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Transcript
Transport Experiments with Dirac Electrons Joseph George Checkelsky a dissertation presented to the faculty of princeton university in candidacy for the degree of doctor of philosophy recommended for acceptance by the department of physics Adviser: N. Phuan Ong June 2010 c Copyright 2010 by Joseph George Checkelsky. All rights reserved. Abstract This thesis presents transport experiments performed on solid state systems in which the behavior of the charge carriers can be described by the Dirac equation. Unlike the massive carriers in a typical material, in these systems the carriers behave like massless fermions with a photon-like dispersion predicted to greatly modify their spin and charge transport properties. The first system studied is graphene, a crystalline monolayer of carbon arranged in a hexagonal lattice. The band structure calculated from the hexagonal lattice has the form of the massless Dirac Hamiltonian. At the charge neutral Dirac point, we find that application of a magnetic field drives a transition to an insulating state. We also study the thermoelectric properties of graphene and find that the states near the Dirac point have a unique response compared to those at higher charge density. The second system is the 3D topological insulator Bi2 Se3 , where a Dirac-like dispersion for states on the 2D surface of the insulating 3D crystal arises as a result of the topology of the 3D bands and time reversal symmetry. To access the transport properties of the 2D states, we suppress the remnant bulk conduction channel by chemical doping and electrostatic gating. In bulk crystals we find strong quantum corrections to transport at low temperature when the bulk conduction channel is maximally suppressed. In microscopic crystals we are able better to isolate the surface conduction channel properties. We identify in-gap conducting iii states that have relatively high mobility compared to the bulk and exhibit weak antilocalization, consistent with predictions for protected 2D surface states with strong spin-orbit coupling. iv Acknowledgments I would like to thank my adviser Phuan Ong for being a deeply generous mentor. He has been generous with his knowledge of science as well as afforded me countless opportunities in and out of the laboratory. Practically every step forward in my work has come as a product of insightful discussions with him. His enthusiasm and attitude regarding research will serve as a lasting model for me. I am also particularly indebted to Minhyea Lee who guided me from the moment I sat down at the microscope. One could not ask for a better teacher nor a more devoted friend. Lu Li (Fig. 2.2) was also a great teacher and compadre. We survived many long weeks at the magnet lab because of his patience and perseverance. I have benefited greatly from the other members of the lab. Shu-Wen Teng, Yoshi Onose, Wei-Li Lee, Phil Casey, and Yufang Wang have all been supportive during my time in Princeton. The collaboration with Professor Cava and his group in chemistry has been vital to my projects. In particular working with Yew San Hor has been fruitful; Yew San is never short of crystals and career advice. Conversations with Professor Hasan, Professor Haldane, Professor Petta, Professor Shayegan and Professor Bernevig have all been extremely important. Support at the National High Magnetic Field Laboratory in Tallahassee, Florida from Scott Hannahs, Tim Murphy, and the support staff is also greatly appreciated. The staff in the physics department has been a great resource. Mike Peloso spent v many hours teaching me skills in the student machine shop that have allowed me to solve many experimental problems. The members of the purchasing department Claude Champagne, Barbara Grunwerg, and Mary Santay helped me day after day. Laurel Lerner and now Kim Fawkes have been knowledgeable resources throughout the years. Jim Kukon and Geoff Gettelfinger were particularly helpful as a flexible resource for liquid helium. I have met many (mostly basement dwelling) friends along the way- Qiucen Zhang, Bart McGuyer, Mike Kolodrubetz, Michael Schroer, Aakash Pushp, Colin Parker, Lukas Urban, Shashank Misra, and Justin Brown to name a few. Above ground, Dima Abanin and Mira Parish have been great theoretical colleagues and Ted Laird helped me make it through the roller coaster that was generals. Finally I wish to thank my family. They have been endlessly supportive and understanding, particularly my parents for tolerating the din of vacuum pumps in the background during our Sunday phone calls. I thank them for this and more. vi Contents Abstract iii Acknowledgments v Table of Contents vii 1 Introduction 1 1.1 Dirac materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 3D Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental setup 12 14 2.1 Cryostats and Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Transport Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 High field ground state of Graphene at the Dirac Point 23 3.1 Electronic Band Stucture of Graphene in H . . . . . . . . . . . . . . 23 3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 vii 3.2.1 Quantum Hall effect and Landau Levels at ν = 0 . . . . . . . 28 3.2.2 Sample Quality Dependence . . . . . . . . . . . . . . . . . . . 33 3.2.3 Device Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.4 Measurements with pW dissipation . . . . . . . . . . . . . . . 41 3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Thermopower and Nernst effect in Graphene in a high magnetic field 4.1 56 . . . . . . . . . . . . . . . . . . 57 4.1.1 Thermoelectric Measurements . . . . . . . . . . . . . . . . . . 57 4.1.2 Girvin and Jonson Theory . . . . . . . . . . . . . . . . . . . . 60 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 S with H = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.2 S and Syx in H . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Thermoelectric Measurements in 2D 5 Quantum Interference in Macroscopic Crystals of Nonmetallic Bi2 Se3 76 5.1 Transport measurements in 3D Topological Insulators . . . . . . . . . 77 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.1 Nernst effect in Cax Bi2−x Se3 : Tuning the Chemical Potential . 81 5.2.2 Nernst effect in Cax Bi2−x Se3 : Low T behavior . . . . . . . . . 85 5.2.3 Transport properties of non-metallic Bi2 Se3 . . . . . . . . . . 89 5.2.4 Angular Dependence of non-metallic crystals . . . . . . . . . . 95 viii 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Evidence for metallic surface states in voltage-tuned crystals of Bi2 Se3 107 6.1 Gate voltage chemical potential control in Bi2 Se3 . . . . . . . . . . . 108 6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7 Concluding Remarks 126 Appendix 129 A Brief Review of Quantum Corrections to Transport 129 Bibliography 135 ix Chapter 1 Introduction 1.1 Dirac materials Dirac electrons are distinct from electrons found in normal metals because of their energy-momentum dispersion relation. Depicted in Fig. 1.1a, electrons in a typical metal such as Cu behave like massive particles so that their energy has a quadratic dependence on momentum, obeying the Schrodinger equation. However, in a Dirac material this dependence is linear (Dirac-like), as in Fig. 1.1b . In other words, the charge carriers behave as zero mass particles like photons or neutrinos, but (in the case of the prototypical Dirac material graphene) with a velocity ∼ 0.3% of the speed of light. Unlike other zero mass particles, however, Dirac electrons carry a charge. Compared to electrons in Cu, these particles are more strongly influenced by magnetic fields and have stronger Coulomb (charge) interactions. Despite these exotic features, a surprising number of materials, some quite familiar to solid state experiments, are proposed to be host to these states. The electrons in graphene, 1 1.1. Dirac materials 2 Figure 1.1: (a) Electrons in ordinary materials have a quadratic dependence of energy E as a function of wavevector k and obey the Schroedinger equation. In the absence of spin-orbit coupling, the bands are spin degenerate as indicated by the arrows. (b) Dirac electrons have a linear dispersion analogous to photons with a modified velocity. In graphene and topological insulators, the bands are pseudo-spin or spin polarized due to symmetries of the crystal. (c) In the 2D quantum spin Hall state, the bulk of the crystal is insulating and currents are carried by counter-propagating, spin-polarized edge modes. bismuth, Bi1−x Sbx , Bi2 Se3 , gray tin (α Sn), and quasiparticles (broken cooper pairs) in d-wave superconductors are all examples. We refer to these as Dirac materials. One example of the unique nature of Dirac electron systems is the ubiquity of quantum spin Hall (QSH) physics. The 2D QSH effect is depicted in Fig. 1.1c. Analogous to the integer quantum Hall effect, the bulk of the system is insulating and currents are carried by edge modes which are protected from scattering. Unlike the quantum Hall case, however, edge modes come in counter propagating pairs. Furthermore, these modes are spin polarized. This is an exciting state of matter because it presents a source of highly robust, spin-filtered states potentially useful for spintronic or quantum computing applications. Proposals for the QSH state arise in a number of Dirac materials. As discussed below, strong spin-orbit coupling leads to the realization of this state in 2D HgTe 3 1.2. Graphene quantum wells and 3D topological insulators [1, 2]. In graphene, the pseudo-spin associated with the underlying lattice predicts the same spin polarized counter propagating modes for the Landau levels near the Dirac point [3]. Whether or not the QSH state is realized in a given Dirac system is an experimentally important question. If the QSH state is absent, it is also important to understand the nature of the true ground state. We introduce the basic electronic properties of two candidate Dirac materials below. 1.2 Graphene Graphene is a single layer of carbon atoms arranged in a hexagonal lattice. Since it was first realized experimentally in 2005 [4], a vibrant subfield of condensed matter physics has developed to probe its experimental, theoretical, and industrially-useful properties (for a review, see [5, 6]). The hexagonal lattice of graphene is shown in Fig. 1.2a. In terms of crystal structure, the carbon atoms form a triangular lattice with two atoms forming the basis at rA and rB in the unit cell (shown in gray). The primitive lattice vectors are a1 = ! a, 0 , a2 = √ ! a 3 , a 2 2 (1.1) where a is the carbon-carbon bond length ∼ 1.42 Å. Of each carbon’s 6 electrons, 2 are tightly bound in 1s2 orbitals and the remaining 4 occupy the 2s and 2px,y,z orbitals. Hybridization of the in-plane orbitals gives the crystal structure great strength in plane, and leaves the remaining pz orbital to weakly bind adjacent layers (in the case of stacks of graphene forming graphite) or to roam in the plane (for single layer 4 1.2. Graphene Figure 1.2: (a) Crystal structure of graphene. There are two atoms in the basis which decorate a triangular lattice. Basis vectors and bond vectors are labeled. The two interpenetrating triangular lattices are shown colored in red and blue. (b) Band structure of graphene using the tight binding calculation in the text. The energy goes to zero at two distinct momenta K and K’. Near these points (shown in (c)) the dispersion is linear. graphene). Such a configuration of orbitals lends itself to a calculation of the electronic band structure in the tight binding approximation. We take as the atomic orbital the pz orbital on each atom and consider hopping between atoms with transfer integral t. Considering nearest neighbor hopping, examination of Fig. 1.2a shows that electrons always hop from sublattice A to B or vice versa (colored red and blue). The corresponding bond vectors for hopping bi are ! √ ! −a 3 b1 = a, 0 , b2 = , a , b3 = 2 2 √ ! −a − 3 , a 2 2 (1.2) A natural choice of basis to perform the tight binding calculation is amplitude of the wavefunction ψ per sublattice. Since each hopping is between sublattices, if we set the on-site (atomic) energy ǫ = 0, the 2 × 2 Hamiltonian H will be purely off-diagonal. 5 1.2. Graphene Assuming the states are Bloch-like, the off diagonal term HAB is HAB = −t 3 X eik·bi "i=1 = −t e ikx a −ikx a/2 + 2e cos √ 3ky a 2 !# (1.3) ∗ and HBA = HAB . Solving for the energy eigenvalues for H by use of the secular equation yields the spectrum shown in Fig. 1.2b. A particularly interesting feature is that at 6 points, the energy E crosses through zero at a sharp point (a Dirac point). In reality, there are only 2 such distinct points labeled K and K’, the others being related by reciprocal lattice vectors. Expanding Eq. 1.3 in δk around one of these √ points K = (0, 4π/3 3a), we have # " 2π √3a 2π iδkx a cos sin − δky HAB = −t 1 + iaδkx + 2 1 − 2 3 2 3 i 3 h = at − iδkx + δky (1.4) 2 Then H is 0 3 H = at 2 +iδk + δk x y −iδkx + δky 0 p− = v 0 p+ 0 (1.5) where in the last step we have explicitly written H in the form of the massless Dirac Hamiltonian with Fermi velocity v = 3at/2 and p± /~ = ±δkx +δky [7]. This dispersion is shown in Fig. 1.2c; the energy E for charge carriers near this point is 1/2 E(k) = ±~v δkx2 + δky2 (1.6) with corresponding eigenfunctions 1 1 ik·r ψ± (k) = √ e 2 ±eiφ (1.7) 1.3. 3D Topological Insulators 6 This electronic structure is at the heart of the interest in graphene and more generally in Dirac materials. Eq. 1.6 shows that (low-energy) charge carriers in graphene have a light-like dispersion relation, yet are Fermionic and carry charge. These massless Dirac fermions have behavior quite distinct from massive electrons, as becomes particularly clear with application of H in Chapter 3. 1.3 3D Topological Insulators 3D TIs are bulk bulk insulating crystals with 2D conducting surfaces. After the prediction [1, 2, 8] and realization [9] of analogous states in 2D, there has been a flurry of activity in confirming the existence of the 3D TIs [10, 11, 12, 13, 14, 15, 16, 17]. To understand the unique nature of the 2D states, it is useful to view the Angle Resolved Photoemission Spectroscopy (ARPES) results that have largely founded the experimental side of this field [13, 14, 15, 16, 17]. Shown in Fig. 1.3a-c are ARPES results for the proposed TI Bi2 Se3 , while Fig. 1.3d shows a schematic band structure based on theoretical expectations. TIs are predicted to have an energy gap between the bulk conduction and valence band within which a 2D surface band resides. The surface band itself is spin-polarized, has a Dirac dispersion, and (as described below) is protected from being destroyed by most kinds of impurities. These elements depicted in the cartoon in Fig. 1.3d are all reproduced in ARPES. For example, Fig. 1.3c shows that there is a 300 meV gap between the bulk valence and conduction band, with the 2D linear dispersion band in the gap. The band is continuous, showing no signs of a disorder induced gap. Spin polarization experiments are shown in Fig. 1.3b, demonstrating that the spin s has 1.3. 3D Topological Insulators 7 Figure 1.3: ARPES results (a-c) for topological insulator Bi2 Se3 from [15] and schematic band structure (d). (a) View along z direction of Dirac cone of surface states. The spin direction is noted with arrows. (b) Spin polarization measurement of electrons. The polarizations are antiparallel on opposite sides of the Dirac cone. (c) Bulk band gap and in-gap surface state Dirac cone. The band gap is 300 meV. (d) Schematic of prediction of band structure for 3D topological insulators. There is a bulk band gap with an odd number of Dirac cones of spin-polarized, linear dispersion states inside. 1.3. 3D Topological Insulators 8 a non-zero polarization and is opposite on each side of the Dirac cone. The nature of this spin polarization is depicted in Fig. 1.3a: s is locked perpendicular to the wave vector k. It is this combination of linear dispersion, spin polarization, and protection that drives interest in these materials. Before outlining the concepts of topological band theory underlying the connection between the 3D band structure and these novel 2D states, some intuition for the roots of these properties may be found by considering electrons on the 2D surface of a 3D crystal. Fig. 1.4 shows a schematic TI, focusing on the electrons on the top surface of the crystal. Inversion symmetry is broken at the surface of the crystal so that the electron experiences a trapping electric field Etrap in ẑ that arises as a gradient of the surface potential [18]. In the reference frame of the electron moving with wavevector k, this Etrap is felt as an effective magnetic field B ∼ k × Etrap . Then, it will be energetically favorable for s to point in the plane and perpendicular to k to minimize the energy s · B. This so-called Rashba contribution to the Hamiltonian is well known in systems with large spin-orbit coupling [19]. One can see immediately that any charge current in the plane would then be accompanied by a spin current. However, the sense of the spin polarization is exactly the opposite on the bottom surface (Etrap is in the opposite direction), so one needs to isolate the current on either crystal face to take advantage of the spin texture. This simple picture can also help us understand in what sense the surface bands are protected. Fig. 1.4a shows a schematic of the surface band structure with the spin configuration discussed above. In these materials there are no magnetic orderings, magnetic fields applied, or related phenomena, so viewing the behavior of the electrons should look the same if the arrow of time is reversed. More mathematically, 1.3. 3D Topological Insulators 9 Figure 1.4: (a) Simplified band structure of surface states demonstrating action of the time reversal operator T . (b) Electrons on the top surface of a topological insulator have their spin s pointed perpendicular to their wavevector k as determined by the effective B. (c) A band structure with an even number of Dirac cones is not protected by Kramers theorem from the opening of a gap. application of the time reversal operator T should leave the states in the system unchanged. If we consider the surface electrons, that means for each electron moving with a particular k, s, and energy E, there must be an electron moving at the same E with −k and −s. In this way, time reversal symmetry is preserved. This preservation is known as Kramers theorem [20]. This is demonstrated schematically in Fig. 1.4a. The texture of spins from the effective B obeys this symmetry. It is interesting to note that if inversion symmetry was preserved E(k, s) = E(−k, s), which together with time reversal symmetry would imply E(k, s) = E(k, −s). Thus, in the bulk of the materials where inversion symmetry is preserved the bands are spin degenerate. For the simplest case here, we have two lines of states defining E(k) (Fig. 1.4a). In principle, all odd powers in k are allowed. Consider the T operation again on the cone of states in Fig. 1.4a, but now acting at k = 0 (the Dirac point) instead of 1.3. 3D Topological Insulators 10 a finite k as before. For time reversal symmetry to be preserved, there must exist 2 states (with opposite s) at this point. While the cones can be deformed in this parameter space, the cones can never be split apart and still retain 2 states at k = 0 at the same E. In this sense, the continuity of the states is a topological property and the surface band is topologically protected from the opening of a gap. If instead we were to have 2 cones as in Fig. 1.4c, this argument would not be true- that structure belongs to a different topological class. There, Kramers theorem can be satisfied even with the opening of a gap (there are still an even number of states at each E). Originally formulated in a model of graphene due to Haldane [21], these ideas evolved with the theoretical prediction by Kane and Mele that graphene would be host to the 2D analog of these materials- a 2D insulating bulk wrapped with counterpropagating, spin polarized edge states [1]. This is an alternate realization of the QSH state proposed for graphene discussed in chapters 3 and 4, though in that case Landau level quantization is involved. However, since graphene is made of carbon and thus has small spin-orbit coupling, experimentally realizing the zero H QSH state proved difficult. It was realized soon after that HgTe based quantum wells would be a more promising candidate for realizing this state [8], and experimental verification soon followed [9]. This verification came from the transport experiment shown in Fig. 1.5. Different samples corresponding to topologically trivial insulators and QSH insulators were prepared. The trivial class was found to be true insulators when the chemical potential was tuned via a gate potential in to the bulk gap (black curve in Fig. 1.5), while those predicted to have conducting edge states showed conductance σxx = 2e2 /h. Importantly, this σxx did not depend on the geometry of the sample; the current was shown to be 1D. Further experiments with non-local 1.3. 3D Topological Insulators 11 transport measurements further demonstrated the existence of conducting edge modes [22]. The theoretical generalization to 3D materials followed soon after [10, 12, 23]. Figure 1.5: Proof of 2D topological insulator in HgTe system from [9]. Device I is a trivial insulator and shows a diverging resistance in the bulk band gap. Devices III and IV are topological insulators and show conductance G = 2e2 /h consistent with edge mode transport. Device II is a non-trivial insulator but too large for the edge modes to persist unscattered. The theoretical framework developed in these works is the concept of topological band theory. In 2D, states can be characterized by a quantity known as the Chern number nC , which may be calculated as a integral over the bulk band structure of a given material [24]. Materials with different nC belong to topologically distinct classes. For example, the vacuum has nC = 0, while the quantum Hall insulator has nC = 1. One important concept in topological band theory is that when materials of different nC are brought in to contact, a bound state is formed at the interface. An example familiar to many is the bound state trapped between the quantum Hall insulator and vacuum- the edge state (skipping orbit) shown in Fig. 1.6a. As was 1.4. Organization 12 Figure 1.6: (a) Depiction of state trapped between two insulators with different Chern number. The skipping orbit is trapped between the quantum Hall insulator and vacuum. (b) A Majorana bound state is trapped between superconductivity and magnetism induced gapped states in a topological insulator. shown by Thouless and co-workers [25], one can understand edge states and the quantum Hall effect that arises as σxy = nC e2 /h. The next step forward was to consider topological classes of materials that, unlike the quantum Hall case with a large magnetic field H, exist in the presence of time reversal symmetry. This new topological class characterized by the so-called Z2 invariant was shown by tricks of symmetry to be straightforward to calculate [11], and soon particular material systems based on Bi alloys were proposed. Experimental verification led by Hasan, Cava, and coworkers by ARPES followed in Bi1−x Sbx [13, 14], followed by the discovery of additional realizations in Bi2 Se3 [16, 15] and Bi2 Te3 [17, 26]. Proposals for realizing exotic states of matter such as Majorana fermions (Fig. 1.6b) have fueled intense interest in these materials [27]. 1.4 Organization We describe the general experimental apparatus in Chapter 2. Further details relevant to each experiment are given at the beginning the appropriate chapter. Chapter 3 describes the zero energy state of graphene in high magnetic field. Chapter 4 1.4. Organization 13 describes the thermoelectric response of the Dirac states in graphene. Chapter 5 presents a materials study of topological insulator Bi2 Se3 , focusing on low temperature transport. Chapter 6 focuses on probing the Dirac states in Bi2 Se3 in a field effect transistor geometry. Finally, we summarize our results in chapter 7. A brief overview of quantum corrects to transport is given in the appendix. Chapter 2 Experimental setup Experiments presented in this thesis are all transport measurements. Electric, thermoelectric, and thermal transport properties were all performed. These measurements are done as a function temperature T , magnetic field H, and electron density n varied by an electrostatic gate or chemical composition. Here we introduce the basic machinery used to vary the parameters T and H. Then we present an overview of measurement techniques. 2.1 Cryostats and Magnets A liquid helium (LHe) cooled superconducting 14 T magnet is the main environment for our experiments (Oxford Instruments). A schematic of the configuration for measurements for T down to 4 K is shown in Fig. 2.1a. A magnet submerged in LHe is enclosed within a vacuum isolated dewar. A power supply (PS-120, Oxford Instruments) supplies current to the magnet. Inside of the bore of the magnet, we place a stainless steel cryostat consisting of an outer vacuum jacket, inner vacuum 14 2.1. Cryostats and Magnets 15 Figure 2.1: (a) Schematic of magnet and He-4 cryostat. The insert (probe) is shielded from a LHe bath by several layers of vacuum. The bath itself is shielded from air by a vacuum jacket. T between 4.2 K and 400 K can be achieved varying the amount of He exchange gas in the vacuum jackets and heating power of the probe. (b) Schematic of He-3 cryostat used for measurements down to 0.3 K. The sample is placed in vacuum in physical contact with the He-3 pot where liquid He-3 is condensed and cooled. (c) View of He-4 probe sample space. The sample is contained within a vacuum chamber made by a taper seal. Electrical connections come from the top of the probe and terminate in contact pads. T is monitored with a cernox thermometer. A resistive heater is used to warm the sample. 2.1. Cryostats and Magnets 16 jacket, and space to put an experimental probe. All three of these spaces are kept at low pressure by a diffusion pump (Edwards Diffstak). The sample is mounted on a home-made experimental probe on a Cu cold finger shown in Fig. 2.1c. Experimental wiring (phosphor bronze wires for AC measurements, Cu wires for DC measurements) comes down a long stainless steel tube that connects to a mating copper taper seal and cap. The top of the probe has a multipin connector which allows connection to measurement equipment. The sample is connected with Au wires to the contact pads and the chamber sealed with vacuum grease. The probe is pumped down to 10−6 Torr with a turbomolecular pump before loading in the cryostat. T is monitored by a cernox resistor (Lakeshore). Temperature control is accomplished by PID control with a Lakeshore 340 temperature controller. Care is taken to minimize the power dissipated in the cernox to avoid unintentional heating, typically using an excitation current of 10 µA. The heating power is supplied by a thin film 25 Ω resistor (Minco flexible heater) mounted on the cold finger, while the cooling power comes from the LHe bath. The amount of cooling power can be greatly varied by introducing different amounts of He gas in to the vacuum chambers. In this way, T can be controlled within 10 mK over the range 10 K to 400 K with the sample in vacuum. In order to go down to the base T of 4.2 K, He gas is added to the sample space. However, this can be problematic for thermal transport measurements as the He gas presents an alternate path for heat current to flow. For measurements below 4 K, we use a closed cycle He-3 cryostat (Oxford Heliox) in place of the 3 chambered He-4 cryostat. A schematic of the He-3 setup is shown in Fig. 2.1b. He-3 is contained at the top of the probe in a large cylinder. When the probe is inserted in to the LHe bath, the He-3 is cryopumped down to the Sorb where 2.1. Cryostats and Magnets 17 Figure 2.2: Experimental setup in the National High Magnetic Field Lab in Cell 5. Labmate Lu Li can be seen loading the He-3 probe in the the cryostat. The cryostat itself is not in direct contact with the magnet to minimize vibrational noise. The maximum H in this magnet is 31.2 T. 2.2. Transport Experiments 18 it is absorbed by charcoal. By heating the charcoal to 45 K, we can force He-3 to flow down toward the He-3 pot, on the outside of which the sample is mounted. Cooling the He-3 via the 1 K pot (LHe from the bath which is cooled to ∼1.5 K via lowering of the vapor pressure with a mechanical pump) condenses the He-3 and allows it to collect in the He-3 pot. Once He-3 liquid is collected, the sorb is cooled to 4 K and begins to slowly reabsorb the He-3 gas evaporating from the He-3 pot. This lowers the vapor pressure of the He-3 and cools the sample down to the base T of 0.3 K. After running this condensing cycle for 1 hour, the hold time is more than 10 hours. Measurements are also performed at the National High Magnetic Field Laboratory in Tallahassee, Florida. A picture of the cryostat and magnet is shown in Fig. 2.2. For this cryostat (He-3 system B in Cell 5), the maximum field is 31.2 T. A long probe with the sample mounted on a 16-pin dip header is inserted in to a cryostat with liquid He-3 inside. The principle of operation is the same as detailed for Fig. 2.1b, but here the sample is immersed in the liquid rather than isolated in vacuum. The largest difference between the experimental setup is that the magnet in Fig. 2.2 is a water cooled, copper magnet instead of a superconducting magnet. The cooling water used to carry heat away from the magnet causes significant mechanical vibration in the system and causes measurements to be noisier. 2.2 Transport Experiments Transport experiments on macroscopic crystals are performed in a configuration similar to Fig. 2.3. We typically measure in-plane (defined as the xy plane with i, j = (x, y)). For experiments involving thermal gradients, the crystal is mounted 19 2.2. Transport Experiments vertically using silver epoxy (Epotek H20E) on a sapphire substrate. A ceramic heater (Film Microelectronics) is mounted to the top of the crystal which has a resistance 1 kΩ, independent of H and T . Gold wires (0.001 inch, California Fine Wire) are attached to the sample using conducting Ag paint (Dupont) to measure electric potentials Vij and inject electrical current I. These wires connect to contact pads on the measurement probe. The sapphire substrate is attached the cold finger in Fig. 2.1b with thermal joint compound (Type 120, EG&G Wakefield Engineering). Care is taken to isolate the wires thermally from the cold bath but suspending them with nylon pillars. The principle behind this design is to force the heat generated by Joule heating of the resistor to flow down the sample to the sapphire, which is in good thermal contact with a Cu cold finger. In this way, we know the heat heating power put in to the sample. Temperature gradients ∂i T are measured by thermocouples (type K, Omega Engineering) for T greater than 5 K and with calibrated miniature RuO2 temperature sensors (Lakeshore) for T below 5 K down to 0.3 K. We can then measure the resistivity tensor ρij , thermoelectric tensor Sij , and thermal conductivity tensor κij through Ei = ρij Jjc (2.1) Ei = Sij ∂j T (2.2) Jiq = κij ∂j T (2.3) where J c is the input charge current, J q is the input heat current and Ei is the electric field determined from the voltage measurement. In principle it is possible to prepare 2.2. Transport Experiments 20 a sample to measure all of these quantities, though not simultaneously. In practice, due to complexity in mounting, only one of Eq. 2.1 to 2.3 is measured in a single experiment. For an example of a purely electrical measurement, see Fig. 5.22. This does not require the sample to be mounted vertically, since no J q are involved. Figure 2.3: Typical bulk crystal (Bi2 Se3 ) prepared for transport measurements. The crystal (shiny surface) is standing vertically with a heater mounted on top of the crystal. Gold wires and thermocouples suspended by nylon pillars are attached to the sample. The pillars and crystal are epoxied to a sapphire substrate. We can measure the same quantities in microscopic samples, except for κ since the sample is not mounted vertically. Fig. 2.4 shows a graphene device patterned for measurement of ρij and Sij . Crystals are cleaved by mechanical exfoliation on to a ntype Si substrate with 300 nm of thermally grown SiO2 . All electrodes are patterned with ebeam lithography followed by thermal evaporation of 5 nm Cr and 30 nm of Au. A heater is patterned near the sample; putting mA order current through the heater generates a mK scale gradient across the sample, which is enough to detect thermoelectric voltages. The details of the Sij measurement are given in Chapter 4. 2.2. Transport Experiments 21 Figure 2.4: (a) Graphene microdevice prepared for thermoelectric and electric measurement. A heater is patterned on the right that generates a temperature gradient across the crystal. The gradient can be monitored by measuring the resistance of the patches of Au trace which are calibrated as thermometers (labeled therm). The remaining leads are used to measure Vxx and Vxy either due to ∇T or an applied current bias between the therm leads. The graphene crystal is outlined with the dashed white line. (b) Field effect transistor geometry used to induce carriers in thin crystals. A voltage is applied between the Si back gate and a Au electrode in contact with the crystal. This generates an electric field pointing vertically though the dielectric which attracts carriers. 22 2.2. Transport Experiments The largest difference experimentally between the two situations is electrostatic discharge problems. Microdevices of the type shown in Fig. 2.4 can be easily destroyed by a discharge of static across the measurement leads or large currents flowing on the ground plane. When the device is not being actively measured, all leads are grounded together using a shorting rotary switch (Electroswitch 04-302). Care is taken to have all instruments share the same ground as well as to ground the operator before touching the experimental apparatus. Another difference is the ability to use the electric field effect (EFE) to vary the density of electrons n in a very thin sample. The setup is shown in Fig. 2.4b. Applying a gate voltage VG changes the chemical potential of the crystal by bringing carriers in from the Au leads. For graphene devices, we find that the number of carriers nG induced in the sample follows the expectation for a parallel plate capacitor geometry, nG = CVG e (2.4) where C is the capacitance per unit area of the gate dielectric 1.14 ×10−4 F / m2 . AC voltages are monitored with Stanford Research (SR) 830 lock-in amplifiers. Using SR560 preamplifiers we can reduce the noise floor below 10 nV. Alternatively, a Keithley 6221 current source can be interfaced with a Keithley 2182A to average positive and negative DC bias measurements (Delta mode). This removes effects of thermal drift and DC offsets. When pure DC measurements are required, Keithley 2001 voltmeters with a Keithley 1801 pre-amplifiers are used. Effects of DC drift are minimized by using all Cu clamped connections with solder free joints. This is particularly important for those connections coming from low T to atmosphere. With these precautions, DC noise levels of 20 nV are possible. Data are collected via GPIB 488.2 connections using a National Instruments data acquisition card. Chapter 3 High field ground state of Graphene at the Dirac Point This chapter focuses narrowly on one basic question: What is the electronic ground state of graphene in a magnetic field H at the Dirac point? The reason for asking this particular question is that in terms of electron density ne and H, this is where interaction effects between electrons should be strongest. We start with a simple introduction to Landau level quantization in graphene, present experimental results demonstrating a unique, insulating ground state [28, 29], and discuss implications for broken symmetry in the ground state. 3.1 Electronic Band Stucture of Graphene in H We begin by introducing H in to the electronic structure calculation from Eq. 1.5. Application of a magnetic induction µ0 H = ∇ × A modifies H by k = −i∇ → −i∇ + eA, where A is the vector potential. Again we work in the vicinity of valley 23 3.1. Electronic Band Stucture of Graphene in H 24 K. Following Ando [30], we define the following operators (−ikx + ky )ℓB √ (3.1) 2 p where ℓB is the magnetic length defined as ℓB ≡ h/eB. Rewriting H with Eq. 3.1, √ 2v 0 a H= (3.2) ℓB a † 0 a≡ (ikx + ky )ℓB √ 2 a† ≡ It follows that a and a† obey the commutation relation a, a† = 1, and that we may use Eq. 3.1 as lowering and raising operators to construct the Landau level wavefunctions ψn by n a† ψn = √ ψ0 n! (3.3) where aψ0 = 0. The action of a and a† together is aa† ψn = nψ. Using this raising and lowering structure, we can see that the structure of the energy level spacing is p v En = sign(n) 2 |n| ℓB (3.4) with the wavefunction given by E n − 1 ψ(n) = E n (3.5) There are several interesting aspects of Eq. 3.4. The energy spectrum of the Landau levels in particular is distinct from the massive electron case using the Schrodinger equation. Whereas the energy level spacing in the latter is that of the simple harmonic oscillator with E ∝ n + 1/2, here the energy level spacing is not uniform in n. Moreover, E0 = 0, meaning that there is a state at zero energy for n = 0. Considering 3.1. Electronic Band Stucture of Graphene in H the wave function in Eq. 3.5, for n = 0 we have 0 ψ(0) = E 0 25 (3.6) meaning that there is no amplitude for the wavefunction on the A sublattice. As Figure 3.1: Schematic representation of electron amplitude (orange) for a single valley at the n = 0 Landau Level. Both valleys are polarized on opposite sublattices. depicted in Fig. 3.1, this would imply that the carriers for valley K are localized on one sublattice only. However, valley K’ contributes exactly the opposite carriers so there is no charge-density wave when valley symmetry is preserved. Nonetheless, this is an indication that interaction effects may produce unique ground states near the Dirac point at high H. One view of this sub-lattice polarization is that for the n = 0 state the valley and sublattice degeneracy are equivalent. This is only true at n = 0 [31]. 3.1. Electronic Band Stucture of Graphene in H 26 Figure 3.2: Breaking the four-fold degeneracy √ of the n = 0 Landau Level. Two scenarios are the exchange energy Eex ∼ B causes the physical spin (a) or the valley spin (b) to split first. More generally, graphene has a four-fold degeneracy of 2 spins and 2 valleys at each Landau level. Upon increasing the Coulomb interaction in a high H, one expects this degeneracy to be lifted. A question related to the charge density wave picture above is whether the spin, valley, or some combination of the two is most sensitive to being lifted in intense H. Figure 3.2 shows two possible scenarios for lifting the degeneracy to produce a broken symmetry ground state. A large amount of theoretical work has been devoted to this issue [32, 31, 33, 34, 35, 36]. Depending on the relative role of long-range and short-range Coulomb interactions, a variety of broken symmetry ground states at n = 0 are predicted to be driven by exchange energy. In one scenario, the exchange energy leads to ferromagnetic polarization of the physical spins [3, 37]. This produces a spin gap in the bulk without affecting the valley degeneracy, as shown in Fig 3.2a. Near the edge of the sample, the residual valley degeneracy is lifted by the edge potential (see Fig. 3.3). In a simple picture, because at n = 0 the valley degeneracy and sublattice degeneracy are equivalent, it may not be surprising that the boundary condition at 3.1. Electronic Band Stucture of Graphene in H 27 Figure 3.3: In the physical spin splitting scenario, the edge potential breaks the symmetry between the two valleys. According to the boundary condition at the edge, this causes the two edge modes to counter propagate (shown in (a)). This leads to the counterpropogating modes to carry opposite spin polarization, as depicted in (b). the edge can distinguish between K and K’. An important consequence of this edge splitting at n = 0 is the existence of spin-filtered counter propagating edge modes. These result in a residual conductance of 2e2 /h regardless of the magnitude of the spin gap in the bulk, and are a realization of the quantum spin Hall effect [38]. If however, exchange polarizes the valley spin first (shown in Fig 3.2b) the CPE modes are not present [39, 40, 41]. Yet another possibility is some spontaneous ordering of the spin described by the valley spin / real spin space [42]. 28 3.2. Experimental Results 3.2 3.2.1 Experimental Results Quantum Hall effect and Landau Levels at ν = 0 Measurements of graphene in large H at temperature T = 0.3 K are shown in Fig. 3.4. The Hall conductivity σxy exhibits quantum Hall plateaus corresponding to the filling ν relation σxy e2 1 4e2 n+ =ν = h h 2 (3.7) where n is the Landau level index, in agreement with the original experimental findings [43, 44]. The factor of 4 is due to the spin and sublattice degeneracy, while the half-integer shift is a result of the non-zero Berry’s phase of the electron wavefunction [44, 45]. The Landau level at filling n = 0 stands out. Whereas the longitudinal resistance Rxx is nearly unchanged for levels n = ±1 for H = 8 T to 14 T, the peak in resistance for n = 0 dramatically increases. Accordingly, in σxy an additional plateau at σxy = 0 appears. This is more evident in Fig. 3.5b. We define R0 to be the resistance at charge neutrality. It is evident from Fig. 3.5a that R0 increases as T decreases. At 14 T, the n = 0 Landau level is split into 2 subbands even at T = 100 K (Fig. 3.5b). It is also instructive to view the T dependence of σxx , shown in Fig. 3.6a. As a function of decreasing T , it is evident that the longitudinal conductivity σxx crosses over from metallic (σxx increasing with decreasing T ) to non-metallic (σxx decreasing with decreasing T ) as H is increased. There is also an obvious saturation at low T ; we address this further below. One noteworthy measure is that the separatrix between metallic and non-metallic conductivity occurs at σxx ∼ 4e2 /πh. 29 3.2. Experimental Results -40 -20 100 0 20 T = 0.3 K 40 (a) K7 80 14 T R xx (k ) 60 11 40 8 20 14 T 11 8 0 8 (b) 11 8 14 T 2 (e /h) 4 xy 0 -4 -8 -40 -20 0 20 40 V ' (V) g Figure 3.4: (a) Longitudinal resistance Rxx of device K7 at T = 0.3 K in the quantum Hall regime. The effect of increasing H on the n = ±1 Landau level is negligible, while the n = 0 level responds with a sharp increase in Rxx . The inset shows a false-color micrograph of the sample. The scale bar is 5 µm. (b) Hall conductivity σxy shows quantum Hall plateaus following the filling structure in Eq. 3.7. A plateau at σxy = 0 develops at the highest fields. 30 3.2. Experimental Results -10 -5 200 0 5 (a) 10 H = 14 T 0.3 K 1 150 ) 2 5 R xx (k 100 10 50 20 40,100 0 3 4 (b) 2 3 xy xy 2 0 xx (e /h) 2 xx 2 (e /h) 1 (0.3 K) -1 100 1 40 -2 20 10 0 -10 0.3-2 K 5 -5 0 V g 5 -3 10 (V) Figure 3.5: (a) Resistance as a function of gate voltage VG near the Dirac point in the quantum Hall regime. Decreasing temperature causes the resistance to grow dramatically. (b) Longitudinal conductivity σxx in the same range as (a). Here, splitting of the central Landau level can be seen up to 100 K. At the lowest temperature 0.3 K, Hall conductivity σxy is also shown. The plateau at σxy = 0 is pronounced. 31 3.2. Experimental Results T (K) 1 10 100 2.0 6 T 7 (e /h) 1.5 8 2 9 10 0 xx 1.0 11 12 0.5 13 (a) 14 0.0 200 0.3 K (b) 2 120 0 (k ) 160 R 5 80 20 40 40 0 5 10 15 H (T) 0 Figure 3.6: (a) Temperature dependence of σxx at the Dirac point. At low fields, metallic behavior is observed below 40 K. However, upon increasing H above 10 T, a non-metallic temperature dependence is observed. Below ∼ 2 K, saturation is seen. (b) Dirac point resistance R0 over the measured field range at selected T . The divergent nature of the R0 (H) profile gets progressively stronger as T is lowered. 3.2. Experimental Results 32 Viewing R0 (H), we see that as T decreases, R0 becomes an increasingly divergent function of H. At 0.3 K and 14 T, R0 reaches a value of nearly 200 kΩ for this device. As we describe below, this behavior of R0 is characteristic for high quality (high mobility µe ) crystals. Figure 3.7: The central Landau level is split with increasing of H from 6 T to 14 T. σxx at charge neutrality is pushed below 0.15 e2 /h at the highest field 14 T. Finally, an interesting way of viewing the H dependence of the charge transport is the profile of σxx across the lowest Landau level with increasing H. As shown in Fig 3.7, H breaks the degeneracy at n = 0, though it is not evident if it is physical or valley spin being separated. 33 3.2. Experimental Results 3.2.2 Sample Quality Dependence An important question regarding the acute dependence of the Dirac point resistance R0 to H is if it is the intrinsic behavior of the electronic structure of graphene or an impurity effect. To answer this, several devices of varying quality were measured. While there are many metrics to characterize devices, the two most appropriate here are µe and the offset gate voltage V0 . While from section 1.2 one would expect the chemical potential to be at the Dirac point in devices to occur in undoped graphene, in practice one finds that in fabrication charge impurities are trapped in devices that require a (usually positive) voltage (V0 ) to be applied to the back gate to reach charge neutrality. This trapped negative charge markedly effects the quality of devices. Fig. 3.8b shows a variety of devices with different V0 . For each device, the width of the resistive peak characterizes the electron mobility through G = neµe , as VG = ne/C where C is the capacitance of the 300 nm SiO2 dielectric (1.14 × 10−4 F / m2 ). The devices with smaller V0 clearly have narrower peaks and thus higher µe . Table 3.1 collects these parameters for several representative devices. units K5 K7 K8 K18 K22 K29 V0 R0 (14 T) µe V kΩ 1/T 3 80 0.3 1 190 1.3 12 15 0.6 20 7.5 0.9 -0.6 > 280 2.5 22.5 7 0.2 Table 3.1: Sample parameters. V0 is the gate voltage required to reach charge neutrality. R0 (14 T) is the resistance measured at charge neutrality at 14 T and 0.3 K. µe is the electron mobility at H = 0. 3.2. Experimental Results 34 Figure 3.8: (a) Annealing of sample K32. The offset gate voltage V0 can be moved from greater than 20 V to below 4 V, but the mobility of the sample does not improve significantly. (b) R(VG ) for several devices measured as-prepared. Those devices with naturally lower V0 have significantly narrower profiles. 3.2. Experimental Results 35 The relevance to electron transport near charge neutrality becomes evident upon plotting the Rxx profile of devices with different V0 in the quantum Hall regime. Fig. 3.9a shows the result for devices K7, K5, K8, and K29. As reported in Table 3.1, lower V0 and higher µe correlates with large R0 . This can be seen in both the VG dependence and detailed H dependence in Fig. 3.9a and Fig. 3.9b, respectively. One method known in the literature for decreasing V0 is annealing. While there are many different methods ranging from baking devices in a Ar/He atmosphere [46] to driving a large current through the device to anneal through Joule heating [47], we find that while annealing can reduce V0 , it does not substantially improve the performance of the device compared to as prepared devices with lower V0 . This is demonstrated in Fig. 3.8a, as progressively annealing a device moves V0 down without substantially narrowing the resistive profile. In extremely intense H > 25 T, however, we can see that increasing sample quality through annealing marginally improves devices. Fig. 3.10 shows a device measured to H = 33 T before and after annealing in a He atmosphere at 250◦ C for 1 hour. Annealing pushes the measured R0 to larger than MΩ values at 33 T after the anneal, whereas before there was not an obvious divergence. This can also be seen in the T dependence of R0 for the same device in Fig. 3.11. Before annealing, Rxx is only weakly dependent on T and H, though it does exhibit a crossover from metallic to non-metallic behavior. After annealing, the device increases in resistance much more dramatically at 33 T upon decreasing T . However, while annealing this device allows us to see that increasing mobility favors the high resistance state, it does not produce a device of high quality to produce the state in H < 20 T. 36 3.2. Experimental Results 200 200 (a) (b) T = 0.3 K H = 14 T (k ) R K22 0 100 K5 R 100 T = 0.3 K xx (k ) K7 K7 K8 K18 K29 0 -10 0 10 V g (V) 20 5 10 15 0 H (T) 0 Figure 3.9: (a) Comparison of resistance profiles of the n = 0 Landau level for 4 samples of varying quality. As the offset gate voltage V0 becomes progressively larger, the peak in resistance becomes lower. All measurements are performed at 14 T and 0.3 K. (b) Dirac point resistance R0 as a function of H for three different samples. Samples of higher mobility and lower V0 diverge at systematically lower H. See Table 3.1 for parameters. For sample K7, the solid curve is a trace taken with fixed Vg while the points represent data obtained from taking the peak resistance from a sweep of VG at fixed H. 3.2. Experimental Results 37 Figure 3.10: (a) Resistance near the n = 0 Landau level in high H for device K60 before annealing. The peak does not reach above 105 Ω even at 33 T. (b) After annealing, V0 has been lowered by ∼ 20 V, and a MΩ peak is seen in resistance. 3.2. Experimental Results 38 Figure 3.11: (a) Crossover from metallic to non-metallic behavior in the relatively low quality device K60 before annealing. A field of 20 T is required to see the crossover. (b) After annealing, the crossover occurs below 10 T. The anneal was performed in a He atmosphere at 250◦ C for 1 hour. 3.2. Experimental Results 3.2.3 39 Device Heating A clear question to ask is how large R0 becomes in large H. In Fig. 3.9b, for example, R0 (H) is only shown to 200 kΩ at ∼ 10 T. The reason for the truncated measurement is that when the device begins to have a divergent resistance, Joule heating becomes a serious problem. Several spurious measurements are shown in Fig. 3.12 measured with the constant current bias indicated. Figure 3.12: (a) Spurious measurement of divergence peak with a fixed current bias of 10 nA in sample K52. Upon increasing H, the peak appears to invert. An experiment that avoids Joule heating shows monotonically increasing resistance profile with H (see Fig. 3.18). (b) Heating of devices K22 and K23 beginning at H near the divergent resistance. Even relatively low current excitation of 1 nA causes a clear break in slope followed by unreliable measurements. More directly, Fig. 3.13 shows that upon increasing the current bias above nA 40 3.2. Experimental Results excitation, the resistive peak is suppressed. This shows that power dissipation as low as pW can be problematic for measuring the high resistance state. While this seems incredibly low, being a single layer of atoms only loosely coupled to the heat bath, it is in fact not unreasonable for such a low excitation to cause thermal runaway problems. 200 T = 300 mK 150 15 nA H = 14 T 2 nA 100 R xx (k ) 0.6 nA 50 0 -3 0 3 V (V) G Figure 3.13: Resistance near the charge neutral point for device K7 measured at 14 T and 0.3 K. The applied current bias is changed between 15 nA, 2 nA, and 0.6 nA. Clearly the sample heating becomes problematic between 2 nA and 15 nA as seen by the suppression of the peak. This is consistent with heating the sample to several K. To measure the high resistance state carefully, then, we have to adopt a technique that allow for ultra-low dissipation in the device. The measurement circuit we use is shown in Fig. 3.14. Instead of applying a constant current, we instead apply a 3.2. Experimental Results 41 constant voltage to the circuit. Then, by measuring the current in the circuit at a low AC frequency as well as the voltage drop across the device, we can do phase sensitive measurements to ∼ 40 MΩ with dissipation dropping down to the aW level. As we still use a four-probe measurement scheme, the input-impedance of our voltage preamplifier’s limits the resistance we can measure below 100 MΩ. Because we will be interested in the detailed functional form of R0 (H), we prefer this method to a 2-probe scheme which could resolve R in to the GΩ regime. 3.2.4 Measurements with pW dissipation Adopting the measurement scheme in Fig. 3.14 and going to extremely high H, we can show that R0 is a divergent function of H. Fig. 3.15 shows that while at higher T R0 increases only mildly with H, at 0.3 K the value diverges to our measurement limit. As we show below, such a sharp increase of R0 cannot not be well characterized with a power law fit in H. Instead, we will have to invoke an exponential dependence. The impact on σxx and σxy is dramatic. Shown in a separate device in similar conditions, in Fig. 3.16 it can be seen that σxx is suppressed to a wide plateau at zero in H > 28 T. Similarly, the σxy zero filling plateau observed in lower fields is extended to a wide range (Fig. 3.17). It is instructive to look at the detailed H dependence of the resistance divergence. Fig 3.18b shows a color plot of Rxx as a function of H and VG at 0.3 K. The high resistance state is confined to the lowest densities. In fact, the divergence is confined to below the filling ν = ±1, as indicated by the dashed lines. 42 3.2. Experimental Results Voltage divider W 100 k p -filter Lock-in A low-pass filter pre-amplifiers p Lock-in V2 -filters Ammeter Lock-in V1 W 1 k Sample J18 sample Figure 3.14: Schematic of the low-dissipation voltage-regulated circuit. Lock-in amplifier A produces a regulated voltage at 3 Hz that is reduced to an amplitude of 40 µV by a 100:1 voltage divider. The signal goes through a π-filter, a buffer resistor of 100 kΩ, and a low-pass filter before entering the dewar. The ac current passing through the graphene sample is measured by a picoammeter (DL 1121), whose output is phase detected by lock-in A. The longitudinal voltage Vxx and Hall voltage Vxy are phase detected by lock-ins V1 and V2, respectively, after transmission through a bank of filters and high-impedance preamplifiers. As shown, all wires entering the dewar are buffered by 1 kΩ nichrome thin-film resistors. The inset shows sample J18. 3.2. Experimental Results 43 Figure 3.15: Resistance at the Dirac point for sample K52 measured to 31.2 T. Below 5 K, the resistance increases from ∼ 10 kΩ to more than 40 MΩ (our measurement limit). The profile at the lowest T is divergent. The inset shows the low R behavior. 44 3.2. Experimental Results 2.5 20 25 J18b 2.0 28 15 T = 1.6 K 31.2 1.5 1.0 xx (e 2 / h) 10 T 0.5 0.0 0 5 10 V G 15 (V) Figure 3.16: σxx for sample J18 for field up to 31.2 T. At 10 T, the n = 0 degeneracy is intact, but it is progressively broken with increasing H. At the highest H, breaking of the remaining degeneracy can also be seen between the split subbands. The conductivity at 31.2 T at and around the Dirac point is nearly zero. 45 3.2. Experimental Results 6 J18b T = 1.6 K 4 xy (e 2 / h) 10 T 15 2 20 25 28 0 31.2 -2 -4 -6 -10 -5 0 5 V G 10 15 20 25 (V) Figure 3.17: Quantum Hall plateaus for sample J18 to 31.2 T. At the highest fields the σxy = 0 plateau extends over more than 5 V. 46 3.2. Experimental Results 30 10 10 B (T) 28 10 26 10 24 10 22 7 W 6 5 4 3 2 10 (b) 20 -10 <10 -5 0 11 n 2D (10 5 -2 cm 10 ) Figure 3.18: (a) Detailed field dependence of the resistance profile in device K52 increasing H from 10 T to 31 T. (b) Color plot of measurements taken from (a). The high resistance state is confined below the lowest filling ν = ±1 (marked with dashed line). The 2D density is calculated assuming n2D = CVG /e, where C is the capacitance per unit area of the 300 nm SiO2 gate dielectric 1.14 × 10−4 F / m2 . 47 3.3. Discussion 3.3 Discussion We now attempt to characterize the divergence in resistance seen at the Dirac point. As mentioned in 3.1.3, the functional form of R0 (H) in Fig. 3.15 cannot be satisfactorily fit by a power law in H. We find instead that empirically we can capture the sharp divergence in R0 with the form √ R0 = Ae2b 1−h (3.8) where h ≡ H/HC , HC being a fit parameter determining the critical field at which the device becomes an electrical insulator. A and b are also fit parameters. The quality √ of this fit is best shown by plotting R0 on a logarithmic scale against 1/ 1 − h, which should yield a straight line. For device K52 measured to high H, we find that over 3 decades in R the fit is extremely good with A = 440, b = 1.54, and HC = 29.1 T. We gain further confidence in this fit by comparing the results of several samples. Those devices taken to high H > 30 T are shown in Fig. 3.20. Each shows the characteristic divergence (see Fig. 3.20a), and upon casting the data in the form of Eq. 3.8 we find that all the data collapse on to a universal curve with the same A and b, but different HC . The fitting results are collected in Table 3.2. Another fact clear from Table 3.2 is that low HC correlates well with low V0 but poorly with the size of the device characterized by the voltage lead separation L. The physical inspiration for Eq. 3.8 is the Berezinskii Kosterlitz Thouless (BKT) transition [48, 49]. In the context of 2D superfluids, this is the notion that fluctuations in the phase of the order parameter determine the nature of the transition. Topological defects (vortices) occur in the phase angle to disrupt long-range order. 48 3.3. Discussion Figure 3.19: Fit of R0 (H) from sample K52 (see Fig. 3.15) to the form in Eq. 3.8. The fit shows excellent agreement over more than 3 decades in resistance. units K22 K7 K52 J18 J24 V0 V -0.6 1 3 20 24 L HC µm T 3 11 ± 1 1.5 18 3.5 29 2.75 32 2 36 Table 3.2: Sample parameters and fit results. V0 is the offset gate voltage. L is the distance between the voltage leads. HC is the critical magnetic field determined by the fit of Eq. 3.8. 49 3.3. Discussion 108 T = 0.3 K ) 107 106 0 ( K52 5 10 J24 J18 104 103 (a) 0 10 20 0 108 30 H (T) 0 ( ) 107 106 105 104 103 (b) 1 2 3 4 -1/2 (1 - H/H ) c Figure 3.20: (a) Resistance divergence in three samples J18, J24, and K52 measured to high H. All the sample reach our measurement limit. All measurements are performed at 0.3 K. (b) Casting the data in (a) to the form of Eq. 3.8. With our measurement error, all the data collapse on to a single curve. The only scaling parameter in the fit to change for the 3 devices is the critical field HC . 50 3.3. Discussion A simple energetics argument gives the BKT transition temperature TBKT , which is lower than the mean field transition temperature. The energy of a single vortex Ev is Ev = η ln R a (3.9) where η is, for example, in the case of a 2D superfluid, the superfluid density, R is the system size, and a is the size of a vortex core. The entropy of a vortex can be calculated from the number of possible possible states Ω as S = kB ln Ω = kB ln R2 a2 (3.10) where kB is Boltzmann’s constant, and we estimate the number of positions of the vortex to by the ratio of the system size πR2 and the vortex size πa2 . The free energy F contribution of a vortex is then F = E − T S = η ln R R − 2T kB ln a a (3.11) so that it becomes energetically favorable to have vortices below TBKT = η/2kB . Below TBKT , the vortices bind together as vortex-antivortex pairs and there is long range order in the system. For 2D superconductors, the BKT transition is only valid when R (and a) is smaller than the 2D magnetic screening length. In this regime, the energy of a vortex follows Eq. 3.9. Experimental evidence for BKT physics in this regime is very strong. Fig. 3.21 shows two independent verifications of the BKT theory [50]. In 3.21a, a non-linear IV curve is measured as a result of pulling apart the vortex-antivortex pairs. In 3.21b, the resistance going through the superconducting transition is plotted √ √ on a log scale against 1/ T − TC ∝ 1/ 1 − t, defining t ≡ TC /T . This relation is 51 3.3. Discussion obeyed over more than 4 decades. In this regime, the resistance scales as the BKT correlation length ξ which is has an exponential dependence on T √ ξ = Aeb/ 1−t (3.12) It is this result we draw an analogy with in Fig. 3.19, replacing the reduced temperature t with reduced magnetic field h. Note that b is not a universal number, but should be of order unity, as observed here. Figure 3.21: Experimental evidence for KT transition in thin film superconductor amorphous indium / indium oxide from [50]. (a) Non-linear IV curves show evidence for unbinding vortex-antivortex pairs with increasing bias. The different curves are taken at different T from 1.94 K (a) to 1.46 K (m). (b) The resistance cast in an analogous form to Eq. 3.8 for T instead of H for the resistance of the film passing through the KT transition. The fit is in good agreement over more than 4 decades in R. While in the 2D superconducting case the state destroyed by vortex-antivortex unbinding is the ordered zero resistance state, here we hypothesis that above HC graphene is in an ordered, insulating state. As H is decreased below HC (analogous 3.3. Discussion 52 to increasing T across TBKT ), the ordered state at large H is destroyed by the spontaneous appearance of defects which increase exponentially in density. The fall of R0 below H0 then reflects the current carried by these vortices as ξ 2 . As a prediction we note that in this regime there should be analogous non-linear IV behavior as seen in the 2D superconducting case in Fig. 3.21a. There are several alternative scenarios. One obvious implication of the insulating state is that the proposed CPEs with conductance 2e2 /h appear to be absent. However, these modes are not protected against magnetic impurities, so it can be argued that localization along the edge of the sample is occurring. This could be driven with increasing H as the edge modes are pushed closer to the boundary. However, as shown in Table 3.2, there is no obvious correlation between sample size and HC . Presumably longer samples would be exponentially more susceptible to this localization. Moreover, there is an obvious correlation between high sample quality (as determined by low V0 ) and low HC . One would expect samples with higher impurities levels to be more easily localized, but we see the opposite trend. Higher quality samples have progressively lower HC which we take to imply the insulating state is a property of the pure system. A subtle point is that the BKT transition is relevant for 2D systems, but this relativistic system should be 2+1 dimensional. A possible solution to this scenario has been proposed by Nomura, Ryu, and Lee [42]. Instead of breaking the spin or valley symmetry with increasing H, they propose that the transition at ν = 0 is due to a spontaneous ordering of the pseudospin in the physical/valley spin plane. This, in analogy to the 2D XY ferromagnet model, would allow for a 2D BKT transition. They also predict a decreasing HC for higher mobility crystals. 53 3.3. Discussion Figure 3.22: Representation of U(1) order proposed in [42] (a) The Kekule bonddensity-wave order with two defects marked by a filled circle. These defects cause a change in the sense of the bond ordering, denoted by the different colors. The defects are charged. (b) Representation of the bond ordering in terms of a U(1) phase. The charge defects cause vortices and anti-vortices in the phase. Both (a) and (b) taken from [42]. The ground state wavefunction produced by this ordering breaks the symmetry between A and B sublattice. The wavefunction is given by 1 † √ ψ0 = aKms + eiφ a†K ′ ms |0i 2 m,s=↑,↓ Y (3.13) where the creation operator is creating an electron on the mth n = 0 Landau level with spin s and in valley K or K’ and φ is the angle in the spin-valley plane. According to [42], this represents a bond density wave (see Fig 3.22a). It has been shown that such a density wave, known as a Kekule ordering, can cause insulating behavior on the hexagonal lattice [51]. Charged impurities are responsible for randomizing φ, as can been seen in Fig. 3.22b where the direction of the Kekule order is depicted. One therefore expects the defects to carry charge with a conductivity proportional to the number of vortices nv times their mobility µv . As nv is proportional to ξ −2 , we thus expect ρ ∼ ξ 2 , as observed. 3.3. Discussion 54 Figure 3.23: Temperature dependence of the Dirac point resistance in device J18 at selected H approaching HC = 32 T. At 20 T, the increase in R0 is relatively weakly T dependent. At the highest H, the profile is progressively more well fit to an activated form R ∼ e∆/T . The dashed lines show an activated curve with ∆ = 14 K for the highest H. 3.4. Summary 55 Finally, we return to the question of T saturation of the R0 below 2 K. Fig 3.23 shows the variation of R0 in device J18 as a function of T for H approaching HC = 32 T. At relative low H, the curves are clearly saturating at low T . However, at higher H the curves begin to fit better an activated form R ∼ e∆/T , where ∆ = 14 K at 31.2 T (the dashed line shows the activated form). This implies that the saturation behavior disappears in the insulating state above the critical field HC and grows as H drops far below HC . 3.4 Summary The ground state of graphene at the Dirac point in high H is an insulator. The scenario that describes the phase transition in H most accurately attributes the broken symmetry to a spontaneous ordering of the valley-spin pseudospin. Very recently, measurements on significantly higher mobility devices made with free-standing graphene have been reported [52, 53]. In these devices with V0 < 0.5 V and mobilities higher than 100,000 cm2 / Vs, the field at which insulating behavior is as low as 5 T. This confirms that the transition to an insulating state is an intrinsic property of graphene. Chapter 4 Thermopower and Nernst effect in Graphene in a high magnetic field The thermoelectric properties of graphene have been investigated far less than the electric properties. Here, we measure the thermopower S and Nernst signal Syx in graphene to high magnetic field H [54]. Of particular interest is the response in the quantum Hall regime. We compare our results to a theory due to Girvin and Jonson (GJ) for edge state contributions to thermopower [55, 56]. It has been successful in describing systems with quadratic dispersion, such as GaAs heterostructures [57, 58, 59]. We find agreement with the model with a correction for Berry’s phase effects for S. The off-diagonal thermoelectric conductivity αxy has peaks at each Landau level consistent with the GJ model, but is anomalously narrow at the Dirac point. We describe the method for thermoelectric measurements in graphene, briefly introduce GJ theory, present experimental results and discuss unique features of thermoelectricity at the Dirac point. 56 57 4.1. Thermoelectric Measurements in 2D 4.1 4.1.1 Thermoelectric Measurements in 2D Thermoelectric Measurements In two dimensions (i = (x, y)), the thermoelectric tensor Sij is defined as Ei = Sij ∂j T (4.1) where Ei is the electric field in the i direction, and ∂j (T ) the temperature gradient in the j direction. The thermoelectric conductivity αij is Ji = αij ∂j T (4.2) where Ji is the electrical current in the i direction. These two quantities are useful for considering thermoelectric experiments with two different types of boundary conditions. If electric current is allowed to flow in the circuit (for example, an infinite boundary condition), then it is useful to think in terms of charge currents and Eq. 4.2 is appropriate. If charge currents are not allowed to flow, then Eq. 4.1 is more useful. In practice, it is more common for calculations of αij to be done with the infinite boundary condition while experiments usually measure voltages in a circuit with an open boundary condition and thus measure Sij . As we show below, by additionally measuring the conductivity tensor σij = Gij , we may calculate αij from measured quantities. Our measurements are performed on devices in the form of sample J10 in Fig. 4.1. The operating principle of the device is borrowed from similar measurements done on carbon nanotubes by Small, Kim and coworkers [60]. The graphene crystal was exfoliated onto a thin layer (300 nm) of SiO2 grown on n-doped Si wafer. A strip of 4.1. Thermoelectric Measurements in 2D 58 Figure 4.1: Image of device J10. A heater is patterned near the sample to supply a temperature gradient ∇T . 4-probe thermometers are patterned on top of the sample (labeled Therm) that also act as current leads. Signal leads are also patterned to measure voltages. The scale bar is 3 µm. gold evaporated near one end of the sample serves as the heat source (labeled “Heater” in Fig. 4.1). To measure the gradient −∇T , we pattern two 4 probe thermometers. Their resistances were measured to 4 significant figures between 10 and 300 K. Using this calibration we can use them as local thermometers with resolution ∼ ±1 mK above 10 K. They are also used as current leads for measurements of σij . The voltage signals Vij are detected by the 4 gold lines labeled “Signal Leads”. For a temperature difference δT ∼10 mK, the uncertainty in the gradient −∇T was ±10%, mostly due to the uncertainty in the voltage lead spacing (∼2 µm ± 200 nm). To improve the signal-to-noise ratio, we used an alternating heater current of frequency ω/2π ∼ 3 Hz to produce an alternating gradient. The thermoelectric response was detected at 2ω, with a phase shift of −90◦ . An applied temperature gradient −∇T ||x̂ (with H||ẑ) generates an electric field E 59 4.1. Thermoelectric Measurements in 2D in the sample with components Ex = Sxx (−∂x T ) and Ey = Syx (−∂x T ). By convention we define the thermopower S and Nernst signal eN as S≡ Ex −∇T , eN ≡ Ey ∇T (4.3) As referenced above, we can calculate αij from this experimental geometry. In Fig. P 4.1 with −∇T ||x̂, H||ẑ, the charge current density J is Ji = j [σij Ej + αij (−∂j T )]. Setting J = 0, we solve for E and obtain Ei = − X ρik αkj (−∂j T ) = k,j X Sij (−∂j T ), (4.4) j with ρij = Rij the 2D resistivity tensor. Inverting Eq. 4.4, we may calculate the tensor αij from measured quantities. We have αxx = −(σxx Ex + σxy Ey )/|∇T | αxy = (−σxy Ex + σxx Ey )/|∇T |. (4.5) This will be useful in comparing our results with the GJ theory. Measurement of S and Syx are often more incisive probes of electronic band structure than purely electrical quantities. One way of understanding this is the Mott relation [61], which relates thermoelectric transport quantities to electric quantities, 2 π 2 kB T dσij Sij = − 3 |e| dE E=EF (4.6) where E is the energy and EF is the Fermi energy. Since Sij is the logarithmic derivative of Gij with respect to energy, effects of Landau level quantization which cause changes to the band structure are amplified in the thermoelectric measurement. 60 4.1. Thermoelectric Measurements in 2D A useful tool is the gate voltage VG , which allows us to directly test Eq. 4.6 by varying E. We use the chain rule to explicitly include the dependence of VG 2 π 2 kB T 1 dσij dVG Sij = − 3 |e| σij dVG dE E=EF (4.7) and from the density of states and capacitive charging Q = CV , dVg = dEF r √ e 2 VG πC ~vF (4.8) where C is the capacitance per unit area of the 300 nm thick SiO2 gate dielectric (1.14 × 10−4 F / m2 ) and vF is the Fermi velocity. 4.1.2 Girvin and Jonson Theory The effect of temperature gradients on edge currents in the quantum Hall state has been considered by Girvin and Jonson [55, 56]. The geometry of the 2D electron system is shown in Fig 4.2a. The sample is infinite along ŷ and of length L along x̂. The field H is applied along ẑ. This boundary condition lends to a calculation of αij , as discussed above. In the quantum Hall state, the longitudinal current Ix = 0, and the Hall current Iy is give by [55] L Iy = 2π Z dk X −e ∂ǫkN N ~L ∂k nF (ǫkN ) (4.9) where k is the wave vector, N the Landau index, ǫ the energy, and nF the Fermi function. Expressing the edge current in this way, it is plain to see the physical origin of S in the quantum Hall state: the difference in nF due to different T on the two edges. As shown in Fig. 4.2a, the left and right edge of the sample are at different T and therefore nF is more smeared out on the hot (left) edge. Thus, an extra current 61 4.1. Thermoelectric Measurements in 2D density Jy flows along ŷ giving Jy ∆T = αxy . For a finite sample, δIy produces a quantized Hall potential VH = (h/e2 )δIy whose gradient lies parallel to (−∇T ). The same reasoning shows that Syx arises due to disorder and should be small. Therefore, when the chemical potential µ is aligned with the LL energy in the bulk En , Sxx displays a large peak whereas the Nernst signal Syx is small. The calculation from GJ is shown in Fig. 4.3. Just as σxy is quantized in the quantum Hall regime, GJ show that the peak value of αxy is also quantized. Following the Kubo formulation from above, one would calculate αxy at each n 6= 0 as, e αxy (µ) = hT Z ∞ En ∂nF dǫ (ǫ − µ) − ∂ǫ , (4.10) max As µ is varied, αxy rises to a narrow peak with peak value αxy = (kB e/h) ln 2 ≃ 2.32 nA/K, attained when µ equals En in the bulk. This is depicted in Fig. 4.2b. The width of the peak broadens linearly with T . This implies that for S in a finite sample we expect the quantized values S= kB ln 2 e(n + 1/2) (4.11) We would expect there to be many issues regarding the application of this theory to graphene. First, the Dirac dispersion of graphene leads to a new quantization condition for σxy ; how does this effect αij and Sij ? Perhaps more interesting is the question of the n = 0 Landau level, which is not treated by the GJ theory. As discussed in chapter 3, there is considerable debate to the edge state structure for n = 0, so measurements of Sij would be informative in that regime. 4.1. Thermoelectric Measurements in 2D 62 Figure 4.2: (a) The effect of −∇T on the edge currents Iy in a system with edge currents. The red line is the energy of the nth Landau level and the broken line the chemical potential µ. En increases sharply at the sample edges. For a system with quantizing magnetic field in +ẑ, Hall currents flow as shown. The result of the temperature gradient in x̂ is to unbalance the Fermi-Dirac distributions on the two edges (shown on each side). This results in a thermoelectric response. (b) GJ theory predicts that at each Landau level crossing with µ there is a peak in the off-diagonal thermoelectric conductivity αxy that reaches a universal value of ≈ 2.31 nA / K (for a singly degenerate level). 4.1. Thermoelectric Measurements in 2D 63 Figure 4.3: (a) Calculation for S and Sxy for a sample with moderate disorder from [56]. Successively higher Landau levels have decreased maxima in S, but Sxy remains unchanged. The calculation makes no prediction for filling ν = 0. (b) With increased disorder, Syx grows considerably and Sxx falls far below the quantized value. 64 4.2. Experimental Results 4.2 4.2.1 Experimental Results S with H = 0 Measurement of S near room temperature with H = 0 are shown in Fig 4.4. Also 200 S S Mott K59 exp 302 K 0.5 x S S ( V / K) 100 Mott xx 0 -100 -200 -30 -20 -10 0 V G 10 - V 0 20 30 (V) Figure 4.4: Measurement S of device K59 in zero field near room temperature. The measured Sexp is compared to the prediction from the Mott relation Eq. 4.7. Agreement is best when Smott is divided by 2. The longitudinal conductance σxx is shown in arbitrary units as a thin dashed line. shown (in arbitrary unites) is σxx . The gate voltage VG axis has been shifted by V0 . σxx shows the typical field effect dependence on VG : a minimum at the Dirac point and a strong increase as either holes (negative VG ) or electrons (positive VG ) are introduced in to the system. The experimentally measured Sexp is also shown. It changes sign at the Dirac point and is positive for holes and negative for electrons, reaching a maximum of ∼ 80 µV / K. Comparison to the Mott relation from Eq. 4.7 65 4.2. Experimental Results is also shown as SM ott . The agreement is poor. Dividing SM ott by 2 yields a more satisfactory result, but there is still disagreement between theory and experiment. It has been suggested that this is due to the use of 4 probe instead of 2 probe conductance in Eq. 4.7 [62], but the issue remains unresolved. 120 280 K 240 200 S ( V / K) 80 40 0 160 80 60 120 -40 -80 J10 H = 0 -10 0 10 V G 20 30 40 (V) Figure 4.5: Thermopower S for various T between 10 K and 300 K in device J10. The peak structure on either side of the Dirac point is steadily scaled down as T drops. The maximum value attained is ∼ 100 µV / K. Detailed T dependence of S(VG ) is shown in Fig. 4.5 and Fig. 4.6 for devices J10 and K59, respectively. As T is lowered, the magnitude of S systematically decreases. Both samples show similar peak structures on either side of the Dirac point. The maximum magnitude of S is shown against T for both samples in Fig 4.7. The dependence is roughly linear, as expected from Eq. 4.6. Sample J10 shows a small 66 4.2. Experimental Results 75 80 K 106 K 50 133 K S ( V / K) 159 K 25 205 K 302 K 0 -25 -50 K59 H = 0 -75 -100 -20 -10 0 10 V G 20 30 40 (V) Figure 4.6: Measurement of device K59 over a limited T range. The data largely reproduce the features in device J10 (Fig. 4.5.) The maximum |S| at 300 K is ∼ 100 µV / K. 67 4.2. Experimental Results anomaly at low T . This seems to be associated with the appearance of large, reproducible fluctuations in S. An example of this is shown in Fig. 4.8 for sample J10 at 10 K. Five different sweeps of S(VG ) are shown to overlap. One surprising aspect of these fluctuations is that they are larger than the baseline magnitude of S. Moreover, the seem too large to be explained by Eq. 4.7. 100 H = 0 60 J10 40 |S max | ( V / K) 80 K59 20 0 0 50 100 150 200 250 300 T (K) Figure 4.7: Maximum value of S measured for devices J10 and K59. Smax (T ) is approximately linear in T , as expected from the Mott relation. There is a small anomaly at the lowest T for sample J10. 4.2.2 S and Syx in H Application of H causes large oscillations to appear in all of the thermoelectric quantities. For device J3, S is shown at H =5, 9, and 14 T. The Dirac point is at ∼ 12 V. Upon tuning away from charge neutrality, we observe a series of peaks in S that 68 4.2. Experimental Results 2 J10 T = 10 K Sweep 1 S ( V / K) 1 2 3 4 5 0 -1 -2 -10 0 10 V G 20 30 40 (V) Figure 4.8: Large fluctuations in S for device J10 at 10 K. The fluctuations are reproducible; 5 successive scans of VG are shown. The fluctuation grows on the electron side to overwhelm the baseline S. 4.2. Experimental Results 69 coincide with successive Landau levels crossing µ. At higher H, the spacing in VG increases as expected for the enhanced degeneracy of each Landau level. As is particularly clear in Fig. 4.9a, the magnitude of S is a decreasing function of the Landau level index n, similar to that predicted by Eq. 4.11. However, at 20 K and 14 T S at n = −1 is ∼ 41 µV /K, exceeding the value predicted 39.8 µV / K. An interesting comparison is between S at 20 K and 50 K (Fig. 4.9c). Shown at 14 T, there is almost no change in the magnitude of the n = 1 peak after the factor of 2.5 change in T . This implies that at 50 K, S is already in the quantum regime. Similarly, comparison between S at 9 and 14 T shown very little difference at 20 K; as for the case of electrical measurements [63], this is further evidence that the Landau level energy spacings are extremely large even at modest H in graphene. For each measurement of S, the longitudinal conductance Gxx is also shown for comparison. A notable trend is that lower n levels seem to have peak values for S and Gxx at different VG . Also, in the region in which Gxx exhibits a peak at the Dirac point, S appears to exhibit a faint reversal in sign before going through zero on each side of the Dirac point. We revisit this feature further below with the calculation of αij . The Nernst effect Syx shows similar effects of Landau level quantization, shown in Fig. 4.10. Here, there is a large peak in Syx at the n = 0 level, followed by a series of peaks at each successive level. The higher n levels appear to have similar peaks in Syx and the signal oscillates between positive and negative values. This is qualitatively consistent with the GJ calculation in Fig. 4.3. As opposed to the saturating dependence of S in high H, the peak value of Syx at the Dirac point continues to grow with H. It reaches above 40 µV / K at 14 T and 20 K, and has a 70 4.2. Experimental Results -20 -10 0 10 20 30 30 (a) 5 T 6 40 J3, T = 20 K 20 10 4 0 2 -10 0 -20 (b) -1 40 20 4 0 G xx 2 (e /h) n=0 -2 2 +1 +2 S ( V / K) 9 T 6 -20 0 6 (c) 14 T 40 50 K 20 4 20 K 0 2 -20 0 -20 -10 0 10 V G 20 30 40 (V) Figure 4.9: Pronounced oscillations due to Landau level quantization seen in device J3 at H = 5 T, 9 T, and 14 T (a, b, and c). The longitudinal conductance Gxx is also shown for comparison. The Landau level index n is marked in (b). Measurements are taken at 20 K, except for the curve shown for comparison at 14 T in panel (c). 71 4.2. Experimental Results -20 6 -10 0 10 20 30 40 (a) 5 T 20 10 4 0 2 J3, T = 20 K 0 10 G 0 2 yx 4 S 20 ( V / K) 30 (b) 9 T xx 2 (e /h) 6 -10 -10 0 (c) 6 14 T 40 30 20 4 10 0 2 -10 0 -20 -10 0 10 V G 20 30 40 -20 (V) Figure 4.10: Nernst signal Syx as a function of VG for H = 5 T, 9 T, and 14 T (a, b, and c). The dominant feature is the peak at the n = 0 Landau level. It grows with increasing H and has a positive value. Additional oscillations of similar size are seen at higher n. 72 4.3. Discussion sign (positive) consistent with the vortex-Nernst effect in superconductors [64]. 4.3 Discussion The magnitude of the quantized peak in S given by Eq. 4.11 is smaller than that measured in Fig. 4.9, as discussed above. One explanation for this is that the derivation of Eq. 4.11 does not include the effects of the Berry’s phase shift that is responsible for the unique quantum Hall structure (Eq. 3.7) in graphene [44]. Taking this shift in to account, we instead would expect S= kB ln 2 en (4.12) This would predict a peak value of 59.6 µV / K, which is significantly higher than the measured value of 41 µV / K. However, it seems likely that samples with less disorder broadening of Landau levels would have an enhanced S (as well as a suppressed Syx ) consistent with the GJ calculation as shown in Fig. 4.3. Further experiments with devices of higher quality will be required to resolve this issue. If one were able to reliably measure this quantization, in principle it represents a method for thermometry calibrated only by fundamental constants. To analyze the results in more detail, we calculate the αxx and αxy by Eq. 4.5. The results are shown for H = 9 T and 14 T in Fig. 4.11a and b, respectively. Shown as a thin green line at the top of each panel is the quantum value predicted by GJ with the appropriate degeneracy (4) for graphene 4kB e/h ln 2. The peaks reach within 20 % of this value. It should be noted that this may be due to the large uncertainty in the calibration of ∇T simply due to the geometric uncertainty in the device leads. 4.3. Discussion 73 Figure 4.11: The diagonal and off diagonal elements of αij calculated from measured Sij and σij at 9 T and 14 T (a and b). αxx has a repeating positive/ negative profile while αxx exhibits a series of peaks at each Landau level where the value reaches near the predicted maximum of (4kB e/h) ln 2 (shown as a dashed line). 74 4.3. Discussion Between the peak values, αxy falls to zero. Within error it appears to be always non-negative. αxx shows a dispersive profile changing from positve to negative values as the chemical potential crosses the center of each Landau level. One surprising feature is that even for n = 0, αxy appears to be reaching near the GJ model’s prediction. However, the edge current model cannot accommodate the structure of the n = 0 level, so it is unclear why it appears to obey the same quantization. One noticeable feature of αxy at the n = 0 level is that it is substantially narrower in VG than the higher n levels. A useful metric for comparison is the area An (T ) for each n from Eq. 4.10, An (T ) = Z dn2D αxy ∼ Nn 2 4c0 kB eT h (4.13) where Nn is the maximum value of the density of states of the Landau level and c0 is a constant ≈3.29. Note that we have performed the integral with respect to the 2D carrier density n2D which is given by CV /e. At a given T , the area under the peak of αxy is directly proportional at Nn . If we naively extend the validity of Eq. 4.10 and 4.13 to the n = 0 Landau level, we would be forced to conclude that Nn is smaller by roughly a factor of 4 compared to the higher n. It is difficult to reconcile this with the basic notion of conservation of states, and presents a puzzle for interpretation at the Dirac point. Whereas in S we noticed what appeared to be a reversal in sign on either side of the Dirac point, in αxy states appear to be missing in the same region. While the reversal in sign of S may hint at the presence of counter-propagating edge modes at n = 0 [3], it is unclear how these modes could describe the anomalous behavior of αxy . Similarly, the source of the large Syx at the Dirac point remains unexplained. 4.4. Summary 4.4 75 Summary Our results reveal that, at 9 T, the thermoelectric response in graphene already falls in the quantum regime at 50 K. Measurement of S indicate that Berry’s phase effects modify the quantization relation. The inferred off-diagonal term αxy is a series of peaks independent of n, B and T . The peak values come close to the predicted value of (4kB e/h) ln 2. The most surprising feature of the thermoelectric response is that at n = 0 the peak in αxy is significantly narrower than for higher n, but also approaches the GJ quantized peak value. Chapter 5 Quantum Interference in Macroscopic Crystals of Nonmetallic Bi2Se3 3D topological insulators (TIs) are a new class of insulators consisting of a 3D insulating bulk wrapped with novel 2D conducting surface states. The surfaces states are host to spin-polarized Dirac electrons with potential use for a broad range of applications [65]. However, a serious materials drawback is that the bulk is a relatively poor insulator. Without achieving a significantly improved insulating state, the bulk of the crystal dominates electrical transport and many of the exciting proposals for potential uses of TIs are unachievable. Here, we show how through chemical doping of Ca in to the large band gap TI Bi2 Se3 we can suppress conduction of the bulk bands. We uncover a unique conduction channel characterized by large quantum corrections to electrical transport at low temperature T [66]. We present a brief introduction to 76 5.1. Transport measurements in 3D Topological Insulators 77 transport in 3D TIs, show experimental results for doping and low T transport, and finally discuss implications for the surface bands in these materials. 5.1 Transport measurements in 3D Topological Insulators There are several specific expectations for the transport properties of 3D topological insulators. 1. There should be conducting, in-gap states as seen in the 2D case in Fig. 1.5. 2. In-gap states should not experience 2kF scattering (backscattering) as that would require a spin-flip and break time reversal symmetry. This implies that they would have relatively high mobility. 3. A magnetic field H should open a gap at the Dirac point due to breaking of timereversal symmetry. The states away from the Dirac point should be quantized in to Landau levels and exhibit 2D Shubnikov de-Haas oscillations. 4. As in graphene, Berry’s phase effects should modify the quantum Hall structure, but here the degeneracy (for a single surface) would be 1/4 of graphene (see Eq. 3.7), σxy = e2 1 n+ ) h 2 (5.1) 5. From the two dimensionality and large spin-orbit coupling, we would expect to see weak anti-localization [67]. 5.1. Transport measurements in 3D Topological Insulators 78 Before any of these surface properties can be measured by transport, however, we must experimentally address the remnant bulk contributions to transport. In Fig. Figure 5.1: (a) ARPES measurement for as-grown Bi2 Se3 from [16]. The surface states reside in the bulk band gap, but the Fermi energy is pinned to the bulk conduction band. (b) Electron carrier density of Bi2 Se3 crystals grown with increasing excess of Se from [68]. Increased Se suppresses the size of the bulk conduction band pocket. 5.1a, ARPES results for crystals of Bi2 Se3 are shown [16]. While the 2D surface band with linear dispersion is evident in the bulk band gap, it is also apparent that the Fermi level EF is located in the conduction band. While a band structure calculation would predict the chemical potential would be in the bulk band gap, it has been found that over time Se escapes from crystals and in the process donates electrons. One proof of this hypothesis is shown in Fig. 5.1b. Crystals in these experiments were grown large amounts of excess Se [68]; crystals with more Se tend to have lower carrier densities n, presumably due to suppression of Se vacancies [69]. We can still measure crystals from Fig. 5.1a, though, and address whether or not we see signatures of 2D transport. 5.1. Transport measurements in 3D Topological Insulators 79 Basic transport experiments characterizing these crystals are shown in Fig. 5.2. The resistivity profile ρ(T ) is that of a metal with a relatively low mobility 500-5000 Figure 5.2: (a) Electrical transport in heavily n-doped Bi2 Se3 . The resistivity profile is metallic, though the low temperature ρ is that of a poor metal. (b) Shubnikov de-Haas oscillations in n-doped Bi2 Se3 . The period of the oscillation in 1/H is only weakly dependent on angle θ. This indicates a bulk conduction pocket that is approximately spherical. cm2 / Vs (depending on the particular crystal). The Hall density nH (not shown) for these crystals is ∼ 1019 e− / cm3 , demonstrating that EF is deep in the conduction band. Measuring the extremal Fermi surface size Ak is possible through Shubnikov de-Haas measurements [70], as shown in Fig. 5.2b. Here, the period of the oscillation in 1/H is proportional to Ak 2πe 1 −1 Ak = ∆ (5.2) ~ H p The resulting Fermi wavevector kF = Ak /π is roughly 0.07 Å−1 , consistent with the conduction band measured by ARPES in Fig. 5.1a. When the field is tilted, the observed Ak changes only by ∼ 30%, consistent with a slightly ellipitcal Fermi 5.1. Transport measurements in 3D Topological Insulators 80 surface as expected for the bulk conduction band [71]. Thus, we conclude that these transport properties are coming from the 3D bands. To measure the 2D states, we will have to move the chemical potential down to the gap. 81 5.2. Results 5.2 5.2.1 Results Nernst effect in Cax Bi2−x Se3 : Tuning the Chemical Potential Following the discovery that Ca can be used to donate holes to Bi2 Se3 by substituting for Bi [72], we have used light amounts (. 0.5%) of Ca to attempt to move the chemical potential in to the bulk band gap. As our metric for progress, we will monitor Ak via the Nernst effect eN . As described in Chapter 4, thermoelectric quantities such as eN are more sensative to changes in the electronic band structure due to Landau level quantization than purely electrical quantities. 5 7.5 K - 40 K 0 60 -10 80 100 -15 125 -20 150 e N ( V / K) -5 -25 175 200 -30 225 250 -35 n H 19 ~ 10 - 275 3 e / cm 300 -40 0 5 10 H (T) 0 Figure 5.3: Measurement of Nernst effect eN for heavily doped Bi2 Se3 crystal M1. The signal crosses over from negative to positive with decreasing T and shows Landau level quantization below 40 K. 82 5.2. Results Typical results for a crystal (sample M1) with relatively high nH (∼ 1019 e− cm−3 ) are shown in Fig. 5.3. At high T , eN is large and negative. As T decreases, eN changes sign and begins to exhibit Landau level oscillations periodic in 1/H. From the temperature dependence of the oscillation magnitude, we estimate the effective mass to be ∼ 0.1me . 10 15 20 30 40 8 50 60 4 80 2 100 e N ( V / K) 6 0 150 -2 200 250 K 18 -4 n ~ 10 0 5 - 3 e / cm 10 H (T) 0 Figure 5.4: Measurement of Nernst effect eN for lightly n-doped Bi2 Se3 crystal M4. A crossover from negative to positive eN is seen, but the negative contribution is suppressed relative to more heavily doped crystals. Landau level quantization is apparent at 20 K and below. Crystals with smaller densities show a similar pattern, but with the negative eN signal suppressed. Fig. 5.4 shows eN at selected T for crystal M4, with nH ∼ 1018 e− cm−3 . The crossover to positive eN occurs near T = 100 K. Landau quantization is still seen, but the period is significantly longer. This corresponds to a smaller conduction band pocket. 5.2. Results 83 Figure 5.5: (a) Fan diagram constructed from Nernst measurements on crystals with a range of Ca dopings. The larger slopes correspond to crystals with smaller extremal Fermi surfaces. The sign of the Landau level is determined by the Hall effect and thermopower. (b) Comparison of Nernst measurement for relatively high and low carrier densities at 15 K. 84 5.2. Results Samples M1 and M4 are compared in Fig. 5.5b for T = 15 K. To make comparisons between samples more clear, we construct the index plot in Fig. 5.5a. We record the H at which eN shows a maximum, and plot the successive peak values assuming each corresponds successive integer Landau levels. The steeper the slope in Fig. 5.5a, the smaller Ak . Note that we choose the Landau levels to be hole-like or electron like (positive or negative) from measurements of thermopower S and the Hall coefficient p RH . The smallest Ak found corresponds to Fermi wavevector kF = A/π ≈ 0.046 Å−1 . Given only this Ak , it is ambiguous if this corresponds to the bulk or surface Fermi surface. However, given that these crystals exhibit metallic ρ(T ) profiles and the lack of significant angular dependence of Ak in Fig. 5.2, it seems much more likely that these are bulk properties. To further test this idea, we can compare kF determined from these measurements to that determined from RH . We assume that the Fermi surface is perfectly spherical so that the density derived from Shubnikov de-Haas oscillations nSdH is nSdH = 2 (4/3)πkF3 kF3 3 = 2 3π 2π (5.3) The result is shown in Fig. 5.6. The scaling between the two is satisfactory, though there is disagreement by an overall factor of roughly 2. This underestimate of the density is likely do to the non-zero eccentricity of the Fermi surface [69]. Also plotted in Fig. 5.6 is the quantity kF ℓ which characterizes the mobility of the crystals. The act of Ca doping is to push the crystals to the right in Fig. 5.6 (toward more hole doping). We note that while kF ℓ is monotonically reduced at higher dopings, it is not dramatically suppressed even from the highest doping. There are also a few crystals grown near the lowest doping that had significantly higher kF ℓ, which we attribute 5.2. Results 85 to successful attempts to suppress Se vacancies. Figure 5.6: Comparison of electron densities calculated from Shubnikov de-Haas oscillations and the Hall effect. The two scale closely, differing by an overall factor of roughly 2. Also shown is the parameter kf ℓ which reflects the mobility of electrons in the system. Even highly doped crystals (those with large hole dopings) do not have a significantly suppressed value. 5.2.2 Nernst effect in Cax Bi2−x Se3 : Low T behavior Before moving on to results for crystals with the chemical potential near the gap, we briefly detail results of eN and S at low T . Fig. 5.7 shows the result of a measurement of sample M3 down to 0.5 K. The Landau level quantization continues to be pronounced, but a new feature that appears is a near H independent eN at low T . This is brought out more clearly by confining the data below 2 T so that the effect of Landau level quantization does not obscure the signal. Casting the signal as eN /T in Fig. 5.8 to remove the T dependence inherent in eN from the Mott Formula (Eq. 86 5.2. Results 10 8.7 K Bi Se 2 3 e N ( V / K) 8 5.5 K 4.3 K 6 2.2 K 4 2 1.7 K 0.5 K 0 0 5 10 0 H (T) Figure 5.7: Nernst effect eN for Bi2 Se3 crystal M3 (low n-type doping) at low T . The most striking feature is the H-independence that appears at the lowest T . 87 5.2. Results 4.6), we see that as T decreases eN /T appears to approach a saturation value of 0.8 µV / K2 at 2 T. 1.0 Bi Se 2 3 1.5 0.9 0.5 2.6 0.8 3.3 5.5 2 e / T ( V / K ) 4.3 0.6 6.4 N 0.4 9.4 10.8 12.5 14.8 16.4 18.5 K 8 0.2 0.0 0.0 0.4 0.8 1.2 1.6 2.0 H (T) 0 Figure 5.8: Measurements of Nernst effect eN in crystal M3 divided by T . At low T , the traces appear to fall on a universal curve within the uncertainty of the measurement. For completeness, we also show the result of the thermopower S to compare to eN . Fig 5.9 shows the result for sample M4 over a large T range. The values of S at zero T are large at high T : above 170 µ V / K at 250 K. Note that since this is an electron-like crystal S is negative. The H dependence of S changes from slightly increasing the magnitude at high T, to suppressing S below 150 K. Starting at 60 K, a sharp low H dependence develops which resembles a large peak in S for zero H. At the lowest T Landau level quantization can be observed. Fig. 5.10 shows S(H) for sample M3 for T below 10 K. The low H peak in S is 88 5.2. Results 200 250 200 -1 x S ( V / K) 160 150 120 100 80 60 80 50 40 30 40 20 15 K 0 -10 -5 0 5 10 H (T) 0 Figure 5.9: Thermopower S for crystal M4. Note that the scale has by multiplied by -1 since this electron-like crystal has a negative S. The H dependence changes as function of T , eventually developing a peak at low H for the lowest T . 89 5.2. Results more pronounced, but then becomes difficult to resolve at the lowest T . Similar to eN , S is largely H independent below 2 K except for 1/H oscillations from Landau quantization. 60 -1 x S ( V / K) 8.7 K 40 5.5 4.3 20 2.2 1.7 1 0.5 0 -10 -5 0 0 5 10 H (T) Figure 5.10: Low T measurement of S in crystal M3. The peak in S(H) is the dominant feature with Landau level quantization also evident. At low T , S becomes H independent. 5.2.3 Transport properties of non-metallic Bi2 Se3 Having found the Ca doping (x = xmin ≈ 0.25%) which resulted in the minumum Ak , a large number of crystals were tested with nominally the same doping to find crystals with the chemical potential in the bulk band gap. Among those crystals that showed measurable Landau level quantization (samples M1-M11), all had metallic resistivity profiles. However, by focusing on crystals at xmin , several crystals were found with 5.2. Results 90 non-metallic profiles (denoted G3-G8). These profiles are shown in Fig. 5.11a. Figure 5.11: (a) Resistivity versus temperature profiles comparing non-metallic and metallic crystals (metallic crystals are multiplied by 20 for clarity). Non-metallic crystals appear to all begin to rise in ρ near 150 K. (b) Fan diagram for metallic samples. Crystals with non-metallic profiles were found by growing with a Ca doping near that which produced the smallest conduction band crystals M11 and M3-M5. The resistivity at low T for G3-G8 is large, reaching above 70 mΩ cm at 0.3 K for sample G5. This is more than 150 times larger than the most resistive metallic crystals. However, up to 31 T (discussed further below in Fig. 5.17), Landau level quantization was not observed. Therefore, we cannot place this crystals on the index plot in Fig. 5.11b. However, when cross correlated with the same batch of crystals in ARPES measurements, we find that xmin is the same doping at which ARPES 91 5.2. Results measures the chemical potential to be in the bulk gap [15]. Figure 5.12: (a) Resistivity ρ profile for crystal G4. ρ rises below 150 K and has a sharp upturn below 5 K. This profile is much weaker than would be expected from a simple activated form (shown in red). (b) Conductance per square as a function of ln T for crystal G4. The ln T dependence persists up to 2 K with a slope of roughly 200e2 /h. Focusing on the low T portion of ρ(T ), we see that the rise occurs starting near 150 K. This is inconsistent with the predicted 300 meV gap. Furthermore, the shape of the increase does not fit an exponential; a comparison of ρ(T ) to an activated form is shown in Fig. 5.12a. This implies that if the chemical potential is indeed in the bulk gap, there is another conductance channel contributing significantly. The conductance G at 0.3 K for sample G4 (the sample which we focus on here) is roughly 8000 e2 /h. It seems unlikely that we can ascribe this to a simple surface conduction mode, which we would expect in the ballistic limit to be given by G = Nchan e2 e2 = wkF h h (5.4) where Nchan is the number of conducting channels which we estimate as the width of 92 5.2. Results the sample w divided by kF . As the crystals are mm in lateral size and µm thick, G is more likely given in the diffusive limit G = kF ℓ e2 h (5.5) where ℓ is the mean free path. In terms of surface conduction, one sign that is more encouraging is the extremely low T (less than 5 K) ρ(T ). Here, we see a sharp pickup of ρ. When plotted as G(ln T ) in Fig. 5.12b, we see that the T dependence is logarithmic. This is a signature of weak localization [67]. We will discuss this in detail below. We find the coefficient of this ln T to be approximately 200 e2 /h. 20 K cm) 1.5 150 15 (m 100 1.0 100 10 cm) 200 50 20 2.0 250 K (m 25 250 50 200 0.5 5 20 150 (a) G3 H||c M5 10 -10 (b) H||c 0 0.0 -10 -5 0 0 5 H (T) -5 0 0 5 10 H (T) Figure 5.13: (a) Magnetoresistance (MR) for non-metallic crystal G3. The profile changes from a positive MR at high T to negative at low T . At 50 K and below, a sharp low H dip is apparent. (b) MR for metallic crystal M5. The MR is always positive and develops Shubnikov de-Haas oscillations at 20 K. It is useful to further compare the non-metallic and metallic crystals. Fig. 5.13 shows the H dependence for crystal G3 and M5 at selected T . The metallic crystal in 93 5.2. Results Fig. 5.13 has a roughly quadratic H dependence consistent with Lorentz force magnetoresistance (MR). As T is lowered, the magnitude of ρ decreases and a Shubnikov de-Haas pattern develops at 20 K. In the non-metallic crystal, ρ(H) is also initially quadratic and positive, but as T drops and ρ rises it develops in to negative MR at high H. Moreover, the dominant feature that appears is a sharp low H cusp with positive MR, seen clearly at 20 K. No Shubnikov de-Haas signatures are seen. H (T) H (T) 0 0 4 8 0.1 12 0 1 32 30 1.15 0.85 0.5 28 28 0.3 K 4 26 (m cm) (b) 1.15 30 (m 0.85 0.3 K 8 2.5 26 1.8 4 K 24 2.5 1.8 0.5 C (B,T) cm) (a) 0.5 24 (c) 1.8 K 0.0 0.85 K 0.3 K -0.5 0 1 2 3 4 5 0 6 7 8 9 10 H (T) Figure 5.14: (a) Magnetoresistance for crystal G4 between 8 K and 0.3 K. There is a strong low H anomaly and obvious fluctuations in ρ at the lowest T . (b) The MR is linearized over a large range in H on a log scale. As T rises, the range becomes smaller. (c) Autocorrelation C at selected T . There is a large oscillation seen over the entire H range. Going to lower T and focusing on non-metallic crystal G4, we see that the low H anomaly begins to dominate the ρ(H) profile as T drops below 4 K. This is shown in Fig. 5.14a. When cast on a log scale (Fig. 5.14b), we see that over a large field range the MR fits to a log H trend (shown as a dashed line). This is further evidence for 5.2. Results 94 weak (anti) localization, discussed below. The slope dG/d ln H is 140 e2 /h. Figure 5.15: Fluctuation in conductance δG obtained by subtracting a smooth background from G(H). Shown in geometries with H out of plane (a) and in-plane perpendicular to the current (b) and parallel to the current (c). Two sweeps for each geometry at 0.3 K are shown. The fluctuations fades away as T approaches 4 K. Another feature evidence in Fig. 5.14 at T below 4 K is that the trace appears to become noisy, fluctuating with magnitude 0.5%ρ. However, we can distinguish this signal from random noise by repeatedly sweeping H. To see the signal more clearly, we remove a smooth background from the traces in Fig. 5.14 and display the fluctuation is as a change in conductance δG. Shown in Fig. 5.15, for each configuration of H, we see reproducible fluctuations in δG that die out as T rises to 4 K. Each configuration has two traces taken at 0.3 K to demonstrate that the features are reproduced in fine detail. It is instructive to view the autocorrelation C(B, T ) defined as 95 5.2. Results C(B, T ) = hδG(B ′ , T )δG(B ′ + B, T )iB ′ , hδG2 i (5.6) which measures how peaks in the signal at a particular B = µ0 H are correlated. Computation of C(B, T ) for these traces at selected T is shown in Fig. 5.14c. The most obvious feature in C is the periodic structure. There appears to be more than one frequency- a fast 1 T period signal and an obvious beating of this signal. Upon observing these features, understanding their origin and relevance to the surface states was the prime goal of experiments. 5.2.4 Angular Dependence of non-metallic crystals Before quantitatively analyzing the results, one straightforward approach to addressing the contribution to magnetotransport from the bulk and surface electrons is to tilt the angle of H in experiments. The results are shown in Fig. 5.16. Focusing on the low H cusp in Fig. 5.16a, we note that for all configurations of H angle θ the peak is seen. However, it is reduced by a factor of ∼ 3 in terms of dG/d ln H. Similarly, the fluctuation, characterized by the rms value over the range 4 to 10 T, has similar dependence on θ. Shown in Fig. 5.16b, the rms δG has a cosine dependence with an θ independent offset of roughly 30%. Fig. 5.16c shows that C has a persistent oscillation for all geometries, but there is no obvious evolution of this periodicity with θ. Doing these experiments to high H = 31 T, we find that the patterns remain the same. Sample G8 was measured at 0.35 K in the same geometry; the result is shown in Fig. 5.17. Notably there are still no Shubnikov de-Haas oscillations seen. To bring out the fluctuation signal, we plot dR/dH for each θ in Fig. 5.18. The 96 5.2. Results 0 0.32 0 H (T) 4 (deg) 8 12 0 20 40 60 80 (a) T = 2.2 K 4 (b) o 13 G ( 0.28 o 33 67 2 37 74 0.26 2 20 =85 z 55 61 47 H rms [ G] (e -1 ) / h) =4 0.30 y 42 0 C (B, ) 0.5 o o =4 o 47 (c) 85 0.0 -0.5 0 1 2 3 4 0 5 6 7 H (T) Figure 5.16: (a) Angular dependence of G(H) at 2.2 K. Both the low H anomaly and conductance fluctuation are strongest with θ near 0◦ , but persist for all angles. (b) rms amplitude of the fluctuation as a function of θ. The fit is a cosine with an overall offset. The offset makes up ∼ 30% of the overall amplitude. (c) Autocorrelation C at selected θ. The oscillation persists for all θ. 97 5.2. Results 50 40 0 31 o 47 (m cm) 59 30 75 89 0 0 0 0 0 20 Ca Bi 0.0025 Se 2-x c 3 H T = 0.35 K 10 a G8 0 -30 -20 -10 0 10 20 30 H (T) 0 Figure 5.17: Angular dependence of magnetoresistance taken to 31 T in crystal G8. The low H anomaly is seen for all geometries. No Shubnikov de-Haas oscillations are observed. 98 5.2. Results fluctuations are strongest when H is perpendicular to the sample plane, but they are evident for all θ. As the fluctuations are roughly the same size over this large H range, this makes it unlikely that the signal is coming from Landau level quantization, which would be small for low H and grow exponentially. c H Ca Bi 0.0025 a 1.0 Se 2-x 3 T = 0.35 K / T) G8 0 o 15 o dR / dH ( 31 0.5 o 47 o 59 o 75 o 0.0 89 0 0 10 20 30 H (T) 0 Figure 5.18: Derivative of R(H) at 0.35 K up to 31 T. The fluctuation in resistance remains relatively unchanged over the full H range and persists for all θ. 99 5.3. Discussion 5.3 Discussion We now attempt to understand the transport signatures of the non-metallic crystals. Upon first viewing δG in Fig. 5.15, the ln H MR in Fig. 5.14b, and the ln T ρ(T ) in Fig. 5.12b, one may be reminded of several phenomena from 2D and mesoscopic physics. We briefly review the relevant phenomena in the appendix. Some prototypical measurements for 2D mesoscopic systems which bear resemblance to our results are shown in Fig. A.1. Samples with non-metallic ρ(T ) profiles invariably show the type of low H cusp in Fig. 5.19. As T is lowered, the peak in G(H) gets sharper. As shown in Fig. 5.14, the H dependence is well described by ln H between 0.02 T and 2 T. Given that H suppresses G, it would on these grounds seem to point to weak anti-localization in a 2D system as described in the previous section. However the coefficient of the ln H (as well as the ln T behavior) measured here is hundreds of times larger than predicted by Eq. A.1. We refer to the coefficient A defined as ∆G = Ae2 /h ln H, and collect it for several samples at 0.3 K in Table 5.1. units G3 G4 G5 G6 G7 G8 ρ mΩcm 30 15 76 18 16 25 c µm 50 50 80 50 25 10 G rms δG e2 /h e2 /h 8000 5.9 1760 0.8 7000 1 4800 0.9 1050 0.6 A nH 1018 cm−3 0.7 178 5 35 1 135 7 63 8 20 5 Table 5.1: Sample parameters. ρ is the resistivity at 0.3 K (except for G4, which was only cooled to 4 K). c is the crystal thickness. G is the conductance per square at 0.3 K. rms δG is reported at 0.3 K. A is the coefficient of the ln H term in ρ(H). nH is the density inferred from the Hall coefficient RH = 1/ne. 100 5.3. Discussion One possibility is that the crystal is broken up to hundreds of layers such that many faces are acting in parallel. However, we have also seen that weak anti-localization should only depend on H perpendicular to the 2D system, but the H rotation experiment showed that the signal persisted for all geometries (Fig. 5.16 and Fig. 5.17). This would imply both an orbital and spin contribution to weak anti-localization. 0.44 40 K G4 H||c 0.40 0.36 20 G ( -1 ) 30 10 5 0.32 1.8 1.1 0.28 -2 0.3 -1 0 1 2 H (T) 0 Figure 5.19: Low field magnetoconductance below 2 T for crystal G4. The low H anomaly grows systematically with decreasing T . Turning to the fluctuations seen in the conductance, we present the T dependence of δG, the autocorrelation C, and the Fourier transform F in Fig. 5.20. Comparing these results to theory of conductance fluctuations rules out a conventional source (see appendix). In Fig. 5.20a, the signal is seen to persist to H = 12 T, ruling out AAS oscillations. Fig. 5.20b shows that C oscillates instead of decaying as a power law, which rules out UCF. The Fourier transform in Fig. 5.20c shows a broad peak 5.3. Discussion 101 corresponding to an AB ring size dAB ≈ 60 nm. This would imply that the crystal surface is tiled with a period array of features that produce AB patterns on the scale of 60 nm. However, we can similarly rule this possibility out as follows, unless LT is extremely large. Fig. 5.21 shows the F for three tilt angles θ, bending H away from normal to the surface. While the amplitude of the Fourier components falls, there is no obvious shift in the peak frequency f . One would expect f to change as the area of the ring normal to H changed. A stronger argument is that of size scales. Fig. 5.22 shows a typical crystal above a ruler with mm rulings. The separation of the voltage contacts is ∼ 1 mm, so that the fluctuations would be suppressed by 1000 for UCF (according to Eq. A.3) and 108 for a tiling of 60 nm AB rings (suppressed as N 1/2 ). Despite this quantitative disagreement, the qualitative similarities between the results for these non-metallic crystals and 2D / mesoscopic transport are striking. The ln H behavior and conductance fluctuation are seen in all non-metallic samples. Results for several samples are shown in Table 5.1 and Fig. 5.23. One additional observation is that the T dependence of the rms δG is faster than the power law decay in T expected for UCF (shown in Fig. 5.23b). There are several possible explanations for these effects. The possibility of the crystal cracking in to many 2D systems seems to be able to explain the large ln H and ln T dependence, but has difficulty explaining the conductance fluctuation. Alternatively, examination of the ARPES data for these materials as in Fig. 5.1a seems to indicate that there is a large amount of bulk Fermi surface remnant in the bulk band gap. One possibility based on this observation is that the signal arises from the hybridization of the bulk and surface electrons. This may explain why the effects are 102 5.3. Discussion 4 6 8 10 (a) 0.3 K 10 12 0.85 G (e 2 / h) 2 0 1.8 -10 C(B) 0.5 (b) 0.0 -0.5 2 4 6 8 10 12 H (T) 0 4 F( B ) (a.u.) (c) 0 0 2 4 B -1 6 8 (T ) Figure 5.20: (a) Fluctuations in conductance at selected T obtained by subtracting a smooth background from G(H). The amplitude of the signal reaches 10 e2 /h. (b) Autocorrelation at the same T shows periodic oscillations containing more than one frequency. (c) Fourier spectrum of the fluctuation shows a broad peak near 1 T−1 . 5.3. Discussion 103 Figure 5.21: Fourier spectrum of conductance fluctuation in crystal G4. The amplitude of the spectra are decreased as H is bent away from the surface normal (larger angle), but the frequency does not dramatically shift. 5.3. Discussion 104 Figure 5.22: Photograph of crystal G3. Au wires are attached to the sample with Ag paint, mounted on a sapphire substrate. A ruler with mm rulings is shown. 105 5.3. Discussion 0.3 K (b) (a) G4 Hc G4 ( 0.25 || ) 6 5 G6 G ( 0.20 Hb G7 G4 ( ) 3 H| a 0.15 0.10 || 2 -1 ) 4 G4 ( 2 | ) G6 G7 G5 / h) H || c rms ( G) (e 0.30 1 G5 Ha G5 ( -2 0 H (T) 0 2 0.0 || ) 0.5 1.0 1.5 0 2.0 T (K) Figure 5.23: (a) Magnetoconductance for crystals G4-G7 at 0.3 K. The low H anomaly is present in all. (b) Comparison of T dependence of the conductance fluctuation in several crystals. The signal decays quickly as T increases for each crystal and measurement geometry. 5.4. Summary 106 so large in terms of G, but does not readily explain all observations. Finally, in regard to the large conductance fluctuations, it has been recently found that in graphene topgated nanostructures Fabry-Perot resonances of the conduction electrons can generate reproducible fluctuations up to δG ∼ 10 e2 /h [73]. We hypothesize that step heights at the crystal surface may act as similar resonator where H can modulate the electron wavevector k through Lorentz force bending. This could in turn modulate scattering at step edges by changing the incident angle of the electrons. Scanning tunneling microscopy experiments would be a sharp test of this hypothesis. 5.4 Summary We have shown that Ca doping can smoothly change Bi2 Se3 from an n-type to ptype conductor as monitored by nH and Ak . For intermediate dopings with high ρ, low T transport qualitatively resembles both 2D and mesoscopic transport, but quantitatively is inconsistent with existing theories. We propose these properties may be due to a novel hybridization of bulk and surface states or resonant scattering of surface electrons off of step edges on the crystal surface. Chapter 6 Evidence for metallic surface states in voltage-tuned crystals of Bi2Se3 To discern the transport properties of surface states in Topological Insulators (TIs), it would be ideal to compare crystals dominated by bulk and surface currents. One method for doing this in-situ is by manipulating the chemical potential µ in a crystal via an electrostatic gate. In this chapter, we present an experiment in which we tune the chemical potential µ of thin crystals of Bi2 Se3 from n-type to p-type parts of the Fermi surface [74]. We show that there are high mobility, conducting in-gap states that undergo weak anti-localization and are host to conductance fluctuations. We estimate the surface state conductance σss to be approximately 6e2 /h, and show that the conductance channel from the surface acts in parallel to the bulk when the chemical potential is tuned in to the bulk conduction band. 107 6.1. Gate voltage chemical potential control in Bi2 Se3 6.1 108 Gate voltage chemical potential control in Bi2Se3 The crystal structure for Bi2 Se3 is shown in Fig. 6.1d [75]. The basic unit in the structure is a 5 layer sandwich of Bi and Se layers, referred to as the quintuple layer. The unit cell is comprised of three of these quintuple layers. The binding between the quintuple layers is weak compared to the intralayer bonding, so these crystals can be easily cleaved. Crystals are cleaved on to heavily doped Si substrates with 300 nm of thermally grown SiO2 (see Fig. 6.1c). It was found that crystals of lateral size up to 10 µm with thickness between 5 nm and 50 nm could be cleaved. Fig. 6.1a shows an Atomic Force Microscope (AFM) image of such a crystal. The crystal is approximately 9 nm thick, corresponding to 9 quintuple layers. It has been predicted [76] and shown experimentally by ARPES [77] that if the top and bottom surface in Bi2 Se3 are closer than ∼ 4 quintuple layers, the surface states band structure is distorted and is eventually destroyed. Therefore, we limit our study to crystals thicker than 5 nm. One important observation that aided in finding thin crystals is that in an optical microscope crystals thinner than ∼ 20 nm show a color that darkens progressively toward the color of the SiO2 layer as the crystal becomes thinner. This is similar to that reported in graphene [4]. By cross correlating optical images with AFM measurements, we can create an approximate color map for layer thickness, useful for estimation within ±2 nm. This is shown in Fig. 6.2. Our measurement geometry is shown in Fig. 6.1b. Electrodes are patterned by ebeam lithography followed by deposition of 10 nm of Cr and 100 nm of Au. It is found empirically that the cleaving and lithographic process pushes µ of the crystal further in the the conduction band (crystals as-grown are n-type). Therefore, we 6.1. Gate voltage chemical potential control in Bi2 Se3 109 Figure 6.1: (a) AFM image of cleaved Bi2 Se3 crystal. It is approximately 9 nm thick, 3 µm long and 1.5 µm wide. It is suitable for making contacts with ebeam lithography. (b) Similar crystal with Cr/Au contacts deposited in to measure resistance and Hall effect. (c) Schematic of field effect geometry used in these experiments. (d) Crystal structure of Bi2 Se3 from [75]. The unit cell is composed of 3 quintuple layers consisting of Se:Bi:Se:Bi:Se stacks. Figure 6.2: Optical image of cleaved Bi2 Se3 crystal showing changing color on different plateaus with approximate thickness labeled in nm, measured separately by AFM. 6.1. Gate voltage chemical potential control in Bi2 Se3 110 work primarily with initially hole-doped crystals with relatively large Ca content (0.5% substiting for Bi). This is roughly twice the amount found to minimize the bulk band contribution to transport for macroscopic crystals in Chapter 5. Figure 6.3: Measurements of few layer graphene using field effect transistor geometry from [4]. (a) The resistance R is tuned by the gate voltage VG and goes through a peak at charge neutrality. (b) The Hall coefficient changes from electron-like to hole-like at ∼ 40 V. The associated band structure and electron filling is shown. The technique of cleaving single crystals in the field effect geometry has been employed successfully for measurement of graphene, as in Chapters 3 and 4. A more appropriate comparison for these relatively thick crystals is the original experiments done on few layer graphene [4]. Measurements of few layer graphene in this device geometry are shown in Fig. 6.3, adapted from [4]. The schematic band structure is 111 6.1. Gate voltage chemical potential control in Bi2 Se3 shown in Fig. 6.3b. By applying positive (negative) gate voltage VG , electrons (holes) are added to the sample. Accordingly, the hall constant RH can be changed between electron-like to hole-like, and at the charge neutral point VN a maximum in R is seen (Fig. 6.3a). 0 800 -50 0 T -100 ) 600 ( R 400 14 T -200 ( xx yx R -150 ) 200 -250 T = 0.3 K -300 0 -100 -50 0 V G 50 100 (V) Figure 6.4: Field effect experiment on a Bi2 Se3 device with thickness greater than 50 nm. The longitudinal resistance Rxx and Hall resistance Ryx can only be marginally changed. From Ryx we estimate -300 V would be required to reach charge neutrality. This is approximately a factor of 2 too large to realize with the current gate dielectric. Similar measurements on early devices of Bi2 Se3 with thickness larger than 50 nm are shown in Fig. 6.4. This device is electron doped when VG = 0. Both the Hall resistance Ryx and longitudinal resistance Rxx can be manipulated with VG , but no change in carrier type is observed. Comparing to Fig. 6.3, we appear to be exploring only the heavily electron doped side (right side) of the band structure (in 112 6.2. Results the conduction band in Fig. 1.3). From the curvature of Ryx , Ryx = H H = ne C(VN − VG ) (6.1) where we assume the induced charge is given by n = CVG /e with C = 1.14 F / m2 for 300 nm of SiO2 . A fit to the curve in Fig. 6.4 yields VN . From this we estimate that -300 V would be required to reach charge neutrality. Ideally, the breakdown electric field Eb for SiO2 is 1 V / nm, but in practice after fabrication is closer to 0.5 - 0.6 V / nm for 300 nm thick dielectric layers. Therefore, we are unable to reach VN in this device. By cleaving thinner crystals, however, we are able to bring the magnitude of VN down as low as -80 V. 6.2 Results Table 6.1 shows parameters for five different devices. Devices of thickness t ≤ 20 nm exhibit similar characteristics. For these devices an applied gate voltage can change the carrier type from electron-like to hole-like at VN listed in Table 6.1. Each exhibits a maximum in the resistance per square Rmax of 2 − 3 kΩ. For thicker crystals such as devices S4 and S5, we reach the threshold for breakdown of the SiO2 gate dielectric (typically -180 V to -200 V) before we are able to empty out the conduction band electrons from the system. A comparison of the R(VG ) profiles for samples S3, S4, and S5 is shown in Fig 6.5. S3 exhibits a peak in resistance for temperatures T below 80 K followed by a change in carrier type (see Figure 6.6a) upon further decreasing VG . The progressively thicker crystals show systematically less response to VG . S4, approximately 30 nm thick, can be tuned by the back gate, but there are too many 113 6.2. Results carriers to empty the conduction band below the breakdown voltage of the insulating SiO2 layer. S5 is thicker than 40 nm and can only be slightly tuned by the back gate. units S1 S2 S3 S4 S5 t VN µCB Rmax 2 nm V cm /Vs kΩ 10 -80 330 2.4 6 -90 830 2.3 20 -170 720 2.8 30 < -175 910 1.9 > 40 < -175 1630 0.3 Table 6.1: Sample parameters. t is the sample thickness estimated from an optical image of the cleaved crystal (see Fig 6.5). VN is the gate voltage required to reach charge neutrality at 5 K. µCB is the Hall mobility in the conduction band, calculated at zero gate voltage. Rmax is the maximum resistance recorded over the measured gating range. Figure 6.6 shows detailed transport data for device S3. The Hall resistance of the device changes sign at VG ∼ −170 V, though this is at the extreme limit of our measurement range. The R(T ) profiles for selected gate voltages are shown in Figure 6.6b. For VG = 0 the R(T ) profile appears as a poor metal below 160 K. Decreasing VG brings non-monotonic R(T ) profiles as µ is brought near the conduction band edge. Finally, upon emptying the conduction band (as determined by Ryx ) a re-entrant metallic profile with a relatively large residual resistivity ratio appears. We now focus on the thinnest devices S1 and S2 measured at He-4 and He-3 temperatures, respectively. A schematic of the band structure under electrostatic gating is shown in the inset of Fig. 6.7. As VG is set to larger negative values, we expect that µ in the crystal will be pulled toward the band gap. Given a crystal that has µ close enough to the gap at VG = 0 and is thinner than the depletion width, we expect that at some VG we can move µ uniformly in to the bulk gap. In Fig. 6.7a, 114 6.2. Results 3.0 3.0 3.0 2.5 2.5 S3 40 2.5 H = 0 80 2.0 2.0 2.0 5 K S4 120 H = 0 ) R (k 1.5 (c) (b) (a) 5 K 1.5 160 50 1.5 100 1.0 1.0 1.0 150 S5 200 0.5 0.5 0.0 0.5 0.0 -150 -100 V G -50 (V) 0 -150 H = 0 5 K 0.0 -100 -50 V G (V) 0 -100 -50 V G 0 (V) Figure 6.5: Resistance R as a function of back gate voltage VG for three different samples (see Table 6.1.) (a) S3 of thickness t = 20 nm at various T . (b) S4 with t = 30 nm at various T . (c) S5 with thickness larger than 40 nm at 5 K. 115 6.2. Results -175 (b) 0 2.4 -165 -165 2.0 0 -175 1.6 -150 yx 1.2 -100 0 R yx ) ( ) R -150 -50 -200 -50 -100 V 4 T (c) -150 -75 V -300 0 0.8 1 T -100 -200 R (k -100 ( ) (a) G 2 3 0.4 S3 T = 10 K (V) 1 0 V S3 0 H = 0 4 0 H (T) 0 50 0.0 100 150 T (K) Figure 6.6: Results for S3. (a) Hall resistance Ryx as a function of magnetic field H for different VG . The overall slope changes from negative to positive for VG ∼ −170 V. The inset shows Ryx at various gate voltages at 1 T and 4 T. (b) Resistance as a function of temperature T at H = 0 at various VG . we see that upon applying ∼ −90 V at 10 K, the resistance per square R(VG ) reaches a maximum and then begins to fall. Combined with the change in Hall sign (shown below) we take this as evidence that we are accessing surface state conduction. This becomes more apparent on closer inspection of the R(T ) profile obtained by taking constant T cuts to R(VG ), show in Fig. 6.7b. For VG = +25 V, µ is in the conduction band. The R(T ) profile is that of a bad metal with a small residual resistivity ratio (RRR). In fact, qualitatively this curve resembles metallic bulk crystals of Bi2 Se3 as in Fig. 5.2a. Both curves have relatively low RRR and exhibit a shallow minimum near 40 K. Taking a T cut at -50 V, we 6.2. Results 116 Figure 6.7: (a) Resistance R at H = 0 as a function of VG for selected T . The traces disperse strongly as VG is tuned beyond the maximum in R. The inset shows band bending driven by the gate potential as well as the depletion length a.(b) Cuts at selected T show significantly different R(T ) profiles at different VG . Inelastic scattering appears to become more important at large negative VG . The inset shows an image of device S1. The scale bar is 4 µm. 117 6.2. Results find a non-monotonic T dependence that resembles the profiles observed in the nonmetallic crystals G3-G7 in Fig. 5.11a. This implies we are near the conduction band edge. Finally, a cut at -100 V reveals a large RRR conduction channel that fits a T 2.5 power law up to 100 K. We have no analog for this profile in bulk crystals. Moreover, the RRR is improved by more than 50% from the bulk channel at VG = +25 V. Measurements of Ryx at 5 T are shown in Fig. 6.8a. At the lowest T , we find that Ryx changes sign at ∼ −80 V. As T increases, Ryx rapidly drops to electron-like values even at the most negative VG . This is consistent with the constant rise in R(VG ) observed in Fig. 6.7a. The shape of Ryx now qualitatively resembles that of few layer graphene in Fig. 6.3. It is noticeable, however, that the hole side of Ryx is suppressed compared to the electron side. This is made more clear when we plot the Hall conductance Gxy and the Hall angle tan θH . Both are large on the electron side, but fall near zero and remain attain only small positive values on the hole side. The strong suppression of the Hall signal in the gap does not imply a sharply reduced mobility, since the T dependence of G shows that the mobility is in fact significantly enhanced over that in the bulk. Under asymmetric gating (as here), Gxy has different values between the top and bottom surfaces. Hence the Hall currents are unbalanced and the Hall resistance RH should be calculated in a multiband model, RH (VG ) = nt µ2t + nb µ2b e(nt µt + nb µb )2 (6.2) where nt,b and µt,b refer to the top and bottom surface state density and mobility, respectively, which all may depend on VG . The multiband Hall response would yield a suppressed RH , but it seems unlikely that the character of the two bands could be 6.2. Results 118 Figure 6.8: (a) Hall resistance Ryx at 5 T. At low T we see a change in sign from electron-like to hole-like response. At elevated T , the entire gating spectrum is electron-like. (b) Hall conductance Gxy at selected T . A noticeable feature is that on the hole side, Gxy is positive but extremely small. (c) Hall angle tan θH shows a similar pattern to Gxy but the fall on the hole side resembles a rounded step function. 6.2. Results 119 so different to accommodate such a strong suppression as in Fig. 6.8. For example, it would require |nt − nb | ≫ |n(0 V)| to produce the observed asymmetry in Ryx (VG ). Unless µb on the hole side was significantly higher than for the electron µt [78], this would be difficult to reconcile with the change in sign of RH . Addition of a third channel from the bulk would involve a significantly lower mobility band and fails to capture the observed response. An interesting possibility is if the 2 surfaces are electrically connected by the side walls, there may be an additional contribution or modification to Eq. 6.2. To test this scenario, one may add a top gate to tune the 2 surfaces independently, or deposit a magnetic film to decouple the 2 surfaces. Plotting G(VG ) at 5 K in Fig. 6.9 brings out some surprising features. First, we can see fluctuations in G across the entire VG range. Subtracting a smooth background, we plot the fluctuation δG in Fig. 6.9b. Here, three separate measurements are shown. The signal is reproducible and of approximately uniform size for all VG . The T dependence of the rms δG is shown in Fig. 6.9c. It fits well to a T −1/2 power law. Application of H uniformly shifts the G(VG ) down by approximately a factor of 2 at 14 T. Surprisingly, the same suppression is seen in δG. Traces of Rxx (H) are shown for several VG at T = 5 K. Both a low H anomaly and fluctuations become more obvious as VG approaches VN . We recognize these features to be qualitatively similar to the quantum corrections to conductance seen in macroscopic Bi2 Se3 in Chapter 5. To study this features in more detail, we cool a second sample S2 with similar VN to 0.3 K. Fig. 6.11a and Fig. 6.11a show the magnetoconductance of device S2 at VG = 0 and VG = VN , respectively. The low H peak is now prominent, and the fluctuations are more pronounced. Additionally, there are longer H scale oscillations developing. 6.2. Results 120 Figure 6.9: (a) Conductance G for device S1 at T = 5 K. There is a broad minimum near charge neutrality. Application of H = 14 T appears to uniformly shift G(H) down by a factor of 2. (b) Fluctuation in conductance δG obtained by subtracting a smooth background. Three separate measurements are shown. The fluctuations stay of the same order across the entire gating spectrum. (c) The rms size of δG decreases as T increases and closely follows a T −0.5 power law. The rms δG is suppressed by a factor of 2 at 14 T at 5 K. 121 6.2. Results 5 T = 5 K S1 4 3 -50 V R xx (k ) -75 V 2 -25 V 0 V 1 0 0 4 8 12 H (T) 0 Figure 6.10: Magnetoresistance at selected VG approaching VN = −80 V for T = 5 K. The magnitude of the MR at both low H and high H grows as electrons are drained from the system. Fluctuations in R(H) are also observed. 122 6.3. Discussion 6.3 Discussion We first analyze the results to determine the basic properties transport properties of the surface states. The metallic profiles of R(T ) for in-gap states in Fig. 6.7b is strong evidence for conducting surface states. As the conduction band has a low RRR, it is unlikely that impurity driven in-gap states would have such a high mobility. From Fig. 6.9a, we can estimate σss assuming that the bulk contribution is minimized. Then, considering 2 surfaces, an upper bound for σxx is approximately 6e2 /h. We can also estimate the mobility µss using σss = neµss µe = σss 4πσss = ne ekF2 (6.3) Taking the Fermi wavevector kF from the ARPES result for the surface states in Fig. 5.1a as kF = 0.05 Å−1 we find µss ≈ 700 cm2 /Vs. This is roughly twice the conduction band mobility µCB found from the Hall effect (see Table 6.1). This together with the high RRR of the in-gap conductance seen in Fig. 6.7b imply the in-gap states have a relatively high mobility. We therefore associate the relatively large MR with the surface states, which given the rigid shift of σ(VG ) with H implies that the surface states act as a parallel conductance channel even when µ overlaps with the bulk conduction band. This is further implied by the uniformity of the conductance fluctuations in Fig. 6.9b across the gating spectrum. We now turn our attention to the low H dependence of the magnetoconductance at 0.3 K. We associate the sharp peaks in Fig. 6.11a and 6.11b with weak antilocalization. We fit the low field magnetoconductance to the standard theory for weak anti-localization 1 H0 e2 H0 ∆Gxx (H) = A ln −ψ + h H 2 H (6.4) 6.3. Discussion 123 Figure 6.11: (a) Magnetoconductance of device S2 at VG = 0 V and T =0.3 K. The low H peak is prominent and fluctuations overwhelm the background G(H). (b) Magnetoconductance at VN . The low H peak grows and further oscillations in G(H) are observed. (c) Fit of low H profile to theory for weak anti-localization (see text). The fit is satisfactory considering the large conductance fluctuations. (d) Results of fitting to Eq. 6.4. The parameter A peaks at charge neutrality near the theoretical value 1/π for 2 conducting surfaces. A peak is seen concurrently in the dephasing field H0 . 6.3. Discussion 124 where A is a number predicted from theory, H0 is the dephasing magnetic field, and ψ is the digamma function. Positive A implies field-suppression of antilocalization in a 2D (two-dimensional) system. For the present system with relatively low mobility and strong spin-orbit coupling, theory predicts A = 1/2π [79], naively implying for the present system a contribution of 2A = 1/π for two conducting surfaces. Confining the fit to H below 0.4 T, two examples are shown in Figure 6.11c at VG = 0 V and -100 V. The fit is satisfactory considering the large conductance fluctuations in the system. Fitting results across the entire gate voltage range are shown in Figure 6.11d. At VN the coefficient A peaks at 0.38, which is within 20% of the theoretical value considering 2 contributing surfaces. It should be noted that our uncertainty in geometric calculation of Ω per square is of roughly this order. We also find a peak in the dephasing field H0 at VN . We may explain this by the increased Coulomb interaction due to the weakining of screening when the density of electrons becomes low. These two facts together with the charge neutrality measured by the Hall effect strongly imply that both surfaces have µ within the bulk band gap at VN . It is noteworthy that even with µ in the conduction band the signature of weak antilocalization persists. Finally, we return to the suppression of conductance fluctuations with H. While AAS oscillations are suppressed on the field scale of weak localization (see the appendix), here the rms δG appears to be suppressed with H over much larger scale. Fig. 6.12 shows a plot of δG(H). It is clear that the amplitude of δG is decreasing as H increases. We hypothesize that the surface state contribution to conductance is disproportionately affected by H compared to the remnant bulk channel. Thus the lessening of δG may reflect a conductance more heavily carried by bulk states. 125 6.4. Summary T = 5 K 0.1 V G = -75 V 2 G (e /h) 0.2 0.0 -0.1 -0.2 4 6 8 10 H (T) 0 Figure 6.12: Fluctuation in conductance δG at VG = −75 V and T = 5 K. The amplitude of fluctuations is suppressed with increasing H. 6.4 Summary We have seen that by electrostatic gating we can isolate transport properties of the surface states in the 3D topological insulator Bi2 Se3 . We are able to tune the crystals to charge neutrality as determined by the Hall effect to focus on the contributions of the surface states to electrical transport. We estimate the surface state conductance to be σss ≈ 6e2 /h, with a mobility enhanced over the bulk as indicated by an increased residual resistivity ratio and estimates of the electron density from ARPES. At low T , we find the H dependence of the conductivity is consistent with field suppression of weak anti-localization for two conducting surfaces at the charge neutral point. We also measure fluctuations in the conductance δG that persist when µ is moved in to the conduction band. This, along with the H dependence of σxx , indicates that the surface acts as a parallel conductance channel with the bulk. Chapter 7 Concluding Remarks We have studied two systems with disparate crystal structure but similar electronic structure described by the Dirac equation. The 2D Dirac (pseudo-)spin polarized states in graphene arise directly from a tight-binding calculation of the band structure, whereas for the 3D topological insulator Bi2 Se3 they arise as a result of time reversal symmetry and the topology of the 3D bands. To conclude, we compare results for the two systems and suggest directions for future work. One basic conclusion from these experiments was that for H = 0, for all their differences, graphene and the 2D surface of Bi2 Se3 both have a conductance of a few times e2 /h. An important parallel in these systems is the prediction of protected transport in a quantum spin Hall (QSH) state. These states arise in the two systems for different reasons. In graphene, symmetry breaking at the lowest Landau level in the quantum Hall regime was predicted to produce the 2D QSH state with counterpropogating edge modes protected from scattering [3]. In Bi2 Se3 , strong spin-orbit coupling and time reversal symmetry were shown to produce a 3D QSH state that has 126 127 been observed in ARPES [14]. Our experiments show that the effect of H on these two proposed QSH states was contrary to expectations. Whereas the graphene QSH state had been proposed to be protected from scattering, we found that in high H the system became an electrical insulator. Moreover, as the insulating state was more robust in cleaner samples, measurements indicate that the CPE modes are absent. For Bi2 Se3 , application of H breaks time reversal symmetry and should therefore destroy the QSH state. However, for H up to 14 T we found no evidence of this even for states at the Dirac point. Future directions for graphene are to further probe the hypothesis of a KT-like transition to the insulating state. Higher quality samples that can be measured without the need for extremely intense H > 14 T would be ideal. Measurements of the angular dependence for H of the transition, IV curves near HC , and detailed T dependence above HC would all be sharp tests of the KT hypothesis. Extending thermoelectric measurements to the insulating regime would also be informative, though perhaps difficult due to the large fluctuation signal observed at low T . For the study of the 3D topological insulator Bi2 Se3 , the main challenge to overcome was isolating the properties of the 2D surfaces. We addressed this issue from a materials perspective, attempting to improve the quality of the insulating bulk state with chemical doping and electrostatic gating. Electrostatic gate control of µ proved to be a powerful tool for isolating the surface band contribution to electronic transport. Provided crystals that are sufficiently thin and uniform, we showed one may overcome the persistent bulk electrons that have thus far hampered experiments in 3D topological insulators. The usefulness of the technique is that it allows in-situ control of µ to compare bulk and in-gap states. Next, one can imagine replacing the 128 Au electrodes here with magnetic or superconducting materials to probe a variety of exotic states that have been proposed [27, 80]. With the characterization of surface state conduction here, realizing topologically non-trivial supercurrents and the physics of spin transport to test the QSH texture in these materials becomes the next step. Appendix A Brief Review of Quantum Corrections to Transport Localization The theoretical framework for localization in electron systems dates back to the original prediction by Anderson that an electron wavefunction could be trapped by disorder [85]. When experiments on 2D systems began to emerge, a ubiquitous feature was a ln T correction to the resistance at extremely low T . Typical data for a Au-Pd thin film is shown in Fig. A.1a below 20 K [81]. Two competing theories were presented to explain this effect. Anderson and coworkers showed that non-interacting electrons would undergo a ln T localization due to constructive interference of time-reversed paths [86], while Altshuler and coworkers showed that Coulomb interaction between electrons predicted the same behavior [87]. One prediction that arose from this debate was that in the same regime in which the conductance of an electron system obeyed a ln T dependence, the contribution 129 130 Figure A.1: (a) Measurement of resistance of Au-Pd thin film from [81]. A strong log T correction appears below 5 K. (b) Magnetoconductance of a Au film from [82]. As T is decreased, the profiles fit to a log H form over increasing range in H. (c) Conductance fluctuation in thin Bi film from [83]. Two traces are shown offset to demonstrate reproducibility. The measurement is performed below 1 K. (d) Comparison of in plane (circles) and out of plane (crosses) magnetoresistance for a thin Cu film from [84]. The MR is flat for the former but large for the latter. 131 from the non-interacting theory should be suppressed in H with a ln H dependence [88]. This effect was later observed; typical experimental data are shown for a Au film in Fig. A.1b [82]. Figure A.2: Real space time reversed paths leading to weak localization from [67]. An electron can explore a closed path 0 → 1 → 2 by elastic scattering off of impurities in the diffusive regime. By symmetry, the time reversed path 0 → 1′ → 2′ must also be allowed. The two paths constructively interfere at the origin, causing the electron to be localized. A real space picture of the mechanism underlying the non-interacting electron picture is shown in Fig. A.2, due to Bergmann [67]. At low T in the diffusive regime, the time between inelastic scattering events τin exceeds that for elastic scattering events τel . Therefore, an electron may scatter several times and retain its wavefunction’s phase. It is possible for an electron to explore a closed path, as shown by the process 0 → 1 → 2 etc. in Fig. A.2. By time reversal symmetry, the opposite path (0 → 1′ → 2 etc. ) must also be possible with the same likelihood. When these 132 two paths meet at the origin, they will constructively interfere since they have have covered the same path. This constructive interference leads to an increased likelihood of the electron to stay in the same position- it is said to be localized. As T goes down and the number of the possible paths grows, the localization grows stronger and the ln T correction dominates the ρ(T ) profile. The prediction of ln H suppression can then be qualitatively understood as the dephasing of the electrons going through these closed paths. The field was considerably enriched with the prediction that spin-orbit coupling would lead to destructive rather than constructive interference [79, 89]. This is referred to as weak anti-localization. The Au film result in Fig. A.1b is actually field suppression of weak anti-localization, as σ drops in H. This theory was confirmed in experiments that measured a crossover from weak localization to weak anti-localization with the deposition of Au on to a Mg film [67]. Important experimental facts about these effects are that: 1. The ln H and ln T dependence of ρ are results of calculations in 2D. 2. These quantum corrections to transport arise only at low T . 3. The field suppression of localization for a single 2D system has the form ∆σ ≈ e2 ln H 2πh (A.1) with factors of order unity. 4. The ln H dependence is only for H perpendicular to the plane since the effect is based on orbital motion. Experimental data consistent with this for a Cu film is shown in Fig. A.1d [84]. 133 5. Both single particle and interaction effects make a contribution to the ln T dependence of σxx at low T , but the contributions may be separated with the application of H. Conductance Fluctuations Figure A.3: (a) AB oscillations arise from modulation of the constructive interference of an electron exploring separate paths about a magnetic flux Φ. The signal that emerges is periodic in h/e. (b) AAS oscillations arise from modulation of the interference of time reversed paths about an enclosed Φ. Here the signal is periodic in h/2e. (c) UCF is generated by modulation of interference of multiple electron paths in a phase coherent region. The signal is aperiodic. In mesoscopic systems 2D systems, several additional corrections to conductance arise from interference of electron paths. Generally, the length scale for these effects is set by the shorter of the phase coherence length Lφ and the thermal length LT Lφ = p Dτin , LT = r D~ kB T (A.2) where D is the diffusion coefficient. Physically LT corresponds to the dephasing of electron paths of conduction electrons from opposite edges of the thermally broadened Fermi function. In a typical system at 1 K, LT ∼ 1 µm. Three important phenomena at these scales are shown in Fig. A.3. 134 The Aharonov Bohm (AB) effect is depicted in Fig. A.3a [90]. In the context of solid state experiments, we consider an electron which can traverse both paths around a region which encloses a magnetic flux Φ . The electron explores both paths, so that Φ modulates the constructive interference when the two paths meet [91]. This fluctuation is seen, for example, in the geometry of a nanoscale Au ring in Fig. A.4 [92]. The field scale of the modulation is h/e. As T grows higher, there is an exponential suppression of this effect as LT becomes shorter than the ring circumference [93]. Periodic arrays of rings can be used to see AB oscillations over a larger length scale, but the signal is suppressed by N 1/2 , where N is the number of rings [94]. The h/e modulation persists to arbitrarily high H. Figure A.4: (a) Oscillations of resistance of 784 nm Au ring (shown in inset) measured at 10 mK from [92]. (b) The Fourier spectrum has peaks corresponding to both the AB and AAS frequencies. 135 A closely related effect is the Altshuler, Aronov, Spivak (AAS) effect shown in Fig. A.3b [95]. Here, instead of an electron exploring each arm of the ring, electrons with time reverse paths have their weak localization modulated by Φ, resulting in a h/2e modulation [96]. The signal only persists to the field scale of weak localization, typically ∼ 0.1 T. Multiply connected rings have been used to generate this effect in arrays as large as 106 [97]. These oscillations are also seen in addition to the AB oscillations in the the Au ring geometry in Fig. A.4. This is evident from the two separate peaks in the Fourier spectrum corresponding to periods h/eA and h/2eA for ring size A for the AB effect and AAS effect, respectively. Finally, in a phase coherent patch electron paths diffuse to form a complex array of overlapping areas. Application of H in a system of this type leads to modulation of G, but it is aperiodic as depicted in Fig. A.3c [98, 99, 100, 101]. Typical experimental data for a thin Bi film is shown in Fig. A.1c [83]. The autocorrelation of these aperiodic fluctuations decays as a power law H d−4 , where d is the dimension of the system. The size of the modulation δG for a coherent patch saturates to e2 /h at low T , which is a universal number for any system where the charge carriers are electrons. For this reason, these fluctuations are referred to as Universal Conductance Fluctuations (UCF). 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