Electronic and Transport Properties of Weyl Semimetals

Home , Electronic entropy, Weyl semimetal

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Timothy M. McCormick, B.S.

Graduate Program in Physics

The Ohio State University

2018

Dissertation Committee:

Professor Nandini Trivedi, Advisor Professor Yuri Kovchegov Professor Mohit Randeria Professor Rolando Valdes Aguilar c Copyright by

Timothy M. McCormick

2018 Abstract

Topological Weyl semimetals have attracted substantial recent interest in condensed matter physics. In this thesis, we theoretically explore electronic and transport properties of these novel materials. We also present results of experimental collaborations that support our theoretical calculations. Topological Weyl semimetals (TWS) can be classified as type-I TWS, in which the density of states vanishes at the Weyl nodes, and type-II TWS, in which an electron pocket and a hole pocket meet at a singular point of momentum space, allowing for distinct topological properties. We consider various minimal lattice models for type-II TWS. We present the discovery of a type II topological Weyl semimetal (TWS) state in pure MoTe2, where two sets of WPs (W2±, W3±) exist at the touching points of electron and hole pockets and are located at different binding energies above EF . Using ARPES, modeling, DFT and calculations of Berry curvature, we identify the Weyl points and demonstrate that they are connected by different sets of Fermi arcs for each of the two surface terminations.

Weyl semimetals possess low energy excitations which act as monopoles of Berry curvature in momentum space. These emergent monopoles are at the heart of the extensive novel transport properties that Weyl semimetals exhibit. We show how the Nernst eﬀect, combining entropy with charge transport, gives a unique signature for the presence of Dirac bands. The Nernst thermopower of NbP (maximum of 800

ii µV·K−1 at 9 T, 109 K) exceeds its conventional thermopower by a hundredfold and is signiﬁcantly larger than the thermopower of traditional thermoelectric materials. The

Nernst eﬀect has a pronounced maximum near TM = 90±20K = µ0/kB (µ0 is chemical potential at T = 0 K). A self-consistent theory without adjustable parameters shows that this results from electrochemical potential pinning to the Weyl point energy at

T ≥ TM , driven by charge neutrality and Dirac band symmetry.

We propose that Fermi arcs in Weyl semimetals lead to an anisotropic magnetothermal conductivity, strongly dependent on externally applied magnetic field and resulting from entropy transport driven by circulating electronic currents. The circulating currents result in no net charge transport, but they do result in a net entropy transport. This translates into a magnetothermal conductivity that should be a unique experimental signature for the existence of the arcs. We analytically calculate the Fermi arc-mediated magnetothermal conductivity in the low-field semiclassical limit as well as in the high-field ultra-quantum limit, where only the chiral Landau levels are involved. By numerically including the effects of higher Landau levels, we show how the two limits are linked at intermediate magnetic fields. This work provides the first proposed signature of Fermi arc-mediated thermal transport and sets the stage for utilizing and manipulating the topological Fermi arcs in experimental thermal applications.

iii To my parents, who inspired my love of science.

iv Acknowledgments

There are many people who have been indispensible during my time in graduate school. First and foremost, I extend my deepest thanks to my advisor, Professor

Nandini Trivedi. Her guidance and support were invaluable during my time here and she was a superb role model for how one should attack an unstructured problem. I can only hope that I picked up some of her physical insight during my time here. I am particularly indebted to her for the many opportunities that I had to collaborate with excellent experimentalists and theorists, as well as to attend several truly amazing conferences in the US and abroad.

My time here would have been much less productive had it not been for the won- derful collaborations that I have been apart of. I thank Professor Adam Kaminski for the opportunity to collaborate on the ARPES discovery of type-II Weyl semimetal

MoTe2. I am deeply grateful to Professor Jos Heremans and his student Sarah Watz- man for our many collaborations on transport phenomena in Weyl semimetals. From them I learned how truly messy yet rewarding studying transport can be. Much of this thesis was inspired by their excellent experiments. I am thankful to Professor Mohit

Randeria, whose precise questions always got to the heart of a matter and were the source of many illuminating discussions on anomalous transport in Skyrmions. I also had the pleasure of collaborating with a very hard-working undergraduate student,

Robert McKay, who was as ideal a mentee as I could ask for.

v The Trivedi and Randeria groups were close to a second family to me during my time here. I follow in the footsteps of Will Cole, Mason Swanson, Eric Duchon,

Onur Erten, and Nganba Meetei, who all helped to shape my physical intuition. I am particularly thankful for their inﬁnite patience in answering the many questions of a new member of the group. I thank Hasan Khan, James Rowland, Tamaghna

Hazra, Po-Kuan Wu, Wenjuan Zhang, Tim Gao, Kyungmin Lee, Kyusung Hwang,

David Nozadze, Mehdi Kargarian, and Sumilan Banerjee for their discussions and companionship. I was also fortunate to have the best oﬃce-mates that I could ask for in Blythe Moreland and Jiaxin Wu.

The condensed matter theory group is particularly lively at Ohio State. My time here was greatly enriched by the many interactions with other students, postdocs, and professors. I extend my thanks to Professor Ilya Gruzberg, Professor Jason Ho, and Professor Yuan-Ming Lu for teaching several excellent classes. I always found their doors open for questions and discussion.

My friends and family provided a backbone of support while I completed my dissertation. I thank Jon Zizka, Andrew Hausman, Jake Kerrigan, Drew Gallagher,

Chris Wolfe, Marci Howdyshell, Adam Ahmed, Simran Singh, Jyoti Katoch, Igor

Pinchuk, Chris Ehemann, Dennis Bazow, and Dante O’Hara for always always being there for me. I thank my wife, Beth McCormick, for her love and support. I also thank my parents for their constant love and encouragement.

Last but certainly not least, I would like to thank the NSF and the Center for

Emergent Materials for funding the majority of my research. I also acknowledge the

OSU presidential fellowship for funding my ﬁnal year.

vi Vita

January 24, 1990 ...... Born - Wilmington, DE, USA

2012 ...... B.S. Physics

Publications

Research Publications

Timothy M. McCormick, Sarah J. Watzman, Joseph P. Heremans, Nandini Trivedi, “Fermi arc mediated entropy transport in topological semimetals” Phys. Rev. B 97, 195152 (2018). Sarah J. Watzman, Timothy M. McCormick, Chandra Shekhar, Shu-Chun Wu, Yan Sun, Arati Prakash, Claudia Felser, Nandini Trivedi, Joseph P. Heremans, “Dirac dispersion generates unusually large Nernst eﬀect in Weyl semimetals” Phys. Rev. B 97, 161404 (2018). Timothy M. McCormick, Robert C. McKay, Nandini Trivedi, “Semiclassical theory of anomalous transport in type-II topological Weyl semimetals” Phys. Rev. B 96, 235116 (2017) Joseph R. Smith, Amber Byrum, Timothy M. McCormick, Nathan Young, Christo- pher Orban, and Christopher D. Porter, “A controlled study of stereoscopic virtual re- ality in freshman electrostatics” Physics Education Research Conference Series 2017, 363 (2017). Timothy M. McCormick, Itamar Kimchi, Nandini Trivedi, “Minimal models for topological Weyl semimetals” Phys. Rev. B 95, 075133 (2017). Lunan Huang, Timothy M. McCormick, Masayuki Ochi, Zhiying Zhao, Michi-to Suzuki, Ryotaro Arita, Yun Wu, Daixiang Mou, Huibo Cao, Jiaqiang Yan, Nandini

vii Trivedi, Adam Kaminski, “Spectroscopic evidence for type II Weyl semimetal state in MoTe2” Nature Materials 15, 1155-1160 (2016). Timothy M. McCormick, Nandini Trivedi, “Tuning the Chern number and Berry curvature with spin-orbit coupling and magnetic textures” Phys. Rev. A 91, 063609 (2015). Fields of Study

Major Field: Physics

viii Table of Contents

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita...... vii

List of Tables ...... xii

List of Figures ...... xiii

1. Introduction ...... 1

2. Topology in Solid State Systems ...... 5

2.1 Berry Phase ...... 6 2.1.1 Berry Curvature ...... 8 2.1.2 Chern Number ...... 10 2.2 Chern Insulators ...... 11 2.2.1 Berry Curvature of a 2-Band Hamiltonian ...... 12 2.2.2 Chern Insulator on the Square Lattice ...... 13 2.2.3 Chiral Edge Modes ...... 15

3. Introduction to Weyl Semimetals ...... 18

3.1 Weyl Fermions ...... 19 3.1.1 Weyl Fermions in Quantum Materials ...... 21 3.2 Lattice Models for Weyl Semimetals ...... 26 3.3 Type II Weyl Semimetals ...... 30

ix 3.3.1 Time Reversal Breaking Model ...... 33 3.3.2 Inversion Breaking Model ...... 42 3.3.3 Surface States: Topological and Track ...... 46 3.3.4 Comparison with Experiments ...... 53 3.3.5 Conclusions ...... 55

3.4 Experimental Discovery of Weyl Semimetal MoTe2 ...... 56 3.4.1 ARPES Results ...... 59 3.4.2 DFT and Topological Analysis ...... 65

4. Thermoelectric Transport in Weyl Semimetals ...... 67

4.1 Boltzmann Transport Theory ...... 68 4.1.1 Definition of Transport Coefficients ...... 68 4.1.2 Boltzmann formalism ...... 71 4.2 Nernst Effect of Isotropic Weyl Nodes ...... 75 4.2.1 Numerical Results ...... 80 4.2.2 Analytic Results ...... 81 4.2.3 Nernst Thermopower in Weyl Semimetal NbP ...... 83 4.3 Anomalous Transport in Type-II Weyl Semimetals ...... 86 4.3.1 Model ...... 88 4.3.2 Anomalous Transport ...... 94 4.3.3 Relation to Measurable Quantities ...... 102 4.3.4 Discussion and Conclusion ...... 103

5. Fermi Arc-Mediated Entropy Transport in Weyl Semimetals ...... 106

5.1 Fermi arc-mediated magnetothermal transport ...... 110 5.1.1 Model ...... 110 5.1.2 Semiclassical Regime ...... 111 5.1.3 Eﬀect of disorder ...... 119 5.1.4 Ultra-quantum Regime ...... 120 5.1.5 Intermediate regime ...... 123 5.2 Comparison with bulk thermal conductivity of WSM ...... 126 5.3 Fermi Arc-Mediated Entropy Transport in Dirac Semimetals . . . . 128 5.4 Discussion and Summary ...... 129 5.5 Conclusion and further directions ...... 133

Appendices 135

A. Isothermal Heat Transport and the Adiabatic Nernst Eﬀect ...... 135

x A.1 Heat transport under isothermal conditions ...... 135 A.2 The adiabatic Nernst eﬀect ...... 137

B. Landau Levels of a Weyl node ...... 140

C. Scattering in Weyl semimetals ...... 143

D. Speciﬁc Heat of Weyl Nodes ...... 145

D.1 Semiclassical regime ...... 145 D.2 Ultra-quantum regime ...... 146 D.3 Crossover regime ...... 147

Bibliography ...... 148

xi List of Tables

Table Page

3.1 Experimental realizations of Weyl semimetals...... 54

3.2 The locations (kx, ky, E) of the Weyl points from DFT and ARPES for weyl semimetal MoTe2...... 66

4.1 Summary of properties of anomalous transport coeﬃcients in type II Weyl semimetals...... 100

5.1 Summary of temperature and magnetic ﬁeld dependence of arc-mediated

κzzz in the semiclassical and ultra-quantum limits as well as the bulk semiclassical magnetothermal conductivity...... 126

5.2 Several topological semimetal candidates for Fermi arc-mediated en-

tropy transport. Since Cd3As2 and Na3Bi are Dirac semimetals, the number of Weyl nodes Np reported is double the number of Dirac nodes. 132

xii List of Figures

Figure Page

2.1 Band structure for the Hamiltonian given in Eqn. (2.19) for B = t and various M: (a) M = −4 where we see the transition between the trivial state and the first topologically nontrivial sector. (b) M = −2 where we see the transition between the two distinct topological regimes. (c) M = 0 showing a gap closing at the final topological tranisition to the trivial state. (d) M = 1 where we see the gapped topological trivial sector. As M increases, this gap grows and the bands flatten...... 14

2.2 Berry curvature (in units of a2 of the lower band and edge states for the Hamiltonian given in Eqn. (2.19) for B = t and various M. Edge states are calculated using strip geometry, ﬁnite in the y-direction with N = 50 layers. In the strip calculations, states localized to the top edge are colored red, states localized to the bottom surface are colored blue, and states in the bulk are colored black. (a) Berry curvature averages to zero over the either band for M = −5 in one of the topologically trivial sectors. (b) Berry curvature in the lower band M = −2.5 where

the Chern number is given by c− = +1. (c) Berry curvature in the lower band M = −1.5 where the Chern number is given by c− = −1. (d) Berry curvature is again zero in the trivial sector where M = 1. (e) Strip geometry calculation for M = −5 in the trivial sector showing a lack of edge states. (f) Edge states for M = −2.5 crossing zero energy

at the kx = ±π point. (f) Edge states for M = −1.5 crossing zero energy at the kx = 0 point. We note that the chirality, or direction, of the edge state on a given surface changes between (e) and (f) due to the change in Chern number. (g) Strip geometry calculation for M = 0 in the trivial sector showing a lack of edge states...... 16

3.1 (a) Continuum dispersion of a Weyl node. (b) Dependence of the chemical potential on temperature µ(T ). We see that on a temperature

scale TW ∼ µ(T = 0)/kB, the chemical potential reaches the Weyl node. 25

xiii 3.2 Band structure for Weyl semimetal in Eqn. (3.19) for tx = 0.5t, m = 2t, k0 = π/2, and lattice constant set to unity a = 1. (a) Bulk energy dispersion with ky = 0. (b) Cut through the Weyl nodes along kx with ky = kz = 0. (c) Constant energy EF = 0.2t cut for a system with slab geometry with N = 50 layers in the y-direction. Surface states are colored red (top) and blue (bottom). We see that small bulk Fermi pockets (shown in black) enclosing the Weyl points (green points) are connected by the Fermi arcs on the top and bottom states...... 28

3.3 Bulk band structure for the “Hydrogen atom” of type-I and type-II Weyl semimetal. a-c The bulk band structure for the Hamiltonian in Eqn. (3.25). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here

we have chosen parameters ky = 0 with parameters k0 = π/2, tx = t, m = 2t for (a) type-I Weyl semimetal with γ = 0, (b) the critical point between type-I and type-II Weyl semimetal with γ = 2t and (c) type-II Weyl semimetal with γ = 3t. The cones comprising the Weyl nodes develop a characteristic tilt of the type-II TWS as γ is increased. d-f

Cuts through the Weyl nodes at ky = kz = 0 for the same parameters as (a-c). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). We see that for a type-I TWS, there are no states at the Fermi energy. At the critical point between a type- I and type-II TWS, we see lines of bulk states appearing between the nodes. These lines open into bulk hole and electron pockets (in the repeated zone scheme) when the system becomes a type-II TWS. . . 35

xiv 3.4 Fermi surface and arc conﬁguration for the “Hydrogen atom” of type-I and type-II TWS. a-c Bulk Fermi surfaces and surface Fermi arcs for a type I TWS with the same bulk parameters as in Fig. 3.3a,d,g calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the following constant energy: (a) E = −0.2t, (b) E = 0, (c) E = 0.2t. We color the states which are exponentially localized to the y = 1 (y = L) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots and marked with pink arrows). We note that at E = 0 the two Fermi arcs are degenerate

along kz = 0 and we color them purple to signify this. d-f Bulk Fermi surfaces and surface Fermi arcs for a type-II TWS with the same bulk parameters as in Fig. 3.3c,f,i calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the same constant energies as above: (d) E = −0.2t, (e) E = 0, (f) E = 0.2t. 37

3.5 Bulk band structure for type-I and type-II TRB model with separate pockets (the “Helium atom”). a-c The bulk band structure for the Hamiltonian in Eqn. (3.26). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown

in green. Here we have chosen parameters ky = 0 with the parameters k0 = π/2, tx = t, m = 2t and γx = t/2 for (a) type-I TWS with γ = 0, (b) type-II TWS with γ = t and (c) type-II TWS with γ = 1.5t. The cones comprising the Weyl nodes again develop a characteristic tilt of the type-II TWS as γ is increased. d-f Cuts through the Weyl nodes

at ky = kz = 0 for the same parameters as (a-c). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). Note that for a type-I TWS, there are no states at the Fermi energy while in the type-II regime, there are two sets of electron and hole pockets on either side of the Weyl nodes. We see that unlike the Hydrogen-atom model, the Helium-atom model has disjoint pairs of electron and hole pockets and a pair of each meet at the two Weyl nodes...... 40

xv 3.6 Fermi surface and Fermi arc conﬁguration for type I and type- II time-reversal-breaking model with separate pockets (the “Helium atom”). a-c Bulk Fermi surfaces and surface Fermi arcs for a type-II Weyl semimetal given by Eqn. (3.26) with the same bulk parameters as in Fig. 3.5b,e,h calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the constant energies: (a) E = −0.2t, (b) E = 0, (c) E = 0.2t. As in Fig. 3.4, we color the states that are exponentially localized to the y = 1 (y = L) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots). We note unlike in Fig. 3.4, each node is isolated in its own hole (a) or electron (c) pocket when the chemical potential is away from E = 0. These pockets are connected by arcs conﬁned to the surface in the y- direction. However, in this type-II TWS the Fermi pockets enclosing a Weyl node can be quite extended, unlike a type-I TWS, the arcs can terminate on a pocket quite far away from the projection of the nodes. d-f Bulk Fermi surfaces and surface Fermi arcs for a type-II TWS with the same bulk parameters as in Fig. 3.5c,f,i calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the same constant energies as above: (d) E = −0.2t, (e) E = 0, (f) E = 0.2t. We see that as the tilt grows, so do the pockets enclosing the nodes. We note that a trivial electron pocket appears

around the (kx, kz) = (π, π) point. This pocket encloses no Weyl nodes and so is not connected via Fermi arcs to any other pockets...... 41

xvi 3.7 Bulk band structure for type-I and type-II inversion breaking TWS. a-c The bulk band structure for the Hamiltonian in Eqn. (3.28). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen param-

eters ky = 0 with the parameters k0 = π/2, tx = t/2, m = 2t for (a) type I TWS with γ = 0, (b) the critical point between a type-I and a type-II TWS with γ = 2t and (c) type-II TWS with γ = 2.4t. The cones comprising the Weyl nodes develop a characteristic tilt of the type-II Weyl node as γ is increased. d-f Cuts through the Weyl nodes

at ky = 0 and kz = −π/2 for the same parameters as (a-c). These cuts are shown as the green lines in (g-i). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). We see that for a type-I Weyl semimetal, there are no states at the Fermi energy. At the critical point between a type-I and type-II TWS, the density of states still vanishes. In the type-II regime, electron and hole pockets form near the Weyl nodes. These pockets enclose the projections of the Weyl nodes when the chemical potential is shifted away from E = 0. Trivial pockets also appear at k = (0, 0, 0) and k = (0, 0, π)...... 45

3.8 Fermi surface and Fermi arc conﬁguration for type-I and type- II inversion-breaking Weyl semimetal. a,b The Fermi surface and Fermi arc conﬁguration for the Hamiltonian given in Eqn. (3.28) in the type-I limit (γ = 0) calculated in a slab geometry with L = 50 layers and with bulk parameters the same as in Fig. 3.7a,d,g. We show this calculation at constant energies: E = −0.25t (a) and E = 0.25t (b).

Here we see that Weyl nodes located at (kx, kz) = (±π/2, ±π/2) are connected by surface states (red and blue lines) to one of opposite

chirality across the Brillouin zone in the kx-direction. c,d The Fermi surface and Fermi arc conﬁguration for the Hamiltonian given in Eqn. (3.28) in the type II limit (γ = 2.4t) calculated in a slab geometry with L = 50 layers and with bulk parameters the same as in Fig. 3.7c,f,i. We show these for the same constant energies as above: (c) and E = 0.25t (d). The locations of the Weyl nodes are marked with pink arrows. We term the exponentially localized surface states that form closed loops “track states”. Fermi arcs are shown as bold lines and connect Weyl

nodes in the kz-direction...... 47

xvii 3.9 Evolution of Fermi surface and Fermi arc conﬁguration for inversion-breaking Weyl semimetal as a function of γ. a-d The evolution of the Fermi surface and Fermi arc conﬁguration in a slab geometry for Eqn. (3.28). Bulk states are down in black, surface states are shown in red and blue. We have chosen the parameters

k0 = π/2, tx = t/2, m = 2t. The calculations are done at constant energy E = −0.25t for γ = 0 (a), γ = 0.8t (b), γ = 1.4t (c), and

γ = 2. (d) shown in an extended Brillouin zone where both kx and kz range from −1.5π to 1.5π. We see that at the critical point between a type-I and type-II (d), the Fermi arcs that previously connected

Fermi pockets in the kx-direction now connect Fermi pockets in the kz-direction and track states have formed on the bottom surface (blue) around the (kx, kz) = (π, π) point...... 48

3.10 Sketch of the three types of surface states in a topological Weyl semimetal. a Two type-I Weyl nodes of opposite chirality connected by a Fermi arc on the top (red) and bottom (blue) surfaces. In an arbitrary type-II TWS at an energy away from the Weyl energy, these arcs would connect Fermi pockets instead of nodes. b A single Fermi pocket enclosing two nodes of opposite chirality. Since no Gaussian surface can be constructed in a region that is both gapped and encloses only one node, the only possible surface states are trivial ones, shown in red and blue at the boundary of the pocket that hybridize with bulk states due to lack of topological protection. c Pairs of Weyl nodes, two of each chirality with each node surrounded by a Fermi pocket. The pockets are connected by Fermi arcs (thinner red and blue contours) as well as track states (thicker blue lines) on the bottom surface. Note that states on opposite sides of a given loop of track states will disperse in opposite directions and so a Gaussian surface enclosing a given Fermi pocket will still have one net surface state of each chirality...... 51

xviii 3.11 Simple model of type II Weyl semimetal described by a two band model given by Eq. 3.28 which exhibits four Weyl nodes. a Electronic band structure for µ = ±0.1t indicated by the blue translucent plane. b,c The topological surface states and Fermi arcs on surface A (in red) and B (in blue) are calculated for a slab geometry conﬁned along the y-direction. The bulk bands are shown in black. When µ = 0 exactly, the electron and hole pockets touch and the arcs terminate on the node (green dot) itself. For Fermi energy below (above) the nodal energy, arcs of surface states connect the Fermi hole (electron) pockets surrounding a node rather than terminating on a node. d,e Energy

dispersion along kz at ﬁxed kx as shown by cuts in panels (b, c). Cut 1 along kx = π/2 shows the bulk electron and hole bands touching at the node and the merging of surface states into the bulk away from

the Weyl node. Cut 2 along kx = 0.63π shows a gap between the bulk bands and a surface state that disperses with opposite velocities at

the projections of the two WPs. The WPs are located at (kx, kz) = (±π/2, ±π/2) indicated by pink arrows pointing to green dots. . . . . 58

3.12 Experimental Fermi surface and band structure of MoTe2. a Constant energy intensity plot measured at EF using 6.7 eV photons for a sample with termination A. The calculated (DFT) positions of Weyl points W2 are marked as pink dots, while experimentally determined locations of

W2 and W3 points are marked as red dots. The chiralities of Weyl points are marked with “+” and “-” and their locations (kx, ky, E) are summarized in Table 3.4.2. b Same as in a above but taken at 10 meV

above EF . c A sketch of constant energy contours of electron and hole bands showing the locations of Weyl points and Fermi arcs. d Constant

energy contour measured at 30 meV above EF using 5.9 eV photons for a sample with termination B. Positions of calculated and measured Weyl points are marked as above. e Same surface termination and

photon energy as d but at 30 meV below EF . f - i Experimental band dispersion along cuts at kx = 0.24, 0.28, 0.32 and 0.36 π/b. j - m Calculated band dispersion for a sample with termination A along kx = 0.24, 0.28, 0.32 and 0.36 π/b...... 60

xix 3.13 Identiﬁcation of Weyl points and Fermi arcs from experimental data.

a Constant energy contour at EF , measured by 6.7 eV photons for surface termination A. DFT predicted locations for Weyl points W2 and measured Weyl points W2, W3 are marked as red and pink dots respectively. b The same panel as a except for surface termination B. c The same panel as b except for using 5.9 eV photons. d - g Energy

dispersion for surface termination A along ky = 0, 0.05, 0.10 and 0.20 π/a. The projections of Weyl points W2 are marked as dots. h - k The same panels as (d - g) except for surface termination B. l - o The same panels as (h - k) except for using 5.9 eV photons. p - s

Calculated band dispersion for surface termination A along cuts at ky = 0, 0.05, 0.10 and 0.20 π/a. Positions of W2 are marked similarly as above. t - w The same as (p - s) except for surface termination B. Bands plotted with darker lines have more surface weights...... 62

3.14 Results of DFT calculations. a Calculated bulk Fermi surface of MoTe2 for kz = 0.6π/c and projections of W2 (kx, ky) = (±0.17 π/b, ±0.06π/a) are marked with pink dots. b Bulk band dispersion along W2-W2 direction (the vertical dashed line in a). DFT predicted positions of

W2 (ky, E) = (±0.06π/a, 0.028 eV) are marked. c The dominant DD DD contribution for the divergence of the Berry curvature (Ωn,yz, Ωn,zx) for the n = N + 1 th band where N is the number of electrons in

the unit cell with kz = 0. Red and blue indicate diﬀerent chiralities of the two Weyl points. d - g Calculated constant energy contours

of MoTe2. Darker bands are surface bands and lighter bands are bulk bands. d, e are at Fermi level for surface termination A and B. f, g are at Fermi level + 28 meV of surface termination A and B, respectively.

h, i Surface band dispersions of termination A and B along W2-W2 direction. j, k Surface band dispersions of termination A and B along

ky = 0.05 π/a direction, which is very close to the ky position of W2 (0.06 π/a). Positions of calculated Weyl points W2 are marked and darker bands have more surface weights in d - k...... 64

4.1 Geometry for measuring the Nernst effect αxyz. Temperature gradient ∇rT , electric field E, and magnetic field B are all mutually perpendicular. 75

xx 4.2 Magnetic field dependence of Nernst thermopower αxyz, and temperature dependence of Nernst coefficient Nxyz (A) Data for the Nernst thermopower plotted as a function of applied magnetic field at the various temperatures indicated. The insert shows the magnetic field dependence of the Nernst voltage measured at 4.92 K in a temperature gradient of 2.17 mK. SdH oscillations are plainly visible that correspond to the periods measured in the magnetization. (B) Nernst coefficient plotted as a function of temperature, with low-field Nernst effect in red and high-field Nernst effect in blue. The low field curve peaks near 50K; the high-field curve peaks near 90K, which is the temperature at which the chemical potential touches the Weyl nodes. Error bars represent a 95% confidence interval on the standard deviation of the systematic error, excluding geometrical error on the sample mount itself...... 79

4.3 Nernst coeﬃcient Nxyz as a function of temperature calculated using Eqn. (4.47) using the temperature dependent chemical potential found by solving Eqn. (3.10) self-consistently. We see agreement with the numerical results above and we note that the temperature

dependence of Nxyz is independent of τ...... 82

4.4 Cuts through the band structure given by the Hamiltonian in Eqn.

(4.48). In (a-d), we show energy versus kz cuts for kx = ky = 0. Here we have chosen m = 3t; tz = t; k0a = π/2; γz = 0.5t for γ = 0 (a), γ = 1.2t (b), γ = 2t (c), and γ = 2.8t (d). In (e-h), we show constant energy cuts for the band structure deﬁned by Eqn. (4.48). (a) and (e) are in the type-I limit; (b) and (f) are in the type-II limit with distinct pockets making up each nodes; (c) and (g) are in the type-II limit after the electron pockets have merged; and (d) and (h) are in the type-II regime where the Weyl nodes share only a single electron and single hole pocket. Thus, as γ is increased we can see the successive Lifshitz transitions described in the text...... 90

xxi 4.5 Each column corresponds to a particular γ with: γ = 0 (a, e, i), γ = 1.2t (b, f, j), γ = 2t (c, g, k), and γ = 2.8t (d, h, l). (a-d) show the net net Berry curvature in the z-direction Ωz (E) deﬁned by Eqn. (4.50). We see that for nonzero γ, the net Berry curvature around the nodes

is of the same sign. (e-h) the density of states for m = 3t, tz = t, k0a = π/2, and γz = 0.5t, for diﬀerent values of the tilt parameter. (i-l) illustrates the temperature dependence of the chemical potential,

µ(T ). Each plot shows three separate values of EF : EF = 0 (blue), EF = 0.1 (green), and EF = 0.2 (red). We see that for smaller values of γ, g(E) has a minimum close to the Weyl energy E = 0, but for larger values of γ, this minimum shifts far from the nodes. This has a strong eﬀect on the shift of the chemical potential with temperature. 93

EE ET 4.6 In (a-c), we plot each anomalous transport coeﬃcient Lxy , Lxy , and TT Lxy for the Hamiltonian given by Eqn. (4.48) with parameters m = 3t, tz = t, k0a = π/2, and γz = 0.5t, as a function of γ for the following temperatures: T = 0.05t (purple), T = 0.1t (blue), T = 0.15t (green), EE ET TT and T = 0.2t (red). In (d-f), we show Lxy , Lxy , and Lxy for the same values as in (a)-(c), with γ = 1.2t, plotted as functions of temperature

for various Fermi energies: EF = 0 (black), EF = 0.1t (magenta), and EF = 0.2t (blue)...... 96

5.1 (a) Mixed real-space and momentum-space depiction of a Weyl semimetal in a slab geometry with thickness L in the z-direction. Bulk Weyl nodes

are labeled with their chirality χ = ± and separated in the kx-direction. The projections of the Weyl nodes on the surface Brillouin zone define the end points of the Fermi arcs on the top and bottom surfaces. (b) Schematic of the “conveyor belt” motion of charge leading to a net heat flux. When B is aligned in the z-direction in (a), charge current density circulates Je in the clockwise direction shown in a mixed real and momentum space orbit. When ∇T is also aligned in the z-direction, this circulation of charge leads to a net flow of heat current density JQ

in the direction shown. Unit tangent vectors et are shown for the arcs. 108

xxii 5.2 (a) Thermal occupations for the minimal model in Eqn. (5.2) with

m = 2t, tz = t, and kW a = π/2 for cuts along ky at ﬁxed kxa = 0.75π (top and bottom) and kxa = 0.5π (bulk). We show the occupations of the top two layers (top), the bottom two layers (bottom) and the other layers (bulk), weighting each point’s color and thickness in the ﬁgure

with the Fermi function f0 at that layer for a temperature gradient of dT 0.8t = at an average temperature of kBT = 0.6t. (b) Orbits for the dz kB a minimal model in Eqn. (5.2) with the same parameters as in (a) for

cuts along kx at ky = 0. We again show the occupations of the top two layers (top), the bottom two layers (bottom) and the other layers (bulk), weighting each point as in (a) with the same temperature and temperature gradient. We show the mixed real- and momentum-space orbits and see that the the states most occupied on the top and bottom are the arcs, while the bulk states merge into the arc and carry heat through the bulk...... 112

5.3 (a) Schematic of the Landau levels. Chiral n = 0 Landau levels are shown in red. The sign of the slope of the n = 0 Landau levels is positive (dashed line) for χ = 1 nodes and negative (solid line) for χ = −1. Non-chiral Landau levels (n 6= 0) are shown in black. (b) Speciﬁc heat for a single pair of Weyl nodes in the intermediate quantum limit as a

function magnetic field. We have set the Landau level cutoff Nmax = 50 and we have considered an electron density such that nB = 1. (c) Mag- netothermal conductivity κzzz for a single pair of Weyl nodes in the intermediate quantum limit as a function of magnetic field. We have

set the Landau level cutoff Nmax = 50 (see Appendix C) and n = 1. We have fixed the temperature such that kB T = 0.02 (blue), kB T = 0.04 ~vF Λ ~vF Λ (green), and kB T = 0.06 (red). At higher temperatures (red), we see a ~vF Λ crossover between the linear low-field behavior and the quadratic field dependence when the ultra-quantum limit is reached. At lower temperatures (green, blue) only the lowest Landau level is populated in the field range shown. (d) Fermi arc-mediated magnetothermal conductivity for a single pair of Weyl nodes in the intermediate quantum limit as a function of magnetic field. Parameters are the same as in (b,c)...... 125

xxiii 5.4 Schematic of the different cases for surface states in a Dirac semimetal with a single pair of Dirac nodes. Green dots represent projections of the Dirac nodes on the surface Brillouin zone. Red contours are surface states on a single surface. Purple filled regions represent projections of bulk Fermi pockets which enclose Dirac nodes. In (a-b), the Fermi energy is at the Dirac nodes, while in (c-d) the Fermi energy is doped away from the Dirac nodes. (a) A pair of surface Fermi arcs terminates on the Dirac nodes on a single surface. (b) A perturbation containing the full symmetry of the full Dirac Hamiltonian which does not shift the bulk Dirac nodes can deform the Fermi arcs into a closed Fermi pocket on the surface, much like a topological insulator. (c) When the Fermi energy is shifted away from the Dirac nodes, the surface states may still form a closed Fermi pocket. (d) Sufficient doping may cause the Fermi pockets to grow large enough for the surface states to merge with the bulk pockets...... 130

xxiv Chapter 1: Introduction

The intersection of topology and the band theory of solids has attracted extensive recent interest in condensed matter physics. Topological insulators [1, 2] opened the door to novel phases of matter with bulk energy gaps and protected, gapless edge states. Weyl semimetals[3, 4, 5] were the ﬁrst topological material that were no longer constrained by a bulk energy gap that protected the topological nature of the energy bands. In this thesis, we predict new electronic and transport properties of

Weyl semimetals. We also present experimental discoveries of the first type-II Weyl semimetal and the first measurement of the Nernst effect in a Weyl semimetal for which we have provided theoretical support. For the remainder of this chapter, we shall outline the historical discovery of Weyl semimetals and provide context and motivation for our work.

Weyl ﬁrst predicted his massless fermions as candidates for the neutrino[6], however the discovery of the neutrino mass[7, 8] eliminated this possibility. It was decades until Weyl fermions were again proposed to exist, this time as low-energy, linearly dispersing modes in pyrochlore iridates[9]. These antiferromagnetic materials with strong electron-electron interactions and large spin-orbit coupling attracted intense experimental interest searching for these Weyl fermions, however to this date, no convincing signatures have been found in this class of materials.

1 Other theoretical predictions for the emergence of Weyl fermions were made for

superlattices of topological insulators and trivial insulators [10], magnetic spinels

[11], and in liquid 3He [12]. Further density functional theory predictions of Weyl

semimetals were made, particularly in the TaAs class of materials[13, 14] where a

Weyl semimetal was discovered experimentally in TaAs [15, 16, 17] by angle-resolved

photo-emission spectroscopy (ARPES).

In a Weyl semimetal, the linearly dispersing Weyl nodes are monopoles of Berry

curvature, which acts like a momentum-space magnetic ﬁeld. The Berry curvature

is responsible for the topological nature of these materials. We devote Chapter 2 to

a pedagogical introduction of topology in quantum materials as well as the Berry

curvature and associated topological quantities such as the Berry connection and

phase. We also introduce the Chern insulator, a 2D gapped phase with nontrivial

topology which we will see plays a key role in understanding Weyl semimetals.

Like topological insulators, Weyl semimetals also have highly nontrivial surface

states. These so-called Fermi arcs are perhaps the most fascinating aspect of these

materials. In Chapter 3, we formally introduce Weyl fermions as solutions to the

Dirac equation and show that, on a lattice, these Fermi arcs must exist. In Chapter

3, we also introduce the tilted type-II class of Weyl semimetals and introduce the ﬁrst

lattice models describing this class of materials. We also present the ﬁrst experimental

evidence for a Weyl semimetal in MoTe2, where our theoretical work plays a central role.

The bulk Berry curvature of Weyl fermions is known to result in a plethora of unique transport phenomena in Weyl semimetals. In parallel magnetic and electric

ﬁelds, Weyl semimetals exhibit negative longitudinal magnetoresistance as a result

2 of the chiral anomaly[18, 19, 20, 21, 22, 23, 24]. In Chapter 4, we predict that the thermoelectric Nernst response of Weyl semimetals contains a key signature that allows for a quantitative determination of the Fermi energy. The Nernst eﬀect is the transverse conversion of a thermal gradient to a voltage and has technological applications [25]. In Chapter 4, we also present the ﬁrst experimental study of the

Nernst thermopower in a Weyl semimetal and show that this matches our prediction quantitatively. The Berry curvature of time-reversal breaking Weyl semimetals is also known to lead to an anomalous Hall coefficient[26]. In Chapter 4, we show the first theoretical predictions for how the tilt of a type-II Weyl semimetal affects the anomalous Hall coefficient, the anomalous transverse thermoelectric coefficient, and the anomalous thermal Hall coefficient. These results also have broad possible technological applications.

The role of Fermi arcs in transport was ﬁrst predicted to lead to quantum oscillations in conductivity that are dependent on the thickness of a ﬁnite slab of Weyl semimetal [27]. Putative signatures of these mixed bulk and surface coherent oscillations have been observed in Dirac semimetal Cd3As2 [28]. However, there are limitations to detecting and utilizing these mixed bulk and arc oscillations since they rely on quantum coherence across the orbits. Motivated by these previous works, in

Chapter 5, we propose a novel Fermi-arc mediated pathway for thermal transport in Weyl semimetals. Unlike the Shubnikov-de-Haas oscillations discussed above, the

Fermi-arc mediated thermal conductivity that we predict does not rely on quantum coherence. Signatures of this heat transport should be more readily detectable due to the characteristic sample thickness independent conductance. Our predictions open

3 the door to the possibility utilizing this pathway in technological applications such as magnetically tunable thermal switches.

4 Chapter 2: Topology in Solid State Systems

Beautiful examples of symmetry abound in the natural world, ranging from closely spherical stars to the spectacular six-fold patterns of snowﬂakes. Symmetry also proves to be a strikingly useful tool in theoretical physics. In condensed matter physics, where one considers vast collections of matter organized in a plethora of states, symmetry plays an especially crucial role in the classication of states of matter.

The Ginzburg-Landau[29] theory of phase transitions characterizes a stable phase of matter with a local order parameter, which is zero in the disordered phase but nonzero in the ordered phase. Diﬀerent ordered phases can be organized by how their local order parameters transform under symmetry operations. Prior to 1980, this phenomenological theory of phase transitions, along with microscopic many-body theories, formed the backbone of the study of condensed matter physics.

The 1980 discovery of the quantum Hall eﬀect (QHE) [30] shattered this paradigm and presented the ﬁrst example of a system that had no spontaneously broken symmetry. Instead, its behavior depends only on a topological invariant that is independent of the system’s geometry or microscopic properties. In the QHE, the transverse current is characterized by a quantized conductivity given by

e2 σxy = cn, (2.1) ~

5 where e is the fundamental unit of electronic charge, ~ is Planck’s constant, and cn is an integer. This integer quantization has been experimentally veriﬁed to one part in 107[31] and is a direct manifestation of nontrivial topology of the bulk electronic wavefunctions. The quantum Hall system possesses cn topologically protected

edge modes, each carrying a quantum of conductance e2 . This correspondence of ~ bulk topology with protected edge modes is characteristic of topological systems in condensed matter physics.

Since the discovery of the quantum Hall eﬀect, many examples of topological states of matter have emerged in condensed matter physics. Topological insulators

[1, 2] extended this paradigm to three dimensional materials. Since then, many exotic states that possess a gapped, nontrivial bulk topology and corresponding gapless edge modes have been theoretically and experimentally discovered, such as topological crystalline insulators[32] and topological superconductors [33, 34].

In this chapter, we show that an electron will in general pick up a geometric phase known as the Berry phase under time evolution. The Berry phase is a global, topological invariant related to a local geometrical quantity known as the Berry curvature.

We derive general facts about these related quantities. Finally, we introduce a simple model for a Chern insulator that will provide key insights in our later discussions of

Weyl semimetals.

2.1 Berry Phase

Consider a system with a time-dependent Hamiltonian H(k) that depends on time through several parameters k = (k1, k2, ...), where kj = kj(t). We are interested in

how the wavefunctions |ψ(t)i vary adiabatically as the system evolves in time. The

6 time evolution of the system is given by

d H(k(t)) |ψ(t)i = i |ψ(t)i . (2.2) ~dt

We introduce the instantaneous eigenstates |n(k)i by solving

When we insert this wavefunction into Eqn. (2.2), we obtain that θ(t) is given by

Z t Z t 1 0 0 0 d 0 0 θ(t) = En(k(t ))dt − i hn(k(t ))| 0 |n(k(t ))i dt . (2.4) ~ 0 0 dt

The ﬁrst term of the phase is the conventional dynamical phase. The negative of the second term is known as the Berry phase. Since the states only depend on time through the parameter k(t), we can write the Berry phase as

Z t Z 0 0 dk 0 γn = i hn(k(t ))| ∇k |n(k(t ))i · 0 dt = i hn(k)| ∇k |n(k)i · dk, (2.5) 0 dt C where C is the path that the electron traverses in time t. We can deﬁne a vector potential called the Berry connection:

An(k) = i hn(k)| ∇k |n(k)i , (2.6) which acts analogous to the vector potential from electromagnetic ﬁeld theory. There- fore, we can write the Berry phase as a line integral of this vector potential such that

Z γn = dk · An(k). (2.7) C

7 Continuing the analogy with real-space electrodynamics, we see that γn plays the role

of a magnetic ﬂux.

The Berry connection is gauge dependent. If the states are transformed by a U(1) rotation |n(k)i → eiξ(k) |n(k)i for some real-valued function ξ(k), then the Berry connection transforms to

∂ A (k) → A (k) − ξ(k). (2.8) n n ∂k

Hence the Berry phase is changed by an amount ξ(k(t)) − ξ(k(0)) over the contour

C. It was originally thought that any such phase factor could be eliminated with a

clever choice of gauge. However, Berry showed that[35] that for closed paths where

k(t) = k(0), since the wavefunctions must be single-valued, it must be true that

ξ(k(t)) − ξ(k(0)) = 2πm, with m an integer. Hence the Berry phase itself cannot be

canceled for closed paths unless it is an integral multiple of 2π.

2.1.1 Berry Curvature

We have shown above that the Berry connection is gauge dependent and it is often

convenient and instructive to consider the gauge independent Berry curvature deﬁned

Ωn(k) = ∇k × An(k). (2.9)

This quantity is analogous to the magnetic ﬁeld and is a similarly observable quantity,

unlike the Berry connection or real-space electromagnetic vector potential. By Stokes’

theorem, we can write the Berry phase as

Z γn = dS · Ωn(k), (2.10)

where dS is the diﬀerential area element of the region enclosed by C.

8 We brieﬂy discuss how Ωn(k) transforms under some basic symmetry operations.

Henceforth, we let k be the wavenumber of a particle p = ~k. A system with no internal degrees of freedom transforms under inversion (or parity) Pˆ by sending each

momentum k → −k. Similarly, a spinless system under time-reversal symmetry Tˆ by the complex conjugation operator composed with k → −k. The gradient operator ˆ ˆ in momentum-space therefore transforms ∇k → −∇k under either P or T . Hence the Berry connection transform as An(k) → An(−k) under inversion and transforms as An(k) → −An(−k) under time-reversal. Hence, using Eqn. (2.9), we obtain the following useful relations

Pˆ Tˆ Ωn(k) −→ Ωn(−k), Ωn(k) −→−Ωn(−k). (2.11)

Hence in the presence of inversion symmetry, the Berry curvature is an even function

of k, while in the presence of time-reversal symmetry it is is an odd function of

k. Therefore in the presence of both, the Berry curvature must be identically zero

everywhere.

In principle, the Berry curvature is straightforward to calculate directly from the

quantum mechanical wavefunctions following Eqns. (2.6) and (2.9). However, in

many cases, it is diﬃcult or impossible to obtain an analytic result for the eigenvec-

tors of a Hamiltonian with Hilbert space dimension ≥ 5. One can naively calculate

the Berry connection and curvature for a generic Hamiltonian whose states have been

numerically solved for, but this can present serious challenges if the numeric methods

used to obtain the spectrum do not consistently choose a gauge for diﬀerent parame-

ters k. Speciﬁcally, if the matrix diagonalization routine chooses diﬀerent numerical

phases for nearby eigenvectors, a simple-minded calculation of the Berry connection

as a ﬁnite diﬀerence will introduce substantial errors in the calculation. In this case,

9 it is best practice to use the more sophisticated method of Fukui, Hatsugai, and

Suzuki[36].

2.1.2 Chern Number

The Berry curvature represents a local, gauge-invariant geometric property of the spectrum for a particular Hamiltonian. Much like the physical Gaussian curvature of a manifold, the integral of the Berry curvature is a quantized topological invariant, known as the TKNN invariant[37] or the Chern number[38]. The Chern number for the nth electronic band is given by

1 Z c = dk · Ω (k), (2.12) n 2π n where k = (kx, ky) is the crystal momentum for a two-dimensional periodic system integrated over the entire Brillouin zone. The Chern number for a single band is well- deﬁned only if there exists a local gap in energy |En(k)−En±1(k)| > 0 for all k. When electronic bands are degenerate at some subset of points in the Brillouin zone, then the Chern number given by Eqn. (2.12) is not well-deﬁned for any of those bands.

It is clear that the Chern number can only be non-zero if time-reversal symmetry is broken.

The integer in Eqn. (2.1) for a QHE system on a lattice is the Chern number given above and, generically, the Hall conductivity of a ﬁlled band is given by Eqn.

(2.1), even for systems with zero external magnetic ﬁeld. Just as in the quantum

Hall system, there exists a number of dissipationless edge modes equal to the sum of Chern numbers for all occupied bands. We shall see in Chapter 4 that even when bands are not completely occupied, the anomalous Hall conductivity can be related to the net Berry curvature of fully and partially occupied bands.

10 There is a rather sophisticated mathematical argument based on homotopy theory by Avron, Seiler, and Simon [39] that for a system with a finite number of bands, the sum of the Chern numbers of all bands must be zero. Since one is often interested in the sum of the Chern number of all occupied bands, this sum rule tells us that the sum of the Chern numbers of all unoccupied bands must therefore be equal and opposite. This sum rule also tells us that a system with only a single band cannot have a non-trivial Chern number. Hence, we must look to systems with at least two bands to find the simplest example of non-trivial topology in the band theory of solids. In the next section, we will discuss a specific example of a two-band system with non-zero Chern number for its bands.

2.2 Chern Insulators

We can write the generic Hamiltonian for any two-band system in the following way 3 X H(k) = d0(k)ˆσ0 + dj(k)ˆσj, (2.13) j=1 where dα(k) is a periodic function of k,σ ˆ0 is the 2 × 2 identity matrix, andσ ˆj is the j-th Pauli matrix. Here, the space over which the Pauli matrices span need not be electronic spin. Instead, “up” and “down” may represent the states of any two-orbital system. H(k) is the kernel of the momentum space Hamiltonian given by

ˆ X ˆ † ˆ H = Ψk ·H(k) · Ψk, (2.14) k ˆ ˆ where Ψk is a spinor of electron creation and annihilation operators given by Ψk =

(†) (ˆck,↑, cˆk,↓) and wherec ˆk,s destroys (creates) an electron in state s.

11 2.2.1 Berry Curvature of a 2-Band Hamiltonian

The energy spectrum of the Hamiltonian in Eqn. (2.13) is straightforwardly found to be q 2 2 2 E(k) = d0(k) ± dx(k) + dy(k) + dz(k). (2.15)

q 2 2 2 We henceforth denote d ≡ dx(k) + dy(k) + dz(k). The eigenstates are found to be 1 dz ± d ψ± = p . (2.16) 2d(d ± dz) dx − idy

Hence, we can directly compute the j-th component of the Berry connection to be

(j) −1 A− (k) = i hψ−| ∂kj |ψ−i = dy∂kj dx − dx∂kj dy . (2.17) 2d(d + dz)

We can similarly calculate the Berry connection for the band with energy E+. We can then write the j-th component Berry curvature for each band compactly as

d(k) · (∂ d(k) × ∂ d(k)) Ω(j)(k) = ∓ kl km . (2.18) ± jlm 2d3(k)

We notice that the ﬁeld strength of the Berry curvature is the Jacobian of the map

d 2 k → d . This Jacobian is a map between 2-D Brillouin zone - the two torus T - and the target manifold given by the 2-D sphere S2 given by d2 = 1. We see that for a two-band Hamiltonian, the Berry curvature ﬁeld strength, integrated over momentum-space, simply counts the number of times d winds wraps the unit sphere.

We deﬁne a Chern insulator as a system where: (1) the Hamiltonian is of the form given by Eqn. (2.13) whose d-vector wraps the unit sphere a non-zero integral number of times and (2) where there exists a nonzero gap between the E+ and E− bands throughout the Brillouin zone. There are several examples of such systems in the literature. Historically, the ﬁrst example was Haldane’s model[40] for the quantum

12 Hall eﬀect without Landau levels on the honeycomb lattice. However, since no system that we will be interested in possesses the C3 symmetry of the honeycomb lattice, we

ﬁnd it more insightful to introduce a Chern insulator on a square lattice.

2.2.2 Chern Insulator on the Square Lattice

Consider the following Hamiltonian

H(k) = t sin (kxa)ˆσx + t sin (kya)ˆσy + B (2 + M − cos (kxa) − cos (kya))σ ˆz, (2.19) where t and B are parameters with the dimension of energy, M is a dimensionless parameter, and a is the spacing of the real-space lattice. We note that this Hamil- tonian reduces to a linear Hamiltonian around kx = ky = 0. Physically, we may think of this as the Hamiltonian for some spinless two-orbital system with one s- type orbital and one p-type orbital (more generally, we require that the orbitals must have diﬀerent parity). Hence, the oﬀ-diagonal terms Eqn. (2.19) represent an L = 1 angular-momentum coupling of the form sin kx + i sin ky and we have included and intraorbital dispersion of the form 2 − cos kx − cos ky, which is allowed by the C4 rotational symmetry of our model.

It is straightforward to see that the energy dispersion for the system described by

Eqn. (2.19) is fully gapped, other than at several values of M: for M = 0, the system is gapless at kx = ky = 0; when M = −2, it is gapless at (kxa, kya) = (π, 0) and

(kxa, kya) = (0, π); and for M = −4, the gap closes at kxa = kya = π. By adiabatic continuity, the system must remain in the same topological phase so long as its gap does not close. For M → ±∞, the system is in the atomic limit. Since a gap does not close as we tune M from +∞ to M = 0 or from −∞ to M = −4, the system is

13 (a) (b) (c) (d)

Figure 2.1: Band structure for the Hamiltonian given in Eqn. (2.19) for B = t and various M: (a) M = −4 where we see the transition between the trivial state and the first topologically nontrivial sector. (b) M = −2 where we see the transition between the two distinct topological regimes. (c) M = 0 showing a gap closing at the final topological tranisition to the trivial state. (d) M = 1 where we see the gapped topological trivial sector. As M increases, this gap grows and the bands flatten.

a trivial insulator for M > 0 or M < −4. We show the band structure for various regimes in Fig. 2.1.

We can explicitly compute the Berry curvature, which we ﬁnd to be

2 (z) 2 Bt (cos (kxa) ((2 + M) cos (kya) − 1) − cos (kya)) Ω± (k) = ±a 3 . 2 2 2 2 2 2 2 t (sin (kxa) + sin (kya)) + B (2 + M − cos (kxa) − cos (kya)) (2.20)

Although a closed form anti-derivative for Eqn. (2.20) does not exist, it is straight-

forward to numerically compute the integral for the Chern number given by Eqn.

(2.12). Doing so, we obtain that for −4 < M < −2, c± = ∓1. For −2 < M < 0,

the Chern numbers for each band are given by c± = ±1. Hence, we see that there

are four separate, topologically distinct sectors: two topological and two trivial. We

show the Berry curvature for all four regimes in Fig. 2.2(a-d). For the topologically

trivial regions, the Berry curvature averages to zero, while for the nontrivial regions,

it integrates to the Chern numbers given above. We note that the Berry curvature in

the non-trivial regions is concentrated near the points where the gap closings occur

14 (see Fig. 2.1). This can be understood intuitively by a careful examination of Eqn.

(2.18): since the denominator d3 is essentially half the energy gap, as the gap shrinks, the Berry curvature is enhanced. The Berry curvature is singular at the gap closings.

2.2.3 Chiral Edge Modes

As discussed above, in topological systems, there exists a “bulk-boundary” correspondence between bulk topological invariants and gapless edge modes. In the case of the Chern insulator given by Eqn. (2.19), a positive-(negative)-definite Chern number corresponds to the net number of right-(left)-moving chiral edge modes on the real-space edge of the system. In order to directly calculate the dispersions of these edge modes, we employ the standard procedure of considering a system periodic in one direction and with open, finite boundary conditions in the other. Without loss of generality, let us consider the Chern insulator in the strip geometry which is periodic in the x-directions and finite in the y-direction with N sites. To do so, we transform the operators in Eqn. (2.14) as follows:

1 ˆ X −ikyy ˆ Ψk ,y = √ e Ψk ,k , (2.21) x N x y ky

2πny where ky = N for ny = 0, 1, ..., N − 1. We obtain a 2N × 2N Hamiltonian at every momentum kx.

In Fig. 2.2(e-h), we show the edge states calculated in the above manner for the Chern insulator with N = 50 layers in the ﬁnite y-direction. For states which are localized exponentially on the top edge, we color them red while we color states on the bottom edge blue. States whose expectation values in the y-direction are predominantly found in the bulk are colored black. We see that in the topologically trivial sectors (M < 4 or M > 0) shown in Fig. 2.2(e) and 2.2(h), there are no

15 (a) (b) (c) (d)

(e) (f) (g) (h)

Figure 2.2: Berry curvature (in units of a2 of the lower band and edge states for the Hamiltonian given in Eqn. (2.19) for B = t and various M. Edge states are calculated using strip geometry, ﬁnite in the y-direction with N = 50 layers. In the strip calculations, states localized to the top edge are colored red, states localized to the bottom surface are colored blue, and states in the bulk are colored black. (a) Berry curvature averages to zero over the either band for M = −5 in one of the topologically trivial sectors. (b) Berry curvature in the lower band M = −2.5 where the Chern number is given by c− = +1. (c) Berry curvature in the lower band M = −1.5 where the Chern number is given by c− = −1. (d) Berry curvature is again zero in the trivial sector where M = 1. (e) Strip geometry calculation for M = −5 in the trivial sector showing a lack of edge states. (f) Edge states for M = −2.5 crossing zero energy at the kx = ±π point. (f) Edge states for M = −1.5 crossing zero energy at the kx = 0 point. We note that the chirality, or direction, of the edge state on a given surface changes between (e) and (f) due to the change in Chern number. (g) Strip geometry calculation for M = 0 in the trivial sector showing a lack of edge states.

16 edge states, reﬂecting the lack of bulk topology. However, we see in Fig. 2.2(f) and

2.2(g), we note the presence of topological edge states. These edge states on a given surface (say, the top shown in red) have disperse linearly and are said to be chiral.

The chirality, or sign of Fermi velocity, of these edge states on a given edge is directly related to the bulk topology. Hence, on the top (bottom) surface the edge states transitions from a left (right) mover for −4 < M < −2 to a right (left) mover for

−2 < M < 0. Hence we see that the chiral edge states are a direct manifestation of the bulk topology. Each of these edge states carries a conductance of e2 [25] and ~ consequently, the Hall conductance of this system is a consequence of these edge modes.

Although there are other nontrivial topological systems in two dimensions, the

Chern insulator is perhaps the simplest to understand. It is also important in our story. In the next chapter, we shall see that although the Chern insulator is a 2D system, it plays a crucial role in 3D topological Weyl semimetals.

17 Chapter 3: Introduction to Weyl Semimetals

In the previous chapter, we showed how a 2-D gapped Hamiltonian that breaks time-reversal symmetry could be classiﬁed by its bulk topology using the Chern number. This bulk topological invariant was a global property of the local geometrical quantity known as the Berry curvature. Using a speciﬁc model for a Chern insulator, we showed that the Chern number corresponded directly to the number chirality of gapless edge modes on the real-space boundary of the system. A fundamental property of the Chern insulator - as well as 3-D topological insulators - is the necessity for a bulk gap to protect the topology of the system.

Weyl semimetals introduced a new paradigm for topological band theory, where the excitations in the bulk are gapless and yet the system still possesses a topological invariant as well as gapless, topologically protected edge modes. The bulk modes of

Weyl semimetals are linearly dispersing Weyl fermions that share the dispersion of

c light, Fermi velocity vF ∼ 100 . The three dimensional nature of this system is crucial. We have seen that for the 2-D Chern insulator given by Eqn. (2.19), the energy bands became degenerate for speciﬁc, ﬁne-tuned parameter regimes.

In this chapter, we will show that Weyl semimetals are broadly characterized by three criteria: (1) linearly dispersing Weyl nodes (or Weyl points), (2) monopoles of

Berry curvature originating at the location of the Weyl points in momentum space,

18 (3) unique boundary modes known as Fermi arcs that connect projections of these

Weyl nodes in the surface Brillouin zone. Using the simplest two-band lattice model

that one can write down by symmetry, we will show how these features are closely

related. A central eﬀort of this thesis has been to explore the consequences of the tilt

of a type-II Weyl semimetal. This class of Weyl semimetals breaks Lorentz invariance

and we show using several lattice models how the connectivity of the Fermi arcs can

be strongly modiﬁed by the tilt of type-II Weyl semimetals. We also show that in

inversion-breaking type-II Weyl semimetals, where there are necessarily multiple pairs

of Weyl nodes, novel surface states that we have termed “track states” may emerge.

We conclude this chapter with a case study of how our relatively simple models were

applied to provide insight into the experimental discovery of the ﬁrst type-II Weyl

semimetal MoTe2.

3.1 Weyl Fermions

We consider the Dirac equation for an electron with mass m in d + 1 dimensions given by

µ (iγ ∂µ − m) ψ = 0, (3.1)

where we have set the speed of light and Planck’s constant to unity ~ = c = 1, and where ψ is the spinor of the electronc and µ runs from 0, ..., d. The d + 1 dimensional gamma matrices satisfy the anticommutation relations {γµ, γν} = 0 for µ 6= ν and

2 2 (γ0) = − (γj) = I, for j = 1, ..., d. The minimum size of the matrices that satisfy these relations depends on the spatial dimension.

Hermann Weyl [6] noticed that in odd spatial dimensions, the Dirac equation can sometimes be written in a simpliﬁed form. In the simplest case of d = 1, we only

19 0 need two 2 × 2 gamma matrices. We choose the following representation: γ =σ ˆz

1 and γ = iσˆy. With some rearrangement, we can write the 1 + 1-D Dirac equation as

0 1 0 i∂tψ = (γ γ px + mγ )ψ, (3.2)

where the momentum in the x-direction is given by px ≡ −i∂x. We can introduce the

0 1 matrix γ5 = γ γ =σ ˆx, where the last equality holds for the particular representation that we have chosen above. When the electron described by Eqn. (3.2) has no mass m = 0, if ψ is an eigenstate of γ5 such that γ5ψ± = ±ψ±, then

i∂tψ± = ±pxψ±, (3.3) which describes a right-moving and left-moving chiral electrons with linear dispersion

E±(px) = ±px. These chiral electrons are precisely the edge states we found at the boundary of the Chern insulator and a single mode of a given chirality may only exist at the boundary of such a topological state. In a truly 1-D system, massless

Dirac electrons must come in pairs of opposite chirality. We see that the mass can be understood as the term causing the chiralities to mix.

In higher odd spatial dimensions d = 2k + 1, where k is a positive integer, one

k 0 1 d can form the Hermitian operator γ5 = i γ γ ...γ . We note that in even directions,

γ5 is given rather trivially by the identity matrix. However, in odd dimensions, γ5 anticommutes with γ0. The Dirac equation in this dimension can be written as

d 0 X 0 j i∂tψ = γ mψ + γ γ pjψ (3.4) j=1 µ From the anticommutation properties of γ , it is straightforward to show that γ5 commutes with the prefactors of the j-th momenta given by γ0γj.

Now we take d = 3. In this case, we can represent each γµ as a 4 × 4 matrix in

0 j 5 the following way: γ = I ⊗ σˆx, γ = iσˆj ⊗ σˆy, and γ = −I ⊗ σˆz. We note that the

20 Kronecker product is between two distinct 2-component subspaces. When m = 0, we

can write Eqn. (3.4) as

i∂tψ± = ∓p · σψˆ ±, (3.5)

where we see that the fermion propagates with its spin parallel or antiparallel to its

momentum. This deﬁnes the chirality of the electron. Hence, we obtain the Weyl

Hamiltonian for the massless electron with chirality χ = ±1, given by

Hχ(k) = χ~vF k · σ,ˆ (3.6)

where we have restored Planck’s constant ~, the speed of the fermion vF , and written the fermion’s momentum as p = ~k.

3.1.1 Weyl Fermions in Quantum Materials

We have seen that Weyl fermions emerge from a massless solutions of the relativistic Dirac equation in three spatial dimensions. Hermann Weyl ﬁrst proposed that neutrinos were described by Eqn. (3.6), however we now understand that this cannot be the case, since it has been shown that neutrinos possess mass[7]. For the remainder of this thesis, we will discuss Weyl fermions in the context of condensed matter physics. While the Dirac equation is the correct description of electrons in vacuum, we know that the Schrodinger equation is adequate to treat electrons with kinetic energy much less than their rest mass, such as in the band structures of metals and semimetals. In this sense, it is rather remarkable that Weyl fermions with a linear relativistic spectrum emerge as the low energy excitations of quantum materials.

It would be too substantial a digression here to recapitulate all of the electronic band theory of solids. Instead, we wish to remind the reader of several facts that will be key to our discussion:

21 1. Bloch’s theorem states that an electron in a periodic potential U(r+R) = U(r)

ik·r is described by a wavefunction ψn,k(r) = e un,k(r), where un,k(r) = un,k(r + R) is a function with the same periodicity as the potential and n is known as the band index.

2. The wavevector k in the Bloch eigenstates plays the same role as the wavevector k = p/~ in Sommerfeld’s theory of the free electron. However, in the case of a periodic potential, the above simple relation between p and k does not hold.

3. For a real-space lattice vector R, a reciprocal lattice vector K satisﬁes eiK·R = 1.

Any k0 can be written as k0 = k + K where k is conﬁned to the ﬁrst Brillouin zone.

4. Since the periodic Schrodinger equation for the Bloch function un,k(r) takes place in a finite volume subject to the boundary conditions given in the first point above, we expect on fairly general grounds that the solutions will take the form of an infinite spectrum of discretely spaced energy levels. It is these levels that we have labeled by the band index n.

5. The eigenstates and eigenvalues are periodic function of k such that ψn,k(r) =

ψn,k+K(r) and En(k) = En(k + K).

For the details, we direct the reader to one of any of the standard texts [41, 42].

In the electronic band structure, if two nondegenerate bands meet at isolated zero- dimensional crossings in the Brillouin zone, then close to the so-called Weyl point at the crossing k = 0, they may be described by the Hamiltonian in Eqn. (3.6). In general, it can be useful to consider the case where the Weyl fermions disperse with diﬀerent velocities in each direction, however for now we restrict ourselves to the isotropic limit. In 2D, where k = (kx, ky), there is no coeﬃcient ofσ ˆz, which is only possible if both time-reversal and inversion symmetry are present. Graphene[43], a two-dimensional hexagonal array of covalently bonded Carbon atoms, possesses

22 the required symmetries and its Brillouin zone contains two doubly degenerate Dirac nodes. The Chern insulator in the previous section illustrates the 2D case where time- reversal symmetry is broken and there exists a massive, gapped Weyl node. Since by our deﬁnition, a Weyl node is comprised of non-degenerate bands, we must break either time-reversal or inversion symmetry to remove the degeneracy of Kramer’s pairs everywhere in the Brillouin zone. Unlike in two-dimensions, when k = (kx, ky, kz), the Weyl node described by Eqn. (3.6) cannot be gapped out by a broken symmetry.

Such a broken symmetry will lead to the introduction of “mass term” M(kx, ky, kz)ˆσz to the Hamiltonian. However, it is simple to show that the presence of any such term in three dimensions will only shift the location of the zero crossing.

For the remainder of this thesis, we will restrict ourselves to the three-dimensional case. Consider a material that breaks either inversion or time-reversal symmetry and that contains Weyl nodes described by Eqn. (3.6). We note that the associated energies are independent of chirality χ and are given by

q 2 2 2 E±(k) = ±~vF kx + ky + kz . (3.7)

This dispersion shown in Fig. 3.1a. The eigenstates around a Weyl node with chirality

χ are given by χ kz ± χk ψ± = p , (3.8) 2k(k ± χkz) kx − iky

p 2 2 2 where k = kx + ky + kz .

Density of states and temperature dependence of chmical potential

The density of states for the Hamiltonian in Eqn. (3.6) is found to be

1 E 2 g(E) = 2 3 , (3.9) π (~vF )

23 for energies E measured from the nodal energy. The density of states in Eqn. (3.9) vanishes at the Weyl nodes and this is responsible for a number of novel phenomena in Weyl semimetals. Later in this thesis, we will discuss how this density of states reveals itself in transport phenomena.

In the usual way, we can calculate the temperature dependence of the chemical potential by ﬁxing the density and requiring

Z g(E) 1 Z E 2 n = dE β(E−µ(T )) = 2 3 dE β(E−µ(T )) , (3.10) 1 + e π (~vF ) 1 + e where β = 1 , n is the density and µ(T ) is the temperature dependent chemical po- kB T tential. We can self-consistently attain the chemical potential by numerically solving for the chemical potential of Eqn. (3.10). We plot the results in Fig. 3.1, though we obtain the same upon carefully performing the continuum calculation. We note that the chemical potential always shifts to the nodal energy µ = 0 on a temperature scale

TW ∼ µ(T = 0)/kB. This has the interesting consequence that no matter the Fermi energy EF ≡ µ(T = 0), the chemical potential will always go to zero for T & TW . This is illustrated in Fig. 3.1b.

Berry curvature of a Weyl fermion

Since the Weyl Hamiltonian given by Eqn. (3.6) is a 2-band Hamiltonian, we can analyze its associated Berry curvature using the formalism developed for the

Chern inslator in the previous chapter. From Eqn. (2.18), we can show that the momentum-space Berry curvature for the n = ± band of a node with chirality χ is given by χ k Ωχ (k) = ± . (3.11) ± 2 k3

24 (a) (b)

Figure 3.1: (a) Continuum dispersion of a Weyl node. (b) Dependence of the chemical potential on temperature µ(T ). We see that on a temperature scale TW ∼ µ(T = 0)/kB, the chemical potential reaches the Weyl node.

Furthering the analogy developed in the previous chapter of the Berry curvature as a magnetic ﬁeld in momentum space, we see that Weyl nodes act as monopoles in the

χ Brillouin zone, with the charge of the monopole given by 2 . We now present a simple argument that the total chirality of Weyl nodes in a condensed matter system must be zero[44]. Hence, if Weyl nodes exist in the band structure of a solid, they must come in pairs of opposite chirality. Consider a system with N > 0 Weyl nodes each with chirality χj, then for momenta around the Weyl nodes, the Berry curvature is given by

N 1 X (k − kj) Ω(k) = χ , (3.12) 2 j k3 j=1 where the j-th Weyl node is at momentum kj. Then the integrated divergence of this

Berry curvature is given by Z N Z N 3 X 3 X d k ∇k · Ω(k) = 2π χj d k δ(k − kj) = 2π χj, (3.13) BZ j=1 BZ j=1

25 where the integrals are over the Brillouin zone. Then, from the Divergence Theorem, it is true that Z Z 3 2 d k ∇k · Ω(k) = d k · Ω(k), (3.14) BZ ∂BZ where ∂BZ denotes the boundary of the Brillouin zone. However, the Brillouin zone is a 3-torus and tori have no boundaries, so the integral above is zero. Hence we have that N X χj = 0. (3.15) j=1 We see that for N > 0, the net chirality of the Weyl nodes must be zero and hence if

Weyl nodes exist in a system, they must come in pairs of opposite chirality.

3.2 Lattice Models for Weyl Semimetals

The deﬁning features of a TWS are the nodal energy crossings in the Brillouin zone and so a minimal lattice model for a TWS must have at least two bands. A general two-band Hamiltonian takes the form

ˆ X † ˆ H = cˆkα H(k) cˆkβ (3.16) αβ k

(†) wherec ˆkα annihilates (creates) an electron at momentum k in orbital α and

ˆ X H(k) = di(k)σ ˆi (3.17) i=0,1,2,3 as was true for the Chern insulator of the previous chapter.

As we saw last chapter, in the presence of both inversion and time reversal symmetry the Berry curvature is identically zero throughout the Brillouin zone. Hence, the presence of Weyl nodes relies on breaking either inversion (henceforth labeled Pˆ) or time reversal symmetry (labeled Tˆ). For spinless fermions, we choose a deﬁnite

26 representation for the Pˆ and Tˆ operators,

ˆ ˆ ˆ P ↔ σˆ1, T ↔ K, (3.18)

where Kˆ is the anti-Hermitian complex conjugation operator. Each of Pˆ and Tˆ also reverse the sign of the momentum such that k → −k. In this paper we investigate

lattice models for Weyl semimetals that break either Tˆ or Pˆ, and using the deﬁnitions

in Eqn. (3.18) it will be straightforward to show this symmetry breaking explicitly

for each model we consider.

Under inversion symmetry, k → −k and σ → σ. Hence a time-reversal breaking

Weyl semimetal (with inversion symmetry present) may have only two nodes. On the other hand, since under time-reversal symmetry k → −k and σ → −σ, a system that breaks time-reversal symmetry but not inversion symmetry must have at least four nodes, since under time-reversal symmetry a node does not change chirality. Hence there must be an additional pair. The simplest model for a Weyl semimetal therefore breaks time-reversal symmetry, preserves inversion symmetry and possesses a single pair of Weyl nodes.

The ﬁrst two-band model that appears in the literature is due to Yang, Lu, and

Ran[45] and is given by

TRB H (k) = − (m[2 − cos (kya) − cos (kza)] + 2tx[cos (kxa) − cos (k0a)])σ ˆx

− 2t sin (kya)ˆσy − 2t sin (kza)ˆσz, (3.19)

where m, tx, and t are parameters with the dimension of energy and a is the lattice constant. This model explicitly breaks time-reversal symmetry and contains a single pair of Weyl nodes at k = (±k0, 0, 0). Around these points, it is straightforward

to see that the linearized Hamiltonian reduces to that of Eqn. (3.6). Each of these

27 (a) (b) (c)

Figure 3.2: Band structure for Weyl semimetal in Eqn. (3.19) for tx = 0.5t, m = 2t, k0 = π/2, and lattice constant set to unity a = 1. (a) Bulk energy dispersion with ky = 0. (b) Cut through the Weyl nodes along kx with ky = kz = 0. (c) Constant energy EF = 0.2t cut for a system with slab geometry with N = 50 layers in the y-direction. Surface states are colored red (top) and blue (bottom). We see that small bulk Fermi pockets (shown in black) enclosing the Weyl points (green points) are connected by the Fermi arcs on the top and bottom states.

Weyl nodes acts as a monopole of Berry curvature as described above. We show an

example of the bulk band structure in Fig. 3.2(a) and 3.2(b) for tx = 0.5t, m = 2t, k0 = π/2, and lattice constant set to unity a = 1.

We note that this model, for ﬁxed kx, is very similar to the model for the Chern insulator given by Eqn. (2.13). The mass term M of this Chern insulator is given t by M(k ) = x [cos (k a) − cos (k a)]. We take what will be typical values for the x m x 0

remainder of this thesis of m = 2t and tz = 0.5t. We see that in the ﬁrst Brillouin

zone, the 2D subspace of the Brillouin zone for constant |kx| < k0 describes a trivial gapped insulator with Chern number zero. However, the 2D subspace of the Brillouin zone for constant k0 < |kx| < π describes a Chern insulator and therefore possesses chiral edge states in a ﬁnite geometry.

28 We can therefore understand the simple model for a Weyl semimetal in Eqn.

(3.19) as a ”stack” (in momentum space) of Chern insulators and a ”stack” of trivial insulators. We know that the boundary of systems with diﬀerent Chern numbers must be gapless and it is precisely at the momenta of the Weyl nodes where the energy gap goes to zero. For a system which is ﬁnite in the y-direction, much like the

Chern insulator in Chapter 2, but periodic in x and z, as kz evolves (k0 < |kx| < π) from one node to the other, the chiral edge states of the Chern insulator at each kx trace out an arc in the surface Brillouin zone. This arc is known as a Fermi arc and is perhaps the most unique property of Weyl semimetals.

In Fig. 3.2(c), we show a constant energy EF = 0.2t cut for a system with slab geometry with N = 50 layers in the y-direction. The states on the surface are colored red (top) and blue (bottom). We see that small bulk Fermi pockets (shown in black) enclosing the Weyl points (green points) are connected by the Fermi arcs on the top and bottom states. The Fermi arcs disperse linearly in kz. Strikingly, we see that these states form open contours in the surface Brillouin zone. These open contours are completely unlike traditional electronic states in solids and reﬂect the bulk topology of the Weyl nodes.

Three features clearly identify Weyl semimetals:

(1) Pairs of linearly dispersing Weyl nodes in momentum space.

(2) Monopoles of Berry curvature centered on the Weyl nodes

(3) Fermi arcs in the surface Brillouin zone connecting projections of Weyl nodes of opposite chirality.

29 In the following chapters, we shall see how each of these features reveal themselves in thermoelectric and thermomagnetic transport. For the remainder of this chapter, we will highlight important aspects of the band structure of Weyl semimetals.

Both the bulk Weyl nodes and the surface Fermi arcs have unique signatures in angle resolved photoemission spectroscopy (ARPES) experiments. Searching for these signatures has proven to be extremely fruitful and several groups[15, 16, 17, 46,

47, 48, 49] have discovered a Weyl semimetal phase in the transition metal pnictide family: TaAs, TaP, NbP and NbAs. These materials were the ﬁrst experimentally discovered Weyl semimetals and, unlike the simple model above, they break inversion symmetry and contain twelve pairs of Weyl nodes. However, both the model above and the transition metal pnictide family are known as type-I Weyl semimetals, where the density of states of the Weyl fermions goes to zero at the Weyl points. In some sense, we can think of these as the limiting case of a direct gap semiconductor where the conduction and valence bands meet at the Weyl nodes. In the next section, we will explore the so-called type-II Weyl semimetals, where the bands comprising the

Weyl nodes have a ﬁnite density of states at the Weyl energy.

3.3 Type II Weyl Semimetals

In a type-I Weyl semimetal, the valence and conduction bands that meet at the

Weyl node have a finite density of states at the Weyl energy. We may picture this system as the limit of an indirect gap semiconductor, where offset bands meet at isolated pairs of points in the Brillouin zone. We will present a formal definition shortly. Type-II Weyl semimetals were recently theoretially predicted in a variety of compounds[50, 51, 52, 53]. Recently, signatures of a type-II Weyl semimetals

30 have been reported[54, 55, 56] in MoxW1−xTe2, stoichiometric MoTe2, and LaAlGe,

opening the door for further experimental study of the type-II Weyl semimetals. Later

in this chapter, we will outline our role in the discovery of MoTe2, the ﬁrst of this

family of materials.

We extend the Hamiltonian given by Eqn. (3.6) to the case of an anisotropic Weyl

node with a momentum-dependent tilt (shift) of both energy bands:

ˆ X X HWP(k) = γikiσˆ0 + kiAijσˆi, (3.20) i=x,y,z i,j=x,y,z whereσ ˆ0 is the 2 × 2 identity matrix. Eqn. (3.20) describes a Weyl fermion with nodes of chirality χ = det(Aij). The energy spectrum for the Hamiltonian in Eqn.

(3.20) is given by v u !2 X u X X E±(k) = γiki ± t kiAij i=1,2,3 j=1,2,3 i=1,2,3 = T (k) ± U(k), (3.21) where T (k) tilts the Weyl cone. The deﬁnition[50, 57] of a type-II Weyl node is one where there exists a direction ek in the Brillouin zone such that

T (ek) > U(ek). (3.22)

The simplest Hamiltonian for a type-II Weyl Fermion is given by

Hχ(k) = γkxσˆ0 + χ~vF k · σ,ˆ (3.23)

where we have chosen the tilt direction to be the x-direction. Eqn. (3.23) describes

a type-II Weyl semimetal so long as γ > ~vF . Although there have been some studies of lattice models for TWS,[58, 59, 60]

much of the theoretical work on topological Weyl semimetals has focused on low

31 energy eﬀective models of single Weyl nodes. In a type-I TWS, where the density of states vanishes at the energy of the Weyl nodes, these eﬀective models capture much of the essential physics including electro- and magnetotransport, [61, 62, 63, 64,

65, 66, 67, 68] thermoelectric properties, [69, 70, 71, 20] magnetic properties,[72] and eﬀects of disorder[73, 74, 75]. In a type-I TWS, when the chemical potential is shifted slightly away from the nodal energy, the Fermi pockets enclosing the projections of the Weyl nodes are very small. However, in a type-II TWS extended pockets of holes and electrons exist already at the node energy. Doping away from the node energy then results in the surface projections of the Weyl nodes, for typical crystal surfaces, becoming enclosed within large Fermi pockets. Understanding the interplay of these large Fermi pockets and any topological properties associated with the type-II nodes can require explicit lattice models, rather than just a low-energy theory. Here we present a study of a few such relatively simple lattice models for type-II TWS.

We begin by discussing models for time-reversal-breaking type-II TWS. We distinguish between two types of basic models: the simplest model (“hydrogen-like model”) has a single pair of Weyl nodes which share a single electron pocket and a single hole pocket. However, we argue that this simplest model fails to capture some important properties. These are instead captured by the next-simplest model (“helium-like model”), with an additional term that splits both the electron pocket as well as the hole pocket into pairs of separate pockets. Each Weyl node is now formed from the intersection of an isolated pair of electron and hole pockets. The hydrogen-like model has no topologically-protected Fermi arcs, though it exhibits relics of them away from the Fermi energy; in the helium-like model, the topological Fermi arcs are restored.

We also study inversion-breaking type-II TWS models, and ﬁnd that even simple toy

32 models support an additional set of surface states (”track states”) which are not topological but nonetheless play a role in how the Fermi arc connectivity changes when either the Fermi energy is changed or when the tilt of the Weyl nodes is changed.

3.3.1 Time Reversal Breaking Model

We begin by investigating a lattice model given by a Hamiltonian Hˆ(k) that hosts

Weyl nodes and breaks time reversal symmetry but preserves inversion symmetry such that

Pˆ†Hˆ(−k)Pˆ = Hˆ(k), Tˆ †Hˆ(−k)Tˆ 6= Hˆ(k). (3.24)

The minimal number of Weyl nodes for such a Hamiltonian is two and we ﬁnd that such a minimal model can be used to investigate a wide range of possible TWS Fermi surface and arc connectivity. We begin by writing down the simplest possible two node time-reversal breaking (TRB) Hamiltonian with a type-II tilt and investigating its band structure. A pair of Weyl nodes are formed from the nodal crossing of exactly one electron band with one hole band. By calculating the band structure for the system in a ﬁnite slab geometry, we investigate the surface Fermi arc behavior.

We then show that this minimal model can be modiﬁed with a term that splits these electron and hole pockets into pairs that exist around each node.

The “Hydrogen atom” for a type II time reversal breaking TWS

The following Hamiltonian

ˆTRB HA (k) = γ cos(kx) − cos(k0) σˆ0 − m(2 − cos(ky) − cos(kz)) + 2tx(cos(kx) − cos(k0)) σˆ1

− 2t sin(ky)ˆσ2 − 2t sin(kz)ˆσ3 (3.25)

33 satisﬁes the symmetry conditions in Eqn. (3.24) and possesses two Weyl nodes at k = (±k0, 0, 0). When γ = 0, this Hamiltonian is known[26] to host nodes of type-I. However, the addition of the term γ cos(kx)−cos(k0) σˆ0 bends both bands and when

γ > 2tx it is simple to see these nodes become type-II as deﬁned by Eqn. (3.22). We see this evolution from type-I to type-II very clearly in Fig. 3.3. When γ = 0, the hole band (blue) touches the electron band (red) at the two Weyl points where the density of states vanishes, as seen in Fig. 3.3a,d,g. When the system is in the type-II regime, the Weyl cones are tilted and this leads to a nonzero density of electron and hole states at the node energy, as seen clearly in Fig. 3.3c,f,i. When γ = 2tx exactly, the system is at a critical point between a type-I and a type-II Weyl semimetal. This is clearly seen in Fig. 3.3b,e,h, where a single line of bulk states connect the Weyl points at E = 0. The states seen in Fig. 3.3h open up into the electron and hole pockets seen at E = 0 for the type II case in Fig. 3.3i.

In a type-II TWS, it is important to consider the net chirality enclosed by the bulk Fermi pockets when determining the Fermi arc connectivity. If one encloses a bulk pocket by a Gaussian surface in a region where the bulk band structure is gapped, the number of Fermi arcs impinging on the Gaussian surface are quantized and equal to the net chirality of Weyl nodes enclosed. When the model in Eqn. (3.25) is in the type-II regime and the chemical potential is shifted away from E = 0, the projections of both Weyl nodes are either enclosed in the electron pocket (E > 0) or the projections are both enclosed in the hole pocket (E < 0). Since the projections of both nodes lie within the same Fermi pocket, we expect that Fermi arcs in this system are not topologically protected in general. Surface states may exist, but their lack of topological protection stems from the fact that there are no isolated Fermi

34 Type I Critical Point Type II

a b c

d e f

g h i

Figure 3.3: Bulk band structure for the “Hydrogen atom” of type-I and type-II Weyl semimetal. a-c The bulk band structure for the Hamiltonian in Eqn. (3.25). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters ky = 0 with parameters k0 = π/2, tx = t, m = 2t for (a) type-I Weyl semimetal with γ = 0, (b) the critical point between type-I and type-II Weyl semimetal with γ = 2t and (c) type-II Weyl semimetal with γ = 3t. The cones comprising the Weyl nodes develop a characteristic tilt of the type-II TWS as γ is increased. d-f Cuts through the Weyl nodes at ky = kz = 0 for the same parameters as (a-c). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). We see that for a type-I TWS, there are no states at the Fermi energy. At the critical point between a type-I and type-II TWS, we see lines of bulk states appearing between the nodes. These lines open into bulk hole and electron pockets (in the repeated zone scheme) when the system becomes a type-II TWS.

35 pockets that enclose Weyl nodes with nonzero net chirality. As a result, the surface

states can hybridize with bulk states and are therefore trivial.

In order to investigate the structure of the Fermi arcs, we introduce an edge

by considering a slab with a ﬁnite thickness in one direction. We partially Fourier

transform the Hamiltonian in Eqn. (3.25) into real space for a L layer system in the

y-direction, while keeping the system inﬁnite in the x- and z-directions. In Fig. 3.4,

we show the results of such a slab calculation for the model given by Eqn. (3.25) in

the type I regime (γ = 0) with the same bulk parameters as in Fig. 3.3a,d,g and in

the type II regime (γ = 3tx) with the same bulk parameters as in Fig. 3.3c,f,i for

L = 50 layers. We calculate the expectation of the ﬁnite position hyi and label the

states as “top” (“bottom”) if they are exponentially localized at hyi = 1 (hyi = L).

We color these top and bottom states red and blue respectively.

As we expect, for the type-I case when γ = 0, a Fermi arc on each surface connects

the Weyl nodes, as seen in Fig. 3.4a-c. This is seen clearly in Fig. 3.4b where two

Fermi arcs connect the two nodes from (kx, kz) = (−π/2, 0) to (kx, kz) = (π/2, 0). At

E = 0, both the top and bottom arcs are degenerate at kz = 0, shown as a purple

line. When we lower the Fermi energy below the node energy, each node is enclosed

in a small isolated Fermi pocket. Since each pocket encloses a net chirality χ = ±1,

the pockets are connected by an arc on each surface, as seen in Fig. 3.4a. The same

is seen at higher energies E > 0 in Fig. 3.4c.

We calculate the band structure in the slab geometry for a type-II TWS (γ = 3tx)

and ﬁnd that there are marked diﬀerences in the surface state behavior (see Fig.

3.4d-f). Since both nodes are formed from a single electron and a single hole pocket,

we cannot construct a simply connected 2D Gaussian surface in the Brillouin zone

36 a b c

Type I

d e f

Type II

Figure 3.4: Fermi surface and arc conﬁguration for the “Hydrogen atom” of type-I and type-II TWS. a-c Bulk Fermi surfaces and surface Fermi arcs for a type I TWS with the same bulk parameters as in Fig. 3.3a,d,g calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the following constant energy: (a) E = −0.2t, (b) E = 0, (c) E = 0.2t. We color the states which are exponentially localized to the y = 1 (y = L) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots and marked with pink arrows). We note that at E = 0 the two Fermi arcs are degenerate along kz = 0 and we color them purple to signify this. d-f Bulk Fermi surfaces and surface Fermi arcs for a type-II TWS with the same bulk parameters as in Fig. 3.3c,f,i calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the same constant energies as above: (d) E = −0.2t, (e) E = 0, (f) E = 0.2t.

37 that encloses a single node. When the energy is lower than the Weyl energy in Fig.

3.4d, we see that the projections of both nodes are enclosed by the same hole pocket.

Although there are two sets of surface states connecting the hole and electron pockets,

they are trivial in a topological sense. When one considers a Gaussian surface that

encloses the central hole pocket, it is pierced by four arcs, two on each real-space

surface. The Fermi velocity of each arc is opposite on a given real-space surface and

so the net chirality of the arcs is zero. We see that as we raise the chemical potential

to the node energy and above, these arcs disappear completely. This is completely

diﬀerent from the type-I case where the arcs exist at all energies since the nodes were

always isolated in separate Fermi pockets.

The “Helium atom” for a type II time reversal breaking TWS

In order to study the physics of type-II Weyl nodes surronded by isolated Fermi

pockets which they do not share, we seek to introduce a term to the Hamiltonian

separates the single pair of pockets possessed by the ”Hydrogen-atom” model. In

particular, this new term must gap out the bulk band structure in the kx = 0 plane

and the kx = π plane. Due to the pairs of electron and hole pockets supported by this

model, we call it the “Helium model” for a type-II time-reversal-breaking TWS in

analogy with the “Hydrogen model” above. We consider the following Hamiltonian

ˆTRB ˆTRB HB (k) = HA (k) − γx(cos(3kx) − cos(3k0))ˆσ1, (3.26)

where we have added to Eqn. (3.25) the term proportional to γx. In general, this model supports up to six Weyl nodes. However, so long as |2tx| > |3γx|, there are only two Weyl nodes in the Brillouin zone. These nodes are located at E = 0 and k = (±k0, 0, 0) and they are type-II if γ > 3γx − 2tx. The addition of the term

38 γx(cos(3kx) − cos(3k0)) gaps out the bulk spectrum along the lines (ky, kz) = (0, 0)

and (ky, kz) = (0, π) at the nodal energy. This leads to a pair of isolated hole pockets touching a pair of isolated electron pockets at the Weyl nodes when the system is type-II. In Fig. 3.5, we ﬁnd that as γ grows relative to 3γx − 2tx, the Fermi pockets grow in size. This is because as the tilt of the nodes gets larger, more electron and hole states exist at the Fermi energy. As we shift the chemical potential away from the node energy, the projections of the nodes are now isolated with each node in a single electron (hole) pocket when the chemical potential is raised (lowered).

We again consider the slab geometry described in the section above in order to investigate the interplay of the bulk pockets and the Fermi arcs for the model given by Eqn. (3.26). Unlike the simpler model described by Eqn. (3.25), we see in Fig.

3.6 that Eqn. (3.26) supports isolated Fermi pockets enclosing the Weyl nodes in the type-II regime when γ = t (Fig. 3.6a-c) and γ = 1.5t (Fig. 3.6d-f). Unlike the

Fermi surfaces and arcs generated by Eqn. (3.25), in Fig. 3.6 we see that each node

is isolated in its own hole (Fig. 3.6a,d) or electron (Fig. 3.6c,f) pocket when the

chemical potential is away from E = 0. We emphasize that this is due to the extra

σˆ1 term in the Hamiltonian in Eqn. (3.26). These isolated pockets in Fig. 3.6 are

connected by arcs conﬁned to the surface in the y-direction. However, in this type-II

TWS the Fermi pockets enclosing a Weyl node can be quite extended and, unlike a

type-I TWS, the arcs can terminate on a pocket quite far away from the projection

of the nodes. We see that as the tilt grows in Fig. 3.6d-f, so do the pockets enclosing

the nodes. We note that a trivial electron pocket appears around the (kx, kz) = (π, π)

point. This pocket encloses no Weyl nodes and therefore it is not connected via Fermi

arcs to any other pockets.

39 Type I Type II Type II

a b c

d e f

g h i

Figure 3.5: Bulk band structure for type-I and type-II TRB model with separate pockets (the “Helium atom”). a-c The bulk band structure for the Hamiltonian in Eqn. (3.26). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters ky = 0 with the parameters k0 = π/2, tx = t, m = 2t and γx = t/2 for (a) type-I TWS with γ = 0, (b) type-II TWS with γ = t and (c) type-II TWS with γ = 1.5t. The cones comprising the Weyl nodes again develop a characteristic tilt of the type-II TWS as γ is increased. d-f Cuts through the Weyl nodes at ky = kz = 0 for the same parameters as (a-c). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). Note that for a type-I TWS, there are no states at the Fermi energy while in the type-II regime, there are two sets of electron and hole pockets on either side of the Weyl nodes. We see that unlike the Hydrogen-atom model, the Helium-atom model has disjoint pairs of electron and hole pockets and a pair of each meet at the two Weyl nodes.

40 a b c

Type II

d e f

Type II

Figure 3.6: Fermi surface and Fermi arc conﬁguration for type I and type-II time-reversal-breaking model with separate pockets (the “Helium atom”). a-c Bulk Fermi surfaces and surface Fermi arcs for a type-II Weyl semimetal given by Eqn. (3.26) with the same bulk parameters as in Fig. 3.5b,e,h calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the constant energies: (a) E = −0.2t, (b) E = 0, (c) E = 0.2t. As in Fig. 3.4, we color the states that are exponentially localized to the y = 1 (y = L) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots). We note unlike in Fig. 3.4, each node is isolated in its own hole (a) or electron (c) pocket when the chemical potential is away from E = 0. These pockets are connected by arcs conﬁned to the surface in the y-direction. However, in this type-II TWS the Fermi pockets enclosing a Weyl node can be quite extended, unlike a type-I TWS, the arcs can terminate on a pocket quite far away from the projection of the nodes. d-f Bulk Fermi surfaces and surface Fermi arcs for a type-II TWS with the same bulk parameters as in Fig. 3.5c,f,i calculated in a slab geometry with L = 50 layers in the y-direction. The slab calculations are done at the same constant energies as above: (d) E = −0.2t, (e) E = 0, (f) E = 0.2t. We see that as the tilt grows, so do the pockets enclosing the nodes. We note that a trivial electron pocket appears around the (kx, kz) = (π, π) point. This pocket encloses no Weyl nodes and so is not connected via Fermi arcs to any other pockets.

41 Although the local linearized Hamiltonian describing the spectrum close to a node in Eqn. (3.26) is identical to the eﬀective Hamiltonian of nodes of the model described by Eqn. (3.25), the full lattice models describe topologically distinct conﬁgurations of bulk Fermi surfaces and surface Fermi arcs. When there is only one electron pocket and one hole pocket with the projections of the Weyl nodes enclosed by the same pocket, the topological protection of the Fermi arcs is lost. However, we see that once each node is enclosed in its own isolated pocket, the topological protection of the Fermi arcs is restored.

Finally, we consider the energy dispersion of the Fermi arcs near a node. Again using the slab geometry as above, we calculate the energy spectrum, this time at a constant kz. We see that for the simplest type-I case (Eqn. (3.25) with γ = 0), the surface arcs do not disperse in kx for a ﬁxed kz. This changes in the type-II case for both the simple Hamiltonian in Eqns. (3.25) and (3.26). At ﬁxed kz, the arcs connecting the node inherit the tilt proportional to γ and now bend. This characteristic bend of the Fermi arc dispersion has been observed in ARPES studies of type-II Weyl semimetal[55].

3.3.2 Inversion Breaking Model

We now turn to a lattice model for a topological Weyl semimetal that breaks inversion symmetry but is invariant under time-reversal. Analogous with Eqn. (3.24), we seek a Hamiltonian Hˆ(k) that satisﬁes the following symmetry conditions

Pˆ†Hˆ(−k)Pˆ = 6 Hˆ(k), Tˆ †Hˆ(−k)Tˆ = Hˆ(k), (3.27) where Pˆ and Tˆ are again given by Eqn. (3.18). Unlike a time-reversal-breaking Weyl semimetal, the minimum number of Weyl nodes for a spinless inversion-breaking (IB)

42 TWS is four. More importantly, the lattice model for an IB TWS exhibits what we term ”track states” that are loops of states that live on the surface of the TWS and are degenerate with the states forming the topological Fermi arcs. However, unlike topological Fermi arcs, these track states form closed contours rather than open ones; they are not topological, but do evolve from the topological arc states upon the transition from type-I to type-II.

It is easy to show that the Hamiltonian

ˆIB H (k) = γ(cos(2kx) − cos(k0))(cos(kz) − cos(k0))ˆσ0

2 − (m(1 − cos (kz) − cos(ky)) + 2tx(cos(kx) − cos(k0)))ˆσ1

− 2t sin(ky)ˆσ2 − 2t cos(kz)ˆσ3 (3.28) satisﬁes the conditions in Eqn. (3.27). When γ = 0, Eqn. (3.28) describes a TWS with four nodes located at kW = (±k0, 0, ±π/2) that breaks inversion but preserves time-reversal symmetry. The term γ(cos(2kx)−cos(k0))(cos(kz)−cos(k0))ˆσ0 causes a diﬀerent shift in both band than those considered in the time reversal breaking cases and results in both bands bending in both the kx- and kz-directions. This can produce isolated Fermi pockets around the Weyl points without having to add an additional

σˆ1 term like in the time-reversal-breaking case in Eqn. (3.26). The inversion-breaking model above also easily generates trivial Fermi pockets that exist in isolation from those that meet at the Weyl nodes.

We show the bulk band structure for Eqn. (3.28) in Fig. 3.7. We see that indeed when γ = 0 (Fig. 3.7a,d,g), the electron band meets the hole band at four isolated type-I Weyl points and the density of states vanishes at the nodal energy. As γ increases, the Weyl nodes begin to tilt in the kz-direction. When γ is tuned to the

43 critical point between the type-I and type-II phases (Fig. 3.7b,e,h), the electron and

hole pockets still meet at the four Weyl nodes with a vanishing density of states, but

we see in Fig. 3.7e that the Weyl nodes are now tilting in the kz-direction. As γ is

further increased into the type-II limit (Fig. 3.7c,f,i), we now see that the nodes are

tilted as seen in Fig. 3.7f and the electron (hole) pockets are shifted below (above)

the node energy. In particular, we see in Fig. 3.7i that there are four electron and

four hole pockets that exist at E = 0 and meet at the Weyl nodes. There is also a trivial hole pocket centered at k = (0, 0, 0) and a trivial electron pocket centered at

k = (π, 0, 0).

In order to study the Fermi arcs, we again construct a slab geometry by trans-

forming the terms dependent on ky in Eqn. (3.28) into real space and considering a

system with L layers in the y-direction and inﬁnite in the x- and z-directions. In the

type-I limit with γ = 0 shown in Fig. 3.8a and b, we ﬁnd that away from E = 0, the

projections of the nodes are enclosed by isolated small Fermi pockets. These pockets

are connected to one another by topological Fermi arcs in the kx-direction. At E = 0,

the top and bottom arcs are degenerate along the lines kz = ±π/2. In a sense, this

type-I (γ = 0) limit in the inversion-breaking model is eﬀectively composed of two

copies of a time-reversal-breaking Weyl semimetal separated by π reciprocal lattice

vectors along the kz direction.

When γ is increased to the type-II limit, the Fermi arc and bulk Fermi surface

conﬁguration in the inversion-breaking case is very diﬀerent from the time-reversal-

breaking model as we see in Fig. 3.8c and d. The projections of the Weyl nodes are

now enclosed by extended hole pockets for E < 0 (Fig. 3.8c) and electron pockets

for E > 0 (Fig. 3.8d). These Fermi pockets are connected by topological Fermi

44 Type I Critical Point Type II

a b c

d e f

g h i

Figure 3.7: Bulk band structure for type-I and type-II inversion breaking TWS. a-c The bulk band structure for the Hamiltonian in Eqn. (3.28). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters ky = 0 with the parameters k0 = π/2, tx = t/2, m = 2t for (a) type I TWS with γ = 0, (b) the critical point between a type-I and a type-II TWS with γ = 2t and (c) type-II TWS with γ = 2.4t. The cones comprising the Weyl nodes develop a characteristic tilt of the type-II Weyl node as γ is increased. d-f Cuts through the Weyl nodes at ky = 0 and kz = −π/2 for the same parameters as (a-c). These cuts are shown as the green lines in (g-i). g-i Constant energy cuts through the nodal energy (E = 0) for the same parameters as (a-c). We see that for a type-I Weyl semimetal, there are no states at the Fermi energy. At the critical point between a type-I and type-II TWS, the density of states still vanishes. In the type-II regime, electron and hole pockets form near the Weyl nodes. These pockets enclose the projections of the Weyl nodes when the chemical potential is shifted away from E = 0. Trivial pockets also appear at k = (0, 0, 0) and k = (0, 0, π).

45 arcs, shown by thick red and blue lines, to pockets containing Weyl nodes of opposite

chirality. Unlike in the type-I limit, here the Fermi arcs connect pockets along the

kz-direction rather than the kx-direction. One might expect that the transition point where the Fermi arcs connect nodes in one direction rather than another is concurrent with the transition point between a type-I and type-II Weyl semimetal and indeed our numerical calculations show that is the case (see Fig. 3.9). Hence we see that for the same model with all other parameters held constant, merely tilting the nodes can lead to a dramatic recombination of the Fermi arcs and a qualitatively diﬀerent pocket connectivity.

In Fig. 3.8c and d, we see that there are many states that are exponentially localized on the surface, however many of them form closed loops. We term these closed loops “track states”; they are degenerate in energy with the Fermi arcs but do not share their topology. Unlike Fermi arcs, track states form closed rather than open contours of surface states. By investigating the evolution of the Fermi arc and Fermi surface conﬁguration as a function of γ (Fig. 3.9), we see that when the Fermi arc connectivity changes from the the kx-direction to the kz-direction, they leave behind

track states around the (kx, kz) = (π, π) point.

3.3.3 Surface States: Topological and Track

We brieﬂy recapitulate the argument[9] for the existence of topologically protected

Fermi arcs in a Weyl semimetals. It can be shown that a Weyl node is a monopole

source of Berry curvature with charge equal to its chirality χ. We enclose an isolated

Weyl node by a closed 2D subspace of the Brillouin zone. It is well known that

the integral of Berry curvature over a 2D manifold is a quantized integer known

46 a b

Type I

c d

Type II

Figure 3.8: Fermi surface and Fermi arc configuration for type-I and type- II inversion-breaking Weyl semimetal. a,b The Fermi surface and Fermi arc configuration for the Hamiltonian given in Eqn. (3.28) in the type-I limit (γ = 0) calculated in a slab geometry with L = 50 layers and with bulk parameters the same as in Fig. 3.7a,d,g. We show this calculation at constant energies: E = −0.25t (a) and E = 0.25t (b). Here we see that Weyl nodes located at (kx, kz) = (±π/2, ±π/2) are connected by surface states (red and blue lines) to one of opposite chirality across the Brillouin zone in the kx-direction. c,d The Fermi surface and Fermi arc configuration for the Hamiltonian given in Eqn. (3.28) in the type II limit (γ = 2.4t) calculated in a slab geometry with L = 50 layers and with bulk parameters the same as in Fig. 3.7c,f,i. We show these for the same constant energies as above: (c) and E = 0.25t (d). The locations of the Weyl nodes are marked with pink arrows. We term the exponentially localized surface states that form closed loops “track states”. Fermi arcs are shown as bold lines and connect Weyl nodes in the kz-direction.

47 a b c d

Figure 3.9: Evolution of Fermi surface and Fermi arc conﬁguration for inversion-breaking Weyl semimetal as a function of γ. a-d The evolution of the Fermi surface and Fermi arc conﬁguration in a slab geometry for Eqn. (3.28). Bulk states are down in black, surface states are shown in red and blue. We have chosen the parameters k0 = π/2, tx = t/2, m = 2t. The calculations are done at constant energy E = −0.25t for γ = 0 (a), γ = 0.8t (b), γ = 1.4t (c), and γ = 2. (d) shown in an extended Brillouin zone where both kx and kz range from −1.5π to 1.5π. We see that at the critical point between a type-I and type-II (d), the Fermi arcs that previously connected Fermi pockets in the kx-direction now connect Fermi pockets in the kz-direction and track states have formed on the bottom surface (blue) around the (kx, kz) = (π, π) point.

as the Chern number[76] when the bulk band structure is gapped over the region of integration. In the case of a surface enclosing a Weyl node, the Chern number calculated in this way is equal to the chirality χ of the node enclosed. By deﬁnition, such a surface enclosing a Weyl node deﬁnes a 2D Chern insulator and therefore possesses |χ| chiral edge modes on its boundary. As we consider various families of such closed surface in the Brillouin zone, these chiral edge modes trace out the open contours of surface states known as Fermi arcs that must terminate on Weyl nodes. In this way, there is a correspondence between the Berry curvature of the Weyl nodes, a topological property of the bulk, and the surface Fermi arcs (see sketch in Fig. 3.10a) that are also topological in nature.

48 Topological Protection of Fermi Arcs in Type-II Weyl Semimetals

The chirality and Berry curvature of a Weyl node are unaﬀected by its type[50].

In the case of the lattice models we consider in the sections above, this can be shown

explicitly by noting that the ith component of the Berry curvature of each band (E+ and E−) is given by ∂dk ∂dk dk · × ∂kj ∂kl Ωk,±,i = ±ijl 3 , (3.29) 4|dk|

where ijl is the rank 3 Levi-Civita tensor and dk ≡ (d1(k), d2(k), d3(k)) as deﬁned in

Eqn. (3.17). Since the type of the Weyl node is determined by d0(k) which does not

enter Eqn. (3.29), the Berry curvature around a node is indeed manifestly invariant

with respect to its type.

The presence of topologically protected Fermi arcs relies on the quantized edge

modes of 2D surfaces enclosing Weyl nodes. We again emphasize that it is necessary

for such 2D surface to exist in a region which is fully gapped in the bulk. If one

constructs such a surface which intersects a bulk pocket, then it no longer describes

a Chern insulator and the quantization of the edge modes is destroyed. It is clear

that the extended pockets around type-II Weyl nodes play an important role in the

nature of the connectivity of the Fermi arcs and the pockets, since by deﬁnition one

necessarily cannot take a gapped 2D surface to lie within these pockets. Therefore,

the presence of Fermi arcs in a type-II Weyl semimetal is only guaranteed by ensuring

that the Gaussian surfaces one constructs in the Brillouin zone enclose Fermi pockets

rather than bare nodes.

We provide a simple counting argument that limits the possible connectivity of

Fermi arcs in a Weyl semimetal of either type:

49 1. If a Weyl node is type-I with chirality χ, then |χ| pairs of Fermi arcs will

terminate on the Weyl point when the Fermi energy lies at the nodal energy.

This well-known result is illustrated for the lattice models in Fig. 3.4b.

2. If an isolated Fermi pocket fully encloses n Weyl node of either type such that

a closed 2D subspace where the bulk band structure is gapped can completely

surround the pocket, then the Fermi arcs on a given surface will have net chiral-

ity χtot and terminate on the pocket. Here χtot is the total chirality of all Weyl Pn nodes enclosed by the Fermi pocket such that χtot = i=1 χi. For type-I nodes, this is illustrated by Fig. 3.4a,c and Fig. 3.8a,b. The lattice models illustrate

Fermi pockets enclosing the projections of isolated type-II Weyl nodes in Fig.

3.6a,c,d,f and Fig. 3.8c,d. We see in Fig. 3.4d,f that when the net chirality

enclosed is zero, Fermi arcs are not present.

3. When the chemical potential lies at the energy of a type-II Weyl node at least

two Fermi pockets are connected at the Weyl node. In this case, it is necessary to

consider the set of all connected Fermi pockets when determining the possible

Fermi arc conﬁguration. When multiple Weyl nodes connect a set of Fermi

pockets such that the only gapped 2D subspace of the Brillouin zone surrounding

it contains a net chirality χtot = 0, then the net chirality of Fermi arcs on a

surface is also zero, even though Weyl nodes are present at the Fermi energy.

This is illustrated in Fig. 3.4e where two Weyl nodes of opposite chirality

connect a single pair of hole and electron pockets and Fermi arcs are absent

even at the nodal energy. However, when the net chirality of nodes connecting

the set of pockets is nonzero, then a set of Fermi arcs with net chirality χtot must

50 a Topological Arcs b Trivial c Track States

Figure 3.10: Sketch of the three types of surface states in a topological Weyl semimetal. a Two type-I Weyl nodes of opposite chirality connected by a Fermi arc on the top (red) and bottom (blue) surfaces. In an arbitrary type-II TWS at an energy away from the Weyl energy, these arcs would connect Fermi pockets instead of nodes. b A single Fermi pocket enclosing two nodes of opposite chirality. Since no Gaussian surface can be constructed in a region that is both gapped and encloses only one node, the only possible surface states are trivial ones, shown in red and blue at the boundary of the pocket that hybridize with bulk states due to lack of topological protection. c Pairs of Weyl nodes, two of each chirality with each node surrounded by a Fermi pocket. The pockets are connected by Fermi arcs (thinner red and blue contours) as well as track states (thicker blue lines) on the bottom surface. Note that states on opposite sides of a given loop of track states will disperse in opposite directions and so a Gaussian surface enclosing a given Fermi pocket will still have one net surface state of each chirality.

satisfy is that they must terminate somewhere on the set of Fermi pockets. This

has the striking consequence that even when the Fermi level lies at the node

energy and topologically protected Fermi arcs are present, the termination of

the surface arcs on the bulk pockets can occur at any point on the surrounding

Fermi pockets. We see this illustrated for type-II Weyl nodes connecting isolated

pairs of electron and hole pockets in Fig. 3.6b,e, where the Fermi level is at the

Weyl energy but Fermi arcs terminate on a bulk pocket a substantial fraction

of a reciprocal lattice vector from the Weyl nodes.

51 Surface ”Track States” in Type-II Weyl Semimetals

Topological Fermi arcs are not the only novel surface states possible in a type-II

Weyl semimetal. We have shown that due to the ﬁnite density of states available at type-II Weyl nodes, new surface states can be appear which we term “track states.”

These track states are degenerate with the Fermi arcs but do not share the topological properties of the arcs; instead track states form closed contours on a given surface which are contractible to points in the Brillouin zone. Although topologically trivial, track states appear to play an important role in determining the locations Fermi arcs may appear in the surface Brillouin zone.

Track states are generated when the connectivity of Weyl nodes changes as we tune the parameters of a system with multiple pairs of Weyl nodes. In Fig. 3.9, we see that by tuning the parameter γ in the Hamiltonian in Eqn. (3.28) through the type-I to type-II transition, the Fermi arcs shift locations. When the nodes are type-I,

Fermi arcs pair up nodes of opposite chirality in the kx-direction; when the nodes are type-II, Fermi arcs pair up nodes in the kz-direction. Because the Berry curvature is invariant with respect to γ, the Chern number of a bulk-gapped 2D subspace of the

Brillouin zone surrounding an isolated node does not change. Although the Fermi arcs can shift locations in the Brillouin zone, the net chirality of modes on a given surface is conserved. When γ = 2t, track states appear at the (kx, kz) points where

Fermi arcs were located in the type-I limit.

In a type-II Weyl semimetal, track states can also appear as the Fermi energy shifts. It is shown in Fig. 3.8c,d that as the Fermi energy changes from below the

Weyl energy in Fig. 3.8c to above the Weyl energy in Fig. 3.8d, the locations of the arcs shift. For E < 0, the arcs on the bottom surface (shown as thick blue contours)

52 connect across the kz = π line while track states are seen as closed blue contours

encircling the points (kx, kz) = (±π/2, 0) points. For E > 0, this pair of track states

have become a single track state encircling the (kx, kz) = (±π, 0) point and a pair of arcs connecting electron pockets across the kz = 0 line. A precisely analogous reconﬁguration of states on the top surface also occurs as shown by the reorientation of the red contours.

We note that these track states can appear very similar to Fermi arcs when track states and arcs lie close together. Caution must therefore be taken when analyzing the surface Fermi state conﬁgurations of type-II Weyl semimetals in DFT calculations or in ARPES data. There is experimental evidence for the existence of track states in

MoTe2[55], WTe2[77, 78, 79], and a recent ARPES study of Ta3As2[80] has revealed closed contours of surface states which are strong track state candidates. The Ta3As2 system is particularly promising as it has been predicted that pressure can tune a type-I to type-II transition where track states are likely to appear.

3.3.4 Comparison with Experiments

In this section, we describe the current state of experimental realizations of topological Weyl semimetals. Our results are summarized by Table 3.1. Although various ab-inito studies have proven useful in the study of the materials in Table 3.1, as well as the prediction of a variety of Weyl semimetals yet to be discovered experimentally, it is clear that there is a distinct need for a set of minimal models which describe the general features of topological Weyl semimetals. From the abundance of type II TWS in Table 3.1, it is particularly evident that our models provide a general framework

53 Material Type Broken symmetry Pairs of Weyl nodes Surface states TaAs[16, 17, 46] I Inversion 12 MoTe2[55] II Inversion 4 track states WTe2[77, 78, 79] II Inversion 4 track states LaAlGe[56] II Inversion 20 Ta3S2[81] II Inversion 4 track states

Table 3.1: Experimental realizations of Weyl semimetals.

for understanding the topological features of type II TWS which is complimentary to

DFT.

In Table 3.1, we note that other than the transition metal monophosphides, all of the Weyl semimetals which have been uncovered by spectroscopic experiments are of type II. Additionally, they all break inversion symmetry with strictly more than the minimum of two pairs of nodes. For this reason, we expect track states may be common in Weyl semimetals. Indeed, we have found that a detailed examination of the spectroscopic results indicate that evidence of track states is found in nearly all of the type II Weyl semimetals so far discovered.

The transition metal dichalcogenides MoTe2 and WTe2 each feature long surface states which begin on bulk electron pockets and terminate on bulk hole pockets.

These bulk pockets each enclose the projections of a net zero chirality of Weyl nodes and, by the arguments in Section 3.3.3 above, cannot have a nonzero net chirality of

Fermi arcs terminating on it. This is borne out in both the ab-initio calculations as well as the ARPES results[77, 78, 79, 55]. The long surface state in WTe2 has been shown[79] to have both topological and trivial character, depending on the material

54 parameters used in the ab-initio calculations and therefore the conﬁgurations of the

Weyl nodes. This is manifestly a characteristic of a track state.

The transition metal pnictide Ta3S2 features[81] 4 pairs of Weyl nodes which are formed from the merging of two hole pockets with an electron pocket. As we have shown in the section above, in such a conﬁguration, there cannot exist a closed and gapped region of the Brillouin zone which encloses a net chirality of Weyl nodes. In this way, all surface states shown in Fig. 4 of Ref. [81] are in fact trivial in a topological sense. Additionally, Ta3S2 has set of surface states that lie close in momentum to the bulk hole pockets. The bulk band structure of Ta3S2 is predicted to be highly tunable and is has been predicted [81] that strain can tune transitions between type I and type II Weyl semimetals as well as between these semimetal phases and a strong topological insulating phase. It is possible that the track state nature of these surface states will be revealed by such an experiment.

3.3.5 Conclusions

The models we present here comprehensively describe the four classes of Weyl semimetals which can be delineated by the type of the nodes and whether they break inversion or time-reversal symmetry. Examples from each class have been predicted by theory and have been experimentally observed in quantum materials. Particular realizations obey point group symmetries diﬀerent in general than those presented here. It is straightforward to extend the models we present here to study a Weyl semimetal with a chosen point group symmetry.

This summary of minimal models for type-I and type-II Weyl semimetals for both time-reversal-breaking and inversion-breaking cases may contribute to future

55 investigations of their properties in applied electric and magnetic ﬁelds. In particular, we expect our models to shed light on the nature of quantum oscillations in type-II

Weyl semimetals. Preliminary calculations[50] show the absence of a chiral zero- energy Landau level when the direction of the applied magnetic ﬁeld lies outside of the tilt cone of the type-II Weyl node. However, these calculations rely on a linearized model for type-II Weyl nodes and a proper treatment should include the full Fermi pockets surrounding the Weyl nodes. The models presented here provide an ideal framework for such a calculation which we leave for future study. These models also provide a foundation for additional eﬀects of repulsive and attractive interactions.

Experimental discoveries of magnetism and superconductivity in Weyl semimetals could provide impetus for such theoretical studies. In the following section, we shall see how the model given by Eqn. (3.28) played a key role in the discovery of the ﬁrst type-II Weyl semimetal MoTe2.

3.4 Experimental Discovery of Weyl Semimetal MoTe2

To set the stage for interpretation of the experimental results, we investigate the two-band lattice model given by Eqn. (3.28) which breaks inversion symmetry but is invariant under time-reversal symmetry. The main lessons learned by examining this model are shown in Fig. 3.11 and summarized here: (1) The minimum number of four

Weyl nodes in this type II TWS occur at E = 0 at the touching point of electron and hole pockets in contrast with a type I TWS that has a zero density of states at E = 0.

The touching of electron and hole bands in our model is similar to the touching of the electron and hole bands in the experimental data shown in Fig. 3.12a and b.

(2) For a slab geometry, constant energy cuts at E = 0 show Fermi arcs on surface

56 termination A and B that connect Weyl points of opposite chirality. In addition there

are what we term “track states” that exist on the surface and pass through the WPs

but, unlike Fermi arcs, form closed loops. For E < 0, the projections of the WPs are

within the hole pocket, and at the surface the arc states connect the two hole pockets

and the track states loop around the electron pockets. The opposite is true for E > 0.

(3) The energy dispersion clearly shows a surface state dispersing separately from the

bulk bands and merging with the bulk bands close to the WP in Fig. 3.11d. This is

corroborated by the experimental data around the Weyl nodes in Fig. 3.12i and Fig.

3.13n where the arc merges with the bulk states.

Henceforth, we set the parameters of Eqn. (3.28) to m = 2t, tx = t/2, k0 = π/2,

and γ = 2.4t. The bulk band structure for this parameter choice was explored in

detail in the previous section and can be seen in Fig. 3.11a which shows hole and

electron pockets touching at the Weyl nodes as well as pockets disconnected from

the nodes. Similar Fermiology is also present in the MoTe2 system and we can gain

insight into this and other related materials by taking advantage of the lattice model’s

simplicity and tunability.

We again consider a slab geometry ﬁnite in the y- direction with L layers but inﬁnite in the x- and z-directions. We label the states as “surface termination B”

(“surface termination A”) if they are exponentially localized at hyi = 1 (hyi = L).

Fig. 3.11 also shows the surface states at µ = ±0.1t overlaid on the bulk band structure. We show constant energy cuts through the band structure of the slab geometry in Fig. 3.11b and c for µ = ±0.1t. When µ < 0, the projections of the

Weyl nodes (shown by green dots) are enclosed by hole pockets. Each of these hole pockets are connected to another pocket containing a node of opposite chirality by

57 a d

Cut 1

b 1 2 c 1 2 e

Cut 2

Figure 3.11: Simple model of type II Weyl semimetal described by a two band model given by Eq. 3.28 which exhibits four Weyl nodes. a Electronic band structure for µ = ±0.1t indicated by the blue translucent plane. b,c The topological surface states and Fermi arcs on surface A (in red) and B (in blue) are calculated for a slab geometry conﬁned along the y-direction. The bulk bands are shown in black. When µ = 0 exactly, the electron and hole pockets touch and the arcs terminate on the node (green dot) itself. For Fermi energy below (above) the nodal energy, arcs of surface states connect the Fermi hole (electron) pockets surrounding a node rather than terminating on a node. d,e Energy dispersion along kz at ﬁxed kx as shown by cuts in panels (b, c). Cut 1 along kx = π/2 shows the bulk electron and hole bands touching at the node and the merging of surface states into the bulk away from the Weyl node. Cut 2 along kx = 0.63π shows a gap between the bulk bands and a surface state that disperses with opposite velocities at the projections of the two WPs. The WPs are located at (kx, kz) = (±π/2, ±π/2) indicated by pink arrows pointing to green dots.

58 one Fermi arc on surface A (B) shown as a thick light red (blue) line. When µ > 0, the projections of the Weyl nodes are enclosed by electron pockets which are similarly connected by Fermi arcs on the surfaces. At precisely µ = 0, because all of the nodes lie at E = 0, all Fermi arcs terminate on the nodes themselves as in a type I TWS.

The slab conﬁguration energy dispersion for ﬁxed kx is shown in Fig. 3.11d and

1e. These cuts are shown as green dashed lines labeled cut 1 and cut 2 respectively.

We can see that at the Weyl nodes, the red surface bands in 1d disappear into the bulk. As we move past the Weyl points in e, we see that these two red bands combine into a single continuous band.

3.4.1 ARPES Results

MoTe2 is a semimetal that crystallizes in a orthorhombic lattice. The Fermi surface of MoTe2 also has two 2-fold symmetry axes, along Γ - X and Γ - Y directions.

The lattice constants are a = 6.33 A,˚ b = 3.469 A.˚ Due to breaking of the inversion symmetry there are two different possible terminations of the cleaved sample surface, referred to as termination “A” and “B” respectively. The two different terminations also have different surface band structures as seen by laser-based angle resolved photoemission spectroscopy (ARPES) and corroborated by DFT calculations.

We identify electron and hole bands in the spectroscopic data shown in Figs. 3.12 and 3.13. The hole bands at the center of the Brillouin zone have a “butterﬂy” shape.

The electron pockets shaped like ovals are located on each side of the butterﬂy. There are also two banana like hole pockets partially overlapping the oval electron pockets.

The conﬁguration of these pockets can be seen at the Fermi energy in Fig. 3.12a and

10 meV above the Fermi energy in Fig. 3.12b and their electron or hole character

59 0.4 a FS, 6.7eV f 40 π / W3exp kx = 0.36 ( b) j + 20 0.2 + 0 a) + -20 (meV)

0.0 f π /

( + W2calc -40 y

k + W2exp -60 -0.2 E - W3exp + -80 π / kx = 0.36 ( b) -100 -0.4 g 40 k = 0.32 (π / b) b FS + 10 meV x k 20 + 0 + a) -20

+ (meV) f

π / -40

( + y

k + -60 E -

-80 π / + kx = 0.32 ( b) -100

h 40 π / kx = 0.28 ( b) l c 0.4 Hole Band (HB) 1 Weyl points 20 HB 2 EB 1 0 0.2 -20 a) (meV) f 0.0 Fermi Arcs -40 π / (

y -60 E - k -0.2 -80 π / kx = 0.28 ( b) EB 2 Arc W2-W3 -100 -0.4 Arc W2-W2 i 40 π / kx = 0.24 ( b) W2exp -0.4 -0.2 0.0 0.2 m 20 π / kx ( b) 0 d 0.4 FS + 30meV FS - 30 meV -20 (meV) e f -40 + + a) 0.2 -60 + E - π / -80 π / ( 0.0 kx = 0.24 ( b) y + + k -100 + + -0.2 0.0 0.2 -0.2 0.0 0.2 -0.2 + π / π / -0.4 -0.2 0.0 -0.4 -0.2 0.0 ky ( a) ky ( a) π / π / kx ( b) kx ( b)

Figure 3.12: Experimental Fermi surface and band structure of MoTe2. a Constant energy intensity plot measured at EF using 6.7 eV photons for a sample with termination A. The calculated (DFT) positions of Weyl points W2 are marked as pink dots, while experimentally determined locations of W2 and W3 points are marked as red dots. The chiralities of Weyl points are marked with “+” and “-” and their locations (kx, ky, E) are summarized in Table 3.4.2. b Same as in a above but taken at 10 meV above EF . c A sketch of constant energy contours of electron and hole bands showing the locations of Weyl points and Fermi arcs. d Constant energy contour measured at 30 meV above EF using 5.9 eV photons for a sample with termination B. Positions of calculated and measured Weyl points are marked as above. e Same surface termination and photon energy as d but at 30 meV below EF . f - i Experimental band dispersion along cuts at kx = 0.24, 0.28, 0.32 and 0.36 π/b. j - m Calculated band dispersion for a sample with termination A along kx = 0.24, 0.28, 0.32 and 0.36 π/b. 60 is easily identiﬁed because hole (electron) pockets shrink (expand) with increasing

energy. A simpliﬁed sketch of constant energy contours of electron and hole bands is

shown in Fig. 3.12c.

The central hole pocket touches the electron pockets at four Weyl points shown

as red dots in Fig. 3.12a-c which we label as W2. The outer banana shaped hole

pockets also touch the oval electron pockets at two other Weyl points labeled as W3.

At surface termination A, Fig. 3.12b, those two types of Weyl points are connected by topological arcs seen as white-gray high intensity areas. For this surface termination there is no strong evidence for arcs connecting positive and negative chirality W2 nor positive and negative chirality W3 points. The situation for surface termination B is more complicated as shown in Fig. 3.12d. There seems to be a sharp contour connecting both sets of W2 and W3 points. Most likely this is a track state discussed above. The examination of constant energy plot at energy of 30 meV below EF

(Fig 3.12e), reveals that there are actually two bands present. In addition to the track state, there is also an arc present that connects positive and negative chirality

W2 points. Although present data does not allow us to deﬁnitely demonstrate a connection between positive and negative chirality W3 points, we can deduce that they are likely connected, so the arcs on surface A between W2-W3 together with arcs

+ − + − on surface B W2 -W2 and W3 -W3 form a closed loop when connected via the bulk of the sample.

We now examine the locations of the Weyl points in the band dispersion. In Fig.

3.12f-i we plot the band dispersion along ky cut for selected values of kx. At ky=0.36

π/b (panel f) two bands are clearly visible: an “M” shaped band at higher binding

energy and a “U” shaped band at slightly lower binding energy. Both bands appear

61 6.7 eV, FS, Top k = 0 (π / a) k = 0.05 (π / a) ky = 0.1 (π / a) k = 0.2 (π / a) a y y y 0.0 d e f g Cut d e f g d - g -0.1 6.7 eV 0.0 W2exp (eV) W2calc A f -0.2 W2exp E - / b) -0.3 π -0.1 (

x W3exp k -0.4 h - k h i j k W2exp 6.7eV 0.0 (eV)

-0.5 f B -0.6 b 0.0 E - 6.7 eV, FS, Bottom -0.1 -0.1 l m n o W2calc l - o 0.0 W2exp -0.2 W2exp 5.9 eV (eV) f

/ b) B -0.3 π E - ( W3exp x

k -0.4 -0.1 0.10 0.25 0.10 0.25 0.10 0.25 0.10 0.25 π π π π -0.5 kx ( /b) kx ( /b) kx ( /b) kx ( /b) 0.1 Band 3 -0.6 p - s p W2calc q r s 0.2 c A

5.9 eV, FS, Bottom (eV) f 0.0 0.1 Band 2

0.0 E - Band 1 -0.1 0.1 / b) -0.1 Band 3 t u v w π W2calc t - w W2calc ( x -0.2 B (eV) k W2exp f 0.0

-0.3 Band 2

W3exp E - Band 1 -0.4 -0.1 0.10 0.25 0.10 0.25 0.10 0.25 0.10 0.25 -0.4 -0.2 0.0 0.2 0.4 π k (π/b) k (π/b) k (π/b) k (π/b) ky ( / a) x x x x

Figure 3.13: Identiﬁcation of Weyl points and Fermi arcs from experimental data. a Constant energy contour at EF , measured by 6.7 eV photons for surface termination A. DFT predicted locations for Weyl points W2 and measured Weyl points W2, W3 are marked as red and pink dots respectively. b The same panel as a except for surface termination B. c The same panel as b except for using 5.9 eV photons. d - g Energy dispersion for surface termination A along ky = 0, 0.05, 0.10 and 0.20 π/a. The projections of Weyl points W2 are marked as dots. h - k The same panels as (d - g) except for surface termination B. l - o The same panels as (h - k) except for using 5.9 eV photons. p - s Calculated band dispersion for surface termination A along cuts at ky = 0, 0.05, 0.10 and 0.20 π/a. Positions of W2 are marked similarly as above. t - w The same as (p - s) except for surface termination B. Bands plotted with darker lines have more surface weights.

62 connected at zero momentum with Dirac-like structure. As we move towards the

zone center, both bands move to lower binding energy and their energy separation

decreases. In panel h, the tips of the “M” shaped band (red dotted line) touches the

EF and form parts of the butterﬂy hole pockets. As these tips move above EF , they

touch merge with wings of the “U” shaped electron band (white dotted line) forming

two Weyl points approximately 20 meV above EF marked by black dots. At each

side of the symmetry line, they form two tilted cones characteristic of a type II Weyl

node. The data along kx direction are shown in Fig. 3.13d-o along with results of calculations (Fig. 3.13p-w) for the two surface terminations. The surface termination

A is characterized by lower binding energy of electron pocket in panels d-g, when compared to the data from surface termination B shown in panels h-k and l-o. The data in panels l-o best illustrates the formation of the W2 points. In panel l, the hole band is marked with red dashed line, while the electron band is marked with white dashed line. As we move away from the symmetry line, the separation between those bands becomes smaller and they merge at a point located ∼20 meV above EF marked by red dot in panel n. For higher values of ky momentum they separate again as seen in panel o. The DFT calculation also demonstrates the energy diﬀerence of the band locations for the two terminations and formation of the W2 Weyl point that agrees with experiment on a qualitative level.

The momentum location of the experimentally determined Weyl points is some- what diﬀerent from DFT predictions (marked as pink dots in 3.12a,b) most likely due to high sensitivity of the band calculation to structural parameters. Table I summarizes the positions of WPs determined from experiment and DFT. Despite the discrepancy between the predicted locations of the Weyl nodes from DFT and where

63 Figure 3.14: Results of DFT calculations. a Calculated bulk Fermi surface of MoTe2 for kz = 0.6π/c and projections of W2 (kx, ky) = (±0.17 π/b, ±0.06π/a) are marked with pink dots. b Bulk band dispersion along W2-W2 direction (the vertical dashed line in a). DFT predicted positions of W2 (ky, E) = (±0.06π/a, 0.028 eV) are marked. DD DD c The dominant contribution for the divergence of the Berry curvature (Ωn,yz, Ωn,zx) for the n = N + 1 th band where N is the number of electrons in the unit cell with kz = 0. Red and blue indicate diﬀerent chiralities of the two Weyl points. d - g Calculated constant energy contours of MoTe2. Darker bands are surface bands and lighter bands are bulk bands. d, e are at Fermi level for surface termination A and B. f, g are at Fermi level + 28 meV of surface termination A and B, respectively. h, i Surface band dispersions of termination A and B along W2-W2 direction. j, k Surface band dispersions of termination A and B along ky = 0.05 π/a direction, which is very close to the ky position of W2 (0.06 π/a). Positions of calculated Weyl points W2 are marked and darker bands have more surface weights in d - k.

64 they are located experimentally, in each case they are at the touching points of the electron and hole bands. In the ky = 0 cuts shown in Fig. 3.13d,h,l,p,t, band 1 is connected to bulk states below the Fermi level, while band 3 dips down and goes into bulk just before it reaches the Weyl point. As we increase ky, band 1 and band 3 merge together. In the ky = 0.1π/a cuts, the two bands merge into one band which goes through the position of the projection of W2. This behavior is exactly the the behavior predicted in Fig. 3.11d and 3.11e.

3.4.2 DFT and Topological Analysis

Fig. 3.14 is the DFT calculation of the band structure of MoTe2 performed by

Ryotaro Arita’s group. Fig 3.14a is the bulk Fermi surface for kz = 0.6π/c and calculated positions of four Weyl points are marked. The shapes of outermost electron and hole bands are very similar to our experiment result in Fig. 3.12b. Pink dots are projections of the calculated Weyl points on the kz = 0 plane from energy +28 meV above Fermi level, thus the electron band is not touching the two Weyl point projections. The surface weighted constant energy contours are shown in 3.14d -

3.14g. Fig. 3.14d and 3.14e are at Fermi surfaces of termination A and B, while

3.14f and 3.14g are at Fermi level + 28 meV, the DFT predicted energy of W2. In the calculations, W2 is not directly connected to another W2 by surface states on the

Fermi surface of termination A calculation while they are connected by weak and short surface states in termination B calculation. However, the W2 points are connected by bulk electron bands in termination A. This is consistent with our experimental results shown in Fig. 3.12a-e. Fig. 3.14b is the bulk band dispersion at W2-W2 direction, as the vertical dashed line shown in 3.14a. The two W2 points from DFT are right

65 kx (π/b) ky (π/a) E (meV)

W2 DFT ±0.17 ±0.06 28 W2 Exp ±0.24 ±0.12 20 W3 Exp ±0.37 ±0.25 30

Table 3.2: The locations (kx, ky, E) of the Weyl points from DFT and ARPES for weyl semimetal MoTe2.

at the touching points of one hole band and one electron band. Fig. 3.14h and 3.14i show termination A and B surface band dispersions along the same direction as in

Fig. 3.14b. The surface bands are to connect bulk states near the positions of the

Weyl points. Fig 3.14j and 3.14k are termination A and B surface band dispersions along ky = 0.05 π/a direction, as the horizontal dashed line shown in 3.14a. We also calculated the Berry curvature on Fermi surface. The bright points in Fig. 3.14c are possible singular points of the Berry curvature and DFT calculated W2 points are marked in red and blue, indicating diﬀerent chiralities of the Weyl points. The summary of energy and momentum locations of Weyl points based on calculations and experiment are provided in Table 3.2.

66 Chapter 4: Thermoelectric Transport in Weyl Semimetals

In the previous chapter, we saw that Weyl semimetals have a truly unique fermiology. In particular, we saw that they feature three deﬁning and interwoven characteristics: linearly dispersing low energy modes, monopoles of Berry curvature in momentum space centered at these nodes, and Fermi arcs on in the surface Bril- louin zones connecting projections of Weyl points. Since the ﬁrst theoretical prediction of Weyl semimetals in solid state systems, various transport formalisms have been used to predict how charges move in these materials. Most notably, Weyl semimetals exhibit negative longitudinal magnetoresistance as a result of the chiral anomaly[18, 19, 20, 21, 22, 23, 24]. This phenomena is due to the chiral Landau levels of the Weyl nodes and relies on sensitive measurements at quite high magnetic

ﬁelds. Although theoretically interesting, we seek other transport quantities that will reveal the unique bulk electronic structure of Weyl semimetals.

The thermoelectric response of a material is in general an excellent way to probe the bulk electronic characteristics. In this chapter, we present the ﬁrst comprehen- sive theory for the Nernst thermopower in Weyl semimetals. We ﬁnd that the linear band structure of a Weyl semimetal with its unique temperature dependent chemical

67 potential (see previous chapter) leads to a distinctly identiﬁable peak in the temperature dependence of the Nernst coeﬃcient. We also present experimental data from collaborators that show remarkable agreement with our prediction.

We also find that the Berry curvature leads to anomalous transverse currents in the absence of a magnetic field. Although this was first predicted in the type-I

Weyl semimetals [82], we extend this formalism to type-II Weyl semimetals for the

first time. In particular, we find that, due to the Berry curvature, the anomalous transverse thermoelectric coefficient is enhanced with increasing tilt in type-II Weyl semimetals.

Thus in this chapter, we identify clear signatures in thermoelectric transport for the deﬁning bulk characteristics of Weyl semimetals. Both of these features can be well-explained by Boltzmann transport calculations and are readily accessible to experimental probes. In the next chapter, we shall also see that the third deﬁning characteristic of Weyl semimetals, the presence of Fermi arcs, leads to a topological thermomagnetic conductivity.

4.1 Boltzmann Transport Theory

In this section, we deﬁne the bulk thermoelectric transport coeﬃcients and introduce the Boltzmann transport formalism. We apply it to the general case of a lattice model with Berry curvature. In subsequent sections, we will apply these results to the lattice models for Weyl semimetals of the previous chapter.

4.1.1 Deﬁnition of Transport Coeﬃcients

We deﬁne the thermomagnetic coeﬃcients in the following way:

68 JE LEE LET E = · , (4.1) JQ LTE LTT −∇T where J is the charge current density, Q is the heat current density, E is the electric

ﬁeld and ∇T is the gradient in temperature. Each of the thermomagnetic transport

coeﬃcients is a tensor of the form

αβ αβ αβ Lxx Lxy L = αβ αβ . (4.2) Lyx Lyy

αβ αβ Due to symmetry, these must be antisymmetric tensors with Lxy = −Lyx .

The Isothermal Nernst Eﬀect

We are interested in calculating the Nernst eﬀect deﬁned by

Ey αxyz = , (4.3) −∇xT

under the isothermal conditions

Jx = Jy = 0 (4.4)

and

∇yT = 0. (4.5)

From Eqn. (4.1) and Eqn. (4.63), it is clear that

J = LEE · E + LET · (−∇T ) = 0. (4.6)

Solving for the electric ﬁeld, we obtain

E = −(LEE)−1 · LET · (−∇T ). (4.7)

Now, due to the isothermal condition in Eqn. (4.64), we have that

X EE −1 ET EE −1 ET EE −1 ET Ey = − (L )yj Ljx (−∇xT ) = − (L )yx Lxx + (L )yy Lyx (−∇xT ) (4.8) j

69 It is straightforward to obtain the inverse of LEE as

EE −1 EE −1 EE EE EE −1 (L )xx (L )xy 1 Lyy −Lxy (L ) ≡ EE −1 EE −1 = EE EE EE EE EE EE . (L )yx (L )yy Lxx Lyy − Lxy Lyx −Lyx Lxx (4.9)

EE EE We now take advantage of the fact that Lyy = Lxx and the antisymmetry property

EE EE Lxy = −Lyx to obtain

EE −1 EE −1 EE EE (L )xx (L )xy 1 Lxx Lyx EE −1 EE −1 = EE 2 EE 2 EE EE . (4.10) (L )yx (L )yy (Lxx ) + (Lxy ) Lxy Lxx

The y-component of the electric ﬁeld now becomes

EE ET EE ET Lxy Lxx + Lxx Lyx Ey = − EE 2 EE 2 (−∇xT ). (4.11) (Lxx ) + (Lxy )

ET ET Finally we use the antisymmetry Lyx = −Lxy to write the isothermal Nernst eﬀect as EE ET EE ET Ey Lxx Lyx − Lxy Lxx αxyz = = EE 2 EE 2 . (4.12) −∇xT (Lxx ) + (Lxy )

Thermal Conductivity

The thermal conductivity is found to be

TT TE EE −1 ET κxj = L − L · (L ) · L xj . (4.13)

We can expand this and ﬁnd

TT X TE EE −1 ET κxj = Lxj − Lxm(L )mnLnj . (4.14) mn Hence, ET 2 ET 2 EE ET EE ET TT (Lxx ) − (Lxy ) Lxx + 2Lxx Lxy Lxy κxx = Lxx − T EE 2 EE 2 , (4.15) (Lxx ) + (Lxy ) and ET 2 ET 2 EE EE ET ET TT (Lxy ) − (Lxx ) ) Lxy + 2Lxx Lxy Lxx κxy = Lxy − T EE 2 EE 2 . (4.16) (Lxx ) + (Lxy ) 70 We will utilize all of the above deﬁnitions in the remainder of this thesis. However, to really make use of these results, we must ﬁrst show how to obtain each component

αβ of the tensor Lij

4.1.2 Boltzmann formalism

We begin by considering the semiclassical equations of motion for an electronic wavepacket in the presence of electromagnetic ﬁelds. Weyl nodes are monopoles of

Berry curvature which acts as a momentum-dependent magnetic ﬁeld. Berry curvature is at the heart of the anomalous transport of a variety of materials and is essential to understanding transport phenomena in a general setting. Crucially, it modiﬁes the semiclassical equation of motion such that the velocity is now given by

dr 1 dk = ∇kE(k) + × Ωk, (4.17) dt ~ dt where k is the crystal momentum, E(k) is the energy dispersion and Ωk is the Berry curvature. In Eqn. (4.17), the first term is the normal velocity and the second term now reflects the so-called anomalous velocity which gives rise to the anomalous Hall effect in ferromagnets and time-reversal symmetry breaking Weyl semimetals. The equation of motion for the momentum is given by the standard Lorentz force

dk e e dr = E + × B, (4.18) dt ~ ~ dt where, E and B are the magnetic ﬁelds as usual. It is straightforward to solve the coupled Eqns. (4.17) and (4.18) to obtain [83] the velocity

dr 1 e e = v + (E × Ω ) + (Ω · v )B (4.19) dt e k k k k 1 + (B · Ωk) ~ ~ ~

71 and the equation of motion for the momentum

dk 1 e e e2 = E + (v × B) + (E · B)Ω . (4.20) dt e k k 1 + (B · Ωk) ~ ~ ~ ~

Here we follow the now standard notation of deﬁning the band velocity as vk =

∇kE(k). The Poisson brackets of diﬀerent position components are no longer zero in

the presence of Berry curvature and this noncommutativity gives rise to a necessary e modiﬁcation of the invariant phase-space volume element of ddxddp → (1 + (B · ~ d d Ωk))d xd p such that Liouville’s theorem still holds with respect to this new volume

element. For convenience, we will follow the now standard notation of D(B, Ωk) =

e −1 (1+ (B·Ωk) . Eqns. (4.19) and (4.20) have been well-studied and the various terms ~ can be understood as giving rise to a variety of novel transport phenomena[CITE]. In addition to the second term in Eqn. (4.19) which we have noted as the anomalous velocity, the third of Eqn. (4.19) gives rise to the so-called chiral magnetoeﬀect in Weyl semimetals and the third term of Eqn. (4.20) is responsible for the chiral anomaly induced negative magnetoresistance which has been putatively veriﬁed experimentally

[84, 85, 86, 87].

Within the relaxation-time approximation, the Boltzmann equation describing the distribution function f(k, r, t) for an electronic wavepacket is given by

∂f dk dr f − f + · ∇ f + · ∇ f = − 0 , (4.21) ∂t dt k dt r τ(k)

β(E−µ) where τ(k) is the momentum-dependent scattering time, f0 = 1/(1 + e ) is

the equilibrium Fermi distribution function with β ≡ 1/kBT and where we have sup-

pressed the dependence of f on position, momentum and time such that f = f(k, r, t).

In this work, we will take the scattering time to be independent of momentum, though

we at times may take it to depend on energy τ = τ(E).

72 We take the distribution function f to not depend explicitly on time, such that we

have that ∂f/∂t = 0. By simple application of the chain rule, it is easy to verify that,

for the real-space term, keeping only the lowest order deviation from equilibrium,

E − µ ∂f ∇ f = (− 0 )∇ T. (4.22) r T ∂E r

We can then substitute Eqns. (4.19,4.20,4.22) into Eqn. (4.21) to obtain

E − µ ∂f0 e e − vk + (E × Ωk) + (Ωk · vk)B · ∇rT T ∂E ~ ~ ∂f0 e + e − (E + e(E · B)Ωk) · vk + (vk × B) · ∇kf ∂E ~ f − f = − 0 . (4.23) D(B, Ωk)τ

We ﬁrst note that, since we are interested in the lowest order response to external perturbations, we can ignore the term which goes like (E×Ωk)·∇rT . As noted above,

the chiral anomaly has been the subject of extensive study in Weyl semimetals and

we will therefore not consider conﬁgurations for which E and B are parallel, allowing

us to also drop the term which goes like (E · B)Ωk. We then are left merely with

E − µ ∂f0 e − vk + (Ωk · vk)B · ∇rT T ∂E ~ ∂f0 e f − f0 + e − E · vk + (vk × B) · ∇kf = − . (4.24) ∂E ~ D(B, Ωk)τ

Solving Eqn. (4.24) for f(k, r, t) allows us to obtain the charge and heat current,

each of which have ordinary contributions from the velocity as well as anomalous

contributions [88, 89, 90, 91, 92] driven by Berry curvatures. We will therefore write

E E E E Q Q them as J = J0 + JA and J = J0 + JA respectively. The ﬁrst part of the charge

73 current can be computed in the usual way

Z 3 E d k e dr J0 = −e 1 + (B · Ωk) f(k, r, t) (2π)3 ~ dt Z d3k e = −e f(k, r, t) vk + (E × Ωk) (2π)3 ~ e + (Ωk · vk)B . (4.25) ~

The anomalous contribution to the current can be understood as arising from the

Berry-phase correction to the orbital magnetization instead of from the anomalous velocity. It has been shown[88, 89] that

Z 3 E kBe d k JA = ∇rT × Ωksk, (4.26) ~ (2π)3 where sk = −f0ln(f0) − (1 − f0)ln(1 − f0) is the entropy density of the electrons. The ordinary contribution to the heat current is well-known to be

Z d3k JQ = f(k, r, t)(E(k) − µ(T )) v , (4.27) 0 (2π)3 k while it has been shown[90, 91, 92] that the anomalous contribution to the heat current can be written as

Z 3 Q ekBT d k JA = (E × Ωk)sk ~ (2π)3 2 Z 3 2 2 kBT d k π E(k) − µ(T ) + ∇rT × 3 Ωk + f0 ~ (2π) 3 kBT ! −β(E(k)−µ(T )) 2 − ln(1 + e ) − 2Li2(1 − f0) , (4.28)

where Lis(z) is Jonquire’s polylogarithm function of order s, deﬁned as

∞ X zk Li (z) = , (4.29) s ks k=1

74 x

y z

Figure 4.1: Geometry for measuring the Nernst effect αxyz. Temperature gradient ∇rT , electric field E, and magnetic field B are all mutually perpendicular.

for complex z such that |z| < 1. Physically, we understand the ﬁrst term in Eqn.

(4.28) as the heat current analog of the anomalous Hall charge velocity of a ferro- magnet. The rest of the terms in Eqn. (4.28) arise from electromagnetic orbital magnetization as well as the gravitomagnetic energy magnetization which emerge as corrections to the heat current in the presence of Berry curvature. From the solution to the Boltzmann equation in Eqn. (4.24) and the currents given by Eqns. (4.25-4.28) we can obtain the transport coeﬃcients in Eqn. (4.1). .

4.2 Nernst Eﬀect of Isotropic Weyl Nodes

We will consider what we will refer to as the Nernst geometry where the temperature gradient ∇rT , electric field E, and magnetic field B are all mutually perpendicular. For concreteness, we will let ∇rT = dT/dx ex ≡ ∇xT ex, E = Eey, and B = Bez. We now note that the Boltzmann equation in Eqn. (4.24) simplifies

75 dramatically to yield

E − µ ∂f v ∇ T + eE v − 0 T x x y y ∂E eB ∂ ∂ f − f0 + vy − vx f = − , (4.30) ~ ∂kx ∂ky D(B, Ωk)τ where we have introduced the standard notation for the components of the band

1 ∂E velocity vj = . The solution of the Boltzmann equation in Eqn. (4.30) was ~ ∂kj obtained by Ref.[93] and we outline that result here for completeness.

We take the following ansatz for the distribution function

f(k, r, t) = f(k) =

E − µ(T ) f − D(B, Ω )τ v ∇ T + eE v 0 k T x x y y ! ∂f − v · Λ − 0 , (4.31) k ∂E

where we have introduced the correction factor Λ necessary in Boltzmann transport

calculations in the presence of magnetic ﬁelds. Then after substituting this ansatz

for f(k), Eqn. (4.30) becomes

eB ∂ ∂ E − µ(T ) vy − vx − D(B, Ωk)τ vx∇xT ~ ∂kx ∂ky T e vk · Λ + Eyvy − vk · Λ = − . (4.32) ~ τ

Since Eqn. (4.31) must be true for arbitrary velocities vk, we see that it must be true

that Λz = 0.

76 ET We will ﬁrst calculate L and so we set Ey to zero and see that Eqn. (4.32)

simpliﬁes to

E − µ(T ) eB∇ T Dτ v m−1 − v m−1 x T x xy y xx −1 −1 + eB vyΛxmxx − vxΛymyy 1 1 = −v Λ − eBm−1 − eBm−1 + , (4.33) x x Dτ xy xy Dτ

−1 1 ∂2E where we have written the inverse mass tensor as m = 2 . Eqn. (4.33) can ij ~ ∂ki∂kj ˜ be solved [93] by introducing the complex variables V = vx + ivy and Λ = Λx − iΛy

and obtaining that " # E − µ(T ) Re eB∇ T Dτm−1 + im−1 x T xy xx " ! # 1 = Re V Λ˜ ieBm−1 − + eBV Λ˜ ∗ m−1 . (4.34) xx Dτ xy

Solving Eqn. (4.34), we obtain the following expressions[93] for Λx and Λy:

" −1 −1 −1 −1 vx (mxy vx − mxx vy eB(mxy vx − mxx vy) − Dτ Λx = eBD 2 2 −1 −1 vx −1 −1 vy eB(mxy vx − mxx vy) − Dτ + eB(mxx vx − mxy vy) − Dτ −1 −1 −1 −1 vy # (mxx vx + mxy vy eB(mxx vx − mxy vy) − Dτ + 2 2 , (4.35) −1 −1 vx −1 −1 vy eB(mxy vx − mxx vy) − Dτ + eB(mxx vx − mxy vy) − Dτ and

" −1 −1 −1 −1 vy (mxy vx − mxx vy eB(mxx vx − mxy vy) − Dτ Λx = eBD 2 2 −1 −1 vx −1 −1 vy eB(mxy vx − mxx vy) − Dτ + eB(mxx vx − mxy vy) − Dτ

−1 −1 −1 −1 vx # (mxx vx + mxy vy eB(mxy vx − mxx vy) − Dτ − 2 2 . (4.36) −1 −1 vx −1 −1 vy eB(mxy vx − mxx vy) − Dτ + eB(mxx vx − mxy vy) − Dτ

It now becomes useful to rewrite the above equations by introducing cx and cy

such that E − µ(T ) Λ = ∇ T τ c . (4.37) j x T j 77 Now from Eqns. (4.35) and (4.36) can fully determine the distribution function f(k) from Eqn. (4.31) which allows us to directly calculate the charge current due to a temperature gradient from Eqn. (4.25). By adding the anomalous term, independent of f(k), given by Eqn. (4.26), we can extract the transport coeﬃcient LET directly from the deﬁnition in Eqn. (4.1). Upon doing so, we obtain

Z d3k E − µ(T ) ∂f LET = e v2 τ − 0 (c − D), (4.38) xx (2π)3 x T ∂E x and ! Z d3k E − µ(T ) ∂f LET = −e v2 c + (c − D)v v τ − 0 xy (2π)3 x y y x y T ∂E Z 3 kBe d k − Ωzsk. (4.39) ~ (2π)3 Now in order to find the electrical conductivity LEE, we can go back to Eqn. (4.30) and consider zero temperature gradient and non-zero electric field. It is straightforward to carry out the subsequent derivation to obtain the correction Λ due to the magnetic field. It is straightforward to show that we obtain the same expressions for cx and cy in this case and utilizing Eqn. (4.25), we can find the thermomagnetic tensor LEE to be given by

Z d3k ∂f LEE = e2 v2 τ − 0 (c − D), (4.40) xx (2π)3 x ∂E x and

Z d3k ∂f LEE = −e2 v2 c + (c − D)v v τ − 0 xy (2π)3 x y y x y ∂E e2 Z d3k − Ωzf0, (4.41) ~ (2π)3 where in Eqn. (4.41), we have included the anomalous term due to the Berry curvature from Eqn. (4.26).

78 (A) (B)

Figure 4.2: Magnetic field dependence of Nernst thermopower αxyz, and temperature dependence of Nernst coefficient Nxyz (A) Data for the Nernst thermopower plotted as a function of applied magnetic field at the various temperatures indicated. The insert shows the magnetic field dependence of the Nernst voltage measured at 4.92 K in a temperature gradient of 2.17 mK. SdH oscillations are plainly visible that correspond to the periods measured in the magnetization. (B) Nernst coefficient plotted as a function of temperature, with low-field Nernst effect in red and high-field Nernst effect in blue. The low field curve peaks near 50K; the high-field curve peaks near 90K, which is the temperature at which the chemical potential touches the Weyl nodes. Error bars represent a 95% confidence interval on the standard deviation of the systematic error, excluding geometrical error on the sample mount itself.

79 4.2.1 Numerical Results

We calculate the thermoelectric tensor components given by Eqns. (4.38)-(4.41)

for the case of constant scattering time τ. This approximation will be illustrative

in examining how the temperature dependence of the chemical potential directly

aﬀects the Nernst response. It is clear that if τ is constant, then αxyz will be weakly

dependent of τ, since the prefactors of τ in each of Eqns. (4.38)-(4.41) will cancel

upon division.

In Fig. 4.2c, we plot the temperature-dependence of the Nernst coeﬀcient Nxyz =

dαxyz with constant τ = 100t−1 for the lattice model in Eqn. (3.28) at several ﬁxed dBz

magnetic fields. We have chosen the lattice parameters γ = 0, m = 2t, tz = t, k0a = π/2 and set the Fermi energy to be EF = 0.2t. In Fig. 4.2b, we see that the the temperature dependence of the Nernst coefficient Nxyz (as well as αxyz) has a maximum at slightly less than TW = EF /kB. As the field is increased, the peak moves closer to TW = EF /kB. We emphasize that, since αxyz very weakly dependent on the magnitude of τ when τ is independent of energy, this temperature dependence arises purely from the temperature dependence of the chemical potential discussed in the previous chapter. Specifically, we recall that on a temperature scale TW = EF /kB,

the chemical potential shifts to the node energy.

The magnetic ﬁeld dependence of the Nernst thermopower αxyz is shown in Fig.

4.2d for the same parameters as above. Generally speaking, we see two distinct regions

at low and high ﬁelds were the slope (ie the Nernst coeﬃcient) changes magnitude. In

the literature, the distinct Nernst coeﬃcient at low ﬁeld is known as the anomalous

Nernst coeﬃcient (not to be confused with an anomalous Nernst thermopower as will

be discussed later in this chapter).

80 4.2.2 Analytic Results

The numerical results in the preceding chapter allow us to investigate the Nernst thermopower for a lattice model. As we have seen throughout this thesis, lattice models allow for a robust investigation of Weyl semimetals without complications which arise from choosing a regularization or cutoff of a continuum model. However, we can gain substantial insight from calculating the transport coefficients for a continuum model. Specifically, we take 2 pairs of opposite chirality, isotropic Weyl nodes given by Eqn. (3.6). Using the results of Ref. [69], we can calculate the various thermoelectric transport coefficients of Eqn. (4.1).

From our numerical results, we know that we may approximate the Nernst thermopower as ET Lxy αxyz ≈ EE . (4.42) Lxx In a time-reversal invariant system with only continuum Weyl nodes, we can write

[69] ! ! 2 Z 3 −1 EE e X d k e 2 ∂f0 1 Lxx = 1 + B · Ωχ vxτ − 2 , (4.43) (2π)2 c ∂E (ωcτ) ~ ~ 1 + e 2 χ (1+ B·Ωχ) ~c and

! Z 3 −1 kBe X d k e E − µ(T ) LET = 1 + B · Ω v2τ xy (2π)2 c χ y k T ~ χ ~ B ! 2 ! ∂f0 ωcτ × − 2 e 2 , (4.44) ∂E (ωcτ) + (1 + B · Ωχ) ~c where we have suppressed the band dependence of n = ± for the velocity, energy and

Berry curvature. For an isotropic Weyl node, at E > 0, the cyclotron mass ωc can be

eBv2 F written ωc = cE at energy E = ~vF k. We note that the only chirality dependent

81 Figure 4.3: Nernst coeﬃcient Nxyz as a function of temperature calculated using Eqn. (4.47) using the temperature dependent chemical potential found by solving Eqn. (3.10) self-consistently. We see agreement with the numerical results above and we note that the temperature dependence of Nxyz is independent of τ.

kj term is the Berry curvature, given by Ωj,χ = ±χ k3 . After explicitly performing the summation over band index and chirality, we are left with formidible integrals that cannot be performed analytically.

However, the situation simpliﬁes dramatically in the low B limit. In this case, we obtain that 2 v2 τ 2 EE 4 e F 2 π 2 Lxx ≈ 2 3 µ (T ) + (kBT ) , (4.45) 5(2π) ~ (~vF ) 3 and 2 v2 τ v2 τ ET 4 π kBe F F Lxy ≈ 2 3 2 kBT, (4.46) 5(2π) 3 ~ (~vF ) lB

82 q ~c where lB = eB is the magnetic length. Dividing to obtain αxyz and taking its derivative with respect to B, we obtain ! 2 k v2 π B F kBT Nxyz ≈ τ . (4.47) 3 c 2 π2 2 ~ µ (T ) + 3 (kBT ) We plot the temperature dependence of Eqn. (4.47) in Fig. 4.3 using the temperature dependent chemical potential found by solving Eqn. (3.10) self-consistently. We see agreement with the numerical results above and we note that the temperature dependence of Nxyz is independent of τ.

4.2.3 Nernst Thermopower in Weyl Semimetal NbP

We report the single-crystal Weyl semimetal NbP isothermal Nernst and Seebeck effects with data on samples with a large, unsaturated magnetoresistance and ultrahigh mobilities. NbP breaks inversion symmetry, preserves time-reversal symmetry, and its band structure consists of 24 Weyl points and several trivial pockets [94]. In the low-temperature limit, the electron or hole charges residing in the trivial bands and/or in chemical defects in the samples, including vacancies or unintentional alio- valent impurities, determine the NbP chemical potential, µ0 ≡ EF . The observation that, with increasing temperatures, the electrons/holes in the symmetric Dirac bands dominate transport as the electrochemical potential µ(T ) moves toward the Dirac point is a key finding. The space-charge neutrality condition dictates the electrochemical potential pinning at the Dirac point when the thermally excited carrier density exceeds that of residual carriers. Hall and Seebeck effects are not sensitive to this, since they probe differences between contributions to transport of different- polarity charge carriers, but the Nernst effect is, constituting the second finding of this manuscript.

83 The Nernst eﬀect, a decreasing function of temperature in classical semimetals

like Bi [95] except in the phonon-drag regime, becomes a non-monotonic function of

temperature with a maximum at TW = EF /kB. The Nernst thermopower of charge carriers in Dirac bands exceeds 800 µV/K at 9T, 109K, and does not saturate within our measurement range. This Nernst thermopower is two orders of magnitude larger than the Seebeck coeﬃcient: the isothermal Seebeck coeﬃcient is |α| < 8µV/K under

the same circumstances; the adiabatic Seebeck coeﬃcient is much larger because it

contains a large contribution from the Nernst eﬀect. Indeed, the Seebeck coeﬃcients

arising in the upper and lower parts of the Dirac band are each ascribed a partial

Seebeck coeﬃcient that are odd functions of the charge-carrier polarity; thus, the

total Seebeck coeﬃcient averages two counteracting contributions. In contrast, the

Nernst coeﬃcient is an even function of the carrier polarity, entering once as in the

Seebeck coeﬃcient and again via the Lorentz force. Therefore, in the total Nernst

coeﬃcient, the opposing-polarity charge carrier contributions add, and the Seebeck

effect is smaller than the Nernst effect in magnetic fields ∼ 1T While the inequality

between Nernst and Seebeck coeﬃcients is common to many semimetals, we show that

the non-monotonic temperature dependence is speciﬁc to Dirac bands: for T < TW

the Nernst coeﬃcient will increase with increasing temperature as µ → 0 , then

decrease, as in Bi [95], at T > TW We provide experimental evidence obtained by Jos

Heremans and Sarah Watzman for this and present a quantitative theory to explain

the data.

NbP single crystals are used from the same source as References [94] and [96]. The

single crystals were characterized using resistivity, Hall eﬀect, thermal conductivity,

speciﬁc heat, de Haasvan Alphen oscillations, and Shubnikovde Haas oscillations, with

84 these characterization results found in the Supplementary Material. With a magnetic

field applied along the z-axis, the isothermal Nernst thermopower αxyz is defined as the transverse electric field measured along the crystals y-axis with a temperature gradient and heat flux applied along the x-axis as outlined in the previous section.

The experimental results of the Nernst eﬀect measurements are shown in Figs.

4.2A (magnetic field dependence of the Nernst thermopower αxyz at discrete temperatures) and 4.2B (temperature dependence of the Nernst coefficient Nxyz). αxyz is an odd function of Bz with a higher slope near zero magnetic field than at high

ﬁeld. We see an unsaturating, large Nernst thermopower, with a maximum exceeding

800 µV/K at 9T, 109K, which is 2-4 times larger than the maximum thermopower of conventional, commercial, thermoelectric semiconductors. The Nernst coeﬃcient

Nxyz temperature dependence taken at low field (Bz < 2T) and high field (magnitude between 3 T and 9 T) is non-monotonic, with a maximum around TW < 50K for low-field Nxyz and TW < 90K for high-field Nxyz. Fig. 2 shows the temperature dependence of αxxz, which is almost two orders of magnitude smaller than the Nernst thermopower αxyz (9T) near 100 K, and its absolute value is much smaller than the high-field Nernst effect at all temperatures. The αxxz temperature dependence has a minimum around TW < 50 K. No magnetic-field dependence is observed for αxxz within instrumentation sensitivity, despite repeated attempts. The properties of the present samples Fermi surface exclude phonon drag as a possible source for observed non-monotonicities at temperatures exceeding 14 K [96].

At low temperature (inset of Fig. 4.2A), the Nernst thermopower exhibits Shubnikov- de Haas oscillations. The periods observed in Shubnikov-de Haas and de Haas-van

Alphen oscillations correspond very well, within the error bars of our measurements,

85 even though they were obtained on diﬀerent samples which may have had diﬀerent

defect densities and thus residual doping levels. These results are also similar to those

found by J. Klotz et al. [97]. The de Haas-van Alphen oscillation periods for H par-

allel to all three axes and the corresponding values for the area of the Fermi surfaces

and values for the fermi wavevector kF calculated assuming that the cross-sections are circles. These periods are used in conjunction with the DFT calculations (see

Methods) to derive the position of the electrochemical potential vis--vis the Weyl points, which is determined to be EF ≈ 8.2 ± 2 meV below the main Weyl bands

Dirac point.

We see a an excellent agreement between theory and experiment in Fig. 4.2. In

the numeric calculations, there are essentially two free parameters: EF and τ. The

Fermi energy is set independently by the quantum oscillations described above. The

scattering time is set from the magnitude of the peak from the analytic result in Eqn.

(4.47). Hence, there are no free parameters for the temperature dependence of the

peak. Remarkably, we see quantitative experimental conﬁrmation that the Nernst

thermopower can be used as a probe for the Fermi level in Weyl semimetals.

4.3 Anomalous Transport in Type-II Weyl Semimetals

In the previous section, we saw that the linear dispersion of the Weyl nodes

reveals itself in the Nernst thermopower. However, the model and the experimental

realization in NbP was for an inversion breaking Weyl semimetal with time-reversal

symmetry present. In Chapter 2, we saw that if time-reversal symmetry is not broken,

the Berry curvature Ω(k) must be an odd function of k. Therefore, the “anomalous”

86 contributions from the Berry curvature to heat and charge transport given by Eqns.

(4.26) and (4.28).

Although there have been some preliminary predictions of transport in type-II

Weyl semimetals[50, 98, 99, 100], it remains comparatively less well-understood.

There have been signatures of the chiral anomaly in WTe1.98[101] as well as evidence of viscous electronic and thermal transport in the type-II Weyl semimetal

WP2[102]. There are also strong candidates for type-II time-reversal breaking Weyl semimetals[?, 103] and two such candidates Mn3Sn and Mn3Ge have shown tantalizing

signatures of a large anomalous Hall eﬀect[104, 105, 106]. Furthermore, experimental

signatures of the anomalous Nernst eﬀect and anomalous thermal Hall eﬀect have

also been detected in Mn3Sn[107].

Ferromagnetic metals are known to possess anomalous transport coeﬃcients in

zero ﬁeld[108, 109, 110], however Mn3(Ge,Sn) is instead a weakly canted antiferro-

magnet. It has been suggested that the real-space magnetic texture can account for

a large anomalous Hall eﬀect in Mn3Ge if the spins are non-coplanar[111], however

experiments have shown that the Mn3(Ge,Sn) system does possess a large anomalous

Hall eﬀect in the planar magnetic phase with a Hall coeﬃcient that is much larger

than its weakly canted moment would suggest[104]. Thus many puzzles remain.

Motivated in part by some of these experimental puzzles, we study anomalous

transport in a lattice model of a time-reversal breaking Weyl semimetal[112]. The

model we use allows for tuning through the type-I to type-II transition as well as

between diﬀerent type-II phases with distinct Fermi surface connectivities. Although

the Berry curvature of a Weyl node is independent of its type, the occupation of

87 states immediately surrounding the Weyl nodes is strongly dependent on the pre-

cise Fermiology of the material. Furthermore, the chemical potential in low density

systems, such as semimetals, is a strong function of temperature[113]. A detailed

understanding of the interplay between the tilt of the nodes, the connectivity of the

Fermi pockets, and the temperature dependence of the chemical potential is required

for a complete theoretical picture of anomalous transport in type-II Weyl semimetals.

One of the central results of our work is the strong enhancement of the transverse

ET thermoelectric transport coeﬃcient Lxy with temperature and with increasing nodal tilt. This term is zero in the strictly type-I limit, but we ﬁnd that by increasing

ET the tilt of the Weyl nodes in the type-II regime, Lxy increases due to the change

ET in the net Berry curvature of occupied states. We also ﬁnd that Lxy is sensitive to many of the Lifshitz transitions that occur throughout the type-II Weyl semimetal regime. This has broad implications for the possible utilization of Weyl semimetals in thermoelectric applications.

4.3.1 Model

We have seen earlier in this Chapter that the continuum model for a type-II Weyl fermion given by Eqn. (3.23) is manifestly unphysical due to the unbounded nature of the electron and hole pockets. Eqn. (3.23) describes an electron and hole pocket that are unbounded and never close for large k. Since various aspects of type-II

Weyl semimetals are strongly dependent on the nature of the extended Fermi pockets comprising the Weyl point at the Weyl energy[112] as we saw in the previous chapter, it is necessary to consider a lattice model. Speciﬁcally, we choose the “Helium atom model” from the previous chapter, but rotate the axes such that the node separation

88 is in the z-direction:

ˆ H = γ(cos(kza) − cos(k0a))ˆσ0 − 2t sin(kxa)ˆσ1 h − 2t sin(kya)ˆσ2 − 2tz cos(kza) − cos(k0a) i + m(2 − cos(kxa) − cos(kya)) + γz(cos(3kza) − cos(3k0a)) σˆ3 (4.48)

where again σ0 is the 2×2 identity matrix,σ ˆj is the j-th Pauli matrix, a is the lattice spacing, t, m, γz and tz are hopping amplitudes, k0 sets the node separation, and γ sets the tilt of the Weyl nodes. We have seen that this model supports two sets of electron and hole pockets, with each Weyl point being comprised of a separate pair of electron and hole pockets. We recall that the type-I to type-II Lifshitz transition happens at γ = 2tz − 3γz.

Lifshitz Transitions

In Fig. 4.4, we plot the energy band structure given by Eqn. (4.48) for the parameters m = 3t, tz = t, k0a = π/2, and γz = 0.5t for several values of γ. For these model parameters, the type-I to type-II transition occurs at γ = 0.5t. The

Fermi surfaces of the initial type-II band structure are shown in Fig. 4.4f, where we see a pair of electron pockets (red) and hole pockets (blue) meeting at a pair of Weyl points. As γ increases, each of these pockets grow in size.

There is also a second set of topological Lifshitz transitions where the separate electron and hole pockets each merge into a single pocket. In Fig. 4.4g, we see that the electron pockets have merged across the kz = π plane at γ ≈ 2t and a hole pocket within the electron pocket also forms for a small range of parameters within the electron pocket. For the model parameters above, the hole pockets similarly merge into a single Fermi pocket across the kz = 0 plane at γ ≈ 2.5t. The resulting

89 =0 =1.2t =2t =2.8t (a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4.4: Cuts through the band structure given by the Hamiltonian in Eqn. (4.48). In (a-d), we show energy versus kz cuts for kx = ky = 0. Here we have chosen m = 3t; tz = t; k0a = π/2; γz = 0.5t for γ = 0 (a), γ = 1.2t (b), γ = 2t (c), and γ = 2.8t (d). In (e-h), we show constant energy cuts for the band structure deﬁned by Eqn. (4.48). (a) and (e) are in the type-I limit; (b) and (f) are in the type-II limit with distinct pockets making up each nodes; (c) and (g) are in the type-II limit after the electron pockets have merged; and (d) and (h) are in the type-II regime where the Weyl nodes share only a single electron and single hole pocket. Thus, as γ is increased we can see the successive Lifshitz transitions described in the text.

90 band structure deeper in the type-II limit is shown in Fig. 4.4h. The tunability

of this model between these four diﬀerent regimes makes it the minimal model to

understand type-II Weyl semimetals. We will show how each regime has its own

signature in anomalous transport.

Berry Curvature

We can recall that we may rewrite the Hamiltonian in Eqn. (4.48) as Hˆ =

d0(k)ˆσ0 + d(k) · σ and upon doing so we recall that the Berry curvature of the n-th band is given by [114]

d · ∂ d × ∂ d Ω (k) = (−1)n kj kl , (4.49) n,i ijl 2|d|3

where ijl is the perfectly antisymmetric tensor. The Berry curvature plays a central

role in the theory of anomalous transport. The parameter γ which determines the

tilt, and therefore the type of the Weyl nodes, is embedded in d0(k), which does not

enter Eqn. (4.49). We note that due to the extended nature of the pockets in type-II

Weyl semimetals, the anomalous transport coeﬃcients are strongly dependent on the

details of the Fermiology.

We can better understand the interplay between band Fermiology and the Berry

curvature by introducing what we call the net Berry curvature deﬁned by

Z X dS Ωn,z(k) Ωnet(E) = , (4.50) z (2π)3 |∇ E| n k

where E is the energy and the integral in Eqn. (4.50) is over surfaces of constant

net energy in the Brillouin zone. We plot Ωz (E) as a function of energy for various values of γ in Fig. 4.5a-d for the model given by Eqn. (4.48). We see that, for γ = 0,

net net Ωz (E) is a strictly odd function of energy. However, as γ increases, Ωz (E) has the

91 same sign for positive and negative energies close to E = 0. As the electron pockets

merge in Fig. 4.5c,g,k, we see that the net Berry curvature peaks. This interplay of

Berry curvature and node tilt has a strong eﬀect on zero-ﬁeld transverse transport.

Temperature dependence of the chemical potential

In metals with high densities of electrons at degenerate temperatures, the chemical

potential is nearly constant with respect to temperature. However, in low density

semimetals, it has been shown that the chemical potential has a strong temperature

dependence at experimentally relevant temperatures[113]. For the lattice model in

Eqn. (4.48), we calculate the temperature dependence of the chemical potential by

self-consistently solving for µ(T ) for ﬁxed density as we did in Chapter 3 for the

simple model of Weyl semimetals:

Z ∞ g(E) n = dE E−µ(T ) , (4.51) −∞ 1 + e kB T where n is the density, T is the temperature and g(E) is the density of states found

through " # 1 X Z d3k g(E) = − Im G (k, E) , (4.52) π (2π)3 n n

where Gn(k, E) is the Green function of the n-th band.

For isolated type-I Weyl points, the minimum of g(E) will generically occur at the

Weyl points[113] due to the symmetry of the particle and hole bands. This results

in the chemical potential shifting to the Weyl points with increasing temperature in

type-I Weyl semimetals. However, in general, the hole and electron pockets are not

symmetric about the Weyl energy in type-II Weyl semimetals, and the minimum of

g(E) will occur above or below the Weyl nodes. In Fig. 4.5e-h we plot the density of

states g(E) of the model in Eqn. (4.48) for several values of the tilt parameter γ at

92 =0 =1.2t =2t =2.8t (a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Figure 4.5: Each column corresponds to a particular γ with: γ = 0 (a, e, i), γ = 1.2t (b, f, j), γ = 2t (c, g, k), and γ = 2.8t (d, h, l). (a-d) show the net Berry curvature net in the z-direction Ωz (E) defined by Eqn. (4.50). We see that for nonzero γ, the net Berry curvature around the nodes is of the same sign. (e-h) the density of states for m = 3t, tz = t, k0a = π/2, and γz = 0.5t, for different values of the tilt parameter. (i- l) illustrates the temperature dependence of the chemical potential, µ(T ). Each plot shows three separate values of EF : EF = 0 (blue), EF = 0.1 (green), and EF = 0.2 (red). We see that for smaller values of γ, g(E) has a minimum close to the Weyl energy E = 0, but for larger values of γ, this minimum shifts far from the nodes. This has a strong effect on the shift of the chemical potential with temperature.

93 energies around the node energy. For γ = 0, we see the density of states is minimum at the Weyl energy. However, for larger values of γ, in the type-II regime, the tilt of the Weyl cones breaks particle hole symmetry and shifts the minimum of g(E) away from the Weyl energy. In the large γ limit where both the electron and hole pockets merge (Fig. 4.5h), the minimum of the density of states shifts far from the Weyl energy. In Fig. 4.5i-l we show how the chemical potential evolves with temperature for each value of γ. For low values of γ in Fig. 4.5i and 4.5j, we see that the chemical

potential shifts to the Weyl energy roughly on a temperature scale of the distance EF

is away from the energy for which g(E) is minimized. However, as the nodes become

increasingly tilted, the temperature scale over which µ(T ) shifts becomes much larger

than the relevant scales in transport that we will consider.

4.3.2 Anomalous Transport

In Eqn. (4.1), we deﬁned the transport coeﬃcients in terms of the currents and

EE ET TT applied fields. We wish to calculate the transverse coefficients Lxy , Lxy , and Lxy in the absence of an external magnetic field. As we saw earlier in this chapter, in this case

the Berry curvature is solely responsible for transverse transport. We have seen that,

in general, the transport coeﬃcients in Eqn. (4.1) must be obtained by solving for the

non-equilibrium distribution function using the Boltzmann formalism. However, in

the absence of a a magnetic ﬁeld, the solution for the transverse components simpliﬁes

dramatically. n this case, the modiﬁed velocity of the n-th band is found to be

e ˙rn = vn(k) + E × Ωn(k) , (4.53) ~

1 e where vn(k) ≡ ∇kEn(k) is the usual group velocity and where E × Ωn(k) is the ~ ~ anomalous velocity due to the Berry curvature[115].

94 In this section, we consider anomalous transport in a two band lattice model for the type-II time reversal breaking Weyl semimetal described in Section the previous

EE subsection. We calculate the anomalous Hall coeﬃcient Lxy , the anomalous trans-

ET TT verse thermoelectric coeﬃcient Lxy , and the anomalous thermal Hall coeﬃcient Lxy .

Anomalous Hall Eﬀect

The anomalous Hall eﬀect has long been studied in the context of Weyl semimetals. For a time-reversal breaking Weyl semimetal, it was shown that the anomalous

EE Hall conductivity Lxy was directly proportional to the net separation between Weyl nodes[?]. In the presence of Berry curvature, we have seen that the equations of motion are modiﬁed and as a result[116, 117, 115, 109], the anomalous Hall coeﬃcient takes the following form

e2 X Z d3k LEE = Ω (k)f (k), (4.54) xy (2π)3 n,z 0 ~ n

−1 E(k)−µ k T where f0(k) = 1 + e B is the equilibrium Fermi-Dirac distribution. Since f0(k) only depends on k through the energy E, we can write Eqn. (4.54) in terms of

net Ωz (E) in the following way

2 Z EE e net Lxy = dE Ωz (E)f0(E). (4.55) ~

The anomalous Hall coeﬃcient in Eqn. (4.54) integrates the Berry curvature over all ﬁlled states. In the un-tilted regime (γ = 0), Eqn. (4.48) will have particle-hole symmetry and since, from Eqn. (4.49), the Berry curvature is opposite in sign but equal in magnitude at a particular point k for two bands. Therefore, equal energy

Fermi surfaces at positive and negative energies will have opposite net integrated

net Berry curvature Ωz (E). Hence, at γ = 0 and EF = 0, the ﬁlled states all contribute

95 (a) (b) (c)

(d) (e) (f)

EE ET TT Figure 4.6: In (a-c), we plot each anomalous transport coeﬃcient Lxy , Lxy , and Lxy for the Hamiltonian given by Eqn. (4.48) with parameters m = 3t, tz = t, k0a = π/2, and γz = 0.5t, as a function of γ for the following temperatures: T = 0.05t (purple), EE T = 0.1t (blue), T = 0.15t (green), and T = 0.2t (red). In (d-f), we show Lxy , ET TT Lxy , and Lxy for the same values as in (a)-(c), with γ = 1.2t, plotted as functions of temperature for various Fermi energies: EF = 0 (black), EF = 0.1t (magenta), and EF = 0.2t (blue).

96 EE to one sign of Lxy . As the tilt γ is increased, the Fermi surfaces become increas-

ingly asymmetric, and the distributions of Ωn,z(k) over the occupied states undergoes

substantial change, as shown in Fig. 4.5a-d.

EE In Fig. 4.6a, we plot the anomalous Hall coeﬃcient Lxy as a function of the tilt

parameter γ for several ﬁxed temperatures at EF = 0. Other than a small increase

EE in Lxy for γ < 1.5t for small temperatures, we note that the curves are substantially

EE similar. We can understand the increase of Lxy by noting that the asymmetry in the Berry curvature distribution enhances the Hall coefficient for small temperatures but that this effect is washed out as the temperature increases and the Fermi function broadens. When the separate electron pockets comprising the Weyl nodes merge into a single electron pocket at γ = 2t, we see that the Hall coefficient uniformly decreases for all temperatures.

In Fig. 4.6d, we show the temperature dependence of the anomalous Hall coeﬃ-

EE cient Lxy as a function of temperature for ﬁxed γ = 1.2t for various Fermi energies

EE EF . We see that for EF = 0, Lxy decreases monotonically as a function of temperature. This is because as the temperature is raised, the Fermi function broadens and electronic states with the opposite sign of Ωn,z(k) begin to become occupied. This is true for the type-I case as well. However, for EF 6= 0, the Hall coeﬃcient will in general be lower than its EF = 0 value at T ≈ 0. For EF > 0 in Fig. 4.6d, we see

EE that Lxy attains a maximum at T > 0. This is due to the movement of the chemical potential with temperature (see Fig. 4.5i-l). As the chemical potential crosses the

Weyl energy, a maximum is attained in the anomalous Hall coefficient as a function of temperature. This effect is seen to a lesser extent in the type-I case, however there are two distinct differences in the type-II regime:

97 (i) In the type-I case, µ(T ) always shifts to the node energy. However, in the type-

II case, the minimum of the density of states g(E) occurs generically at a diﬀerent

energy, causing µ(T ) to cross the Weyl energy rather than approach it asymptotically.

EE This leads to a sharper rise of Lxy with temperature. (ii) Due to the higher density of states around the nodes, where the Berry curvature is

stronger, the anomalous Hall coeﬃcient is more sensitive to the shift of the chemical

potential.

At higher temperatures, the chemical potential reaches the energy where the den-

EE sity of states is minimized and the temperature dependence of Lxy is uniform across

each value of EF . Although here we have only shown the results for a single value of

γ, the broad conclusions that we have drawn remain true. There is a peak in temperature where µ(T ) crosses the node energy. In the type-II regime, this increase over the T = 0 value is, in general, greater than that in the type-I regime for other similar model parameters. However, we note that for values of γ such that the electron (or

the hole pockets) have merged, the temperature dependence becomes much weaker.

Anomalous Transverse Thermoelectric Eﬀect

In the presence of Berry curvature, the electronic wavefunction acquires an orbital

magnetization that is responsible for nontrivial anomalous thermoelectric properties

[88]. It is found that in this case, the anomalous transverse thermoelectric coeﬃcient

is given by Z 3 kBe X d k LET = Ω (k)s(k), (4.56) xy (2π)3 n,z ~ n where s(k) is the electronic entropy given by

s(k) = −f0(k) ln(f0(k)) − (1 − f0(k)) ln(1 − f0(k)), (4.57)

98 and where f0(k) is the equilibrium Fermi-Dirac distribution deﬁned above. The entropy, as deﬁned in Eqn. (4.57), is sharply peaked around µ(T ). Therefore, unlike

ET the anomalous Hall coeﬃcient, the anomalous thermoelectric coeﬃcient Lxy is only

EE sensitive to the Berry curvature around the chemical potential. As in the case of Lxy

net above, we can also write Eqn. (4.56) in terms of Ωz (E) since s(k) is also only a function of momentum through the energy. Doing so, we obtain

Z ET kBe net Lxy = dE Ωz (E)s(E). (4.58) ~

The tilt γ of the model for a type-II Weyl semimetal can lead to a change in the

net distribution of net Berry curvature that is large enough to ﬂip the sign of Ωz (E) above or below the Weyl node in energy, for energies small compared with the bandwidth

E t, as shown in Fig. 4.5b-d. Hence, at energies just above and below the

net Weyl energy, Ωz (E) has the same sign for a tilted Weyl cone. This leads to a large enhancement of the anomalous thermoelectric coeﬃcient for energies around

ET the node. We see this in Fig. 4.6b, where Lxy is plotted as a function of γ for various

ET temperatures. In the type-I case, Lxy is small in magnitude and positive in sign. After the Lifshitz transition to the type-II regime at γ = 0.5t, we see a change in sign

ET net of Lxy and substantial increase in its magnitude. This is precisely because Ωz (E)

ET has the same sign and a large value. The anomalous thermoelectric coeﬃcient Lxy attains a maximum at γ ≈ 2.3t after the electron pockets have merged and just as the hole pockets are about to merge (with increasing γ). Hence, we see that measurable

ET quantities, such as the Nernst eﬀect, which depend on Lxy may be quite sensitive to Lifshitz transitions between various regimes of Fermi pocket connectivity in type-II

Weyl semimetals.

99 Sensitive only near µ Enhanced by γ Enhanced with T EF dependence EE Lxy No No For T . EF Yes ET Lxy Yes Yes For T . EF Yes TT Lxy No No Yes No

Table 4.1: Summary of properties of anomalous transport coeﬃcients in type II Weyl semimetals.

ET In Fig. 4.6e, we plot the anomalous thermoelectric coeﬃcient Lxy as a function of temperature for various Fermi energies EF at γ = 1.2t. Here we see that there is an

ET increase with temperature in the magnitude of Lxy for small T . This occurs because,

ET as the entropy s(k) broadens, Lxy is enhanced from the entropy including a larger range of energies in the integral in Eqn. (4.56). At a temperature on the order of

ET the energy for which the density of states g(E) attains its minimum, Lxy reaches its maximum absolute value. For Fermi energies EF farther from the node energy, we see

ET that the maximum absolute value of Lxy increases. This occurs for the same reason as in the anomalous Hall coeﬃcient above; for a type-II Weyl semimetal the chemical

net potential µ(T ) will pass over the Weyl energy where Ωz (E) is enhanced. Finally, we also note that since the Berry curvature is symmetric in magnitude but opposite in sign for an un-tilted type-I Weyl node, as shown in Fig. 4.5a, at

ET ET exactly EF = 0, Lxy must vanish. Hence, at precisely EF = 0, Lxy can only take a nonzero value for a type-II Weyl semimetal where the shifted occupation of states can lead to a nonzero net Berry curvature when Eqn. (4.56) is evaluated at the node energy.

100 Anomalous Thermal Hall Eﬀect

As in the anomalous transverse thermoelectric effect above, similar effects of orbital magnetization in the presence of Berry curvature also lead to an anomalous thermal Hall effect in the absence of magnetic field[118, 119, 120]. In the presence of

Berry curvature, the anomalous thermal Hall coeﬃcient is given by

k2 T X Z d3k π2 (E − µ)2 LTT = B Ω (k) + f (k) xy (2π)3 n,z 3 (k T )2 0 ~ n B ! (4.59) E−µ(T ) − k T −ln 1 + e B + 2Li2 1 − f0(k) ,

where f0(k) is again the equilibrium Fermi-Dirac distribution and where Lim(z) is the polylogarithm function of order m deﬁned by

∞ X zk Li (z) = . (4.60) m km k=1

Recognizing that, aside from Ωn,z(k), the integrand of Eqn. (4.59) only depends on the energy, we can write

2 Z 2 2 TT kBT net π (E − µ) Lxy = dE Ωz (E) + 2 f0(E) ~ 3 (kBT ) ! (4.61) E−µ(T ) − k T −ln 1 + e B + 2Li2 1 − f0(E)

Like the anomalous Hall coefficient, the anomalous thermal Hall coefficient defined by Eqn. (4.59) integrates over many states below the chemical potential. However, unlike the simple Fermi distribution f0(k), the kernel of the integrand multiplying

Ωn,z(k) in Eqn. (4.59) has a broader inﬂection point at E = µ(T ) than f0(k) and

TT EE hence, Lxy is sensitive to Berry curvature over a wider range than Lxy . In Fig. 4.6c,

TT we plot Lxy as a function of γ and we see that it is quite similar with the anomalous Hall coeﬃcient plotted in Fig. 4.6a. However, we note that the anomalous thermal

101 Hall coeﬃcient increases with increasing γ for all temperatures up to the Lifshitz transition where the electron pockets merge. We also see that after the Fermi pockets

TT EE do merge, the decrease of Lxy is less rapid than Lxy with increasing γ. We also note

TT that the temperature dependence of Lxy is increasing rather than decreasing as in

EE TT Lxy at large T . Finally, the dependence of Lxy on EF is essentially negligible due to the broadened kernel in the integrand in Eqn. (4.59).

4.3.3 Relation to Measurable Quantities

In the previous section, we studied transverse transport in zero magnetic ﬁeld

EE ET TT and calculated the transport coeﬃcients Lxy , Lxy , and Lxy . From Eqn. (4.1), we

EE immediately recognize Lxx as simply the electrical conductivity. Unders isothermal conditions, the Nernst eﬀect is deﬁned by

Ey αxyz = , (4.62) −∇xT under the conditions

Jx = Jy = 0 (4.63)

and

∇yT = 0. (4.64)

From Eqn. (4.1), we can ﬁnd αxyz to be given by

EE ET EE ET Ey Lxx Lxy − Lxy Lxx αxyz = = EE 2 EE 2 . (4.65) −∇xT (Lxx ) + (Lxy )

EE We note that Eqn. (4.65) depends not only on the transverse coeﬃcients Lxy and

ET EE ET Lxy , but also the longitudinal coeﬃcients Lxx and Lxx . We can ﬁnd the transverse thermal conductivity to be

ET 2 ET 2 EE EE ET ET TT (Lxy ) − (Lxx ) ) Lxy + 2Lxx Lxy Lxx κxy = Lxy − T EE 2 EE 2 . (4.66) (Lxx ) + (Lxy ) 102 Similar to the Nernst thermopower given above, we note that the thermal conductivity

TT κxy depends on Lxy as well as the second, explicitly temperature dependent term which we recognize as the ambipolar thermal conductivity that originates from a

Seebeck eﬀect generated by a Peltier current.

So far, we have only considered the transverse transport coeﬃcients in zero applied

αβ external magnetic ﬁeld. However, in general, each Lij will be nonzero. Furthermore all transverse coeﬃcients will have additional contributions at nonzero applied mag-

netic ﬁeld. Full expressions for the transport coeﬃcients of lattice models for Weyl

semimetals can be found in Ref. [70]. All contributions other than the anomalous

terms discussed in the previous section will depend explicitly on the transport scat-

tering time. From Matthiessen’s rule, the scattering rate from independent processes

can be written as a sum of the scattering rates of each process. We expect that

impurity scattering, electron-electron scattering, and phonon-induced scattering will

all play a role in general. The various contributions to the scattering time in Weyl

semimetals has been investigated previously[121, 122, 123, 124], however the nature

of scattering in type-II Weyl semimetals remains an open problem. We have limited

the scope of our work to the Berry curvature induced anomalous transport and so we

leave the eﬀects of scattering for future investigations.

4.3.4 Discussion and Conclusion

We have calculated how the anomalous transport coeﬃcients reveal signatures of

type-II Weyl semimetals. Our results are summarized in Table 4.1. In particular, the

ET anomalous transverse thermoelectric coeﬃcient Lxy is greatly enhanced by tilting the Weyl nodes. However, we have also seen that even deeper in the type-II phase,

103 the various anomalous coeﬃcients are sensitive to changes in Fermi surface topology

EE TT even quite far from the Weyl nodes. Each of Lxy and Lxy show a marked decrease

ET in magnitude as the electron pockets merge, while Lxy peaks just after they merge

ET and Lxy decreases for γ large enough that the hole pockets similarly merge. Previous calculations have shown[112] that tuning through the various Lifshitz transitions within the type-II Weyl semimetal phase is associated with changing connectivities of the topological Fermi arcs. The bulk Fermi surface Berry curvature is deeply linked to the topology of the bulk band structure, and it is enlightening to see the anomalous transport coeﬃcients reﬂect this bulk-boundary correspondence.

In some inversion-breaking Weyl semimetals[80], it has been shown that strain may be possible to tune between a type-I to type-II phase transition. Although this has not yet been demonstrated in a time-reversal breaking Weyl semimetal, it should be possible in principle. This would allow for an experimental veriﬁcation of our results as well as allowing for deeper explorations of connections between anomalous transport and changes in Fermi arc topology. We note that the coeﬃcients that we have calculated in this work are manifestly independent of scattering time. However, the experimentally measurable quantities, such as the Nernst thermopower, will necessar-

EE ET ily involve the longitudinal transport coeﬃcients, such as Lxx and Lxx which depend on scattering times.

We have also seen that the temperature dependence of the anomalous transport

EE ET TT coefficients is quite distinct for each of Lxy , Lxy , and Lxy . The temperature dependence of the chemical potential has already been demonstrated to lie at the heart of the ordinary Nernst thermopower in the Weyl semimetal NbP[113]. However, similar effects in zero field anomalous transport have remained, until this point, unexplored.

104 We have demonstrated that, through the location of the Fermi energy EF , the tem-

EE perature at which Lxy attains its maximum can be tuned, as can the strength of the

ET anomalous thermoelectric coefficient Lxy . The generation of a transverse response to an applied electric field or thermal gradient unifies the zoo of the various Hall effects. The ability to generate such a response in the absence of an externally applied magnetic field opens the door to a wide variety of technological applications. Weyl semimetals have been predicted to generate various anomalous transport phenomena in zero field due to their Berry curvature. However, the lack of experimental realizations of time-reversal breaking

Weyl semimetals have stymied their application. Recently it has been proposed that time-reversal breaking Weyl semimetal candidates are type-II in nature. Our calculations serve not only to guide future experiments but also demonstrate that through changing various experimentally accessible properties, it may be possible to tune the anomalous transport coeﬃcients in type-II weyl semimetal to an extent not possible in the type-I case.

105 Chapter 5: Fermi Arc-Mediated Entropy Transport in Weyl Semimetals

In addition to the plethora of novel transport phenomena exhibited by WSMs, these monopole charges of Berry curvature are responsible for topological Fermi arcs in WSMs, perhaps their most fascinating feature. Fermi arcs form open contours of surface states that terminate on the projections of Weyl nodes, or in the case of doped

WSMs, terminate on the projections of Fermi pockets enclosing Weyl nodes. The dispersion around the bulk nodes is linear in three dimensions. The Fermi arc states on the surface have a linear dispersion in the two dimensional surface Brillouin zone and are chiral, so that the velocity of electrons on one of the surfaces is unidirectional and is opposite on the other surface. These gapless surface modes provide the key signature for WSMs in spectroscopy experiments.

Fermi arcs in WSMs are known to lead to exotic quantum oscillations involving mixed real and momentum space orbits[27, 125, 28, 126] as well as resonant transparency[127]. However, the eﬀect of Fermi arcs on thermal transport so far remains an unexplored frontier. Preliminary studies of thermal transport in WSMs have so far only considered contributions from the bulk Weyl fermions[69, 70, 128,

129, 129, 130]. One of the major challenges we address in this chapter is: Can we

ﬁnd signatures of Fermi arcs in thermal transport? How can we design an experiment

106 that isolates the eﬀect of the arcs from all of the other contributions, such as the bulk monopoles, ambipolar transport from electron and holes around the Weyl points, and all of the other trivial pockets?

We predict that Fermi arc-mediated entropy transport and consequently the anisotropic magnetothermal conductivity discussed in this paper provides a unique signature of the topological Fermi arcs in WSMs. Speciﬁcally we obtain the following results:

(i) Even when transport in a magnetic ﬁeld is incoherent i.e. ωcτ 1 where ωc is the cyclotron frequency and τ is the elastic intra-valley scattering time, it is possible to get coherent entropy transport in an applied thermal gradient without any charge transport. The charge circulates like a ﬂuid in a “conveyor belt” from the Fermi arc on the top surface, into the bulk via the Weyl node, down to the arc on the other surface and back up to the top surface via the second Weyl node. The precise geometry is described below.

(ii) Using the Landauer formalism we calculate the thermal and charge conductance of the “conveyor belt” channel in the semiclassical and ultra-quantum regimes and show that in both regimes the arc-mediated conductances dominate over the bulk contributions. We find that the Wiedemann-Franz law is obeyed in both regimes, albeit with different Lorenz numbers, because of the different density of states. We note that although there is heat transport in the absence of charge transport, the

Wiedemann-Franz law is obeyed in the “conveyor belt” mechanism. This is strikingly diﬀerent in materials with strong correlations where separation of electronic degrees carrying heat and charge leads to the expected breakdown of the Wiedemann-Franz law[131, 132, 133, 134, 135, 136].

107 3 (iii) In the low-ﬁeld regime, the thermal conductivity κzzz ∝ T BL [see Eqn. (5.11) for the exact expression]. Note, the three directions in the tensor κ pertain to the thermal gradient, the heat current and the magnetic ﬁeld, all taken to be along z and perpendicular to the separation between Weyl points along x. In the ultra-quantum

Q 2 Q limit, κzzz ∝ GthB L [see Eqn. (5.21) for full expression], where Gth is the quantum of thermal conductance[137, 138] as deﬁned in the sections below.

(a) (b)

Figure 5.1: (a) Mixed real-space and momentum-space depiction of a Weyl semimetal in a slab geometry with thickness L in the z-direction. Bulk Weyl nodes are labeled with their chirality χ = ± and separated in the kx-direction. The projections of the Weyl nodes on the surface Brillouin zone define the end points of the Fermi arcs on the top and bottom surfaces. (b) Schematic of the “conveyor belt” motion of charge leading to a net heat flux. When B is aligned in the z-direction in (a), charge current density circulates Je in the clockwise direction shown in a mixed real and momentum space orbit. When ∇T is also aligned in the z-direction, this circulation of charge leads to a net flow of heat current density JQ in the direction shown. Unit tangent vectors et are shown for the arcs.

To put our results in perspective, all semimetals have ambipolar (i.e. electron and hole) contributions to thermal conductivity[139]. As an example, in bismuth ambipolar conduction contributes 40% of the total thermal conductivity at T = 200K[140].

108 However, we emphasize that ambipolar conductivity is a Peltier current which itself is driven by a Seebeck eﬀect and does not involve the conveyor belt motion that we propose here. The conveyor belt motion of entropy that we propose is a topological property of the Fermi arcs of Weyl and Dirac semimetals. Its driving mechanism is a combination of the Lorentz force from the magnetic ﬁeld on the Fermi arcs and from conservation of charge which necessitates motion through the bulk.

In the low-field semiclassical limit, it is shown that a Lorentz force on the electrons comprising the Fermi arcs necessarily leads to a bulk flow of electrons, carrying a net energy current in the presence of a temperature gradient. When the field strength reaches the ultra-quantum limit, we find that the electrons from the arcs can only hybridize with the chiral Landau levels in the bulk. In the intermediate quantum regime, where several quantized Landau levels become involved in Fermi arc-mediated magnetothermal transport, we calculate the specific heat and the magnetothermal conductivity for various temperatures and magnetic fields.

We conclude by comparing the bulk contribution to the thermal conductivity with the arc-mediated contribution and show that the latter dominates. We discuss the role of elastic, intra and intervalley, and inelastic scattering as well as phonon drag and argue that because of the limited phase space, the “conveyor belt” mechanism for entropy transport is robust. Our results have broad experimental implications for both type I and type II WSMs as well as Dirac semimetals. In particular, our predictions provide possibly the most deﬁnitive transport signature of Fermi arcs in topological semimetals. This work also sets the stage for utilizing and manipulating topological Fermi arcs in experimental magnetothermal applications and in novel tools for thermal energy conversion technology, such as magnetically driven heat switches.

109 5.1 Fermi arc-mediated magnetothermal transport

5.1.1 Model

We consider a linearized model of Weyl fermions in the continuum limit. For a given Weyl node of chirality χ = ±1, the Hamiltonian near the node is given by:

ˆ Hχ = χ~vF kxσˆx + kyσˆy + kzσˆz , (5.1)

where vF is the Fermi velocity of the Weyl electrons and the Pauli matricesσ ˆj span either spin or orbital degree of freedom. These Weyl nodes come in pairs of opposite chirality; we consider Np pairs of nodes. We take all Weyl nodes to lie at the same energy E = 0, but our calculations generalize to cases where sets of Weyl nodes lie at diﬀerent energies.

For certain parts of the analysis discussed below, it is useful to consider the following lattice model[82, 112] for a Weyl semimetal HLatt (k) = − m(2 − cos(kya) − cos(kza))+ 2tx(cos(kxa) − cos(kW a)) σˆx (5.2)

−2t sin(kya)ˆσy − 2t sin(kza)ˆσz, where a is the lattice spacing; kW sets the separation of the Weyl nodes; and t, tz and m are other parameters which determine the exact dispersion at and away from the Weyl nodes. The lattice model in Eqn. (5.2) reduces to the linearized model in

Eqn. (5.1) for momenta below a cutoﬀ Λ. We consider a slab geometry by Fourier transforming Eqn. (5.2) in the z-direction. This lattice model proves to be useful in understanding the heat ﬂow in the mixed real- and momentum-space description of the WSM.

110 5.1.2 Semiclassical Regime

In this section, we calculate the heat current ﬂow in response to a thermal gradient

and an externally applied magnetic ﬁeld whose coeﬃcient gives the magnetothermal

Q conductivity Jµ = κµνλ∇νT and λ defines the direction of the magnetic field. In particular, we focus on the Fermi arc mediated entropy transport. We show that this heat current depends on the particle flux from the arcs through the bulk driven by a magnetic field and on the heat capacity of the bulk states. The particle current through the bulk is calculated from the continuity of charge and the Lorentz force of an external magnetic field on the arcs. We show that the particle flux results in no net charge transport, but produces a heat current in the presence of a thermal gradient.

Our results for the thermal and electronic conductances can be cast transparently in the Landauer framework and show that the Wiedemann-Franz law is obeyed.

We consider a Weyl semimetal with Np pairs of Weyl nodes in a slab geometry

with the dispersion of the Fermi arcs in the surface Brillouin zone (kx,ky). In the

presence of a perpendicular magnetic ﬁeld B = Bez, a wavepacket of electrons on a

Fermi arc follows the trajectory governed by the Lorentz force,

dk e e = (vk × B) = vF Bet, (5.3) dt ~c ~c

where et is the unit tangent vector to the Fermi arc, and we have assumed for simplicity that the magnitude of the velocity vF of the linearly dispersing Fermi arc states at diﬀerent points along the arc is the same. The Lorentz force on the arcs has been predicted[27] to give rise to unusual Fermi arc-mediated quantum oscillations in Weyl semimetals, but similar eﬀects remain unexplored in thermal transport.

111 (a) (b)

Top

Bulk

Bottom

Figure 5.2: (a) Thermal occupations for the minimal model in Eqn. (5.2) with m = 2t, tz = t, and kW a = π/2 for cuts along ky at ﬁxed kxa = 0.75π (top and bottom) and kxa = 0.5π (bulk). We show the occupations of the top two layers (top), the bottom two layers (bottom) and the other layers (bulk), weighting each point’s color and thickness in the ﬁgure with the Fermi function f0 at that layer for a temperature dT 0.8t gradient of = at an average temperature of kBT = 0.6t. (b) Orbits for the dz kB a minimal model in Eqn. (5.2) with the same parameters as in (a) for cuts along kx at ky = 0. We again show the occupations of the top two layers (top), the bottom two layers (bottom) and the other layers (bulk), weighting each point as in (a) with the same temperature and temperature gradient. We show the mixed real- and momentum-space orbits and see that the the states most occupied on the top and bottom are the arcs, while the bulk states merge into the arc and carry heat through the bulk.

112 We next consider the collective motion of electrons comprising the Fermi arcs.

The change in the number N of electrons on a given surface with one Fermi arc is

dN dn dE dk = A , (5.4) dt dE dk dt

dn k0 where A is the area of the surface on which the Fermi arcs reside; dE ≡ gA(E) = is ~vF the density of states of the Fermi arcs, related to the magnitude of the Fermi velocity

dE dk e vF and the length of the Fermi arc k0; = vF ; and = vF B from Eqn. (5.3). dk ~ dt ~c Thus the total rate of electrons moving along each arc is given by

dN e = A k0vF B. (5.5) dt ~c

We have considered the arcs to lie in a plane perpendicular to the direction of the applied magnetic ﬁeld. In an inversion-breaking Weyl semimetal with time-reversal symmetry present, there is only one such plane. However, in a time-reversal breaking

Weyl semimetal, arcs may exist simultaneously perpendicular and co-planar with respect to the applied magnetic field. In this case, the arcs co-planar with the field will not experience a force in the plane of the arcs. Hence, we expect that only the arcs perpendicular to the field will result in a particle flux give by Eqn. (5.5).

In Fig. 5.1a, we show a mixed real-space and momentum-space diagram of a

Weyl semimetal in a slab geometry. We see the bulk Weyl nodes, labeled with their chirality χ = ±1, and separated in the x-direction. The magnetic ﬁeld induced ﬂow of electrons along the arcs is from left to right along the Fermi arc on the top surface in

Fig. 5.1b and similarly from right to left along the Fermi arc on the bottom surface.

In the absence of electric ﬁelds or temperature gradients, the only way the system can maintain a steady-state circulation of electrons is by transporting electrons via

113 the bulk Weyl nodes. This circulation of current occurs for arbitrarily small magnetic

ﬁeld.

We assume that the chemical potential lies at the Weyl nodes. In a real material,

this is not necessarily the case, and the bulk Fermi surface will be a pocket of radius

µ kF = . At ﬁnite temperatures, the Fermi surface will broaden over the energy scale ~vF

kB T kBT , and the width of the bulk channel available will be ∆k ∼ . Our analysis ~vF remains valid in both of these cases.

The projections of these Fermi pockets in the surface Brillouin zone occur at the end points of the Fermi arcs. Thus, by the continuity equation, through each bulk pocket surrounding a node with chirality χ, we must have a real-space current density

χ JB given by e dN Jχ = χ e , (5.6) B A dt z

dN where dt is given by Eqn. (5.5). These circulating currents are the low-ﬁeld analog of those explored by the authors of Ref.[27]. Furthermore, the electronic currents given

by Eqn. (5.6) are zero when summed over all pairs of nodes and chiralities. Hence

there is no Seebeck eﬀect associated with these arc-mediated currents. We note that

a Seebeck eﬀect will still occur in the usual way from the bulk pockets.

We have shown that a magnetic ﬁeld will cause a particle current through each

bulk Weyl node and we will now show that this arc-mediated particle ﬂux has a

highly nontrivial eﬀect in the presence of a thermal gradient. However, we brieﬂy

comment on the eﬀects of a thermal gradient, independent of a magnetic ﬁeld. It

is well known that in the presence of a thermal gradient, the bulk Weyl nodes will

contribute to the thermal conductivity. We will return to this bulk contribution later

in this paper. Since the Fermi arcs lie in a plane perpendicular to the ﬁeld, the Fermi

114 velocity of electrons residing in the arcs is normal to the direction of the applied

thermal gradient. We note that in the absence of a magnetic ﬁeld, the arcs will not

contribute to the heat current in the z-direction due to their two dimensional nature.

Since Weyl nodes come in pairs of opposite chirality, the circulating currents in the

steady state do not cause any net current ﬂow along the z-direction in the absence

of external potentials. However, in the presence of magnetic ﬁeld and an applied

dT temperature gradient ∇T = dz ez, the diﬀerence in temperature creates a diﬀerence

L in the energy per particle of electrons on the top surface E T (z = 2 ) and those on

L the bottom E T (z = − 2 ) . The temperature diﬀerence generates additive entropy contributions to the heat current arising from the circulating charge currents in a

“conveyor belt” involving the Fermi arcs and Weyl points.

The heat current density for Np pairs of Weyl nodes at some point z in the bulk

is given by 1 dN L L J Q = N E T ( ) − E T ( ) . (5.7) z p A dt 2 2

Here E(T (z)) is the thermal energy per particle at the layer z of a slab of thickness

L. In Eqn. (5.7), the ﬁrst term in brackets comes from electrons moving from the

hot surface through the bulk around one node and the second term is from electrons

moving in the opposite direction from the cold side through the other node. In order

to obtain to the total heat current, we have summed these contributions. We expand

L the thermal energies per particle E(T (z = ± 2 )) to obtain

N L dN dE dT J Q = p , (5.8) z 2A dt dT dz

dE dU where N dT ≡ dT is the total electronic heat capacity of the bulk Weyl pockets. We note that, unlike the charge current, the ﬁeld-induced, arc-mediated ﬂow of electrons

115 results in a non-zero net heat current. The thermally excited electrons from the

surfaces must traverse one of the Weyl pockets as they journey to the other surface

and then back again via the other Weyl pocket.

In Appendix D, we provide the details of the calculation of the heat capacity of

the bulk Weyl nodes, and ﬁnd that for kB T 0.4, it is given by (see Eqn. (D.4)) ~vF Λ . 2 4 3 dE 7π kBT = 3 , (5.9) dT 15 nB(~vF )

N where nB = V is the density. Since the momentum cutoﬀ Λ provides the only length

3 scale in this continuum limit, the density nB ∼ Λ and we obtain

2 3 Q 14π Np 2 kBT L k0 dT Jz = (kBvF Λ ) 2 , (5.10) 15 ~vF Λ lB lBΛ dz q ~c where lB = eB is the magnetic length. From Eqn. (5.10), we obtain the desired result, the contribution of the Fermi arcs to the thermal conductivity (the superscipt S

stands for “semiclassical” to distinguish it from “ultra-quantum” that we will consider

later), 3 S 14Np 2 kBT L k0 κzzz = (kBvF Λ ) 2 . (5.11) 15 ~vF Λ lB lBΛ We point out that since there is no Seebeck eﬀect from the arc-mediated channels,

the thermal conductivity given by Eqn. (5.11) does not contain any ambipolar con-

tributions. The dependence of κ on the thickness L indicates that given the one-

dimensional nature of the arc-mediated transport it is more appropriate to deﬁne a

thermal conductance. This insight motivates us to explore the Landauer formalism

for further interpretation of our results.

We note that Eqn. (D.4) has the same T 3 dependence as contributions to the

speciﬁc heat from phonons, but the thermal conductivity of phonons should be in-

dependent of magnetic ﬁeld, providing a route for separating phonon contributions

116 from the electronic contribution of Weyl nodes. We also contrast our results here

with classical Drude theory for a metal where κ ∼ T and is independent of the mag-

netic ﬁeld. We have so far neglected a discussion of scattering. In the Discussion and

Summary section below, we justify neglecting internodal scattering and show that

intranodal scattering does not disrupt the Fermi arc-mediated channels.

In Fig. 5.2, we give a “heat map” which highlights the states that are involved in

momentum space as heat ﬂows from bottom to top in real space. In Fig. 5.2a, we show

three cuts along ky for the top and bottom two layers (at ﬁxed kya = 0.75π) and for the

bulk (at kya = 0.5π). We weight each point’s size and color with the expectation value E−µ −1 0 k T hzi in the ﬁnite direction multipled by the Fermi function f (Ek) = 1 + e B (we

have again taken µ = 0) and to show its occupation in space as well as its thermal

occupation. We have taken a ﬁxed temperature gradient applied around an average

temperature. We see that on the “hot” end, states at higher energies have a higher

thermal occupation, while on the “cold” end, states on the Fermi arcs have a lower

thermal occupation. In Fig. 5.2b, we show three cuts along kx at ky = 0 for the top two layers, the bottom two layers, and all other layers with each point weighted as in Fig. 5.2a. In Fig. 5.2b, we also trace out the orbits shown in Fig. 5.1, this time overlaying them on the slab coordinates. We see that the states most occupied on the top and bottom are the arcs, while the bulk states merge into the arc and carry heat through the bulk (see supplement).

Landauer formalism in the semiclassical regime

We use the Landauer formalism[141, 142, 143, 144] to calculate the thermal and electrical conductances of the arc-mediated channel and from their ratio show that the

A Wiedemann-Franz law holds. The thermal conductance K = L κ of the arc-mediated

117 channel is given by [25]

K k2 Z E − µ2 ∂f = 2 B dE − 0 M (E) T (E) , (5.12) T ~ kBT ∂E where T (E) is the transmission coeﬃcient and M (E) is the number of propagating

modes. We expand M (E) T (E) in a power series, and, by evaluating Eqn. (5.12) and

γs comparing it with Eqn. (5.11), we ﬁnd that it takes the form M (E) T (E) = Γs E .

Upon determining the coeﬃcients Γs and γs, we ﬁnd that

~ dN M (E) T (E) = NP gB (E) , (5.13) nB dt

where NP and nB are the number of pairs of Weyl nodes and the bulk density re-

1 E2 dN spectively, the bulk density of states is given by gB (E) = 2 3 and is given by π (~vF ) dt Eqn. (5.4).

A The electrical conductance G = L σ of the arc-mediated channel can be similarly determined by calculating

e2 Z ∂f G = 2 dE − 0 M (E) T (E) . (5.14) ~ ∂E Using M (E) T (E) given by Eqn. (5.13), we obtain

2 S Np e ~ dN G = gB (E) . (5.15) 3 ~ nB dt From Eqns. (5.11) and (5.15), we can ﬁnd the Lorenz number

S 2 2 S κzzz 7π kB L = S = (5.16) T σzzz 5 e for the arc-mediated channels. We note that the dimensionless prefactor of the Lorenz

number diﬀers from its usual value of π2/3 due to the massless Dirac nature of the

Weyl nodes. We note that although the Wiedemann-Franz law holds, heat ﬂows in

the presence of a thermal gradient without a net ﬂow of charge due to the circulating

currents.

118 5.1.3 Eﬀect of disorder

We have assumed in the discussion above that the ﬂow of electrons from the Fermi

arcs is topological in nature and have therefore neglected scattering. Here, we justify

that assumption in some detail. The discussion so far has been in the ballistic regime,

where the mean free path is at least as large as other relevant length scales. However,

in the diﬀusive regime, we can separate eﬀects of disorder into scattering of electrons

on the Fermi arcs as well as internodal and intranodal scattering in the bulk. We

ﬁnd that although scattering between Fermi arcs or between nodes can disrupt the

“conveyor-belt” motion of charge discussed in the section above, these mechanisms

can be neglected in many cases. On the other hand, it can be justiﬁed that intranodal

scattering does not disrupt the topological ﬂow of electrons.

1 P 0 2 On the arcs, the scattering rate (k) ∼ 0 g (E 0 )|hk|V|k i| , where V (r) is τarc k A k

k0 the impurity potential and the density of states on the arc is given by gA(Ek) = . ~vF Hence, we see that scattering on the arcs is suppressed by the small density of states

of the linearly dispersing Fermi arcs. If there are multiple pairs of Weyl nodes, there

will be more than one Fermi arc on each surface. In this case, scattering between

arcs is possible, and, for materials with closely spaced arcs in the surface Brillouin

zone, such scattering could suppress the arc-mediated thermal transport. Thus, ideal

materials are those with single pairs of Weyl nodes, or those with well-isolated Fermi

arcs in the surface Brilllouin zone.

At ﬁnite temperatures, the Fermi surface broadens over the energy scale kBT , so

the width of the bulk channel available is ∆k ∼ kB T . Since the ﬂow of electrons in ~vF Eqn. (5.5) is topological in nature, intranodal scattering will not impede the conveyor belt motion of charge since any short range scattering (such as from impurities [145])

119 will not aﬀect the drift in the z-direction necessary for continuity of the ﬂow from the arcs. Internode (or interpocket) scattering at low temperatures can only occur from short ranged impurities as they can provide the large momentum transfers between the nodes. Internode scattering from slowly varying potentials or phonons is suppressed at low temperatures kBT ~ωD because only small momenta phonons can be excited at these temperatures, and would typically contribute to inelastic scattering within a given pocket. This phase space argument also rules out phonon drag contributions, since for most known Weyl semimetals, the Fermi energy is small compared to the

Debye energy EF ~ωD, suppressing the number of states available for phonon scattering and phonon drag as pointed out by Stockert et al.[96].

5.1.4 Ultra-quantum Regime

We next investigate the heat current ﬂow in response to an applied temperature gradient in strong applied magnetic ﬁelds, where the Fermi arcs drive the “conveyor belt” motion of charge through only a single chiral Landau level in the bulk. In this limit, the chiral Landau levels provide single channels for entropy transport. We show that the arc-mediated magnetothermal conductivity in this regime allows for a direct probe of the quantum of thermal conductance.

For large magnetic ﬁelds along the z-direction, the Weyl node energies split into discrete Landau levels shown in Fig. 5.3, given by

q −2 2 En(kz) = ±~vF 2|n|lB + kz , (5.17) for an integer n 6= 0 and

E0(kz) = χ~vF kz, (5.18)

120 for n = 0. The sign of the slope of the n = 0 Landau level depends on the chirality

of the Weyl node. Due to the linear dispersion around the Weyl nodes, the Landau q ~c levels are not evenly spaced. Here, lB = eB is the magnetic length, and vF is the magnitude of the Fermi velocity.

As in the semiclassical case, an electron follows the semiclassical motion of Eqn.

(5.3) along a Fermi arc. From Ref. [27], we know that, in the high ﬁeld limit, electrons

travel along mixed momentum and real space orbits that traverse the bulk, parallel to

the direction of the magnetic ﬁeld. As an electron moves along the arc, it terminates

on the projection of a Weyl node, which, in the presence of a large magnetic ﬁeld,

~vF is described by Landau levels in Eqns. (5.17-5.18). For kBT , the only state lB available is the zeroth Landau level given by Eqn. (5.18). Because of the chiral

nature of these states, electrons from the arc traverse the bulk without dissipation

and emerge on the other surface. After it reaches the other side, it moves along

another Fermi arc until it reaches the bulk Weyl node with the opposite chirality. It

then traverses the bulk in the opposite direction and completes the loop.

Once again, we consider a thermal gradient in the same direction as the magnetic

ﬁeld along z. The conveyor belt motion of the electrons enhances the magnetothermal

conductivity. However, unlike the semiclassical regime, now the n = 0 Landau levels

provide a single quantum channel for heat transport. The heat capacity of a pair

dU0 dE0 opposite chirality n = 0 Landau levels is given by dT = N dT where E0 is the energy per particle of the n = 0 Landau levels. The continuity of charge leads to a particle

dN current dt deﬁned by Eqn. (5.5) through the bulk, though the energy current is only carried by the chiral n = 0 Landau level. As in Eqns. (5.7-5.8), by summing over the nodes, we obtain a nonzero net heat current.

121 The general expressions for the heat capacity of the zeroth Landau level are derived k T in Appendix D. In the low temperature limit B 1, the heat capacity behaves ~vF Λ as, dU 2π V k2 0 ≈ B TB, (5.19) dT 3 ~vF Φ0 hc where Φ0 = e . This results in the thermal conductivity

2 UQ π kB Lk0 κzzz = Np (kBT ) 4 . (5.20) 3 h lBnB π2 k (k T ) In units of the quantum of thermal conductance [137, 138] GQ = B B we can th 3 h rewrite Eqn. (5.21) to be

UQ Q κzzz = GthNch, (5.21) where the number of channels Nch is given by

Lk0 Nch = Np 4 (5.22) lBnB

In the ultra-quantum limit, we see that the chiral Landau levels provide individual

channels of thermal transport, allowing for a direct probe of the quantum of thermal

conductance.

Unlike the semiclassical case, we see that the thermal conductivity in the ultra-

quantum regime is quadratic in ﬁeld and has a linear temperature dependence so long

~vF as kBT . As the upper limit is approached, higher Landau levels become ther- lB mally populated, allowing them to also engage in Fermi arc-mediated magnetothermal

transport.

Landauer formalism in the ultra-quantum regime

As in the semiclassical regime above, we can use the Landauer framework to

ﬁnd the electrical conductance in the ultra-quantum limit. We again evaluate Eqn.

122 (5.12), taking M (E) T (E) to have an arbitrary series expansion. We ﬁnd that only

γUQ one term survives, such that M (E) T (E) = ΓUQ E . By comparing our result with

Eqn. (5.21), we ﬁnd that

NP 1 dN M (E) T (E) = 2 . (5.23) π nB vF lB dt

We use Eqn. (5.23) to also evaluate the electrical conductance given by Eqn. (5.14) to obtain 2 UQ 2NP e 1 dN G = 2 . (5.24) π ~ nB vF lB dt As in the semiclassical regime, we can again calculate the Lorenz number in the

ultra-quantum regime UQ 2 2 UQ κzzz π kB L = UQ = . (5.25) T σzzz 3 e We see that in the ultra-quantum limit, the conventional dimensionless prefactor of

π2 UQ 3 is found in L which reﬂects occupation of only the chiral n = 0 Landau level.

5.1.5 Intermediate regime

In order to investigate the crossover from the low-ﬁeld semiclassical limit to the

ultra-quantum limit only involving chiral Landau levels, we include quantum eﬀects

of nonchiral Landau levels. In this intermediate ﬁeld regime, the density of states

used to calculate speciﬁc heat must include the higher Landau levels, so that for a

pair of Weyl nodes in a magnetic ﬁeld,

Z Λ B dkz gLL(E) = δ(E − |E0|) + δ(E + |E0|) Φ0 −Λ 2π X + 2 δ(E − |En|) + δ(E + |En|) , (5.26) n

123 where En is given by Eqn. (5.17). We numerically evaluate the total energy as the sum

of the analytically evaluated contribution in Eqn. (D.7) and the additional internal

energy obtained from Eq. (D.10) as described in Appendix D.

Fig. 5.3b shows the total heat capacity with contributions from all Landau levels,

1 dUtot dutot cv = V dT ≡ dT , as functions of temperature and magnetic ﬁeld. The total heat

q 1 L dN dutot dT current for all Landau levels is given by Jz = Np . The magnetother- 2 nB dt dT dz mal conductivity is given by

1 L dN dutot κzzz = Np , (5.27) 2 nB dt dT shown in Fig. 5.3c as a function of ﬁeld for several temperatures. At high temperature,

we see a crossover between linear ﬁeld dependence to quadratic ﬁeld dependence

as the ﬁeld is increased and only the lowest Landau level becomes occupied. In

Fig. 5.3d, we plot κzzz as a function of ﬁeld and temperature. We see that the

magnetothermal conductivity has a linear dependence on temperature at high ﬁeld

but a cubic temperature dependence at low ﬁeld. We also see that the ﬁeld dependence

is linear for the temperature ranges shown here, where multiple Landau levels are

involved in thermal transport. This matches the calculations which we obtained

semiclassically. We summarize the ﬁeld and temperature dependence of the arc-

25nm mediated κzzz in Table 5.1. The magnetic length is given by lB = √ . Hence, B[Tesla] 5 m for a typical Weyl semimetal with Fermi velocity vF = 10 s , the ultraquantum limit

~vF p is accessed at temperatures kBT ∼ 10meV B[Tesla]. lB We note that when the chemical potential does not lie at the Weyl nodes, the speciﬁc heat will contain quantum oscillations just as is experimentally observed in the bulk κxxz response of NbP by Stockert et al.[96]. However, these quantum oscillations arise purely from the bulk Weyl pockets and will also be present in the

124 (a) (b)

Figure 5.3: (a) Schematic of the Landau levels. Chiral n = 0 Landau levels are shown in red. The sign of the slope of the n = 0 Landau levels is positive (dashed line) for χ = 1 nodes and negative (solid line) for χ = −1. Non-chiral Landau levels (n 6= 0) are shown in black. (b) Specific heat for a single pair of Weyl nodes in the intermediate quantum limit as a function magnetic field. We have set the Landau level cutoff Nmax = 50 and we have considered an electron density such that nB = 1. (c) Magnetothermal conductivity κzzz for a single pair of Weyl nodes in the intermediate quantum limit as a function of magnetic field. We have set the Landau level cutoff Nmax = 50 (see Appendix C) and n = 1. We have fixed the temperature such that kB T = 0.02 (blue), kB T = 0.04 (green), and kB T = 0.06 ~vF Λ ~vF Λ ~vF Λ (red). At higher temperatures (red), we see a crossover between the linear low-field behavior and the quadratic field dependence when the ultra-quantum limit is reached. At lower temperatures (green, blue) only the lowest Landau level is populated in the field range shown. (d) Fermi arc-mediated magnetothermal conductivity for a single pair of Weyl nodes in the intermediate quantum limit as a function of magnetic field. Parameters are the same as in (b,c).

125 T -dependence B-dependence S 3 S Arcs (S) κzzz ∼ T κzzz ∼ B UQ UQ 2 Arcs (UQ) κzzz ∼ T κzzz ∼ B bulk 3 bulk 2 Bulk κzzz ∼ T κzzz ∼ B

Table 5.1: Summary of temperature and magnetic ﬁeld dependence of arc-mediated κzzz in the semiclassical and ultra-quantum limits as well as the bulk semiclassical magnetothermal conductivity.

speciﬁc heat and thermal conductivity in other orientations of applied magnetic ﬁeld and temperature gradient. We emphasize that it is the non-oscillatory component of the magnetothermal conductivity that will contain signatures of Fermi arc-mediated entropy transport. We point out that in Eqns. (5.11), (5.21), and (5.27), the thermal conductivity is proportional to L, meaning that the ratio κ/L, the areal conductance, is really the intrinsic quantity that does not depend on the sample’s dimension along the z-axis.

5.2 Comparison with bulk thermal conductivity of WSM

The bulk thermal conductivity for a Weyl node of chirality χ is given by [69]

Z d3k v2 κbulk = D (B, Ωχ) τ F zzz (2π)3 k 3 (E − µ)2 ∂f × 1 + l−2|Ωχ|2 − 0 , (5.28) B k T ∂E

χ −1 χ B·Ωk where D (B, Ω ) = 1 + 2π , τ is the scattering time, lB is the magnetic k Φ0

χ χ k length, f0 is the equilibrium Fermi distribution, and Ωk = 2 k3 is the chirality- dependent Berry curvature of the bulk Weyl nodes. To obtain the full magnetothermal conductivity of the bulk, we sum over pairs of nodes of opposite chirality. At zero

126 ﬁeld and for µ → 0, we obtain the bulk thermal conductivity from Eqn. (5.28) to be

2 3 bulk 7π 2 kBT κzzz (B = 0) = Np(kBvF Λ )(vF τΛ) , (5.29) 45 ~vF Λ as shown in table I. Here, instead of the cutoﬀ Λ we can equally well use the Brillouin zone size 2π/a, but we prefer to leave it in terms of the cutoﬀ to compare with the arc contribution below.

v For small values of ﬁeld ~ F 1, the bulk thermal conductivity is lB kB T N κbulk (B) = p (k v Λ2)(v τΛ) zzz 3π2 B F F ! k T 3 7π4 v 4 × B + ~ F . (5.30) ~vF Λ 15 lBkBT

bulk The enhancement of κzzz in a magnetic ﬁeld is related to the chiral anomaly, and is a consequence of the negative magnetoresistance in Weyl semimetals. By combining the bulk and arc contributions, we obtain the total thermal conductivity,

3 " 2 2 kBT 7π κ(B) = (kBvF Λ ) (vF τΛ) + ~vF Λ 45 (5.31) 4 # 2 ! 1 ~vF 14π L k0 2 + 2 . 3π lBkBT 15 lB lBΛ We recognize the third term to be from the arcs given by Eqn. (5.11). The ﬁeld independent term can be easily separated, and the term from the bulk proportional

v 4 to B2 is suppressed by a factor of ~ F . Therefore, we expect the arc-mediated lB kB T term to be the dominant magnetic ﬁeld-dependent term.

It is useful to compare the thermal conductivity of the WSM with that of free

free 2 electrons at low temperatures for which κ ∼ cvvF ` ∼ kBT g(F )vF `, where cv is the free electron speciﬁc heat and ` is the temperature-independent mean free path due to elastic scattering. Substituting for the density of states g(F ) we ob-

free 2 −1 −1 tain, κ ∼ 10 (Wm K )(kBT/F ). The scale for κ in a bulk WSM is set by

127 2 2 −1 −1 5 m 3 kBvF (2π/a) ≈ 10 Wm K where we have used vF ∼ 3 × 10 s ∼ c/10 and the lattice constant a ∼ 0.3nm. As an aside, notice that this scale is close to the prefactor of κfree. Now the other dimensionless factors contributing to κbulk are p vF τ(2π/a) ≈ 200 and ~vF /lBkBT ≈ 10 meV B(T )[Tesla]/kBT , where we have used p the magnetic length estimate lB ≈ 25nm/ B(T )[Tesla]. We therefore estimate the

bulk 2 −1 −1 3 bulk κzzz ≈ 10 (Wm K )10Np(kBT/4eV) . Against this bulk term, we compare the additional arc contribution which is enhanced by the factor (L/lB) × (1/lBΛ) and should be experimentally measurable. We also point out that the bulk thermal conductivity in Eq. (5.30) is intrinsically volume-independent, whereas the arc conductivity is a surface property: therefore, in Eq. (5.31), the dependence of heat transport on the dimension of the samples along the z-axis will diﬀerentiate the different contributions experimentally.

5.3 Fermi Arc-Mediated Entropy Transport in Dirac Semimet- als

Dirac semimetals are topological semimetals where Weyl nodes of opposite chirality coexist at the same momenta but are protected from annihilation with each other by symmetry [146]. The Dirac semimetal state has been predicted [147, 148] and subsequently observed [149, 150] in Na3Bi and Cd3As2. Like the Weyl semimetals discussed above, Dirac semimetals also contain Fermi arcs. However, because each Dirac node is comprised of a pair of opposite chirality Weyl nodes, each real space surface of a Dirac semimetal has a pair of Fermi arcs for each pair of Dirac nodes. It has been shown that these double Fermi arcs are not topologically protected in general[151], except on planes in the surface Brillouin zone which preserve

128 time-reversal invariance. We show schematic cartoons of the diﬀerent cases for sur-

face states of Dirac semimetals in Fig. 5.4. In Ref. [151], it is shown that, although

Fermi arcs may terminate on projections of Dirac nodes (Fig. 5.4a), it is possible

for perturbations which break no symmetries of the Dirac semimetal to deform the

surface Fermi arcs into closed Fermi pockets on the surface (Fig. 5.4b). Doping the

system away from the Dirac point may not change the nature of these closed Fermi

pockets (Fig. 5.4c) unless the doping is large enough for the bulk Fermi pockets to

grow and mix with the surface states restoring the arcs (Fig. 5.4d).

In cases (b) and (c) above, the topologically trivial nature of the surface states

negates the possibility of Fermi arc-mediated entropy transport discussed in this work.

The cases (a) and (d) are more interesting. In these cases, we have two copies of the

conveyor belt orbits shown in Fig. 5.1. Hence, we obtain the same results for the arc-

mediated magnetothermal conductivity of a Dirac semimetal as in Eqn. (5.11) and

(5.21) by setting Np to be twice the number of pairs of Dirac nodes. Case (d) above may be the most physically relevant. In quantum oscillation experiments on thin ﬁlms of Cd3As2, there is evidence that the observed Shubnikov-de Haas oscillations arise from combinations of bulk Landau levels and surface Fermi arcs [27, 28]. This opens the door for studies of Fermi arc-mediated entropy transport in Dirac semimetals.

5.4 Discussion and Summary

Comparison with arc-mediated quantum oscillations:

As we noted previously, there have been several studies of exotic quantum oscillations involving mixed real and momentum space orbits[27, 125, 28, 126] in topological semimetals. Although the Fermi arc-mediated entropy transport we have discussed

129 a b

c d

Figure 5.4: Schematic of the different cases for surface states in a Dirac semimetal with a single pair of Dirac nodes. Green dots represent projections of the Dirac nodes on the surface Brillouin zone. Red contours are surface states on a single surface. Purple filled regions represent projections of bulk Fermi pockets which enclose Dirac nodes. In (a-b), the Fermi energy is at the Dirac nodes, while in (c-d) the Fermi energy is doped away from the Dirac nodes. (a) A pair of surface Fermi arcs terminates on the Dirac nodes on a single surface. (b) A perturbation containing the full symmetry of the full Dirac Hamiltonian which does not shift the bulk Dirac nodes can deform the Fermi arcs into a closed Fermi pocket on the surface, much like a topological insulator. (c) When the Fermi energy is shifted away from the Dirac nodes, the surface states may still form a closed Fermi pocket. (d) Sufficient doping may cause the Fermi pockets to grow large enough for the surface states to merge with the bulk pockets.

130 above shares some similarities with these predictions of quantum oscillations, there are key diﬀerences that demonstrate the complimentary role of our work. Crucially, the previous work in Ref. [27] relies on the Onsager quantization from quantum coherence of the electrons across the entire bulk-arc orbits. Rather than oscillations of the density of states from the Fermi arcs, in this paper we predict that the presence of Fermi arcs will result in distinct channels of entropy transport. The experimental results in Ref. [28] highlight the challenges of separating contributions of quantum oscillations from bulk-arc orbits from those of only the bulk.

Since Fermi arc-mediated entropy transport does not rely on quantum coherence across the orbits, the magnetothermal conductivity driven by the arcs should be robust with respect to scattering as discussed below. Because of the disparate origin of the “conveyor-belt” transport of entropy that we predict, experimental signatures of

Fermi-arc mediated thermal conductivity should distinct from purely bulk transport.

Additionally, the arc-mediated entropy transport is highly tunable by changing the magnetic ﬁeld, opening the door for technological applications not possible using only the charge sector.

Experimental implications:

We summarize the transport properties of some candidate materials for Fermi arc-mediated entropy transport in Table 5.2. Although weak spin-orbit coupling compared to the tantalum monopnictides causes the Weyl nodes to be closer together in NbP and NbAs, they still appear to be promising cadidates with high Fermi velocities. The Dirac semimetal Cd3As2 has been shown to possess putative Shubnikov-de

Haas oscillations that arise from combinations of bulk Landau levels and surface Fermi

131 m Material vF s Np TaAs[23] 3 × 105 12 TaP[152, 47] 3 × 105 12 NbAs[46, 153] 6 × 105 12 NbP[94] 4.8 × 105 12 6 Cd3As2[154, 155] 1.5 × 10 2 5 Na3Bi[156] 7.5 × 10 2

Table 5.2: Several topological semimetal candidates for Fermi arc-mediated entropy transport. Since Cd3As2 and Na3Bi are Dirac semimetals, the number of Weyl nodes Np reported is double the number of Dirac nodes.

arcs [27, 28]. We predict that Fermi arc-mediated entropy transport would be an even stronger transport signature of topological Fermi arcs in these materials.

We have considered orientations of magnetic ﬁelds perpendicular to the surface

Brillouin zone in which Fermi arcs reside. In systems which preserve time-reversal symmetry, there must be only a single pair of such surfaces that have Fermi arcs.

On all of the other surfaces, Weyl nodes of opposite chirality will project on top of one another and therefore not lead to Fermi arcs. When the magnetic ﬁeld is applied along such a direction, there will be no contribution to the magnetothermal conductivity from the Fermi arcs and no such conveyor-belt transport of entropy.

Although the bulk electronic pockets as well as phonons may contribute to the thermal conductivity in these other magnetic field configurations, we have argued above that the contribution of arc-mediated entropy transport in the cofiguration shown in Fig.

5.1b will likely be stronger than these other contributions. This will lead to a clear anisotropy between directions with Fermi arcs and without Fermi arcs.

Applications:

132 We point out that conveyor-belt thermal transport could potentially ﬁnd appli-

cations in magnetically actuated all-solid-state thermal switches. Presently, most

thermal switches are mechanical or involve exchange gases, because all-solid-state

switches have either low switching ratios or work over only a limited temperature

range. Here we suggest that the amplitude or the direction of an external magnetic

ﬁeld can aﬀect a change in a Weyl semimetal from an “on” state where arcs contribute

to entropy transport to an “oﬀ” state where they do not. Given the highly directional

nature of the entropy transport, we expect it to be relatively immune from scattering.

We further expect to achieve high switching ratios and a large operating temperature

range by adjusting the sample length, as long as the phonon contribution to the total

conductivity is not too large.

5.5 Conclusion and further directions

In conclusion, we have shown that in the presence of a magnetic ﬁeld and tem-

perature gradient, each applied perpendicular to the surface Brillouin zone of a Weyl

semimetal, the Lorentz force on the Fermi arcs leads to a conveyor belt motion of

charge and a net ﬂow of heat. This heat ﬂow leads to a highly anisotropic magne-

tothermal conductivity that has distinct behaviors in the semiclassical and quantum

regimes. For relatively high temperatures and small ﬁelds, the Fermi arc-mediated

magnetothermal conductivity κzzz is found to be linear in magnetic ﬁeld and cubic in temperature.

In the ultra-quantum limit where the magnetic ﬁeld is strong and temperatures are low such that only the chiral n = 0 Landau level is involved, we ﬁnd that the

Fermi arc-mediated magnetothermal conductivity is instead linear in temperature

133 and quadratic in ﬁeld. The diﬀerence in temperature dependences can be understood

by noting that in the semiclassical regime, there are many more degrees of freedom in

the bulk available to thermal transport. Once the system is the ultra-quantum limit,

S the sole degrees of freedom available are the chiral n = 0 Landau levels, and κzzz is much less sensitive to changes in temperature. In the semiclassical limit, the linear

magnetic ﬁeld dependence comes only from the Lorentz force on the arcs. In ﬁelds

suﬃciently strong enough, the quantization of the Landau levels results in a density of

states which also depends on the magnetic ﬁeld. In the ultra-quantum limit, the lone

chiral n = 0 Landau levels result in a speciﬁc heat which is quadratic in ﬁeld. In this

high-ﬁeld regime, the chiral n = 0 Landau levels provide single channels of thermal

conductivity, allowing for a direct probe of the quantum of thermal conductance. As

UQ more Landau levels become populated at higher temperatures, κzzz again becomes linear in ﬁeld and tends toward the semiclassical limit. We summarize our results for

bulk the diﬀerent regimes and compare them with the bulk κzzz in Table 5.1.

134 Appendix A: Isothermal Heat Transport and the Adiabatic Nernst Eﬀect

In this appendix, we present derivations for heat transport in a generic system under isothermal conditions. Using this result, we derive the adiabatic Nernst thermopower.

A.1 Heat transport under isothermal conditions

Using Eqn. (4.1), we can ﬁnd the heat current density to be

Q = LTE · E + LTT · (−∇T ). (A.1)

Now we can use Eqn. (4.7) for the electric ﬁeld, we can write the heat current density as

Q = LTE · (−(LEE)−1 · LET · (−∇T )) + LTT · (−∇T )

= LTT − LTE · (LEE)−1 · LET · (−∇T ). (A.2)

Noting again the isothermal conditions given by Eqn. (4.64), we ﬁnd that ! TT X TE EE −1 ET Qi = Lix − Lij (L )jmLmx (−∇xT ). (A.3) jm

135 Expanding the sum, we ﬁnd

TT TE EE −1 ET TE EE −1 ET Qi = (Lix − (Lix (L )xx Lxx − Liy (L )xy Lxx

TE EE −1 ET TE EE −1 ET + Lix (L )xy Lyx + Liy (L )xx Lyx ))(−∇xT ). (A.4)

Onsager’s reciprocal relation tells us that LTE = T LET and so we can write the components of the heat current density as

TT ET EE −1 ET ET EE −1 ET Qi = (Lix − T (Lix (L )xx Lxx − Liy (L )xy Lxx

ET EE −1 ET ET EE −1 ET + Lix (L )xy Lyx + Liy (L )xx Lyx ))(−∇xT ). (A.5)

We ﬁrst work out the x-component as

TT ET EE −1 ET ET EE −1 ET ET EE −1 ET Qx = (Lxx − T (Lxx (L )xx Lxx + Lxy (L )xx Lyx − 2Lxx (L )xy Lxy ))(−∇xT ) ET EE ET ET EE ET ET EE ET ! TT T (Lxx Lxx Lxx + Lxy Lxx Lyx − 2Lxx Lxy Lxy ) = Lxx − EE 2 EE 2 (−∇xT ). (A.6) (Lxx ) + (Lxy )

ET ET Again using the property that Lxy = −Lyx , we ﬁnd that the x-component of the heat current density is

ET 2 ET 2 EE ET EE ET ! TT (Lxx ) − (Lxy ) Lxx + 2Lxx Lxy Lxy Qx = Lxx − T EE 2 EE 2 (−∇xT ). (A.7) (Lxx ) + (Lxy )

We work out the y-component in a similar way and ﬁnd

TT ET EE −1 ET ET EE −1 ET Qy = (Lyx − T (Lyx (L )xx Lxx − Lyy (L )xy Lxx

ET EE −1 ET ET EE −1 ET + Lyx (L )xy Lyx + Lyy (L )xx Lyx ))(−∇xT ). (A.8)

ET ET Recalling the isotropy condition Lyy = Lxx , we write

TT EE −1 ET ET EE −1 ET 2 ET 2 Qy = (Lyx − T (2(L )yy Lyx Lyy + (L )xy ((Lyx ) − (Lyy ) )))(−∇xT ). (A.9)

136 Again using the form of the inverse of LEE given by Eqn. (4.10), we ﬁnd that the

heat current density in the y-direction

ET 2 ET 2 EE EE ET ET ! TT (Lyx ) − (Lyy ) ) Lyx + 2Lyy Lyx Lyy Qy = Lyx − T EE 2 EE 2 (−∇xT ). (A.10) (Lxx ) + (Lxy )

A.2 The adiabatic Nernst eﬀect

We deﬁne the adiabatic Nernst eﬀect as

ad Ey αxyz = , (A.11) −∇xT

under the conditions

Jx = Jy = 0, (A.12)

and

Qy = 0. (A.13)

We can again write the electric ﬁeld as we did in Eqn. (4.7) as

E = −(LEE)−1 · LET · (−∇T ). (A.14)

Expanding this equation component-wise, we obtain

EE −1 ET EE −1 ET Ex = − (L )xx Lxx + (L )xy Lyx (−∇xT )

EE −1 ET EE −1 ET − (L )xx Lxy + (L )xy Lyy (−∇yT ) (A.15)

and

EE −1 ET EE −1 ET Ey = − (L )yx Lxx + (L )yy Lyx (−∇xT )

EE −1 ET EE −1 ET − (L )yx Lxy + (L )yy Lyy (−∇yT ) (A.16)

137 Now, to ﬁnd the Nernst eﬀect, we would like to eliminate (−∇yT ) from Eqn. (A.16).

From Eqn. (4.1), we can ﬁnd the heat current density to be

Q = LTE · E + LTT · (−∇T ). (A.17)

Due to the adiabatic condition in Eqn. (A.13), we have that

TE TE TT TT 0 = Lxx Ex + Lxy Ey + Lxx (−∇xT ) + Lxy (−∇yT ). (A.18)

We now solve Eqn. (A.18) for Ex

−1 TE TT TT Ex = TE Lxy Ey + Lxx (−∇xT ) + Lxy (−∇yT ) , (A.19) Lxx

and equate this with Eqn. (A.15) to obtain

1 TE TT TT TE Lxy Ey + Lxx (−∇xT ) + Lxy (−∇yT ) Lxx EE −1 ET EE −1 ET = (L )xx Lxx + (L )xy Lyx (−∇xT )

EE −1 ET EE −1 ET + (L )xx Lxy + (L )xy Lyy (−∇yT ). (A.20)

We can now solve Eqn. (A.20) for (−∇yT ), obtaining

TE TT TE EE −1 ET EE −1 ET L Ey + L − L (L ) L + (L ) L (−∇xT ) (−∇ T ) = xy xx xx xx xx xy yx , (A.21) y TE EE −1 ET EE −1 ET TT Lxx (L )xx Lxy + (L )xy Lyy − Lxy or using Eqn. (4.10),

TE EE 2 EE 2 L (L ) + (L ) Ey (−∇ T ) = xy xx xy y TE EE ET EE ET TT EE 2 EE 2 Lxx Lxx Lxy − Lxy Lyy − Lxy (Lxx ) + (Lxy ) TT EE 2 EE 2 TE EE ET EE ET L (L ) + (L ) − L L L − L L (−∇xT ) + xx xx xy xx xx xx xy yx , (A.22) TE EE ET EE ET TT EE 2 EE 2 Lxx Lxx Lxy − Lxy Lyy − Lxy (Lxx ) + (Lxy ) Again using Eqn. (4.10), we now rewrite Eqn. (A.16) as

EE 2 EE 2 EE ET EE ET Ey (Lxx ) + (Lxy ) = − (Lxy Lxx + Lxx Lyx (−∇xT )

EE ET EE ET − Lxy Lxy + Lxx Lyy (−∇yT ). (A.23)

138 We now use Eqn. (A.22) to eliminate (−∇yT ) from Eqn. (A.23), resulting in

EE 2 EE 2 EE ET EE ET Ey (Lxx ) + (Lxy ) = − Lxy Lxx + Lxx Lyx (−∇xT ) EE ET EE ET TE EE 2 EE 2 L L + L L L (L ) + (L ) Ey − xy xy xx yy xy xx xy TE EE ET EE ET TT EE 2 EE 2 Lxx Lxx Lxy − Lxy Lyy − Lxy (Lxx ) + (Lxy ) EE ET EE ET TT EE 2 EE 2 TE EE ET EE ET L L + L L L (L ) + (L ) − L L L − L L (−∇xT ) − xy xy xx yy xx xx xy xx xx xx xy yx . TE EE ET EE ET TT EE 2 EE 2 Lxx Lxx Lxy − Lxy Lyy − Lxy (Lxx ) + (Lxy ) (A.24)

We now have an equation only in terms of the thermoelectric coeﬃcients along with

Ey and (−∇xT ) and can solve for the adiabatic Nernst eﬀect to obtain

ad Ey Γxxκxx αxyz = = αxyz − , (A.25) −∇xT κxy

where αxyz is the isothermal Nernst eﬀect and we have introduced the thermal conduc-

tivity coefficient κ and the Ettinghausen-Nernst coefficient Γxx. From Eqn. (A.24), we can find

EE −1 ET X EE −1 ET EE −1 ET EE −1 ET Γxx = −((L ) · L )xx = − (L )xj Ljx = −((L )xx Lxx + L )xy Lyx ), j (A.26)

which, using Eqn. (4.10), can be simpliﬁed to

EE ET EE ET Lxx Lxx + Lxy Lxy Γxx = EE 2 EE 2 . (A.27) (Lxx ) + (Lxy )

139 Appendix B: Landau Levels of a Weyl node

We begin with a simple Weyl node described locally by a Hamiltonian of the form

HI = ±k · ~σ, (B.1) and we align a magnetic ﬁeld B along the z-direction. Then we can solve for the spectrum in a gauge independent way by introducing the canonical momenta

1 † −i † kx → πˆx = √ (ˆa +a ˆ), ky → πˆy = √ (ˆa − aˆ), (B.2) 2lB 2lB wherea ˆ(†) is the Landau level lowering (raising) operator. These obey the usual commutation relations

[â, aˆ†] = 1, [â, aˆ] = [â†, aˆ†] = 0. (B.3)

It is then clear thatπ ˆx andπ ˆy obey the following commutation relations

−i † † −i † † † † [ˆπx, πˆy] = 2 [â +a, ˆ aˆ − aˆ] = 2 [â , aˆ ] + [â, aˆ ] − [â , aˆ] − [â, aˆ] 2lB 2lB −i † † −i † −2 = 2 [â, aˆ ] − [â , aˆ] = 2 2[â, aˆ ] = −ilB = −ieB, (B.4) 2lB 2lB √ where we have defined a magnetic length as lB = 1/ eB and e is the fundamental unit of charge.

Now in the presence of a magnetic ﬁeld the Hamiltonian has become

HI = ±(ˆπxσx +π ˆyσy + kzσz), (B.5)

140 where kz remains a good quantum number. We can square this Hamiltonian to

diagonalize this in the pseudospin-space (here representing the valence and conduction

bands which cross at the Weyl node) as follows

2 2 2 2 HI =π ˆx σxσx +π ˆy σyσy + kz σzσz +π ˆxπˆy σxσy +π ˆyπˆx σyσx

+ kzπˆx σzσx + kzπˆx σxσz + kzπˆy σyσz + kzπˆy σzσy. (B.6)

2 Now we use that σi = σ0 and σασβ = iαβγσγ, where αβγ is the perfectly antisymmetric tensor, to obtain

2 2 2 2 HI = (ˆπx +π ˆy + kz )σ0 + i[ˆπy, kz]σx + i[kz, πˆx]σy + i[ˆπx, πˆy]σz (B.7)

and then using the fact that kz is a c-number and commutes with all operators, we

obtain

2 2 2 2 HI = (ˆπx +π ˆy + kz )σ0 + i[ˆπx, πˆy]σz. (B.8)

Now it is very clear that the square of our Hamiltonian is diagonal in the σz-basis.

2 2 † We can work out the termπ ˆx +π ˆy and we ﬁnd that it is diagonal in then ˆ =a ˆ aˆ

2 2 basis. We begin by working outπ ˆx andπ ˆy individually to obtain

2 1 † † 1 † † † † πˆx = 2 (â +a ˆ)(â +a ˆ) = 2 (â aˆ +a âˆ +a ˆ aˆ +a âˆ ), (B.9) 2lB 2lB and

2 −1 † † 1 † † † † πˆy = (â − aˆ)(â − aˆ) = (−aˆ aˆ − aâˆ +a ˆ aˆ +a âˆ ). (B.10) 2lB 2lB Adding these together, it is clear that we obtain

2 2 2ˆn + 1 πˆx +π ˆy = 2 . (B.11) lB

−2 The coeﬃcient of σz is just a constant i[ˆπx, πˆy] = −lB . Then we can write 2 2ˆn + 1 2 1 HI = 2 + kz σ0 + 2 σz, (B.12) lB lB 141 and practically read oﬀ the energies:

2 2n + 1 ± 1 2 En(kz) = 2 + kz . (B.13) lB

Taking the square root, we allow n to range over all of the integers rather than just the natural numbers:

p 2 En(kz) = ±vFsgn(n) 2eB|n| + kz . (B.14)

We have restored the Fermi velocity vF at the Weyl node through dimensional analysis.

The zeroth Landau level has a linear dispersion which depends on the chirality of the

Weyl node:

E0(kz) = ±vFkz. (B.15)

142 Appendix C: Scattering in Weyl semimetals

It is clear that the scattering time τ plays an important role in the behavior of the distribution function. The energy dependence of the scattering time has been calculated for Weyl semimetals[157] and we recap these results here for completeness.

Matthiessen’s rule tells us that, for independent scattering mechanisms each with a rate 1 , the total scattering rate is just the sum τj

1 X 1 = . (C.1) τ τ j j

0 For scattering due to impurities with potential hVi(k, k )i in a system with linear dispersion, Fermi’s golden rule yields the following scattering rate

Z 3 2 1 2πni d k 0 2 (1 − cos (θ)) 0 = 3 |hVi(k, k )i| δ (E(k) − E(k )) . (C.2) τi ~ (2π) 2

For short-range neutral impurity scattering with impurity scattering potential

0 hVs(k, k )i = V0, the integral in Eqn. (C.2) is simply evaluated[157]) to obtain the scattering rate 2 2 1 nsV0 k = 2 , (C.3) τs 3π~ vF where ns is the density of short-range neutral impurities and vF is the Fermi Fermi velocity such that E = ~vF k.

143 For long-range Coulomb scattering due to charged impurities, we have that

2 0 4πe 1 hVs(k, k )i = 2 2 , (C.4) κ q + qs

0 where q = |k − k |, qs is the screening wave vector and κ is the dielectric constant of the background lattice. With this form of the scattering potential, Eqn. (C.2) can again be evaluated and the scattering time in the long-wavelength Thomas-Fermi approximation is found to be

1 2 vF = 4πnlα 2 It(q0), (C.5) τl k

e2 where α = is the modiﬁed ﬁne structure constant, nl is the density of long-ranged κ~vF p gα impurities and q0 = 2π with g given by the number of Weyl nodes. The function

It(q0) in Eqn. (C.5) is given by

2 1 1 It(q0) = q0 + ln 1 + 2 − 1. (C.6) 2 q0

We note that since Eqns. (C.3) and (C.5) only depend on the magnitude of k.

This allows us to write each of the scattering rates as purely dependent on energy.

Doing this, we obtain that 2 2 1 nsv0E = 3 . (C.7) τs 3π~ vf and 2 3 1 4πnlα ~vF It(q0) = 2 . (C.8) τl E Surprisingly, we ﬁnd that, empirically, a constant scattering time explains trans-

port data very well. We have included the above energy dependencies in our calcula-

tions and we ﬁnd that, for most parameter regimes, they do not qualitatively aﬀect

our results.

144 Appendix D: Speciﬁc Heat of Weyl Nodes

D.1 Semiclassical regime

The energy density u = U/V of Weyl nodes is:

v Λ Z ~ F E g(E) u = dE 1 + eβ(E−µ(T )) −~vF Λ v Λ 1 Z ~ F E 3 = dE , (D.1) 2 3 E−µ(T ) π ( vF ) − v Λ k T ~ ~ F 1 + e B

where V is the volume and Λ is a momentum regularization which is set by the sep-

aration of the Weyl nodes in momentum-space. It is useful to deﬁne a dimensionless

temperature Θ = kB T in units of the energy cut oﬀ. For the case when µ(T ) → 0 ~vF Λ

1 Z Θ 3 1 3 x u = 2 (kBT )Θ dx x . (D.2) π 1 1 + e − Θ

This integral can be evaluated in terms of the polylogarithm function of order s, zk Li (z) = P∞ for complex z such that |z| < 1. The heat capacity is given by: s k=1 ks

3 ! du Λ kB 3 7 1 = − Θ + Li −e Θ dT π2 15 4

2 1 1 Θ + 1 − 8ln 1 + e Θ 1 + e Θ ! 1 2 1 − 24ΘLi2 −e Θ + 48Θ Li3 −e Θ . (D.3)

145 For temperatures such that Θ . 0.4, it can be shown that

du c ≈ 0 Λ3k Θ3, (D.4) dT π2 B

7π4 where c0 = 15 is a purely numerical constant. We can also rewrite it in terms of the

dE energy per particle as: dT from Eqn. (D.4) to be

3 dE kBΘ = c0 2 , (D.5) dT π nB

N where nB = V is the density.

D.2 Ultra-quantum regime

In the ultraquantum regime, only the n = 0 Landau level participates in entropy

transport. In Eq.D.1 the internal energy is given in terms of g0, the density of states

of the zero energy Landau levels

Z Λ B dkz g0(E) = δ(E − |E0|) + δ(E + |E0|) , (D.6) Φ0 −Λ 2π

hc where E0 is the energy of the chiral Landau level. Here Φ0 = e is the quantum of magnetic ﬂux. The internal energy density from both nodes is:

Z Λ B dkz kz u0 = vF ~ β(~vF kz−µ) Φ0 −Λ 2π 1 + e ! k − z , (D.7) 1 + eβ(−~vF kz−µ)

which can be evaluated to obtain

du0 kBΛB 4 1 1 = − 8kBΛln 1 + e Θ − 1 dT πΦ0 Θ 1 + e Θ 2 ! π 1 + 2Θ + 4Li − e Θ , (D.8) 3 2

146 where Lis(z) is the polylogarithm function of order s.

In the low temperature limit Θ 1, we ﬁnd that the speciﬁc heat is

du 2π Λk 0 ≈ B ΘB · (D.9) dT 3 Φ0

D.3 Crossover regime

From Eqn. (5.26), the additional internal energy can be calculated by

Z Λ B vF X dkz u = 2 ~ h(k , n) e Φ l 2π z 0 B n −Λ 1 1 − , (D.10) 1 + e(β~vF /lB )(h−µ) 1 + e−(β~vF /lB )(h−µ)

p 2 2 where h(kz, n) = |n| + kz lB. The total energy density is now given by the sum of

Eqns. (D.7) and (D.10) to obtain utot = u0 + ue. Unlike Eqn. (D.7), the expres-

sion above for ue cannot be evaluated analytically. Instead, we evaluate Eqn. (D.10)

numerically by introducing a regularization Nmax which cuts oﬀ the sum over Lan-

dau levels. We take the system to be at charge neutrality where µ = 0. We can then calculate the speciﬁc heat by numerically evaluating derivatives with respect to temperature.

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