Quantum transport in topological insulator-based van der WAALS heterostructures

The discovery of three-dimensional (3D) topological insulators (TIs) has offered the unprecedented candidates for the study of topological phases in quantum states of matter. Over the past decade, the research on 3D TI devices has been focused on the isolation of surface transport from the bulk and the elimination of bulk carriers for the convenient of probing their surface states. This thesis research continues the trend of development in 3D TIs and tackles their shortcomings from two aspects, namely the crystal growth and device structure, to achieve high-quality TI based devices. With optimization in growth conditions, Bi2-xSbxTe3-ySey (BSTS) 3D TI single crystals with significant improvement in surface mobility were grown. These together foster the development of surface integer quantum Hall effect (QHE) at low magnetic field. Next, to realize more advanced TI devices, we introduce van der Waals (vdW) heterostructures in the form of TI/insulator/graphite (Gr) to effectively control chemical potential of the topological surface states (TSS). The hBN/Gr gating in the QH regime shows improved quantization of TSS by suppression of magnetoconductivity of massless Dirac fermions. The TI vdW heterostructures also provide the local gates necessarily for quantum capacitance measurements, which allow a quantitative evaluation of the surface states' LL energies in 3D TI. With quantum capacitance studies, the top and bottom TSS can be separated and individually probed by applying excitation voltages to the gates coupled capacitively to different surfaces. Furthermore, in the studies of variable thickness 3D TI devices towards iv 2D thin limit, we establish a tunable capacitive coupling between the top and bottom TSS and study the effect of this coupling on QH plateaus and LL fan diagram via dual-gate control. We observe a splitting of N= 0 LL in magnetic field for the thin devices indicates intersurface hybridization possibly beyond single-particle effects. The studies are extended to the intersurface hybridization regime where a thermally-activated transport gap clearly resolved at the Dirac point for film thickness less than 10 nm. The layer-dependent hybridization gaps scale up exponentially with decreasing layer thickness. The perpendicular electric field and magnetic field responses to the hybridization gap are further analyzed and discussed. These works show the promise of the vdW platform in creating advanced high-quality and tunability 3D TI-based devices.

QUANTUM TRANSPORT IN TOPOLOGICAL INSULATOR-BASED VAN DER WAALS HETEROSTRUCTURES by Su Kong Chong A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah May 2020 Copyright © Su Kong Chong 2020 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Su Kong Chong has been approved by the following supervisory committee members: Vikram V. Deshpande , Chair 11/7/2019 Date Approved Oleg A. Starykh , Member 11/7/2019 Date Approved Jordan Gerton , Member 11/7/2019 Date Approved John Belz , Member 11/7/2019 Date Approved Taylor D. Sparks , Member 11/6/2019 Date Approved and by Christoph Boehme the Department/College/School of , Chair/Dean of Physics and Astronomy and by David B. Kieda, Dean of The Graduate School. ABSTRACT The discovery of three-dimensional (3D) topological insulators (TIs) has offered the unprecedented candidates for the study of topological phases in quantum states of matter. Over the past decade, the research on 3D TI devices has been focused on the isolation of surface transport from the bulk and the elimination of bulk carriers for the convenient of probing their surface states. This thesis research continues the trend of development in 3D TIs and tackles their shortcomings from two aspects, namely the crystal growth and device structure, to achieve high-quality TI based devices. With optimization in growth conditions, Bi2-xSbxTe3-ySey (BSTS) 3D TI single crystals with significant improvement in surface mobility were grown. These together foster the development of surface integer quantum Hall effect (QHE) at low magnetic field. Next, to realize more advanced TI devices, we introduce van der Waals (vdW) heterostructures in the form of TI/insulator/graphite (Gr) to effectively control chemical potential of the topological surface states (TSS). The hBN/Gr gating in the QH regime shows improved quantization of TSS by suppression of magnetoconductivity of massless Dirac fermions. The TI vdW heterostructures also provide the local gates necessarily for quantum capacitance measurements, which allow a quantitative evaluation of the surface states’ LL energies in 3D TI. With quantum capacitance studies, the top and bottom TSS can be separated and individually probed by applying excitation voltages to the gates coupled capacitively to different surfaces. Furthermore, in the studies of variable thickness 3D TI devices towards 2D thin limit, we establish a tunable capacitive coupling between the top and bottom TSS and study the effect of this coupling on QH plateaus and LL fan diagram via dual-gate control. We observe a splitting of N= 0 LL in magnetic field for the thin devices indicates intersurface hybridization possibly beyond single-particle effects. The studies are extended to the intersurface hybridization regime where a thermally-activated transport gap clearly resolved at the Dirac point for film thickness less than 10 nm. The layer-dependent hybridization gaps scale up exponentially with decreasing layer thickness. The perpendicular electric field and magnetic field responses to the hybridization gap are further analyzed and discussed. These works show the promise of the vdW platform in creating advanced high-quality and tunability 3D TI-based devices. iv TABLE OF CONTENTS ABSTRACT....................................................................................................................... iii LIST OF TABLES ............................................................................................................ vii ACKNOWLEDGMENTS ............................................................................................... viii Chapters 1. INTRODUCTION .......................................................................................................... 1 1.1 Topological Invariants and Topological Insulators ............................................. 2 1.2 Topological Surface States and Their Characteristics ......................................... 6 1.3 Evidence of Topological Surface States ............................................................ 14 1.4 Unusual Electromagnetic Phenomena and Novel Quantum States ................... 21 2. CONTROL GROWTH OF A THREE-DIMENSIONAL TOPOLOGICAL INSULATOR ............................................................................................................... 27 2.1 Single Crystal Growth Techniques .................................................................... 29 2.2 Characterization Techniques .............................................................................. 31 2.3 Electrical Transport ............................................................................................ 40 3. TOPOLOGICAL INSULATOR-BASED VAN DER WAALS HETEROSTRUCTURES ............................................................................................. 51 3.1 Device Fabrication ............................................................................................. 52 3.2 Topological Insulator/Normal Insulator ............................................................ 55 3.3 Topological Insulator/Ferromagnetic Insulator ................................................. 60 4. LANDAU LEVELS OF TOPOLOGICAL SURFACE STATES PROBED BY QUANTUM CAPACITANCE..................................................................................... 72 4.1 Capacitance Devices and Measurements ........................................................... 73 4.2 Landau Levels Development in Magnetic Field ................................................ 75 4.3 Dual-Gated Magneto-Capacitance ..................................................................... 77 4.4 Electronic Compressibility and Landau Level Spacings ................................... 82 5. CAPACITIVE-COUPLING BETWEEN TOPOLOGICAL SURFACE STATES ..... 91 5.1 Variable Thickness Topological Insulator Devices ........................................... 92 5.2 Low-Temperature Dual-Gated Transport .......................................................... 92 5.3 Surface States Coupling in the Quantum Hall Regime ...................................... 95 6. TOPOLOGICAL PHASE TRANSITIONS IN HYBRIDIZED SURFACE STATES ..................................................................................................................... 111 6.1 Intersurface Hybridization ............................................................................... 112 6.2 Hybridization Gap Analysis ............................................................................. 113 6.3 Normal and Inverted Hybridization Gap States ............................................... 117 6.4 Magnetic Field-Induced Phase Transition ....................................................... 119 6.5 Electric Field-Induced Phase Transition .......................................................... 122 7. CONCLUSION ........................................................................................................... 127 7.1 Concluding Remarks ........................................................................................ 127 REFERENCES ............................................................................................................... 130 vi LIST OF TABLES 1.1 Candidates of topological materials. ............................................................................. 7 2.1 The elemental composition measured using EDS and ICP-MS for different methods prepared BSTS samples................................................................................................ 33 2.2 Crystallographic data for BiSbTeSe2. ......................................................................... 38 2.3 Atomic coordination for BiSbTeSe2. .......................................................................... 41 5.1 The thickness of the BSTS (d), top (ht) and bottom (hb) hBN flakes are measured using a Bruker Dimension Icon atomic force microscopy. The width to length aspect ratio (W/L) of the BSTS devices was estimated by the optical images. The errors indicate the accuracy of the thickness measurements from the atomic force microscopy. .................................................................................................................. 93 5.2 Ctg and Cbg are calculated from parallel plate capacitor relation with a dielectric constant of hBN fixed as 3. The CBSTS is estimated from the slope (S) of the Vtg versus Vbg plots using the relation (5.7). The percentage of errors are obtained from the average of linear fitting errors. The dielectric constant of the BSTS can thus be estimated from the CBSTS and d. ................................................................................. 107 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Prof. Vikram V. Deshpande, for his guidance and support in my Ph.D. research. I owe him enough for the training, experience, and knowledge sharing in all aspects throughout my studies. I faithfully thank my Ph.D. supervisory committee, Prof. Oleg A. Starykh (theoretical condensed matter physics), Prof. Jordan Gerton (experimental optical physics), Prof. John Belz (atmospheric physics), and Prof. Taylor D. Sparks (crystal growth & materials science), for their time and valuable pieces of advice given to me. My appreciation also goes to our collaborators: Dr. Kyu Bum Han and Dr. Akira Nagaoka from Prof. Taylor D. Sparks’s group (MSE) who helped to grow the BiSbTeSe2 single crystals; Renlong Liu from Prof. Changgu Lee’s group (Sungkyunkwan University) for preparing the Cr2Ge2Te6 single crystals; Dr. Ryan McLaughlin and Dr. Haoliang Liu from Prof. Valy Vardeny’s group (Physics) for their assistance in Sagnac Interferometry measurement; Prof. Dima Pesin (Physics) for the discussion and theoretical inputs to support our data; Haoxin Zhou from Prof. Andrea Young’s group (UCSB) for the discussion on capacitance measurements; and Dr. Lizhe Liu from Prof. Feng Liu’s group (MSE) for the theoretical simulations to support our data. To my fellow labmates at Deshpande’s lab, especially Dr. Ryuichi Tsuchikawa, Shuwan Liu, Jared Harmer, and Isaac Martin, I am sincerely thankful for their involvement in this project. I also thank Neda Lotfizadeh, Rachael Morris, Rohit Kumar, Chuankun Liu, Jamie Berg, and Deric Session for the precious time and all the fun we have had together. I wish to thank the Utah Nanofab Cleanroom staff process engineers, Dr. Brian Baker, Dr. Steve Pritchett, and Dr. Tony Olsen; and the Surface Analysis Lab scientist, Dr. Brian van Devener, Dr. Paulo Perez and Dr. Randy Polson, for the training on nanofabrication and instrumentation support. Also, I would like to thank the National High Magnetic Field Laboratory user support scientists, Dr. Jan Jaroszynski, Dr. Hongwoo Baek, Glover Jones, and Dr. Alimamy Bangura, for their assistance on my measurements at DC Field Cell 9 and SCM2. I gratefully thank the Department of Physics for awarding the Swigart Graduate Scholarship in 2019-2020 academic year as a recognition of my academic and research performance. I thank the support from the University of Utah Graduate School for the four years teaching assistantship and Graduate Student Travel Grant. I also thank American Physical Society (APS) for awarding the APS DCMP division travel award for travel support to the APS March meeting 2019. Also, I acknowledge funding agencies Materials Research, Science, and Engineering Center (MRSEC) for their financial support to our research. My acknowledgments are extended to the journal editors, Dr. Yusuke Kozuka (Scientific Reports), Prof. Klaus Ensslin (Nano Letters), Dr. Donavan Hall (Physical Review Letters) and Prof. Ajay Sood (ACS Nano), for handling our manuscripts. Last but not the least, I would like to express my deepest gratitude to my Family: My respectful parents, Teck Chong and Geok Kheng Khoo; my beloved wife So Yen Yit and my lovely daughter Yu Han Chong. I would not have completed my Ph.D. studies without their love and supports. ix CHAPTER 1 INTRODUCTION Both topological invariants and quantum Hall effects (QHE) are the benchmarks in condensed matter physics which relate the fundamental topological order to quantum mechanical phenomenon in matter. Both concepts apply to an important class of quantum materials, known as three-dimensional topological insulators (3D TIs). The former is related to the bulk state of the 3D TI due to a bandgap inversion, while the latter characterizes the topological surface states (TSS) due to their nontrivial Berry’s phase and Chern number. Because of the topological and symmetry protection, the TSS behaves very differently from the surface states in an ordinary (nontopological) metal or insulator. For example, the TSS with a linear dispersion band structure can host massless Dirac fermions with unique spin texture. In this chapter, I introduce the origin, characteristics, and realizations of the TSS at the boundary of 3D TIs. As the earlier theoretical framework of 3D TI is based on Z2 topological invariant in 2D TI, the chapter begins with the 2D TI and then generalizes to 3D. The section continues by reviewing the unique electronic band structure and properties of the TSS in 3D TI, followed by the discussions on experimental tools for probing the TSS with a focus on the quantum transport measurements. The peculiar behavior of TSS can be used to probe the exotic particles, composites, and unconventional electromagnetic 2 activity. Thus, the chapter ends with a brief discussion of novel 3D TI device configurations for those studies. 1.1 Topological Invariants and Topological Insulators The TI with a topologically protected boundary state is a manifestation of its topological invariants, which may be understood in terms of robust protection against perturbations. The topological invariants arise from a concept of band inversion, where the conduction and valence bands are inverted at the bandgap [1,2]. This band inversion can change the topology of the system to a nonzero quantum number or topological invariant and results in a gapless state at the boundary. This is because the nontrivial topology is a discrete characteristic of inverted gap states, and the topology cannot change as long as the inverted bulk gap remains open. Thus, in order for the topology to change across the boundary into a trivial one, the gap must close at the interface. This bulk-boundary correspondence guarantees gapless boundary states. Therefore, the band inversion mechanism is also called a topological phase transition. An important ingredient for the band inversion is spin-orbit coupling (SOC) [1,2]. SOC arises from the interaction between spin (of electrons) and its orbital motion (of the nucleus) as explained by a general form of Hamiltonian [3] as HSOC = λL∙S, where  is the SOC strength, and L and S are the orbital and spin angular momentum operators, respectively. It is known that spin-orbit interaction is a relativistic effect that can cause a distortion in electronic band structures. For small bandgap element (or compound made of elements) with large SOC, the coupling strength is strong enough to flip the band structure at the gap, causing a negative gap. When this mechanism is associating with a 3 transformation in topological order, topologically protected boundary states can exist. Importantly, although the SOC can shift the band structures, it does not break the existing time-reversal symmetry (TRS) [4]. An example of SOC driven band inversion is the CdTe/HgTe quantum well, known as the first discovered 2D TI [5]. The prediction was made theoretically [1] based on the inverted band structure of HgTe with zero bandgap at the Γ point due to the strong SOC. The gap is opened by constructing a quantum well with CdTe to break the cubic symmetry. Detailed calculations show that the CdTe/HgTe becomes a 2D TI when the quantum well thickness exceeds a critical thickness. Below the critical thickness, the quantum well remains as an ordinary insulator. The gapless edge states of 2D TI are characterized by the Z2 topological invariant with a topological quantum number, ν, of 1 for the nontrivial topological phase [6]. A consequence of the band inversion with nonzero topology is a pair of crossing edge states, which then give rise to a quantum spin Hall (QSH) insulating state. The QSH effect can be explained as a pair of QH effect for up and down spin components, each undergoes an edge conduction channel. In QSH state, the spin species are split by an external electric field into helical edge states (as shown in Figure 1.1(c)), and thus the QSH effect can exist without a magnetic field. The helical edge states were verified by probing the four-probe conductance (Gxx) in CdTe/HgTe quantum well at low temperatures. A quantized conductance of 2e2/h independent to the channel lengths was observed at zero magnetic fields when the chemical potential was tuned into the bulk bandgap [7], giving solid evidence of the gapless edge states. The prefactor of two is a consequence of the degeneracy of the two spin species. Therefore, the 2D TI is also known as a QSH insulator. 4 Figure 1.1. Topological invariants and band structures of TIs in 2D and 3D. Z2 indices presented as TR-invariant momenta for the (a) 2D and (b) 3D Brillouin zone. Real-space images of (c) 2D and (d) 3D TIs for illustration of edge and surface conduction states, respectively. Dispersion relation features of (e) 2D and (f) 3D TIs in the Brillouin zone. 5 This was the first experimental confirmation of the TR-invariant TI characterized by the Z2 topology. Theoretically, the Z2 invariant in 2D TI with topologically-protected edge states can be generalized to infinite dimensions [4]. Namely, an N-dimensional material has (N−1)-dimensional boundaries, which can host topologically-protected conducting gapless state at the boundary. The success in 2D TI has stimulated the interest in 3D version by using the theoretical frameworks of topological invariant established in 2D TI. Figure 1.1 shows a comparison between the topological edge and surface states in 2D and 3D TI, respectively, in real and energy-momentum spaces. The topology of the 3D TI can be fully characterized by four Z2 topological invariants (ν0; ν1 ν2 ν3) [8-10] in contrast with the single Z2 invariant (ν) in 2D TI. Figure 1.1(b) presents the physical origin of the Z2 invariant in a 3D Brillouin zone with eight TRinvariant momenta at the vertices of the cube. The ν1, ν2, and ν3 invariants can be interpreted as Miller indices in the reciprocal space. The ν0 invariant indicates the number of TRinvariant momenta enclosed by the surface Fermi arcs. 3D TIs with nontrivial topological invariants can be categorized by ν0= 1 as “strong” and ν0= 0 for any nonzero ν1, ν2, and ν3 as “weak” TIs. The strong 3D TI is robust against nonmagnetic disorders, whereas weak 3D TI is susceptible to disorder. A consequence of the topological invariants is the fully gapped 3D bulk states bridged by topologically protected surface states. The surface states dispersions cross at a TR-invariant momentum and present a single Dirac cone dispersion, hence the physics of relativistic Dirac fermions become relevant. To date, many other materials/compounds with different classes of topology have been identified and verified experimentally. It is now clear that the 2D and 3D TIs or more 6 precisely named as topological band insulators are only one-class among the large group of topological materials. The topological phases are also found on high symmetry crystal surfaces, which allow the metallic surface states with quadratic band degeneracy on the crystal plane, therefore named as topological crystalline insulator (TCI) [11-13]. Different from the topological band insulator, the TCI band structures are characterized by new topological invariants called mirror Chern numbers (𝑛𝑀 ), and their surface states are protected by crystal (including rotation, reflection, and mirror) symmetries. The discovery of the TCI also reflects that SOC is not the only ingredient for the topological phase transition. Another class of TIs called topological Kondo insulator (TKI) is proposed in a Kondo insulator where the topological phases arise from the strong electron correlations between orbitals [14]. On the other hand, topological phases have also been observed in metals, such as topological Dirac semimetals (DSM) and Weyl semimetals (WSM). Different from the edge or surface states in TIs, topological semimetals exhibit topologically-protected Fermi arcs at their edges [15]. Table 1.1 summarizes different classes of topological materials and their representative candidates discovered to date. 1.2 Topological Surface States and Their Characteristics An important consequence of the nontrivial topology in 3D TIs is that the gapless surface states necessarily emerge when the insulator is physically terminated by an ordinary insulator (including vacuum). These gapless surface states are protected by a series of topological protections [16]. The fundamental Z2 topological invariants guarantee the existence of gapless surface states as long as the bulk gap stays open and inverted. As the spin eigenvalues at opposite momentum states (k and -k) are exactly opposite, electrons 7 Table 1.1 Candidates of topological materials. Topological system Topology Symmetry Candidates 2D TI Z2 TRS HgTe, InAs/GaSb 3D TI (strong) Z2 TRS Bi1-xSbx, Bi2Se3, Bi2Te3, Insulator Sb2Te3, BiTeSe, Bi2-xSbxTe3-ySey 3D TI (weak) Z2 No TRS Bi14Rh3I9, Bi2TeI TCI (symmorphic) 𝑛𝑀 CS SnTe, PnSnTe TCI (nonsymmorphic) 𝑛𝑀 MS KHgSb TRS SmB6 Metal TKI DSM CFS TRS and/or IS Na3Bi, Cd3As2, ZrTe5 WSM (type I) CFS Breaking TRS TaAs WSM (type II) CFS and/or IS WTe2, MoTe2 *CFS= Fermi surface Chern numbers, CS= crystal symmetry, MS= mirror symmetry, IS= inversion symmetry 8 with k state cannot be backscattered into the -k state or vice versa because of the spin mismatch. Therefore, the TSS is protected from backscattering. In addition, the linear dispersive TSS can host massless Dirac fermions with π Berry’s phase, which prevents weak localization through destructive interference paths. Besides, the TSS is also protected against nonmagnetic disorders by the TRS. 1.2.1 Linear Dispersion Relation One of the most striking characteristics of the TSS is the linear energy-momentum relationship. When two spin eigenstates of TSS forming a Kramers pair cross each other at a TR-invariant momentum, the energy dispersion near this crossing point display a linear dispersion (as shown in Figure 1.1). This dispersion relation can be described as the massless limit of the Dirac energy equation [17]. Thus, the dispersion is also known as Dirac dispersion, and the electrons are called to behave as massless Dirac fermions. The TSS Hamiltonian can be written as a 2D massless Dirac model as [4,18,19]: ⃗ × 𝜎) ∙ 𝑛̂ = 𝑣(𝜎𝑥 𝑘𝑦 − 𝜎𝑦 𝑘𝑥 ) 𝐻 = 𝑣(𝑘 (1.1) where v is the effective velocity, and k and σ denote the wave vector and Pauli matrices, respectively. A prominent property of the surface Dirac fermions is that they carry a nonzero Berry’s phase [20,21]. The surface Dirac fermions acquire a π Berry’s phase after completing a closed trajectory adiabatically around the Fermi surface. The π Berry’s phase has a significant effect on magnetoelectric and quantum transport of the 3D TIs. In magnetoelectric transport, the π Berry’s phase causes the Dirac fermions to interfere destructively along the time-reversed scattering paths, leading to the weak antilocalization 9 (WAL) effect in TSS [22,23]. This can happen because the TSS conduction channel lies in the quantum diffusion regime where the phase-coherent length much longer than the charges’ mean free path, and thus the Dirac fermions maintain their phase coherence upon scattering. The WAL effect leads to the quantum enhancement in electrical conductivity at very low temperatures [24]. The quantum interference gradually ceases with magnetic field, giving rise to a negative magnetoconductivity. Moreover, the π Berry’s phase has significant impact in quantum transport; namely the Landau level (LL) filling factor (ν) for Dirac fermions is not proportional to the LL index (N) but follows a proportionality of 1 𝜈 = 𝑔 (𝑁 + 2) (1.2) where g is the degeneracy. This leads to the half-integer QHE in Dirac TSS, which is clearly distinguishable from the bulk states’ LLs. 1.2.2 Helical Spin States The helical spin polarization in TSS, which is also known as spin-momentum locking, is distinct from the ordinary surface states. The spin in the surface is always pointing to the direction perpendicular to the momentum vector and flipped with opposite momentum. The spin species exhibit a left-handed helicity above the Dirac point, and flip to right-handed below the Dirac point [16], as shown in Figure 1.1(f). Also, the spin degeneracy in TSS Dirac fermions is lifted because the spin polarization is locked to momentum. This makes the TSS very different from the well-studied Dirac material, graphene, as the latter one has both spin and valley degeneracies. Therefore, 3D TI is also ascribed as 1/4 graphene, which offers a simpler platform to study Dirac physics. The spin-momentum locking induced helical spin states naturally gives rise to 10 various interesting spin-related physics. One of emerging research is the searching of Majorana fermions on TSS in the presence of proximity-induced or bulk doping superconductivity [25,26]. Another field is 3D TIs based spintronics on generation and detection of the surface spin-polarized current in 3D TI/ferromagnet structure [27]. 1.2.3 Symmetry Protection and Breaking The TSS is protected from nonmagnetic perturbations by TRS, while magnetism is essential to break the TRS. When a 3D TI comes into contact with a ferromagnetic layer (or is chemically doped with magnetic atoms), the spontaneous magnetization induces an exchange field (or exchange coupling), which modifies the electronic structure of the surface states. The modified Hamiltonian by the magnetic exchange interaction can be described as [28,29]: 𝐻𝑚 = 𝑚𝜎𝑧 (1.3) where m is the mass term signifies a mass gap proportional to JM, and where J and M are the effective exchange coupling and magnetization on the ferromagnetic layer (or magnetic atoms), respectively. The exchange interaction due to the presence of magnetic ordering opens a magnetic gap at the Dirac point (Figure 1.2) in the energy spectrum of the TSS, and causes the surface Dirac fermions to become massive. The magnetic gap size is proportional to the magnetization and the exchange field strength from the magnetic layer, as shown by the resulting energy dispersion as [30]: 𝐸 = ±√(𝑣ℏ𝑘𝑥 + 𝐽𝑀𝑦 2 2 ) + (𝑣ℏ𝑘𝑦 + 𝐽𝑀𝑥 2 2 ) +( 𝐽𝑀𝑧 2 2 ) (1.4) For a ferromagnetic layer with out-of-plane magnetization, the induced exchange gap size is ½JMz. The theoretical calculations estimate the magnetic gap is typically in the range of 11 Figure 1.2. Gapped topological surface states. (Left) The gapped TSS and manifestation of massive Dirac fermions by magnetic exchange coupling, and intersurface hybridization. (Right) The LLs developed in both cases of the gapped TSS by applying perpendicular magnetic field. 12 several to tens of meV [28,31]. The magnetic exchange gap can modulate the Berry’s phase () as [32]: 𝐽𝑀 𝛾 = 𝜋 (1 − 2𝐸 𝑧 ) 𝐹 (1.5) where EF is the Fermi energy measured from the Dirac point. This causes the Berry’s phase deviation from π and weakens the associated destructive interference in charge transport. A direct consequence of the reduced Berry’s phase is the constructive quantum interference in the electronic paths of massive Dirac fermions, leading to the weak localization (WL) effect. The competition between WL and the preexisting WAL can lead to an opposite trend in ∆σxx with magnetic field in magneto-transport [33,24]. The induced magnetism in TSS by proximity effect can also generate an anomalous Hall effect (AHE) in ρxy, which can be expressed as [34,35]: 𝜌𝑥𝑦 = 𝜌𝐻 𝐵 + 𝜌𝐴𝐻 (1.6) where ρH and ρAH are the Hall and anomalous Hall resistivities, respectively. In the quantization regime, the induced magnetic gap causes the N= 0 LL pinning to either top valence or bottom conduction bands, depending on the sign of magnetization [36]. 1.2.4 Finite Size Effect Each surface state in a 3D TI is associated with its own Dirac cone wavefunction. The TSS breaks down at a finite film thickness when the wavefunctions between top and bottom surface states overlap. The intersurface hybridization leads to a mass gap open at the Dirac point, called hybridization gap (Figure 1.2). The Hamiltonian of the TSS is modified by the perturbation due to the hybridization gap as [37]: 𝐻𝑝 = ∆ℎ 𝜎𝑧 (1.7) 13 where the ∆h represents the hybridization matrix element. The hybridization gap size scales up exponentially with a decrease in layer number of 3D TI due to the strong tunneling between surfaces [38,39]. Similar to the magnetic exchange gap, Berry’s phase in TSS is reduced from π due to the hybridization gap, which can be expressed as [32,40]: ∆ 𝛾 = 𝜋 (1 − 𝐸ℎ ) 𝐹 (1.8) The hybridization gap opening in the Dirac dispersion can lead to a decline of the metallic surface transport and loss of topological protection in the 3D TI. Again, a crossover from the WAL to WL behavior can be observed in the hybridization gap regime as a result of the Berry’s phase modification. In the QH regime, the hybridization gap can cause a splitting of the N= 0 LLs, residing at the electron and hole edges of the gap [41,42]. Different from the conventional Landau quantization in Dirac fermions, the LL energies of the N= 0 states deviate from the square root dependence on magnetic field [43]. Also, the two N= 0 LLs can turn highly asymmetric when the Zeeman energy exceeds the hybridization energy [44]. More strikingly, in the hybridization regime, an oscillatory mode alternating between topologically trivial and nontrivial 2D states as a function of the layer thickness is predicted by analyzing the total parity of the surface state’s subband energy levels at the  point [45]. The crossing between the first electron and hole subbands at a critical thickness range will flip the total parity of the system and cause a topological phase transition between a normal insulator and 2D TI. This happens when the first hole subband is at the higher energy level than the first electron subband, and thus the total parity reads negative, similar to the description of the band inversion and negative gap in bulk band of 14 3D TIs. An important consequence of the topological phase transition is the existence of the topological edge states in the critical thickness range that can host a QSH state. The crossing points (when electron and hole subbands energy are equal) and oscillation period between QSH insulator and ordinary insulator depend on both the magnitude and sign of the bulk band gap and the hybridization gap in the bulk band inversion and finite-size regimes of the 3D TIs [45]. Therefore, the oscillatory behavior differs for different 3D TI materials. 1.3 Evidence of Topological Surface States The existence of the linear-dispersive and spin-polarized surface states in 3D TIs was first evidenced in spectroscopies, and then the surface conduction is further verified in magneto-electrical transport measurements. These discoveries have opened a brand-new research area to study the surface states’ topology and their related quantum phenomena. 1.3.1 Photoemission and Tunneling Spectroscopies Soon after the prediction, the first 3D TI was identified experimentally in Bi1-xSbx alloy by mapping its bulk and surface band structures using angle-resolved photoemission spectroscopy (ARPES) [46]. The energy-momentum spectrum of Bi0.09Sb0.91 mapped by ARPES (Figure 1.3(a)) shows the bulk energy bands associated with the nearly linear ̅ points in the Brillouin zone. Dirac-like dispersion surface bands formed between Γ̅ and Μ The odd number of crossings between two time-reversal invariant momenta indicates that these surface states are topologically protected. The results confirmed Bi0.09Sb0.91 as a strong TI with topological invariants (ν0;ν1ν2ν3) of (1;111). 15 Figure 1.3. Evidence of topological surface states. (a) Color map of ARPES spectrum for Bi0.91Sb0.09 [46]. Copyright 2008, Springer Nature. (b) Fourier transform of STS spectrum dI/dV map at Fermi level for Bi0.92Sb0.08 [48]. Copyright 2009, Springer Nature. (c) SdH oscillation signifies the surface states in Bi2Te3 [54] Copyright 2010, The American Association for the Advancement of Science. (d) QHE of the TSS observed at a high magnetic field in BiSbTeSe2 [56]. Copyright 2014, Springer Nature Physics. 16 To further probe the spin texture of the TSS and its correlation with momentum, a follow-up experiment was carried out on the Bi0.09Sb0.91 3D TI using a spin-resolved ARPES [47]. The observations demonstrated solid evidence of the spin polarization in the surfaces, and thus verified their spin nondegeneracy. Furthermore, the spin texture signified a 2π rotation around the central Fermi surface with left-handed chirality, providing a clue of the π Berry’s phase associated with their topological classification [47]. The TSS is forbidden from the backscattering between Kramer pairs of opposite spin and momentum due to the TRS protection. This characteristic was verified in a scanning tunneling spectroscopy (STS) study (Figure 1.3(b)) by analyzing the interference pattern due to scattering at the surface of a Bi0.92Sb0.08 film [48]. By comparing the Fourier transform of the observed interference pattern to the analysis from ARPES spectrum, it has been shown that despite strong atomic-scale disorder, k to -k (or vice versa) backscattering is absent in the TSS. Despite a series of evidence confirming the TSS in Bi1-xSbx alloys, their complicated bulk spectrum motivated the search for 3D TIs with a simpler bulk band structure and a larger bulk gap. The joint efforts of theoretical computation [3] and experimental [49] research led to the discovery of the prototypical 3D TI, Bi2Se3. The simple direct gap band structure of bulk Bi2Se3 results in a clean single Dirac cone surface state. Moreover, the relatively large bulk gap of ~0.3 eV (3600 K) of Bi2Se3 offers the potential for realization of the topologically protected behavior at room temperature (RT). Concurrent theoretical and experimental analyses on the electronic structure further identified Bi2Te3 and Sb2Te3 as large bulk band-gap and single Dirac cone 3D TIs [3,50,51]. These works further showed that the Bi2Se3 family of 3D TIs are associated with 17 a band inversion at k= 0, which belong to the (1;000) topological invariants, in contrast to the (1;111) class of Bi0.09Sb0.91. 1.3.2 Magneto and Quantum Transport Although the 3D TI with topologically protected surface states had been realized in spectroscopies almost instantly after the prediction, it has taken much longer for transport experiments to demonstrate the most fundamental signature of surface states, namely the ambipolar metallic electronic transport and their unique Landau quantization. A key challenge holding the progress is the significant bulk carriers, which impede the surface conduction. Thus, it is crucial to distinguish between the transport behaviors of the surface Dirac fermions and the bulk conduction electrons. A prominent property of surface Dirac fermions is the Landau quantization of their energy states in the presence of a perpendicular magnetic field. A classical picture of the Landau quantization can be viewed as the surface Dirac fermions confined in the cyclotron orbits with finite energy states known as Landau levels (LLs). When a 3D TI is subjected to a magnetic field, the density of states (DoS) is periodically modulated as a function of magnetic field due to the change of DoS across different LLs. This leads to an oscillating feature in longitudinal resistivity (or conductivity) known as Shubnikov–de Haas (SdH) oscillations. The SdH oscillations provide an effective way to quantitatively characterize the TSS even if it overlaps with the 3D bulk states. In SdH oscillations regime, the longitudinal conductivity (σxx) follows the relation as [52]: 𝐵 1 0 ∆𝜎𝑥𝑥 = 𝜎𝑥𝑥 cos [2𝜋 ( 𝐵𝐹 − 2 + 𝛽)] (1.9) 18 where B is the external magnetic field, BF is the oscillation frequency, and  is the phase factor (0 =  < 1). The parameter  is related to the Berry’s phase  by a factor of 2π. The Berry’s phase is zero for a parabolic energy dispersion (= 0) and, as already noted, π for Dirac fermions with linear energy dispersion (= 1/2). For bulk conducting 3D TIs, the phase factor directly reflects the Berry’s phase of the system, which can be used to distinguish the Landau quantization from the 2D electron gas from the bulk states and Dirac fermions from the TSS. The phase factor  in the SdH oscillations can be experimentally determined from an analysis of the LL fan diagram, in which the Nth minima in σxx are plotted against their corresponding reciprocal values of a magnetic field (1/BN). From the ∆σxx relation in (1.8), the Nth minima occur when the argument of the cosine term equals (2N-1)π, which simplifies the relation to 𝐵 𝑁 = 𝐵𝐹 + 𝛽 𝑁 (1.10) Therefore, the plot of N versus 1/BN makes a straight line with a slope of BF corresponding to the oscillation frequency. The linear fit to the LL fan diagram can be extrapolated to zero limits of 1/BN, and the intercept on the N-index axis gives the phase factor . Depending on the  value obtained from the fan diagram, one can conclude the origin of the conduction channels to the SdH. Besides that, the SdH oscillations also contain various useful information for determining the Fermi surface, quantum scattering time, effective mass, and Fermi velocity of the Dirac fermions [53,16]. The first experimental report on surface states SdH oscillations studies on Bi 2Te3 (Figure 1.3(c)) [54] and Bi2Se3 [55] confirmed their conduction origin from the surface of the 3D TIs. The 2D nature of the SdH oscillations was confirmed by taking the angular 19 dependence of the oscillation frequency [54]. The Fermi velocity extracted from the SdH oscillations showed a good agreement with the ARPES data. The observations of surface states quantum oscillations were attributed to the minimization of naturally-doped bulk carriers in Bi2Te3 and Bi2Se3 based 3D TIs, but still, the surface transport accounted for only less than 1% of the total conductance. In fact, the significant bulk doping in Bi2Se3 and Bi2Te3 film had limited the access of Landau quantization near their surface states Dirac points, and thus prevented the realization of surface QHE. This further stimulated the exploration of bulk-insulating 3D TIs by mixing the naturally n-type Bi2Se3 & Bi2Te3, and p-type Sb2Te3 to form ternary or quaternary tetradymite such as (Bi1-xSbx)2Te3 (BST) and Bi2-xSbxTe3-ySey (BSTS) 3D TIs. Eventually, the surface QHE was realized at strong magnetic field in the truly bulk insulating BTS and BSTS 3D TIs (Figure 1.3(d)) [56,57]. Similar to SdH oscillations, QHE is a manifestation of LLs for the Dirac fermions confined in 2D surface states. The difference is that SdH oscillations typically detect the large Nth LLs, whereas QHE probes the lower indices LLs due to the larger energy spacings. QHE is a quantized version of the Hall effect, which can be probed by measuring the transverse charge flows in a perpendicular magnetic field. The 2D conductivity tensor is derived from the inverse of the resistivity tensor as: 𝜎𝑥𝑥 (𝜎 𝑦𝑥 𝜎𝑥𝑦 𝜌𝑥𝑥 ) = ( 𝜎𝑥𝑥 𝜌𝑦𝑥 𝜌𝑥𝑦 −1 𝜌𝑥𝑥 1 ) = ( 2 2 𝜌𝑥𝑥 𝜌𝑥𝑥 +𝜌𝑥𝑦 𝜌𝑦𝑥 𝜌𝑥𝑦 𝜌𝑥𝑥 ) (1.11) where σxy and ρxy are the Hall conductivity and resistivity. In the QH regime, when the Fermi level is controlled in between two neighboring LLs, the σxx reduces to a minimum as the DoS vanishes inside the LL gap, while the σxy develops into a Hall plateau, which quantizes at an integer factor of e2/h. When the Fermi level is inside an LL, the σxx takes a finite value and the σxy changes toward the next filling level. 20 QHE in surface states of 3D TIs exhibits several unique behaviors owing to their π Berry’s phase and zero spin degeneracy. As discussed in the relation (1.2), ν takes a halfinteger value in the QH states. The top and bottom surface states with two independent Dirac fermions form a pair of edge mode conducting channels at the surfaces in perpendicular magnetic field. Thus, the fully quantized Hall plateaus at integer number are attributed to the consequence of two states of half-integer QHE from top and bottom surface states, and the σxy can be expressed as [56,57]: 𝜎𝑥𝑦 = (𝜈𝑏 + 𝜈𝑡 ) 𝑒2 ℎ 1 1 = [(𝑁𝑏 + 2) + (𝑁𝑡 + 2)] 𝑒2 ℎ (1.12) where the Nt and Nb are the LL indices of top and bottom TSS, respectively. The νt or νb change sign when crossing the Nt= 0 or Nb= 0 LLs. An interesting quantum state in 3D TIs is the ν= 0 QH state, which emerges as the top and bottom surfaces fill the LLs with the same magnitude and opposite sign filling factors, 𝜈𝑡 = −𝜈𝑏 . This implies that the ν= 0 states in 3D TI have clearly different origin from graphene as the former one is a result of their counterpropagating edge channels, whereas the latter is related to electrons correlation state [58]. Also, unlike the other integer QH states in TSS that carry dissipationless edge modes, the ν= 0 state is a dissipative state; namely the ρxx tends toward maximization (instead of vanishing) in magnetic field. Following the conductivity tensor calculation from relation (1.11), both the σxy and σxx tend to develop towards zero when ρxx>>ρxy. LLs of the TSS follow a Dirac energy equation relation as [55,59]: 𝐸𝑁 = 𝑠𝑔𝑛(𝑁)𝑣𝐹 √2𝑒ℏ|𝑁|𝐵 (1.13) where N is the LL index, and vF is Fermi velocity. The LL energy spacing changes as a square root of N, as opposed to constant energy spacing in ordinary metals and insulators. Similarly, different from the linear proportionality with the magnetic field in ordinary 21 insulator, the TSS’s LL spacing scales as a function of square root of magnetic field. Another important signature is the N= 0 LL, which is pinned to the Dirac point of the TSS and independent to the magnetic field. This generates an LL fan with symmetrical appearance of square root of N states on both the positive (electron) and negative (hole) energy sides of the Dirac point. Besides that, it is important to note that the LL energy is indexed to N instead of ν, meaning that relation (1.13) is applied to individual surface LLs. 1.4 Unusual Electromagnetic Phenomena and Novel Quantum States Since the discoveries of 3D TIs, tremendous numbers of proposals have been made by theorists to realize new quantum phenomena using the novel TI edge/surface states as platforms. Among those, the most striking effects are proposed based on the proximity induced exchange interactions when coupling the TSS to a ferromagnet or an s-wave superconductor to realize the peculiar quantum states or electromagnetic activity. This section briefly reviews three quantum phenomena, namely topological magnetoelectric effect, Majorana edge states, and topological exciton condensation. We note that the realization of these effects requires a basic quantization of TSS, namely in the QH or QAH regime to serve as a ground for the studies. 1.4.1 Topological Magneto-Electric Effect A general description of the topological magnetoelectric effect (TME) is an electric (magnetic) field that generates a topological contribution to the magnetization (electric polarization) in the same direction, with a universal constant of quantization in units of e2/2h. The electromagnetic response in 3D TIs can be described by a modified Lagrangian 22 (LTI) that includes an axion E·B and a dimensionless topological  terms in addition to the conventional Maxwell term [8,60]: 𝑒2 𝜃 𝐿𝑇𝐼 = 2ℎ 𝜋 𝐸 ∙ 𝐵 (1.14) The unconventional axion electrodynamics leads to the derivation of the relations between the observable quantities that relate the electric (P) and magnetic (M) polarizations to the magnetic (B) and electric (E) fields, respectively, as [61]: 𝑒2 𝜃 𝑃 = 2ℎ 𝜋 𝐵 𝑒2 𝜃 𝑀 = 2ℎ 𝜋 𝐸 (1.15) These relations provide a clearer picture of the quantized version of the magnetoelectric responses due to the nontrivial topology in 3D TIs. Under time-reversal invariant, the topological θ term is equal to π (modulo 2π) for TI and zero in a vacuum or ordinary insulator. When TRS is broken under the presence of magnetic order, the opening of the gap in the surface state allows a change in θ from vacuum or ordinary insulator to TI (or vice versa), producing the exotic electromagnetic responses. Figure 1.4(a) shows an experimental proposal by Wang et al. [62] to realize the TME in a ferromagnetic insulator (FMI)/3D TI heterostructure where the TI is sandwiched by two FMI layers with different coercive fields. The FMIs in this configuration play two important roles, namely the induction of the magnetic exchange gap and quantum anomalous Hall (QAH) states on the TSS. By modulating a small magnetic field to tune the top and bottom FMI layers into antiparallel magnetization, the 3D TI is brought into a zeroth plateau QAH state. By tuning the Fermi level into the magnetic gap, an in-plane electric field (Ey) applied induces the Hall currents on the top (𝐽𝑥𝑡 ) and bottom (𝐽𝑥𝑏 ) surfaces. 23 Figure 1.4. Proposed device structures for probing the (a) topological magneto-electric effect, (b) Majorana edge states, and (c) topological exciton condensation. 24 𝑡 The induced Hall currents 𝐽𝑥𝑡 = −𝐽𝑥𝑏 due to the half-quantized Hall conductance 𝜎𝑥𝑦 = 𝑏 −𝜎𝑥𝑦 form a circulating current loop on TI surface. This current circulation is equivalent to the surface-bound current, which can give rise to a quantized magnetization (My) in parallel to the applied Ey. The induced quantized My is an indication of TME, which can be probed by the magnetoelectric susceptibility measurements such as Faraday or Kerr rotation and superconducting quantum interference device (SQUID). 1.4.2 Majorana Edge States The superconductivity in TSS of a 3D TI is of great interest as it provides a promising route to realize a nontrivial topological superconducting state. The topological superconductivity stems from the Cooper pairing of spin-polarized surface state electrons when proximitized by a conventional s-wave superconductor, and such a “spinless” superconductor can host a self-conjugate electron called Majorana fermion [60,63]. The localized Majorana fermions in vortices of topological superconductors form Majorana zero-modes, which obey non-Abelian statistics and are useful for topological quantum computation [64]. Starting with the original proposal by Fu and Kane (Figure 1.4(b)) [25], numerous schemes to couple topological materials with superconductors have been forwarded to seek the signature of Majorana fermions. Majorana fermions behave as charge-neutral particles with zero energy in a topological superconductor. Their existence can be experimentally probed as a zero-bias conductance peak and modulated external electrical and magnetic fields [65,66]. However, it is difficult to exclude the contributions from other effects such as Kondo correlations, Andreev bound states, and weak antilocalization. Alternatively, the 25 presence of Majorana bound states can be verified by a 4π periodicity in Josephson junctions made in superconductor-TI-superconductor configuration [67-69]. Nevertheless, the transparency of the superconducting-TI junctions is relatively low, and the supercurrent can also be carried by the conventional Andreev bound states from the bulk TI [70]. Instead of Majorana bound states, one can probe Majorana edge states by proximately coupling a topologically nontrivial QH system to a superconducting reservoir [71,72]. The Majorana edge state is topologically equivalent to a chiral topological superconductor with a topological number. Compared to the QH system, QAH insulator forms chiral edge modes without magnetic field is more practical for realization the chiral topological superconductor. An important transport signature of the chiral Majorana edge states in the QAH insulator/superconductor heterostructure is the half-integer plateau in two-terminal conductance (σ12) within the coercive field regime of the QAH insulator. The half-quantized plateau signifies an equal probability of backscattering between the normal and Andreev processes in the chiral Majorana edge state. 1.4.3 Topological Exciton Condensation The previous two activities require the intrinsic 3D TI interfaced with either a ferromagnet or a superconductor. The third quantum phenomenon discussed here is a quantum coherence gapped state, which involves only the interlayer Coulomb interaction, known as topological exciton condensation [73,74]. When the top and bottom surfaces are independently doped to induce electrons in one layer and holes in the other, the interlayer electron-hole pairs form indirect excitons, which can support the topological exciton condensate at low enough temperature. This correlated state can occur when a TI film is 26 thinned to a thickness where the top and bottom surface states are coupled strongly yet not strong enough for their wave functions to hybridize. This usually appears at 2D surface ℏ separations in an order of magnetic length, 𝑙𝐵 = √𝑒𝐵 [38]. The relatively flat LL band serves as a ground for the topological excitonic condensate, while the ν= 0 QH state with the widest separation from other LLs is ideal for the realization of this effect. The topological exciton condensate exhibits unique properties from the quantum well Bose-Einstein condensate and interlayer bilayer graphene exciton superfluid. The resulting odd number of valleys associated with each surface state due to its nontrivial topological phase can support a vortex with an exact zero-energy mode. This zero-mode quantized vortex carries a fractional charge of ±e/2 [73]. Several experiments have been proposed to probe the topological exciton condensation in 3D TIs, such as Coulomb drag [75], optical scattering experiments [76], and heat transport [77]. CHAPTER 2 CONTROL GROWTH OF A THREE-DIMENSIONAL TOPOLOGICAL INSULATOR Control of stoichiometry of the bulk insulating Bi2-xSbxTe3-ySey (BSTS) is crucial to control the chemical potential of the surface states to be within the bulk gap. Despite many studies [78-85], the single crystal growth of stoichiometric BSTS remains challenging. This chapter presents a systematic study on the crystal growth of BSTS using three different methods, namely melting growth (MG), vertical Bridgman growth (BG), and two-step melting-Bridgman growth (MBG). The MG and BG parameters are illustrated in Figure 2.1(a) and (b), respectively. MBG begins with growing an ingot via MG (possibly single crystal) and then follows with recrystallization to form single-crystal BSTS by BG. The composition homogeneity, crystal structure, and low-temperature magneto-transport of the three types of BSTS samples are characterized. We show that the structural and electrical transport properties depend strongly on the growth method. The crystallinity of the BSTS, identified by structural properties, is correlated to the surface mobility and the quantum transport properties. 28 Figure 2.1. The measured temperature profiles of the (a) melting and (b) vertical Bridgman furnaces. The schematics of ampoules in (a) and (b) illustrate the crystal growth mechanism for the respective methods. 29 2.1 Single Crystal Growth Techniques 2.1.1 Ampoule Preparation The metal trace of Bi, Sb, Te, and Se (Sigma-Aldrich Co., purity 5N grade) were used. The quartz tubes with an outer diameter of 1.4 cm and 0.1 cm thickness of wall were used for ampoule preparation. The raw materials were weighed to the molar ratio of 1:1:1:2 for Bi:Sb:Te:Se and mixed into a mixture with total weight of 5 g. The quartz tube was sealed on one end into a conical shape. The inner wall of the tube was coated with an inert carbon layer via pyrolysis of acetone to prevent the reaction of the materials with the tube. The mixture in the quartz tube was flushed with argon gas a few times to displace the air out from the tube. This followed by an evacuation of the tube to a pressure of below 10-6 torr to ensure a high vacuum environment for the crystal growth. The tube was then sealed by a torch into an ampoule with length of 6-8 cm. 2.1.2 Melting Growth Melting growth BSTS single crystals were prepared in a muffle furnace (F30438CM, Fisher Scientific, Waltham, MA). The ampoule was placed at the center of the muffle furnace. The temperature of the furnace was increased to 850oC at a slow rate of 0.1oC/min and was held for 48 h. A few times of intermittent mixing were performed by gently shaking the ampoule at the temperature. The ampoule was then cooled down to 550oC at a rate of 0.1oC/min and was annealed at the temperature for 96 h. After that, the sample was slowly cooled down to room temperature (RT) at a rate of 0.1oC/min. The typical settings for temperature as a function of time for the melting growth method is illustrated in Figure 2.1(a). 30 The single crystal growth methodology for the melting method (Figure 2.1(a)) generally involves the following procedures: (i) Melting and mixing the source materials in a sealed ampoule at a temperature above the melting temperature. (ii) The molten materials are annealed at a temperature just above the melting point for a long period to allow atomic diffusion and nucleation of a crystalline lattice. (iii) The materials are cooled down to the RT at a very slow rate to minimize additional nucleation sites in favor of growing the existing single crystal. Obviously, melting growth can cause polycrystalline ingot as there is no fixed single nucleation site. On the other hand, the vertical Bridgman method applies a vertical motion to translate the materials along a temperature gradient from hot to cold zones at a very slow rate, as shown in Figure 2.1(b). The source materials are gradually melted and solidified from the bottom to the top of an ampoule to yield single crystal growth. The crystal quality greatly depends on the translation rate as the slow solidification encourages the sample homogeneity and minimizes the crystal defects. 2.1.3 Bridgman Growth Bridgman growth was carried out in a vertical Bridgman furnace with three coil heaters. The ampoule was held by a thin string at one end and placed vertically at the level above the first heating zone. While the other end of the string was attached to a motion controller. The temperatures were set to 670oC, 770oC, and 500oC for warm, hot, and cold zones, respectively, with the sequence from top to bottom heaters. The temperature profile was illustrated in Figure 2.1(b). The first heater acted as a preheater to prevent the deposition of Se vapors on the wall of the ampoule. The ampoule was translated vertically downward through the furnace at a very slow rate of 0.6 cm/day. 31 2.1.4 Two-Step Melting and Bridgman Growth The single crystal BSTS was first grown in the muffle furnace by following the steps discussed in the melting growth process. The as-grown single crystal BSTS was placed in a new carbon coated quartz tube with a quartz rod placed on top of the sample. The second growth was carried out in the vertical Bridgman furnace in the same processes described in the Bridgman growth method. 2.2 Characterization Techniques The as-grown crystals were cleaved and exfoliated along the crystalline surface for energy-dispersive X-ray spectroscopy (EDS) characterization. EDAX EDS (equipped in FEI Quanta 600 field emission scanning electron microscopy), operating at an acceleration voltage at 15 kV, was utilized to obtain EDS signals. Elemental compositions of the crystal were also studied using an inductively coupled plasma mass spectrometry (ICP-MS, Agilent 7500 series, Agilent Technologies Inc.). For specimen preparation, the BSTS flakes were dissolved into a mixed nitric acid (HNO3, 1.2 M) and hydrochloric acid (HCl, 0.3 M) solution. Bruker D2 Phaser X-ray diffractometer with Cu Kα radiation and a zerobackground holder (G130706, MTI Co.) was employed for crystal plane and powder Xray diffraction (XRD) data acquisition. The full-pattern Rietveld refinement was performed using GSAS-II software to determine the crystal structure. A small crystal was picked from the BSTS sample and examined on a Bruker PLATFORM single-crystal diffractometer equipped with a SMART APEX II CCD area detector with Gr-monochromated radiation (Mo K,  = 0.70296 Å). Intensity data were collected using  scans at 7 different  angles with a frame width of 0.3° and an exposure time of 15 s per frame. 32 2.2.1 Stoichiometry Study The elemental composition of the as-grown BSTS samples was investigated using both energy-dispersive X-ray spectroscopy (EDS) and inductively coupled plasma mass spectrometry (ICP-MS). The results are presented in Table 2.1. The EDS data were taken from the homogeneous part of the crystals (excluding the side surfaces). Typically, the crystal formed a cone shape with about 1 cm in diameter and about 1-2 cm in height. The region of the EDS study is about 0.5×0.5 cm2, near the center of the crystal. The EDS results are presented in the ratio of Bi:Sb:Te:Se converted from the atomic percentage of the quantitative data. The BSTS crystals grown by both MG and BG methods showed similar composition ratios which differed from MBG method. The composition of the MBG sample was found to be ~10% closer to the stoichiometric ratio (1:1:1:2 for Bi:Sb:Te:Se) compared to the MG and BG samples. To further confirm the composition results, we utilized the more accurate ICP-MS for the elemental characterization. Consistent with the EDS analysis, both MG and BG samples showed similar composition in the ICP-MS spectrum and MBG sample was again found to more closely match the desired stoichiometric BiSbTeSe2. Although the MBG sample was grown from the MG crystal, which is showing slightly off stoichiometry in the center part of the crystal. We emphasize that the second growth step was done by using the whole MG crystal where we expect the off stoichiometry is balanced by side area of the crystal. 2.2.2 Crystallinity Evaluation The crystal structure and crystallinity of the BSTS samples were studied by X-ray diffraction (XRD). Figure 2.2(a) shows the XRD patterns collected from the (00l) crystal 33 Table 2.1 The elemental composition measured using EDS and ICP-MS for different methods prepared BSTS samples. Growth method EDS (±2%) ICP-MS (±3%) MG Bi1.00Sb0.90Te0.95Se2.15 Bi0.75Sb1.25Te0.56Se2.44 BG Bi0.92Sb0.95Te0.98Se2.15 Bi0.79Sb1.21Te0.62Se2.38 MBG Bi0.96Sb1.02Te0.97Se2.05 Bi0.89Sb1.11Te0.88Se2.12 34 Figure 2.2. Crystallinity study. (a) XRD patterns scanned at (00l) crystal surface for BSTS grown by MG, BG, and MBG methods. The (b) (006) diffraction peaks and their (c) FWHMs for the three BSTS growth methods. The FWHM is the mean value of the fitting of (006) diffraction peak from three representative XRD patterns for each sample. The error bars in (c) present the standard deviation of FWHMs of the (006) peak from three samples for each growth. 35 surface. The BSTS thin flakes were cleaved from the center part of the crystal using a razor blade. The crystallographic planes of the corresponding diffraction peaks were wellindexed in the figure with no evidence of polycrystalline grains. The highest intensity (006) diffraction peak for the BSTS samples grown by different methods is compared in Figure 2.2(b). The MG BSTS showed a broad diffraction peak with two shoulders at both sides of the peak. The shoulder broadening could be due to the inhomogeneous compositions resulting in antisite or vacancy defects in the crystals [79,81]. The diffraction peak of BG BSTS revealed only high angle broadening, suggesting a more homogeneous composition due to the zone melting and solidification process in the Bridgman method. Similar observations had been found for the vertical Bridgman growth in Bi2(Te1-xSex)3 system [79]. The MBG BSTS showed a more symmetric Gaussian curve and narrower peak width, which indicated a better crystallinity and homogeneity compared to the MG and BG BSTS. For a quantitative comparison, the full-width at half maximums (FWHM) of the (006) diffraction peaks for the three BSTS growth methods are plotted in Figure 2.2(c). The smaller FWHM generally implied the improvement in crystal lattice arrangement as observed in MBG BSTS. 2.2.3 Crystal Structure Investigation The crystal structures of the BSTS have examined from the powder XRD patterns as presented in Figure 2.3(a). The upper panel presents the references for the Rietveld refinement (ICOD codes: 00-040-1211, 01-073-7366, and 01-089-2007). The solid dots and orange color lines are then measured and the refined powder XRD intensity. The green color lines show the difference between the measured and the refined XRD. The Rietveld 36 Figure 2.3. Crystal structure analysis. (a) Powder XRD patterns of the BSTS samples grown by MG, BG, and MBG methods. (b) The extended view of the powder XRD patterns near the (108) peak to compare their characteristic diffraction peaks. (c) The I(0012 ̅̅̅) /I(018) and I(107) /I(018) of BSTS crystal grown by different methods. The solid dots and error bars in (c) present the mean values and standard deviation of the means calculated from three representative powder XRD patterns for each sample. 37 refinements from the diffraction data were well-matched to the rhombohedral structure in space group of R3̅m. The weighted residual (WRP) indicates the residual difference between observed and calculated intensities. The results agreed with the reported XRD data for BSTS 3D TIs [81,80]. Figure 2.3(b) compares the characteristic diffraction peaks of BSTS type chalcogenide [80] located at 2θ ~32.5o and 36.5o for MG, BG, and MBG BSTS. These ̅̅̅) crystal planes, respectively, where their peaks are corresponding to the (107) and (0012 appearances indicated the occupation of Se atoms in the center of the quintuple layer in our ̅̅̅) crystal planes BSTS [80,86]. The peak intensities of their corresponding (107) and (0012 have been used to characterize the Bi(Sb)/Te antisite and Se vacancy defects in the BSTS ̅̅̅) diffraction peaks in MBG crystal [78,80,85]. The higher intensities of (107) and (0012 BSTS indicated the highly-ordered structure with relatively low composition-related defects compared to the MG and BG BSTS. The integrated peak intensity ratios of (107) ̅̅̅) referred to (018), labeled as I(0012 and (0012 ̅̅̅) /I(018) and I(107) /I(018) , respectively, for the three types of BSTS samples are compared quantitatively and plotted in Figure 2.3(c). To further study the mosaicity of the BSTS crystal, we performed XRD in  scan mode as shown in Figure 2.4. Structure solution and refinement (as presented in Table 2.2) were performed with the use of the SHELXTL (version 6.12) program package. Faceindexed numerical absorption corrections were applied, since geometry of the crystal was 140 × 90 × 20 μm3 plate (Figure 2.4(a)). From the Laue symmetry, the rhombohedral space group R3̅ m was deduced. The structure type, as expected, was found to adopt Bi2Te3-type structure. Atomic mixing between Bi–Sb and Te–Se pairs was considered. Refined occupancies are within a reasonable agreement with the loaded composition, considering 38 Figure 2.4. Single crystal X-ray diffraction. (a) SEM image of a single crystal BSTS flake. (b) rocking curve, -scan of 006 reflection. 39 Table 2.2 Crystallographic data for BiSbTeSe2. Formula (refined occupancies) Formula mass (amu) Space group Bi1.12Sb0.88Te1.36Se1.64 616.25 R3̅ 𝑚 (No. 166) a (Å) c (Å) V (Å3) Z 4.187(2) 29.648(16) 450.0(4) 3 6.822 calcd (g cm–3) T (K) Crystal dimensions (mm) 273(2) (Mo K) (mm ) 0.14  0.09  0.02 50.530 Transmission factors 0.0553–0.4632 2 limits Data collected 11.334–61.233 –1 No. of data collected No. of unique data, including Fo2 < 0 No. of unique data, with Fo2 > 2(Fo2) No. of variables R(F) for Fo2 > 2(Fo2) a Rw(Fo2) b Goodness of fit ()max, ()min (e Å–3) –6  h  5, –6  k  6, –37  l  41 923 201(Rint = 0.0309) 179 12 0.0471 0.1043 1.112 3.994, –1.816 R(F) = ∑||Fo| – |Fc|| / ∑|Fo|. b Rw(Fo2) = [∑[w(Fo2 – Fc2)2] / ∑wFo4]1/2; w–1 = [σ2(Fo2) + (Ap)2 + Bp], where p = [max(Fo2,0) + 2Fc2] / 3. a 40 highly anisotropic geometry of the crystal, which greatly affects intensity data. Residual density is within typical values, considered heavy element composition. Mosaicity of the diffracted crystal is found to be 0.73°. Rocking curve plot for the -scan of 006 reflection shows FWHM to be 0.88° (Figure 2.4(b)). The value of the FWHM for 006 reflection is used to compare crystallinity of the sample, and 0.88° result is comparable to nonepitaxially grown Bi2Te3 crystals [87]. The value of FWHM is rather low, considering high atomic disorder within the BiSbTeSe2 sample. Furthermore, the atomic coordination, Wyckoff, and site occupancy of the MBG BSTS obtained from the Rietveld refinement of the powder XRD pattern are analyzed in Table 2.2. BSTS has the tetradymite structure unit, consisting of the stacked Te(Se)1Bi(Sb)1-Te(Se)2-Bi(Sb)1-Te(Se)1. Since Se atom has more electronegativity than Te atom, the Se atom tends to occupy the central layer in the BSTS structure unit. The almost unity of Se2 occupancy (and much smaller Te2 occupancy) as presented in Table 2.3 clearly elucidate this effect. 2.3 Electrical Transport The BSTS flakes were exfoliated by using scotch tape on a polydimethylsiloxane (PDMS, Sigma-Aldrich Co.) substrate and transferred onto the prepatterned gold leads in a Hall bar configuration. Three representative devices fabricated from the BSTS crystals grown by MG, BG, and MBG methods were compared. The thicknesses of the MG, BG, and MBG BSTS devices are about 120 nm, 80 nm, and 70 nm, respectively. The transport measurements were carried out at various temperatures from RT down to 1.5 K, and magnetic field up to 9 T. The Hall measurements were performed by an SR830 DSP lock- 41 Table 2.3 Atomic coordination for BiSbTeSe2. Wyckoff x y z Ueq (Å2) a U11 = U22 position (Å2) Bi 1 0.56(6) 6c 0 0 0.39654(4) 0.0259(5) 0.0206(5) Sb 1 0.44(6) 6c 0 0 0.39654(4) 0.0259(5) 0.0206(5) Te 1 0.65(8) 6c 0 0 0.21296(7) 0.0284(6) 0.0228(6) Se 1 0.35(8) 6c 0 0 0.21296(7) 0.0284(6) 0.0228(6) Te 2 0.06(8) 3a 0 0 0 0.0194(9) 0.0170(10) Se 2 0.94(8) 3a 0 0 0 0.0194(9) 0.0170(10) a Ueq is defined as one-third of the trace of the orthogonalized Uij tensor. Atom Occupancy U33 (Å2) 0.0365(8) 0.0365(8) 0.0395(12) 0.0395(12) 0.0241(17) 0.0241(17) 42 in amplifier, operating at frequency of 17.777 Hz and the constant AC excitation current of 100 nA. The DC gate voltage (Vg) was applied to the gate electrodes by a Keithley 2400 source measure unit. 2.3.1 Temperature-Dependent Transport The electrical transport properties of the BSTS were studied to compare the quality of the crystals grown by different methods. Insets in Figure 2.5(a), (c), and (e) show the typical BSTS devices made by mechanical exfoliation and transfer onto prepatterned electrodes on a Si/SiO2 substrate. The gate-dependent four-probe resistances (Rxx) of the BSTS measured at RT and 1.5 K are compared for the MG, BG, and MBG BSTS samples. All three BSTS samples display a broad resistance change over the Vg range at RT. The MG BSTS shows a larger change at positive Vg and nearly constant at negative Vg; while the BG and MBG BSTS reveals a clear ambipolar signature at RT. The broad resistance peaks indicate a combination of bulk and surface conductions. The bulk contribution is more significant in the hole-conduction (constant Rxx region in negative Vg) in MG BSTS. As the samples cooled down to base temperature (1.5 K), a sharp resistance peak was revealed in all samples. The distinct resistance peak is attributed to the surface transport due to the Dirac dispersion nature of the TSS as evidenced by angle-resolved photoemission spectroscopy in the stoichiometric BiSbTeSe2 compound [56]. To further investigate the temperature-dependent transport, the color maps of the Rxx as a function of temperature and Vg for MG, BG, and MBG BSTS are shown in Figure 2.5(b), (d), and (f), respectively. For MG BSTS, the Rxx increased as the temperature decreases, and the Rxx peak maximized at ~30 K. This insulating behavior is attributed to 43 Figure 2.5. Gate-dependent resistances of MG (a), BG (c), and MBG (e) grown BSTS measured at RT and base temperature of 1.5 K. Optical images of the corresponding BSTS devices are inserted in (a) and (c) (scale bar= 50 m). 2D color plots of the Rxx as a function of temperature and Vg for MG (b), BG (d), and MBG (f) grown BSTS samples. 44 the bulk dominating conduction due to its insulating nature (bulk gap). The insulating trend terminated at about 30 K, suggesting the suppression of the BSTS bulk conduction at the temperature [88]. A gradual decrease in Rxx was observed by further reducing the temperature. As the surface states are gapless, this metallic behavior indicates prominently surface contribution to transport in this temperature range [88,89]. A similar temperature dependence profile was observed for the BG and MBG BSTS with resistance peak reaches maximum at ~50 K and ~100 K, respectively. To provide quantitative analyses on the bulk and surface conduction, we fit the four-probe resistivity (ρxx) as a function of temperature for all three devices (as shown in Figure 2.6(a)) using a parallel conductance model [56] as: 1 ρxx = G = G bulk +Gsurf tot Gbulk = 1 t ∆ (2.1) (2.2) ρ3D ekB T Gsurf = ρ 1 2D +AT (2.3) where Gtot, Gbulk, and Gsurf are the total, bulk, and surface conductance, respectively. The Gbulk and Gsurf are calculated from the relations (2.2) and (2.3), where t, ∆, A, ρ2D, ρ3D, and kB are the thickness, activation energy, parameter for electron-phonon scattering, sheet resistance due to impurity scattering, bulk resistivity, and Boltzmann constant, respectively. Figure 2.6(b) shows that the Gbulk is negligibly small, and the Gsurf is dominating the transport at a temperature below ~100 K for all the BSTS samples. Additionally, all three BSTS devices showed a visible shift in Rxx peak position with temperature near the region when the Rxx reaching its maximum. This implied a shift of chemical potential happened as the bulk conduction was suppressed and total conduction 45 Figure 2.6. Temperature dependent conduction. (a) Plots of ρxx as a function of temperature for the MG, BG, and MBG BSTS at Vg away from Dirac point (high charge density regions). The hollow dots are the experimental data, and the solid lines present the fitting of ρxx from a parallel conductance model as described in the main text. (b) The Gsurf to Gtot ratio (Gsurf/Gtot) as a function of temperature for the MG, BG, and MBG BSTS. 46 developed into the surface conduction [89,90]. 2.3.2 Surface Mobility Study The low-temperature magnetoelectric transport properties of three types of BSTS samples, grown by MG, BG, and MBG methods, are compared in Figure 2.7. Figure 2.7(a) shows the ρxx of the MG, BG, and MBG BSTS devices as a function of Vg measured at base temperature (1.5 K) and zero magnetic fields. Both BG and MBG BSTS reveal sharper resistivity peak and greater change in ρxx compared to the MG BSTS. This indicates a better quality of the BSTS grown by vertical Bridgman furnace. The resistivity peak width of the MBG is about 33% narrower than the BG BSTS. This is consistent with the observation of the narrower peak width from the XRD analyses. The Hall resistance (Rxy) plots of the BSTS devices as a function of Vg measured at 2 T are compared in Figure 2.7(b). The sign change in Rxy at the Dirac point confirms the ambipolar charge transport for all the BSTS devices. The Rxy of MBG BSTS (~2.7 kΩ/T) and BG BSTS (~2.0 kΩ/T) develop much faster than MG BSTS (~0.3 kΩ/T) in magnetic field. The Rxy of BG BSTS develops slower in the electron conduction region, while the MBG BSTS shows higher symmetry in both hole and electron conduction regions. Hall mobility of the BSTS devices was calculated using the relation as: μH = ρxy 1 ρxx B (2.4) from the linearly increasing region of Rxy versus B (0–2 T). Figure 2.7(c) and (d) compare the μH of the three BSTS devices at different charge density regions tuned by Vg. The μH of the MG BSTS maximized at ~1000 cm2/Vs in a low electron density region. Both BG and MBG BSTS samples showed significantly higher μH , which are about 3800 and 4400 47 Figure 2.7. Gate tunability and mobility. (a) xx and (b) Rxy of the MG, BG and MBG BSTS as a function of Vg measured at magnetic field of (a) 0 T and (b) 2 T. (c) Hall mobilities of the three methods grown BSTS devices at different carrier densities at base temperature of 1.6 K. (d) Hall mobilities as a function surface charge density for BSTS grown by three different methods. 48 cm2/Vs, respectively, in low hole density region. The mobility of our BG BSTS is comparable to values reported in the literature [56,88]. However, the μH in the electron region of BG BSTS is about five times smaller than the hole conduction region. The electron mobility of the BG BSTS is markedly improved by recrystallizing the single crystal, as revealed by MBG BSTS. The extremely high surface mobility in MBG BSTS is attributed to the enhanced crystallinity of its parent crystal as identified by XRD. 2.3.3 Quantum Transport The quantum magneto-transport of the three BSTS devices are compared in Figure 2.8. The ρxy, σyx, and ρxx, σxx of the MG, BG, and MBG BSTS as a function of a magnetic field are presented in Figure 2.8(a) and (b), respectively. For MG BSTS device, the ρxy and ρxx increase monotonically with magnetic field, which indicates that the surface states are in the normal Hall effect regime. For BG BSTS, the ρxy approaches the quantum limit (25.8 kΩ) of the integer QHE at 9 T, together with the decreasing of ρxx at magnetic field above 4 T. Meanwhile, the ρxy of MBG BSTS reaches QH regime at magnetic field about 7 T as indicated by the saturation of ρxy and the suppression in ρxx. The σyx are plateauing at +1e2/h at about 7 and 5 T for BG and MBG BSTS devices, respectively, as shown in Figure 2.8(a). The corresponding σxx (Figure 2.8(b)) vanishes to <0.1e2/h. The development of IQHE in both BG and MBG BSTS suggests a better-quality BSTS crystal grown by Bridgman as compared to melting method. To further confirm the IQHE in the BG and MBG BSTS, we investigate the quantization states by controlling the Vg. Figure 2.8(c) and (d) present the σxy and σxx plots of the MG, BG, and MBG BSTS as a function of Vg. The MG BSTS shows no sign of σxy 49 Figure 2.8. Quantum Hall effect. Plots of the (a) ρxy and σyx, and the (b) ρxx and σxx as a function of a magnetic field for the MG, BG, and MBG BSTS devices. We noted that the MG BSTS is plotted in ρxy and σxy as its electron mobility is higher than the hole mobility. The (c) σxy and (d) σxx as a function of Vg at 9 T and 1.5 K for three methods grown BSTS devices. 50 plateau formation at any integer and the σxx is far from vanishing (>2e2/h). Whereas the BG BSTS displays clear LL formation in σxy with filling factor, ν= -1, 0 and +1 by tuning the gate near the Dirac point. The LL states show different minima in σxx as indicated by the arrows. The IQHE has been observed in BG BSTS by several groups [36,56,88]. The IQHE is explained as a combination of half-integer QHE arising from the top and bottom surface states with the LL filling factor (ν) follows the relation (1.12). CHAPTER 3 TOPOLOGICAL INSULATOR-BASED VAN DER WAALS HETEROSTRUCTURES The realization of exotic phenomena in TSS requires fine control of the chemical potential near the Dirac point [91,92]. This can be achieved effectively by performing electrostatic gating using the highly configurable vdW platform [93]. The 2D insulator hexagonal boron nitride (hBN) serves as a better gate dielectric owing to its relatively inert, surface-charge-free and atomically-flat surface [94]. The substrate-induced disorder due to hBN has been measured in graphene to be an order magnitude smaller than from SiO2 [95]. Another interesting set of candidates is the ferromagnetic insulators (FMI) as they can possibly break time-reversal symmetry (TRS) on the TSS by exchange coupling proximity effect [60]. The layered chalcogenide Cr2Ge2Te6 (CGT) can couple well with Bi-based TIs as it exhibits similar crystal structure and small lattice mismatch from the Bi-based TI [34]. Moreover, the out-of-plane magnetization in CGT can provide a stronger ferromagnetic exchange coupling to the TI surface [34] than most oxide and sulfide-based in-plane magnetization FMIs. More recently, graphite (Gr) gates have been used to reduce charge inhomogeneity in case of graphene devices [96]. Despite these advantages, 2D materials have not been widely explored in conjunction with TIs. This chapter presents BSTS-based vdW heterostructures in TI/insulator/Gr configuration, using hBN and FMIs as gate 52 dielectrics and Gr as the gate, to achieve and control high-quality TI surface states. 3.1 Device Fabrication The device fabrication process by mechanical transfer technique is illustrated in Figure 3.1(a). To effectively induce charges on the surface of the BSTS, our devices are typically constructed from ≤20 nm hBN as dielectric and Gr layer as gate electrode. The transfer is carried out carefully at a temperature close to the vaporization temperature of water (~95oC) to avoid the water vapors trapping between hBN and Gr layers, which can adversely affect electrical transport in BSTS. Atomic force microscopy image as shown in Figure 3.1(b) reveals the relatively flat and clean vdW assembly of the BSTS/hBN/Gr device. A step height trace over the device shows the thicknesses of each layer. 3.1.1 Interface Study of the vdW Layers Transmission electron microscopy (TEM) was utilized to investigate interfaces of the heterostructures. A cross-sectional TEM micrograph of a representative BSTS/hBN/Gr stack is shown in Figure 3.2(a), where the layers are clearly revealed by contrast. Atomically sharp interfaces are formed at the boundaries, signifying good interface quality between the layers. This confirms the formation of highly ordered heterostructures using our dry transfer method. The quintuple layer structure of the BSTS is visibly resolved in the high-resolution TEM micrograph recorded at the interface of BSTS and hBN (Figure 3.2(b)). The lattice spacings of hBN and BSTS quintuple layers are determined from the contrast line profile taken across the adjacent lattice planes (Figure 3.2(c)). EDX elemental mapping analyses (Figure 3.2(d)) confirm the cleanliness of each layer. 53 Figure 3.1. Device fabrication. (a) Schematic of the mechanical transfer of 2D materials on prepatterned electrodes. (b) Atomic force microscopy image of the BSTS/hBN/Gr stack device and the height profile along the horizontal axis. 54 Figure 3.2. Transmission electron microscopy analysis. (a) Cross-section high-resolution TEM micrograph of a mechanically transferred BSTS/hBN/Gr stack. (b) Magnified TEM micrograph at the BSTS and hBN interface (marked as white frame). (c) Contrast line profile along the perpendicular plane across the BSTS and hBN interface. (d) EDX mapping images of the BSTS/hBN/Gr stack. 55 3.1.2 Electrical Transport Figure 3.3(a) shows the optical image of a typical dry transferred BSTS/hBN/Gr device. The Rxx of the BSTS as a function of top-gate voltage (Vtg) applied through the Gr electrode is shown in Figure 3.3(b). The BSTS device exhibits ambipolar transport with resistance maximum of ~9.3 kΩ located at charge neutrality point (CNP) at Vg= -3.3 V, indicating the top surface is electron-doped. The ambipolar resistance is a signature of surface transport in a TI. The induced charge density on the top BSTS surface is calculated as: n= Cg|Vg–VD| (3.1) where Cg is the gate capacitance (≈ 140 nF/cm2 for 18 nm hBN). This configuration allows us to induce surface charge density >3×1012 /cm2 in electron conduction. In the highdensity regime, the Rxx falls sharply, resulting in a change of ~7 and ~4 kΩ in electron and hole conduction regions, respectively. The significant resistance change of the BSTS indicates an effective field-effect gate tuning due to the good coupling of the 2D materials. Inset in Figure 3.3(b) compares field-effect mobility (μFET) of the top and bottom surface states gated by hBN/Gr and SiO2/Si, respectively. The BSTS gated by hBN/Gr displays a clear enhancement in μ compared to the SiO2/Si gate over the range of charge densities. 3.2 Topological Insulator/Normal Insulator 3.2.1 Dual-Gated Transport The quality of our BSTS/hBN/Gr devices is confirmed by magneto-transport measurements. Our device configuration allows us to apply a Vtg and Vbg through the hBN and SiO2 dielectrics, respectively, to individually modulate the charge densities on top and 56 Figure 3.3. VdW heterostructure device. (a) Optical image of a transferred BSTS/hBN/Gr device with the top-gate measurement scheme. (b) Rxx and xx as a function of Vtg applied between BSTS and Gr layers measured at cryogenic temperature of 1.6 K. Inset in (b) is the μFET versus n plots extracted from the hBN/Gr and SiO2/Si gate-dependence resistivity. 57 bottom surfaces. Figure 3.4(a) shows the 2D color map of the σxx of a BSTS dual-gated device as a function of dual-gate voltages measured at zero magnetic fields. The Dirac fermions in top and bottom surface states are tuned independently to the chemical potential near the (top and bottom) Dirac cones. The conductivity minima of the top and bottom surfaces intersect at Vtg= -3.3 V and Vbg= +3 V, which corresponds to the overall center CNP resulting from the top and bottom TSS combined. The color map can be divided into four quadrants with hole-hole (h-h), electron-electron (e-e), hole-electron (h-e), and electron-hole (e-h) conduction regions for charge carriers in top-bottom TI surfaces. 3.2.2 Two Surfaces Integer Quantum Hall Effect With an external magnetic field of 9 T applied perpendicular to the device, the σ xx (Figure 3.4(b)) develops into four well-resolved quadrants (framed by the dashed white line) with uniform rectangular areas forming near the Dirac points of the top and bottom surfaces. In both h-h and e-e conduction regions, the σxx reaches a minimum value < 0.01 e2/h. Vanishing of σxx is an indication of the LL formation in the TSS. This is confirmed by the corresponding quantization in Rxy in the same parameter spaces as shown in the color map in Figure 3.4(c). The Rxy forms perfectly symmetric plateaus saturating ~-25 kΩ (~+25 kΩ) in the h-h (e-e) conduction. The Rxy plot (red curve in Figure 3.4(e)) along the dashed red line (i) in Figure 3.4(c) is extracted by plotting it as a function of sum of the top and bottom surface charge densities, where the quantized Rxy= ±h/νe2 are labeled. The fully quantized Rxy= ±h/e2 is highly antisymmetrical in opposite magnetic field of -9 T (Figure 3.5). Conversely, the Rxx develops oppositely in the h-e and e-h conduction regions. Instead of vanishing, the Rxx enhances toward maximization (refer to the Rxx dual-gates color map) 58 Figure 3.4. Quantum transport of the BSTS/hBN/Gr dual-gated device. 2D map of the σxx of the BSTS as a function of Vtg and Vbg at magnetic field of (a) 0 T and (b) 9 T at cryogenic temperature of 1.6 K. (c) Rxy of the BSTS as a function of Vtg and Vbg at magnetic field of 9 T at temperature of 1.6 K. The horizontal and vertical dashed lines in (a), (b), and (c) denote the top and bottom Dirac points, respectively. (d) σxy map of the BSTS as a function of dual-gate voltages. The dashed line rectangles in (b), (c), and (d) mark the QH plateaus formed near the center Dirac point. (e) Line profiles of the Rxy extracted at (i) nb–nt=0 and (ii) nb+nt=0. (f) Renormalization group flow diagram (σxy versus σxx) plots of the BSTS gated by hBN (top-gate) and SiO2 (bottom-gate) dielectric layers. 59 Figure 3.5. Magnetic field dependent QHE. (a) Rxx & Rxy, and (b) σxx & σxy as a function of a magnetic field for the BSTS/hBN/Gr device. The Rxx (σxx) and Rxy (σxy) are highlysymmetrical and antisymmetrical, respectively, in opposite magnetic fields. The σxy approaches quantum limit of +e2/h (-e2/h), together with the vanishing in σxx, at magnetic field greater than +6T (-6T). 60 in conjunction with suppression in Rxy in the region. Line profile of Rxy across the e-h and h-e regions (dashed black line (ii) in Figure 3.4(c)) shows two average-to-zero (-0 and +0) plateaus in the low-density regime, as plotted in Figure 3.4(e). These dissipationless and dissipative QH states in 3D TIs have been observed in the literature [56,57,88]. To further analyze the QH states, we plot the σxy as a function of dual-gate voltages in color scale as depicted in Figure 3.4(d). The rectangular σxy plateaus correspond to the addition of LL indices of the top (Nt) and bottom (Nb) surfaces are indexed in the color plot. The IQHE with LL filling factors (ν= νt+νb) of ν= 0, ±1 and ±2 is resolved in the figure. In addition, we compare the quantization on the top and bottom surfaces by extracting the σxx and σxy with Vtg (Vbg) swept at Vbg (Vtg) fixed to the bottom (top) Dirac point. Figure 3.4(f) illustrates the renormalization group (RG) flow [97] (σxy versus σxx) plots of the top and bottom quantized TSS. As the top and bottom surfaces are tuned separately by hBN and SiO2 dielectric, the quantized states are determined solely by the quality of the dielectric materials. The σxx of the QH states for the surface with hBN dielectric is suppressed to a greater extent, and the σxy is closer to the quantized value, as compared to the surface with SiO2 dielectric. This is indicative of better development of the QH state using hBN in comparison with the higher disorder SiO2 dielectric. 3.3 Topological Insulator/Ferromagnetic Insulator 3.3.1 Ferromagnetic Insulator as Gate Dielectric The well-developed QHE in BSTS/hBN/Gr heterostructures at low magnetic field is practical for advanced studies owing to the effectiveness of the vdW platform for TIs. We further study electrical transport in BSTS interfaced with FMI by replacing the normal 61 insulator hBN with a 2D FMI. CGT appears to be an appropriate FMI candidate as it can be mechanically exfoliated and integrated with BSTS devices. Kerr rotation of circularly polarized light from the CGT flakes measured using Sagnac interferometry confirms the ferromagnetic phase at temperatures below 60 K (Figure 3.6). Temperature-dependent resistance plot (Figure 3.7(a) and (b)) shows the insulating behavior of CGT with resistance exceed GΩ at cryogenic temperatures. To ensure that CGT is a good dielectric at low temperature, we check the conductivity of CGT at high bias voltage and gate-voltage as presented in Figure 3.7(c) and (d), respectively. No turn-on current signal is detected, indicating that CGT can be used as a dielectric. 3.3.2 Anomalous Hall Effect Coupling of CGT ferromagnet can induce magnetism in BSTS via proximity effect. To verify this effect, we performed Hall measurement analysis at a low magnetic field on a BSTS/CGT/Gr device, as presented in Figure 3.8. A nonlinear slope at low magnetic field range of -0.3 to +0.3 T (antisymmetrical in opposite field) is observed in the Rxy for the BSTS/CGT/Gr device (Figure 3.8(c)). As comparison, the control measurement performed on a BSTS/hBN/Gr device shows a linear field-dependent Rxy at low magnetic field. We attribute this to the AHE resulting from the proximity induced magnetization in BSTS, which is consistent with the AHE realized in (BixSb1-x)2Te3/YIG system [98]. The switching of the Rxy to the ordinary Hall effect (OHE) at magnetic field >0.3 (and <-0.3) T agrees with out-of-plane (easy axis) saturation field of CGT measured from SQUID measurement [99]. In addition, we note that the nonlinearity in Rxy is insensitive to the Vg (Figure 3.8(b)). The ∆Rxy (obtained from the difference between Rxy in the BSTS/hBN and 62 Figure 3.6. Kerr rotation studies of a CGT flake. The magnetic properties of an 80 nm thick CGT flake (a) were studied using a Sagnac interferometer at cryogenic temperature. 2D spatial mapping of Kerr rotation angles (θKerr) for the CGT device measured at (b) 0 G, (c) ~+200 G, and (d) ~-200 G out of plane magnetic field. The large θKerr detected and the antisymmetrical of θKerr in the opposite magnetic field confirm its ferromagnetic behavior. 63 Figure 3.7. Transport properties of a single crystal CGT flake. (a) Source-drain current (I) of a 120 nm CGT device (shown in the inset) as a function of temperature, measured at a fixed source-drain voltage of +100 mV. (b) The resistance (R) of the CGT device as a function of temperature shows the insulating behavior with R exceeding 2 GΩ at temperature below 20 K. (c) The I versus V of the CGT device measured at temperature of 4 K. Negligible I detected in the range of Vsd applied (±5V), showing the truly insulating of the CGT at cryogenic temperature. (d) I as a function of Vg applied through the Si/SiO2 gate. Negligible I detected indicates that the CGT is in the electronic bandgap region within the range of Vg applied. 64 Figure 3.8. Anomalous Hall effect of a BSTS/CGT/Gr device. Color maps of the Rxy of the (a) BSTS/hBN/Gr, and (b) BSTS/CGT/Gr devices as a function of a magnetic field and Vbg. The OHE and AHE regions were labeled in the figure. (c) Comparison of the Rxy at low magnetic field (±0.5 T) for the BSTS/hBN/Gr and BSTS/CGT/Gr devices. The different slope of Rxy versus B in BSTS/CGT/Gr device plot at low magnetic field range of about ±0.3 T is a result of the AHE. (d) The difference between Rxy of BSTS/hBN/Gr and BSTS/CGT/Gr (∆Rxy) as a function of magnetic field. 65 BSTS/CGT devices) ascribed to the AH conduction saturates at ~0.1 kΩ (-0.1 kΩ) (Figure 3.8(d)). However, we also note that the AHE observed in our system is qualitatively less profound compared to the data reported in reference [34] for the epitaxial grown Bi2Te3 on the freshly cleaved CGT surface. This might also indicate the lattice alignment between 3D TI and ferromagnet is critical to achieving the optimum magnetic coupling effect. 3.3.3 Magnetic Gap In theory, the TSS in TIs can be gapped by proximity coupling to the FMI. Thus, electrostatic gating using CGT as a dielectric to tune the chemical potential of the gapped TSS is highly desirable. Using dual-gated BSTS/CGT/Gr devices, we have achieved ambipolar transport using CGT dielectric and Gr gate (Figure 3.9(a)). The gate-dependence Rxx using CGT dielectric with thickness ~250 nm reveals similar gating behavior as the 18 nm hBN, suggesting that the dielectric constant of CGT is about 20 times greater than hBN. Moreover, we noted that the gate leakage current is negligibly small (< 1 nA due to the sensitivity of our voltage source meter) in the applied Vg range for both the CGT and hBN gate-dielectrics. Similar to BSTS/hBN/Gr devices, the Rxx can be tuned into four quadrants (Figure 3.9(b)) and magnetotransport develops into well-formed QH states in an external magnetic field of 9 T. However, in contrast to typical BSTS/hBN/Gr devices (Figures 3.10(c) and 3.11(a)), the four-quadrant QH plateaus in BSTS/CGT/Gr devices at the center CNP (Figures 3.10(d) and 3.11(b)) are highly asymmetric, with the quantization of the top surface state between Nt= -1 and 0 LLs extended over a longer range in Vtg compared to the plateau between Nt= 0 and 1 LLs. This asymmetry is attributed to a gap opening due to TRS-breaking in the surface state by magnetic proximity effect [100]. The corresponding 66 Figure 3.9. Gate-dependent resistance of a BSTS/CGT/Gr device. (a) Rxx of the BSTS as a function of Vtg applied through a CGT dielectric (thickness ~250 nm) measured at a temperature of 1.6 K. Inset in (a) is the optical images of the device. (b) The color plot of the Rxx of the BSTS as a function of dual-gate voltages with the top-gate as CGT/Gr and back-gate as SiO2/Si. Similar to the BSTS/hBN/Gr device, the top and bottom surface states can be tuned into the four-quadrant conductions as labeled in (b). 67 Figure 3.10. Estimation of the magnetic gap from dual-gated transport. Dual-gated color maps of Rxx for (a, c) BSTS/hBN/Gr_2, and (b, d) BSTS/CGT/Gr_2 devices measured at magnetic field of 0 and 9T. The thickness of the hBN and CGT layers is about 20 nm and 200 nm, respectively. Different from the hBN/Gr gating, the top surface tuned by CGT/Gr shows an irregular LL spacing between Nt= -1 to 0 and Nt= 0 to +1. The magnetic exchange gap (∆m) induced by the 2D ferromagnet CGT is evaluated from the difference between E0,-1 and E1,0 LL spacing. The ∆m is estimated to be about 18 meV. 68 Figure 3.11. Gating effect and magnetoelectric transport of the BSTS/CGT/Gr device. 2D maps of the Rxx as a function of the Vtg and Vbg at the magnetic field of 9 T for (a) BSTS/hBN/Gr and (b) BSTS/CGT/Gr devices. Schematic of the LL band diagrams of the top gapless TSS and gapped TSS corresponding to the Rxx maps in (a) and (b), respectively, are shown on the right side. (c) σxx and σxy of the BSTS as a function of Vtg (Vbg) at Vbg= -43 V (Vtg= -2.7 V) and magnetic field of 9 T. Inset in (c) is the map of the σxy of the BSTS as the function of dual-gate voltages. The black and red arrow lines represent the Vtg and Vbg where the line profiles in (c) were obtained. The (Nt, Nb) are indexed in the color map. (d) Renormalization group flow plots of the BSTS gated by CGT and SiO2 dielectric layers. 69 energy band diagrams with the LLs formed in the linear and gapped Dirac dispersions are illustrated. An equal LL energy spacing between Nt= -1, 0 and +1 is observed when the chemical potential is tuned across the Dirac point by hBN/Gr gating. In contrast, for BSTS/CGT, the magnetic gap opened at the Dirac point causes an upshift [29,101] in Nt= 0 LL, which resides at the bottom of the conduction band [36,102]. As a result, the band separation between Nt= -1 and 0 LLs is larger than the Nt= 0 and +1 LLs due to the presence of magnetic ordering at the BSTS/CGT interface. This observation is consistent with theoretical calculations and experimental reports for magnetic-doped TI [29,101], magnetic-doped/undoped TI bilayer [102], and Co-decorated/BSTS [36] systems. For clean BSTS/CGT interfaces (no impurities or trap states), the magnetic-gap size (∆m) can be semiquantitatively estimated from the relation, ∆𝑚 = E0,-1 − E1,0 (3.2) where E0,-1 (E1,0) is the LL spacing of E0 & E-1 (E1 & E0). The E1,0 is calculated from the theoretical expression for LL energy as in relation (1.13) with N= 0, 1, and the Fermi velocity ~3.2×105 m/s [56]. The E0,-1 is evaluated from the asymmetry in LL spacing extracted from the gate-dependent magnetoresistance plots (Figure 3.10(d)). ∆m is evaluated from the asymmetry in LL spacing to be ~18 meV at 9T, which is comparable to the magnetic-doped TIs [103] and much higher than Zeeman spin-splitting ~1-3 meV [104], indicating the magnetic mass origin of this gap. 3.3.4 Half-Quantization Plateau The consequences of the gap are even more pronounced in the value of the quantization. Figure 3.11(c) (inset) displays a map of the σxy plateaus developed near the 70 center CNP of BSTS/CGT/Gr. Figure 3.11(c) shows the line profiles of σxx and σxy of the top surface as a function of Vtg (with Vbg tuned to hole conduction and fixed at Nb= -1 (or νb= -e2/2h) LL in the bottom surface state). The corresponding plots for the bottom surface are also included for comparison. Surprisingly, σxy of the magnetized top surface is quantized in steps of e2/2h. The corresponding RG flow plot in Figure 3.11(d) clearly illustrates half-quantization steps with total ν= 0, -½, and -1. In comparison, fixing the top surface state in the hole conduction region, the bottom Dirac surface state (gated using the SiO2/Si back-gate) shows the normal QHE with integer e2/h steps, similar to the observation in Figure 3.4(f). We note that half-quantization is an observable signature of the so-called parity anomaly of Dirac fermions [105]. A single half-quantized conductance value was observed recently in cobalt-decorated BSTS system [36]. However, their TI surface could not be gated due to screening from cobalt nanoparticles, preventing further exploration of this state. It is well-known that the existence of a true Hall plateau with half-quantized Hall conductivity is forbidden in a noninteracting system, as it violates gauge invariance. It is unlikely that electronic correlations play any significant role in our system, hence we are not dealing with a true half-quantized Hall plateau. This is signaled by the fact that the feature in σxy is accompanied by a substantial value of σxx (a fraction of e2/h), much larger than that for the true Hall plateau at σxy= -e2/h (see Figure 3.11(c)). The quantitative description of the transport properties of our sample necessarily considers both 2D bulk transport on topological surfaces, and edge conduction on the sample’s sides. The edge transport in the QAH effect on topological surfaces is complicated by the existence of chiral and nonchiral edge states [106]. In our system, further complexity is added by the 71 proximity to a ferromagnet (CGT), which strongly modifies the band structure of surface and 3D bulk states, and the importance of the disorder, as signaled by the fact that only a few lowest Hall plateaus corresponding to the largest cyclotron gaps are developed. However, at the qualitative level, it appears clear that the quasiplateau at σxy≈ -e2/2h develops close to the σtxy = -e2/2h → +e2/2h transition at the top surface, which corresponds to zeroth LL of the top surface crossing the chemical potential of the system, while the bottom surface is kept at σbxy = -e2/2h. The fairly constant value of σxx in the quasiplateau region points either to the importance of edge transport in that regime, or to the fact that (localized) 3D bulk states participate in accommodation of electric charge induced by the gate electrodes, which would further promote zeroth LL pinning to the chemical potential, and stabilization of the quasiplateau. CHAPTER 4 LANDAU LEVELS OF TOPOLOGICAL SURFACE STATES PROBED BY QUANTUM CAPACITANCE Quantum capacitance (CQ) is an ideal measure of the thermodynamic density of states (DoS) for low carrier density and gate-tunable materials [58,107,108]. CQ can also be exploited to study the LLs in TSS electronically [58,107-109]. Different from quantum transport, which is only sensitive to edge modes, CQ provides extra merit to probe the localized charge states in the surfaces. Nevertheless, it is relatively less explored in 3D TIs as compared to quantum transport primarily due to shortcoming of proper 3D TI candidates. Previous studies of CQ in Bi2Se3 reveals the Dirac-like surface states [110] and quantum oscillations in magnetic field [111]. However, the Bi2Se3 is known to exhibit substantial carrier doping in bulk, and extra trapped states induced by intrinsic defects [110,112]. More recently, CQ in high mobility 3D TI based on strained HgTe was reported [113], showing more intrinsic bulk band and clear LL quantization. Yet, the narrow bulk bandgap in strained HgTe (~20 meV), which is equivalent to lowest LL energy spacing at ~4 T, thus limits the LLs study at high magnetic field. Alternately, quaternary Bi-based TI compounds in the form Bi2-xSbxTe3-ySey exhibit a relatively large bulk bandgap (~0.3 eV) [56,91], and fully suppressed bulk conduction at low temperature, which serves as ideal candidates for probing the surface state LLs in CQ. 73 4.1 Capacitance Devices and Measurements 4.1.1 Device Configuration Figure 4.1(a) shows an optical image of a BSTS vdW five-layer heterostructures device for both quantum transport and capacitance measurements. The device studied here is made of a thin BSTS flake (~17 nm), where the bulk capacitance is large enough, yet the hybridization of the surfaces is insignificant. The devices are fabricated with Hall bar configured contact electrodes and encapsulated by hBN/Gr layers for both transport and capacitance measurements in a single device. 4.1.2 Low-Temperature Transport Figure 4.1(c) shows Rxx of the BSTS as a function of Vbg measured at 0.3 K. The Rxx peak at Vbg~-0.5 V is assigned to CNP, where the peak position indicates a lightly ndoped BSTS. By applying dual-gate voltages, the Rxx is controlled to its maximum, corresponding to the center CNP (interception of top and bottom CNPs) of both surfaces (inset in Figure 4.1(c)). The diagonal feature of Rxx center maximum indicates a strong capacitive coupling between the top and bottom surface states [92,114]. 4.1.3 Zero-Field Capacitance The BSTS devices based an assembly of BSTS/hBN/Gr sandwiched layers [115]. Such a configuration acts as three parallel plate capacitors in series, where chemical potentials of the TSS can be controlled by applying voltages through individual Gr layer. Thin hBN and BSTS flakes are used in device fabrication to achieve large geometry and TI bulk capacitances, such that the measured signals will be dominated by CQ. Figure 4.1(b) 74 Figure 4.1. Device for capacitance measurement. (a) Optical image of a vdW heterostructure of BSTS/hBN/Gr device fabricated on an undoped Si substrate. The scale bar is 10 μm. (b) Schematic diagram of the dual-gated quantum capacitance measurement. (c) Rxx, and (d) CS as a function of Vbg for the BSTS device measured at temperature and magnetic field of 0.3 K and 0 T, respectively. Insets in (c) and (d) are the 2D color maps of the Rxx and CS, respectively, as a function of dual-gate voltages. Color scales in (c) and (d) insets are in units of kΩ and fF, respectively. 75 shows a schematic of configuration for capacitance measurements. The top (bottom) surface of the BSTS forms a parallel plate capacitor with the top (bottom) Gr conducting layer, in which hBN acts as a dielectric layer. Symmetric capacitance (CS) signals of the top and bottom surface states are measured by applying two in-phase AC excitation voltages to the top and bottom Gr layers, and detecting the unbalanced current (δI) through a capacitor bridge [116]. Figure 4.1(d) presents the CS of the same BSTS device (Figure 4.1(a)) as a function of Vbg, measured in a separate cooling cycle. The CS is tuned to center minimum coincide with the Rxx peak at the CNP. The minimum value of CS is associated with the low DoS near the CNP [58,108,117]. The CS tends to saturate at large Vbg as the contribution of CQ to the total capacitance decreases at high charge densities. The dualgated map of CS inserted in Figure 4.1(d) shows the same diagonal feature as Rxx map. The diagonal line splits into two corresponding to CNPs in each of the parallel surfaces as the charge density tuned away from the overall CNP. 4.2 Landau Levels Development in Magnetic Field Figure 4.2 inspects the magnetotransport and magnetocapacitance of the BSTS device as a function of a magnetic field. The vanishing Rxx and quantized Rxy at integer of h/νe2 in magnetic field are a consequence of LL formation. The linear tracelines diverging from the CNP as shown in the Rxx and Rxy color maps signify a direct proportionality of magnetic field (B) versus bottom surface charge density (nb) relation. This is an indication of the Dirac dispersion of the LLs residing in TSS [3,82]. The slopes of the B versus nb are extracted from the Rxx color map as 3.5×10-11/ν T/cm2 (hole) and -3.3×10-11/ν T/cm2 (electron), which are close to theoretical relation of dB = dn h νe ≈ 4.1×10-11/ν T/cm2 [118,119]. 76 Figure 4.2. LLs development in magnetic field. Color maps of (a) Rxy, (c) Rxx, and (e) CS of the BSTS vdW heterostructure device as functions of magnetic field and Vbg measured at 0.3 K. The white dashed lines in (a), (c), and (e) trace the onset of the LLs about the Dirac point. (b) Rxy, (d) Rxx, and (f) CS curves as a function of Vbg at different magnetic fields. 77 To further confirm the origin of the formed LLs, we plot the LL index (N) versus 1/B curves at different gate voltages (Vbg-VD), and extract the y-intercept () as presented in Figure 4.3. The converging curves are linearly fitted and yield an  ~0.48±0.04, which provides strong evidence of π Berry phase arising from the Dirac surface states [59,120]. Consistent with transport, the CS develops into symmetric dips that coincide with the minima and plateaus in Rxx and Rxy, respectively, about the CNP. The sharp CS dips developed at the high magnetic field are a strong indication of the diminishing DoS due to large cyclotron gaps between the discrete LLs [58,108,117]. To further study the evolution of capacitance in the magnetic field, we plot the magnitude of Rxy, Rxx, and CS as a function of the magnetic field for ν= +1 and -1 LLs in Figure 4.4. A noticeable change in CS prevails as the Rxy approaches quantum limit of ±h/e2 and the Rxx diminishes to zero. The magnitude of CS gradually deviates from a linear function with a further increase in magnetic field, denoting a nonlinear proportionality of quantum capacitance to magnetic field. 4.3 Dual-Gated Magneto-Capacitance 4.3.1 Dual-Gated Magnetotransport Color maps of magnetotransport and magnetocapacitance of the BSTS device as a function of dual-gate voltages are compared in Figure 4.5. The dual-gating provides independent control of chemical potentials of the top and bottom surfaces. By tuning the two surface states across CNPs, four-quadrant rhombus-like plateaus develop around the center CNP. Rxx (Figure 4.5(a)) forms well-defined minima (maxima) in the parallel (counter)-propagating quadrants separated by the clear onset of the LLs. The boundaries around the center CNP are traced by dashed lines and labeled with the LL indices of top 78 Figure 4.3. LL fan diagram of topological surface states. (a) A plot of LL indices (N) as a function of the inverse magnetic field (1/B) for N minima of Rxx at different Vbg-VD. The dashed lines are linear fits of the Rxx minima. The y-intercept of the fits can be expressed as N±, where N= 0 at the intercept and a positive (negative) sign indicates electron (hole) LL. (b) A plot of  as a function of Vbg-VD from the linear fits in (a). The average value  is calculated to be ~0.48±0.04, indicating a π Berry phase originated from the Dirac TSS. 79 Figure 4.4. Magnetic field dependence quantum transport and capacitance. Plots of (a, d) Rxy, (b, e) Rxx, and (c, f) CS as a function of a magnetic field for the development of ν= +1 (black dots) and -1 (red dots) LLs. The dashed lines in (a) and (d) are quantum limit of Rxy at ±h/e2 for ν= ±1, while in (b) and (e) are the Rxx= 0. The dashed lines in (c) and (f) guide to the eye of the deviation of CS from linear with magnetic field for ν= +1 and -1 LLs. 80 Figure 4.5. Dual-gated magnetotransport and capacitance. Top: 2D color maps of (a) Rxx, (b) σxx, (c) σxy, (d) CS, (e) CA, and (f) δDS of the BSTS device as a function of dual-gate voltages measured at the temperature and magnetic field of 0.3 K and 18 T, respectively. Bottom: Line profiles of (a) Rxx, (b) σxx, (c) σxy, (d) CS, (e) CA, and (f) δDS as a function of Vbg extracted at the center CNP (Vtg= -0.3V) from the 2D maps. Labels and tracelines in the color map (b) represent the LL indices of the top (black) and bottom (red) surfaces. Dashed lines in σxx plots in (b) indicate zeros on the y-axis in each plot. Indices in color map (c) are the top (red) and bottom (black) LL filling factors (νt, νb). Color scales in (a), (b)-(c), and (c)-(e) are plotted in units of kΩ, e2/h and fF, respectively. 81 (Nt) and bottom (Nb) surfaces in σxx maps in Figure 4.5(b). σxy (Figure 4.5(c)) develops into rhombus-shaped QH plateaus at integer LL filling factors (ν), corresponding to the sum of half-integer of each surface (ν= νt+ νb = (Nt+½)+(Nb+½)), in the σxx minima regions. The four quadrants are assigned to two symmetric ν= 0 (white); and antisymmetric ν= -1 (blue) & +1 (red) QH plateaus. The σxy line profiles as a function of Vbg swept across the different Vtg reveal the quantized steps of height of e2/h as indexed in the curves in Figure 4.5(c) (bottom). 4.3.2 Symmetric and Antisymmetric Magneto-Capacitance The dual-gated quantum capacitance grants the access of LLs in the parallel paired TSS. The dual-gated CS reveals well-formed capacitance dips about the center CNP as shown in Figure 4.5(d). The rhombus-shaped capacitance dips coincide with the fourquadrant QH plateaus in σxy. The most prominent dips at lowest LL filling (Nb, Nt= +1 and -1) are clearly resolved in the color map. The line profiles of CS as a function of Vbg cutting through the three different Vtg show nearly identical capacitance dips emerging for the ν= 0 and ±1 plateaus, suggesting a similar cyclotron gap size formed in these states. An effective way to confirm the LLs induced CQ dips is by measuring the out of phase current signals, known as dissipation. Symmetric dissipation (DS) depicted in Figure 4.5(f) detects pronounced dissipative signals at the onset of LLs, and the signals vanish inside the LL gaps. Such Ds features were also observed in graphene LLs [96,116]. The relatively large background dissipative signals in capacitance are attributed to the resistive nature of the BSTS surface states in the entire density regions (refer to zero field transport data). By flipping the sign of AC excitation voltage applied to the top Gr layer, dual-gated 82 antisymmetric capacitance (CA) were measured (Figure 4.5(e)). The opposite phase AC excitation voltage is adjusted to null the signal at high charge densities. As the CA is probing the antisymmetric combination of the top and bottom surface capacitance, the CA color map shows the opposite sign of the background signal across the zero electrical polarization. The CA color map reveals an antisymmetric feature between ν= 0 (red) and ν= ±1 (blue) QH plateau states, which accounts for the difference between the quantum capacitances of the respective LLs. 4.4 Electronic Compressibility and Landau Level Spacings 4.4.1 Quantum Capacitance of Individual Topological Surface States The total capacitance (C) can be expressed as: 1/C= 1/Cg+1/CB+1/CQ (4.1) where Cg, CB, and CQ are geometric, BSTS bulk, and quantum capacitances, respectively [111,121,122]. Note that we neglect the parasitic capacitance from the measuring lines as the local Gr gates on resistive intrinsic Si substrate are used. The CB strongly depends on the thickness of the BSTS, which can be estimated from dual-gated Vtg versus Vbg plots using capacitor charging model [91]. The effective surface area of the BSTS flake of about 594 m2 (estimated from the optical image) yields a CB of ~23 pF, about two orders of magnitude larger than the measured C. Thus, we can disregard the contribution of CB in the total capacitance. The measured C in the form of CS and CA is equivalent to the sum and difference between capacitances of top (Ct) and bottom (Cb) surfaces, respectively. The Ct and Cb are solved algebraically from the CS and CA to further study the individual surface states. Subsequently, quantum capacitance of top (CQt) and bottom (CQb) surface states can 83 be extracted from the Cb and Ct using the relations as: 1/Cb(t)= 1/Cgb(gt)+1/CQb(Qt) (4.2) where the geometric capacitance (Cgb(gt)) is calculated from the parallel plate capacitor relation, Cg = εhBN εo A/dhBN (4.3) with the dielectric constant of hBN being fixed to 3. The CQb and CQt dual-gated maps are plotted in Figure 4.6(a) and (b), respectively. 4.4.2 Chemical Potential-Charge Density Relation CQ is related to electronic compressibility as [58,107-109]: dn CQ = Ae2 dμ (4.4) where A is the effective surface area, n and μ are the surface charge density and chemical potential, respectively. μ(n) can be obtained by integrating the electronic compressibility (dμ/dn) with respect to n [58,107]. In our calculations, we applied the charge density relation of the bottom surface (nb) based on the derivations in literature [91] as: C nb =Cbg ∆Vbg + (1+ Cbg ) Ctg ∆Vtg B (4.5) where the first term is the density controlled by the back-gate, and the second term encounters the capacitive couplings to the top-gate and top surfaces. The same derivation is used for the charge density relation of top surface (nt) by switching the back- and topgating terms. The chemical potential (μb) as a function of charge density (nb) of the bottom surface at different magnetic fields from 12 to 18 T is plotted in Figure 4.7(a). The jumps in μb around the integer Nb indicate the corresponding gaps between Nb= +1, 0 and -1 LLs 84 Figure 4.6. Individual surfaces quantum capacitance. 2D maps of (a) CQb and (b) CQt as a function of dual gate voltages. Black and white dashed lines in (a) and (b) trace the LLs of the bottom and top surface states, respectively. Line profiles of (c) CQb and (d) CQt as functions of Vbg and Vtg, respectively, for different LLs, ν= (i) 0± , (ii) +1, (iii) -1, and (iv) 0∓ (as labeled in the color map). 85 Figure 4.7. Surface states’ LL energy. (a) The chemical potential of the bottom surface state (μb) as a function of bottom surface density (nb) at different magnetic fields. The μb= 0 is manually matched to Nb= 0 LL. The full set of Nb is labeled in (a). (b) Plots of Eb versus magnetic field for Nb= +1 and -1 LLs. The dashed line in (b) is the calculated LL energy from the theoretical relation at the different magnetic fields, where the Fermi velocity is fixed at 3×105 m/s. 86 developed in the bottom surface state. The LL energies of bottom surface state (Eb) for LL indices of Nb= +1 and -1 as a function of magnetic field are extracted from the step heights and plotted in Figure 4.7(b). We note that Eb for B< 12T is omitted as the capacitance dips are not well-resolved due to overlap of the dips with the minimum DoS at CNP. The transport data (σxy) as a function of magnetic field for the corresponding v= +1 and -1 plateaus are included in Figure 4.8(c). The Eb approach the theoretical model of LL energy of TSS as expressed in relation (1.13) at high magnetic field as the σxy reaches the complete development of ±1 e2/h. The deviation of Eb from the Dirac LL energy equation at low magnetic field is attributed to the disorder broadening in the LL bands. To analyze this effect, we extracted the disorder broadening parameter, SdH= /q [123] from the quantum lifetime (q) based on temperature and magnetic field dependence Shubnikov-de Haas (SdH) oscillation amplitude analyses as presented in Figure 4.8. The obtained SdH of ~7.1 meV agrees to the LL broadening at low magnetic field. We note that the Nb= ±1 LL chemical potentials of ~40-42 meV are quantitatively more accurate compared to the thermal activation energy (~6 meV) of the same LL states at 31 T as reported by Xu et al. [56], and comparable to the localized DoS probed by scanning tunneling microscopy [118]. This again verifies the capability of CQ in quantifying the DoS of LLs in TSS. 4.4.3 Landau Level Spacings of Quantum Hall Plateaus We turn to discuss the LL-modified chemical potentials of two (top and bottom) surface states at a constant magnetic field. Figure 4.9(a) presents a color map of μb as a function of top and bottom charge densities at 18 T. The dual-density μb map is generated 87 Figure 4.8. Quantitative analysis of the disorder broadening in LLs of the topological surface states. (a) A plot of the normalized ∆ρxx(T)/ ρxx(0) as a function of temperature at magnetic field of 18 T and dual-gate voltages tuned to ν= +1 QH state. These data are fitted to the standard Lifshitz-Kosevich theory [54,124] and extracted the cyclotron mass (mc) to be about 0.047 me. (b) A plot of ∆ρxx(T)/ ρxx(0) versus reciprocal of magnetic field at base temperature ~400 mK. These data of In ∆ρxx(T)/ρxx(0) versus 1/B are linearly fitted, together with the mc obtained from (a) to estimate the q (~92 fs). The disorder broadening parameter (SdH) is extracted to be ~7.1 meV. (c) A plot of σxy versus magnetic field for the ν= +1 and -1 LLs at base temperature of 400 mK. (d) A plot of σxy versus temperature for the ν= +1 and -1 LLs at magnetic field of 18 T. Dashed lines in (c) and (d) are the guide to the eye to σxy= +1 and -1 e2/h. 88 Figure 4.9. LL energy spacings. (a) 2D color map of μb as a function of charge density of top (nt) and bottom (nb) surfaces. The dotted line in (a) represents the Nb= 0 LL. (b) Line profiles of μb as a function of nb at different nt, as labeled in (a) color map. Inset in (b) is a schematic of the energy (E) versus density of states (DoS) diagram for the respective LLs of the bottom surface. The LL energy spacing, ∆+1,0 =E+1 -E0 , is labeled in the figure. (c) Plots of Eb and Et versus Nb and Nt for ν=+1 (square), -1 (rhombus), 0± (triangle), and 0∓ (circle). Dashed lines in (c) represent the average energy of Eb and Et for different ν. (d) Plots of Et as a function B1/2 for different LL filling factors. The color lines are guided to the eye of the linear fit to LL energy relation. Inset in (d) is the Et as a function of temperature. 89 by integrating the reciprocal of CQb over nb for the entire nt range, with zero μb (white area) manually fixed at Nb= 0 LL (black dashed line) as drawn in the figure. We focus on the μb in charge density regions corresponding to the four-quadrant QH plateaus around the overall CNP. The μb versus nb data for LL filling factors ν= (i) 0± , (ii) +1, (iii) -1, and (iv) 0∓ (as labeled in the color map) are plotted in Figure 4.9(b). The notations of 0± and 0∓ are used for (νt, νb) of (+½,-½) and (-½,+½), respectively, to distinguish the two ν= 0 states. The comparable gaps for the 0± (0∓ ) and -1 (+1) states indicate that the chemical potential of bottom surface lies in the same LL i.e. Nb= -1 (+1). By separating Eb and Et from our dual-gated capacitances, we obtain the LL energies of individual surfaces at different QH states. The results of Eb (black dots) and Et (red dots) for the corresponding Nb and Nt are plotted in Figure 4.9(c). In ν= +1 (-1) states, the Eb and Et display the same sign for both Nb and Nt because of the chemical potentials of both surfaces reside in electron (hole) LLs. This is consistent with the origin of parallel-propagating states in conduction. Conversely, the Eb and Et reveal nearly equal magnitude and opposite sign for the ν= 0± (0∓ ) states due to the opposite occupation of Nb= -1 (+1) and Nt = +1 (-1). The LL energies from the top and bottom surfaces balance out and give rise to the ν= 0 states. In 3D TIs, the ν= 0 quantum states are a consequence of two parallel surfaces occupying opposite LL fillings, corresponding to a counter-propagating state in conduction. Magnetotransport shows that these are dissipative quantum states as indicated by finite Rxx (Figure 4.5(a)) and residual σxx (Figure 4.5(b)) in the states. The CA map (Figure 4.5(e)) captures the difference in quantum capacitance (or DoS) between the dissipative ν= 0 states and dissipationless v= ±1 QH states. To further analyze the difference between these states, we plot the Et as a function of √B for different ν states in 90 Figure 4.9(d). The Et is linearly fitted to the LL energy relation, which gives the Fermi velocity (vF ) of ~3.2-3.4×105 m/s for ν= ±1, and ~2.6-2.9×105 m/s for ν= 0 states. The slight difference in vF for the equivalent QH states could be related to the counterpropagating nature of the ν= 0 states, which enhance scattering between surfaces [88] and reduces the vF in the states. These vF of BSTS are consistent with the similar Bibased 3D TI compounds [40,56,119,125], despite the magnitude being smaller than vF obtained from angle-resolved photoemission spectroscopy [126]. In addition, Et as a function of temperature measured at 18 T for the different ν are presented in Figure 4.9(d). Et for ν= 0 and ±1 states decrease with increase in temperature and eventually reach the same energy level beyond 10 K, indicating a slightly different thermal activation between these states. The temperature dependence of Et is significantly different from the σxy of BSTS where the quantization is preserved up to 50 K as shown in Figure 4.8(d), consistent with literature [56]. The stability of QH states at higher temperatures suggests that the LL bands remain robust up to 50 K. Therefore, the large temperature response of Et in the lowest LL states could originate from a different origin such as midgap impurity states in TI surfaces [78] or possibly developing states, which are correlated to the smaller vF [127]. Nevertheless, higher quality BSTS devices are required to resolve the extra features in these LLs. CHAPTER 5 CAPACITIVE-COUPLING BETWEEN TOPOLOGICAL SURFACE STATES Time-reversal symmetry guarantees the pairing of top and bottom Dirac (linear dispersion) surface states in 3D TIs [2,128]. The half-integer QHE with LL filling factors vt,b=(Nt,b+1/2) for top and bottom surfaces is a signature of the Dirac surface states [56,57,129-131]. Fine control of chemical potentials of the paired top and bottom surface states has been central to research in 3D TIs [82,132,133]. Dual electrostatic gating is an effective method as it provides an additional degree of tuning on both surfaces than a single gate configuration [88,91,92,134,135]. Independent gate control of the decoupled top and bottom surface states in the QH regimes has been reported [88,134]. However, the study of the QHE in capacitively-coupled surface states, which serves as the starting point for intriguing quantum states such as topological exciton condensates [73,75], is still lacking. Several groups have demonstrated dual-gate control of the bulk insulating Bi2-xSbxTe3-ySey based 3D TI in weak and moderate surface-state coupling regimes [91,92,135]; however, the quality of those devices prevented the observation of QH states. This chapter investigates the effect of capacitive coupling of paired TI surface states as functions of TI flake thickness and charge density in QH regime. 92 5.1 Variable Thickness Topological Insulator Devices Variable thickness BSTS devices from 89 nm down to 10 nm were fabricated for this study. The complete device structure consists of vdW five-layer heterostructures of Gr/hBN encapsulated BSTS. The top and bottom Gr and hBN layers serve as the gate electrode and gate dielectric, respectively. The vdW heterostructure is effective in controlling the charge density of the TI surface states [115]. The device specifications were summarized in Table 5.1. 5.2 Low-Temperature Dual-Gated Transport Color maps of dual-gated Rxx of the respective BSTS devices are shown in Figure 5.1. By tuning the Vtg and Vbg, the top and bottom surface states are tuned separately to the two independent ambipolar transport behaviors. The red and black dashed lines in the 2D color maps illustrate the Dirac points of the top and bottom surface states, respectively. The two Dirac points intersect and form an Rxx maximum at the intersection. These two lines divide the Rxx map into four quadrants, corresponding to hole-hole (h-h), electronelectron (e-e), hole-electron (h-e), and electron-hole (e-h) conduction of the (top-bottom) surface as labeled in the figure. For 89 nm BSTS (Figure 5.1(a)), the Rxx color map shows nearly equal-sized four conduction quadrants versus the dual gate voltages. The top and bottom surface states are tuned independently by the top-gate and back-gate, implying negligible coupling between the surface states due to the relatively large spatial separation in the bulk. As the thickness of BSTS is reduced, the top and bottom Dirac points (traced by red and black lines) tend toward a diagonal direction as shown in Figure 5.1(b)-(d). The diagonal feature of the Rxx 93 Table 5.1 The thickness of the BSTS (d), top (ht) and bottom (hb) hBN flakes are measured using a Bruker Dimension Icon atomic force microscopy. The width to length aspect ratio (W/L) of the BSTS devices was estimated by the optical images. The errors indicate the accuracy of the thickness measurements from the atomic force microscopy. Device d (±5% nm) W/L ht (±5% nm) hb (±5% nm) 1 2 3 4 5 6 7 8 9 89 47 34 31 24 16 13 12 10 0.367 0.357 0.447 0.300 0.367 0.300 0.500 0.591 1.20 13 20 18 14 19 17 20 17 23 16 13 18 25 24 22 25 20 29 94 Figure 5.1. Thickness-dependent dual-gated transport. 2D color maps of Rxx as a function of Vtg and Vbg for the respective BSTS devices with flake thickness of (a) 89 nm, (b) 31 nm, (c) 16 nm, and (d) 10 nm. (e)-(h) Plots of Rxx as a function of Vbg extracted from the corresponding 2D maps in (a)-(d) at different Vtg indicated by the arrows. Inset in (e) is a schematic of the cross-sectional illustration of the BSTS device. 95 maximum is a result of the strong capacitive-coupling between the top and bottom surface states, as observed in similar compounds [92,135]. A consequence of the coupling is the increase of parallel-propagating (h-h and e-e) parameter space in Figure 5.1(a)-(d), together with squeezing of the counter-propagating (h-e and e-h) space into the diagonal area. Another important feature arising from the coupling is the overlapping of both surface Dirac points in transport as a function of any one gate voltage. Line profiles of Rxx as a function of Vbg for the 89 nm BSTS taken at three different Vtg across the top Dirac point (indicated by the arrows in Figure 5.1(e)) show the peak position of Rxx (bottom Dirac point) is nearly constant at different densities of the top surface. Whereas for the 31 nm BSTS, the Rxx line profiles (Figure 5.1(f)) show a downshift in the bottom Dirac point with an increase in Vtg. In addition, the Rxx shows a broadening at both high-density holes (blue curve) and electron (black curve) conductions, corresponding to the top surface Dirac point as indicated by the arrows in Figure 5.1(f). This feature is more apparent in thinner sample as in 16 nm BSTS, where the bottom and top Dirac points shift oppositely toward each other, resulting in the broad double Rxx peaks in Vbg (Figure 5.1(g)). The double Dirac points eventually overlap to form a single Rxx peak with higher resistance value as shown in 10 nm BSTS (Figure 5.1(h)). 5.3 Surface States Coupling in the Quantum Hall Regime 5.3.1 Thickness Dependent Magnetotransport Dual-gated magneto-transport at a perpendicular magnetic field of 18 T for different thickness BSTS is studied in Figure 5.2. The well-developed QH plateaus at high magnetic field give rise to clear QH boundaries in 2D maps of σxx and σxy versus dual-gate 96 Figure 5.2. QH plateaus development at a magnetic field. 2D color maps of σxx and σxy as a function of dual-gated voltages for the (a, b) 89 nm, (d, e) 31 nm, (g, h) 16 nm, and (j, k) 10 nm BSTS devices measured at magnetic field of 18 T. The LL indices for top and bottom surfaces, Nt and Nb are labeled in y-axis and x-axis, respectively. The Nt and Nb are determined by fixing the 0 LLs at the top and bottom Dirac points. Line profiles of σxx and σxy as a function of Vbg extracted from the corresponding color maps near the overall CNP as indicated by the arrows for the (c) 89 nm, (f) 31 nm, (i) 16 nm, and (l) 10 nm BSTS devices. The vertical black dashed line and the grey highlight in (c), (f), (i), and (l) denote the overall CNP and ν= 0 QH plateau, respectively. The LL filling factors (νt, νb) and ν= νt+νb are indexed in the figure. 97 voltages. The dashed lines in the color maps are tracelines at the boundaries of QH steps in the parameter space of two gate voltages. For 89 nm BSTS (Figure 5.2(a) and (b)), the QH boundaries traced with vertical and horizontal straight lines imply that the LLs formed at the top and bottom surface states are completely independent of each other. The line profiles (Figure 5.2(c)) reveal equally-spaced QH plateaus in σxy as tuned by Vbg into integer increments of e2/h, along with minimum σxx in the QH regimes. The coupling effect between surface states results in the development of a narrower ν= 0 plateau near the overall CNP due to the squeezing of counter-propagating regions (in the 2D map) in thinner BSTS (Figure 5.2(d)-(e)). The (νt, νb) indexed in 31 nm (Figure 5.2(f)) reveals a sign change in νt due to the crossing of Nt= 0 level, as compared with the constant νt in 89 nm BSTS (Figure 5.2(c)). The strongly-coupled surface states (≤16 nm) reveal a further squeezing in ν= 0 LL due to the diagonal tendency of both Nt and Nb= 0 lines (Figure 5.2(g)-(l)). The LL fan diagram for any surface transforms from asymmetric into highly symmetric between the hole and electron LLs with ν= 0 plateaus residing at the Dirac point of the surface as shown in Figure 5.3. 5.3.2 Splitting of N= 0 Landau Levels In the low carrier density region, we observe a splitting in both Nt and Nb= 0 LLs at the overall CNP as the thickness of BSTS reduces to 16 nm (Figure 5.2(g)). The line profile of σxx reveals a dip with zeroth-plateau formed in σxy as presented in Figure 5.2(i). A similar feature is observed in 10 nm BSTS (Figure 5.2(l)) with more pronounced 0 LL peak splitting. The displacement field (D) and total charge density (n) are calculated using the relations [133,136] as: 98 Figure 5.3. LL fan diagrams. 2D color maps of σxy and σxx as a function of B and Vbg for (a, b) 89 nm and (c, d) 13 nm BSTS devices. Dashed black lines in (a) and (c) trace the ν= 0 LL spacing, while the dashed white lines trace the integer nonzero QH states. The LL filling factors (ν) are indexed in the maps. 99 1 εg ∆Vbg εg ∆Vtg - ℎ ) ℎb t D= (ε ) ( (5.1) n= Cbg ∆Vbg +Ctg ∆Vtg (5.2) o The D versus n maps of σxx and σxy are plotted in Figure 5.4 to extract the total charge density of the 0 LL splitting (∆n) for the thin BSTS. The n= nb+nt is calculated from the field-effect transistor relation as expressed in relation (3.1). A direct interpretation of the 0 LL splitting is a consequence of the hybridization between the top and bottom surface states [41-44]. As the thickness reduces to the thin limit of 3D TI, the intersurface tunneling due to proximity of the surface states leads to an energy gap at CNP [137]. A key signature of such intersurface hybridization is the degeneracy lifting of the N= 0 levels in Landau quantization, which is consistent with our observation. The asymmetry in hole and electron N= 0 sublevels (Figure 5.2(l)) in 10 nm BSTS could be explained by the presence of the Zeeman effect [42,44]. However, the intersurface hybridization in our 16 nm and 10 nm BSTS seems to contradict the well-known 2D limit of 5 nm for Bi2Se3 [138]. One possibility is the limitation of angleresolved photoemission spectroscopy in resolving the sub-meV energy scale. This is evidenced by the sub-meV hybridization gap observed in 12-17 nm Bi2Se3 from phasecoherent transport [40]. A gap this small can easily be obscured by disorder and may be observable only in relatively clean 3D TIs. To check this, we examined the weak antilocalization (WAL) effect by measuring magneto-conductivity (∆σxx) for the capacitively-coupled BSTS devices at the low magnetic field, as presented in Figure 5.5. The 31 nm clearly shows weak antilocalization (WAL) behavior with the sharp ∆σxx cusp at zero magnetic fields. For thinner BSTS of 16 nm, we observe symmetric ∆σxx broadening about ±0.5 T (indicated by black arrows) consistent with the 12 QL Bi2Se3 reported in [40], 100 Figure 5.4. Displacement field (D) versus total charge density (n). 2D color maps of σxx (top row), σxy (bottom row) as a function of D and n for (a, e) 31 nm, (b, f) 16 nm, (c, g) 13 nm, and (d, h) 10 nm BSTS devices. The dotted lines are tracelines of QH boundaries, and the corresponding ν of the QH plateaus are indexed in the σxy maps. In (a)-(d), the top and bottom LL indices, Nt= 0 and Nb= 0, are indicated by the red and black arrows, respectively. The splitting of both Nt and Nb at 0 LLs in parameter space of total charge density, ∆n, are indicated by the white arrows in (a)-(d). 101 Figure 5.5. Weak antilocalization effect. (a) Plots of ∆σxx as a function of a magnetic field for different thickness BSTS ranging from 31 nm down to 8 nm. The magnetic field sweeps were taken by fixing the dual-gate voltages, Vtg and Vbg close to the overall CNP. (b) Fittings of the normalized ∆σxx (open dots) to the HLN formula (solid lines). 102 indicating the first crossover of intersurface hybridization. Further reducing the BSTS thickness to 13 nm and 10nm, the zero-field WAL cusp gets broadened due to the competition between WAL and weak localization (WL) at low magnetic field [40,124,139]. Eventually, both WAL and WL contributions are greatly suppressed in the insulating region at BSTS of 8 nm, suggesting the second crossover when the hybridization gap is noticeable. The normalized ∆σxx were fitted to the Hikami-Larkin-Nagaoka (HLN) formula as [40,124,139]:  lB2 1   e2  lB2  xx ( B) =  ln −   2 +    h  l2  l 2   (5.3) where Ψ, lB, and l are digamma function, magnetic length ( / eB ), and phase coherence length (fixed at about 200-300 nm), respectively, for the different thickness BSTS. We observed a consistent transition in WAL [40], indicating the first crossover of 2D limit in 16 nm BSTS, and suppression of WAL by stronger hybridization for even thinner BSTS (8 nm) [124,139]. The fitted prefactor α approaches ½ for 16 nm to 10 nm, indicating a transition to single 2D channel conduction [40]. The ∆n as a function of BSTS thickness (d) inserted in Figure 5.6(a) supports the WAL effect. The sub-meV gap size can easily be obscured by disorder, therefore not visible in our zero magnetic field transport. The fact that the gap feature shows up in the form of N= 0 LL splitting at strong magnetic field indicates a mechanism of field-dependent hybridization gap as discussed in [43]. Higher magnetic fields will be required for us to check the magnetic-field-proportionality of the gap size. Nonetheless, we cannot exclude the possibility of a many-particle gap developed by the topological exciton condensate as it is also predicted in the same regime [73]. 103 Figure 5.6. Capacitive coupling and chemical potential in surface states. (a) A plot of VtDVD as a function of nb for different thickness BSTS devices at 18 T. Inset in (a) plots the splitting of Nt and Nb at overall CNP in total charge density (∆n) as a function of flake thickness d. (b) A plot of ntD (as the nb is tuned from bottom Dirac point to 1×1012 cm-2) and εBSTS as a function of d. The black and blue dashed lines in (b) are the fittings of ntD and εBSTS , respectively, with d. Inset in (b) is a schematic of the top and bottom Dirac surface states. (c) A plot of μb as a function of nb for 16 nm BSTS device at magnetic field of 0T and 18T. The color highlights in (c) display the LL indices of bottom surface nb from ‐2 (leftmost) to +2 (rightmost) formed at 18 T. Inset in (c) is the ∆μb= μb(18T)-μb(0T) as a function of nb. (d) A plot of LL gaps (∆μb) for nb= ±1 as a function of magnetic field. Inset in (d) is the ∆±1 as a function of B1/2. 104 5.3.3 Charge Density Dependent Coupling In the high-density region, the Nt and Nb= 0 tracelines intercross at different angles, resulting in the nonlinear QH boundaries as shown in Figure 5.2(d) and (e). We assign the nonlinearity to the charge density-dependent capacitive-coupling between the top and bottom surface states, where screening in the bulk of the sample is weaker at low charge density. This leads to a pronounced bending of Nt= 0 and Nb= 0 tracelines near the overall CNP as observed in thinner BSTS (16 nm and 10 nm in Figure 5.2(g, h) and (j, k)). The above nonlinear QH boundaries features in dual-gated transport are further analyzed to study the coupling effect of top and bottom surface states in thin 3D TIs. Figure 5.6(a) presents the difference between top surface Dirac point and overall CNP voltages (VtD-VD) as a function of nb extracted from Rxx maps for various thickness BSTS at 18 T. The 89 nm and 47 nm BSTS show a nearly linear relation of VtD-VD with nb. In contrast, the nonlinear feature manifests in all other (thinner) BSTS. Schematics of the linear band structure of the top surface in Figure 5.6(a) illustrate the top surface charge density tuned from the overall CNP by Vbg. The change in the value of VtD is solely controlled by Vbg as the chemical potential of the top surface is fixed at its Dirac point. The charge density corresponding to the change of top surface Dirac point from the overall CNP (ntD ) can then be calculated as: ntD = Ctg (VtD -VD ) (5.4) The ntD as a function of d with nb fixed at 1×1012 cm-2 is plotted in Figure 5.6(b). The thickness dependence of ntD is studied at the high density of nb to circumvent the nonlinear bending effect near the overall CNP. ntD increases monotonically with the reduction in d, which indicates a constant increase of the electric field penetrating through 105 the bottom surface state and the interior bulk layer as the thickness is reduced. The linear extrapolation (black dashed line) of the data points intercepts with the y- and x-axes at 1×1012 cm-2 and ~60 nm, respectively. The y-intercept at ntD = nb means the bottom gate tunes an equal amount of density in top and bottom surface states, which is the zerothickness limit (ignoring the hybridization of the surface states). The x-intercept suggests the thickness of the BSTS where the surface states are decoupled capacitively, consistent with observations from literature [140,141]. 5.3.4 Dielectric Constant Considering the top and bottom surface states of a thin 3D TI to be a parallel plate capacitor with an interior bulk insulating layer (as illustrated in inset of Figure 5.6(b)), together with the top-gate and back-gate layers forming a series of three parallel-capacitors, we implement the capacitor charging equations for dual-gated surface states as formulated in [91,92]: e∆nb = Cbg (∆Vbg e∆nt = Ctg (∆Vtg - ∆μb e ∆μt e ∆μb ∆μt ) -CBSTS ( e - e ∆μt ∆μb ) -CBSTS ( e - e ) ) (5.5) (5.6) where ∆μb(t) is the change in chemical potential of the bottom (top) surface state. These two equations are simplified to two linear relations under the condition where the top surface state stays at its Dirac point, meaning ∆nt and ∆μt have both vanished. The outcome of relation (5.5) is the linear dependence of ∆Vtg versus ∆Vbg, and the slope (S) can be expressed as: 1 C C BSTS S= (C ) (C bg+C ) tg bg BSTS (5.7) 106 The slopes of ∆Vtg versus ∆Vbg in zero magnetic field 2D maps of Rxx for different BSTS thickness as shown in the red lines in Figure 5.1(b)-(d) are used to estimate the geometric capacitance of the BSTS. This is to ensure that the slope is not affected by localized states formed in a magnetic field. The details of the fitted S, Cbg, Ctg, and CBSTS are listed in Table 5.2. An average dielectric constant of BSTS (εBSTS ) of about 28 is obtained from different thickness BSTS as shown in Figure 5.6(b) (right y-axis), which is comparable to similar compounds [91,142]. 5.3.5 Chemical Potential and Landau Level Spacing The chemical potential of the bottom surface state (μb) can be evaluated from the ∆Vtg by using the relation derived from relation (5.6) as: Ctg μb =- C BSTS e∆Vtg (5.8) where the ∆Vtg is the difference between VtD and the overall CNP position (VD). Figure 5.6(c) shows the plots of μb as a function of nb for the 16 nm BSTS at magnetic field of 0T and 18T. The μb plot at 18T is symmetric about the zero density of the bottom surface. The increase in μb in magnetic field is related to the localized electronic states in the bottom surface due to the LL formation. The color shaded regions in Figure 5.6(c) emphasize density regions in terms of LL indices of the bottom surface Nb from ‐2 (light blue), -1 (blue), +1 (red), and +2 (light red). The energy gap sizes of the LLs can be assessed from ∆μb= μb(18T)-μb(0T) as plotted in the inset of Figure 5.6(c). The LL gaps (∆±1) are estimated from the ∆μb in the density region Nb= ±1 (blue and red shading). Figure 5.6(d) displays the ∆±1 as a function of magnetic field for the 16 nm BSTS. The details of the estimated of the ∆±1 at different magnetic fields are given in Figure 5.7. The black dashed 107 Table 5.2 Ctg and Cbg are calculated from parallel plate capacitor relation with a dielectric constant of hBN fixed as 3. The CBSTS is estimated from the slope (S) of the Vtg versus Vbg plots using the relation (5.7). The percentage of errors are obtained from the average of linear fitting errors. The dielectric constant of the BSTS can thus be estimated from the CBSTS and d. d (nm) 12 16 S (±2%) -0.800 -0.718 Ctg (nF/cm2) 156 156 Cbg (nF/cm2) 133 121 CBSTS (±2% nF/cm2) 2130 1556 24 -0.714 140 111 1007 31 -0.445 189 106 732 34 -0.833 148 148 740 108 Figure 5.7. Chemical potential plots at different magnetic fields. Dual-gated Rxx color maps (left) and the change in chemical potential, ∆μb= μb(B)–μb(0) (right) for the 16 nm BSTS device at magnetic field of (a) 3T, (b) 5T, (c) 7T, and (d) 9T. The dashed white lines in color maps trace the Nt= 0 LL index at different Vbg. The blue and red highlights in ∆μb plots represent the density of the Nb= ‐1 and +1 QH plateaus, respectively. 109 curve in Figure 5.6(d) serves as a comparison between ∆±1 and the theoretical model obtained from the relation (1.13), with the Fermi velocity (vF ) of the BSTS taken to be ~3×105 m/s. The inset in Figure 5.6(d) shows a nearly linear fitting of ∆±1 with the square root of magnetic field (√B), indicating a good match with the theory even though there is a ~30% deviation from theoretical values. 5.3.6 Quantum Capacitance Despite the agreement of the ∆±1 obtained from dual-gated transport to the theoretical calculation, the effect of the magnetic field on the capacitance of BSTS cannot be ruled out from the study. As in the relation (5.8), μb is inversely proportional to the CBSTS, the reduction in CBSTS can cause an increment in μb as well. To verify this, we perform the dual-gated capacitance measurement on the same device by using a capacitance bridge method [108,116]. Dual-gated maps of quantum capacitance (CQ) of the 16 nm BSTS measured at magnetic field of 0 T and 9 T are shown in Figure 5.8, respectively. A similar diagonal zero-field dual-gated map again verifies the coupling effect of top and bottom surface states. In a magnetic field of 9T, the CQ forms dips in the QH regimes corresponding to electronic density of states dips developed in those regimes. Figure 5.8(a) compares the line profiles of CQ as a function of Vtg at 0 T and 9 T at a fixed Vbg. The capacitance values are nearly constant with magnetic field (CQ at 9 T is slightly bigger), which indicates an insignificant effect of magnetic field on the BSTS bulk capacitance, in agreement with the analysis in [91]. In addition, the LL gap of ∆±1 calculated from the CQ at 9 T is about 19 meV (Figure 5.8(b)), which is close to the value obtained from the dual-gated transport. 110 Figure 5.8. Quantum capacitance. (a) CQ as a function of Vtg of the 16 nm BSTS device measured at the magnetic field at 0 and 9 T. (b) Plot of chemical potential (μ) as a function of charge density (n) at 9T. The μ is calculated from CQ using the relation (4.4). The N= ±1 LL gap of the 16 nm BSTS is estimated to be ~19 meV at 9T, which is close to the estimation from the charging model equation (~22 meV). CHAPTER 6 TOPOLOGICAL PHASE TRANSITIONS IN HYBRIDIZED SURFACE STATES As the thickness of a 3D TI reduces to a sufficiently 2D thin limit, the top and bottom surface states couple strongly, causing an overlap and hybridization between the surface state wavefunctions. This subsequently induces a gapped energy spectrum at the Dirac point, where the topological protection breaks in the gap. The feature of intersurface hybridization gap has been systematically probed in ultrathin Bi2Se3 using ARPES, where a thickness of 6 QL (~6 nm) is known as a switching point for the hybridization [138]. However, this 3D to 2D cross-over thickness is debatable as the ARPES spectrum is limited by its few to ten meV resolution. On the other hand, the electrical transport gap for a ~3.5 nm Bi2Se3 exceeds 250 meV [143], which is equivalent to the hybridization gap probed by ARPES for 1 QL of Bi2Se3 [138]. Moreover, by analyzing the phase coherence magnetotransport, a sub-meV hybridization gap was extracted for Bi2Se3 as thick as 12 nm [40]. The nature of TSS changes significantly as the hybridization gap opens. Despite losing its topological protection in surface states, the hybridization gap in 2D surface can host a topological edge state like a QSH insulator [45]. This can occur if the electron and hole subband of the hybridization gap preserves the parity of the inverted bulk band. Theoretical calculation predicts an oscillatory trend between the trivial insulator gap and 112 the QSH gap as a function of thickness of the 3D TI [45,144]. This chapter studies the transport signature of the hybridization gap in ultrathin BSTS 3D TI as a function of film thickness. The hybridization gap is further analyzed by introducing a perpendicular magnetic field to study the N= 0 LL splitting feature. In addition, the hybridization gap size is manipulated by applying an electric field across the sample to realize the topological phase transition. 6.1 Intersurface Hybridization 6.1.1 Theoretical Models When the thickness of a 3D TI becomes comparable to the penetration depth of the surface states, tunneling between the top and bottom surfaces becomes apparent, leading to a crossover to 2D (gapped) surface states. The Hamiltonian of the surface states can be accounted for by adding a hybridization term as in (1.7) to their original Hamiltonian as in (1.1) [37]. The induced hybridization gap ∆h at the  point can be approximated by [39]: ∆ℎ ∝ 𝑒 −𝜆𝐿 1 ∆ℎ ∝ 𝐿2 (6.1) (6.2) where  and L are the characteristic length and film thickness, respectively. Relation (6.1) indicates the gap size decays exponentially as a function of L. However, as L approaches its zero limits, the gap size is better estimated by (6.2). In a perpendicular magnetic field, the hybridization gap in the surface states essentially lifts the degeneracy of the N= 0 LL, resulting in the splitting into electron and hole N= 0 sublevels. The definition of the hybridization gap can thus be revised as the energy difference between the sublevels of N= 0 LLs as E0 [43,44]. The E0 obtained from 113 the analytical formula in [145] can be expressed as: 𝑒𝐵 𝜇 𝑔𝐵 𝐸0 = 𝐶̃0 + ℏ 𝐶̃2 − 𝐵2 (6.3) where 𝜇𝐵 and g are the Bohr magneton and effective g-factor; 𝐶̃0 and 𝐶̃2 are the parameters related to the Hamiltonian of the surface states. This relation suggests a linear function of the E0 with a magnetic field. The hybridization can also be tuned by applying an electric field perpendicular to an ultrathin 3D TI. In the presence of an external perpendicular electric field (Ez), the Hamiltonian of the hybridized surface states is further modified by an electric potential energy U as [146,147]: 𝐻𝑈 = 𝑈𝑧 = 𝑒𝐄𝑧 ∙ 𝐳 (6.4) The applied electric field can cause a structure inversion asymmetry, leading to the possibility of topological phase transition [148,149]. While a necessary condition for the topological phase transition is a gap-closing point to allow the change in the topological invariants. The theoretical calculations show that the electric field can induce Rashba-type splitting, resulting in a close and reopen of the hybridization gap, provided the gap size is not too large. 6.2 Hybridization Gap Analysis To obtain a quantitative analysis of the intersurface hybridization of BSTS 3D TI, we employ three different approaches to evaluate the hybridization gap for different thicknesses. The first one is the thermal activation energy by fitting the temperature dependence conductivity. The second method is the analysis of the nonlinearity currentvoltage curve at low bias voltage (Vb) by measuring the differential conductance (dI/dV). 114 The third approach is on the calculation of μ(n) by integrating the reciprocal quantum capacitance, 1/CQ, with respect to n. 6.2.1 Thermal Activation Energy Figure 6.1(a) shows the temperature-dependent Rxx as a function of Vbg at different temperatures for a 9 nm BSTS device. The Rxx at CNP increases monotonically with temperature and dramatical increase at temperature below 10 K. Such an insulating behavior implies a gap state at the CNP due to hybridization of the opposite surface states. The Rxx greatly suppressed at high density when the chemical potential is tuned by Vbg into the conduction regime. The inset of Figure 6.1(a) displays the Gxx data taken at CNP on Arrhenius plot. The strongly activated temperature-dependent conductance is observed; straight lines are fitted exponentially to the relation [143] as: Gxx= G0xx exp(EA/kT) (6.5) where EA, k and G0xx are the activation energy, Boltzmann’s constant and constant prefactor, respectively. The linear fit gives an EA of about 6.5 meV for the 9 nm BSTS. 6.2.2 Differential Conductance Figure 6.1(b) (inset) shows a dI/dV map as a function of Vb and Vbg for the 9 nm BSTS. The dI/dV conduction probes the nonlinear characteristic in current-voltage curve originated from the electronic gap state [150]. As the gate-voltage sweeps across the CNP, the dI/dV yields a distinct diamond-shaped minimum elucidate the nature of this gap-like feature. The diamond feature arises from charge transport across the surface states when the chemical potential aligned to or detuned from the hybridization gap at the CNP. In the 115 Figure 6.1. Hybridization gap analysis. (a) Rxx versus Vbg plots at different temperatures. The inset in (a) is an Arrhenius plot of Gxx versus T-1 for Vbg fixed at the CNP. (b) dI/dV versus Vb curve at insulating region (Vbg~ -1.5 V). The inset in (b) is a color map of dI/dV as functions of bias voltage (Vb) and Vbg. (c) C and μ(n) as a function of charge density (n) induced by Vbg with Vtg fixed at the overall CNP. The inset in (c) is optical image of an ultrathin BSTS vdW heterostructure device. Scale bar in (c) is 10 m. (d) ∆h as a function of BSTS flake thickness obtained from the three different methods. 116 gap region, the dI/dV reaches a minimum conductance and turns on as the Vb exceeds the electronic gap. The width of the hybridization gap is determined from the line profile of dI/dV versus Vb (Figure 6.1(b)), which yields a gap size of ~18 meV. 6.2.3 Quantum Capacitance CQ is related to electronic compressibility (dn/dμ) by the relation (4.4), and the integral of reciprocal of the electronic compressibility gives the μ(n). The details in derivation for the CQ and μ(n) from our capacitance measurements can be referred to Section 4.4 in Chapter 4. The minimum capacitance at the CNP (Figure 6.1(c)) indicates a minimum DoS corresponding to the insulating state. The μ(n) as a function of n reveal a step feature, suggesting the hybridization gap formed at the CNP. The gap size is determined from the step height as ~21 meV for the 9 nm BSTS. 6.2.4 Thickness Dependent Hybridization Gap The hybridization gap size as a function of BSTS flake thickness extracted from the three different methods are summarized in Figure 6.1(d). The ∆h increases exponentially with the decrease in BSTS thickness for all three methods, consistent with the reported thickness dependence hybridization gap measured by ARPES [138]. However, we observed the hybridization gap opens in BSTS thickness below 10 nm, indicating a higher 2D crossover point than the ARPES analysis (below 6 nm). The wider range of gapped state provides more accessible layers of 3D TI to probe the possible topological phase transition and QSH state in the gap region. The gapped state in the ultrathin BSTS is confirmed by the consistent observation in all three gap size analyses. The thermal 117 activation gap in general shows a smaller gap size compare to the differential conductance and electronic compressibility deduced gap because of its sensitivity to smearing effect by disorders [151]. 6.3 Normal and Inverted Hybridization Gap States Low-temperature transport of the hybridized surface states is studied by probing the four-terminal resistance. Figure 6.2(a) plots of the Rxx maximum at CNP as a function of temperature for the 9 nm and 10 nm BSTS. The 9 nm BSTS displays a monotonically increasing function in the Rxx maximum with decrease in temperature, particularly below 50 K. Whereas the Rxx maximum saturates at ~12 k (~h/2e2) at temperature below 50 K for the 10 nm BSTS. The weak temperature response in 10 nm devices at low temperature suggests possibly the existence of the edge state conduction in the hybridization gap regime [45]. The full data set of Rxx at all temperatures studied for the 9 nm and 10 nm BSTS are presented as color maps in the inset of Figure 6.2(a). Figure 6.2(b) compares the gate dependent Rxx at temperature of 1.5 K for the 9 nm and 10 nm BSTS. The chemical potential is tuned by the gate voltage into the hybridization gap (minimum conductivity) regime as shown by the dI/dV color map inserted in Figure 6.2(b). For the 9 nm BSTS, the Rxx reaches a maximum value of ~500 k (corresponding to a Gxx ~0.05e2/h) in the hybridization gap regime, indicating a normal insulating behavior in this sample. While the 10 nm BSTS exhibits a finite resistance of ~h/2e2 even when the chemical potential lies inside the insulating gap of ~9 meV. This again implies that the hybridization gap for the 10 nm BSTS is in an inverted gap regime, consequently the gap is bridged by a linear-dispersive edge state, which can host edge mode of QSH effect. 118 Figure 6.2. Trivial and nontrivial gap states. (a) Rxx as a function of temperature for the 10 nm and 9 nm BSTS. The inset in (a) are color map of Rxx as functions of T and Vbg for the 10 nm (right) and 9 nm (left) BSTS. (b) Rxx as a function of Vbg-VD for the 10 nm and 9 nm BSTS. The inset in (b) is a color map of dI/dV as function of Vb and Vbg for the 10 nm BSTS. 119 6.4 Magnetic Field-Induced Phase Transition 6.4.1 Normal Hybridization Gap To further confirm the hybridization nature of the electronic gap at the Dirac point, we perform the magneto-transport measurements on the ultrathin BSTS devices. Figure 6.3(a) plots the σxx and σxy as a function of Vbg measured at a magnetic field of 18 T for the 9 nm BSTS. The two N= 0 LL bands developed in the σxx plot are assigned to the electron and hole band edges of the hybridization gap as illustrated by schematic band diagram in the inset of Figure 6.3(a). This splitting of N= 0 LLs is a key feature of the Landau quantization for intersurface hybridization, which has been theoretically predicted in the references [41,42,44]. The established ν= 0 plateau in the σxy plot within the N= 0 LL onsets, together with the development of ν= -1 and +1 QH plateaus symmetrically about zeroth plateau, further supports the LLs of the hybridized surface states. Moreover, the formation of QH plateaus is indicative of high quality of our BSTS devices. The color map of σxx in Figure 6.3(b) illustrates the development of the N= 0 LLs as a function of magnetic field. Unlike the square-root dependence of LL energy on magnetic field in the Dirac system, the N= 0 LL energy in the hybridized surface states is expected to depend linearly on the magnetic field [43]. To further verify magnetic field dependent zeroth LL energy, we perform the CQ measurements on the same device to extract the (n) versus n plots at different magnetic fields. The N= 0 LL energy spacing (E0) as a function of magnetic field is extracted from the step height of the (n) plots. Agreeing with the theoretical model, the E0 can be fitted to a linear relation with magnetic field as expressed in (6.3). The fitting yields 𝐂̃𝟐 of ~70 and ~90 eV Å2 for the 9 and 8 nm BSTS, respectively, which is comparable to the fitting parameter from [145]. 120 Figure 6.3. Evolution of the zeroth LLs. σxx and σxy plots as a function of Vbg at the magnetic field of 18 T for the (a) 9 nm and (c) 10 nm BSTS. The inset in (a) is a schematic of the LLs form in a hybridized surface state. Black and red arrows in (a) and (c) present the two N= 0 LL bands residing at the hole and electron band edges, respectively. The color maps of the σxx as functions of magnetic field and Vbg for the (b) 9 nm and (d) 10 nm BSTS. The white dashed lines in (b) and (d) trace the development of N= 0 LLs in electron and hole subbands with magnetic field. 121 6.4.2 Inverted Hybridization Gap The σxx and σxy plots as a function of Vbg for the 10 nm BSTS measured at the magnetic field of 18 T are presented in Figure 6.3(c). Similar to the 9 nm BSTS, the N= 0 LLs splitting is observed except the zeroth LL plateau is relatively narrow in charge density compare to the 9 nm BSTS. This is consistent with its smaller hybridization gap size as indicated by the hybridization gap measurements. The development of the N= 0 LLs with magnetic field is presented in σxx color map as functions of magnetic field and Vbg (Figure 6.3(d)). The N= 0 LLs traced by the dashed lines along the σxx minimum corresponding to the hybridization gap reveal a crossing feature between the two N= 0 LLs, symmetrically about the opposite perpendicular magnetic field. In the presence of strong magnetic field, the hybridized surface states develop into QH states in the normal (noninverted) gap regime as revealed by the 9 nm BSTS. For the system with inverted band as inferred to our 10 nm BSTS, switching from the inverted gap into the normal gap state is required. The applied magnetic field causes the two N= 0 LLs to approach and then merge each other. With the further increase in magnetic field, the two N= 0 LLs split again and develop into the normal QH states in hybridized surface states with the N= 0 LLs interchanged. This is equivalent to closing and reopening of the hybridization gap for the system to transform from topologically nontrivial into trivial gap state. This topological phase transition mechanism is similar to the band crossing for the QSH effect in HgTe/CdTe quantum well [7] and more specifically discussed in the theoretical models for ultrathin 3D TIs [152]. The level crossing point at critical magnetic field (Bc) of ~10 T is equivalent to a Zeeman energy of EB =gμB Bc ~10 meV for g factor about 20 [153]. 122 6.5 Electric Field-Induced Phase Transition 6.5.1 Dual-Gated Transport The ultrathin BSTS were fabricated into vdW heterostructures devices with Gr/hBN encapsulation for dual-gating studies. Figure 6.4(a) and (c) show the Rxx color maps in dual-gate voltages for the 9 and 10 nm BSTS, respectively. The diagonal feature in dual-gated Rxx color maps for the ultrathin BSTS justifies the strong coupling between the top and bottom surface states. The Rxx plots versus Vbg at the center CNP (black curves) for the 10 and 9 nm BSTS (Figure 6.4(b) and (d), respectively) reveal a significant change in Rxx signals when gate-tuned into the hole and electron conductance regions, again verifying the hybridization gap feature at the surface Dirac point. As shown in the color maps, the CNPs of both BSTS samples located at nearly zero Vtg, inferring a minimum electric field acting to the samples prior to the Vtg applied. With the applied Vtg across the CNP, we study the response of the hybridization gap in a displacement field (D) via dualgating with opposite sign of gate voltages applied (as illustrated by the schematic device configuration in Figure 6.5(a)). The dual-gated Rxx maps for the 9 nm and 10 nm BSTS display very different features with the applied D. The 10 nm BSTS reveals insignificant change in Rxx maximum to the small applied D, and the Rxx maximum gradually decreases at higher D. Whereas for the 9 nm BSTS, the Rxx maximum responses greatly to the applied D as supported by the substantial reduction in Rxx peak with D. The Rxx versus Vbg line profiles taken at different Vtg (Figure 6.4(d)) clearly elucidates this feature. The more than an order of magnitude reduction in Rxx implies possibly a hybridization gap-closing signature by the perpendicular electric field. 123 Figure 6.4. Dual-gating effect. Color maps of Rxx as a function of dual-gate voltages at temperature of 1.5 K for the (a) 9 nm and (c) 10 nm BSTS. The dashed line arrows in (a) point along the direction of the D and n. Plots of Rxx versus Vbg at different Vtg for the (b) 9 nm and (d) 10 nm BSTS. 124 Figure 6.5. Electric field induced gap closing. (a) Schematic of an ultrathin BSTS in a displacement field (D) induced by the opposite sign of dual-gate voltages. (b) Color maps of Rxx as functions of D and n for the different thickness BSTS in the order of (i) 10 nm, (ii) 9 nm, (iii) 8 nm, and (iv) 7 nm. (c) Rxx as a function of D for the different thickness BSTS. (d) Rxx as a function of temperature at different D for the 9 nm BSTS. Inset in (d) plots Gxx versus T-1 for the different D. (e) Plots of chemical potential (μ) versus n calculated from the quantum capacitance (CQ) for different D. The inset (top) in (e) is a dual-gated color map of C for the 9 nm BSTS. The inset (bottom) in (e) is the plots of ∆h versus D for the 8 and 9 nm BSTS. 125 6.5.2 Displacement Field To further understand the response of the hybridization gap to the applied D, we converted the dual-gate voltages to D versus total charge density (n) plots using the relations (5.1) and (5.2). Figure 6.5(b) shows the Rxx color maps of as functions of D and n for the ultrathin BSTS with thickness decreases from 10 nm to 7 nm. The dI/dV maps of the respective BSTS devices are inserted in the figure to compare the D dependent Rxx feature to the hybridization gaps. As discussed in dual-gated transport, the Rxx for the 10 nm BSTS responds weakly to the D. Whereas for the 9 nm BSTS, the Rxx at the CNP (n= 0) decreases greatly even at small D (tens of mV/nm). This effect is not affected by the polarity of D as the Rxx change is highly symmetric about the opposite D. However, the D responses to Rxx become less pronounced as the hybridization gap increases. To study the D responses to the hybridization gap, we plot the Rxx maxima as a function of D taken at n= 0 for the different thickness BSTS in Figure 6.5(c). As shown in the figure, the Rxx falls significantly with D and tends to saturate at large D (> 150 mV/nm) to a value close to the order of h/e2, suggesting a closing of hybridization gap by applying large D. This gapclosing effect is less intense as indicated by the smaller change in Rxx with D for the thinner BSTS with larger hybridization gap. For the BSTS with thickness of 6nm and below (not shown), the Rxx maxima remain significant even at large D, indicating a trivial D response for the large hybridization gap. To provide quantitative analysis on the gap-closing feature, we investigate the temperature dependence Rxx at different D for the 9 nm BSTS. Figure 6.5(d) shows a strong thermal activation in Rxx with the decrease in temperature at zero D, and the activation behavior reduces abruptly as the strength of D exceeds 100 mV/nm. The thermal 126 activations at different D are analyzed by plotting the Gxx versus T-1 as depicted in inset of Figure 6.5(d). The nearly constant in Gxx at low temperature for D >100 mV/nm supports the gap closing behavior at large D. Alternatively, we analyze the (n) versus n relation extracted from the dual-gated CQ map (Figure 6.5(e), top inset) for the 9 nm BSTS. Again, the great reduction in CQ dip intensity at large D supports the suppression of the hybridization gap size. The (n) curves for different D obtained from the diagonal CQ dips along zero charge density are plotted in Figure 6.5(e). Likewise, the hybridization gap size at different D is estimated from the step height of the (n) plots and summarized in Figure 6.5(e) (bottom inset). In this figure, the decreasing trend of D dependent ∆h for the 8 and 9 nm BSTS is comparable to the analytical results in the references [148,149]. The perpendicular electric field-induced gap-closing in the hybridized surface states provides a strong indication of a topological phase transition. As the role of perpendicular electric field is to invert the valence and conduction band states regardless of the initial band topology [148], the phase transition can occur in both ways, either from trivial to nontrivial topological state or vice versa. In our 9 nm BSTS, we anticipate a transition from the normal insulating into a topological state through the gap-closing mechanism by the perpendicular electric field. This is supported by the saturated finite resistance observed at large D for the 9 nm BSTS. Whereas for the 10 nm BSTS, the gap closing feature at large electric field can be inferred from the gradual loss of its quantization as Rxx deviates from h/2e2 at large D. CHAPTER 7 CONCLUSION 7.1 Concluding Remarks Based on the experimental observations, we draw the following conclusions: (i) The quality of the BSTS crystal is strongly affected by the growth method. As the MG method solidifies from numerous nucleation sites along the wall of the ampoule, the as-grown BSTS crystal is more likely to form grain boundaries. Whereas the BG crystal nucleates from the crystal seed at the bottom tip of the ampoule and solidify along the vertical direction. Crystal defects such as antisites, vacancies, and other crystal dislocation are greatly reduced in BG and MBG BSTS. This minimizes the electron scattering in transport due to the defect sites and therefore results in a nearly four times enhancement in surface mobility of the MBG BSTS. The enhancement in surface mobility led to the Landau quantization in surface states of the MBG BSTS as a manifestation of IQHE at low magnetic field. (ii) In the study of the magnetoelectric transport of BSTS vdW heterostructures devices, we achieved fully developed LL filling factor ν= 0, ±1, and developing ν= ±2 and ±3 QH states via hBN/Gr gating at magnetic fields below 9 T. This was attributed to the effective gate tuning using the vdW platform and our clean device fabrication process. Also, the vdW epitaxial hBN/Gr gate led to better development of the QHE than SiO 2/Si 128 gate, as indicated by a larger suppression of σxx and better quantization in σxy. We also showed the ability to use the vdW platform for fabricating heterostructures with CGT/Gr gate, which allowed access and tuning of gapped TSS. The observation of half-quantized Hall conductance steps for the first time in a system where electron-electron correlations do not play a significant role provides a clear manifestation of massive Dirac fermion physics on the magnetized TI surface. The effective control of the TI surface states using the 2D layered materials paves the way for epitaxial stacking, promising improved quality and tunability of TI devices. (iii) The local 2D gates in BSTS vdW heterostructures provide a platform to study the capacitance of TSS in a dual-gated configuration. By integrating the inverse DoS over a charge density range, we obtained the energy spacings of the lowest LLs corresponding to the fully-quantized QH states. The largest cyclotron gap sizes of ~44-46 meV at 18 T agree with the theoretical relation of Dirac LL energies for N= ±1 LLs. The dual-gated capacitance with independent excitation source applied separately to two surfaces allows the individual probing of surface states. The opposite LL energies for top and bottom surfaces give rise to the ν= 0 quantum states. We observed lower Fermi velocity and thermal activation in ν= 0 states as compared to the conventional ν= ±1 states. Capacitance measurement is promising for further study of exotic gapped quantum states in topological surface states. (iv) The direct correlation between flake thickness and coupling of the top and bottom surface states gives rise to a tunable coupling of topological surface states. A direct manifestation of the capacitive coupling between surface states is the diagonal feature in dual-gating conduction maps. In a perpendicular magnetic field, the strongly-coupled 129 surface states develop into a series of elongated rhombus-shaped QH plateaus versus the dual-gates, and the LL fan diagram transforms into an electron-hole symmetric LL series about the ν= 0 LL. A careful analysis reveals the nonlinear (bending) LL transitions near the center CNP. From the analysis of the nonlinearity in Nt= 0 LL index with Vbg, we estimated a physical quantity: dielectric constant of BSTS, and an electronic quantity: LL energy separation. The study of surface state coupling effects in QH regimes is believed to pave the path to exotic quantum phenomena such as topological exciton condensation and charge fractionalization. (v) The finite-size effect of ultrathin BSTS 3D TI in the quantum tunneling regime is studied by probing the hybridization gap. We observed a consistent exponentially decay in the hybridization gap size with the increase of flake thickness in three analytical methods, namely thermal activation, differential conductance, and quantum capacitance. The responses of the hybridization gap in external magnetic and electric fields were studied. We observed the development of electron and hole subbands N= 0 LL onsets in perpendicular magnetic field. In the normal QH states, the N= 0 LLs spacing increases linearly with magnetic field in contrast to the square-root dependence of Dirac LL spacing. For inverted hybridization gap state, the crossing of two N= 0 LL bands implies the topological phase transition from the QSH state to the normal QH states. In addition, we observed a gap-closing feature in the normal insulating state of the hybridized 3D TI with moderate gap size in perpendicular electric field. The results suggest an electric field induced topological phase transition can occur in the hybridization gap range. 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