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Quantum transport in 2D topological insulators Sajadi Hezave, Seyyed Ebrahim 2020

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Quantum Transport in 2D Topological InsulatorsbySeyyed Ebrahim Sajadi HezaveM.Sc., University of Stuttgart, 2012B.Sc., Isfahan University of Technology, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)March 2020© Seyyed Ebrahim Sajadi Hezave, 2020The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Quantum Transport in 2D Topological Insulatorssubmitted by Seyyed Ebrahim Sajadi Hezave in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Physics.Examining Committee:Joshua Folk, University of British Columbia, (Physics and Astronomy)SupervisorDoug Bonn, University of British Columbia, (Physics and Astronomy)Supervisory Committee MemberMark Halpern, University of British Columbia, (Physics and Astronomy)University examinerShahriar Mirabbasi, University of British Columbia, (Electrical and ComputerEngineering)University examinerGleb Finklestein, Duke University, (Physics)External examinerAdditional Supervisory Committee Members:Marcel Franz, University of British Columbia, (Physics and Astronomy)Supervisory Committee MemberCarl Michal, University of British Columbia, (Physics and Astronomy)Supervisory Committee MemberiiAbstractTopological insulators (TI) have been the subject of intense theoretical and ex-perimental investigation due to their distinct electronic properties compared toconventional electronic systems. This thesis investigates electronic properties oftwo topological insulators, InAs/GaSb double quantum wells and monolayer WTe2,through transport measurements at ultra-low temperatures.Using double gate geometry, InAs/GaSb quantum wells can be tuned betweentopological and trivial states. Previous works have reported the existence of robusthelical edge conduction in the inverted regime. Here, we found an enhancededge conduction in the trivial state with superficial similarity to the observed edgeconduction in those reports. However, using various transport techniques and samplegeometries, the edge conduction in our samples was found to have a non-helicalorigin.Another topological insulator that is studied in this thesis is monolayer WTe2.Here, we report that monolayer WTe2, already known to be a 2D TI, becomes asuperconductor by mild electrostatic doping, at temperatures below 1K. The 2DTI-superconductor transition can be easily driven by applying a small gate voltage.Furthermore, we observed peculiar features such as enhancement of parallel criticalmagnetic field above the Pauli limit possibly, from spin orbit scattering.iiiLay SummaryIn general, materials that conduct electricity are conductors, and those that do notare insulators. Around 30 years ago, it was discovered that there is yet anotherclass of materials that are insulating in their interior region while having conductingsurface/edge states. These conducting states are robust against material impurities.Materials in this new class are referred to as topological insulators (TIs).Though conducting states are the hallmark of TIs, non-topological electronicsystems may also possess these states. In this thesis, we show an experimentalprocedure to investigate the nature of edge conduction in an electronic system thatwas proposed to be a TI.Combining the conducting states in TIs with superconductivity would providea platform to study the interplay between these exotic states of matter. Here, wereport that superconductivity can be induced in a monolayer of WTe2, an electronicsystem that is already known to be a TI.ivPrefaceThis thesis describes my scientific contributions as a graduate student at UBC inexploring the electronic properties of 2D Topological Insulators. The followingdetails the ways in which I and others contributed to the publications described inchapters 3 and 5.• Chapter 3 is associated with the following publication: F. Nichele, H. J.Suominen, M. Kjaergaard, C. M. Marcus, E. Sajadi, J. A. Folk, F. Qu, A. J.A. Beukman, F. K. d. Vries, J. v. Veen, S. Nadj-Perge, L. P. Kouwenhoven,B. Nguyen, A. A. Kiselev, W. Yi, M. Sokolich, M. J. Manfra, E. M. Spanton,and K. A. Moler, Edge transport in the trivial phase of InAs/GaSb New J.Phys. 18 (2016) 083005.This work was a large collaboration between the groups of C. M. Marcus atNiels Bohr Institute, J. A. Folk at UBC, K. A. Moler at Stanford University,L. P. Kouwenhoven at Delft University of Technology, M. Sokolich at HRLlaboratories, and M. J. Manfra at Purdue University. F. Qu, A. J. A. Beukman,F. K. d. Vries, J. v. Veen, and S. Nadj-Perge performed a preliminarymeasurement, the result of which is published in [1]. Upon this result, theconsecutive experiments were designed by F. Nichele, C. M. Marcus, L. P.Kouwenhoven, and M. J. Manfra. All samples studied in this project weregrown using molecular beam epitaxy (MBE) by B. Nguyen, A. A. Kiselev,W. Yi, M. Sokolich at HRL. Two types of Hall bar devices were studiedin this work. The macroscopic Hall bars were fabricated by B. Nguyen,A. A. Kiselev, W. Yi, and M. Sokolich at HRL. The microscopic Hall barswere fabricated by F. Nichele, H. J. Suominen, M. Kjaergaard at Niels BohrvInstitute.I performed non-local (section 3.3.2), in-plane magnetic field (section 3.3.3),and temperature dependence measurements (section 3.3.4) on macroscopicHall bar samples. I analyzed the data presented in sections 3.3.2, 3.3.3, andthe data on macroscopic Hall bar samples in section 3.3.4.In addition, to have the complete story of this work, I have presented thedata measured by other groups. In particular, data in section 3.3.1, and 3.4was measured by F. Nichele, H. J. Suominen, and M. Kjaergaard. Data onCorbino measurements in section 3.3.4 was measured by E. M. Spanton. Themanuscript was written by F. Nichele with input from all other authors.• A version of chapter 5 is published in the following publication:E. Sajadi, T. Palomaki, Z. Fei, W. Zhao, P. Bement, C. Olsen, S. Luescher, X.Xu, J. A. Folk, D. H. Cobden, Gate-induced superconductivity in a monolayertopological insulator, Science, 23 November 2018 (10.1126/science.aar4426).This work was a collaboration between the group of J. A. Folk at UBC andthe group of D. H. Cobden and X. Xu at the University of Washington. Theexperiment was conceived by T. Palomaki, X. Xu, and D. H. Cobden. Bothsamples studied in this work were fabricated at the University of Washingtonby T. Palomaki, Z. Fei, and W. Zhao. The cryostat in which the measurementswere performed was developed by me, S. Luescher, and J. A. Folk. Themeasurement in cryostat was carried out by me, Z. Fei, P. Bement, and C.Olsen. The data were analyzed by me and P. Bement, with input from allother authors. The manuscript for the paper was written by me, J. A. Folk,and D. H. Cobden.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Coexistence of TI and superconductivity . . . . . . . . . . . . . . 41.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Theoretical and Experimental Background . . . . . . . . . . . . . . 72.1 Topology of the band structure . . . . . . . . . . . . . . . . . . . 72.2 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Quantum Spin Hall Effect . . . . . . . . . . . . . . . . . . . . . . 102.4 Experimental realization of a 2D TI . . . . . . . . . . . . . . . . 112.4.1 Transport Measurements in HgTe QW . . . . . . . . . . . 132.5 InAs/GaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20vii3 Edge Conduction in the Trivial Phase of InAs/GaSb . . . . . . . . . 253.1 Previous Reports of the Edge Conduction . . . . . . . . . . . . . 253.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Transport In Macroscopic Samples . . . . . . . . . . . . . . . . . 283.3.1 Magnetotransport Measurements . . . . . . . . . . . . . . 283.3.2 Non-local measurements . . . . . . . . . . . . . . . . . . 313.3.3 In-plane magnetic field . . . . . . . . . . . . . . . . . . . 323.3.4 Temperature dependence . . . . . . . . . . . . . . . . . . 343.4 Transport In Microscopic Samples . . . . . . . . . . . . . . . . . 363.4.1 Length Dependent Measurements . . . . . . . . . . . . . 363.4.2 Quantization In Microscopic Hall bar . . . . . . . . . . . 383.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Exotic phases of WTe2 Semimetal . . . . . . . . . . . . . . . . . . . 424.1 WTe2, Bulk Electronic Band Structure . . . . . . . . . . . . . . . 434.2 Experimental observations in bulk WTe2 . . . . . . . . . . . . . . 454.3 QSH in Monolayer WTe2 . . . . . . . . . . . . . . . . . . . . . . 475 Gate Induced Superconductivity in a Monolayer Topological Insulator 515.1 Experimental Details of Device M1 . . . . . . . . . . . . . . . . 525.2 Device Fabrication of M2 . . . . . . . . . . . . . . . . . . . . . . 535.3 Emergence of 2D Superconductivity . . . . . . . . . . . . . . . . 545.3.1 Doping Effect . . . . . . . . . . . . . . . . . . . . . . . . 585.3.2 Perpendicular Magnetic Field Dependence . . . . . . . . 595.3.3 In-plane magnetic field dependence . . . . . . . . . . . . 615.3.4 Anomalous Metal Phase . . . . . . . . . . . . . . . . . . 635.4 Quantum Critical Point . . . . . . . . . . . . . . . . . . . . . . . 655.5 The Edge Gap in Device M1 . . . . . . . . . . . . . . . . . . . . 675.6 Coexistence of Superconductivity and TI . . . . . . . . . . . . . . 685.7 Measurement Details . . . . . . . . . . . . . . . . . . . . . . . . 725.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Conclusion on TIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75viii7 Measurement Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.1 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.1.1 Filtering Modules . . . . . . . . . . . . . . . . . . . . . . 807.2 Electronic Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A Method to extract critical exponent α in Fig. 5.8 . . . . . . . . . . . 98B Calculation of coherence length, and the criterion for identifyingcritical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C Contact Resistance of Device M1 . . . . . . . . . . . . . . . . . . . . 103D Nonlinear I−V characteristics in the superconducting region . . . . 105E Strontium Vanadium Oxide . . . . . . . . . . . . . . . . . . . . . . . 107E.1 SVO Electronic Structure . . . . . . . . . . . . . . . . . . . . . . 109E.2 Experimental Background . . . . . . . . . . . . . . . . . . . . . 109E.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . 111E.4 Temperature dependence measurements . . . . . . . . . . . . . . 112E.5 Magnetoresistance measurements . . . . . . . . . . . . . . . . . 115E.6 Hall measurements . . . . . . . . . . . . . . . . . . . . . . . . . 117E.7 Conclusion on SVO . . . . . . . . . . . . . . . . . . . . . . . . . 118ixList of TablesTable 5.1 Thicknesses of hBN as measured by AFM, and capacitances perunit area for bottom (cb) and top gates (ct), for M1 (top row) andM2 (bottom row). . . . . . . . . . . . . . . . . . . . . . . . . 54xList of FiguresFigure 1.1 Topology of the geometrical objects . . . . . . . . . . . . . . 2Figure 1.2 Band structure of an ordinary insulator and topological insulator 3Figure 1.3 Schematic of a 2D topological insulator in the real and momen-tum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 2.1 Realization of the QHE in a transverse magnetic field . . . . . 8Figure 2.2 Schematic diagrams for QHE . . . . . . . . . . . . . . . . . 9Figure 2.3 Schematic of the graphene lattice structure, with two atoms Aand B in the unit cell . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.4 Analogy between the QHE and QSH . . . . . . . . . . . . . . 11Figure 2.5 HgTe Band diagram . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.6 Schematic of ballistic transport in a wire with two conductingchannels. Voltage bias creates the chemical potential differencebetween the left and the right contacts. The middle graphrepresents the energy band diagram of two subbands in the 1Dwire, where the empty and filled circles show available andoccupied states in the subbands respectively. . . . . . . . . . . 15xiFigure 2.7 (a,b) Schematic of devices in the Hall bar geometry in the QH(a) and QSH (b) regimes. Gray regions represent the 2DEG, andyellow boxes are the metal contacts to the 2DEG. (a) In the QHstate, the presence of metallic contacts doesn’t effect the edgestates. (b) The spin-polarized edge states that are propagatingin the opposite direction at each edge of the QSH sample, afterentering the metallic contact interact with many modes presentin the contact. There is an equal probability of transmitting bothcounter propagating edge states from the metal contacts. (c)The resistance network equivalent of the edge state in the QSHsystem in (b). Each resistor represents an edge state betweentwo metal contacts with the quantum of resistance h/e2. . . . 17Figure 2.8 Schematic of Hall bar geometry with 6 terminals in QSH samples 19Figure 2.9 Schematic of the band alignment in InAs/GaSb heterostructure 21Figure 2.10 Schematic of the band alignment in InAs/GaSb heterostructure 22Figure 2.11 Effect of the external electric field on the overlap between twosubbands at InAs/GaSb QWs . . . . . . . . . . . . . . . . . . 23Figure 2.12 Transition between the trivial and topological phase in InAs/-GaSb QWs, by tuning the top and back gate voltages . . . . . 24Figure 3.1 Schematic of the device geometry and the measurement schemeused for the measurements in (b-d) . . . . . . . . . . . . . . . 29Figure 3.2 4-terminal resistivity measurement in local and non-local con-figurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 3.3 Longitudinal resistance (Rxx) along the dotted line in figure 3.2 33Figure 3.4 Conductance measurements in the Corbino disk in the trivialregime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.5 Schematic of the long bar along with the gated regions andelectrical set up used to measure the length dependent resistance 37Figure 3.6 Schematic of the H-bar along with contact numbering used inthe measurement presented in (c) . . . . . . . . . . . . . . . . 39xiiFigure 4.1 Crystal structure of TMDs with chemical formula MX2, in 1H(A), 1T (B) and 1T’ (C) phases . . . . . . . . . . . . . . . . . 43Figure 4.2 Calculated electronic band structure of WTe2 bulk . . . . . . 44Figure 4.3 MR versus external magnetic field applied perpendicular to theWTe2 planes and along crystallographic C axis . . . . . . . . 45Figure 4.4 Schematic of phase transition in WTe2 under external pressurealong with the measured data. . . . . . . . . . . . . . . . . . 46Figure 4.5 Calculated band structure of monolayer WTe2 . . . . . . . . . 48Figure 4.6 Temperature dependence of edge conductance in a linear trans-port measurement with 100 µV ac bias . . . . . . . . . . . . 50Figure 5.1 (a) Optical image of device M1. (b) AFM image of device M1along with electrical measurement setup. The boundaries ofWTe2 are indicated by red lines. Scale bars indicate 5µm. . . 53Figure 5.2 (a) AFM image of device M2 along with metal contacts usedfor the electrical measurements. (b) Optical image of deviceM2. Scale bars indicate 5µm. The boundaries of WTe2 areindicated by red lines in (a) and (b). . . . . . . . . . . . . . . 54Figure 5.3 Characteristics of monolayer WTe2 device M1 at temperaturesbelow 1K. (A) Optical image (the white scale bar indicates5µm) and schematic device structure, showing current, voltagecontacts and ground configuration for measuring the 4-proberesistance Rxx. Inset: schematic of the atomic structure ofmonolayer WTe2. (B) Rxx as a function of electrostatic doping(ne) at a series of temperatures. Inset: variation of Rxx at 20 mKwith top and bottom gate voltages,Vt andVb, indicating the axescorresponding to doping ne and transverse displacement fieldD⊥ Rxx depends primarily on ne and only weakly on D⊥. Themeasurements in the main panel for ne > 0 and ne < 0 weremade separately, sweeping Vb along the two colored dashedlines in the inset to avoid contact effects. (C) Phase diagramconstructed from measurements in this chapter, as explained inthe main text. . . . . . . . . . . . . . . . . . . . . . . . . . . 56xiiiFigure 5.4 Rxx on log scale vs temperature T at a series of positive-gatedoping levels ne [20, 12, 8.5, 6.7, 6.1, 5.6, 5, 4.6 1012 cm−2]showing a drop of several orders of magnitude at low T forlarger ne. Inset: location of sweeps on the phase diagram. . . . 58Figure 5.5 (A) Effect of the perpendicular magnetic field, B⊥ on resistanceat the highest ne in Fig. 5.3B (Demagnetization effects areneglected in light of the finite resistivity of the sample.) Inset:characteristic temperatures T1/2 obtained from these temper-ature sweeps, as well as characteristic fields B1/2 measuredfrom field sweeps under similar conditions. (B) Sweeps of B⊥showing rise of resistance beginning at very low field. . . . . 59Figure 5.6 (A) Effect of in-plane magnetic field, B|| on resistance.(B|| = 0data are for ne= 19×1012cm−2 and the rest of the data are forne = 18× 1012). Inset shows reduction of T1/2 with B||, fit tothe expected form for materials with strong spin-orbit scattering(solid line). The Pauli limit BP, assuming g = 2, is indicated bythe dashed line. (B) Sweep of B|| showing zero resistance up to∼ 2.4 T in contrast to prediction from Pauli limit. Inset: Datafrom (A) on a linear scale. . . . . . . . . . . . . . . . . . . . 62Figure 5.7 (A,B) Data from Fig. 5.4 and Fig. 5.5A replotted vs inversetemperature to highlight the saturation of Rxx as a function ofcarrier density and magnetic field at low T respectively. (C)Resistance as a function of temperature for two different dis-placement fields with the same carrier density, demonstratingthat the electronic temperature continues to decrease with mix-ing chamber temperature even below 50mK. . . . . . . . . . . 64Figure 5.8 Scaling analysis of the transition. Main panel: Multiple Rxx vsdoping traces, taken at different temperatures, cross at a criticaldoping level ncrit ≈ 5× 1012cm−2. Upper inset: dashed lineslocate these sweeps on the phase diagram. Lower inset: samedata presented on a scaling plot, taking critical exponent α = 0.8. 66xivFigure 5.9 (A). The 2D colorscale plot of differential conductance as afunction of DC bias voltage and doping level, showing a 600µeV gap in this density range for M1. (B). In-plane magneticfield dependence of the edge gap in M1, taken for a differentsetting of gate voltages where the zero-field gap was somewhatsmaller than in panel A. (T = 20 mK for both panels.) . . . . 67Figure 5.10 (A). AFM image of device M2. (B). Optical image of deviceM2. (C). Optical image of device M2. The region of WTe2with higher conductivity due to the combined effects of top andbottom gates is shown in light red (see text). (D). Schematicof the region of interest in M2, showing the contacts used formeasurements in Fig. 5.11 and highlighting in red the regionof WTe2 that dominates the conductance between those twocontacts. Note that the physical edge of the flake that connectsthose two contacts, shown by a red dashed line, is not above thebottom gate. However, based on the detailed properties seenin the data in Fig. 5.12 we deduce that the conduction at lowne is dominated by one or more internal cracks, not visible inthe images above, that exist in the region above the bottom gate.All scale bars indicate 5µm. . . . . . . . . . . . . . . . . . . 69xvFigure 5.11 Evidence for the presence of both edge conduction and super-conductivity in device M2. The main panel shows the linearconductance between two adjacent contacts vs gate dopingat the temperatures and perpendicular magnetic fields noted.Schematics indicate the state of edge and bulk conduction at dif-ferent points, the bulk being colored to match the phase diagramreproduced above, and red indicating a conducting edge state.Superconductivity occurs for ne > 5×1012cm−2 at B = 0; thezero resistance state, disguised by contact resistance in this fig-ure, was confirmed in a separate 4-wire measurement of R vs T(Fig. 5.12); edge conduction dominates for ne < 2×1012cm−2but appears to be present at all ne. Inset: color-scale plot ofdifferential conductance vs dc voltage bias and doping level,revealing a gap of around 100 µeV that fluctuates rapidly as afunction of doping level. . . . . . . . . . . . . . . . . . . . . 71Figure 5.12 A. Appearance of superconductivity for device M2 in a four-wire measurement of resistance vs temperature. The gate-induced carrier density corresponding to these voltages is ne =8.4×1012cm−2 in the region covered by both top and bottomgates. B. AFM image of the device M2 along with schemat-ics representing the contacts used for the measurement in A.Measurement is performed in an unconventional geometry, dueto the many poor or broken contacts in this device. The redsolid line shows the boundary of WTe2. The layout of the areacovered by the top and bottom gates can be seen in Fig. 5.10C.Scale bar indicates 5µm. . . . . . . . . . . . . . . . . . . . . 72Figure 7.1 Schematic of the cryogenic setup . . . . . . . . . . . . . . . . 79Figure 7.2 Schematic of the electronic setup . . . . . . . . . . . . . . . . 82xviFigure A.1 Explanation of the procedure used to determine an optimal scal-ing exponent. A, B, C: Rxx(ne) data for temperatures from 100mK to 1 K, plotted with the re-scaled horizontal axis indicated,for three values of the scaling exponent α . Panel B representsthe best-fit exponent, also shown in the main text (Fig. 5.8inset). D. Collapse error quantifies the failure of the multiple-temperature datasets to collapse onto metallic and insulatingbranches, as described above. The minimum collapse error(best scaling) is found for α ∼ 0.8 ± 0.1, where the error baris determined qualitatively from the rounding of the collapseerror dependence. . . . . . . . . . . . . . . . . . . . . . . . . 99Figure B.1 Temperature dependence of the characteristic field, B⊥0.X(T ),comparing characteristic fields defined using different frac-tions of the normal-state resistance [0.1,0.3,0.5,0.8,0.9]. Twodifferent densities are shown (A. ne = 15× 1012cm−2 and B.ne = 19×1012cm−2). From this temperature dependence, char-acteristic temperatures in the zero field limit and characteristicfields in the zero temperature limit can be estimated by extrapo-lation, as can the slope dB⊥0.XdT |Tc . The table of values above thegraphs indicates coherence lengths extracted from these data,via the two approaches described above. Also shown is thegap-based coherence length in the dirty limit, obtained from thediffusion constant and characteristic temperature as describedin the chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . 102xviiFigure C.1 A. Contact resistance measurement for a particular pair of con-tacts (in this case, those to which the current bias and ground areconnected, labelled I and G). B. Although the data is plotted vsne, the curves represent sweeps ofVt for various fixedVb. Whenthe density is high (ne > 12× 1012cm−2), contact resistanceis low independent of relative top- and bottom- gate voltages.For lower densities (ne < 8×1012cm−2), however, the contactresistance is much lower for strongly negative bottom gate, thatis, where the top gate is very positive. Conversely, less positivetop gate voltages (corresponding to more positive bottom gatevoltages) give very high contact resistances at lower density. . 104Figure D.1 Two (A) and four (B) terminal differential resistance measure-ments for sample M1 in the superconducting regime, for theset of contacts investigated in the main text. The measurementwas made with a small AC current bias (2 nA) on top of the DCcurrent bias on the horizontal axis, using a locking at the ACfrequency to measure dV/dI. . . . . . . . . . . . . . . . . . . 106Figure E.1 Metal insulator transition in vanadium oxides . . . . . . . . . 108Figure E.2 crystal structure of SVO . . . . . . . . . . . . . . . . . . . . 109Figure E.3 PES spectra of various SVO thin films represents a gap openingat Fermi energy for 1-2 MLs . . . . . . . . . . . . . . . . . . 110Figure E.4 (a) Schematic of LSAT/STO/SVO stack along with their latticeconstant denoted as (a). (b) Optical image of 10ML SVO on achip carrier with aluminum bond wires on its perimeter. Scalebar is 1mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure E.5 (a),(b) Schematic of SVO sample with contacts shown by redsquare on its perimeter along with electrical configuration formeasuring longitudinal resistance RAD,BC in (a) and transverseresistance in (b). . . . . . . . . . . . . . . . . . . . . . . . . 112xviiiFigure E.6 (a) Resistivity of different SVO thickness versus temperaturesshows a metallic state down to 3ML with an upturn in theresistivity at low temperature. (b) The upturn in Resistivity for3ML sample is replotted as conductance vs temperature in logscale. The solid line is the fit to the equation E.2. . . . . . . . 114Figure E.7 Resistivity of two SVO samples, S1 (red lines) and S2(bluelines) vs temperature measured in two orthogonal orientationsshown in schematics (right). Inset in the main panel shows aresistance vs temperature of a 2ML sample along with the fit(blue line) to VRH model. . . . . . . . . . . . . . . . . . . . 115Figure E.8 (a) MR of various SVO thin films at 4K. (b) Magnetoconduc-tance of 3ML SVO plotted vs magnetic field shows WL fortemperatures below 600mK. (c) Extracted τ−1φ s from the mea-surements in (b) vs temperature .The blue and brown solidlines are fits to the power law τ−1φ ∝ Tp with exponent 1 and 3respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure E.9 Sheet carrier density and electron mobility on the left and rightaxes respectively plotted vs number of layers. Sheet carrierdensity shows a straight linear trend vs SVO thicknesses downto 5 monolayers. A deviation from the linear trend appearswhen the SVO thickness is less than 5 monolayers.The solidblack line is a guide to the eye. . . . . . . . . . . . . . . . . . 117xixDedicationThis thesis is dedicated to my loving mother, and my father whosadly passed away years ago, for their endless love, support andencouragement.xxChapter 1IntroductionIn standard solid state theory, materials are classified as conductors and insulatorsbased on their ability to conduct electricity. However, around 30 years ago, itwas found that there exists another class of materials where a non-trivial topologyof their band structure gives rise to an insulating state in their interior regionwhile having conducting surface/edge states protected against disorder [2, 3]. Thisintriguing property of these materials might open up a door for new types ofelectronic and storage devices. This discovery led to a new paradigm for classifyingelectronic materials based on the concept of topology [4]. To understand therole of topology in this classification, an analogy with geometrical topology ishelpful. Topology of geometrical objects is the property that is invariant undertheir continuous deformation. For example, as shown in Fig. 1.1 a ball can becontinuously transformed into a bowl, or a coffee cup into a doughnut. On the otherhand one can’t deform a ball continuously into a doughnut without introducing ahole on its surface.1Figure 1.1: Topology of the geometrical objects. A ball can continuouslydeform into a bowl (a), and a coffee cup into a doughnut (b). However, itis not possible to deform a ball into a doughnut without making a holeon its surface. Image in (b) is taken from [5].For electronic materials, the underlying topology is related to the electronicband structure [4]. Similar to the geometrical objects, where a ball and a bowl aretopologically the same, simply because there in no hole on their surface, conven-tional insulating materials are topologically equivalent. On the other hand, like adoughnut, which has a hole on its surface, thus has a different topology than a bowland a ball, these newly discovered materials are topologically distinct (non-trivial)from the conventional insulating materials (trivial). This new class of materials wascalled topological insulators (TIs).1.1 Topological InsulatorsAs we know from standard band theory, an insulator is a material where all availableelectronic states (bands) are completely filled, therefore electrons cannot movefreely inside an insulator. Moreover, there is an energy gap between the highestoccupied band (valence band) and the lowest empty band (conduction band) [6].There is always a natural ordering for a pair of orbitals that makes the conductionand valence band of the electronic systems [7]. In many cases these are s and porbitals, where the conduction band stems from s-orbital and the valence band from2p-orbital (Fig. 1.2(a)). However, the conduction and valence bands stemmed fromother orbital pairs are also possible [7]. Now, if in an insulator the natural orbitalordering of the bands in the vicinity of the high symmetry points in the momentumspace as depicted in Fig. 1.2(b) gets reversed, the material might possess a non-trivial topology [7, 8]. For the case of an insulator with conduction and valencebands from s and p orbitals respectively, the reverse ordering means a conductionband with p-orbital character and a valence band with s-orbital character. Similarto the geometrical objects such as a ball and a doughnut where it is not possibleto continuously deform a ball into a doughnut without introducing a hole into thesurface of the ball, materials with different topology also cannot be transformedinto each other without an abrupt change in their band structure. In insulators,this change corresponds to the gap closing. Therefore, at the interface betweeninsulators with different topologies, the energy gap must vanish.Figure 1.2: (a) In normal band insulator, conduction band lies on the top ofthe valence band. Here the conduction band stems from s orbital andvalence band from p orbital. (b) In a topological insulator, a normalband ordering gets reversed in vicinity of high symmetry points in themomentum space. Here the inversion of s and p orbitals are depicted.In 3D systems, such a closing of the energy gap at the boundary would createconducting surface states and in 2D systems, 1D edge channels [8]. As depictedin Fig. 1.3(a), at each edge of a 2D TI, there exist two edge states, which arepropagating in the opposite direction along the sample perimeter with the spin of3an electron locked to its momentum. This can also be observed in the energy banddiagram representation as shown in Fig. 1.3(b). Here, two linearly dispersing linesare crossing the bulk band gap. The yellow/green lines with positive/negative groupvelocity corresponds to the yellow/green edge states that are counter propagatingin the real space representation. The time-reversal symmetry in TIs is preserved[8], since under time-reversal both the direction of momentum and spin of theelectron will reverse and interchange the two counter-propagating edge modes. Thisimplies that these edge states are protected against backscattering by non-magneticimpurities. The bulk of this thesis is dedicated to investigating the electronicproperties of two 2D topological insulators.Figure 1.3: Schematic of a 2D topological insulator in real and momentumspace. (a) In the real space representation, two edge states, with spinslocked to their momentum counter propagates at the boundary of thesample. (b) In the momentum space, these edge states with a lineardispersion cross the bulk gap of the 2D topological insulator. Conductionand valence bands are denoted as CB and VB respectively. The Fermilevel is shown by the dotted line.1.2 Coexistence of TI and superconductivityAlthough topological insulators are by themselves interesting, systems in whichtopology can be combined with superconductivity are even more interesting, because4they may host quasiparticles that behave like Majorana fermions (MFs).In 1937, Ettore Majorana predicted the existence of a fermionic particle thatis its own antiparticle. It was originally proposed by Majorana that a neutrinomight be such a particle [9], but so far there is no evidence of the neutrino beingits own antiparticle. Nonetheless, over the last decade, it has been postulatedthat quasiparticle excitations in certain electronic materials may behave like MFs[9]. These quasiparticles might be harnessed as building blocks of topologicalquantum computers. In particular, it has been predicted that MFs may emergewhen superconductivity is induced in topologically protected edge states [10]. Thisapproach has been followed experimentally by proximitizing a TI with an ordinarysuperconductor [11]. The success of this hybrid scheme depends on the quality ofthe interface between the superconductor and the TI, and this interface is, in generaldifficult to control. Therefore, it would be more appealing if both superconductivityand TI phases could coexist in one material. We have observed evidence forthe coexistence of TI and superconductivity in monolayer WTe2, which will beexplained later in this thesis.1.3 Thesis StructureThe content of this thesis is organized into the following chapters:Chapter 2 describes the theoretical and experimental background of TIs. The-oretical concepts and definitions relevant to understand the physics of TIs willbe explained at a simple level. It will be shown briefly how one can get precisequantization in quantum Hall effect (QHE) and 1D edge conduction based on thetheoretical concepts. The Landau Bu¨tikker formula, as a mathematical model toexplain the ballistic transport at the edge of QHE and quantum spin Hall (QSH)samples, will be described. The proposed theoretical electronic systems and theirexperimental realization will also be addressed in this chapter. In particular, 2DTI phase in HgTe and InAs/GaSb quantum wells will be outlined. The structureof InAs/GaSb will be described in more details as a preparation step for the nextchapter.Chapter 3 is dedicated to the experiment that I did on 2D TI InAs/GaSb asa part of a collaborative project that was planned with the Marcus group at the5Niels Bohr Institute. In the previous experimental reports, the edge conduction thatwas observed in the InAs/GaSb was interpreted as a signature of the QSH state inthis material. However, the signature of QSH state in those experiments was notconvincing enough to rule out other possibilities for the observed edge conduction.We have used the unique possibility of tuning between the trivial and topologicalstate in this electronic system to probe the signatures of QSH state. The set ofmeasurements performed in this experiment aimed to shed light on whether theedge conduction is helical as expected for the QSH system or has other origins.Chapter 4 addresses the electronic structure of WTe2 and the various phasesthat have been observed in bulk and thin layers, such as large non saturatingmagnetoresistance and pressure-induced superconductivity in the bulk. Recently,it has been observed that WTe2 in a single layer limit is a QSH insulator. Here, abrief review of both theoretical proposal and experimental findings of the TI phasein WTe2 will be reviewed.Chapter 5 is an adaptation of our work published on gate tunable supercon-ductivity in monolayer WTe2 with a small modification to fit the structure of thethesis.Chapter 6 summarizes the TI part of this thesis.Chapter 7 explains the measurement-setup used to acquire the data in thisthesis.6Chapter 2Theoretical and ExperimentalBackgroundThis chapter presents a theoretical overview of 2D topological insulators (TIs).What makes an electronic system a TI will be explained on a basic level. Thereafter,different electronic systems, wherein a 2D TI state has been proposed will bediscussed along with their experimental realization.2.1 Topology of the band structureTopological classification is based on a property of the system that is invariantunder continuous deformation. This property is called the topological invariant. Forexample, consider the case of the number of holes for a geometrical object. Since,this number must be an integer, (zero for sphere and one for doughnut), it cannot bechanged under continuous deformation and is regarded as the topological invariantof a geometrical object.The concept of topology can be extended to the realm of solid state physics,where all materials can be classified as trivial or topological based on a set oftopological invariants. The discovery of the quantum Hall effect (QHE) was thefirst instance where a topologically non-trivial state was realized.72.2 Quantum Hall EffectThe QHE occurs when a layer of electrons referred to as a 2D electron gas (2DEG)is placed in an external magnetic field that is perpendicular to the plane of electrons(Fig. 2.1(a)). As shown in Fig. 2.1(b), transport measurements revealed thatthe Hall resistance in the QHE is quantized with a remarkable precision that isindependent of the microscopic details of the sample under study. In 1982, Thouless,Kohmoto, Nightingale and den Nijs (TKNN) showed that the astonishing precisionof conductance quantization in the QHE is a reflection of its topological nature [3].Figure 2.1: Realization of the QHE in the transverse magnetic field. (a)Schematic of a device containing a 2DEG in a Hall bar geometry alongwith an electronic setup to measure the Hall resistance Rxy. (b) The solidline shows that the Hall resistance of the 2DEG in the presence of aperpendicular magnetic field develops plateaus, whose values are equalto the quantum of resistance h/e2 divided by an integer, C. The dashedline illustrates the Hall resistance expected in the classical regime thatincreases linearly with the magnetic field. “Republished with permissionof Royal Society, from [2]; permission conveyed through CopyrightClearance Center, Inc”.As depicted in Fig 2.2(a), due to the external magnetic field, electrons in the bulk ofthe sample move in a cyclotron orbit. The energy levels of electrons in the QHE, theso-called Landau levels, are quantized (Fig. 2.2(b)). Similar to an ordinary insulatorwhere the Fermi energy lies between the filled valence and the empty conductionband, in the QHE when the Fermi level is located between the filled and empty8Landau levels, the bulk of the sample becomes insulating.Figure 2.2: Schematic diagrams to explain the QHE. (a) Electrons in the bulkof the QHE sample orbit in a closed path due to the strong magnetic field.(b) Filled energy levels (blue lines) are separated from empty ones (redlines) by the quantized cyclotron energy. (C) Formation of edge states atthe boundary of the QHE sample and a surrounding ordinary insulatorwith a different Chern number, adapted from [8].TKNN found that in the QHE sample, as electrons move in a lattice in a presenceof a periodic potential and an external magnetic field, the electron wavefunctionacquires a phase, which is an integer multiple of 2pi . In contrast to QHE samples,for ordinary insulators, this phase is zero. This integer, which is called the Chernnumber, is the topological invariant for the QHE systems. They showed that thequantized Hall conductance, σxy, (inverse of the quantized Hall resistance) is relatedto the Chern number, C, as σxy =C e2h [3].Samples featuring the QHE have a different Chern number than ordinary insula-tors. Thus, at the boundary of the QHE sample and an ordinary insulator such as avacuum, the Chern number of the system (which is an integer) cannot change con-tinuously. As a result, the band gap must close within the boundary, and conducting1D edge channels will appear as illustrated in Fig. 2.2(c). The number of these 1Dedge channels are equal to the Chern number of the QHE sample.It was further understood that in 2D electronic systems, a distinct topologicalstate of matter can be realized even in the absence of an external magnetic field,called the quantum spin Hall (QSH) effect [12, 13].92.3 Quantum Spin Hall EffectIn 2005, Kane and Mele proposed that a topological state of matter could be realizedin graphene due to intrinsic spin-orbit coupling (SOC) [12]. Graphene has twoatoms in each unit cell, denoted by A and B as illustrated in Fig. 2.3(a). The valenceand conduction bands of graphene in K-space touch at two distinct points (the Diracpoints) as shown in Fig. 2.3(b).Figure 2.3: (a) Schematic of the graphene lattice structure, with two atomsA and B in the unit cell. (b) Representation of graphene band structure.At points denoted as K and K’, the valence and conduction bands touchand form the Dirac points. Figure (b) is reprinted by permission fromSpringer Nature: NPG Asia Materials, Ref. [14], Copyright(2009).In this model, spin-orbit coupling (SOC) creates the band inversion and opens agap in the Dirac band structure of graphene. This would give rise to an insulatingstate in the bulk and 1D edge states at the boundary, as in the QHE. However, unlikein the QHE, the edge states would not be chiral, but helical: the spin of an electronwould be locked to its momentum (Fig. 2.4(b)). At each edge of the QSH sample,there are two counter propagating helical edge states.One of the salient features of the edge states in electronic systems with a non-trivial topology is that they are protected from backscattering. In the QHE, thisarises from the fact that the edge state can move only in one direction at eachedge of the sample (chiral edge states), which is defined by the external magneticfield. Therefore, electrons cannot be backscattered on these chiral edge states (Fig.102.4(a)).Figure 2.4: Analogy between the QHE and QSH. (a) In the QHE sample, thetop edge has the right-mover only and the bottom edge the left-moveronly. Edge states are robust against backscattering, thus they will movearound an impurity. (b) In QSH there are 4 degrees of freedom due to thetwo-fold spin degeneracy and right and left movers. At each edge, thereare two channels where the spin is locked to the momentum. (Inspiredfrom [15]).2.4 Experimental realization of a 2D TIGraphene was the first 2D electronic system that was proposed to observe the QSHeffect. However, the experimental realization of the TI phase in graphene turnedout to be difficult. The induced band gap at the charge neutrality point (Dirac point)of graphene resulting from the intrinsic SOC is tiny, 1-50 µeV [16, 17], whilethe Dirac point broadening as a result of charge inhomogeneity is several ten’s ofmeV [18]. The search for topological materials, containing elements with strongspin-orbit coupling, led to the discovery that a 2D TI phase can be observed in thequantum well (QW) of a HgTe/CdTe heterostructure [19, 20].11Figure 2.5: HgTe and CdTe band diagrams. (From [19]. Reprinted withpermission from AAAS). (a) Bulk band diagrams of HgTe and CdTe.In CdTe, the Γ6 band, which has s-type character, lies above the Γ8band which has p-type character and a conventional semiconductor banddiagram results. In HgTe a reverse ordering of these bands leads to aninverted band structure. (b) In the HgTe/CdTe QW, when the HgTe isthinner than the critical thickness (dc), subband E1 sits above H1 andforms a trivial state. (C) In heterostructures, where HgTe is thicker thandc, the reverse ordering of the subbands leads to a topological state.In the HgTe/CdTe QW, the narrow band gap (10meV) semiconductor HgTe issandwiched between the wide band gap (1.5eV) semiconductor CdTe. As a result,charge carriers in the HgTe will be confined to move in the plane perpendicular tothe growth direction. Since the motion of charge carriers in the growth directionis confined, they will have discrete energy levels called subbands. The energy ofthese subbands is inversely proportional to the square of the width of the narrow12band gap semiconductor:En =h¯pi2n2m∗d2(2.1)Here, h¯ is the reduced Planck constant, m∗ is the effective mass, which is positivefor electrons and negative for holes, and d is the width of the potential well.As shown in Fig. 2.5(a), CdTe has a normal band ordering with an s-typeband lying above a p-type band. In the case of HgTe, this is reversed, whichleads to an inverted band ordering. In the model proposed by Bernevig, Hughes,and Zhang, (BHZ) [20], a trivial or topological insulating state will occur in thisheterostructure by changing the thickness of HgTe. As depicted in Fig. 2.5(b,c),there is a critical thickness (dc) of HgTe, above which, the QW subbands followthe bulk HgTe inverted band ordering (whereupon the hole subband lies abovethe electron subband). On the other hand, when HgTe is thinner than dc, the QWsubbands follow the bulk CdTe band ordering forming a trivial state. This canbe understood from equation (2.1), where the electron subband goes to higherenergies as the width of the well decreases, whereas the hole subband moves tolower energies due to negative effective mass of the holes. Hence, below somecritical thickness, the electron subband will sit above hole subband giving rise totrivial band ordering.The existence of the QSH state in the HgTe QWs was confirmed by chargetransport measurements [20].2.4.1 Transport Measurements in HgTe QWThe charge transport along 1D channels of the QSH is ballistic for a length shorterthan a characteristic length scale (ls), determined by any source of the electronspin-flip such as magnetic impurities. For the QSH in HgTe QW, ls is found to be afew microns [20]. In the ballistic regime, the conductance of a 1D edge channel isquantized in units of e2/h, as seen from the following argument.First, we should find the conductance of a 1D electronic system, with a length Lin the ballistic regime (L << ls) connected to the electron reservoirs as shown inthe band structure representation of a 1D wire in Fig. 2.6. Due to the transverseconfinement, energy levels of the 1D system are quantized into a set of discrete13energy levels, called 1D subbands as depicted in Fig. 2.6. A bias voltage createsa chemical potential difference between the left and right contacts, which resultsin the flow of a net current by populating and depopulating the available electronicstates in 1D subbands.The net current passing from the left contact to the right one is given by [21]:I =MeL∫ URULD(E)vg(E)dE (2.2)where M is the number of subbands participating in the transport, e is the elementarycharge, vg is the group velocity, D(E) is the density of states in the wire, UL and URare chemical potentials of the left and right contacts respectively. For a 1D electronicsystem, we have: D(E) = gsL2pidKdE and vg =1h¯dEdK . The factor gs in the density of states(D(E)) accounts for the spin degeneracy of the subbands. Substituting these back toequation (2.2) and writing the chemical potential difference between left and rightcontacts based on the applied voltage bias (Vbias): UL−UR = e(Vbias), one finds that:I =M(gse2h )(Vbias).14Figure 2.6: Schematic of ballistic transport in a wire with two conductingchannels. Voltage bias creates the chemical potential difference betweenthe left and the right contacts. The middle graph represents the energyband diagram of two subbands in the 1D wire, where the empty and filledcircles show available and occupied states in the subbands respectively.Thus the conductance (G) of the wire would be quantized in units of the quantum ofconductance, e2/h, that is:G=Mgse2h(2.3)The factor gs in equation (2.3) is 2 for spin degenerate subbands and 1 for the spinpolarized edge states in the QSH state. Equation (2.3) can be written in terms ofa transmission matrix, which is known as the Landauer formalism [22]. In thisframework, the probability of transmission of an electron through the 1D wireis G=M(e2/h)T, where T is the probability of transmission of an electron fromone side to the other side. Later on, Bu¨ttiker [23] generalized this formula for amulti-probe conductor, which is known as the Landauer-Bu¨ttiker formula. Based onthis formula, the current in terminal i would be:Ii =e2h ∑jTi j [Vi−Vj] (2.4)15where Ti j is the transmission probability from the i to j terminal andVi andVj are thevoltage at the terminal i and j respectively. The summation goes over all terminalsexcept terminal i. By employing the Landauer-Bu¨ttiker formula, one can find theexpected two or four terminal resistance for a device in the quantum Hall (QH)or QSH regimes, with well defined geometries such as Hall bars depicted in Fig.2.7(a,b).It is important to note the difference between the expected four-terminal resis-tance for a device in the QH vs QSH regime. As shown in Fig. 2.7(a), in the QHstate, the edge states propagate only in one direction at each edge of the sample,and the presence of a metal contact in their path doesn’t create any backscatteringprocess, which implies that there is no voltage drop at the metal contacts. Thereforethe four-terminal resistance should be zero. On the other hand, in the QSH state(Fig. 2.7(b)), when the two counter propagating edge states enter a voltage probe,they will lose their spin polarization as a result of interaction with multi-modespresent in metal contacts. Hence, there will be a voltage drop at the metal contacts,and the four-terminal resistance will not be zero. Using equation (2.4), the expectedfour-terminal resistance can be calculated as follows. Much simplification happensby considering that in the QSH edge states, the only non-zero transmission coeffi-cients (Ti j) in equation (2.4) are those for the adjacent contacts, i.e: Ti,i+1 and Ti+1,i.Moreover, since the probability of transmission to and from the contact is the same,Ti,i+1 = Ti+1,i. Applying these conditions, the equation (2.4) can be written as:Ii =e2h(Ti−1,i(Vi−1−Vi)+Ti,i+1(Vi+1−Vi))Now, if contact i serves as a voltage probe, then Ii is zero. By writing the aboveequation for all the contacts, and solving the resulting linear equations, the two andfour-terminal resistances for various measurement configurations can be obtained.As an example, let’s consider the four-terminal resistance measurement of the QSHdevice in Fig. 2.7(b). In the case where the current is applied between terminals 1and 4, and voltage measured between terminals 2 and 3, we have I1 =−I4, and forthe rest of the contacts, Ii will be zero. Now if we measure the voltage drop betweencontacts 2 and 3, the four-terminal resistance, R14,23 will be:16I1 =2e2h(V2−V3) and R14,23 = h2e2Alternatively, the QSH system can also be modeled as a network of resistances.Since each spin polarized edge state at each side of the QSH system carries aresistance of he2 , each segment of the edge channel between the two metal contactswould have a resistance of he2 . Therefore, the multi-terminal QSH system can beviewed as a network of resistances connecting the adjacent contacts by the quantumof resistance he2 as shown in Fig. 2.7(c). This way, the two and four-terminalresistance in any measurement configuration can be easily calculated by analyzingthe distribution of the current in the resistance network.Figure 2.7: (a,b) Schematic of devices in the Hall bar geometry in the QH (a)and QSH (b) regimes. Gray regions represent the 2DEG, and yellowboxes are the metal contacts to the 2DEG. (a) In the QH state, thepresence of metallic contacts doesn’t effect the edge states. (b) The spin-polarized edge states that are propagating in the opposite direction ateach edge of the QSH sample, after entering the metallic contact interactwith many modes present in the contact. There is an equal probabilityof transmitting both counter propagating edge states from the metalcontacts. (c) The resistance network equivalent of the edge state in theQSH system in (b). Each resistor represents an edge state between twometal contacts with the quantum of resistance h/e2.Figure 2.8(a) depicts a schematic of HgTe QW device in a Hall bar geometryused to verify the presence of quantized edge conduction in the QSH regime throughthe transport measurement [20]. Figure 2.8(b) represents a 4-terminal resistancefor QWs with HgTe thickness less than the critical thickness (dc) labeled as I andthose with HgTe thickness greater than (dc) labeled as II, III, and IV. Using a topgate the Fermi level was tuned in and out of the gapped region and the four-terminal17resistance (R14,23) measured by sending a current between contacts 1 and 4 andmeasuring the voltage drop between contacts 2 and 3 as shown in Fig. 2.8(b). Fordevice I, R14,23 in the gap region is much higher than the quantum of resistance,illustrated by the black trace in Fig 2.8(b), indicating the trivial state in these QWs.On the other hand, for devices II, III and IV, where HgTe thickness is greater thandc, R14,23 drops by more than two orders of magnitude compared to the deviceI, as evident by the blue, green and red traces respectively. Among devices withan inverted band structure, only those (III, IV) with edge length shorter than thescattering length display the quantized conductance (red and green traces) expectedfor the helical edge states in QSH state, while the one with longer edge length (II)illustrates a higher resistance (blue trace), as the length exceeds the spin scatteringlength (ls).18Figure 2.8: (a) Schematic of Hall bar geometry with 6 terminals in QSHsamples. (From [20]. Reprinted with permission from AAAS). Counter-propagating edge states at each edge of the sample are shown by solidand dotted lines pointing in opposite directions. (b) Four-terminal resis-tance of HgTe QWs with different thicknesses of HgTe as a function ofgate voltage that tunes the position of the Fermi level. The black tracecorresponds to the normal insulating phase of HgTe QW (d < dc) andthe other traces are for the inverted band structure in HgTe QW. Whenthe edge length becomes shorter than characteristic scattering length(red and green traces), edge conduction in four-terminal measurementapproaches the expected theoretical value of (2e2h ). The inset displaysthe quantized conductance at the elevated temperature of ∼1.8K.Unfortunately, studying the 2D TI phase in HgTe quantum well poses significantexperimental challenges. For example, molecular beam epitaxy (MBE) growth ofthese quantum wells is a very difficult process, which is accessible to only a fewgroups in the world. In the next section, a more conventional electronic system,InAs/GaSb QWs, to explore 2D TI state will be discussed.192.5 InAs/GaSbAs an alternative to HgTe QWs, hybrid QWs formed in the InAs/GaSb system weresuggested, with a band structure that could be tuned from trivial to topological byan external electric field [24]. The heterostructure of these QWs consists of threeclosely lattice matched semiconductors, InAs, GaSb, and AlSb. While AlSb with awide band gap of ∼1.6 eV acts as a potential barrier, the other two semiconductors,host a 2D electron (InAs) and hole (GaSb) gas. In this band alignment, the top ofthe valence band of GaSb lies above the bottom of the conduction band of InAs asshown in Fig 2.9(a). Figure 2.9(b) depicts the relative position of Fermi energiesin InAs and GaSb. Due to the discontinuity of the Fermi levels at their interface,electrons will transfer from the valence band of GaSb to the conduction band of InAs.This creates an internal electric field, causing a band bending and the formation oftriangular potential wells containing electron and hole subbands in InAs and GaSbrespectively as depicted in Fig. 2.9(c).20Figure 2.9: (a) Schematic of the band alignment in InAs/GaSb heterostructurealong with the band gap of its constituent semiconductors, the unit ofnumbers are electron volts. (b) Transfer of electrons from the GaSbvalence band into the InAs conduction band until the equilibrium setsin by having a common Fermi level across the two semiconductors. (c)Band bending at the interface of InAs and GaSb, owning to the built-inelectric field as a result of the charge transfer. Electron and hole subbandsforms at the resultant potential wells in InAs and GaSb respectively.Figure 2.10(a) represents the situation where only the first electron and holesubbands in InAs and GaSb are occupied. As shown in the electronic band dispersion21of these subbands in Fig. 2.10(b), the conduction band in InAs and valence bandin GaSb overlap. At the crossing point of the two bands, due to the finite couplingbetween the electron and hole wavefunctions, the degeneracy is lifted and a minigap is opened. The spatial separation of electron and hole gases at InAs/GaSb QWs,facilitate a transition between inverted and normal band structure by applying anexternal field as shown in Fig. 2.11.Figure 2.10: (a) Schematic of the band alignment in InAs/GaSb heterostruc-ture at zero external electric field. Electron (E1) and hole (H1) sub-bands are spatially separated. (b) Band dispersion of InAs/GaSb QW.Reprinted from [25], with the permission of AIP Publishing. Overlapbetween InAs and GaSb subbands causes the crossing of two bands atKcr as shown with dotted black lines. The hybridization of electronsand holes lift the degeneracy at Kcr(solid lines).22Figure 2.11: Reprinted from [25], with the permission of AIP Publishing.Effect of the external electric field on the overlap between two subbandsat InAs/GaSb QWs. Moving toward positive values of the electric fieldcause the increase of the overlap and shift the position of Kcr.Similar to HgTe QWs, the inverted band alignment in these QWs is expectedto have a non-trivial topological state [24]. Moreover, as shown in Fig. 2.12, thegate tunability between the trivial and topological states in these QWs make them a23Figure 2.12: Reprinted figure with permission from [24]. Copyright (2008)by the American Physical Society. Transition between the trivial andtopological phase in InAs/GaSb QWs, by tuning the top and back gatevoltages. Using both gates make it possible to change independentlyboth positions of the Fermi level (in and out of the gap), as well astuning the band structure from the normal ordering into the invertedalignment. Black dotted lines shows the border between normal (blue)and inverted (red) regimes. Electron and hole doped regions in eitherof two alignments are labeled as the e-doped and h-doped respectively.The hashed area represent the alignment when the Fermi energy placesin the gap.unique test-bed to study various interesting physics underlying QSH. Next chapterdescribes a set of measurements, which applies this gate tunability to study the edgeconduction in the trivial phase of InAs/GaSb double QWs.24Chapter 3Edge Conduction in the TrivialPhase of InAs/GaSbThis chapter will describe a set of measurements, which were designed to addressthe nature of the edge conduction in InAs/GaSb QWs. Here, electrostatic gatingwas employed to tune between the trivial and topological state in this material.Combining different experimental techniques, we found significant edge conductionoccurs in the trivial state of InAs/GaSb QWs, with remarkable similarity to previousreports, where the observed edge conduction was interpreted as helical. The workthat is explained in this chapter is an adaption of our published paper in [26].Specifically, I will explain the measurements that I have done in this work, whichare in-plane magnetic field, temperature dependence and non-local measurementson macroscopic Hall bar devices. In addition, I will briefly overview the other twosections of the paper for a better understanding of the whole picture.3.1 Previous Reports of the Edge ConductionFirst efforts to realize QSH in InAs/GaSb succeeded in observing some signaturesof edge conduction, albeit the observed edge conduction was accompanied by bulkresidual conductivity [27]. Further attempts tried to suppress the bulk conductionby adding Si impurities at the interface of InAs and GaSb [28, 29, 1] or using Gasource with a reduced purity [30, 31]. Measurements on these samples showed25compelling evidence of a robust edge conduction as a signature of the QSH state.For the mesoscopic devices with an edge length ∼ 1µm, the conduction approachedthe expected quantized conductance based on Landauer-Bu¨ttiker formalism forQSH samples. Further experiments using non-local measurements [32, 33, 34]onmesoscopic samples and scanning superconducting quantum interference devices(SQUIDs)[29] also confirmed the existence of the edge conduction in the invertedregime of the InAs/GaSb double quantum wells. One of the puzzling aspects of theseexperiments were the temperature and magnetic field dependence of the observededge conduction.The dependence of helical edge conduction on temperature can be determined byidentifying the possible spin-flip scattering mechanisms in these edge states. Elasticbackscattering, which results in a temperature-independent conduction is forbiddenin the helical edge states on account of being protected by time-reversal symmetry.On the other hand, inelastic scattering mechanisms, such as scattering by chargepuddles [35] could result in a spin-flip between the counter-propagating helicaledge states while preserving the time-reversal symmetry. The inelastic scatteringmechanisms are expected to create a power-law dependence of the edge conductanceon temperature [36, 37]. Considering that the only possible scattering mechanismin these helical edge states is the inelastic scattering, one would expect to observe atemperature independent behaviour for edge lengths shorter than inelastic scatteringlength, and a power-law temperature dependence for the longer samples. Contraryto these expectations, previous reports for both small and large samples, show atemperature independent behaviour over a large range of temperatures. [28, 1, 38].Only in one study [39], which used a very low bias current, a power-law temperaturedependence was observed for the edge conduction in mesoscopic samples. This wasattributed to the strong electron-electron interaction in 1D edge states in InAs/GaSb,owing to its small Fermi velocity [39].The other ambiguity regarding the helical nature of the edge conduction inthese reports is its weak in-plane magnetic field dependence. One would expectthat the magnetic field breaks time-reversal symmetry, which would result in anenhancement of the edge channel resistivity due to the backscattering processes.Different justifications, such as a small g-factor of the edge state due to their lowFermi velocity [28], and the fact that the topologically protected crossing point of26the counter-propagating edge states might be hidden in the bulk valence band inthe inverted regime [40, 41] were given to explain this weak in-plane dependence.On the other hand, as discussed in section 3.3.3, the trivial edge states in thenon-inverted regime also show a weak in-plane magnetic field dependence.In summary, the existing reports show a robust edge conduction in InAs/GaSbQWs, which are interpreted as a signature of helical edge modes. However, thetemperature and magnetic field dependence of these edge channels seem not to agreewith the expected picture of the helical edge conduction. Moreover, one shouldalso note that significant edge conduction could also occur in the trivial state dueto other effects such as band bending [42, 43, 44]. Therefore, it becomes crucialto use the distinct possibility of electrostatic gate tuning between the trivial andtopological state in this material to address the concern regarding the nature of theedge conduction. To do so, a collaborative experiment was planned with the Marcusgroup at the Niels Bohr Institute.3.2 Experimental detailsAll devices studied in this work were fabricated by HRL laboratories. I have donethe non-local, in-plane magnetic field, and temperature dependence measurementson the macroscopic samples.The samples investigated in this study were grown by molecular beam epitaxy(MBE) on a conductive GaSb substrate, which served as the global back gate [45].From the substrate to the surface, all structures consist of a GaSb/AlSb insulatingbuffer, a 5nm GaSb QW, a 10.5nm InAs QW, a 50nm AlSb insulating barrier and a3nm GaSb capping layer. In the following paragraph adapted from ref. [22], thedevice fabrication procedure employed by HRL laboratories is explained.Similar fabrication recipes as in previous edge channels studies in InAs/GaSbwere adapted [32, 46, 47]. Devices were patterned by conventional optical and elec-tron beam lithography and wet etching. Devices shown in Fig. 3.1(a) were etchedusing a sequence of selective etchants [48], the other devices with a conventionalIII-V semiconductor etchant [45]. The two recipes gave consistent results. Ohmiccontacts were obtained by etching the samples down to the InAs QW and depositingTi/Au electrodes. Top gates were defined by covering the samples with a thin (80nm)27Al2O3 or HfO2 insulating layer grown by atomic layer deposition and a patternedTi/Au electrode. Special care was taken during the entire fabrication process notto accidentally create or enhance spurious edge conductance in the samples. Inparticular, it is known that antimony compounds react with oxygen and opticaldevelopers giving rise to amorphous conductive materials [49, 50]. We thereforestore the samples in nitrogen, never heat the samples above 180°C and deposit theinsulating oxides immediately after the wet etching, serving as a passivating layer.In many devices, the backgate leaked when more than 100 mV was applied,presumably due to damage during processing. These leaky devices were onlyoperated at zero backgate voltage, where the resistance to the backgate was at least10 GΩ. Except where specified, transport experiments were performed in dilutionrefrigerators at a temperature of less than 50 mK with standard low frequencylock-in techniques. Further details regarding the electronic setup and cryogenics arediscussed in chapter 7.3.3 Transport In Macroscopic Samples3.3.1 Magnetotransport MeasurementsAs the first step to investigate the nature of edge conduction in InAs/GaSbdouble QWs, the phase diagram was mapped out by measuring the longitudinalresistance (Rxx) of a Hall bar device (Fig. 3.1(a)), as a function of top gate (VTG)and back gate (VBG) voltages. This double gate geometry enables independenttuning of the band alignment and Fermi level as shown in Fig. 3.1(b). The red axiscorresponds to changing the electric field (band alignment) at a constant density.The normal band ordering (trivial state) can be achieved by moving towards theright side of the red axis by applying a positive Vbg and a negative Vtg, which lowersthe valence band in GaSb and raises the conduction band in InAs, as proposed in[24] and observed in [51]. On the other hand, on the left side of the red axis, whichis achieved by moving towards negative Vbg and more positive Vtg, the state isexpected to have an inverted band ordering. For either of the trivial or invertedstates, electron or hole doped regions can be obtained by tuning the Fermi level at afixed band alignment by moving to the right or left side of the blue axis respectively28(Fig. 3.1(b)).Fig. 3.1(c) represents the observed phase diagram for the device configurationshown in Fig. 3.1(a). Placing the Fermi level in the energy gap of the inverted/trivialstate results in high resistance regions I/II in Fig. 3.1(c) respectively. The resistancein region I reaches a few hundreds of kΩ and in region II to around 40 kΩ. The lowresistance regions III and IV indicate electron and hole dominated transport wherebythe Fermi level is positioned in the conduction and valence bands respectively.Figure 3.1: (a) Schematic of the device geometry and the measurement schemeused for the measurements in (c-e). (b) Independent tuning of the electricfield (band alignment) and Fermi level using double gate geometry. Thered (blue) axis corresponds to the change of the electric field (Fermilevel) at a constant density (electric field). (c) Top and backgate voltagedependence of Rxx (bias current I = 5 nA). The Hall resistivity (ρxy(B⊥))is measured at each of the locations marked by circles along the lines Rand L, shown in (d) and (e) respectively (bias current I = 10 nA). Part(a),(c),(d) and (e) are taken from [26].29Figures 3.1(d) and (e) represent Hall measurements along lines R and L in thephase diagram of Fig. 3.1(c) as a way of verifying the nature of the two gappedregions. As can be seen in Fig. 3.1(d), along the line R in the trivial state, the Hallresistivity (ρxy) shows linear magnetic field dependence on both sides of the trivialgap, with positive (negative) slope corresponding to the electron (hole) dominatedtransport in the conduction (valence) band. Similar behavior can be observed in Fig.3.1(e) along line L in the inverted region when the Fermi energy is tuned into theconduction or the valence band. This behaviour contrasts with the Hall resistancein the gapped regions I and II. In the gapped region of the trivial state (region I)such as point 4 in Fig. 3.1(d), ρxy shows large fluctuations with no net slope, due toa vanishingly small number of charge carriers. On the other hand, in the invertedgap region (points 3 and 5 in region II) ρxy displays a non-monotonic behaviorindicating the simultaneous presence of the electron and hole carries in the invertedgap [51]. It should also be mentioned that for a few points along the line L in theelectron doped side, such as point 7, ρxy shows oscillations with a periodicity of1/B, suggestive of Shubnikov-de Haas oscillations [52].As the hallmark of the QSH effect is the presence of the helical edge state in theinverted gap, we check for the signature of edge conduction in the gapped regions ofthe phase diagram. In our device with edge length ∼ 100 µm, which is much longerthan the ballistic limit in InAs/GaSb double QWs (of the order of a few microns[28, 1]), the resistance in the hybridization gap in region II should be at least anorder of magnitude higher than he2 . However, as evident in Fig. 3.1(c) the resistancein the region II is ∼ 40 kΩ, which perhaps is due to the presence of residual bulkconduction in addition to the edge conduction. This bulk dominated conductionis also clear in the non-local measurement shown in the next section. As a result,exploring the edge conduction in the inverted region of the phase diagram becomesdifficult.The other gapped region I, which shows non-zero conductance is inside thetrivial part of the phase diagram. We see in the next section that the finite resistancein the region I of the phase diagram in Fig. 3.1(c) comes from the conductingstates along the edge of the sample, not the residual bulk conductivity in this region.Despite being in the trivial state, these edge states illustrate superficial similarity asdiscussed in the upcoming sections with those in the previous reports where their30origin was claimed to be helical. We will show the length dependent measurementsin section 3.4 that confirm the non-helical origin of the edge states in our samples.3.3.2 Non-local measurementsIn order to know the origin of the finite resistance in the region I in Fig. 3.1(c) non-local measurements are employed. Non-local measurements are a common approachfor distinguishing between edge dominated transport and bulk conduction. The ideaof the measurement is to find the voltage drop far from what would be the bulkcurrent path in a conventional Ohmic material. Figure 3.2 shows the schematic of theelectrical setup used for the four-terminal (a) local and (c) non-local measurements.Theoretical calculations have shown that in the non-local configuration the bulkcontribution to the non-local signal is suppressed by e−piS, where S is the numberof squares between the current and voltage probes obtained by dividing the lateralseparation between the voltage probes by the width of the device (S is 5 for ourdevice). On the other hand, a current that is passing along the edge of the samplegives rise to a sizeable signal by passing directly through the non-local voltageprobes. Figures 3.2 (b) and (d) show the four-terminal measurement of resistivity inlocal (Rxx) and non-local (RNL) configurations respectively of a microscope Hall barsample nominally identical to the one investigated in Fig. 3.1(c). However, due to alower bias current, 10 pA, compared to 5nA applied for the measurement of Fig.3.1(c), Rxx is an order of magnitude higher in region I of Fig. 3.2(b) compared toFig. 3.1(c). As can be seen for the regions II, III, IV, RNL is∼ 4 orders of magnitudesmaller than Rxx, which implies the bulk dominated transport in these regions. Onthe contrary, for the region I, the non-local signal is within an order of magnitude ofRxx, which means the transport in this region is mostly due to the edge conduction.31Figure 3.2: 4-terminal resistivity measurement in local and non-local config-urations. (a,c) Schematic of Hall bar geometry in local and non-localmeasurements along with electronic setup used to measure the longitudi-nal resistance Rxx (b) and non-local resistance RNL (d). Dotted lines in(b) represents back gate voltages where measurements of the in-planemagnetic field (Fig. 3.3) and temperature dependence (Fig. 3.4) aretaken. Taken from [26].In the next two sections, we characterize the edge states in the region I of thephase diagram through the in-plane magnetic field and temperature dependencemeasurements.3.3.3 In-plane magnetic fieldThe in-plane magnetic field measurement is in fact a way to distinguish between thehybridization gap in the inverted region and the normal gap in the trivial region. Fora device lying in the x-y plane, the application of a magnetic field parallel to thex-axis, B||, will cause a relative momentum shift of the electron and hole subbandsin the y direction: ∆Ky = eB<z>h¯ as shown in Fig. 3.3(a,b). Here e is the electroncharge, h¯ is the reduced Planck constant and z is the separation between 2D electrongas in InAs and 2D hole gas in GaSb [53, 54]. This would result in quenching ofthe hybridization gap in the inverted regime, whereas it almost has no effect on32Figure 3.3: (a,b) Schematic of the effect of the in-plane magnetic field on theband structure of InAs/GaSb in the inverted regime in zero field (a) and inthe finite field (b). (c,d) Schematic of the effect of the in-plane magneticfield on the band structure of InAs/GaSb in the trivial regime in zerofield (c) and in the finite field (d). Longitudinal resistance (Rxx) along thedotted line in Fig. 3.2(b) in the presence of the in-plane magnetic field inthe inverted (e) and trivial (f) regimes. The resistance peak correspondsto the onset of the conduction band in the hole regime in (e) and thetrivial gap with edge conduction in (f). The direction of the appliedin-plane magnetic field is shown in Fig. 3.2(a). Parts (e) and (f) are takenfrom [26].the insulating gap in the trivial regime (Fig. 3.3(c,d)). The experimental signaturewould be a decrease in the resistivity peak associated with the hybridization gap asa function of the in-plane magnetic field and almost no effect on the resistivity peakcorresponding to the normal gap in the trivial regime.Figure 3.3(e,f) shows such a measurement for both regimes. In the invertedregime (Fig. 3.3(e)) contrary to the expectation, we find that the resistivity peak at33zero magnetic field increases with the application of an in-plane magnetic field. Thismeans that the observed resistivity peak does not correspond to the hybridizationgap, which here is suppressed by the residual bulk conduction apparent in Fig.3.1(c). The observed peak however correspond to placing the Fermi level at thebottom of the conduction band in the hole regime as represented in the inset of theFig. 3.3(e), consistent with the analysis in [51].In the trivial regime, a weak dependence of the edge states on the in-planemagnetic field is observed. Only for the most resistive sample, we saw a resistancedrop by a factor of more than two (Fig. 3.3(f)). For all other devices measured in thisstudy, the reduction in the resistance in the presence of an in-plane magnetic fieldwas less than 10%. This sample-to-sample variation is consistent with an extrinsicorigin for the edge states in region I. It should be noted that in the previous transportexperiments on InAs/GaSb, the edge conduction illustrated a weak dependence onthe in-plane magnetic field [28, 1].3.3.4 Temperature dependenceThis section describes measurements that are done in order to find the energy gap ofthe bulk state in the region I of Fig. 3.2(b) as well as the temperature dependence ofthe edge states in that region.To estimate the bulk gap in region I, one can do temperature dependence mea-surement in a Corbino geometry, which is a device consisting of a ring-shapedsample contacted by two concentric contacts. As a result, in the transport measure-ment via this geometry, the current flows exclusively through the bulk and no edgesconnect the two contacts. Figure 3.4(a) shows such a measurement performed byour collaborator on a Corbino device with internal and external radii of 50 mmand 120 mm respectively. The temperature dependence of the conductivity of theaforementioned Corbino device fits very well (Fig. 3.4 (b)) to the Arrhenius law, i.e: σxx ∝ exp(−∆/kBT ), where, σxx is the bulk conductivity, 2∆ represents the bulkenergy gap and kB is the Boltzmann constant. The extracted energy gap for VBG =0V, which is well inside the region I in figure 3.2(b) was found to be ∼ 500µeV .We also investigated the temperature dependence of the macroscopic Hall barsamples in the region I of Fig. 3.2(b), which is dominated by the edge conduction.34Figure 3.4: (a) Conductance measurements in the Corbino disk in the trivialregime. The inset is a schematic of the Corbino disk. (b) Blue dotsrepresent the temperature dependence of the conductance of a particularpoint on the trace in (a) vs the inverse of temperature. Data in (b) fitsvery well (solid line) to a simple activated behaviour based on Arrheniuslaw. (c) The inverse of the four-terminal resistance vs topgate voltagein a macroscopic Hall bar geometry (inset) taken on a line in the trivialregime similar to the (a). Note that the minimum of the conductancehere is around four orders of magnitude higher than the one in (a),which shows the edge dominated transport in the trivial regime. (d)Temperature dependence of the inverse four terminal resistance vs inverseof temperature square for a point in the middle of the minima plateau in(c). The solid line is a fit to a variable range hopping model. Taken from[26].35Fig. 3.4(c) shows the inverse of longitudinal resistance (R−1xx ) vs VTG at VBG = 0,which is well inside the trivial regime as represented by the red dotted line in Fig.3.2(b). The temperature dependence measurements of the Hall bar device revealedan insulating temperature dependence for the edge conduction in this region asshown in Fig. 3.4(d). To find out the mechanism behind this insulating tendency,different models were investigated. Since the Arrhenius law did not show a goodfit to our data, applying a more general fitting expression as Rxx ∝ exp(ξ/kBT )[1/α]with ξ and α as the fitting parameters resulted in α = 2.0± 0.5, which is inagreement with variable range hopping in one dimension or Coulomb dominatedhopping in one or two dimensions [55].What we have presented so far is qualitatively in agreement with the previousreports, where the edge conduction in InAs/GaSb QWs was claimed to be helical.However, the edge conduction in our samples is in the trivial regime based on the2D map of resistivity presented in Fig. 3.1(c). Nevertheless, it is crucial to testwhether the edge states in our samples resemble any helical signature as expectedfor edge modes in QSH state. To address this question, the next two sections presentthe length dependent measurements and transport measurements of microscopicHall bar samples that are carried out by our collaborators in Niels Bohr Institute.3.4 Transport In Microscopic Samples3.4.1 Length Dependent MeasurementsAs we discussed in chapter 2, the conductance of helical edge modes are quantizedfor edge lengths shorter than the spin scattering length, which is found to be a fewmicrons for InAs/GaSb [28, 1]. To test the conductance quantization of the edgestates in our samples, two devices in the shape of a long bar with a width (W ) of1µm and 2µm were fabricated. Across the long bar, multiple gates with length(L) ranging from 300 nm to 30µm were patterned as shown in Fig. 3.5(a). Whenno bias voltage was applied to both top and back gates, the entire bar was in then-doped regime and highly conductive. By tuning the bias voltage on one gate at atime and monitoring the end to end resistance of the bar, the area underneath thatgate was brought into the trivial insulating regime. Thus, we were able to measure36the length dependent resistance in a single device. We obtained identical results fortwo bars with different widths (1µm and 2µm). This independence of resistanceon the width is another piece of evidence for the edge dominated conductance inthe trivial regime. Figure 3.5(b) shows the two terminal resistance measurement forone of the devices (W = 2µm) vs the top gate voltage. Each trace corresponds tothe two terminal resistance of the whole bar as one of the gates tunes the bar fromthe highly conductive state into the insulating regime. Figure 3.5(c) presents theresistance change (∆R) in each trace, which is taken as a difference between theresistance peak in the insulating state and its minimum in the conducting state vsthe length of the corresponding gates.Figure 3.5: (a) Schematic of the long bar along with the gated regions andelectrical set up used to measure the length dependent resistance. (b)Two terminal resistance of the long bar with W = 2µm vs applied biasvoltage on top gates of different length (L). (c) Resistance change inthe two-terminal device as a function of gate length for the W = 1µm(crosses) and W = 2µm (pluses) together with a linear fit (black line).In the fitting function, λ is the resistance per unit length of one edgechannel, the factor 1/2 takes into account two edge channels that conductin parallel, and R0 is the resistance minimum in the short-channel limit.Circles and squares indicate the edge resistances measured in the Hbarand µHall bar respectively, as discussed in the next section. Taken from[26].37One would expect that if the edge states were helical, for the edge length shorterthan the critical length the resistance approaches the expected quantized value basedon the Landauer-Bu¨ttiker formalism. On the contrary, as can be seen in figure 3.5(c), ∆R illustrates a linear dependence on the length (L). It should be mentioned thata more recent report [59], using the same procedure as explained in this section, alsofound a linear length dependent resistance for the edge conduction in InAs/GaSbQW.It is notable that for the edge length close to the one in the previous reportsof the edge conduction in InAs/GaSb [28, 1], we got also a resistance around thereported value. Thus, these results cast doubt on the claimed helical nature of theedge states in the previous reports.3.4.2 Quantization In Microscopic Hall barConductance saturation with a value close to the expected quantized conductanceof the helical edge conduction has been provided as an evidence for the QSH statein InAs/GaSb for the specific geometry and edge length in the previous reports.Though in the last section, we showed the evidence for the non helical nature of theedge states in our samples, here we present that for the specific device length andgeometry (Fig. 3.6(a,b)) similar to the ones used in those reports [28, 1], we alsoobserve a conductance saturation within a few percent of the expected quantizedconductance of the helical edge conduction. Figure 3.6(c) displays 4-terminalresistance measurements on H-bar geometry (Fig. 3.6(a)) as a function of thetopgate voltage for various configurations of the current and voltage probes. Oncethe Fermi level is placed in the band gap, the conductance shows a value veryclose to the quantized conductance expected for the helical edge state based on theLandauer-Bu¨ttiker formula. However, as seen in the last section, the conductancein our samples in the trivial regime cannot have a helical nature, as a result, theapparent agreement with the quantum of conductance is accidental.This accidental nature of conductance saturation is further verified in the mea-surements on a more conventional Hall bar geometry with a few micron separationbetween adjacent contacts, called µHall bar geometry (Fig. 3.6(b)).38Figure 3.6: (a) Schematic of the H-bar along with contact numbering used inthe measurement presented in (c). The length of the H bar covered by thetop gate is 3.8µm, and the width of the H bar arms is 1µm. (b) Schematicof the µHall bar along with contact numbering used in the measurementin panel (d). The electrical set up used to do the measurement is alsodisplayed. The width of µHall bar is 1µm, and the separation betweencontacts 2 and 3 as well as 3 and 4 is 2.4µm, and separation between4 and 5 is 4.8µm. (c,d) Four-terminal longitudinal resistance measuredas a function of the VTG for various contact configurations. Dottedlines represent the expected quantized resistance values based on theLandauer-Bu¨ttiker formula. Taken from [26].The 4-terminal longitudinal resistance as a function of VTG is measured bypassing current between end to end contacts (1 and 6) and measuring the voltagedrop between a pair of lateral contacts (Fig. 3.6(d)). The four-terminal resistanceshows saturation when the top gate voltage drives the Fermi level into the insulatingbulk state. In the case where the voltage drop is measured between the contacts 2and 3 (red trace), with a separation of 2.4µm, the saturated resistance is close to theexpected quantized resistance based on the Landauer-Bu¨ttiker analysis. However,when the separation between the voltage probes is doubled, the saturated resistancemarkedly differs from the expected quantized resistance, as shown by the greentrace, which is a measurement between voltage probes 4 and 5 with a separationof 4.8µm. Moreover, the measurement done between contact 2 and 4 (blue trace),39with the same separation of 4.8µm, but with an intervening contact between them,resulted in the same saturated value as the green trace. This is in contrast to whatwe expect for a helical edge mode, where the intervening contacts are expected toact as a dephasing center between counter-propagating helical edge states and raisethe resistance by a factor of two. Altogether, the result of the Hall bar geometrymeasurements also points out that our observed saturated resistance with a valueclose to the expected quantized resistance of the helical edge states are coincidental.3.5 ConclusionIn summary, we have presented a unique set of measurements in investigating theedge conduction in InAs/GaSb QWs. Using the interesting possibility of tuningbetween trivial and inverted regimes in the InAs/GaSb QWs, we found a robustedge conduction in the trivial phase, which showed identical features to the onein previous reports of helical edge conduction. However, our length dependentmeasurement as well as microscopic Hall bar measurements ruled out the helicalnature of the edge conduction in the trivial phase of our samples. One interpretationfor the observed edge conduction in the trivial regime could be band bending inthese heterostructures with a small band gap [42, 43, 44]. Our findings in this workhighlight the importance of detailed characterization of the edge conduction inInAs/GaSb QWs, before taking the observed results as a definite signature of thehelical edge conduction and set a platform for doing such measurements.Finally, we should emphasize that our results don’t rule out the possibility ofthe helical edge conduction in the inverted region of InAs/GaSb. Moreover, theprevious reports on the edge conduction might indeed correspond to the helical edgestates. However, to definitely and without any ambiguity ascribe the observed edgeconduction in those reports as a helical edge state, further studies should be done.These should include mapping the complete phase diagram using double gate geom-etry in order to confirm that the edge conduction is indeed occurring in the invertedregion. Another crucial measurement is to do the length-dependent measurementsto ensure that the edge resistance doesn’t extrapolate to zero resistance as the edgelength approaches zero limit. Furthermore, the edge resistance should saturate tothe value close to the expected quantized resistance for the edge length shorter than40the spin scattering length. Moreover, since trivial edge states might coexist with thehelical edge states in the inverted region, it is crucial to suppress any trivial edgestates, in order to solely study the helical edge states in the inverted region.The experimental challenges in realizing a robust helical edge conduction inthe topological state of QW systems, such as those presented in this chapter forthe InAs/GaSb double QWs, prompted the search of QSH states in other electronicsystems. In the next chapter, WTe2 will be introduced as an example of a van derWaals 2D material, where 2D TI state has recently been reported.41Chapter 4Exotic phases of WTe2 SemimetalThe other electronic systems where a 2D topological state have been predicted toexist are the transition metal dichalcogenides (TMDs) [61]. Owing to their layeredstructure and ease of device fabrication, the experimental realization of a QSH stateis less challenging in these systems as compared to MBE grown quantum wellsystems such as InAs/GaSb. In this chapter, we review the electronic structure of amember of the TMDs family, WTe2, which recently was discovered to be a 2D TI inits monolayer limit [60, 61]. Moreover, as revealed in the next chapter, monolayerWTe2 can also be turned into a 2D superconducting state under a mild electrostaticgating.Transition metal dichalcogenides (TMDs) have a chemical formula MX2, whereM is a transition metal (Mo or W) and X is a chalcogenide atom (S, Se or Te). Asingle layer of TMD consists of a plane of metallic atoms sandwiched betweentwo identical planes of chalcogen atoms. The metallic and chalcogen atoms withina layer are bonded together via covalent bonding. These layers are stacked viainterlayer van der Waals force in bulk TMDs crystals. The relative position of themetal and chalcogen atoms within each layer as well as the stacking pattern of theselayers would give rise to a few different structural phases with different electricaland optical properties. The two common phases are called the 1H and 1T phases,in which TMDs with 1H phase are semiconductors and those with 1T phase aremetallic.As shown in Fig. 4.1(a), in the 1H phase, the M (metal) and X (chalcogen)42Figure 4.1: From [61]. Reprinted with permission from AAAS. Crystal struc-ture of TMDs with chemical formula MX2, in 1H (A), 1T (B) and 1T’(C) phases. Top row represents the structural order in each case andthe middle and bottom rows illustrate the top and cross sectional viewrespectively.atoms form a trigonal prismatic structure, where three atomic planes in each layerorder in ABA stacking pattern, i.e., the X atoms in bottom and top planes take thesame position in the in-plane direction. On the other hand, in the 1T phase, M andX atoms crystallize in an octahedral coordination, where the three atomic planesin individual layers form an ABC stacking as depicted in Fig. 4.1(b). For some ofthe TMDs, the stable thermodynamic phase is a distorted octahedral structure ofthe 1T phase known as the 1T’ phase (Fig. 4.1(c)). One of the TMDs, with the 1T’structure in monolayer form, is tungsten ditelluride (WTe2), which has been foundto possess a variety of exotic phases.4.1 WTe2, Bulk Electronic Band StructureIn terms of the electronic band structure, WTe2 is a semi-metal, meaning that thereis an overlap between the conduction and the valence bands (Fig. 4.2(a)). Theconduction and the valence bands in WTe2 arise from tungsten 5d and tellurium 3p43orbitals respectively.Figure 4.2: Taken from [62]. (a) Calculated electronic band structure of WTe2bulk. (b) Schematic of Fermi surface in the vicinity of the Γ point. Purpleand blue represents hole and electron pockets respectively.It has been found both from density functional theory (DFT) calculations (Fig.4.2(a)) and experimental data obtained from angle-resolved photoemission spec-troscopy (ARPES) [68] that there are electron and hole pockets of approximatelyequal size close to the Fermi level as shown in Fig 4.2(b) [62]. This equal size ofelectron and hole pockets could result in a perfect charge compensation in vicinity ofFermi level [62, 63, 68], which might explain the experimental report of extremelylarge magnetoresistance in bulk WTe2 [63].444.2 Experimental observations in bulk WTe2Owing to its unique band structure and structural phase, WTe2 possesses a varietyof exotic phases. When the bulk crystal WTe2 is placed in an external magneticfield perpendicular to the Te-W-Te sandwich planes, an extremely large magnetore-sistance (XMR) with no sign of saturation up to 60 Tesla was observed as shown inFig. 4.3.Figure 4.3: Reprinted by permission from Springer Nature: Nature [63], Copy-right (2014). MR versus external magnetic field applied perpendicularto the WTe2 planes and along crystallographic c axis. MR is defined asρ(H)−ρ(0)ρ(0) , whereρ(H) and ρ(0) are measured resistivity at zero magneticfield and magnetic field H respectively.The observed MR was highly anisotropic with a negligible in-plane magnetic fielddependence. The XMR is attributed to the semimetallic character of WTe2 and anearly perfect charge compensation of the electron and hole charge carriers [63].45Figure 4.4: Taken from [64]. Schematic of phase transition in WTe2 underexternal pressure along with the measured data.The large magnetoresis-tance state (LMR) is suppressed under high pressure and superconductingstate (SC) appears at high pressures. Empty circles are the characteristictemperature, where the LMR state turns on. Filled circles representcritical temperature (Tc) of superconducting state obtained from differentruns of electrical measurements. Tc from susceptibility measurementsare shown by filled triangles.It was also found that bulk WTe2 crystal becomes superconducting under theapplication of a high pressure [64, 66]. The appearance of the superconductingstate is accompanied by suppression of the XMR effect. It was proposed that theapplied pressure reconstructs the Fermi surface and disturbs the perfect balance ofthe electron-hole pockets, which leads to the vanishing of the XMR and emergenceof the superconducting phase as shown in Fig. 4.4.So far, we have discussed WTe2 in the bulk limit. As the thickness of the crystalapproaches a few layers, other exotic phases appear. In a single layer limit, it wasproposed and later on experimentally verified that monolayer WTe2 is a quantumspin Hall (QSH) insulator [60, 61]. In the next section, we overview the theoreticalproposal and the experimental verification of QSH state in the monolayer WTe2.464.3 QSH in Monolayer WTe2Recently it was proposed by Qian et al. that monolayer TMDs in the 1T’ phase hosta QSH state [61]. The conduction and valence bands in most TMDs are comprisedmainly of d-orbitals of the transition metal and the chalcogen p-orbitals respectively.In 1T’ phase, the conduction band energy around Γ point is lower than the valenceband, which results in a band inversion, thus leading to a non-trivial topologicalstate. Due to the strong spin-orbit coupling a band gap would open at the crossingpoints of the inverted bands and a QSH state will be realized [61]. The theoreticalcalculations showed contradicting results regarding the existence of a bulk gap forthe monolayer WTe2. One study revealed that WTe2 is a semimetal with negativeband gap as shown in Fig. 4.5(a) [62]. Therefore, to realize a QSH state, it wassuggested to use tensile strain to lift the negative gap. On the other hand, in anothertheoretical calculation it was shown that monolayer WTe2 possesses a positive gapof ∼ 100meV [65].Experimentally, the evidence for a QSH state in WTe2 was found in ARPESand transport measurements [60, 67, 69]. In one of the transport studies as shownin Fig. 4.5(b) the two terminal conductance was measured as a function of gatevoltage, which tunes the Fermi level and hence the charge carrier density. Themeasurement was done over a set of temperatures from 300K down to 1.6K. Asseen in Fig. 4.5(b), the conductance of a monolayer WTe2 device starts dropping asthe temperature is lowered until 100K, after which it doesn’t go down any furtherbut develops a plateau. Though the conductance of the plateau was almost halfof the expected value for the helical edge conduction in the QSH sample, otherexperimental verifications such as collapse of the edge states with the applicationof an external magnetic field that break the time reversal symmetry were stronglysuggestive of the QSH state in monolayer WTe2.47Figure 4.5: (a) Calculated band structure of monolayer WTe2 shows an overlapbetween conduction (red line) and valence (blue line) bands. Takenfrom [62]. (b) Two terminal conductance measurements for varioustemperatures versus back gate voltage that tune the position of the Fermilevel. Reprinted by permission from Springer Nature: Nature Physics[67], Copyright(2017). The corresponding change in the carrier densityas a result of the applied gate voltage is displayed on the top axis. Notethat zero gate voltage position corresponds to the zero energy level in(a), despite that here the temperature dependence represents a gap in thebulk.Another transport study using local bottom gates showed one would indeed seea value very close to the quantized conductance for an edge length shorter than spinscattering length. [60]. Hence both transport studies revealed that the bulk state isgapped. The estimated bulk gap from the ARPES measurements on a MBE grownmonolayer WTe2 is ∼ 50meV [69]. However, the nature of this bulk state, whetherit is a single particle gap or arises from strong correlations is still not clear. A morerecent study, employing scanning tunneling spectroscopy and DFT calculationsconcluded that the origin of the bulk gap in monolayer WTe2 is a Coulomb gap,possibly arising from the electron-hole interaction near the Fermi level [70].Overall, the following observations led to a conclusion that monolayer WTe2is indeed a 2D topological insulator. First it is observed that edge conductionapproaches the expected quantized value for the helical edge states over the edge48length shorter than spin scattering length [60]. Second, the edge conductance issuppressed upon applying magnetic field, which breaks the time reversal symmetry,and a Zeeman gap opens at the edge states as expected for the helical edge states[60, 67].It should be mentioned that the observed conductance plateau in Fig. 4.5(b) isobtained under a finite dc bias of 3meV. In the linear transport regime as shown inFig. 4.6(a) the conductance in the plateau region is almost zero at low temperatures.This implies the presence of an edge gap that blocks the edge transport. At firstglance, the presence of this gap contradicts the expected gapless edge states in theQSH system. However, the following arguments support that the monolayer WTe2is a 2D TI, despite the presence of this edge gap. First, this edge gap seemingly iscreated as a result of an imperfect interface between the metal contacts and the edgestates in the monolayer WTe2. In a more recent study [60], using a combinationof bottom local gates and top graphene gate, gapless edge states were observed asexpected for a TI system. Second, since the size of the edge gap is much smallerthan the estimated bulk gap (∼ 50meV), upon applying a small dc bias (∼ 3meV),the edge transport is recovered as shown in Fig. 4.5(b), with characteristics of thehelical edge states as discussed in [67].Fig. 4.6(b) shows the differential conductance vs dc bias voltage for a set oftemperatures at Vg = −0.68V . As can be seen in Fig. 4.6(b) a small dc bias ofaround 3meV is sufficient to overcome the edge gap. The edge transport can alsobe thermally activated as evident from temperature dependence of the differentialconductance in Fig. 4.6(b). Referring to the differential conductance at zero biasvoltage in Fig. 4.6(b), by increasing the temperature, due to the thermal broadeningof the edge states the edge gap decreases, and at 30K (red trace) the edge gapbecomes completely suppressed.49Figure 4.6: Reprinted by permission from Springer Nature: Nature Physics[67], Copyright(2017). (a) Temperature dependence of edge conductancein a linear transport measurement with 100 µV ac bias. Here the edgeconductance at low temperatures is almost zero representing the pres-ence of an energy gap. (b) Temperature dependence of the differentialconductance measurement at Vg=-0.68V.In the next chapter, we describe our discovery of another exotic phase ofmonolayer WTe2. We found that superconductivity can be induced in this mono-layer 2D TI by mild electrostatic doping, at temperatures below 1K. The 2D TI-superconductor transition can be easily driven by applying a small gate voltage. Thisdiscovery offers possibilities for gate-controlled devices combining superconduc-tivity and non-trivial topological properties and could provide a basis for quantuminformation schemes based on topological protection.50Chapter 5Gate Induced Superconductivityin a Monolayer TopologicalInsulatorThe work presented in this chapter is an adaption of our published work in [71].Materials that combine non-trivial topology with superconductivity have beenthe subject of active investigation in recent years [10, 72, 73]. For example, hybridstructures that couple an s-wave superconductor to a 2D TI have also been proposedas a platform for Majorana modes [10], whose non-abelian exchange propertiesmight be harnessed for qubits [72] with coherence times far longer than those builton conventional platforms. There are also topological superconductors, in whichvortices or boundaries can host Majorana modes [73].This chapter reports that monolayer WTe2, recently shown to be an intrinsic2D TI (as we discussed in the last chapter), turns superconducting under moderateelectrostatic gating. Several other non-topological layered materials superconductin the monolayer limit, either intrinsically or under heavy doping using ionic liquidgates [74, 75, 76, 77, 78, 79, 80, 81, 82]. In monolayer WTe2, however, the phasetransition to a superconducting state is from a 2D topological insulator and occursat such low carrier density that it can be readily induced by a simple electrostaticgate. The discovery may lead to gateable superconducting circuitry and offers thepotential to develop topological superconducting devices in a single material, as51opposed to the hybrid constructions currently required [10].We present data from two monolayer WTe2 devices, M1 and M2, with con-sistent superconducting characteristics. Each contains a monolayer flake of WTe2encapsulated along with thin platinum electrical contacts between hexagonal boronnitride (hBN) dielectric layers.5.1 Experimental Details of Device M1Figure 5.1 shows optical and AFM images of device M1. The making of device M1began with exfoliated few-layer hBN (lower hBN) on thermally grown SiO2 on ahighly doped Si substrate. Seven metal contacts in a row were defined on the lowerhBN using electron beam lithography (EBL) and metalized (∼7 nm Pt) in an e-beamevaporator followed by acetone lift-off and annealing at 200°C. A second exfoliatedfew layer hBN (upper hBN) flake was picked up using a polymer-based dry transfertechnique [83]. Flux-grown WTe2 crystals (from Jiaqiang Yan, Oak Ridge NationalLab) were exfoliated in the glovebox (O2 and H2O concentrations <0.5 ppm) andmonolayer pieces were optically identified. After identification, the monolayerWTe2 was picked up on the upper hBN and transferred to the Pt/hBN/SiO2/Si stack.Only after fully encapsulating the WTe2 the device was removed from the glovebox.After dissolving the polymer a few-layer graphene (FLG) flake was added as thetop gate. Finally, contact pads consisting of 70/7 nm Au/V were added.52Figure 5.1: (a) Optical image of device M1. (b) AFM image of device M1along with electrical measurement setup. The boundaries of WTe2 areindicated by red lines. Scale bars indicate 5µm.5.2 Device Fabrication of M2Device M2 began with few-layer hBN (lower hBN) covering a FLG bottom gateon thermally grown SiO2 prepared using the same dry transfer technique and thenannealed at 400 C. As in M1, contacts were patterned on the lower hBN and 7 nmPt was evaporated, but now in a Hall bar configuration, and the upper hBN flakewas prepared. Monolayer WTe2 was picked up with upper hBN and put down onthe Pt contacts in the glove box. Finally, for this device, a Hall bar-shaped top gatewas patterned with EBL together with the contact pads, using ∼7/70 nm evaporatedAu/V, followed by lift-off in acetone. The full device structure is shown in Fig. 5.2.53Figure 5.2: (a) AFM image of device M2 along with metal contacts used forthe electrical measurements. (b) Optical image of device M2. Scale barsindicate 5µm. The boundaries of WTe2 are indicated by red lines in (a)and (b).The thickness of each hBN flake was determined using an AFM (Bruker Di-mension Edge), and is listed in table 5.1 along with the calculated geometric arealcapacitance for the top and bottom gates for M1 and M2, given by c= ε0εrd , wherewe take εr = 4 for hBN and 3.9 for SiO2.upper hBN (nm) lower hBN (nm) SiO2 (nm) ct (nF/cm2) cb (nF/cm2)5.8 18 285 611 11.48 29 285 443 122Table 5.1: Thicknesses of hBN as measured by AFM, and capacitances perunit area for bottom (cb) and top gates (ct), for M1 (top row) and M2(bottom row).5.3 Emergence of 2D SuperconductivityFigure 5.3A shows an image of M1, together with a side view and a schematicshowing the configuration used to measure the linear 4-probe resistance, Rxx =dV/dI. Details of the electronic setup are explained in chapter 7. Top and bottomgates, at voltages Vt and Vb and with areal capacitances ct and cb respectively, canbe used to induce negative or positive charge in the monolayer WTe2, producing an54areal doping density given by ne = (ctVt + cbVb)/e, where e is the electron charge.Note that we do not interpret this as a carrier density because the insulating statemay be of correlated nature (as in, for example, an excitonic insulator); in addition,Hall density measurements are challenging owing to the 2D TI edge conduction.Figure 5.3B illustrates the electrostatic tuning of M1 from p-doped conductingbehavior at negative gate voltage, through an insulating state, to an n-doped highlyconducting state at a positive gate voltage. M1 is the same device whose insulatingstate was investigated in [67], and there demonstrated to be a 2D TI [67, 60]; atne= 0, Rxx is more than 107Ω owing to a meV-scale gap that blocks edge conductionbelow 1K (see section 5.5). For ne above ncrit ≈ +5× 1012cm−2, however, theresistance drops dramatically when the sample is cooled, indicating the appearanceof superconductivity.The emergence of a superconducting phase in direct proximity to a 2D TI phase,and at a doping level achievable with a single electrostatic gate, is the primary resultof this chapter.The transition from an insulating to a metallic/superconducting T dependence−thecrossing of Rxx lines in Fig. 5.3−occurs at 2.4 kΩ. This corresponds to a squareresistivity ρ ≈ 20 kΩ, with a substantial uncertainty because the precise distributionof current in the device is not known. The estimated square resistivity is based onthe assumption that the current flows primarily in the rectangle between the sourceand drain current contacts indicated in Fig. 5.1B, with minimal spreading. Then,the conversion from resistance to resistivity requires only the aspect ratio of theWTe2 between voltage probes, which is estimated to be 10 (width/length) from theAFM image in Fig. 5.1B.55Figure 5.3: Characteristics of monolayer WTe2 device M1 at temperaturesbelow 1K. (A) Optical image (the white scale bar indicates 5µm) andschematic device structure, showing current, voltage contacts and groundconfiguration for measuring the 4-probe resistance Rxx. Inset: schematicof the atomic structure of monolayer WTe2. (B) Rxx as a function ofelectrostatic doping (ne) at a series of temperatures. Inset: variation ofRxx at 20 mK with top and bottom gate voltages, Vt andVb, indicating theaxes corresponding to doping ne and transverse displacement field D⊥Rxx depends primarily on ne and only weakly on D⊥. The measurementsin the main panel for ne > 0 and ne < 0 were made separately, sweepingVb along the two colored dashed lines in the inset to avoid contact effects.(C) Phase diagram constructed from measurements in this chapter, asexplained in the main text.56Figure 5.3C is a hypothetical phase diagram constructed based on the measure-ments presented in Fig. 5.3C and Fig. 5.4. The red dots in the phase diagramrepresent the critical temperatures of the superconducting state, extracted fromthe temperature-dependent resistance data, as explained in the next section. To-wards the right side of the density axis, in the highly electron doped region, forthe temperatures below the critical temperatures (red dots), the superconductingstate lies as depicted in purple shading. Above the red dots, the superconductingstate gradually disappears, and a normal metallic state appears as shown by thegrey shading. The middle part of the phase diagram, which is shown by the blueshading corresponds to the 2D TI state. The transition from the superconductingstate (purple region) and 2D TI state (blue region) is gradual. This is due to anintervening metallic phase, which occurs around the ncrit , represented by a greyshading. Finally, towards the left side of the TI state, in the highly hole dopedregion, a conductive state gradually appears, which is depicted by the grey shading.As the next step, we investigated the superconducting phase through the magneticfield and gate dependence measurements, as explained in the following sections.575.3.1 Doping EffectThe evolution of the T dependence with the doping density (ne) is illustrated in Fig.5.4. For all densities shown, the collapse of Rxx with temperature is gradual, as0.11101001000Rxx (Ω)8006004002000T (mK)ne= 4.6x1012 cm-220x1012 cm-2T (K)ne (1012 cm-2)0.11155-5SCTImetalA1000Figure 5.4: Rxx on log scale vs temperature T at a series of positive-gatedoping levels ne [20, 12, 8.5, 6.7, 6.1, 5.6, 5, 4.6 1012 cm−2] showing adrop of several orders of magnitude at low T for larger ne. Inset: locationof sweeps on the phase diagram.expected for materials where the normal state 2D conductivity is not much greaterthan e2h . We define a characteristic temperature, T1/2, at which Rxx falls to half its1 kelvin value. Although this specific definition is somewhat arbitrary, it does notsignificantly affect any of our conclusions as discussed in appendix B. Measuredvalues of T1/2 are shown as red dots on the phase diagram in Fig. 5.3C to indicatethe boundary of superconducting behavior. As we will see in the upcoming sections,measurements performed in the presence of magnetic field revealed some peculiarfeatures of the superconducting phase.585.3.2 Perpendicular Magnetic Field DependenceBased on the microscopic theory of superconductivity, BCS, superconductivityoccurs as a result of the condensation of quasi-particles called Cooper pairs attemperatures below the critical temperature of the superconductor. Each Cooperpair consists of two electrons with opposite spin and momentum. A perpendicularmagnetic field through orbital effect destroys superconductivity at a critical magneticfield, B⊥c2 [84, 85, 86].Figure 5.5: (A) Effect of the perpendicular magnetic field, B⊥ on resistance atthe highest ne in Fig. 5.3B (Demagnetization effects are neglected in lightof the finite resistivity of the sample.) Inset: characteristic temperaturesT1/2 obtained from these temperature sweeps, as well as characteristicfields B1/2 measured from field sweeps under similar conditions. (B)Sweeps of B⊥ showing rise of resistance beginning at very low field.Figure 5.5A shows the evolution of the superconducting to the normal phase in a4-terminal resistivity measurement versus temperature upon increasing the strengthof the perpendicular magnetic field (B⊥). Here, we have defined a characteristicfield, B1/2(T ), where Rxx falls to half its normal state resistance, as an analogueof the critical magnetic field (B⊥c2(T )), in line with a common convention for 2Dsuperconductors. The dependence of T1/2 on B⊥ (Fig. 5.5A, inset) in the low-field limit is consistent with the linear B⊥c2(T ) expected from Ginzburg-Landau(GL) theory for thin films that is typically used to analyze B⊥c2(T ) data for 2D59superconductors. In this model, the functional form of B⊥c2(T ) near Tc is given by:B⊥c2(T ) =Φ02piξGL(0)2(1− TTc) (5.1)where Φ0 is the magnetic flux quantum and ξGL(0) is the extrapolation of theGL coherence length to zero temperature. The GL coherence length (ξGL) is acharacteristic length of a superconductor that is a measure of the spatial variationof the Cooper pair density. Following equation 5.1, we can use high (T → Tc) orlow (T → 0) temperature limits of B1/2(T ) data in Fig. 5.5A (inset) to extract ameasured coherence length. The high temperature approach is based on the lineardependence of the data near Tc, where we can use the slope of B1/2(T ) near Tc thatgivesξslope =√√√√ −Φ02piTcdB⊥1/2dT |TcWithout counting on the precise applicability of equation 5.1 over the full tem-perature range, it is also straightforward to estimate a value for ξGL in the lowtemperature limit, from the extrapolation of B⊥1/2(T ) to zero temperature, givingξBC0 =√Φ02piB⊥1/2(T → 0)Because the coherence lengths extracted by these two approaches are verysimilar to each other, well within the error bar of each measurement (which itselfderives primarily from the choice of the characteristic fraction), we refer to a singlevalue of coherence length simply as ξmeas. We have obtained ξmeas = 100± 30 nmfrom the data in Fig 5.5A (inset) using both methods.The fact that ξmeas is significantly larger than the estimated mean free pathλ = h/(e2ρ√gsgvpine)≈ 8nm suggests that the system is in the dirty limit (λ <<ξ ). Here e is the electron charge and h is the Planck constant. To calculate λwe use spin and valley degeneracies gs = gv = 2, and density and normal-stateresistivity reflecting the conditions for Fig. 5.5A, ne = 20×1012cm−2 and ρ ≈ 2kΩrespectively. The coherence length expected in the dirty limit is ξ =√h¯D/∆0,for zero-temperature superconducting gap ∆0 = 1.76kBTc and diffusion constant D,60where h¯ is the reduced Planck constant and kB is the Boltzmann constant. Indeed,using T1/2 = 700 mK for Tc, and D = 2pi h¯2/gsgvm∗e2ρ ≈ 12cm2s−1 (from theEinstein relation) with effective mass [62] m∗ = 0.3me gives ξ ≈ 90nm, where meis the electron mass, consistent with ξmeas.5.3.3 In-plane magnetic field dependenceFor in-plane field, the atomic thinness of the monolayer makes orbital effect small.In the absence of spin scattering, superconductivity is then suppressed when theenergy associated with Pauli paramagnetism in the normal state overcomes thesuperconducting condensation energy. This is referred to as the Pauli (Chandrasekar-Clogston) limit [87], and gives a critical field BP = 1.76kBTc/g1/2µB, where µB isthe Bohr magneton. Assuming an electron g-factor of g = 2 and taking Tc = 700 mKgives BP ≈ 1.3 T. However, the data in Fig. 5.6, A and B, indicate superconductivitypersisting to B||1/2 =3T.Similar examples of B||1/2 exceeding BP have recently been reported in othermonolayer dichalcogenides, MoS2 and NbSe2, but the Ising superconductivitymechanism [75, 82] invoked in those works cannot explain an enhancement of B||1/2here because WTe2 lacks the required in-plane mirror symmetry. One possibleexplanation in this case is a high spin-orbit scattering rate τ−1so .The effect of various pair-breaking mechanisms on critical temperature can beencapsulated by an implicit equation for Tc(B||) that depends on a pair-breakingenergy α [85, 88, 89]:lnTc(B||)Tc0= ψ(12)−ψ(12+α(B||)2pikBTc(B||)) (5.2)where kB is the Boltzmann constant, Tc0 is Tc at zero magnetic field and ψ is thedigamma function. Considering in-plane magnetic fields in atomically thin films(that is, with minimal orbital contribution), α takes the form: α ≈ (gµBB||)2τso/2h¯in the case of the strong spin-orbit scattering characterized by a time τso [88, 89].61Figure 5.6: (A) Effect of in-plane magnetic field, B|| on resistance.(B|| = 0data are for ne = 19× 1012cm−2 and the rest of the data are for ne =18× 1012). Inset shows reduction of T1/2 with B||, fit to the expectedform for materials with strong spin-orbit scattering (solid line). The Paulilimit BP, assuming g = 2, is indicated by the dashed line. (B) Sweep ofB|| showing zero resistance up to ∼ 2.4 T in contrast to prediction fromPauli limit. Inset: Data from (A) on a linear scale.The data in Fig. 5.6A inset, T1/2 (effectively Tc) plotted against B||, are fit tothe implicit function above. Since we are considering the high spin-orbit scatteringlimit here, the fit parameter is τso and g is assumed to be 2. The fit is carried out byminimizing the function F(B||,Tc) = ln( TcTc0 )−ψ(12 +α(B||)2pikBTc(B||)) evaluated for datapairs (B||,T1/2), with respect to τso. The best-fit line is shown in Fig. 5.6A inset,yielding an estimate for τso of around 500 fs. It is also possible that the Pauli limitis not actually exceeded, but that the effective g-factor in WTe2 is smaller than 2owing to the strong spin-orbit coupling.It should be mentioned that in the presented measurements of the in-plane mag-netic field, in order to cancel any out-of plane component of the applied magneticfield, due to the possible misalignment of the sample plane and the main magnetaxis, the following procedure was carried out:In addition to the main magnet, our setup is equipped with a small homemade magnetcapable of applying up to 200mT. The axis of this small magnet is perpendicularto the sample plane. At zero B||, the center of Rxx vs B⊥ trace is located at zero.Then, for a finite B||, this centre shifts to a value (Bsh), which corresponds tothe perpendicular component of the applied B||. To find the Bsh component for62each applied B||, the B⊥ is swept from 100mT to -100mT. Then for each B||, theperpendicular magnetic field is set to minus Bsh to compensate for its perpendicularcomponent.The gate (Fig. 5.4) and magnetic field dependence (Figs. 5.5, 5.6) data displayseveral other features worthy of mention. First, even at the lowest temperature,Rxx rises smoothly from zero as a function of B ⊥ (Fig. 5.5B) whereas the onsetof measurable resistance as a function of B|| is relatively sudden, occurring above2T (Fig. 5.6B). Second, an intermediate plateau is visible in the Rxx−T data atB = 0 over a wide range of ne (Fig. 5.4). It is extremely sensitive to B⊥, almostdisappearing at only 2mT (Fig. 5.5A), whereas it survives in B|| to above 2T (Fig.5.6A and inset to Fig. 5.6B). A similar feature has been reported in some otherquasi-2D superconductors [90, 91, 92], but its nature, and the role of disorder,remain unresolved. Third, at intermediate magnetic fields, the resistance approachesa T -independent level as T → 0 that is orders of magnitude below the normal-stateresistance. The data from Fig. 5.5A are replotted vs 1/T in Fig. 5.7B to highlightthe behavior below 100 mK. This exotic phase has been called ”Anomalous metalphase” [93], which we discuss in more details in the next section.5.3.4 Anomalous Metal PhaseIt is believed that a ground state of a 2D electronic system is either superconductingor insulating [56, 93, 126]. The transition between these two phases could occuras a result of varying external parameters such as carrier density or magnetic field.However, it has been reported in a variety of 2D electronic systems that there existsan intervening metallic phase with saturating low temperature resistivity much lowerthan their normal state resistivity [94, 97, 98, 99]. The theoretical understanding ofthis so called “anomalous metal phase” is still under debate [93].We have illustrated the anomalous metal phase both as a function of magneticfield and doping density. Figures 5.5A and 5.7B demonstrate that, for small per-pendicular magnetic fields, the resistance is nearly independent of temperaturebelow 100 or 200 mK. Similar effects were observed at zero perpendicular field,for densities slightly above ncrit as seen in Fig. 5.4. The lack of temperature de-pendence at the lowest temperatures, for densities near ncrit , can be more clearly63seen when replotted vs 1/T (Fig 5.7A). For different densities, the temperaturebelow which the resistance saturates varies from 150 mK (5.6×1012cm−2) to 50mK (6.1×1012cm−2), although no consistent variation with density is observed.Figure 5.7: (A,B) Data from Fig. 5.4 and Fig. 5.5A replotted vs inversetemperature to highlight the saturation of Rxx as a function of carrierdensity and magnetic field at low T respectively. (C) Resistance asa function of temperature for two different displacement fields withthe same carrier density, demonstrating that the electronic temperaturecontinues to decrease with mixing chamber temperature even below50mK.Because the resistance saturation occurs at such low temperature, one might betempted to ascribe the saturation to a failure of the electronic system to cool belowsome elevated temperature. A strong counterargument can be made based on the64data in Fig. 5.7C, showing a dramatic difference in temperature dependence for datawith nearly identical normal resistance and taken under very similar conditions.The data in Fig. 5.7C highlights another observation that was not exploredfurther in this experiment but will be subject of future investigation: The resistancesaturation appeared to depend not only on the effective density but also on therelative setting of the top and bottom gate voltages (that is, on D⊥). For thisparticular set of curves, the data corresponding to lower contact resistance (redcurve) had a higher low-T resistance despite having a slightly lower resistance at1K, compared with the data with higher contact resistance (light blue curve).5.4 Quantum Critical PointThe high tunability of this 2D superconducting system invites comparison withtheoretical predictions for critical behavior close to a quantum phase transition.Figure 5.8 shows how Rxx depends on doping at a series of temperatures, alongthe dashed lines in the inset phase diagram. The T dependence changes sign atncrit ≈ 5×1012cm−2 with essentially no temperature dependence.65Figure 5.8: Scaling analysis of the transition. Main panel: Multiple Rxx vsdoping traces, taken at different temperatures, cross at a critical dopinglevel ncrit ≈ 5×1012cm−2. Upper inset: dashed lines locate these sweepson the phase diagram. Lower inset: same data presented on a scalingplot, taking critical exponent α = 0.8.In the inset we show an attempt to collapse the data onto a single function of|1− ne/ncrit |T−α . The scaling analysis is described in more detail in appendixA. The procedure is somewhat hindered by the fluctuations, which can be seen tobe largely reproducible. The best-fit critical exponent α = 0.8 is similar to thatreported for some insulator-superconductor transitions in thin films [100], althoughwe note that the anomalous behavior near ncrit mentioned in the previous section isnot consistent with such a scaling.Superconductivity induced by simple electrostatic gating in a monolayer ofmaterial that is not normally superconducting is intriguing, but perhaps even moreinteresting is that the ungated state is a 2D TI. This prompts the question of whetherthe helical edge channels remain when the superconductivity appears, and if so, how66strongly they couple to it. In principle, Rxx includes contributions from both edgesand bulk. In device M1 the edge conduction freezes out below 1K as seen in Fig.5.3B. In the next section, measurements characterizing the edge gap in device M1are presented.5.5 The Edge Gap in Device M1As discussed in section 4.3, a peculiar characteristic of all devices made by our team,following the procedure described in section 5.1, is the existence of a small energygap that blocks edge transport. This gap was typically two orders of magnitudesmaller than the bulk energy gap in the 2D TI state. As a result, thermally-activatededge conduction was measurable over a broad range of temperatures in the 2DTI state. Edge transport in M1 was visible from the 1K scale, below which edgeconduction was blocked, up to around 100K, above which bulk transport wasallowed, see Ref. [67] (M1 was the same device investigated in Ref. [67]). Ref. [67]also clearly demonstrated the suppression of edge conduction with the magneticfield.Figure 5.9: (A). The 2D colorscale plot of differential conductance as a func-tion of DC bias voltage and doping level, showing a 600 µeV gap in thisdensity range for M1. (B). In-plane magnetic field dependence of theedge gap in M1, taken for a different setting of gate voltages where thezero-field gap was somewhat smaller than in panel A. (T = 20 mK forboth panels.)67The edge gap can be characterized by non-linear conductance measurementsat low temperature across the 2D TI state: the conductance is nearly zero at lowbias, then at the edge of the gap the conductance rises sharply to a level that isroughly bias-independent for higher bias. Fig. 5.9A shows the edge gap for M1 tobe around 600µeV with conductance fluctuations but no overall dependence on neacross the gapped region. These fluctuations, which are completely reproducible,are also visible in Fig. 5.8. However, the origin of these fluctuations is not clear. Thecollapse of edge conductance as time reversal symmetry is broken by the magneticfield, observed in Ref. [67] at higher temperatures, apparently is associated with anincrease in the edge gap with the magnetic field. The data in Fig. 5.9B present thegrowth of the edge gap in M1 with the in-plane field up to 3T, together with linecuts through the colorscale data at 0T and 2T.The fact that edge conduction in M1 was only visible above 1K, whereassuperconductivity turns on only below 1K, made it difficult to investigate thecoexistence of superconductivity with 2D TI character in this device. Luckily, theedge gap was significantly smaller in M2 (∼ 100µeV ), so some edge conductionremained even down to base temperature (20 mK) and it was relatively strong at200 mK. Next section will discuss the coexistence of the edge conduction andsuperconductivity in M2.5.6 Coexistence of Superconductivity and TIAs is explained in the last section, in order to investigate the combination of edgechannels and superconductivity we turn to another device, M2, in which edgeconduction persists to lower temperatures. Figure 5.10 (A,B) show AFM and opticalimages of M2. Due to the combination of a patterned top gate and a bottom gatethat covered only part of the WTe2 flake, gate-defined conducting regions werecomplicated in M2. The two contacts used for the conductance measurements inFig. 5.11 are indicated in Fig. 5.10D. For the data in Fig. 5.11, the graphene bottomgate was used to induce n-type conductivity between the contacts, while the topgate (shaped like a Hall bar) was fixed at -1.5 V to suppress conduction in the centerof the flake. As a result, the conductance between the two active contacts wasdominated by the pink region of WTe2 highlighted in Fig. 5.10D, in parallel with68edge conduction on the diagonal edge that connects the two contacts (red dashedline in Fig. 5.10D).Figure 5.10: (A). AFM image of device M2. (B). Optical image of deviceM2. (C). Optical image of device M2. The region of WTe2 with higherconductivity due to the combined effects of top and bottom gates isshown in light red (see text). (D). Schematic of the region of interestin M2, showing the contacts used for measurements in Fig. 5.11 andhighlighting in red the region of WTe2 that dominates the conductancebetween those two contacts. Note that the physical edge of the flake thatconnects those two contacts, shown by a red dashed line, is not abovethe bottom gate. However, based on the detailed properties seen in thedata in Fig. 5.12 we deduce that the conduction at low ne is dominatedby one or more internal cracks, not visible in the images above, thatexist in the region above the bottom gate. All scale bars indicate 5µm.69Figure 5.11 shows measurements of the conductance, G, between adjacentcontacts in M2 as a function of gate doping. The figure includes schematicsindicating the inferred state of the edge (red for conducting), as well as the bulk state(colored to match the phase diagram). Consider first the black trace, taken at 200 mKand B⊥ = 0. At low ne the bulk is insulating and edge conduction dominates, albeitwith large mesoscopic fluctuations. For ne > 2× 1012cm−2, G increases as bulkconduction begins, then once ne exceeds ncrit it increases faster as superconductivityappears, before leveling out at around 200 µS owing to contact resistance. Thisinterpretation is supported by warming to 1K (red dotted trace), which destroys thesuperconductivity and so reduces G for ne > ncrit , but enhances the edge conductionat low ne toward the ideal value of e2/h = 39 µS. (We note that this T dependenceof the edge is associated with a gap of around 100 µeV, visible in the inset map ofdifferential conductance vs bias and doping). A perpendicular field B⊥ of 50mT(green trace) also destroys the superconductivity, causing the conductance to fall forne > ncrit but barely affecting it at lower ne. High magnetic fields have been shown(9) to suppress edge conduction in the 2D TI state by breaking the time reversalsymmetry. This effect can be clearly seen in the B⊥ = 1T data (orange trace in Fig.5.11) as G falls to zero at low ne. Importantly, comparing the green (B⊥ = 0.05T) andorange (B⊥ = 1T) traces, G falls by a similar amount at higher ne, consistent with ascenario in which the edge conduction supplies a parallel contribution, implyingthat helical edge states persist when ne > ncrit and at temperatures below Tc.Although, Fig. 5.11 shows data in support of the coexistence of the super-conductivity with the edge state, these are two-wire measurements and there issubstantial resistance in the superconducting regime owing to contact resistance.To demonstrate the presence of the superconductivity in device M2 in the highlyelectron doped region, we show a 4-wire temperature dependence in Fig. 5.12. Themeasurements were hindered by relatively poor contact quality; nevertheless, a largedrop in the 4-wire resistance is seen here consistent with the similar measurementson device M1 presented in Fig. 5.4.70Figure 5.11: Evidence for the presence of both edge conduction and supercon-ductivity in device M2. The main panel shows the linear conductancebetween two adjacent contacts vs gate doping at the temperatures andperpendicular magnetic fields noted. Schematics indicate the state ofedge and bulk conduction at different points, the bulk being coloredto match the phase diagram reproduced above, and red indicating aconducting edge state. Superconductivity occurs for ne > 5×1012cm−2at B = 0; the zero resistance state, disguised by contact resistance in thisfigure, was confirmed in a separate 4-wire measurement of R vs T (Fig.5.12); edge conduction dominates for ne < 2×1012cm−2 but appears tobe present at all ne. Inset: color-scale plot of differential conductancevs dc voltage bias and doping level, revealing a gap of around 100 µeVthat fluctuates rapidly as a function of doping level.71Figure 5.12: A. Appearance of superconductivity for device M2 in a four-wiremeasurement of resistance vs temperature. The gate-induced carrierdensity corresponding to these voltages is ne = 8.4×1012cm−2 in theregion covered by both top and bottom gates. B. AFM image of thedevice M2 along with schematics representing the contacts used for themeasurement in A. Measurement is performed in an unconventionalgeometry, due to the many poor or broken contacts in this device. Thered solid line shows the boundary of WTe2. The layout of the areacovered by the top and bottom gates can be seen in Fig. 5.10C. Scalebar indicates 5µm.5.7 Measurement DetailsThe measurements done in this chapter used the low-temperature resistivity mea-surement techniques explained in chapter 7. However, there are a few subtleties inthe presented measurements in this chapter, which will be explained in this section.The resistivity measurement in the superconducting phase is done in a four-terminal geometry, details of which are described in chapter 7. The first step in thesemeasurements was to figure out the critical current (IC) of the superconducting state,to ensure that the applied bias current doesn’t exceed IC leading to a transition to thenormal state. To do so, the resistivity of the sample was measured as a function of thebias current, as shown in Fig. D.1 in appendix D for ne = 12×1012cm−2. The sameprocedure was applied for other densities in the superconducting state. Although72determining the IC as explained in appendix D turned out to be difficult, the selectedbias current (5nA) for the resistivity measurements in the superconducting statedidn’t affect the observed minimum resistance as evident for example in Fig. D.1for ne = 12×1012cm−2.One question is about the role of strain on the observed superconducting and TIphases, created as a result of a bent due to laying down of the WTe2 flake on themetallic contact used for the measurement. While this effect certainly might havean influence on the TI state, determining its role is beyond the scope of this thesis.On the other hand, for the superconducting phase, since it is a bulk effect, the localstrain due to the metal contacts, should have a negligible effect on the observedsuperconducting phase.Another concern of the discussed superconducting phase in monolayer WTe2 isabout Berezinskii-Kosterlitz-Thouless (BKT) transition [95, 96]. This transition isexpected to occur for 2D superconductors as a result of breaking of vortex-antivortexbinding above a critical temperature denoted as TBKT [95, 96]. One approach todetermine this transition temperature is fitting the resistivity data vs temperature ofthe 2D superconductor to the BKT model [96]:R ∝ e−( 1(T−TBKT )1/2)(5.3)However, as can be seen in Fig. 5.4, and discussed in the text, a plateau likefeature interfere in the normal to the superconducting transition of our Rxx vs T data.Thus, it wasn’t possible to fit our data to the BKT model to obtain the TBKT .5.8 DiscussionThis discovery raises compelling questions for future investigation. It is likely thatthe helical edge modes persist when the superconductivity is restored by reducingthe magnetic field to zero. Other techniques, such as scanning tunneling microscopyor contacts discriminating edge from bulk, may be needed to probe the edgesseparately from the bulk. The measurements presented here cannot determine thedegree or nature of the coupling between superconductivity and edge conduction.One question concerns the nature of the superconducting order. It is strikingthat ncrit corresponds to only ∼ 0.5% of an electron per W atom, which is about73ten times lower than the doping level needed to observe superconductivity in othertransition metal dichalcogenide monolayers [78]. Many-layer WTe2 is semimetallic[63, 101, 102, 103] under ambient conditions, with near-perfect compensation ofelectrons and holes, but becomes superconducting as the ratio of electrons to holesincreases at high pressure [64, 66]. Some related materials, such as TiSe2, areknown to switch from charge-density-wave to superconducting states at quite lowdoping [104] or under pressure [105]. We therefore speculate that doping tips thebalance in monolayer WTe2 in favor of superconductivity, away from a competinginsulating electronic ordering.Another key question is whether the edge states also develop a superconductinggap, in which case they could host Majorana zero modes [10]. To verify the inducedsuperconductivity in the proximitized helical edge states, an experiment using thegate-tunability of monolayer WTe2 can be done. In the proposed experiment, twolocal gates can be used to induce superconductivity in the two sides of a WTe2flake, where the middle part not covered by the local gates will be in the 2D TIstate. Then, the formation of the induced superconducting gap in helical edges canbe verified by a differential conductance measurement. Once the the proximitizededge state in WTe2 is experimentally realized, the next step would be to search forMajorana zero mode in these edges. One approach can be based on the theoreticalproposal where the topological superconducting edge states are proximitized bya ferromagnet in order to localize Majorana zero modes at the boundary of theferromagnet [131, 132]. Such a scheme is already experimentally verified for thehelical edge states in a 2D TI Bi film proximitized with a s-wave superconductorand a ferromagnet [133].Lastly, it should also be mentioned that the observation of induced superconduc-tivity in the monolayer WTe2 has been reported simultaneously and independentlyby Fatemi. et al. [122].74Chapter 6Conclusion on TIsThis chapter concludes the TI part of this thesis with a summary of the mainfindings and remarks on the open questions and possible future works to addressthem. Overall, we discussed the transport properties of two TI electronic systems,InAs/GaSb QWs and monolayer WTe2.In chapter 3, we described a comprehensive study of the edge conduction inInAs/GaSb double QWs. We found that, in spite of the previous reports regard-ing the observation of the helical edge conduction in this system, significant edgeconduction is present in the trivial phase. We observed a similar characteristic forthe edge conduction in our samples compared to the one in the previous reports.However, two experimental observations led to the conclusion that the edge conduc-tion present in our samples doesn’t have helical nature. First, using the double gategeometry, we mapped out the complete phase diagram as is shown in Fig. 3.1(c).It was observed that the edge conduction occurs in the non-inverted region of thephase diagram, where the helical edge modes are not expected. Second, the lengthdependent measurement in section 3.4.1 revealed that the conductance of the edgechannels in our samples scales linearly with the length. For edge lengths shorterthan scattering length, which is found to be ∼ 4.8µm [1] for InAs/GaSb QWs, onewould expect to see a quantized resistance of h/e2 ∼ 25KΩ for the helical edgeconduction, whereas in our samples for the 300nm edge length, the conductance isaround 6 KΩ.The experimental findings of our work showed that in a material with a broken75band gap, such as InAs/GaSb QWs, significant edge conduction could be presentin the trivial regime. This work also demonstrated an experimental procedure toidentify the origin of the edge conduction in these materials. It is worth mentioningthat despite the clear observation of enhanced edge conduction in the trivial phase,the existence of the helical edge conduction in the inverted phase cannot be ruledout in our samples. Due to the residual bulk conduction in the inverted region,investigation of the nature of edge conduction was difficult. Therefore, it is crucialto implement techniques to suppress the bulk and trivial edge conduction in order toidentify any possible helical edge conduction in InAs/GaSb QWs.Chapter 5 investigated another 2D TI, monolayer WTe2. We studied the inducedsuperconductivity in monolayer WTe2, which is already known to be a 2D TI, withmild electrostatic doping. It was shown that the transition between the two statescan be achieved by applying a small gate voltage. We furthermore, characterizedthe transport properties of the induced superconductivity, where some peculiarfeatures were found. It was illustrated that the temperature evolution of the normalto the superconducting state is very broad and accompanied by a plateau in theintermediate range of the transition. The plateau feature is vanishing in a few mTof the perpendicular magnetic field while surviving up to the high parallel fieldaround 2.5T. We saw also a temperature independent resistance both as a function ofdensity and magnetic field, in the low limit of the temperature. The resistance of thisphase, which is called anomalous metal phase is two or three orders of magnitudesmaller than the normal resistance. We found that the critical parallel magnetic fieldis beyond the Pauli limit by a factor larger than 2. This enhancement is possibly dueto the increase in the pair breaking energy, caused by the strong spin-orbit scatteringrate in the monolayer WTe2.Finally, we explored the possibility of the coexistence of superconductivity andedge conduction in monolayer WTe2. Although our measurements cannot provethis coexistence, they showed preliminary evidence for the parallel coexistence ofthese two phases. Further studies are required to confirm the nature of the couplingbetween edge states and superconductivity in monolayer WTe2. If the helical edgestates develop a superconducting gap, then they may host Majorana zero modes[10]. Another great possibility that this electronic system provides is the potentialto interface monolayer WTe2 with a 2D ferromagnet and therefore investigating76the interplay between superconductivity, QSH state, and magnetism in the resultantheterostructure.77Chapter 7Measurement Set-upThis chapter describes the measurement set-up used to acquire the data presented inthis thesis.7.1 CryogenicsAll the the measurements performed in this thesis were done in a commercial3He/4He oxford dilution fridge with a base temperature below 20mK. Fig. 7.1shows the schematic of the part of the dilution fridge below the mixing chamberplate along with the radiation shield and vacuum can, inserted in a dewar filled withliquid helium (LHe). The dewar contains superconducting magnets, which can applya magnetic field up to 12T perpendicular or parallel to the sample plate dependingon the sample orientation. There is a secondary small magnet that is glued to thevacuum can, and able to apply up to 200mT of magnetic field perpendicular to thesample plane when the sample is mounted parallel to the primary coil. This smallfield is also used to cancel any perpendicular component of the main magnet dueto an accidental misalignment between the sample plate axis and the main magnetaxis. The coldest part of the dilution fridge is the mixing chamber, which gets below20mK. The sample connects to this part via a custom made part, which is called acold finger. Two of these cold fingers are made identically, except that in one casethe sample can be mounted parallel and in the other one perpendicular to the axis ofthe main magnet.78The sample is mounted on a ceramic chip carrier, which connects to a socketat the bottom of the cold finger. The cold finger then is covered by two radiationshields (only one of them is shown in Fig. 7.1), which blocks the backgroundthermal radiations.Figure 7.1: Schematic of the cryogenic setup. (1) Liquid helium. (2) Vacuumcan. (3) Mixing chamber. (4) Radiation shield. (5) Cooling wire module.(6) Attunator plate. (7) RC filter. (8) Superconducting magnet. (9) smallmagnet. (10) Sample.Then finally a vacuum can covers all parts, which can be inserted into the LHebath after being pumped out.The electrical measurements are done with wires, which on one side are con-79nected to the measuring instruments (at the room temperature) and on the otherside reaching down to the mixing chamber. This means that heat can propagatefrom the room temperature through these wires and overheat the sample. Though,the finite resistance of the measurement wires (∼ 100Ω) limits the heat transfer, tocompletely block the heat transfer to the samples, the wires are glued to the variousstages of the dilution fridge acting as a heat sink. These same wires are the mainway of cooling the sample at mK range of temperatures. Thus, the electrical wiringpasses through a copper plate that is screwed to the bottom of the mixing chamberbefore connecting to the sample, to have them as close as possible to the mixingchamber temperature.Besides the heat flow, electromagnetic interferences can also propagate throughthe measuring wires and overheat the sample. Therefore, a set of filtering modulesto block these parasitic signals are installed in the fridge, which will be explainedbriefly in the next section.7.1.1 Filtering ModulesTo prevent the radio frequency (RF) noise from reaching the sample, two filteringmodules were installed in the fridge. First, the signal wires are passed througha resistor-capacitor (RC) filter board, which is connected on top of the mixingchamber plate, with a cutoff frequency of 1MHz. By 100 MHz, the attenuation levelof the RC filter is 10−3. Toward higher frequencies (above 1GHz) the attenuationlevel decreases, due to the parasitic inductance of the capacitors in the RC filter[128]. In order to filter the GHz range of the RF noise, an attenuator plate, which iscomposed of a loop of constantan wires, sandwiched between two copper tapes isscrewed on to a copper post attached to the bottom of the mixing chamber plate. Themeasurement wires connect through this attenuator plate to the chip carrier socketat the bottom of the cold finger. Filtering of GHz frequencies in the attenuator plateis based on the skin effect, which forces the AC signals to propagate through theannulus of the width δ ∝√ρf , which is called skin depth, of the measurement wires[128, 130]. Here, ρ is the bulk resistivity of the wire, and f is the frequency. As aresult, although the resistance of the wires in the attenuator plate is relatively low atlow frequencies, it becomes very large at GHz range, as a result of the much smaller80effective surface area of the wires, due to the decrease in the skip depth. Thus, theattenuator plate forms a lossy transmission line [130], which attenuates the highfrequency signals. Further details on the filtering module used in this setup can befound in [127].7.2 Electronic Set-upThe electrical measurements in this thesis are done in both two and four-terminalgeometries. In case of the two terminal configuration as shown in Fig. 7.2(a), ameasurement is done between one pair of contacts, where a voltage (V ) is appliedfrom lock-in amplifier passed through a voltage divider into a one contact and theother contact is grounded through current-amplifier. The voltage divider is added tothe circuit to be able to apply voltage less than the minimum output of the lock-inamplifier, which is 4meV. This is necessary in order to avoid heating the sample inthe mK range temperature. The output of the current-amplifier fed into the input ofthe lock-in amplifier to measure the current (I) through the device. Then, dividingthe applied voltage to the measured current, will give the two terminal resistance,which is the sample resistance plus contact and measuring wires resistances. Bysubtracting the known measurement wires resistance, shown as RW in Fig. 7.2(a),the resultant resistance will be the sample resistance (Rs) plus contact resistance(RC).If we instead perform a four-terminal measurement, then we will only obtain thesample resistance, excluding the wire and contact resistances. Figure. 7.2(b) showsthe schematic of the four-terminal measurement in the bias current configuration.In order to have a constant current through the measurement circuit a resistor,which has much larger resistance (denoted as RL in Fig. 7.2(b)) than the sampleresistance is placed in a series with the sample. Then the voltage drop across thesample (Vxx) is measured by another pair of contacts along the sample. A voltageamplifier is used to amplify the measured voltage, where its output is fed into thelock-in input. The current through the sample is also monitored with another lock-inamplifier. Recording the current output is useful in the measurements where thesample resistance (Rs) is changing from low resistance to high resistance limit asa function of external parameter like the gate voltage, such as the measurement81presented in Fig. 5.8. In the low resistance limit, where RL >> Rs, it is safe toconsider that the output current is defined by RL. However, in the high resistancelimit, where Rs can be up to a few percent of RL (depending on the value of RL), theoutput current will be defined by both resistances, and therefore to know the exactcurrent through the sample, it is important to read the output current.The four-terminal resistance (Rxx) is then obtained by dividingVxx by the currentthrough the sample, Rxx =Vxx/I.Figure 7.2: Schematic of the electronic setup in two-terminal (a), and four-terminal (b) measurements. RW , RC, and RS denote wire, contact andsample resistance respectively.82Bibliography[1] L. Du, I. Knez, G. Sullivan, and R.-R. Du, “Robust Helical Edge Transportin Gated InAs/GaSb Bilayers,” Phys. Rev. Lett., vol. 114, p. 096802, Mar2015. v, 25, 26, 30, 34, 36, 38, 75[2] J. Weis and K. von Klitzing, “Metrology and microscopic picture of the in-teger quantum Hall effect,” Philosophical Transactions of the Royal SocietyA: Mathematical, Physical and Engineering Sciences, vol. 369, no. 1953,pp. 3954–3974, 2011. 1, 8[3] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “QuantizedHall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett.,vol. 49, pp. 405–408, Aug 1982. 1, 8, 9[4] C. Kane, “Chapter 1 - Topological Band Theory and the 2 Invariant,”in Topological Insulators (M. Franz and L. Molenkamp, eds.), vol. 6 ofContemporary Concepts of Condensed Matter Science, pp. 3 – 34, Elsevier,2013. 1, 2[5] K. Crane, “Robust Fairing via Conformal Curvature Flow.” [On-line]. Available from: https://www.cs.cmu.edu/∼kmcrane/Projects/ConformalWillmoreFlow/. 2[6] N. Ashcroft and N. Mermin, Solid State Physics. Cengage Learning, 2011. 2[7] A. Bansil, H. Lin, and T. Das, “Colloquium: Topological band theory,” Rev.Mod. Phys., vol. 88, p. 021004, Jun 2016. 2, 383[8] M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev.Mod. Phys., vol. 82, pp. 3045–3067, Nov 2010. 3, 4, 9[9] F. Wilczek, “Majorana returns,” Nature Physics, vol. 5, pp. 614 EP –, 092009. 5[10] L. Fu and C. L. Kane, “Superconducting Proximity Effect and MajoranaFermions at the Surface of a Topological Insulator,” Phys. Rev. Lett., vol. 100,p. 096407, Mar 2008. 5, 51, 52, 74, 76[11] S. Hart, H. Ren, T. Wagner, P. Leubner, M. Mu¨hlbauer, C. Bru¨ne, H. Buh-mann, L. W. Molenkamp, and A. Yacoby, “Induced superconductivity in thequantum spin Hall edge,” Nature Physics, vol. 10, pp. 638 EP –, 08 2014. 5[12] C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,” Phys.Rev. Lett., vol. 95, p. 226801, Nov 2005. 9, 10[13] F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels:Condensed-Matter Realization of the “Parity Anomaly”,” Phys. Rev. Lett.,vol. 61, pp. 2015–2018, Oct 1988. 9[14] T. Ando, “The electronic properties of graphene and carbon nanotubes,” NpgAsia Materials, vol. 1, pp. 17 EP –, 10 2009. 10[15] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev.Mod. Phys., vol. 83, pp. 1057–1110, Oct 2011. 11[16] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H.MacDonald, “Intrinsic and Rashba spin-orbit interactions in graphene sheets,”Phys. Rev. B, vol. 74, p. 165310, Oct 2006. 11[17] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, “Spin-orbit gap ofgraphene: First-principles calculations,” Phys. Rev. B, vol. 75, p. 041401,Jan 2007. 11[18] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watan-abe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride84substrates for high-quality graphene electronics,” Nature Nanotechnology,vol. 5, pp. 722 EP –, 08 2010. 11[19] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum Spin Hall Effectand Topological Phase Transition in HgTe Quantum Wells,” Science, vol. 314,no. 5806, pp. 1757–1761, 2006. 11, 12[20] M. Ko¨nig, S. Wiedmann, C. Bru¨ne, A. Roth, H. Buhmann, L. W. Molenkamp,X.-L. Qi, and S.-C. Zhang, “Quantum Spin Hall Insulator State in HgTeQuantum Wells,” Science, vol. 318, no. 5851, pp. 766–770, 2007. 11, 13, 17,19[21] S. Datta, Electronic Transport in Mesoscopic Systems. Cambridge Studiesin Semiconductor Physics and Microelectronic Engineering, CambridgeUniversity Press, 1995. 14[22] R. Landauer, “Electrical transport in open and closed systems,” Zeitschriftfu¨r Physik B Condensed Matter, vol. 68, pp. 217–228, Jun 1987. 15[23] M. Bu¨ttiker, “Four-Terminal Phase-Coherent Conductance,” Phys. Rev. Lett.,vol. 57, pp. 1761–1764, Oct 1986. 15[24] C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, “Quantum SpinHall Effect in Inverted Type-II Semiconductors,” Phys. Rev. Lett., vol. 100,p. 236601, Jun 2008. 20, 23, 24, 28[25] Y. Naveh and B. Laikhtman, “Bandstructure tailoring by electric field ina weakly coupled electronhole system,” Applied Physics Letters, vol. 66,no. 15, pp. 1980–1982, 1995. 22, 23[26] F. Nichele, H. J. Suominen, M. Kjaergaard, C. M. Marcus, E. Sajadi, J. A.Folk, F. Qu, A. J. A. Beukman, F. K. de Vries, J. van Veen, S. Nadj-Perge,L. P. Kouwenhoven, B.-M. Nguyen, A. A. Kiselev, W. Yi, M. Sokolich, M. J.Manfra, E. M. Spanton, and K. A. Moler, “Edge transport in the trivial phaseof InAs/GaSb,” New Journal of Physics, vol. 18, p. 083005, jul 2016. 25, 29,32, 33, 35, 37, 3985[27] I. Knez, R.-R. Du, and G. Sullivan, “Evidence for Helical Edge Modes inInverted InAs/GaSb Quantum Wells,” Phys. Rev. Lett., vol. 107, p. 136603,Sep 2011. 25[28] I. Knez, C. T. Rettner, S.-H. Yang, S. S. P. Parkin, L. Du, R.-R. Du, andG. Sullivan, “Observation of Edge Transport in the Disordered Regimeof Topologically Insulating InAs/GaSb Quantum Wells,” Phys. Rev. Lett.,vol. 112, p. 026602, Jan 2014. 25, 26, 30, 34, 36, 38[29] E. M. Spanton, K. C. Nowack, L. Du, G. Sullivan, R.-R. Du, and K. A. Moler,“Images of Edge Current in InAs/GaSb Quantum Wells,” Phys. Rev. Lett.,vol. 113, p. 026804, Jul 2014. 25, 26[30] C. Charpentier, S. Flt, C. Reichl, F. Nichele, A. Nath Pal, P. Pietsch, T. Ihn,K. Ensslin, and W. Wegscheider, “Suppression of bulk conductivity inInAs/GaSb broken gap composite quantum wells,” Applied Physics Letters,vol. 103, no. 11, p. 112102, 2013. 25[31] S. Mueller, A. N. Pal, M. Karalic, T. Tschirky, C. Charpentier, W. Wegschei-der, K. Ensslin, and T. Ihn, “Nonlocal transport via edge states in InAs/GaSbcoupled quantum wells,” Phys. Rev. B, vol. 92, p. 081303, Aug 2015. 25[32] K. Suzuki, Y. Harada, K. Onomitsu, and K. Muraki, “Edge channel trans-port in the InAs/GaSb topological insulating phase,” Phys. Rev. B, vol. 87,p. 235311, Jun 2013. 26, 27[33] K. Suzuki, Y. Harada, K. Onomitsu, and K. Muraki, “Gate-controlledsemimetal-topological insulator transition in an InAs/GaSb heterostructure,”Phys. Rev. B, vol. 91, p. 245309, Jun 2015. 26[34] F. m. c. Coue¨do, H. Irie, K. Suzuki, K. Onomitsu, and K. Muraki, “Single-edge transport in an InAs/GaSb quantum spin Hall insulator,” Phys. Rev. B,vol. 94, p. 035301, Jul 2016. 26[35] J. I. Va¨yrynen, M. Goldstein, and L. I. Glazman, “Helical Edge ResistanceIntroduced by Charge Puddles,” Phys. Rev. Lett., vol. 110, p. 216402, May2013. 2686[36] T. L. Schmidt, S. Rachel, F. von Oppen, and L. I. Glazman, “Inelastic ElectronBackscattering in a Generic Helical Edge Channel,” Phys. Rev. Lett., vol. 108,p. 156402, Apr 2012. 26[37] S. Groenendijk, G. Dolcetto, and T. L. Schmidt, “Fundamental limits to heli-cal edge conductivity due to spin-phonon scattering,” Phys. Rev. B, vol. 97,p. 241406, Jun 2018. 26[38] I. Knez, R. R. Du, and G. Sullivan, “Finite conductivity in mesoscopic Hallbars of inverted InAs/GaSb quantum wells,” Phys. Rev. B, vol. 81, p. 201301,May 2010. 26[39] T. Li, P. Wang, H. Fu, L. Du, K. A. Schreiber, X. Mu, X. Liu, G. Sullivan,G. A. Csa´thy, X. Lin, and R.-R. Du, “Observation of a Helical LuttingerLiquid in InAs/GaSb Quantum Spin Hall Edges,” Phys. Rev. Lett., vol. 115,p. 136804, Sep 2015. 26[40] R. Skolasinski, D. I. Pikulin, J. Alicea, and M. Wimmer, “Robust helicaledge transport in quantum spin Hall quantum wells,” Phys. Rev. B, vol. 98,p. 201404, Nov 2018. 27[41] C.-A. Li, S.-B. Zhang, and S.-Q. Shen, “Hidden edge Dirac point and robustquantum edge transport in InAs/GaSb quantum wells,” Phys. Rev. B, vol. 97,p. 045420, Jan 2018. 27[42] D. C. Tsui, “Observation of Surface Bound State and Two-DimensionalEnergy Band by Electron Tunneling,” Phys. Rev. Lett., vol. 24, pp. 303–306,Feb 1970. 27, 40[43] M. Noguchi, K. Hirakawa, and T. Ikoma, “Intrinsic electron accumulationlayers on reconstructed clean InAs(100) surfaces,” Phys. Rev. Lett., vol. 66,pp. 2243–2246, Apr 1991. 27, 40[44] L. O. Olsson, C. B. M. Andersson, M. C. Ha˚kansson, J. Kanski, L. Ilver, andU. O. Karlsson, “Charge Accumulation at InAs Surfaces,” Phys. Rev. Lett.,vol. 76, pp. 3626–3629, May 1996. 27, 4087[45] B.-M. Nguyen, W. Yi, R. Noah, J. Thorp, and M. Sokolich, “High mobil-ity back-gated InAs/GaSb double quantum well grown on GaSb substrate,”Applied Physics Letters, vol. 106, no. 3, p. 032107, 2015. 27[46] I. Knez, Transport Properties of Topological Phases in Broken Gap IndiumArsenide/Gallium Antimonide Based Quantum Wells. PhD thesis, Rice Uni-versity, 2012. 27[47] A. N. Pal, S. Mller, T. Ihn, K. Ensslin, T. Tschirky, C. Charpentier, andW. Wegscheider, “Influence of etching processes on electronic transport inmesoscopic InAs/GaSb quantum well devices,” AIP Advances, vol. 5, no. 7,p. 077106, 2015. 27[48] M. J. Yang, K. A. Cheng, C. H. Yang, and J. C. Culbertson, “A nanofab-rication scheme for InAs/AlSb heterostructures,” Applied Physics Letters,vol. 80, no. 7, pp. 1201–1203, 2002. 27[49] C. Gatzke, S. J. Webb, K. Fobelets, and R. A. Stradling, “In situRamanspectroscopy of the selective etching of antimonides in GaSb/AlSb/InAsheterostructures,” Semiconductor Science and Technology, vol. 13, pp. 399–403, apr 1998. 28[50] R. Chaghi, C. Cervera, H. At-Kaci, P. Grech, J. B. Rodriguez, and P. Christol,“Wet etching and chemical polishing of InAs/GaSb superlattice photodiodes,”Semiconductor Science and Technology, vol. 24, p. 065010, may 2009. 28[51] F. Qu, A. J. A. Beukman, S. Nadj-Perge, M. Wimmer, B.-M. Nguyen, W. Yi,J. Thorp, M. Sokolich, A. A. Kiselev, M. J. Manfra, C. M. Marcus, andL. P. Kouwenhoven, “Electric and Magnetic Tuning Between the Trivial andTopological Phases in InAs/GaSb Double Quantum Wells,” Phys. Rev. Lett.,vol. 115, p. 036803, Jul 2015. 28, 30, 34[52] D. Schoenberg, Magnetic Oscillations in Metals. 2009. 30[53] M. J. Yang, C. H. Yang, B. R. Bennett, and B. V. Shanabrook, “Evidence of aHybridization Gap in “Semimetallic” InAs/GaSb Systems,” Phys. Rev. Lett.,vol. 78, pp. 4613–4616, Jun 1997. 3288[54] K. K. Choi, B. F. Levine, N. Jarosik, J. Walker, and R. Malik, “Anisotropicmagnetotransport in weakly coupled GaAs-AlxGa1−xAs multiple quantumwells,” Phys. Rev. B, vol. 38, pp. 12362–12368, Dec 1988. 32[55] A. L. Efros and B. I. Shklovskii, “Coulomb gap and low temperature con-ductivity of disordered systems,” Journal of Physics C: Solid State Physics,vol. 8, pp. L49–L51, feb 1975. 36, 115[56] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,“Scaling Theory of Localization: Absence of Quantum Diffusion in TwoDimensions,” Phys. Rev. Lett., vol. 42, pp. 673–676, Mar 1979. 63[57] M. Imada, A. Fujimori, and Y. Tokura, “Metal-insulator transitions,” Rev.Mod. Phys., vol. 70, pp. 1039–1263, Oct 1998. 107[58] M. Brahlek, L. Zhang, J. Lapano, H.-T. Zhang, R. Engel-Herbert, N. Shukla,S. Datta, H. Paik, and D. G. Schlom, “Opportunities in vanadium-basedstrongly correlated electron systems,” MRS Communications, vol. 7, no. 1,p. 2752, 2017. 108[59] S. Mueller, C. Mittag, T. Tschirky, C. Charpentier, W. Wegscheider, K. En-sslin, and T. Ihn, “Edge transport in InAs and InAs/GaSb quantum wells,”Phys. Rev. B, vol. 96, p. 075406, Aug 2017. 38[60] S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, andP. Jarillo-Herrero, “Observation of the quantum spin Hall effect up to 100kelvin in a monolayer crystal,” Science, vol. 359, no. 6371, pp. 76–79, 2018.42, 46, 47, 48, 49, 55[61] X. Qian, J. Liu, L. Fu, and J. Li, “Quantum spin Hall effect in two-dimensional transition metal dichalcogenides,” Science, vol. 346, no. 6215,pp. 1344–1347, 2014. 42, 43, 46, 47[62] H. Y. Lv, W. J. Lu, D. F. Shao, Y. Liu, S. G. Tan, and Y. P. Sun, “Perfectcharge compensation in WTe2for the extraordinary magnetoresistance: Frombulk to monolayer,” EPL (Europhysics Letters), vol. 110, p. 37004, may2015. 44, 47, 48, 6189[63] M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang,N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. Cava, “Large,non-saturating magnetoresistance in WTe2,” Nature, vol. 514, pp. 205 EP –,09 2014. 44, 45, 74[64] D. Kang, Y. Zhou, W. Yi, C. Yang, J. Guo, Y. Shi, S. Zhang, Z. Wang,C. Zhang, S. Jiang, A. Li, K. Yang, Q. Wu, G. Zhang, L. Sun, and Z. Zhao,“Superconductivity emerging from a suppressed large magnetoresistant statein tungsten ditelluride,” Nature Communications, vol. 6, pp. 7804 EP –, 072015. 46, 74[65] F. Zheng, C. Cai, S. Ge, X. Zhang, X. Liu, H. Lu, Y. Zhang, J. Qiu,T. Taniguchi, K. Watanabe, S. Jia, J. Qi, J.-H. Chen, D. Sun, and J. Feng, “Onthe Quantum Spin Hall Gap of Monolayer 1T-WTe2,” Advanced Materials,vol. 28, no. 24, pp. 4845–4851, 2016. 47[66] X.-C. Pan, X. Chen, H. Liu, Y. Feng, Z. Wei, Y. Zhou, Z. Chi, L. Pi, F. Yen,F. Song, X. Wan, Z. Yang, B. Wang, G. Wang, and Y. Zhang, “Pressure-driven dome-shaped superconductivity and electronic structural evolutionin tungsten ditelluride,” Nature Communications, vol. 6, pp. 7805 EP –, 072015. 46, 74[67] Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney,X. Xu, and D. H. Cobden, “Edge conduction in monolayer WTe2,” NaturePhysics, vol. 13, pp. 677 EP –, 04 2017. 47, 48, 49, 50, 55, 67, 68[68] I. Pletikosic´, M. N. Ali, A. V. Fedorov, R. J. Cava, and T. Valla, “ElectronicStructure Basis for the Extraordinary Magnetoresistance in WTe2,” Phys.Rev. Lett., vol. 113, p. 216601, Nov 2014. 44[69] S. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H.-Z. Tsai, C. Jia, B. Moritz,M. Claassen, H. Ryu, S. Kahn, J. Jiang, H. Yan, M. Hashimoto, D. Lu,R. G. Moore, C.-C. Hwang, C. Hwang, Z. Hussain, Y. Chen, M. M. Ugeda,Z. Liu, X. Xie, T. P. Devereaux, M. F. Crommie, S.-K. Mo, and Z.-X. Shen,“Quantum spin Hall state in monolayer 1T’-WTe2,” Nature Physics, vol. 13,pp. 683 EP –, 06 2017. 47, 4890[70] Y.-H. Song, Z.-Y. Jia, D. Zhang, X.-Y. Zhu, Z.-Q. Shi, H. Wang, L. Zhu,Q.-Q. Yuan, H. Zhang, D.-Y. Xing, and S.-C. Li, “Observation of Coulombgap in the quantum spin Hall candidate single-layer 1T’-WTe2,” NatureCommunications, vol. 9, no. 1, p. 4071, 2018. 48[71] E. Sajadi, T. Palomaki, Z. Fei, W. Zhao, P. Bement, C. Olsen, S. Luescher,X. Xu, J. A. Folk, and D. H. Cobden, “Gate-induced superconductivity in amonolayer topological insulator,” Science, vol. 362, no. 6417, pp. 922–925,2018. 51[72] S. D. Sarma, M. Freedman, and C. Nayak, “Majorana zero modes and topo-logical quantum computation,” Npj Quantum Information, vol. 1, pp. 15001EP –, 10 2015. 51[73] M. Sato and Y. Ando, “Topological superconductors: a review,” Reports onProgress in Physics, vol. 80, p. 076501, may 2017. 51[74] A. W. Tsen, B. Hunt, Y. D. Kim, Z. J. Yuan, S. Jia, R. J. Cava, J. Hone,P. Kim, C. R. Dean, and A. N. Pasupathy, “Nature of the quantum metalin a two-dimensional crystalline superconductor,” Nature Physics, vol. 12,pp. 208 EP –, 12 2015. 51[75] X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forro´, J. Shan,and K. F. Mak, “Ising pairing in superconducting NbSe2 atomic layers,”Nature Physics, vol. 12, pp. 139 EP –, 11 2015. 51, 61[76] Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S. Zhang, W. Li, H. Ding,Y.-B. Ou, P. Deng, K. Chang, J. Wen, C.-L. Song, K. He, J.-F. Jia, S.-H. Ji,Y.-Y. Wang, L.-L. Wang, X. Chen, X.-C. Ma, and Q.-K. Xue, “Interface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Filmson SrTiO3,” Chinese Physics Letters, vol. 29, p. 037402, mar 2012. 51[77] L. J. Li, E. C. T. O’Farrell, K. P. Loh, G. Eda, B. O¨zyilmaz, and A. H.Castro Neto, “Controlling many-body states by the electric-field effect in atwo-dimensional material,” Nature, vol. 529, pp. 185 EP –, 12 2015. 5191[78] Y. Fu, E. Liu, H. Yuan, P. Tang, B. Lian, G. Xu, J. Zeng, Z. Chen, Y. Wang,W. Zhou, K. Xu, A. Gao, C. Pan, M. Wang, B. Wang, S.-C. Zhang, Y. Cui,H. Y. Hwang, and F. Miao, “Gated tuned superconductivity and phononsoftening in monolayer and bilayer MoS2,” npj Quantum Materials, vol. 2,no. 1, p. 52, 2017. 51, 74[79] Y. Saito, T. Nojima, and Y. Iwasa, “Gate-induced superconductivity intwo-dimensional atomic crystals,” Superconductor Science and Technology,vol. 29, p. 093001, jul 2016. 51[80] K. Ueno, S. Nakamura, H. Shimotani, A. Ohtomo, N. Kimura, T. Nojima,H. Aoki, Y. Iwasa, and M. Kawasaki, “Electric-field-induced superconductiv-ity in an insulator,” Nature Materials, vol. 7, pp. 855 EP –, 10 2008. 51[81] D. Costanzo, S. Jo, H. Berger, and A. F. Morpurgo, “Gate-induced supercon-ductivity in atomically thin MoS2 crystals,” Nature Nanotechnology, vol. 11,pp. 339 EP –, 01 2016. 51[82] J. M. Lu, O. Zheliuk, I. Leermakers, N. F. Q. Yuan, U. Zeitler, K. T. Law,and J. T. Ye, “Evidence for two-dimensional Ising superconductivity in gatedMoS2,” Science, vol. 350, no. 6266, pp. 1353–1357, 2015. 51, 61[83] P. J. Zomer, M. H. D. Guimares, J. C. Brant, N. Tombros, and B. J. vanWees, “Fast pick up technique for high quality heterostructures of bilayergraphene and hexagonal boron nitride,” Applied Physics Letters, vol. 105,no. 1, p. 013101, 2014. 52[84] R. A. Klemm, “Pristine and intercalated transition metal dichalcogenide su-perconductors,” Physica C: Superconductivity and its Applications, vol. 514,pp. 86 – 94, 2015. Superconducting Materials: Conventional, Unconventionaland Undetermined. 59[85] M. Tinkham, Introduction to Superconductivity. McGraw Hill, ed.2, 1996.59, 61[86] Y. Saito, T. Nojima, and Y. Iwasa, “Highly crystalline 2D superconductors,”Nature Reviews Materials, vol. 2, pp. 16094 EP –, 12 2016. 5992[87] A. M. Clogston, “Upper Limit for the Critical Field in Hard Superconductors,”Phys. Rev. Lett., vol. 9, pp. 266–267, Sep 1962. 61[88] R. A. Klemm, A. Luther, and M. R. Beasley, “Theory of the upper criticalfield in layered superconductors,” Phys. Rev. B, vol. 12, pp. 877–891, Aug1975. 61[89] K. Maki, “Effect of Pauli Paramagnetism on Magnetic Properties of High-Field Superconductors,” Phys. Rev., vol. 148, pp. 362–369, Aug 1966. 61[90] Y. Cheng and M. B. Stearns, “Superconductivity of Nb/Cr multilayers,”Journal of Applied Physics, vol. 67, no. 9, pp. 5038–5040, 1990. 63[91] L. Z. Deng, B. Lv, K. Zhao, F. Y. Wei, Y. Y. Xue, Z. Wu, and C. W. Chu, “Ev-idence for defect-induced superconductivity up to 49 K in (Ca1−xRx)Fe2As2,”Phys. Rev. B, vol. 93, p. 054513, Feb 2016. 63[92] S.-G. Jung, S. Shin, H. Jang, P. Mikheenko, T. H. Johansen, and T. Park,“Effects of magnetic impurities on upper critical fields in the high-T c super-conductor La-doped CaFe2As2,” Superconductor Science and Technology,vol. 30, p. 085009, jul 2017. 63[93] A. Kapitulnik, S. A. Kivelson, and B. Spivak, “Colloquium: Anomalousmetals: Failed superconductors,” Rev. Mod. Phys., vol. 91, p. 011002, Jan2019. 63[94] D. Ephron, A. Yazdani, A. Kapitulnik, and M. R. Beasley, “Observationof Quantum Dissipation in the Vortex State of a Highly Disordered Super-conducting Thin Film,” Phys. Rev. Lett., vol. 76, pp. 1529–1532, Feb 1996.63[95] M. R. Beasley, J. E. Mooij, and T. P. Orlando, “Possibility of Vortex-Antivortex Pair Dissociation in Two-Dimensional Superconductors,” Phys.Rev. Lett., vol. 42, pp. 1165–1168, Apr 1979. 73[96] B. I. Halperin and D. R. Nelson, “Resistive transition in superconductingfilms,” Journal of Low Temperature Physics, vol. 36, pp. 599–616, Sep 1979.7393[97] C. Christiansen, L. M. Hernandez, and A. M. Goldman, “Evidence of Collec-tive Charge Behavior in the Insulating State of Ultrathin Films of Supercon-ducting Metals,” Phys. Rev. Lett., vol. 88, p. 037004, Jan 2002. 63[98] Y. Qin, C. L. Vicente, and J. Yoon, “Magnetically induced metallic phasein superconducting tantalum films,” Phys. Rev. B, vol. 73, p. 100505, Mar2006. 63[99] M. A. Steiner, N. P. Breznay, and A. Kapitulnik, “Approach to asuperconductor-to-Bose-insulator transition in disordered films,” Phys. Rev.B, vol. 77, p. 212501, Jun 2008. 63[100] A. M. Goldman, “SUPERCONDUCTOR-INSULATOR TRANSITIONS,”International Journal of Modern Physics B, vol. 24, no. 20n21, pp. 4081–4101, 2010. 66[101] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A.Bernevig, “Type-II Weyl semimetals,” Nature, vol. 527, pp. 495 EP –, 112015. 74[102] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park,and D. C. Ralph, “Control of spin-orbit torques through crystal symmetry inWTe2/ferromagnet bilayers,” Nature Physics, vol. 13, pp. 300 EP –, 11 2016.74[103] V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, “Magnetoresistance and quantum oscillations of an electrostaticallytuned semimetal-to-metal transition in ultrathin WTe2,” Phys. Rev. B, vol. 95,p. 041410, Jan 2017. 74[104] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose,T. Klimczuk, A. P. Ramirez, N. P. Ong, and R. J. Cava, “Superconductivityin CuxTiSe2,” Nature Physics, vol. 2, pp. 544 EP –, 07 2006. 74[105] A. F. Kusmartseva, B. Sipos, H. Berger, L. Forro´, and E. Tutisˇ, “PressureInduced Superconductivity in Pristine 1T−TiSe2,” Phys. Rev. Lett., vol. 103,p. 236401, Nov 2009. 7494[106] Y. Tokura and N. Nagaosa, “Orbital Physics in Transition-Metal Oxides,”Science, vol. 288, no. 5465, pp. 462–468, 2000. 108[107] C. Taranto, M. Kaltak, N. Parragh, G. Sangiovanni, G. Kresse, A. Toschi, andK. Held, “Comparing quasiparticle GW+DMFT and LDA+DMFT for thetest bed material SrVO3,” Phys. Rev. B, vol. 88, p. 165119, Oct 2013. 109[108] R. Sakuma, P. Werner, and F. Aryasetiawan, “Electronic structure of SrVO3within GW+DMFT,” Phys. Rev. B, vol. 88, p. 235110, Dec 2013. 109[109] J. M. Tomczak, M. Casula, T. Miyake, and S. Biermann, “Asymmetry inband widening and quasiparticle lifetimes in SrVO3: Competition betweenscreened exchange and local correlations from combined GW and dynamicalmean-field theory GW +DMFT,” Phys. Rev. B, vol. 90, p. 165138, Oct 2014.109[110] H. Tanaka and M. Misono, “Advances in designing perovskite catalysts,”Current Opinion in Solid State and Materials Science, vol. 5, no. 5, pp. 381– 387, 2001. 109[111] K. Yoshimatsu, T. Okabe, H. Kumigashira, S. Okamoto, S. Aizaki, A. Fu-jimori, and M. Oshima, “Dimensional-Crossover-Driven Metal-InsulatorTransition in SrVO3 Ultrathin Films,” Phys. Rev. Lett., vol. 104, p. 147601,Apr 2010. 108, 109[112] M. Gu, S. A. Wolf, and J. Lu, “Two-Dimensional Mott Insulators in SrVO3Ultrathin Films,” Advanced Materials Interfaces, vol. 1, no. 7, p. 1300126,2014. 110[113] Q.-R. Li, M. Major, M. B. Yazdi, W. Donner, V. H. Dao, B. Mercey, andU. Lu¨ders, “Dimensional crossover in ultrathin buried conducting SrVO3layers,” Phys. Rev. B, vol. 91, p. 035420, Jan 2015. 110, 112, 117[114] W. Price, “Electric potential and current distribution in a rectangular sampleof anisotropic material with application to the measurement of the prin-cipal resistivities by an extension of van der Pauw’s method,” Solid-StateElectronics, vol. 16, no. 7, pp. 753 – 762, 1973. 11195[115] P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev.Mod. Phys., vol. 57, pp. 287–337, Apr 1985. 113, 116[116] J. Kondo, “Resistance Minimum in Dilute Magnetic Alloys,” Progress ofTheoretical Physics, vol. 32, pp. 37–49, 07 1964. 113[117] S. Hikami, A. I. Larkin, and Y. Nagaoka, “Spin-Orbit Interaction andMagnetoresistance in the Two Dimensional Random System,” Progress ofTheoretical Physics, vol. 63, pp. 707–710, 02 1980. 116[118] C. Castellani, P. Schwab, and M. Grilli,On Localization Effects in Underdoped Cuprates, pp. 361–367. Boston,MA: Springer US, 2002. 113[119] B. ALTSHULER and A. ARONOV, “CHAPTER 1 - Electron-Electron In-teraction In Disordered Conductors,” in Electron-Electron Interactions inDisordered Systems (A. EFROS and M. POLLAK, eds.), vol. 10 of ModernProblems in Condensed Matter Sciences, pp. 1 – 153, Elsevier, 1985. 113,116, 118[120] N. F. Mott, “Conduction in non-crystalline materials,” The PhilosophicalMagazine: A Journal of Theoretical Experimental and Applied Physics,vol. 19, no. 160, pp. 835–852, 1969. 115[121] J. L. Olsen, Electron Transport in Metals. Interscience, New York and Lon-don, 1962. 115[122] V. Fatemi, S. Wu, Y. Cao, L. Bretheau, Q. D. Gibson, K. Watanabe,T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, “Electrically tunable low-density superconductivity in a monolayer topological insulator,” Science,vol. 362, no. 6417, pp. 926–929, 2018. 74[123] C. Eaton, J. A. Moyer, H. M. Alipour, E. D. Grimley, M. Brahlek, J. M.LeBeau, and R. Engel-Herbert, “Growth of SrVO3 thin films by hybrid molec-ular beam epitaxy,” Journal of Vacuum Science & Technology A, vol. 33,no. 6, p. 061504, 2015. 110, 11196[124] K. Byczuk, W. Hofstetter, and D. Vollhardt, “Mott-Hubbard Transition versusAnderson Localization in Correlated Electron Systems with Disorder,” Phys.Rev. Lett., vol. 94, p. 056404, Feb 2005. 110[125] Y. Zhou and S. Ramanathan, “Correlated Electron Materials and Field Ef-fect Transistors for Logic: A Review,” Critical Reviews in Solid State andMaterials Sciences, vol. 38, no. 4, pp. 286–317, 2013. 110[126] N. P. Breznay and A. Kapitulnik, “Particle-hole symmetry reveals failedsuperconductivity in the metallic phase of two-dimensional superconductingfilms,” Science Advances, vol. 3, no. 9, 2017. 63[127] M. B. Lundeberg, Phase coherence in graphene. PhD thesis, Vancouver,2013. 81[128] J. D. Teufel, Superconducting Tunnel Junctions as Direct Detectors forSubmillimeter Astronomy. PhD thesis, USA, 2008. 80[129] L. Zhaoliang, The effects of spatial confinement and oxygen stoichiometryon complex metal oxides. PhD thesis, 2012. 118[130] L. Spietz, J. Teufel, and R. J. Schoelkopf, “A Twisted Pair Cryogenic Filter,”2006. 80, 81[131] J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, “Splitting of a CooperPair by a Pair of Majorana Bound States,” Phys. Rev. Lett., vol. 101,p. 120403, Sep 2008. 74[132] L. Fu and C. L. Kane, “Probing Neutral Majorana Fermion Edge Modes withCharge Transport,” Phys. Rev. Lett., vol. 102, p. 216403, May 2009. 74[133] B. Ja¨ck, Y. Xie, J. Li, S. Jeon, B. A. Bernevig, and A. Yazdani, “Observationof a Majorana zero mode in a topologically protected edge channel,” Science,vol. 364, no. 6447, pp. 1255–1259, 2019. 7497Appendix AMethod to extract criticalexponent α in Fig. 5.8The transition from superconducting to insulating temperature dependence, as afunction of ne passes through a point (ne = ncrit) with essentially no temperaturedependence, see for example Figs. 5.3B and 5.8 in the main text. To explore thepossibility that this is a quantum critical point, a scaling analysis is presented in theinset to Fig. 5.8. That scaling analysis is described in more detail below.The goal of this analysis is to find an exponent α such that multiple data setsRxx(ne) taken at different temperatures collapse onto a pair of two curves (oneinsulating, the other metallic) when the horizontal axis is re-scaled from ne to|ne−ncrit |.Tα . We take the following approach, illustrated in Fig. A1.1. For a given α , data from multiple temperatures between 100 mK and 1 K isreplotted with the re-scaled horizontal axis described above. Figures A1(A-C) showa set of data re-scaled for three different α , as an example. In order to effectivelyevaluate data that varies over orders of magnitude in resistance, we consider not Rxxitself but lnRxx.2. For each re-scaling of the data, best fit lines are obtained to the insulatingand metallic branches (red lines in Figs. A1(A-C)).3.The integrated error between the re-scaled data (more precisely, the naturallog of the data) and the best fit lines is then calculated, giving a quantity we referto as collapse error for each α: collapse error Σni=0(a+ bXi−Yi)2 where a and b98are intercept and slope of the best fit line respectively, Yi is the natural log of theresistance for point i and Xi is the re-scaled x-value for point i. The collapse errorsin the insulating and superconducting branches are then added up, to get the totalcollapse error for a specific value of the α . 4.This procedure is repeated for a rangeof α and a plot of collapse error versus α is obtained (Fig. A1D). From this plot,we find that the minimum collapse error occurs for α ∼ 0.8 ± 0.1.Figure A.1: Explanation of the procedure used to determine an optimal scal-ing exponent. A, B, C: Rxx(ne) data for temperatures from 100 mK to 1K, plotted with the re-scaled horizontal axis indicated, for three valuesof the scaling exponent α . Panel B represents the best-fit exponent,also shown in the main text (Fig. 5.8 inset). D. Collapse error quan-tifies the failure of the multiple-temperature datasets to collapse ontometallic and insulating branches, as described above. The minimumcollapse error (best scaling) is found for α ∼ 0.8 ± 0.1, where the errorbar is determined qualitatively from the rounding of the collapse errordependence.99Appendix BCalculation of coherence length,and the criterion for identifyingcritical points.The thrust of chapter 5 was the phenomenological observation that superconductivitycan be induced by mild electrostatic gating in a material that is also a quantum spinHall insulator. We do not aim for a precise characterization of the superconductivityitself: the small size of our samples, and limited set of contact arrangements, makeit difficult to extract detailed parameters (such as critical fields, temperatures, andcoherence lengths) with the level of accuracy that is possible in bulk systems.Nevertheless, estimates can be made for each of these parameters; our procedurefor doing so is laid out below. A first challenge is to determine the analogueof critical fields, or temperatures, in samples where the resistance, Rxx, changesgradually as superconductivity emerges or is suppressed in the sample. To doso, it is necessary to define a particular fraction of the normal state resistance,Rxx/RN = 0.X , that delineates the transition to superconductivity, but the choiceof this fraction is somewhat arbitrary. We refer to parameters extracted throughthis fraction as characteristic parameters, rather than critical parameters, in order tohighlight the gradual transition and any impact that might have on further analysis.Figures in the chapter 5 show parameters extracted using the characteristic fraction0.X = 0.5, in line with a common convention for 2D materials. Fig. B.1 compares100the temperature dependence of characteristic out-of-plane magnetic fields, B⊥0.X(T ),extracted using fractions 0.X ranging from 0.1 through 0.9.Superconducting coherence lengths are then extracted from the data in Fig. B.1,for various fractions, following the two approaches mentioned in the main text. Inparticular, we can use either high or low temperature limits of B⊥0.X(T ) to extracta measured coherence length, ξmeas. Because the coherence lengths extracted bythese two approaches are very similar to each other, well within the error bar ofeach measurement (which itself derives primarily from the choice of characteristicfraction), we refer to a single value of coherence length simply as ξmeas in thechapter 5.The higher temperature approach to determining coherence length is based onthe linear dependence of B⊥0.X(T ) data near the critical temperature Tc in Figs. 5.5or B.1, which is consistent with the Ginzburg-Landau (GL) model for thin filmsthat is typically used to analyze Bc2 data for 2D superconductors. In this model, thefunctional form of B⊥c2(T ) near Tc is given by:Bc2 =Φ02piξGL(0)2(1− TTc),where Φ0 is the magnetic flux quantum and ξGL(0) is the extrapolation of theGL coherence length to zero temperature. The slope of B⊥0.X(T ) near Tc then givesξslope =√−Φ02piTcdB⊥0.XdT |Tc. Without counting on the precise applicability of the GLexpression over the full temperature range, it is also straightforward to estimate avalue for ξ in the low temperature limit, from the extrapolation of B⊥0.X(T ) to zerotemperature, giving ξBc0 =√Φ02piB⊥0.X (T→0).The tables above Fig. B.1 show that coherence lengths extracted by these twoapproaches are very similar to each other, and moreover that the range 0.1 to 0.9 forthe critical fraction only results in a variation of ±30%, leading to the value quotedin the main text: ξmeas = 100±30nm. As well, the coherence length, ξd , that mightbe expected in the dirty limit based purely on the critical temperature and diffusionconstant is very similar to both ξslope and ξBc0.101Figure B.1: Temperature dependence of the characteristic field, B⊥0.X(T ), com-paring characteristic fields defined using different fractions of the normal-state resistance [0.1,0.3,0.5,0.8,0.9]. Two different densities are shown(A. ne = 15×1012cm−2 and B. ne = 19×1012cm−2). From this temper-ature dependence, characteristic temperatures in the zero field limit andcharacteristic fields in the zero temperature limit can be estimated byextrapolation, as can the slope dB⊥0.XdT |Tc . The table of values above thegraphs indicates coherence lengths extracted from these data, via thetwo approaches described above. Also shown is the gap-based coher-ence length in the dirty limit, obtained from the diffusion constant andcharacteristic temperature as described in the chapter 5.102Appendix CContact Resistance of Device M1The WTe2 flakes in both devices lay on top of the Pt contacts, so top and bottomgates affected the bulk carrier density and contact resistance differently. The Ptcontacts screen the electric field from the bottom gate, so the bottom gate voltage(Vb) only affects the WTe2 carrier density away from the contacts. The top gatevoltage, on the other hand, affects the WTe2 everywhere and therefore has a strongereffect on the contact resistance. In order to keep the contact resistance low whilechanging the carrier density, the voltage applied to the top gate (Vt) was fixed ata large positive value for n-type transport -or a large negative value for p-typetransport- while varying the voltage applied to the bottom gate (Vb). Figure C.1Aillustrates the effects of both Vt and Vb on contact resistances for device M1. Thisfigure shows a contact resistance measurement for one particular pair of contacts,as an example for a behavior that was generic for all contacts. In this case, wemeasured the contact resistance of the contacts, labeled I and G respectively, towhich the current bias and ground are connected in the schematic. The device wascurrent biased and two voltage drops were measured simultaneously: V1 records,effectively, the voltage drop within the WTe2 film between contacts I and G, whereasV2 includes the voltage drop across the contacts. Thus the contact resistance per 4contact can be estimated from Rc = (V2/I−V1/I)/2. In Fig. C.1B this quantity isplotted versus ne for fixed Vb, although the data was taken by sweeping Vt for fixedVb.As seen in the figure, and described in more detail in the caption, maintaining103lower contact resistances at moderate to low ne mandates working at more positivetop gate voltages, that is, at more negative back gate voltages for a given density,for the n-doped regime shown here.Figure C.1: A. Contact resistance measurement for a particular pair of contacts(in this case, those to which the current bias and ground are connected,labelled I and G). B. Although the data is plotted vs ne, the curvesrepresent sweeps of Vt for various fixed Vb. When the density is high(ne > 12×1012cm−2), contact resistance is low independent of relativetop- and bottom- gate voltages. For lower densities (ne < 8×1012cm−2),however, the contact resistance is much lower for strongly negativebottom gate, that is, where the top gate is very positive. Conversely, lesspositive top gate voltages (corresponding to more positive bottom gatevoltages) give very high contact resistances at lower density.104Appendix DNonlinear I−V characteristics inthe superconducting regionOne common approach to characterizing superconducting systems is the measure-ment of nonlinear I-V characteristics, that is, the measurement of dV/dI for elevatedbias currents. The curves obtained from measurements of this type (see e.g. Fig.D.1) had sharp peaks reminiscent of critical current measurements in a more con-ventional superconductor. Unfortunately, developing a clear interpretation of datalike this in the present samples was not possible due to the relatively high contactresistances, low critical temperatures, and small sample sizes. Together, these fac-tors made it too difficult to distinguish heating effects due to elevated bias currentsfrom critical phenomena unrelated to elevated temperatures, including the variousfeatures often associated with mesoscopic superconducting samples such as multipleAndreev reflection.105Figure D.1: Two (A) and four (B) terminal differential resistance measure-ments for sample M1 in the superconducting regime, for the set ofcontacts investigated in the main text. The measurement was made witha small AC current bias (2 nA) on top of the DC current bias on thehorizontal axis, using a locking at the AC frequency to measure dV/dI.106Appendix EStrontium Vanadium OxideThe main body of this thesis, as explained in the previous chapters, were focused onthe transport properties of the electronic systems with non-trivial topology. In thisappendix, we describe the transport properties of a strongly correlated electronicsystem with a trivial topology.The interplay between spin and orbital degrees of freedom, which gets mani-fested as strong spin-orbit coupling (SOC) played a crucial role in the observed ex-otic phases of the electronic systems investigated in the previous chapters. Stronglycorrelated materials are other examples where the interplay between spin, orbital,charge and lattice degrees of freedom leads to exotic states such as colossal mag-netoresistance and high TC superconductivity, etc. [57]. One of the electronicsystems where the strong correlations play a central role in identifying their phaseare transition metal oxides (TMOs). A well known example are the cuprates inwhich strong electronic correlations give rise to high TC superconductivity. Anothermember of the TMOs family, in which the strong correlation has a dramatic effectare the vanadates. They include both binary compounds of vanadium and oxygen,and ternary compounds, involving a third element such as strontium (Sr) or calcium(Ca). In the binary compounds, the strong correlations lead to a sharp change in theresistance (more than four orders of magnitude, Fig. E.1) and an associated metal-insulator transition (MIT), driven by an external stimulant such as temperature. Inthe following paragraph, we discuss in a simple language, how strongly correlatedsystems are different from conventional electronic systems.107Figure E.1: Reproduced with permission from [58]. Metal insulator transitionin binary compounds, VxOy, of vanadates.In solid state band theory, materials are classified as insulators or metals basedon the filling of their electronic bands. When the highest occupied band is partiallyfilled, the material is considered to be a metal, otherwise, it is an insulator. However,when the Coulomb repulsion between charge carriers is strong, the above definitionof metal/insulator fails, and other theoretical models that include strong correlationsneed to be considered. Many transition metal oxides (TMOs) with partially filledd-orbitals are among the examples where the ratio between Coulomb (U) andkinetic energy (t) define the state of the material [106]. The metallic state of theseelectronic systems arises in a regime where the on site Coulomb repulsion (U) issmaller than kinetic energy (t), which results in a conducting state. The strengthof the correlations can be enhanced in different ways, which might lead to metalto insulator transition (MIT) in these materials. For example, thickness drivenenhancement of electronic correlation has been proposed [111] as a way to induce aMIT.In the experiment described in this appendix, we have used techniques similarto those used in chapter 3 and 5 in investigating the electronic properties of variousthicknesses of strontium vanadium oxide (SVO), which is a member of the transition108metal oxides family.E.1 SVO Electronic StructureOwing to its simple electronic structure, SVO has been employed widely as a test-bed for the theoretical study of the correlation effects in TMOs [107, 108, 109]. Ithas a crystal structure of a simple cubic perovskite as shown in Fig. E.2. The rareelement atom strontium (Sr) with a bigger atomic radius is placed at the center ofthe cubic structure and coordinated by 12 oxygen atoms, while the transition metalatom vanadium (V) sits at the corner and coordinated by six oxygen atoms.Figure E.2: “Reprinted from [110], with permission from Elsevier.” Crystalstructure of SVO. A and B represents strontium and vanadium ionsrespectively.The electronic configuration of SVO is 3d1 with one electron per unit cell.E.2 Experimental BackgroundIt has been reported that SVO is a paramagnetic metal in the bulk with a resistivityof around 10−5−10−3Ω.cm at room temperature [111]. Investigating the variationof the electronic structure of SVO thin films near Fermi energy (EF ) using in-situphotoemission spectroscopy has revealed a MIT as a function of the film thickness[111]. A gap opening was observed at the Fermi energy for SVO thickness of 1-2monolayer (ML) in the photoemission spectra, thus confirming a thickness drivenMIT.109Subsequent transport measurements, as shown in Fig. E.3(a) on SVO filmsgrown on an insulating substrate (LaAlO3)0.3(Sr2AlTaO6)0.7 (LSAT) reported acrossover thickness of around 17 ML at temperature ∼ 50K [112]. As the filmthickness is lowered down, the transition temperatures were shifted toward highervalues, where at 8ML it was found to be ∼ 200K. However, a more recent transportstudy [113] reported a metallic state for SVO thin films encapsulated by lanthanumvanadium oxide (LVO) down to 3ML (Fig. E.3(b)). These contradictory results onthe crossover MIT thickness may arise due to the different quality of grown SVOfilms, apparent from their extracted mobilities. Specifically, it should be consideredthat impurities or defects can have a significant effect in MIT in ultra thin films [124]as shown for the case of VO2 thin films [125]. Therefore, to realize the mechanismbehind the MIT in ultrathin films of SVO, it is crucial to have samples with higherqualities than previous reports. This chapter reports the transport properties ofmultiple SVO thin films grown by hybrid molecular beam epitaxy (from the Engel-Herbert group) [123], which have around two orders of magnitude higher mobilitythan other reports.Figure E.3: (a),(b) Resistivity vs temperature of SVO thin films measured attwo different studies. In (a) a MIT was demonstrated for thickness below6.5nm (∼ 17MLs), while in (b) SVO films were metallic down to 3ML,with ln(T) enhancement of resistivity at low temperature for 3ML. ( (a)and (b) are taken from [112], [113] respectively.)110E.3 Experimental DetailsHere we present the transport properties of various thicknesses of SVO thin filmsfrom 10ML down to 2ML, which were encapsulated by 10nm of strontium titaniumoxide (STO) to protect them from degradation in the ambient condition. Thesandwich stack of STO/SVO/STO was grown by the Engel-Herbert group usinghybrid molecular epitaxy [123] on an insulating LSAT substrate. Both LSATsubstrate and STO protection layer have a cubic structure with very close latticeconstants to SVO (Fig. E.4(a)), which is essential to avoid introducing any strain onSVO thin films. Figure E.4(a,b) shows a schematic of SVO heterostructure alongwith the optical photo of 10ML SVO thin film.Figure E.4: (a) Schematic of LSAT/STO/SVO stack along with their latticeconstant denoted as (a). (b) Optical image of 10ML SVO on a chipcarrier with aluminum bond wires on its perimeter. Scale bar is 1mm.To carry out the transport measurements, aluminum wires were bonded directly onthe perimeter of the SVO thin films as shown in Fig. E.4(b). The resistivity of thesample is obtained by an extended version of van der Pauw method (vdP), which isalso applicable to anisotropic samples, using the following equation [114]:RAD,BC =−8ρpid ln∞∏n=0{tanh[LADLBC(2n+1)pi2]}with d as the thickness of the SVO thin film and ρ is the square resistivity.RAD,BC represents a four-terminal resistance of the sample, as shown in Fig. E.5(a)where a current is sent between point A and D with length LAD and a voltage ismeasured between point B and C with a length of LBC, giving RAD,BC =VBC/IAD. Bycomparing the square resistivity in orthogonal orientations, the amount of anisotropy111in various SVO thin films was identified. It was found out that down to 3ML allSVO samples were isotropic, while for 2ML sample an anisotropy of ∼ (20-30%)was detected, which will be explained in more details later on.Figure E.5: (a),(b) Schematic of SVO sample with contacts shown by redsquare on its perimeter along with electrical configuration for measuringlongitudinal resistance RAD,BC in (a) and transverse resistance in (b).E.4 Temperature dependence measurementsMeasurement of resistivity vs temperature for various thickness of SVO thin filmsrevealed a metallic state down to 3ML with a logarithmic (ln(T )) increase in theresistivity at low temperatures in agreement with [113] as can be seen in Fig.E.6. The ln(T ) enhancement of resistivity at low temperature can be attributed toquantum corrections to the classical resistivity of the metallic systems. We discussin the following paragraph the possible origin of the ln(T ) enhancement.Electron transport in a metallic system is normally diffusive, meaning thatelectrons become scattered many times across the device by impurities, phononsand other electrons. A characteristic time scale of electron scattering is momentumscattering time (τm), which is a measure of the time that an electron travels beforegetting scattered. The conductivity of a metallic system (σ ) can be defined by aclassical Drude model as a function of electron density (n) and τm:σ =ne2τmm∗(E.1)where e and m∗ are charge and effective mass of electron respectively. At high112temperature, the main scattering mechanism is the electron-phonon scattering,which becomes less effective at low temperature, due to a decrease in the number ofphonons present in the systems. At sufficiently low temperature, the electron-phononscattering becomes negligible, and the scattering of the electron by other electronsand impurities becomes dominant. At such low temperatures, a quantum mechanicalphenomena known as weak localization (WL), which is a coherent backscatteringof time reversed electron waves, contributes to a classical Drude conductivity asln(T ). Beside WL effect, electron-electron interaction (EEI) also becomes importantat low temperature, which also contributes to conductivity as ln(T ) dependence[115, 119]. The ln(T ) correction of the conductivity at low temperatures may alsostem from Kondo effect, which is due to scattering of electron spin by magneticimpurities [116]. To single out the dominant mechanism, one can compare the ln(T )enhancement both at zero magnetic field and finite magnetic field as well as themagnetoresistance (MR) [115]. Both WL and Kondo effect give rise to a negativeMR. Looking at MR data in Fig. E.8(a), we see a positive MR for all measured SVOsamples of different thicknesses. Therefore the ln(T ) enhancement in our sample ismost likely due to EEI. Following [119] the electron-electron interaction correctionto conductivity is given by:δe−e =−A e22pi2h¯ln(KBTτmh¯) (E.2)where the prefactor A is a measure of the EEI strength whose value is between 0(weak interaction) and 1 (strong interaction). Fitting the data for the 3ML sample tothe equation E.2, as shown in Fig. E.6(b), we found prefactor A∼ 1, which indicatesa strong interaction, however for 4ML and 5ML samples, this coefficient was foundto be larger than one by a factor of 3-5. The origin of this inconsistency betweenthe theory and our experimental finding is not clear. However, we noted that forunder-doped cuprates which are also strongly correlated electronic systems, a largeprefactor contrary to the theory has been found [118].113Figure E.6: (a) Resistivity of different SVO thickness versus temperaturesshows a metallic state down to 3ML with an upturn in the resistivityat low temperature. (b) The upturn in Resistivity for 3ML sample isreplotted as conductance vs temperature in log scale. The solid line isthe fit to the equation E.2.Although SVO thin films down to 3ML showed a metallic state down to thelowest measured temperature (∼ 20 mK), 2ML SVO samples were insulating at lowtemperature (Fig. E.7). The other dramatic difference between 2ML SVO samplesfrom other thicknesses was a large transport anisotropy between two orientationsin the vdP geometry as shown in Figure 7.7. One explanation for the observedanisotropy is the existence of easy axis (lower resistivity orientation) and hard axis(higher resistivity orientation) in 2ML SVO films due to the growth of 2ML SVO ina terrace-like pattern. The other possibility of the present anisotropy might be dueto the nonuniformity of 2ML samples.114Figure E.7: Resistivity of two SVO samples, S1 (red lines) and S2(bluelines) vs temperature measured in two orthogonal orientations shownin schematics (right). Inset in the main panel shows a resistance vstemperature of a 2ML sample along with the fit (blue line) to VRHmodel.The insulating state of the 2ML sample fits well with a variable range hopping(VRH) model as shown in the inset of Fig. E.7. Based on this model [120] theconduction at low temperature arises from phonon assisted hopping of electronsbetween localized states, which has an exponential form:R= R0exp(T0T)α (E.3)Fitting the data for 2ML sample with equation E.3 results in an exponent α ∼ 0.5,which is consistent with VRH model including electron-electron interaction, knownas Efros-Shklovskii variable range hopping (ES-VRH) model [55].E.5 Magnetoresistance measurementsCharge carriers in a metallic system experience a Lorentz force in the presence of aperpendicular magnetic field, which causes a quadratic increase of the resistance inthe low field limit, µB< 1, and a saturation in the high fields (µB> 1), where µ isthe charge carrier mobility [121]. Figure E.8(a) shows magnetoresistance (MR) of115SVO samples at 4K, which is defined as R(B)−R(0)R(0) ×100%, where R(B) and R(0)are resistances at finite and zero magnetic field respectively. The observed MR at4K shows a weak positive B2 dependence for 10ML. However, for thin films lessthan 6ML this dependency deviates from B2 and moves toward a linear dependency.Converting the resistance to conductance, the change in the magnetoconductanceof the SVO thin films less than 6ML was found to be ∼ 10-15% of the quantum ofconductance (e2/h), which is in the range to observe weak localization effect (WL).The magnetic field breaks the time reversal symmetry and suppresses WL effect,which results in a negative MR. Nevertheless, the negative MR is not illustrated inour samples at 4K. MR measurements for 3ML at sub-kelvin temperatures revealedWL effect (Fig. E.8(b)). The phase coherence time τφ in WL, which is a measure ofa time that two electron waves stay phase coherent, can be obtained by fitting themagnetoconductance curve to the following WL model [117]:∆σxx =e2pih[ln(τφ/τB)+Ψ(12+(τφ/τB)−1)] (E.4)In equation E.4 τB = h¯4eDB , where D is a diffusion constant.Figure E.8: (a) MR of various SVO thin films at 4K. (b) Magnetoconductanceof 3ML SVO plotted vs magnetic field shows WL for temperatures below600mK. (c) Extracted τ−1φ s from the measurements in (b) vs temperature.The blue and brown solid lines are fits to the power law τ−1φ ∝ Tp withexponent 1 and 3 respectively.The decoherence rate at different temperatures is expected to follow a powerlaw dependence as a function of temperature, τ−1φ ∝ T−p [115, 119]. The exponentP determines the main mechanism of the phase breaking, which is 3 for electron-116phonon scattering and 1 for electron-electron scattering. Fig. E.8(c) shows theextracted τ−1φ s vs temperature for the 3ML SVO sample. As shown in Fig. E.8(c),although the data don’t match well with either of the power law dependence forthe electron-electron or electron-phonon interactions, they show better agreementwith the fitting line corresponding to the electron-electron interaction. This mightindicate that the dephasing rate for the 3ML samples at temperatures below 600mKis predominately electron-electron interactions.E.6 Hall measurementsWe extracted the mobility (µ) and sheet carrier density (ns) for different thicknessof SVO samples from the measurement of the transverse resistance vs perpendicularmagnetic field at 4K as shown in Fig. E.9. It is worth mentioning that in the previousreport [113], the mobility of the 40nm SVO sample was much lower (∼ 3cm2V 2s−1) compared to the samples studied in this work.Figure E.9: Sheet carrier density and electron mobility on the left and rightaxes respectively plotted vs number of layers. Sheet carrier densityshows a straight linear trend vs SVO thicknesses down to 5 monolayers.A deviation from the linear trend appears when the SVO thickness isless than 5 monolayers.The solid black line is a guide to the eye.Assuming a constant bulk volume carrier density, the sheet carrier density shouldscale linearly with the film thickness, whereas the measured sheet carrier densities117showed a drop from the linear trend for thicknesses below 5MLs. The decrease inthe density for thinner layers can be attributed to the formation of a dead layer ofthe SVO [129].E.7 Conclusion on SVOThe measurements presented in the last chapter describe the transport properties ofSVO thin films with the highest charge carrier mobilities reported so far for thismaterial. Study of the MIT transition as a function of temperature revealed that2ML samples become insulating, with a transition temperature around 150K. Theresistivity in the insulating state follows the Coulomb dominated variable rangehopping. It was also found that the resistivity of 2ML samples are highly anisotropic,which might be due to the inhomogeneity in these ultra thin films.We also found that SVO thin films are metallic above 2ML, with a ln(T ) en-hancement of the resistivity at low temperatures. This enhancement is most likelydue to EEI, due to the positive MR of these thin films at low temperatures. Fur-thermore, MR measurements at temperatures below 1K revealed weak localizationeffect for 3ML samples. Although the decoherence rates extracted from thesemeasurements didn’t fit well to the power law dependence for electron-electron orelectron-phonon interaction, the data show a closer tendency to the fitting line forthe electron-electron interaction. Finally, the extracted sheet carrier density from theHall measurement versus SVO thickness demonstrated a deviation from the lineartrend for thicknesses below 5 monolayers as displayed in figure E.9. The decreasein the density for thinner layers is possibly due to the formation of a dead layer ofthe SVO.At the end of these measurements, there are some questions worth to explorefurther. As explained in section E.4, the obtained values for the strength of e-einteraction for the 4 and 5 ML samples is larger than unity, which is inconsistentwith the theory [119]. To find out whether this is an intrinsic property for thesesamples, the temperature dependence measurement needs to be done more precisely.Extending the temperature dependence measurement down to sub kelvin rangewould allow to fit the equation (E.3) to a bigger range of temperature and obtain amore accurate result. Doing measurements in samples with Hall bar geometry rather118than the vdP would also enhance the accuracy for extracting the phase coherenttime in WL measurements.119

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