it shouldbe noted that there is a degree of uncertainty associated with the value of nscalculated this way[14, 21], which arises on account of the presence of local-ized states and difficulty in determining the exact dielectic thickness. Thisadds some degree of uncertainty to the calculated θm. A more accurate valueof ns (and hence θm) can be obtained with the help of Hall measurements.314.1. Twist angle measurementsTable 4.1: The target angle (θt) and measured angle (θm) of all the devicesfabricated during this work. T1 and T2 corresponds to the temperatures atwhich the stamp touches the substrate and the stamp progresses until theedge of hBN/cutting line respectively. Devices CS3 and TS4 indicated by(SL) are twisted bilayer graphene devices included for the sake of complete-ness.Device Temperature Target Angle Measured Angle |θt − θm|T1 T2 (θt) (θm)“Cut-and-stack” devicesCS1 50 65 1.4 1.14 0.26CS2 50 65 1.4 1.39 0.01CS3(SL) 50 65 1.28 1.17 0.11“Tear-and-stack” devicesTS1 50 60 1.5 0.68 0.82TS2 50 60 1.4 1.12 0.28TS3 70 80 0.9 0.95 0.05TS4(SL) 80 95 1 0.87 0.13TS5 120 140 1.2 1.87 0.67TS6 120 140 1.12 No SL peak –TS7 120 140 1.46 No SL peak –TS8 120 140 1.52 No SL peak –In order to determine the effectiveness of our fabrication techniques, wemeasured the twist angles of all the fabricated devices. The summary ofthe target (θt) and measured angles (θm) of all the devices fabricated inthis work are presented in Table 4.1. As mentioned in section 3.2.2 we fab-ricated devices using both “cut-and-stack”(CS1-CS3) and “tear-and-stack”(TS1-TS8). While the “tear-and-stack” devices were made at various T1and T2 temperatures (refer section 3.2.2), “cut-and-stack” devices were allfabricated at same T1 and T2. Based on the reliability of fabrication andtwist angle control, we can make three observations:• “Cut-and-stack” devices showed the least deviation (|θt− θm|avg ∼ 0.13◦)between θt and θm of all the fabricated devices .• “Tear-and-stack” devices fabricated at temperatures T1, T2 < 100◦C(TS1-TS4) showed an average deviation of ∼ 0.3◦• “Tear-and-stack” devices (TS5-TS8) fabricated at temperatures T1, T2324.1. Twist angle measurements> 100◦C showed a large deviation between θt and θm. As seen fromthe Table 4.1, three of the four devices fabricated did not show anysuperlattice peak making it impossible to determine the twist angle.The better quality of the “cut-and-stack” devices arises from the factthat the pre-cut graphene precludes unintentional strain generation on thefabricated devices, which would affect the moire´ pattern and hence the twistangle. Additionally, the pre-cut graphene can be picked up at much lowertemperatures, the significance of which is apparent from Table 4.1. At highertemperatures there is a tendency for the graphene flakes to be mobile on thestamp resulting in unreliable twist angle[26]. Since tearing graphene at lowtemperatures presented additional problems during pick up, we recommendthe “cut-and-stack”technique for reliable control of the twist angle for thefabricated devices.Fig. 4.2 gives the Rxx maps of three different twisted double bilayergraphene devices, TS1 (θm = 0.68◦), TS5 ( θm = 1.87◦) and CS2 (θm =1.39◦), and their corresponding line cuts at D = 0 plotted as a function ofcarrier density (n). We observe various moire´ superlattice features for thesethree different twist angles.For the lowest angle device, TS1, peaks corresponding to the full filling ofhigher moire´ bands such as 2ns, 3ns, and 6ns were observed (Fig. 4.2 (a) and(b)). This is because for small twist angles, a number of moire´ superlatticebands are present within the experimentally accessible carrier densities[22].Additionally, we observe a peak Rxx value at the charge neutrality point forD = 0. This observation is supported by band structure calculations, whichshows the gap at the charge neutrality point to open for θ > 1.4 or θ < 1.1[46].On the other hand, for the highest angle device, TS5, we only observeRxx peaks corresponding to the full filling of the lowest moire´ bands atn = ±ns and the charge neutrality point for D = 0 (Fig. 4.2(c) and (d)).This gap at CNP closes and opens again with increasing D. The gap closingand opening with the electric field at large twist angles (θm ∼ 2◦) have beenobserved previously [19] and may be due to the overlap and separation ofthe conduction and valence bands under a displacement electric field. Forfinite D values, no additional features appear in both TS1 and TS5.Finally, device CS2 only display Rxx peaks at n = ±ns for D = 0 (Fig.4.2 (e) and (f)). Contrary to both TS1 and TS5 which display features thatcan be explained within the single particle picture, well developed insulatorstates at half filling of the moire´ band, n = ns/2, were observed in deviceCS2, at finite displacement electric fields. These correlated insulating states334.1. Twist angle measurementsappear at half filling only for a range of twist angles (lowest θ = 1.01◦ in[20] and the highest θ = 1.53◦ in [23]) where the moire´ band is flat andisolated[47]. In the next section, we present the study of these correlatedinsulator states and the surrounding low resistance regions in detail.344.1. Twist angle measurements !" #""#"!" $%&'() ! " ! *%&'()+,-.!/"01231"45"4 #"4"4""/"1" !" #" "#66&'78) -" 9"9-" $%&'()2!" ! 2 *%&'():,#.&!/-05;3""/"4"4<#"/ "4"/"#" -9 -" 9 "#66&'78) -" 9"9-" $%&'() -" " -" *%&'()+,9.&!/-02=3"/"4"/ "4"/""-"" "#66&'78)-"21!##66&'78) -" 9 " 9 -""&'-"-#>? #) -" 9 " 9 -""<"4#"-9-"9"#66&'78) -" 9 " 9 -""&'-"-#>? #) # - " - #"<"4-""2"1"!"#""#66&'78) -" 9 " 9 -""&'-"-#>? #) - " -"<"4'@) '$)'>)'A)'B)'C)Figure 4.2: . Map of Rxx vs V tg and V bg and their corresponding line cutsat D = 0 (n axis) plotted as a function of carrier density (n) for device TS1,θm = 0.68◦, (a) and (b), TS5, θm = 1.87◦, (c) and (d) and CS2, θm = 1.39◦,(e) and (f) at 4.2K.354.2. Correlated Insulator States for device CS24.2 Correlated Insulator States for device CS2 !" #"#!" $%&'()*+,-" - , + * .%&'()!/ !/!/0-!1""" "2-#&"2-#& "2#&"2#&&'(034)#1!2567# , 5 - ! "$88&'9:)%1!+&4;Figure 4.3: Map of Rxx vs V tg and V bg for device CS2 at T = 16mK. Thediagonal axis gives the value of D. The peaks corresponding to n = |ns|,n = ns/2, and the CNP(n = 0) are indicated.In order to study the correlated insulating states more thoroughly, deviceCS2 was cooled down for measurements in a dilution refrigerator. Fig. 4.3gives the map of Rxx vs V tg and V bg for this device at 16 mK. Similar to theprevious section, we observe resistance peaks at n = |ns| for D = 0V/nmsignifying full filling of the moire´ band (four electrons per moire´ unit cell).These single particle gaps exist for the whole range of D accessed during thismeasurement on the electron side (n = ns) but vanish for |D| > 0.5V/nm onthe hole side (n = −ns). The CNP becomes insulating for |D| > 0.25V/nm.In addition to this, we observe fully developed resistance states at ns/2 fora narrow range of |D| (0.3 ≤ |D| ≤ 0.59 V/nm). These gaps on a half-filled moire´ superlattice band are believed to occur as a result of electronic364.2. Correlated Insulator States for device CS2ΔCNPΔnsΔ-ns|D|=0 V/nm |D|=0.2 V/nm |D|=0.36 V/nmK1K2D≠0K1 K2ΔCNPKssMsss Ks Msss Ks Msss Ks Ms(a) (b)(c)K1 K2 K1 K2D=0 D=0wFigure 4.4: (a) BZ of twisted double bilayer graphene for a twist angle,θ. Red (blue) coloured hexagon indicate the individual BZ of top (bot-tom) Bernal stacked bilayer graphene. Moire´ BZ is also displayed (blackhexagon).(b) Illustration of electric field tunability of twisted double bilayergraphene. Uncoupled (left) and coupled graphene bilayers (centre) at D = 0.w is the hybridization gap. With the application of D (right), a gap opens atthe charge neutrality point ∆CNP further tuning the band structure and en-abling flat band formation. (c) The calculated band structures of a twisteddouble bilayer graphene device with θ = 1.3◦ taken from [21] for variousvalues of |D|.correlations. On the hole side, we do not observe an analogous resistancepeak at −ns/2, but a cross-like feature centered at D = 0 is observed.These observations are in line with previous experimental and bandstructure calculations[19, 21, 22] where an isolated flat band appears in theconduction band only for a finite range of electric fields. Fig. 4.4(a) showsthe Brillouin Zone of the two Bernal stacked bilayer graphene sheets rotatedby an angle θ. The twist angle displaces the K points of the individualBernal stacked layers and results in the formation of the mini BZ.As explained in section 2.2, bilayer graphene has a parabolic dispersion374.2. Correlated Insulator States for device CS2near the K point. When the two graphene layers are coupled (Fig. 4.4(b)centre panel), it hybridizes the bands at their crossings, opening a gap,resulting in the miniband formation. With the application of a displacementelectric field(Fig. 4.4(b) right panel), further gap openings can be inducedat the charge neutrality point, tuning the band structure and enabling flatband formation. Fig. 4.4(c) presents the calculated band structure of atwisted double bilayer graphene device with θ = 1.3◦ (adapted from [21])for various values of displacement electric fields. At |D = 0|, the systemis gapless at the charge neutrality point with the lowest superlattice bandseparated from the higher energy conduction and valence bands through thesingle particle gaps ∆ns and ∆-ns respectively. With the application of anelectric field, the gap at the charge neutrality point (∆CNP) widens whilethe superlattice band gaps ∆ns and ∆-ns become narrower. Over a range ofintermediate values of |D|, an isolated flat band appears in the conductionband, separated by ∆CNP and ∆ns.The appearance of the correlated half filling insulating states in transportcoincides with the formation of this flat band. As the electric field valuesare further increased, the band width of this flat band increases and thesuperlattice gaps ∆ns and ∆-ns closes making the band no longer isolated.This accounts for the absence of the correlated insulating state at low valuesof D (where there is no manifest gap at CNP) and large values of D (wherethe superlattice gap closes). The absence of correlated insulating states onthe hole side can also be explained by the band structure calculations, as noflat bands appear in the valence band. Additionally, the band structure oftwisted double bilayer graphene also shows electron-hole asymmetry as seenby the difference in values of ∆ns and ∆-ns (Fig. 4.4(c)). Experimentallythis is observed as different D values above which the single particle gapcloses for ns and −ns.Figure 4.5(a) gives a zoomed-in image of Rxx around the ns/2 correlatedinsulating state. For very narrow ranges of D, we are able to see the n =ns/4 and 3ns/4 correlated insulating states. To study the the temperaturedependence of all three states, ns/4, ns/2, and 3ns/4, a comparison withthe 4.2K data for the D = −0.43 V/nm (white dashed line in Fig. 4.5(a))was made. Most of the 4.2K features remain visible down to 16mK butthe insulating peaks become sharper, and the metallic states move to lowerresistance (see Fig. 4.5(b)). If, instead of raising the temperature, oneadds an in-plane magnetic field, all the correlated insulating states becomemore prominent and extend further in D (Fig. 4.5(c)). Fig. 4.5(d) showsthe evolution of the correlated states as a function of in-plane field alongthe D = −0.43 V/nm line (dashed line in Fig. 4.5 (a)). The states at384.2. Correlated Insulator States for device CS2-4-3-2-10Vbg (V)76543V tg (V)nsns/2n=03ns/4ns/45 4 3 2 1 0Rxx (k )B∥ = 0T-4-3-2-10Vbg (V)76543V tg (V)ns/2 3ns/4ns/4nsn=05 4 3 2 1 0Rxx (k )B∥ = 4T43210B∥ (T)50n (1012cm-2)ns/4 3ns/4 nsns/25 4 3 2 1 0Rxx (k )543210Rxx (k)420n (1012 cm-2)0.0 0.5 1.0n/ns 4.2 K 16 mK(a) (b)(c) (d)Figure 4.5: Map of Rxx around thens2 CI state for negative D values atT = 16mK for (a) B‖ = 0T and (c) B‖ = 4T. The white dashed line in (a)corresponds to D = −0.43V/nm line. The small square in (a) is the regionwhere temperature dependence of Rxx was measured in Fig. 4.6.(c) Linecut of Rxx at D = 0.43V/nm (white dashed line in 4.5(a)) as a function ofcarrier density, n at T = 16mK &4.2K. (d) Evolution of the CI states as afuntion of in-plane magnetic field at T = 16mK.n = ns/4, 3ns/4 grow in resistance, especially above B‖ = 1T. This is likelya result of these states being spin-valley polarized. This was also observedin past studies[19, 21], where the gap at n = ns2 widens with the applicationof B‖ in line with Zeeman splitting (∆ = gµBB‖, g ∼ 2, µB is the Bohr394.2. Correlated Insulator States for device CS2 !"#$ !"%& !"%# !"$' ()*+,-&".$&"&$ /)*+,-0$$ $!11*+2-%&*34 !"#$ !"%& !"%# !"$' ()*+,-&".$&"&$ /)*+,-0$$ $!11*+2-%$$34Figure 4.6: Rxx map of the region indicated in Fig. 4.5(a) at 16 mK and100 mK. The Rxx value is saturated from −50− 500Ω.magneton). It should be noted that this behaviour differs from magic angletwisted bilayer graphene, where the gap decreases as a function of B‖[14].One of the most intriguing questions surrounding the phase diagram oftwisted double bilayer graphene is the nature of the metallic region sur-rounding the ns/2 correlated insulator state. This metallic region is oftenenclosed by a ring-shaped high resistance region, as seen in Fig. 4.5(a).Fig. 4.6 compares the resistance map of the small region close to the edgeof this ring (indicated by a square in Fig. 4.5(a)) at base temperature and100mK, using a saturated color scale. We see that, at 16mK, the Rxx valuesat several points in this region are extremely low (zero and even negativevalues), which rises to ∼ 100Ω at 100mK. This is qualitatively similar toprevious observations where the resistivity of these metallic regions rapidlydrops to zero (or a non zero residual value) with temperature[19, 21–23].This is at variance of what we expect for an ordinary metal where the lowtemperature resistivity often saturates (for impurity dominated) or has a T xdependence (where the value of exponent x depends on the dominant scat-tering mechanism, x = 2 for electron-electron scattering etc.[48, 49]). Also,this rapid drop in resistance is particularly interesting considering its sim-ilarity to twisted bilayer graphene system where these regions surroundingthe correlated insulator states are superconducting.In order to further elucidate the properties of these low resistance regions,we measured the differential resistance dV/dI as a function of Idc at variouspoints inside (Fig. 4.7(b)) and outside (Fig. 4.7(c)) the ring. The points404.2. Correlated Insulator States for device CS2-6-4-20Vbg (V)87654321V tg (V)1.5 0.0Rxx (k )8004000dV/dI ()40200-20-40I (nA)800400040200-20-40I (nA)(a)(b) (c)Figure 4.7: (a) Rxx map of device CS2 near the ns/2 peak indicating thedata points where the dV/dI measurements were done. dV/dI measurementsat various points (b) inside and (c) outside of the ring.where the measurements were done are represented in V tg, V bg space in Fig.4.7(a). We see that, inside the ring, all dV/dI values increase as a functionof |Idc| and saturate for |Idc| > 20nA. Even though the dV/dI curves insidethe ring show a high degree of symmetry and non-linearity reminiscent ofa superconductor, we cannot extract a well defined critical current. dV/dItraces outside the ring in Fig. 4.7(a) are seen to be highly asymmetric (Fig.4.7(c)). This lineshape may be a result of bad electrical contacts.414.2. Correlated Insulator States for device CS2We observe similar regions with bad electrical contacts in the highlydoped electron and hole side as well as within the gaps where the Rxx valuegets noisy and becomes negative (Fig. 4.3). In order to explain this, letus consider the device structure and optical image of CS2 given in Fig.4.8(a) and (b). The black outline gives the extent of the graphite back gateelectrode. In the figure, we can see that there is a very small region ofthe hall bar (red outline in 4.8(b)) which is gated only by the back gateelectrode.When the gate voltages are increased, the carrier density of the abovementioned singly-gated region varies differently from the double-gated deviceregion of interest (underneath the top gate, i.e., the golden hall bar). Sincethe singly-gated region is also composed of twisted double bilayer graphene,the resistance of this region peaks at carrier densities |ns|. As the currentflows from the contacts, through the singly-gated region, into the double-gated part of interest, our contact resistance becomes high every time thecarrier density of this region becomes ns. This manifests in the map oftwo probe resistance across the voltage leads as large resistance stripes for5 < |V bg| < 10 in Fig. 4.8(c).As for the regions where the contact region is no longer gapped, theobserved bad contacts can be explained by the differing band populationof the contact and device region. Fig. 4.8(d) gives the schematic of thedensity of states of twisted double bilayer graphene devices under a finiteelectric field for the contact (left) and device region (right panel). At finiteelectric fields, as mentioned previously, a gap opens at the CNP point, anda flat band develops in the conduction band (C0). The spin splitting of thisband results in the formation of the correlated half filling insulating states.In order to illustrate the band population, we can consider three separatepoints in the region marked by the square (red), circle (blue) and triangular(pink) points where the four probe resistance is noisy and negative. At thecircle, the Fermi level lies in the band V 0 for the contacts region and C0 forthe device region and hence the polarity of the carriers in the contact andthe device are opposite to each other resulting in formation of a p-n junctioncausing a large contact resistance [19].Similar argument holds true at the square, where the Fermi level lies inband V 1 for the contact region and V 0 for the device region. Lastly, at thetriangle, the Fermi level lies in band C0 for the contact region and C1 for thedevice region. These two situations where there is a transition between thedifferent bands as we move from the contact to the device region may alsobe vaguely similar to that of a p-n junction. Such a transition could be thecause for the bad contacts in these regions. A comparison of the different424.2. Correlated Insulator States for device CS2regions in Fig. 4.3 shows that the regions where Rxx gets noisy and negativecoincide either with the regions where the two probe resistance becomes highor where the contact and device region populates differently and hence theiroccurrence can be explained as a consequence of bad electrical contacts.SiO2Top gate (Au)Twisteddouble layergrapheneGraphite back gateTi/AuContacts(c)(a) (b)(d)-10-50510Vbg (V)840-4-8V tg (V)ns-nsCNPns/2100 80 60 40 20 0R (k )ns-ns2CNPC1V1V0C0DOSEFigure 4.8: The (a) device structure and (b) optical image of device CS2.The purple outline marks the boundary of the twisted double bilayer regionbefore etching. The graphite back gate electrode is shown as the blackoutline and the red outline gives the region of the hall bar which is gatedonly by the back gate electrode. The scale bar is 5µm. (c) Two proberesistance map across the voltage leads in Fig. 4.3. The resistance valueis saturated to 100kΩ. (d) The schematic of the density of states for thecontact region (left) and the device region (right). The dashed lines indicatethe Fermi level at various points indicated in the 2D plot.Finally, though dV/dI behaviour in conjunction with the rapid drop inRxx for regions inside the ring is consistent with a superconducting state[19],a separate explanation stemming from spontaneous symmetry breaking ispossible [23]. In this scenario, as the temperature is lowered, the formation434.2. Correlated Insulator States for device CS2of the correlated insulating state at half filling lifts a single degeneracy. Thissymmetry breaking is realized as long as the device is tuned within the regionenclosed by the ring surrounding the correlated insulator state [23]. As thehalf-filled insulator state is spin-polarized, slight doping within the ring mayresult in the formation of a ferromagnetic metallic state. The rapid drop inRxx within this ring could be due to the suppression of various scatteringmechanisms on account of magnetic ordering[50]. The observed non-linearI−V characteristics on the other hand could be a result of Joule heating[23].It is to be noted that there is a great disagreement in literature about thesaturation value to which the resistivity drops at lowest temperature betweendifferent groups and samples. Kim group[19] observes the value approachingzero in one of their device with θ = 1.26◦, but in the more recent literatureit is a few hundred ohms[23].Another plausible explanation for the observed low resistance could beballistic conduction[4]. Such behaviour has previously been observed inmonolayer[4] and bilayer[51, 52] graphene devices at low temperatures overlength scales of several micrometers resulting in a negative resistance regionclose to CNP. Slight deviations from the flat band condition in our samplesmay result in a ballistic transport causing these low (negative) Rxx regions.But a simple scenario of ballistic conduction involving electron-phonon scat-tering alone cannot explain for the observed rapid drop in resistance withtemperature[53].The present results leave the nature of the metallic region inside thering altogether an open question and hence require further experimentaland theoretical efforts in order to verify the nature of this region. The pres-ence of ferromagnetic order can be directly probed with the help of opticalspectroscopy techniques like Polar magneto-optical Kerr effect (MOKE) inconjunction with anomalous Hall measurements[54, 55]. It should also bementioned that these observations are in stark contrast to twisted bilayergraphene where unambiguous signatures of superconductivity are univer-sally observed such as zero resistance, clear critical current behaviour andphase coherent Fraunhofer interference patterns in a magnetic field[26, 45]arising from sample inhomogeneity.44Chapter 5ConclusionsIn this thesis, we were able to develop a reliable method for the fabrication oftwisted graphene devices. Using modified dry transfer techniques of “tear-and-stack” and “cut-and-stack”, we fabricated several devices and measuredtheir twist angles using electrical transport. We find that tearing grapheneusing an hBN flake at temperatures < 100◦C presents several problemsnecessitating high temperature tearing. However, the devices fabricated bytearing at high temperatures showed large deviations of the measured twistangle from the target angle. Hence, we recommend the “cut-and-stack”technique for reliable fabrication and a better twist angle control of thefabricated devices. Further improvements in the twist angle control mightbe possible with the use of an alternative cutting technique using a laser.As the surgical needle has a finite diameter it could sometimes physicallydisplace the graphene flakes from the substrate/fold them at their the edgescausing an additional strain. Though it is difficult to quantify the extent ofimprovement, use of a laser can preclude this from happening and could bea subject of future studies.In order to replicate the electrical phase diagram of twisted double bilayergraphene, we studied the transport properties of a device with θm = 1.39◦in a dilution refrigerator. With the help of an external displacement electricfield, we were able to tune the flat band structure and observe correlatedinsulating states at half filling for a wide range of electric field values onthe electron doped side of the device. We also observed ns/4 and 3ns/4correlated insulating states in an extremely narrow range of electric field.The resistance values of all the correlated insulating states grow and ex-tends further in D space with the application of an in-plane magnetic fieldsuggesting the states being spin/valley polarized. Furthermore, the metallicregion surrounding the ns/2 correlated insulating state has a strong temper-ature dependence at dilution temperatures and shows symmetric non-lineartransport. While this rapid drop in resistance could be reminiscent of asuperconductor, a transition to a ferromagnetic metallic state could also re-sult in the above behaviour. 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B, vol. 101,p. 245436, Jun 2020.52[54] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov,“Magnetic 2d materials and heterostructures,” Nature Nanotechnology,vol. 14, pp. 408–419, May 2019.[55] B. Huang, M. A. McGuire, A. F. May, D. Xiao, P. Jarillo-Herrero, andX. Xu, “Emergent phenomena and proximity effects in two-dimensionalmagnets and heterostructures,” Nature Materials, Sep 2020.53Appendix AResistance maps of all thefabricated devicesFigure A.1 gives the Rxx maps of all the “cut-and-stack” devices at 4.2K. !" #""#"!" $%&'() *" + " + *" ,%&'()-.*/!01*2*!3#" *+ *" + ""44&'56) *" +"+*" $%&'()7!" ! 7 ,%&'()-.#/&!01*2893#" *+ *" + ""44&'56) *" +"+*" $%&'() 7 ! " ! 7 ,%&'()+" !" 8" #" *""44&'56)-.8/!01*2*:3Figure A.1: V tg vs V bg resistance maps of all the “cut-and-stack” twisteddouble bilayer devices at 4.2K. CS3 is twisted bilayer graphene device.54Appendix A. Resistance maps of all the fabricated devicesFigure A.2 gives the Rxx maps of the “tear-and-stack” devices fabricatedat T 1, T 2 < 100◦at 4.2K. !""!" #$%&'( ! " ! )$%&'(*" + , ! -!..%&/0(1-2"3"4,+56782%"93"4:;5 !" -""-"!" #$%&'( ! " ! )$%&'(67*2"93"4,+5," !" -" "!..%&/0( -""-" #$%&'( ! " ! )$%&'(+ , ! - "!..%&/0(67!2"93"4+<5 !" -""-"!" #$%&'( *" ; " ; *" )$%&'(-" *; *" ; "!..%&/0(67-2"93*4*-5Figure A.2: V tg vs V bg resistance maps of the “tear-and-stack” twisteddouble bilayer devices fabricated at T 1, T 2 < 100◦at 4.2K. TS4 is a twistedbilayer graphene device.55Appendix A. Resistance maps of all the fabricated devicesFigure A.3 gives the Rxx V tg vs V bg maps of the “tear-and-stack” devicesfabricated at T 1, T 2 > 100◦at 4.2K. !" #""#"!" $%&'() * ! " ! * +%&'(), ! - # .!//&'01)234 ." ,",." $%&'() ." " ." +%&'()23,5&"67.8*9:#7#;#7 #;#7"#."" "!//&'01) ." ,",." $%&'() ." " ." +%&'()."" "!//&'01)239 !" #""#"!" $%&'() ." " ." +%&'()#, #" ., ." , "!//&'01)23*Figure A.3: V tg vs V bg resistance maps of the “tear-and-stack” twisteddouble bilayer devices fabricated at T 1, T 2 > 100◦at 4.2K.56
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Electronic transport in twisted double bilayer graphene
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