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# Introduction to Quantum Mesoscopic Transport and Topological Matter

Mark Oliver Goerbig

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## To cite this version:

Mark Oliver Goerbig. Introduction to Quantum Mesoscopic Transport and Topological Matter. Engineering school. France. 2023, pp.334. [hal-04248649v2](https://hal HAL Id: [hal-04248649](https://hal HAL Id: [hal-04248649](https://doi.org/10.5285/epub-10000000-1-24))

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# PHY 52001 EP – Introduction to Quantum Mesoscopic Transport and Topological Matter

Mark Oliver GOERBIG

Lecture Notes, September 2024
Foreword

The present lecture notes are meant to accompany the classes “Introduction to Quantum Mesoscopic Transport and Topological Matter”, which I am lecturing since 2019/20 at École Polytechnique, Palaiseau. They are built on the grounds of lectures by Gilles Montambaux, who initiated third-year physics students at École Polytechnique to the research field of quantum mesoscopic physics, which deals with quantum electronic transport at the mesoscopic scale relevant in the understanding of nano-devices. This field emerged as a major research topic in the 1980ies and 1990ies with some ground-breaking experiments, such as the discovery of the quantum Hall effect in 1980 by von Klitzing (Nobel Prize in 1985), which turned upside-down our understanding of electronic conduction based on Ohm’s law and the Drude model. Indeed, electric conduction turned out, under certain circumstances, to be given in terms of the universal constant $G_0 = e^2/h$, independent of sample geometry or even inevitable disorder in the devices. Namely, ballistic transport needed to be seriously taken into account.

This discovery came along with the accessibility of high-mobility two-dimensional electron gases in modulation-doped semiconductor (GaAs) heterostructures that are nowadays used as the basic material to fabricate high-performance electronic devices at the nanoscale, where the size of the device can be smaller than the average distance between two scattering processes, that is the core of ballistic transport.

Later, since 2005, a new class of materials has been and still is heavily investigated in condensed-matter physics. It is the class of relativistic and topological materials that contains graphene (two-dimensional graphite), two- and three-dimensional topological insulators and superconductors, as well as three-dimensional Weyl semimetals. Not only do they display novel forms of quantum transport described in terms of a relativistic Dirac equation, usually employed in high-energy rather than in condensed-matter physics, but they also show a certain number of phenomena similar to the quantum Hall effect, such as the quantum anomalous or the quantum spin Hall effects.

These lecture notes are meant to provide an introduction to this field of modern condensed-matter physics. While one may think that they are mainly written for students who intend to pursue a career in (more or less) fundamental research, some of the concepts are equally relevant – already today – for engineers who design electronic devices at the nano-scale, e.g. in quantum technologies. While this is less the case for topological materials,
the discussion of which constitutes the second part of the lecture notes, their
high potential for future quantum technologies justifies their integration into
the third-year program for physics students at École Polytechnique, be they
interested in the more fundamental aspects of modern research in quantum
matter or oriented towards the engineering aspects of the latter and its ap-
plications. Finally, the concepts of topological band theory (such as Chern
numbers and Berry curvature), which are used in the theoretical description
of topological materials, find an application also in other fields of physics,
such as quantum optics and high-energy physics.

As mentioned above, these classes are based on lectures by Gilles Montambaux whom I want to thank most warmly in the first place. Namely the chapters 3 and 4 unambiguously bear his signature and are heavily inspired by his own lecture notes (in French). I am deeply indebted to him who taught me many aspects of quantum mesoscopic transport, both in his classes and in daily exchange during common research projects ever since I joined Laboratoire de Physique des Solides, Orsay, as a CNRS researcher in 2005. Most of our common research interest was indeed concerned with graphene, which occupies a prominent place in the present classes. I also need to mention Jean-Noël Fuchs with whom I have not only collaborated scientifically, but I also taught some first classes on graphene with him (in the Doctoral School of Université Paris-Sud, in 2008, and during a Summer School in Cargèse, also in 2008). While the later chapters of my lecture notes are based on classes I gave on the quantum Hall effect in a Les Houches Summer School (Singapore, 2009) and at Université Paris-Sud in 2011 on topological matter, they benefited largely from exchange with Jean-Noël Fuchs and his own classes on this topic. Finally I would like to mention and acknowledge my students at École Polytechnique, who allowed me to experiment these classes in 2019 and 2020 and whose feedback was extremely important for the present lecture notes. I would like to thank particularly Clément Brochet, Mathieu Lamblin and Mathieu Lizée for a careful reading of the later chapters of these notes and their fruitful comments and suggestions.

Mark Oliver Goerbig, November 2024
# Contents

## 1 Introduction: quantum phenomena at the mesoscopic scale 1

1.1 Classical electric transport – Ohm and Drude 2

1.2 A hint of quantum mechanics in classical transport 4

1.3 From classical to quantum transport 8

1.3.1 Elastic vs. inelastic scattering: mean free path and coherence length 9

1.3.2 The mesoscopic scale 11

1.4 Quantum and topological matter 14

1.5 Materials and sample fabrication 17

1.5.1 Fabrication of nano-circuits via lithography 19

1.5.2 Fabrication of two-dimensional electron systems 21

1.6 Outline of the lecture notes 26

## 2 Bloch's theorem and tight-binding models – a reminder of band theory 29

2.1 Translation symmetry in Bravais lattices 30

2.1.1 Bravais lattice 31

2.1.2 General lattice 33

2.2 Bloch's theorem 35

2.2.1 Bloch's theorem for a Bravais lattice 36

2.2.2 Bloch's theorem for an arbitrary lattice 44

2.2.3 Periodic Hamiltonian and Bloch Hamiltonian 50

2.3 Band structure of graphene and boron nitride 55

2.3.1 Low-energy model of graphene and boron nitride – the emergence of 2D Dirac fermions 60

## 3 Conduction and conductivity: Ohm's law and its limits 65

3.1 Density of states 66
<table><tr><td>3.2</td><td>Statistical model for conductivity</td><td>73</td></tr><tr><td>3.2.1</td><td>Einstein relation and link with diffusion processes</td><td>77</td></tr><tr><td>3.2.2</td><td>Back to the Drude model</td><td>78</td></tr><tr><td>3.3</td><td>Conductivity and conductance</td><td>79</td></tr><tr><td>3.3.1</td><td>What is electric conductance and what is a conductance measurement? A new paradigm</td><td>80</td></tr><tr><td>3.3.2</td><td>Classical ballistic vs. diffusive transport – Sharvin conductance</td><td>84</td></tr><tr><td>3.4</td><td>Conductance through a 1D wire and the Landauer formula</td><td>89</td></tr><tr><td>3.4.1</td><td>Single-channel conductance</td><td>90</td></tr><tr><td>3.5</td><td>Scattering and transfer matrices</td><td>99</td></tr><tr><td>3.5.1</td><td>From the Landauer formula to Anderson localisation in a 1D conductor</td><td>101</td></tr><tr><td>3.5.2</td><td>From the Landauer formula to Ohm's law</td><td>104</td></tr><tr><td>3.6</td><td>Multi-channel conductance</td><td>105</td></tr><tr><td>3.6.1</td><td>Conductance through a quantum point contact</td><td>108</td></tr><tr><td>3.6.2</td><td>Experimental evidence for a crossover from ballistic to diffusive transport</td><td>110</td></tr></table>

# 4 Conduction in multi-terminal quantum devices: Landauer-Büttiker formalism 113

<table><tr><td>4.1</td><td>Landauer-Büttiker formalism for multi-terminal systems</td><td>114</td></tr><tr><td>4.1.1</td><td>Reference potential and current conservation</td><td>116</td></tr><tr><td>4.1.2</td><td>The role of a magnetic field and Onsager reciprocity relations</td><td>118</td></tr><tr><td>4.1.3</td><td>Some other properties of the conductance matrix</td><td>122</td></tr><tr><td>4.2</td><td>Conductance through a three-terminal device</td><td>124</td></tr><tr><td>4.3</td><td>Conductance through a four-terminal device</td><td>128</td></tr><tr><td>4.3.1</td><td>Test of the Onsager reciprocity relations</td><td>130</td></tr><tr><td>4.3.2</td><td>Negative four-terminal resistances</td><td>131</td></tr><tr><td>4.3.3</td><td>Non-invasive four-terminal device</td><td>133</td></tr></table>

# 5 Quantum Hall effects 139

<table><tr><td>5.1</td><td>Classical Hall effect</td><td>142</td></tr><tr><td>5.2</td><td>Landau quantisation</td><td>147</td></tr><tr><td>5.2.1</td><td>Peierls substitution and quantisation</td><td>149</td></tr><tr><td>5.2.2</td><td>Landau levels of electrons in a parabolic band</td><td>152</td></tr><tr><td>5.2.3</td><td>Landau levels of 2D Dirac fermions</td><td>153</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>5.2.4</td>
      <td>Degeneracy of Landau levels</td>
      <td>160</td>
    </tr>
    <tr>
      <td>5.2.5</td>
      <td>Hamiltonian and wave functions in the Landau gauge</td>
      <td>166</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Extended and localised states in an electrostatic potential</td>
      <td>168</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>Guiding-centre dynamics</td>
      <td>171</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Confinement potential and edge states</td>
      <td>174</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Understanding the quantum Hall effect within the Landauer-Büttiker formalism</td>
      <td>176</td>
    </tr>
    <tr>
      <td>5.4.1</td>
      <td>Appearance of plateaus in the Hall resistance and percolation picture</td>
      <td>180</td>
    </tr>
    <tr>
      <td>5.4.2</td>
      <td>The role of the quantum Hall effect in metrology</td>
      <td>186</td>
    </tr>
    <tr>
      <td>5.4.3</td>
      <td>The role of the spin in the quantum Hall effect and appearance of plateaus at v = 2n + 1</td>
      <td>187</td>
    </tr>
    <tr>
      <td>5.4.4</td>
      <td>Relativistic quantum Hall effect in graphene</td>
      <td>189</td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>Introduction to topological matter</b></td>
      <td><b>193</b></td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Topology in condensed-matter physics – defect classification</td>
      <td>195</td>
    </tr>
    <tr>
      <td>6.1.1</td>
      <td>Characterisation of topological defects in order parameters</td>
      <td>196</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Back to the quantum Hall effect</td>
      <td>201</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Berry-ology – Berry phase, connection and curvature</td>
      <td>202</td>
    </tr>
    <tr>
      <td>6.3.1</td>
      <td>Adiabatic transport in reciprocal space</td>
      <td>204</td>
    </tr>
    <tr>
      <td>6.3.2</td>
      <td>Berry curvature</td>
      <td>206</td>
    </tr>
    <tr>
      <td>6.3.3</td>
      <td>Chern number</td>
      <td>211</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Berry curvature of a massive Dirac fermion and “half Chern number”</td>
      <td>213</td>
    </tr>
    <tr>
      <td>6.4.1</td>
      <td>Description a topological phase transition in terms of a half Chern number</td>
      <td>220</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Adiabatic pumping and Karplus-Luttinger anomalous velocity</td>
      <td>222</td>
    </tr>
    <tr>
      <td>6.5.1</td>
      <td>Quantum anomalous Hall effect</td>
      <td>225</td>
    </tr>
    <tr>
      <td><b>7</b></td>
      <td><b>Two-dimensional topological insulators</b></td>
      <td><b>229</b></td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Time-reversal symmetry</td>
      <td>231</td>
    </tr>
    <tr>
      <td>7.1.1</td>
      <td>Kramers’ theorem</td>
      <td>236</td>
    </tr>
    <tr>
      <td>7.1.2</td>
      <td>Berry curvature in time-reversal-symmetric systems</td>
      <td>241</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Haldane model – a topological insulator without time-reversal symmetry</td>
      <td>244</td>
    </tr>
    <tr>
      <td>7.2.1</td>
      <td>Tight-binding model</td>
      <td>246</td>
    </tr>
    <tr>
      <td>7.2.2</td>
      <td>Low-energy model</td>
      <td>253</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>7.2.3</td>
      <td>Experimental realisation of the Haldane model with cold atoms</td>
      <td>256</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Topological insulators with time-reversal symmetry</td>
      <td>260</td>
    </tr>
    <tr>
      <td>7.3.1</td>
      <td>Kane-Mele model – restoration of time-reversal symmetry</td>
      <td>260</td>
    </tr>
    <tr>
      <td>7.3.2</td>
      <td>Bernevig-Hughes-Zhang model</td>
      <td>265</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>Bulk-edge correspondence</td>
      <td>270</td>
    </tr>
    <tr>
      <td>7.4.1</td>
      <td>Massive Dirac fermions and mass change at an interface</td>
      <td>271</td>
    </tr>
    <tr>
      <td>7.4.2</td>
      <td>Chiral edge states in a topological insulator without time-reversal symmetry – quantum anomalous Hall effect</td>
      <td>278</td>
    </tr>
    <tr>
      <td>7.4.3</td>
      <td>Helical edge states of a topological insulator with time-reversal symmetry – quantum spin Hall effect</td>
      <td>284</td>
    </tr>
    <tr>
      <td><b>8</b></td>
      <td><b>Three-dimensional topological materials</b></td>
      <td><b>291</b></td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>A bit of history – the 3D Dirac equation</td>
      <td>292</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>3D topological insulators and surface states</td>
      <td>298</td>
    </tr>
    <tr>
      <td>8.2.1</td>
      <td>Angle-resolved photoemission experiments</td>
      <td>299</td>
    </tr>
    <tr>
      <td>8.2.2</td>
      <td>Bulk-boundary correspondence in 3D</td>
      <td>301</td>
    </tr>
    <tr>
      <td>8.2.3</td>
      <td>Relativistic quantum Hall effect at the surface</td>
      <td>306</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Weyl semimetals</td>
      <td>309</td>
    </tr>
    <tr>
      <td>8.3.1</td>
      <td>Berry curvature of a Weyl fermion and magnetic monopoles</td>
      <td>311</td>
    </tr>
    <tr>
      <td>8.3.2</td>
      <td>Weyl semimetals and Fermi arcs</td>
      <td>313</td>
    </tr>
    <tr>
      <td>8.3.3</td>
      <td>Chiral anomaly</td>
      <td>320</td>
    </tr>
  </tbody>
</table>
# Chapter 1

## Introduction: quantum
phenomena at the mesoscopic
scale

Electric transport in materials is doubtlessly one of the central issues in modern technology-based civilisations. Metals, such as copper, are used in electric wires that provide our households with the necessary electricity to light our rooms and to run basic appliances. Semiconductors are the basis of high-performance electronic devices, e.g. our computers or smartphones. A deep understanding of the electronic properties of materials and devices is therefore not only a field of research and knowledge shared by few university scholars but also necessary for students, be they interested in modern condensed-matter physics or plan they to become technicians and engineers that conceive, construct and maintain such devices.

As fundamental as it is, electronic transport is also an extremely complex field of condensed-matter physics that appeals to different (length) scales. One may intuitively suppose that macroscopic transport, e.g. in household appliances, needs to be treated differently from the transmission of electronic signals in computer chips. And indeed, the size of the latter becomes so small that quantum effects turn out to be relevant even if it is at an intermediate length scale (some tens or hundreds of nanometers) larger than the microscopic size of atoms. While the link between the microscopic and the macroscopic worlds, with their different laws, is provided by statistical
physics, these intermediate mesoscopic<fnref n="1" /> length scales require a special treatment, which is a major motivation for these classes and lecture notes.

## 1.1 Classical electric transport – Ohm and Drude

The probably first quantitative account of electric transport in conductors was given by G. S. Ohm in 1827, whose famous law

$$I = GV \qquad (1.1)$$

showed a linear dependence of the electric current *I* on the applied voltage *V*. The proportionality constant between the two is nothing other than the conductance *G* = 1/*R*, which is the inverse of the electric resistance *R*. Naturally, all these quantities are macroscopic ones, and Ohm also realised that the conductance depends on the geometry of the metallic rod that he was investigating (see figure 1.1). Indeed, the conductance

$$G = \sigma \frac{S}{L} \qquad (1.2)$$

is proportional to the cross section *S* of the conductor and inversely proportional to its length *L*. It is important to stress that Ohm's law is a purely empirical one obtained experimentally. Again, *σ* is a proportionality constant called conductivity. Contrary to the conductance, it is a material-specific quantity that accounts for the conduction properties in a material and its chemical composition – to give an example, the conductivity of copper is different from that of iron.

While physics is primarily an experimental science, it is natural to be curious about the theoretical models that provide understanding of the observed phenomena. In the present situation, one may then wonder about the underlying (microscopic) mechanisms of Ohm's empirical law (1.1). A major breakthrough in this direction was the discovery of the electron in 1897, an elementary particle carrying the elementary charge –*e*.<fnref n="2" /> Three years later based on this discovery, P. Drude proposed a first microscopic theory of the electric conductivity, using the recently developed framework of the kinetic

<footnote n="1">¹From the Greek “mesos” which means “middle”.</footnote>

<footnote n="2">²Throughout these notes, we use a convention where the elementary charge *e* is positive.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 1.1: Sketch of a conductor of length L and cross section S. The applied voltage V = V_A - V_B induces a current I, according to Ohm's law. On the microscopic level, the electric conduction is characterised by the electric field E and the current density j in the conductor.</figcaption></figure>

theory of gases – a prototype of statistical physics. The microscopic approach is based on Newton’s law,

$$ \frac{d\mathbf{p}}{dt} = -e\mathbf{E}, \qquad (1.3) $$

according to which the variation in time of a particle’s momentum p is given by a force, here the electric force $-e\mathbf{E}$, which a particle of charge $-e$ experiences in an electric field E.

How can this microscopic law be used to obtain a current? We first need to make a connection between the microscopic variables (here the momentum p of the particle) and the macroscopic quantity, which is the current and that is related to the current density j by $I = j_{\perp}S$, where $j_{\perp}$ is the component of the current density perpendicular to the cross section of the conductor (see figure 1.1). At this point, the attentive reader should protest for the first time: while p is the momentum of an individual particle, j as a current density is an average quantity obtained by taking the average over a large number of charged particles. Indeed, we have tacitly made a statistical assumption that all charged particles behave in the same manner and independently from one another. We may then relate the current density

$$ \mathbf{j} = -en_{el}\langle\mathbf{v}\rangle = -en_{el}\left\langle\frac{\mathbf{p}}{m}\right\rangle \qquad (1.4) $$

to the average momentum $\langle\mathbf{p}\rangle = m\langle\mathbf{v}\rangle$ or the average velocity v of particles of mass m. Furthermore, $n_{el}$ is the particle density.

Again you should protest loudly “Hold on! There is a problem here!” We have supposed, via Newton’s law, a proportionality relation between the current density, which is itself proportional to the current, and the average momentum of the particles. Since Ohm’s law stipulates that the current should be proportional to the voltage drop $V = -e|\mathbf{E}|L$, the current density
must thus grow linearly with the electric field **E**, i.e. **j** ~ **E**. However, based on Newton's law, one would expect a proportionality between the time derivative of the current density **d**j/dt and the electric field and not of the current density **j** itself. In order to heal this problem, Drude needed to introduce a relaxation process characterised by a relaxation time τ, e.g. due to scattering that the charged particle experiences while passing through the conductor, and Newton's law should therefore be modified to

$$ \frac{d\mathbf{p}}{dt} = -e\mathbf{E} - \frac{\mathbf{p}}{\tau}. \qquad (1.5) $$

If we now search for a stationary solution, d**p**/dt = 0, we obtain the searched-for equation that relates linearly the current density to the electric field,

$$ \mathbf{j} = \frac{e^2 n_{el} \tau}{m} \mathbf{E} = \sigma_0 \mathbf{E}, \qquad (1.6) $$

where σ₀ = e²n₀eτ/m is the so-called Drude conductivity. The expression (1.6) is sometimes also called Ohm's law – it is actually its microscopic version – and the macroscopic version is retrieved with the help of **E** = V/L and **I** = |**j**|**S** neglecting the vectorial character of the quantities.

## 1.2 A hint of quantum mechanics in classical transport

While the Drude model has been extremely successful and is still used to characterise the electronic properties of conducting materials, it implies severe conceptual problems. The major one is the assumption of free electrons that underlies the validity of Newton's law: to obtain the Drude conductivity in Eq. (1.6), we have supposed that electrons, even if they are displaced in a material – in a crystal formed out of atoms –, follow an energy-momentum relation ϵ = p²/2m of electrons in free isotropic space. This point is tacitly understood if we use the equation **v** = ∇**p**ϵ = **p**/m relating the particle's velocity to its momentum. The validity of this approach is highly questionable for the electronic motion in materials and, to spoil parts of the following notes, it is even utterly wrong in many circumstances.

However, before discussing the limits of the Drude model, let us see how physicists argued during several decades why this model remains valid. The
<figure><img src="image_5.png" /><figcaption>Figure 1.2: (a) Parabolic energy-momentum (dispersion) relation of a free classical particle. (b) Energy bands (energy as a function of the lattice momentum) for electrons in silicon. The capital letters on the x-axis indicate different points in reciprocal space that are connected by straight lines to allow for a two-dimensional plot of the energy bands while the lattice momentum **q** is a three-dimensional vector. The dark grey region indicates an energy window in which there are no electronic states. This energy window is called the gap.</figcaption></figure>

argument is based on a quantum-mechanical treatment of the electronic mo-
tion in a periodic potential that mimics the underlying atomic lattice felt by
the electrons. As it is discussed in great detail in Chap. 2, the periodicity of
the lattice shows discrete translational invariance. Remember the continu-
ous translation symmetry of free space, which is certainly better known and
that is used for a quantum-mechanical treatment of free particles in terms
of plane waves. Indeed, this symmetry allows us to classify the electronic
states and their energies by their momentum **p**, via the above-mentioned
energy-momentum relation ϵ = p²/2m. From a more formal point of view,
the eigenstates of the momentum operator are also eigenstates of the Hamil-
tonian and thus allow us to easily diagonalise the latter; and its eigenvalues
are good quantum numbers.

In 1928, F. Bloch showed that these general arguments may also be
adapted to the discrete translation symmetry of a periodic lattice, in which
case one may also define a momentum, that is sometimes called *lattice mo-
mentum* or *quasi-momentum* **q** and that remains a good quantum number.
There is however a price to pay: the energy-momentum relation, which we
will henceforth call dispersion relation or band E(q), is no longer as simple
as in the case of free particles. First, there is generally not only one band but
as many bands as electronic states in the unit cell, which is the elementary
brick of the periodic lattice. Second, the bands are periodic in the so-called
reciprocal space spanned by the lattice momenta. As an example, the elec-
tronic bands of silicon are shown in figure 1.2(b) in comparison to the disper-
sion relation of a free particle with the parabolic energy-momentum relation
ϵ = p²/2m [figure 1.2(a)]. Notice that the lattice momentum in a three-
dimensional crystal is itself three-dimensional, and it is therefore challenging
to represent the bands graphically.<fnref n="3" /> As a convention, one uses particular
straight paths between high-symmetry points in reciprocal space, which are
marked by captical letters in figure 1.2(b), and plots the evolution of the
energy along these paths.

Bloch's theory of electronic bands in crystals turned out to be a milestone
in the understanding of the electronic properties of materials. Most saliently,
it allows us to understand why some materials are conducting (metals) while
others do not conduct electricity (insulators or semiconductors). In order
to appreciate this point, remember the quantum-statistics chapter of your
statistical-physics classes. Electrons are fermions that are constrained by
Pauli's exclusion principle which states that each quantum state can only
be occupied by a single fermion.<fnref n="4" /> Therefore, fermions occupy an energy
spectrum successively from below, *i.e*. they occupy first the states of lowest
energy such as to minimise the total energy. In the case of our electrons in
a crystal, the bands are similarly filled from below until the *Fermi level* (or
*Fermi energy*), which is the energy of the last occupied state and that de-
pends therefore on the electronic density. Now, two different situations may
arise: first, the Fermi level crosses a band so that the last occupied state
is infinitesimally close in energy to the first unoccupied one. In this case,
electrons can easily travel through the crystal, and one obtains a metal. In
the second case, the Fermi level resides in an energy window, where no quan-
tum states are present, *e.g*. in the dark grey region shown in figure 1.2(b).

<footnote n="3">³I do not know about your cognitive capacities, but personally I have difficulty in visualising a four-dimensional object.</footnote>

<footnote n="4">⁴Here, a quantum state is a state in which all quantum numbers are fixed, be they orbital or internal degrees of freedom such as the electronic spin. Sometimes one encounters another convention in the literature, where only orbital degrees of freedom define the quantum state, which can then host two electrons, one with spin up and one with spin down.</footnote>
An electron from the occupied states must therefore overcome this energy window, which is also called the (*band*) gap, in order to reach the lowest unoccupied state. As we will discuss in detail in Chap. 3, the electronic motion is then frozen at low temperatures when $k_B T$ is smaller than the band gap,<fnref n="5" /> and no current can flow. This is the typical situation of an insulator or a semiconductor. Notice that there is no clear physical distinction between an insulator and a semiconductor, but only a practical and, honestly, rather arbitrary one: if the band gap is larger than 1 eV, one speaks of an insulator, while a semiconductor has a band gap lower than 1 eV. From a technological point of view the latter can more easily be doped, *e.g.* via chemical substitution by other elements, such that the system becomes metallic.

Let us now get back to the Drude model and Ohm's law. As we have just seen, it might seem almost suicidal to use the expression (1.6) obtained for free electrons with a parabolic dispersion relation to mimic the electronic behaviour in crystals with its complex band structure. Consider, however, the bands in silicon of figure 1.2(b) and place the Fermi level a tiny bit above the band gap, *e.g.* if the system is slightly electron-doped.<fnref n="6" /> In this case, the additional electrons start to fill the formerly unoccupied band at its minimum in the vicinity of the point $X_1$. As you easily see, the band in the vicinity of this point is parabolic to lowest order in the wave vector around this point. Naturally, the parabolicity of the band is not given by the bare electron mass that enters the energy-momentum relation of a free electron, but dimensionally one may describe the parabolicity of the slightly filled band, $\epsilon_{X_1}(\mathbf{q}) = \mathbf{q}^2/2m_B$, by another mass $m_B$, which is called the *band mass*. This allows us to use Ohm's law in the form of Eq. (1.6) also for the electronic conductivity in some materials if we replace the bare electron by the band mass, $m \rightarrow m_B$.

Notice, however, that this replacement is yet not generally valid. As one may easily see in figure 1.2(b), the bands are not always well approximated by a parabolic dispersion. One may also encounter band crossings that are often linear rather than parabolic. If the Fermi level is placed in the vicinity of such a linear band crossing, one cannot attribute a band mass to the bands, and Eq. (1.6) needs to be revisited then. This situation arises, *e.g.*, in graphene – a one-atom-thick graphite layer that is still heavily investigated

<footnote n="5">⁵This is usually the case even at room temperature since a gap on the order of one electron-volt corresponds to temperatures in the 10 000 K range.</footnote>

<footnote n="6">⁶This can easily be achieved in silicon by phosphorus substitution. Phosphorus has one valence electron more than silicon and shares it easily with the rest of the crystal.</footnote>
due to its amazing electronic properties and reduced dimensionality – where
two band cross linearly at the Fermi level. The energy bands of graphene are
extensively discussed in Chap. 2, and we will show in Chap. 3 how Ohm’s
law is modified in this case and more general ones.

1.3 From classical to quantum transport

We have seen, in the preceding section, that a quantum-mechanical treat-
ment of electrons in a crystal is necessary to understand the basic transport
properties of materials. Quantum mechanics appears in two facets: it pro-
vides us with the energy bands – the generalised energy-momentum relation
of free particles – and one needs to take into account the fermionic nature
of the electrons to place the Fermi level in this band structure. However,
even in this case, the electric transport is essentially *classical* since the elec-
trons can be considered as point-like particles subject to scattering processes
that are taken into account via the relaxation time τ in Ohm’s microscopic
law (1.6). Now remember your basic quantum-mechanics classes in which
you were taught that one of the motivations for establishing the theory of
quantum mechanics was the particle-wave duality according to which par-
ticles, such as electrons, can behave as waves. This particle-wave duality
has been illustrated with free particles there, but should the wave character
not survive also in the case of electrons in a crystal? After all, the wave
functions associated with the eigenstates in periodic lattices are *Bloch waves*
that should have the same power to interfere as plane waves.

The answer to the question whether one should consider electrons in a
crystal as waves or particles is as simple or complex as in basic quantum
physics, e.g. in atomic physics or quantum optics: it is related to the phe-
nomenon of *decoherence*. Indeed, beyond a certain distance, which is pre-
cisely the *phase-coherence length*, the phase of the electronic wave function
is no longer correlated with its value at the origin such that the electron
can no longer interfere with itself along two different paths. For this to hap-
pen, one requires a particular type of scattering events, as one may see from
the following qualitative argument. The phase of the wave function has two
contributions: one takes into account the static plane waves ~ exp(iq·r/ħ)
in terms of the position **r** and the lattice momentum **q** that is commonly
used to describe interferences. The second one is the time-dependent con-
tribution ~ exp(iet/ħ) in terms of the electron's energy ε. In order to allow
<figure><img src="image_6.png" /><figcaption>Figure 1.3: Sketch of a conductor of length *L* and cross section *S*, as in figure 1.1. The green dots represent elastic scatterers in the form of lattice defects separated by the mean free path *l*<sub>e</sub>, and the big spiky red dots show inelastic scattering events spaced by the phase-coherence length *L*<sub>φ</sub>(*T*), which generally depends on the temperature *T*. The blue line indicates a possible path of the electron.</figcaption></figure>

for interferences along different paths, one must make sure that the latter
phase remains well-defined, *i.e*. the energy must not change erratically in
the scattering processes.

1.3.1 Elastic vs. inelastic scattering: mean free path and coherence length

The electron’s energy is indeed conserved as long as the scattering processes
are *elastic* that is the electron does not transmit energy to its scatterer. To
visualise such an elastic scattering process, you may think of a tennis ball
that hits a wall and bounces back. While its momentum has changed in
the process, the modulus of its velocity and thus its kinetic energy remain
the same. Such elastic scattering events are eventually the most likely ones
in materials: they occur in the form of electron scattering on static lattice
defects that are ubiquitous in matter. Notice a very important point here
that has often be the source of confusion: the atoms of the underlying lattice
are not scatterers! As we have seen, they provide – in the absence of defects
– a perfectly periodic potential for the electrons, and the lattice momentum
$q$, which is the quantum number associated with the discrete translation
symmetry, is therefore conserved. That is why we need to appeal to lattice
defects, which break this discrete translation symmetry, in order to scatter
electrons. Between two elastic scattering events, the electron moves *ballis-
tically* in the crystal and one may associate a characteristic length with the
average distance between two such events, the *mean free path* $l_e$, which varies
typically between 10 nm (in usual metals) and 1 µm in the case of very clean electron gases in semiconductor heterostructures (see figure 1.3). The mean free path is directly related to the relaxation time $\tau$ by

$$l_e = v_F \tau, \qquad (1.7)$$

where $v_F$ is the average (Fermi) velocity of mobile electrons in the vicinity of the Fermi level.

As mentioned above, the electrons maintain phase coherence in these elastic scattering events since their energy is preserved. This is no longer the case when we consider inelastic scattering, in which case the electron is scattered by an obstacle that absorbs part of the electron's energy or alternatively transmits energy to the electron. Such inelastic scattering occurs for example due to the interaction of the electron with the collective lattice vibrations or their quanta, called phonons. Furthermore, an electron can be scattered by another electron due to their mutual interaction, very much like a billiard ball that is hit by another one. These inelastic scattering events occur typically at larger length scales than the elastic scattering events, and this length scale is precisely the phase-coherence length $L_\phi$. Contrary to the mean free path, the phase-coherence length is strongly temperature-dependent. Indeed, the acoustic phonons are gapless excitations and their density is thus an algebraic function of temperature, while the density of static impurities is given by an activated behaviour $n_{\text{imp}} \sim \exp(-\Delta_{\text{imp}}/k_B T)$, where $\Delta_{\text{imp}} \sim 1$ eV is the energy to create a defect and $k_B$ is the Boltzmann constant.<fnref n="7" />

Similarly to the mean free path, one may also associate a characteristic time scale to the phase-coherence length that is called the coherence time $\tau_\phi$. However, the relation between length and time is not as simple as in the expression (1.7) for the mean free path because the electron undergoes a certain number of elastic scattering events between two inelastic ones, as it is shown in figure 1.3. The transport between two inelastic scattering events is therefore intrinsically diffusive – as opposed to the ballistic transport between two elastic scattering events – and one obtains

$$L_\phi = \sqrt{D\tau_\phi}, \qquad (1.8)$$

<footnote n="7">⁷Notice, however, that generally the mean free path does not diverge when one approaches zero temperature. Even if the density of static defects should then tend to zero, they are quenched, i.e. the time it takes for the lattice to reach the thermodynamic ground state with zero defects diverges itself. This is also the sense of the term “static” here – they dynamics of the defects is negligibly slow as compared to the typical time scales associated with the electronic degrees of freedom.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 1.4: Mesoscopic conductors, as defined by the length scales $l_e$ (mean free path) and $L_\phi$ (phase-coherence length). The regime $l_e \ll L \ll L_\phi$ defines the quantum diffusive regime in which electrons are elastically scattered throughout the conductor, but do not encounter an inelastic scatterer. The electron maintains its phase coherence and thus its capacity to interfere quantum-mechanically. The regime $L \ll l_e$ is that of ballistic quantum transport, where the electrons are not scattered at all during their passage through the conductor. The conductor can then be viewed as a wave guide, similarly to that for light in optics.</figcaption></figure>

in terms of the diffusion constant $D = v_F^2\tau/d$, which depends on the (elastic) relaxation time $\tau$ and the spatial dimension $d$, as it will be discussed in more detail in Chap. 3. The typical order of magnitude for the phase-coherence length is 1 µm in metals, and it can reach up to some 10 µm in semiconductor heterostructures.

### 1.3.2 The mesoscopic scale

We are now armed to appreciate under which conditions one needs to take into account the quantum-mechanical wave nature of the electrons in condensed-matter systems: it is all a question of size! As long as we are concerned with large macroscopic systems the size $L$ of which exceeds both mean free path and phase-coherence length, $L \gg L_\phi \gg l_e$, we do not need to worry about the wave nature of the electrons – the only effect of their wave nature is the formation of electronic bands, but the electric transport in itself does not depend on it. This is the classical ($\gg L_\phi$) diffusive regime in which one can safely use Ohm’s law (1.6) or a variant of it that takes into account the specific form of the bands in the vicinity of the Fermi level. However, as mentioned, high-tech devices depend on the miniaturisation of their electronic components, which can reach the nanometer scale, whence the term
<figure><img src="image_8.png" /></figure>

Figure 1.5: Experimental evidence for the violation of Kirchhoff’s laws and the limit of validity for Ohm’s law in the mesoscopic regime, $l_e \ll L \ll L_\phi$. The electrons can travel around either of the arms of the copper rings if they are injected on the left hand side. Classically, one would expect the total conductance $G$ to be simply the sum of the conductances $G_1$ and $G_2$ in each of the arms (Kirchhoff’s law for conductances in parallel). Quantum-mechanically, however, the electron can be transmitted as a wave simultaneously in both arms, in which case one expects quantum interferences, as it has been measured by Webb et al. (see the lower right panel). The interferences can be triggered by the flux $\phi$ in the ring (Aharonov-Bohm effect), and the interference term is then given by the red part in the displayed formula, $g \cos(2\pi\phi/\phi_0)$, in terms of the flux quantum $\phi_0 = h/e$. The amplitude of the interference is given by the quantum of conductance $g \sim e^2/h$. Adapted from R. Webb et al., Phys. Rev. Lett. **54**, 2696 (1985).

*nanotechnologies*. While they are not microscopic$^8$, they are not macroscopic either. They are on an intermediate, mesoscopic length scale, which we now define as the scale set by the mean free path and the phase coherence-length.

More precisely, we shall distinguish two main mesoscopic regimes, which are schematically represented in figure 1.4.

1. In the diffusive quantum regime ($l_e \ll L \ll L_\phi$), the system size is smaller than the phase-coherence length but larger than the mean free path. One therefore needs to take into account the quantum-mechanical nature of the electronic wave function even if the electrons are repeatedly scattered elastically on their way through the system, whence the term *diffusive* to denote this regime. In this regime, one encounters a large number of surprising quantum-mechanical phenomena that were heavily investigated in the 1980s and 1990s in the emergent field of quantum mesoscopic physics, which is still very active today. One may cite, e.g. weak localisation of electrons due to the destructive quantum

<footnote>$^8$Remember that the microscopic world of atoms is governed by the Angström scale, Å= $10^{-10}$ m.</footnote>
interference of their wave functions on various paths through the system or the Aharonov-Bohm effect, which may be viewed as the condensed-matter analogue of Young's double split experiment. Similarly to the latter, the system consists of a ring connected to a source and a drain so that an electron on its way from the source to the drain has the choice to pass the upper or the lowed half ring (see figure 1.5). While one would expect, according to Kirchhoff's laws that are directly based on Ohm's law, that the conductances of the upper and lower path then simply add up because they are simply conductances in parallel, this is not the case due to interferences caused by the disorder in the ring. The latter is modified by a magnetic flux inside of the ring the variation of which yields periodic minima and maxima in the total conductance. Generally, the conductance in this regime is on the average governed by Ohm's law, albeit with *universal conductance fluctuations* that are on the order of the quantum of conductance $e^2/h$.

2. In the ballistic quantum regime ($L \ll l_e \ll L_\phi$) one does not only have to take into account the quantum-mechanical nature of the electrons but also the fact that their transport is now ballistic, i.e. the probability to be scattered while passing through the system tends to zero. The relaxation time therefore ceases to be a relevant quantity because the time the electron spends in the conducting device is much shorter than the intrinsic relaxation time of the material. It is therefore natural to admit that in this regime Ohm's law is no longer valid. One would expect an infinite conductance, and even the concept of conductivity as a "microscopic" physical quantity ceases to be relevant. The electrons behave purely as waves in this regime, and the novel concepts which are required to describe the electronic transport in the ballistic quantum regime were strongly influenced by the transmission of optical waves, e.g. in wave guides. As we will discuss in detail in Chap. 4, the Landauer-Büttiker formalism for quantum transport may indeed be viewed as a bridge between transport in condensed-matter physics and wave transmission in optical media, and we will see that the conductance is given by the Landauer formula,

$$G = \frac{e^2}{h} \mathcal{T}, \qquad (1.9)$$

in terms of the quantum of conductance $e^2/h$ and a transmission coefficient $\mathcal{T}$.
The Landauer-Büttiker formalism for quantum transport will serve us to
understand the quantum Hall effect, in which current passes through chiral edge states formed at the borders of a two-dimensional electron system
submitted to a perpendicular magnetic field (Chap. 5) as well as in two-dimensional topological insulators (Chap. 7).

1.4 Quantum and topological matter

In the preceding sections, we have argued qualitatively that quantum me-
chanics is relevant for electronic transport in two aspects. First, in macro-
scopic systems, the behaviour of the electrons is governed by Bloch's theory
of electronic bands and the Pauli principle; and second, the wave nature of
the electrons needs to be taken into account if the system size is reduced
to the mesoscopic scale, given by the phase-coherence length and the mean
free path. In addition to this, quantum mechanics also plays a relevant
role in the understanding of collective phenomena that arise due to electron-
electron interactions, such as the different forms of magnetism or charge-
ordered states, or electron-phonon interactions such as superconductivity.
However, the study of collective N-particle phenomena is largely beyond the
scope of the present classes, where we stick to one-particle physics.

The above picture of the role of quantum mechanics in electronic systems
was heavily challenged with the advent of topological materials at the be-
ginning of the years 2000. As milestones one may mention the isolation of
graphene, which has already been mentioned above, in 2004/05. Contrary
to most semiconducting materials used in mesoscopic quantum physics, the
low-energy electronic properties in this material are no longer described in
terms of a Schrödinger equation with an effective band mass, but the elec-
trons may be viewed as *massless ultra-relativistic* particles that are governed
by the Dirac equation, which is normally used in high-energy physics to de-
scribe relativistic quantum mechanics. Another milestone was the prediction
in 2005 and its discovery in 2006 of (two-dimensional) topological insulators
that display a *quantum spin Hall effect*, an effect similar to the quantum Hall
effect. Losely speaking, electrons with spin up show a quantum Hall effect
generated by a magnetic field in one direction perpendicular to the system,
while electrons with spin down are exposed to a magnetic field in the opposite
direction.

Two-dimensional topological insulators are introduced in Chap. 7. Their
understanding is based on what may now be called *topological band theory*, which deals not only with the band spectrum but also with the Bloch states, i.e. the quantum-mechanical eigenstates. For a long time, the importance of these states has been underestimated, but they have quite unusual geometric properties that are encoded in what is now known as the *Berry curvature*. These geometric properties, which are discussed in Chap. 6, provide the electronic bands with a topological invariant in the form of an integer, the Chern number, which has profound physical consequences. As its name indicates, the Berry curvature is a curvature that describes a local quantity in some space. This space turns out to be the Hilbert space spanned by the Bloch states, and the Berry curvature can be viewed as the evolution of the Bloch states upon local variation of the lattice momenta, which span the basis state. Topology arises when one considers a global quantity, which is obtained by integrating the local Berry curvature over the entire basis space of lattice momenta. Since both spaces are compact, as it will be discussed in detail in Chap. 2 for the space of lattice momenta and in Chap. 6 for that of the Bloch states, a basic theorem of mathematical topology indicates that all continuous deformations of the system can be characterised by an integer, which is called *topological invariant*. This point is perhaps rather well known in the Gauss-Bonnet theorem, which the reader may have encountered in previous mathematics classes. The theorem deals with the surface *S* of an object or, more formally, a compact two-dimensional (Riemannian) manifold with a Gaussian curvature *K(r)*, which is our local geometric quantity. If one integrates the curvature over the full surface ∫*S* *dσ(r)* *K(r)*, where *dσ(r)* is the infinitesimal surface element at the position **r**, one obtains a global, topological quantity that is related to the number of holes of the object, called genus *g*,

$$g = 1 - \frac{1}{4\pi} \int_S d\sigma(\mathbf{r}) K(\mathbf{r}). \qquad (1.10)$$

The impatient reader may now ask: “Well, this is all nicely abstract, but what does this have to do with the physics of electrons in condensed matter, or even in physics in general?” Imagine that a physically measurable quantity could depend only on this topological invariant. This will then indeed be a very robust quantity: even upon local changes of the environment, say the landscape of lattice defects or charged impurities that should generally affect the electronic dynamics, such a quantity remains unaltered. While these local changes can be described as continuous deformations, as in the example illustrated in figure 1.6, the global topological invariant (in the example, the
<figure><img src="image_9.png" /><figcaption>Figure 1.6: Continuous deformation of a coffee mug into a donought and into the sculpture in the “Grand Hall” of École Polytechnique. All objects belong to the same homotopy class, characterised by the genus $g = 1$, which counts the number of holes here. This means that all different objects (with a single) hole have the same global topological invariant even if they have locally different surfaces given by the geometric curvature $K(\mathbf{r})$.</figcaption></figure>

genus $g$) being an integer cannot be changes continuously. All states that can be obtained by such continuous deformations, i.e. without changing this integer, fall into the same homotopy class, which is precisely given by the topological invariant.

It turns out that, under certain circumstances, the conductance is precisely such a robust quantity protected topologically. It reads

$$G = \frac{e^2}{h} n, \qquad (1.11)$$

where the integer $n$ is just the topological invariant mentioned above. This is the case for the (Hall) conductance in the quantum Hall effect, which is nowadays viewed as the paradigmatic prototype of a two-dimensional topological insulator, but also for that in two-dimensional topological insulators such as those studied in HgTe/CdTe quantum wells, where the quantum spin Hall effect was discovered in 2006.

Most saliently, these topological materials – not only two-dimensional topological insulators, but also topological superconductors or the three-dimensional topological insulators and Weyl semimetals discuss in Chap 8 – display an intriguing *bulk-boundary correspondence* according to which the bulk invariant determines the presence (and the number) of very special conducting states at the boundary. These boundary states happen to be protected topologically and cannot be removed. Even if one cleaves the material into two parts, the newly generated surface happens to be conducting. To illustrate this highly unusual situation, imagine a tablet of chocolate wrapped in a sheet of aluminium that represents the conducting boundary: in order to get the chocolate, which you are usually more interested in than in the
aluminium sheet, break the tablet along with the aluminium cover into two part. If the tablet were a topological insulator, you would encounter the frustrating situation to never get access to the chocolate because the liberated section of the chocolate would immediately be covered by aluminium again. In two spatial dimensions, a topological insulator is thus accompanied by robust conducting edge states which can be viewed as one-dimensional metals. It is precisely these edge channels that conduct electricity in the quantum Hall effect. Moreover the direction of transport is imposed by the orientation of the magnetic field (see Chap. 5) so that the transport in these channels is perfectly ballistic: each channel contributes then a quantum of conductance $e^2/h$ to the conductance that is at the origin of the precise conductance quantisation as stipulated by Eq. (1.11) in this system.

To conclude this section, we notice that topological materials, in contrast to the more conventional metals and semiconductors used in nanotechnologies, do not require a miniaturisation down to system sizes below the phase-coherence length or the mean free path in order to display true quantum transport. It is the bulk-boundary correspondence which ensures the formation of ballistic channels, e.g. in two-dimensional topological insulators. Most saliently the associated quantised conductance (1.11) indicates clearly that Ohm's law is no longer valid in this case: the conductance is completely independent of the geometric character of the conductor. It does neither depend on the length of the conductor nor on a cross section as it is stipulated by Eq. (1.2). On the contrary, the associated ballistic transport is described to great precision by the Landauer formula (1.9), in which case the transmission coefficient $T$ is equal to one per conducting channel.

## 1.5 Materials and sample fabrication

The request for new concepts to describe transport at the mesoscopic scale or in topological materials, as those mentioned in the previous section, did not pop up out of the blue. It is closely related to material synthesis in two and three dimensions on the one hand side and device fabrication on the other side. This section is therefore devoted to these two aspects on a very qualitative level. Since we are dealing with conducting systems, the devices are based on metals or semiconductors (see figure 1.7). Generally, metals have rather short mean free paths due to intrinsic disorder in the material. They are on the order of 10 nm, far below the system size. The latter can
<figure><img src="image_10.png" /><figcaption>Figure 1.7: Characteristics of nanocircuits (a) The mean free path in metallic devices is usually in the 10 nm range so that they are mainly in the diffusive regime, but the phase-coherence length can be on the order of 1 µm and exceed the system size. (b) Larger mean free paths can be achieved in semiconducting devices and graphene (not shown here) where the mean free path $l_e \sim 1...10$ µm can be on the same order of magnitude as the phase-coherence length. These devices can therefore be either in the diffusive or ballistic regime.</figcaption></figure>
nevertheless be below the phase-coherence length, which is in the micron range at low temperatures, so that one may achieve the quantum diffusive regime. The ballistic regime can be obtained in semiconducting devices based on modulation-doped semiconductor heterostructures where mean free paths in the 1...10 µm range can be reached, similarly to graphene-based devices.

### 1.5.1 Fabrication of nano-circuits via lithography

Before discussing how to obtain high-mobility semiconductor devices, let us first take a look at the nanofabrication techniques used to obtain quantum circuits at the nano scale. They are generally based on electron lithography, whose principle is the following (see figure 1.8). The insulating substrate of the future nanocircuit is first covered by a polymer (PMMA) layer that is sacrificed in the process. The polymers are locally bombarded by a focused electron beam that destroys the polymer bonds. This is the proper lithographic process that gives its name to the full procedure. The electron beam can be moved laterally in space and thus allows one to draw the future circuits. The destroyed polymer bonds are then etched away with the help of a chemical solvant that leaves the non-bombarded polymer mask unharmed, and the insulating substrate is thus locally uncovered at the bottom of the trenches. This polymer mask can be viewed as the negative of a photo, and the trenches are the future metallic circuits. The full system is then covered with a metallic film with the help of a (hot) source where the metal is evaporated, before it is put into a bath of another chemical solvant in the lift-off process. This solvent removes the originally unharmed polymer mask but does not attack the metallic parts at the bottom of the trenches that are in direct contact with the substrate. One thus obtains metallic circuits or wires on an insulating substrate that have been designed by the electronic beam. In principle, one may also use focused light for the lithographic process (optical lithography), but electron lithography has the advantage to be more precise thus allowing for a smaller wire width.

Notice that this lithographic technique is also used in semiconducting devices or graphene. In this case, the circuit is not directly etched in the material, but the electron system is first covered by an insulating (cap) layer on which metallic gates are designed with the help of the same lithographic process exposed above. These gates are then used to define electrostatically the circuit in the 2D electron system, e.g. be depleting the regions below the metallic gates. An example is shown in the right panel of figure 1.7(b),
<figure><img src="image_11.png" /></figure>

Figure 1.8: (a) Illustration of electron lithography. In a preparatory step, an insulating substrate (for example Si) is covered by a polymer (PMMA) layer. The polymer bonds are then locally destroyed, in the lithographic process, by a focused electron beam that can be controlled in the *xy*-plane such as to design the future circuits [see inset (b)]. The destroyed polymers are then removed in an etching process with the help of a chemical solvent that leaves the non-exposed parts of the polymer layer intact. This “negative” is the mask for the nanocircuits, which are now visible in the form of trenches up to the (Si) substrate. The systems is now covered with metal within an evaporation from some source. The metal layer covers both the bottom of the trenches and the mask, which is then sacrificed by a chemical solvent that lifts off the mask while it does not affect the metallic parts at the bottom of the trenches. One thus obtains a metallic nanodevice directly on the substrate that can then be contacted by larger metallic contacts (usually via micro-welding). (c) Photo of the apparatus used at C2N, Palaiseau (France), to perform the electron lithography. (d) and (e) Examples of lithographically obtained nanocircuits (from Institut Néel, Grenoble. (d) Nanowires connected to metallic contacts (light grey regions). (e) Square lattice of nanowires. The width of the nanowires is on the order of 50 nm and the lateral size of a square is 640 nm.
which shows a photo of a quantum dot defined electrostatically by the grey
metallic gates.

1.5.2 Fabrication of two-dimensional electron systems

As seen in the previous paragraph, the nanocircuits can be obtained litho-
graphically, but in the case of semiconducting devices, one first needs to fab-
ricate the 2D electron gas which serves as the basis of the device. There are
several semiconductors used to obtain 2D electron gases, the most popular
of which is the AlGaAs/GaAs heterostructure fabricated via molecular-beam
epitaxy (MBE). The 2D electron gas is then formed at the interface between
two types of semiconductor with different band gap.

Before explaining why the interface between two semiconductors can host
a metallic 2D electron gas, let us first see how an atomically sharp interface
can be obtained. Indeed, these systems are grown artificially by MBE the
principles of which are sketched in figure 1.9(a). The crystal is grown atomic
layer by atomic layer on a substrate in an ultra-high vacuum that is required
to avoid accidental contamination by particles in the air. The growth is
ensured by ovens in which the elements constituting the final compound
(here Ga, As, Al and Si) are evaporated and ejected towards the substrate.
Shutters are used to control the atomic flux and can be opened and closed
during the process. In the case of the AlGaAs/GaAs heterostructure, which is
shown in figure 1.9(b), the growth starts with only two elements (Ga and As)
in equal amount to obtain a first layer of GaAs. After the desired thickness,
the Al oven is opened to continue with a layer of AlGaAs of thickness *s*. This
thickness is determined by the experimental use one wants to make of the
final compound and typically varies from some 10 nm to some 100 nm. After
this spacer layer is achieved, some Si atoms are added that serve as electron
dopants. Indeed, Si has one additional electron as compared to Ga and As
and can share it with the AlGaAs host matrix. The process is terminated
by shutting the Si and Al ovens to obtain a protective GaAs layer. During
the process, the substrate is heated from behind in order to ensure a highly
controlled growth of the crystal and to avoid any kind of lattice defects, such
as domain walls or dislocations.

Here, we have illustrated the MBE growth of a typical AlGaAs/GaAs het-
erostructure, but other elements can be and are commonly used to obtain dif-
ferent types of compounds. As an example, one may mention CdTe/HgTe/CdTe
quantum wells that are the prototype of a 2D topological insulator displaying
<figure><img src="image_12.png" /><figcaption>Figure 1.9: (a) Principles of molecular-beam epitaxy (MBE) in ultra-high vacuum. The elements that constitute the final compound are heated in ovens, and their flux towards the substrate is regulated by shutters. The crystal is grown (atomic) layer by layer on the substrate that is heated from behind such as to ensure a controled growth. (b) Sketch of the potential in the upper (conduction) band of a semiconductor heterostructure that consists of GaAs (for $z > 0$), an AlGaAs spacer at $z < 0$ of thickness $s$ and $n$-type doped AlGaAs of width $d$. The structure is then protected by a GaAs cap layer. (c) Schematic cut of the MBE chamber and its ovens. (d) Photo of the MBE apparatus at C2N, Palaiseau (France).</figcaption></figure>
the quantum spin Hall effect, which we discuss in detail in Chap. 7. However, let us concentrate on the AlGaAs/GaAs heterostructure and discuss how the 2D electron gas is formed at the interface between GaAs and the AlGaAs spacer. First, one needs to know that GaAs and AlGaAs, in spite of their structural and electronic similarity, have a different band gap that separates the conduction from the valence band. The energy minimum of the conduction band is depicted in the upper part of figure 1.9(b). It is slightly lower in the GaAs part than in the AlGaAs part and would lead to a step. However, the Si atoms in the *n*-doped part of AlGaAs yield dopant levels below the conduction-band minimum there. Even if they are spatially separated from the GaAs part (over a distance *s*), it is energetically favourable for the extra electrons provided by the Si dopants to populate the conduction band of GaAs situated below the dopant levels. This charge separation (extra electrons in the GaAs part and positively ionised Si atoms in the AlGaAs part) yields a constant electric field *E* over the spacer and thus a linear potential offset *eEz* in addition to the energy step in the conduction band. This provides a triangular confinement potential in the GaAs part at the interface with the AlGaAs spacer over a distance that is itself – by electrostatics – on the order of the spacer thickness *s*. This confinement potential is at the origin of quantised electronic states at the GaAs/AlGaAs interface the lowest level of which is represented by the characteristic wave function *u*₀(*z*). Usually, the Fermi level resides in this lowest level, which is separated from other levels by the confinement gap so that one is confronted with a quenched electron dynamics in the *z*-direction. Since the motion in the *xy*-plane remains that of free particles, one obtains an effective 2D electron gas. The potential in the dopant layer then varies quadratically as it is shown in the figure, but this is no longer of interest for the electronic properties of the 2D electron gas. These so-called *modulation-doped semiconductor heterostructures* have been at the origin of a true technological revolution in semiconductor physics and the fabrication of high-mobility nano-devices. It is namely the spacer level that allows one to push the dopants, which are a necessary ingredient but that also act as undesired scatterers, far away from the interface where the 2D electron gas is formed. The resulting high mobility is at the origin of the large mean free paths in these systems.

Once a wafer of sufficient size is obtained by MBE, quantum circuits can be defined via the above-mentioned lithographic processes. Furthermore, the crystals can be cleaved prior to the patterning of the circuits in order to contact the 2D electron gas directly by lateral Ohmic contacts.
<figure><img src="image_13.png" /><figcaption>Figure 1.10: (a) Exfoliation of graphene. A graphite chip is thinned by repeated folding and unfolding of the scotch tape on which the chip is placed, and a last exfoliation takes place by putting the scotch tape on an insulating substrate on which some graphitic layers are thus deposited. (b) Search for graphene layers by optical microscopy. The thinner the layer, the lower is the contrast between the covered and the empty parts of the substrate.</figcaption></figure>

While these semiconductor heterostructures are still commonly used in
the fabrication of electronic nanodevices, another type of 2D electron systems
is heavily investigated ever since the advent of graphene in 2004/05. These
systems consist of truly two-dimensional crystals whose width is just that
of a single atom, or several atoms in the case of more complex unit cells.
Originally these crystals have been obtained with the help of the so-called
exfoliation technique. It consists of peeling off a layered three-dimensional
crystal, such as graphite, where the atomic layers weakly stick together by
the van-der-Waals force, while the atoms inside each layer are held together
by strong covalent bonds. This is the reason why we can easily draw with a
pencil that has a graphite lead: each time we draw a line on a piece of paper,
we peel off graphitic layers that stick then to the paper. It is this property
that A. Geim and K. Novoselov used in 2004 to isolate graphene with the help
of a simple scotch tape (Nobel Prize in 2010). The receipe is quite simple:
place a thin graphite chip on the right part of a piece of scotch tape and glue
the left part on the right part.<fnref n="9" /> Then separate the two parts of the scotch
tape again. Due to the relatively weak van-der-Waals interaction between
the graphitic layers as compared to the cohesive force of the scotch tape,
the chip is cleaved into two parts. Repeat this procedure about ten times so
that the chip is exfoliated into thinner and thinner graphitic layers. In a last

<footnote n="9">⁹If you prefer, you may inverse right and left here... It does not really matter in this case.</footnote>
<figure><img src="image_14.png" /><figcaption>Figure 1.11: (a) Examples of exfoliated 2D materials beyond graphene. (b) Nanodevice based on an $MoS_2$ monolayer encapsulated in boron-nitride (BN) layers.</figcaption></figure>

exfoliation process, glue the dirty scotch tape on an insulating substrate<fnref n="10" /> and gently lift off the scotch tape [see figure 1.10(a)]. Some graphitic layers are thus transferred to the insulating substrate, which is then investigated via a microscope to search single graphene layers. A typical photo is shown in figure 1.10(b). The optical contrast, which happens to be particularly good on the originally used $SiO_2$ substrates, allows one to identify single-layer from bilayer and multi-layer graphene, and the desired graphene sample can then be contacted by electric contacts.

Ever since the first exfoliation of graphene samples on $SiO_2$ substrates, the technique has been used with great success in the fabrication of other 2D materials [see figure 1.11], such as the 2D transition-metal dichalcogenides $MoS_2$ and $NbSe_2$ or boron nitride. The latter material is particularly important because it is nowadays used as the standard insulating substrate to host graphene samples. It shares with graphene the same lattice structure – the atoms are arranged in a honeycomb lattice – and it is almost commensurate

<footnote n="10">¹⁰At the beginning, most people used insulating $SiO_2$ wavers whose back side consisted of doped Si. This doped part later served as a metallic backgate that allows one to control the electronic density in the graphene devices by applying a voltage between the backgate and the graphene layer.</footnote>
with graphene. Graphene samples encapsulated in boron-nitride layers are
today's highest-mobility samples in which mean free paths of several tenths
of microns can be achieved. Furthermore, exfoliated samples can nowadays
be transferred with great precision so that one can fabricate heterostructures
of layers with different 2D crystals. Encapsulation with insulating boron-
nitride layers finally allows for a long life time of the samples, which can
then be contacted and patterned by metallic gates in a similar manner as the
semiconductor heterostructures, as it is shown in figure 1.11(b) for a MoS<sub>2</sub>
sample.

1.6 Outline of the lecture notes

The remainder of these lecture notes follow this introduction, which was
simply intended to serve as an appetizer. The author does not expect the
reader to have understood all mentioned concepts which have not yet been
defined nor explained in detail. The main course is the following. Chapter 2 is
a reminder of the basic band theory following Bloch's quantum-mechanical
treatment of electrons in a periodic potential. This sets the stage for a
(quantum) statistical treatment of the conductivity in materials with a band
structure that is not necessarily described in terms of "free" electrons, where
we simply replace the bare mass by a band mass. This statistical treatment
yields a generalised version of Ohm's law in terms of the *density of states*,
rather than a band mass (Chap. 3). Furthermore, we review Bloch's band
theory in a language that anticipates topological band theory, which will be
introduced in Chap. 6 along with some very basic aspects of topology in
physics. However, before, we discuss the limits of Ohm's law and its variants
in systems at the mesoscopic scale, where the electronic transport needs to
be treated within an alternative approach that is given by the Landauer-
Büttiker formalism (Chap. 4). Chapter 5 is a central one of the present
notes: it yields an introduction to the quantum Hall effect. On the one hand
side, transport in the quantum Hall effect is an excellent illustration of the
Landauer-Büttiker formalism that describes to great accuracy the ballistic
electron transport, which occurs at the chiral sample edges. On the other
hand, it is the prototype of a topological insulator and its bulk-boundary
correspondence so that it is a bridge to the following chapters.

After the introduction to topological band theory in Chap. 6, Chap. 7 is
devoted to the study of two-dimensional topological insulators. The central
role of time-reversal symmetry is also discussed there, and we will illustrate
two classes of topological insulators with the help of specific models: the
Haldane model, which was proposed in 1988 long before the advent of topo-
logical insulators, illustrates the class of insulators with broken time-reversal
symmetry, and topological insulators with preserved time-reversal symme-
try are presented with the help of the Kane-Mele (2005) and the Bernevig-
Hughes-Zhang (2006) models. The bulk-boundary correspondance is equally
introduced in this chapter, and we will use the Landauer-Büttiker formalism
to describe theoretically the (ballistic) transport properties in these systems
that give rise to the quantum anomalous and the quantum spin Hall ef-
fects. Chapter 8 is finally devoted to three-dimensional topological materials
– both topological insulators and Weyl semimetals – after a short historical
introduction to the relativistic Dirac equation in three spatial dimensions.

These notes cover a series of nine lectures (along with exercise classes) and
can therefore not cover the entire field of quantum mesoscopic transport and
topological materials. They do not mean to be complete – a both necessary
and frustrating situation – but the author has tried to give them a certain
coherence via the question of electric transport. Even this field is not fully
covered, and many phenomena of quantum coherent transport in the diffusive
regime ($l_e \ll L \ll L_φ$), such as weak localisation or universal conductance
fluctuations, are only alluded to or discussed on a very qualitative level. Nat-
urally, the characterisation of the electronic properties in condensed matter
is not restricted to electric transport. Much insight, e.g. in the topological
structure of quantum matter, is obtained by spectroscopic means, be it opti-
cal (absorption and reflection, angle-resolved photoemission,...) or electronic
spectroscopy (scanning-tunneling spectroscopy, electronic Mach-Zehnder in-
terferometry,...). These spectroscopic techniques are not or only sparsely
covered in the present notes. Also a full mathematical description of topol-
ogy in Bloch bands is missing: homotopy classes and topological invariants
are only qualitatively introduced, mathematical proofs are merely sketched,
and the theory of fibre bundles is completely absent.

Another, albeit huge, field that is missing in the present classes is that
of correlated matter, that is matter in which the electrons can no longer
be described in terms of independent or *quasi-independent* particles that
do not interact among one another. This is naturally not true as you all
know: electrons carry charge and thus mutually repell each other due to the
Coulomb interaction. While in many cases these interactions are screened
and not sufficiently strong to invalidate the picture of independent particles –
apart perhaps from yet another renormalisation of there basic properties such
as their mass in Landau's theory of Fermi liquids<fnref n="11" /> – this is far from being
the general situation encountered in condensed-matter systems. Due to these
interactions, the electrons can form novel states of matter, such as different
magnetic or superconducting phases. They can even form an electron crystal,
called the Wigner crystal, that turns out to be a collective insulator, a state
of matter theorised by E. Wigner in 1934 and only recently (in 2021) observed
directly in two-dimensional transition-metal dichalcogenides. These states of
matter are the subject of a very active field of research in today's condensed-
matter research, and some correlated phases are also topological. This is
the case, for example, in the fractional quantum Hall effect. However, as
fascinating as they are, they cannot be covered in these lecture notes. Instead,
they would merit a full series of advanced classes along with the necessary
theoretical techniques to describe them that are beyond the scope of the
present classes.

<footnote n="11">¹¹Whence the term *quasi-independent*.</footnote>
Chapter 2

Bloch's theorem and
tight-binding models – a
reminder of band theory

The present chapter is meant to be a short reminder of band theory and
Bloch's theorem. The adaption of quantum mechanics to electrons in a pe-
riodic potential that is formed by the underlying regular lattice of (ionised)
atoms, is at the heart of the understanding of the electronic and also mag-
netic properties of solids. Covering all its aspects would therefore require a
book on its own, and an introduction to this field of physics is given in the
class "Electrons in Solids" (PHY552A, see also H. Alloul's book, in French).
Here, we will review only its essential aspects to provide the necessary basis
for both the quantum-statistical description of (classical) electronic transport
in materials with an arbitrary band structure (Chap. 3) as well as that for
Chap. 6, where topological band theory is introduced. In short, while a con-
ventional introduction to band theory is mainly concerned with the spectral
aspects, i.e. the electronic bands, and their related density of states respon-
sible for the thermodynamic properties of electrons in solids, the topological
aspects are encoded in quantities beyond these spectral properties. They
are encoded in the Bloch wave functions, i.e. the eigenstates of the periodic
Hamiltonian, and the present chapter therefore emphasizes the structure of
these eigenstates more than perhaps a conventional introduction to band
theory.
## 2.1 Translation symmetry in Bravais lattices

In order to describe the electronic motion in a crystal, let us briefly remember some basic aspects of quantum mechanics. We all know that quantum states are represented in terms of Hilbert-space vectors while a physical quantity – an observable – is described in terms of a Hermitian operator acting in this Hilbert space. In addition to these basic quantities, symmetry plays an essential role in quantum mechanics. A symmetry is associated with a unitary operator $U$ acting on the Hilbert-space states $|\psi\rangle$ and $|\phi\rangle$, $|\psi'\rangle = U|\psi\rangle$ and $|\phi'\rangle = U|\phi\rangle$, while leaving their scalar product invariant, $\langle\psi'|\phi'\rangle = \langle\psi|\phi\rangle$. The unitary operator $U$ represents a symmetry when not only the scalar product but also the Hamiltonian $\hat{H}$ of the system remains invariant, that is

$$U\hat{H}U^{-1} = \hat{H}. \qquad (2.1)$$

Quite generally, this unitary operator can be written as the exponential $U = \exp(i\varphi\hat{A})$ of an observable $\hat{A} = \hat{A}^{\dagger}$, which is called the *generator of the symmetry*, and a real (scalar) quantity $\varphi$. Equation (2.1) is then satisfied if the generator commutes with the Hamiltonian,

$$[\hat{A}, \hat{H}] = 0. \qquad (2.2)$$

As a consequence, the average value $\langle\hat{A}\rangle$ is a constant of motion, i.e. it is conserved in time, as it may be seen with the help of Ehrenfest's theorem

$$i\hbar \frac{d}{dt} \langle \hat{A} \rangle = \langle [\hat{A}, \hat{H}] \rangle = 0, \qquad (2.3)$$

where the average is taken over an arbitrary state $|\psi\rangle$, $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$.<fnref n="1" /> Moreover, the commutation between the generator and the Hamiltonian indicates that there exists a common basis that diagonalizes both the Hamiltonian and the generator of the symmetry. This is not only a profound mathematical statement but of great practical importance in quantum mechanics: instead of diagonalising directly the Hamiltonian, one may first want to find the

$$i\hbar \frac{d}{dt} |\psi\rangle = \hat{H} |\psi\rangle.$$

<footnote n="1">¹Here, we consider an observable that does not depend explicitly on time (Schrödinger picture), with $\partial\hat{A}/\partial t = 0$, while the state satisfies the Schrödinger equation</footnote>
eigenvalues and eigenstates of the generator. The latter are identical with the eigenstates of the Hamiltonian, which is therefore already diagonalised for free (at least in parts, as we may see below), and the eigenvalues are good quantum numbers that specify the spectrum of the Hamiltonian.

Let us illustrate this point with the help of a simple and well-known example that is translation symmetry for a free particle. Indeed, a translation by a vector **a** is represented by the unitary operator

$$ \mathcal{T}_{\mathbf{a}} = e^{i\mathbf{a} \cdot \hat{\mathbf{p}}/\hbar}, \qquad (2.4) $$

which acts on the wave function $\psi(\mathbf{r}) = \langle \mathbf{r} | \psi \rangle$ and the position operator $\hat{\mathbf{r}}$ as

$$ \mathcal{T}_{\mathbf{a}} \psi(\mathbf{r}) = \psi(\mathbf{r} + \mathbf{a}) \quad \text{and} \quad \mathcal{T}_{\mathbf{a}} \hat{\mathbf{r}} \mathcal{T}_{\mathbf{a}}^{\dagger} = \hat{\mathbf{r}} + \mathbf{a}, \qquad (2.5) $$

respectively. Here, the momentum operator $\hat{\mathbf{p}}$ is the generator of spatial translations, and its average value $\langle \hat{\mathbf{p}} \rangle$ is naturally conserved for translationally invariant systems. Furthermore, its eigenstates $|\mathbf{p}\rangle$, associated with the plane-wave states in a volume $V$

$$ \psi_{\mathbf{p}}(\mathbf{r}) = \langle \mathbf{r} | \mathbf{p} \rangle = \frac{1}{\sqrt{V}} e^{i \mathbf{p} \cdot \mathbf{r}}, \qquad (2.6) $$

form a convenient basis of the system, i.e. they diagonalize the translationally invariant Hamiltonian of a free (relativistic or non-relativistic) particle.

### 2.1.1 Bravais lattice

Here, the reader may ask how this illustration could help us to understand electrons in solids. Naturally, these electrons are far from being free since they are subject to a potential generated by the atoms forming the underlying lattice. Naturally, the crystal is overall charge-neutral so that the potential vanishes on the average. However, since the electrons responsible for the electronic properties of a material stem from the valence electrons in the outer orbitals of the atoms, we consider these electrons to be (partially) ionised for the moment and put back the electrons later on. The periodic potential, which is felt by an electron and that we consider here, is thus the potential created by these ionised atoms. While continuous translation symmetry is clearly broken by the potential, the latter is periodic if we consider a perfect lattice<fnref n="2" /> and therefore possesses a discrete translation symmetry. In contrast

<footnote n="2">²In the following, we will make this bold assumption and neglect the inevitable lattice defects or surfaces unless stated explicitly.</footnote>
<figure><img src="image_15.png" /><figcaption>Figure 2.1: Decomposition of the *kagomé* lattice. *Left*: allowed lattice translations (violet arrows) – these translations reproduce the lattice as one may see from the inset (violet box). The translation displaces the dashed violet box to the box delimited by the continuous violet line. The latter box is perfectly superposed with the original lattice pattern, as well as for all other translations that connect sites of the same color. *Right*: forbidden translation connecting sites of different color. The translated box clearly shows that the lattice periodicity is not conserved by this translation since the pattern in the box is not superposed with that of the original lattice. The underlying Bravais lattice, formed by sites of the same color, is a triangular lattice, and the basis (unit cell) consists of three sites of different color connected by the vectors $\mathbf{\delta}_1$ and $\mathbf{\delta}_2$ (in the circle of the left panel).</figcaption></figure>
to a continuous symmetry, where all translations by an arbitrary vector **a** are allowed, this discrete symmetry means that only translations by multiples of some basis vectors **a**_i_ are permitted and leave the system invariant. In three dimensions the lattice is then periodic in all *lattice vectors*

$$ \mathbf{R}_j = j_1 \mathbf{a}_1 + j_2 \mathbf{a}_2 + j_3 \mathbf{a}_3, \qquad (2.7) $$

where the subscript j represents the three integer numbers (positive or negative) j_1, j_2 and j_3, and the *elementary* or *primitive* lattice vectors **a**_i_ need to be non-coplanar, or mathematically the mixed product must not be zero, **a**_1_ · (**a**_2_ × **a**_3_) ≠ 0. The periodicity is illustrated in three dimensions, but it is immediately generalisable to an arbitrary dimension d: j is then a set of d integers, and there are d elementary lattice vectors **a**_i_.

Notice that a perfectly periodic lattice in which all sites or nodes can be obtained by multiples of the elementary lattice vectors is called *Bravais lattice*. Bravais lattices fall into different classes and are characterised by their symmetry. This is the object of crystallography, which we do not revisit here; instead we only mention that there are 14 different Bravais lattices in three spatial dimensions and 5 in two dimensions. The Hamiltonian associated with an electron in a Bravais lattice thus reads

$$ H(\mathbf{p}, \mathbf{r}) = \frac{\mathbf{p}^2}{2m} + V(\mathbf{r}), \qquad (2.8) $$

where the potential is periodic in all lattice vectors, V(**r** + **R**_j_) = V(**r**), and so is the full Hamiltonian H(**p**, **r** + **R**_j_) = H(**p**, **r**). As mentioned above, the discrete translation by a lattice vector

$$ \mathcal{T}_{\mathbf{R}_j} = e^{i\mathbf{p} \cdot \mathbf{R}_j} \qquad (2.9) $$

is a symmetry operation for this system, and it therefore commutes with the Hamiltonian (2.8),

$$ [\mathcal{T}_{\mathbf{R}_j}, H(\mathbf{p}, \mathbf{r})] = 0. \qquad (2.10) $$

### 2.1.2 General lattice

Before exploiting discrete translation symmetries within Bloch's theorem, it is important to mention that not all lattices encountered in Nature are Bravais lattices. In order to illustrate this point, let us consider the so-called *kagomé* lattice in two spatial dimensions depicted in figure 2.1 and let us
test whether all nodes of this lattice can be obtained from a single node via
multiples of two primitive lattice vectors **a**₁ and **a**₂. Let us try, in a first step,
the two vectors **δ**₁ and **δ**₂, which are shown in the left panel of the figure
and that connect the two nearest neighbours of an arbitrary node that we
paint in red just for illustration reasons. If the lattice were periodic in these
basis vectors, any linear combination with integer prefactors of these vectors
should indicate the position of a lattice node. This is evidently not the case
for the combination **δ**₁ + **δ**₂, which denotes a point in the centre of a hexagon
that is not occupied by a node of the lattice. Second try: let us choose the
two violet vectors shown in the left panel of the figure. In this case, one
immediately sees that any combination **R**<sub>j</sub> = j₁**a**₁ + j₂**a**₂ reproduces a node
of the lattice. However, if we fix an origin, *e.g.* the blue node at the origin
of the two violet vectors, only the blue nodes of the lattice are generated by
these combinations, but not the red and green ones. In order to reach these
other nodes, we need to appeal first to the translations **δ**₁ and **δ**₂, even if
they are not (!) symmetries – one needs to insist on this point –, to reach a
nearest neighbour of a different colour and then translate this new node (say
the green one) by the lattice vectors **R**<sub>j</sub> = j₁**a**₁ + j₂**a**₂.

To summarise the structure of a general lattice and to fix the terminology
used in the remainder, there are two types of vectors connecting the nodes.
The vectors **R**<sub>j</sub> = j<sub>1</sub>**a**<sub>1</sub> + j<sub>2</sub>**a**<sub>2</sub> (or their generalisation in *d* dimensions) are
the lattice vectors, which determine the perfectly periodic underlying Bra-
vais lattice and that define the symmetry operations in terms of the discrete
translation operators (2.9). In spite of their importance, these lattice vectors
are not sufficient to span the full lattice but only one of the *sublattices* repre-
sented, here, by a specific colour. The two *primitive lattice vectors* **a**<sub>1</sub> and **a**<sub>2</sub>
span the *elementary unit cell* of the lattice, which contains possibly several
inequivalent nodes that are connected to the origin (usually an arbitrarily
chosen node) by the vectors **δ**<sub>1</sub> and **δ**<sub>2</sub>. These vectors **δ**<sub>*m*</sub> define the *basis*
(also called the *motive*) of the lattice. If there are *N*<sub>basis</sub> inequivalent nodes
in the primitive cell, one has *N*<sub>basis</sub> sublattices each of which has **δ**<sub>*m*</sub>, with
*m* = 1, ..., *N*<sub>basis</sub>, as its origin.<fnref n="3" /> In general, an arbitrary lattice can always be
decomposed into a basis and a Bravais lattice. The general lattice can then
be fully obtained by displacing the entire basis by the lattice vectors **R**<sub>*j*</sub>.

In our illustrative example of the kagomé lattice in figure 2.1, the un-

<footnote n="3">³If one uses one arbitrary node as the origin of the full lattice, one of these vectors is zero, and in our example of the kagomé lattice one formally has **δ**<sub>3</sub> = 0.</footnote>
derlying Bravais lattice spanned by the violet vectors is a triangular lattice. The basis contains three nodes ($N_{\text{basis}} = 3$) represented by the three different colours, and each set of nodes of like colour forms a sublattice, which reproduces the Bravais lattice. Another manner to show that the kagomé lattice is itself not a Bravais lattice is to displace an arbitrary patch of the lattice and to superpose it with the original lattice. This is represented in figure 2.1, where the patch is the violet rectangle. If the patch is displaced by a Bravais lattice vector $\mathbf{R}_j$, such as in the left panel, the overlap is perfect. However, if we use an arbitrary vector that connects nodes of different colour (see right panel of the figure), the patch does not fit the original lattice.

## 2.2 Bloch's theorem

Let us now see how the above symmetry considerations allow us to understand electronic bands in a lattice. We have already remembered the fact that since the discrete translation operators $T_{\mathbf{R}_j}$ commute with the periodic Hamiltonian $H(\mathbf{p}, \mathbf{r} + \mathbf{R}_j) = H(\mathbf{p}, \mathbf{r})$, one might first want to diagonalise the translation operator to simplify the diagonalisation of the Hamiltonian. There exist different solutions of this problem, which are not all reviewed here. Instead, we concentrate on the so-called *tight-binding model*, which is based – as indicated by its name – on electrons that are tightly bound to their individual atoms. This is indeed a reasonable assumption in many cases. Indeed, even if the atoms are placed in a crystal, an electron on a lattice site is most affected by the potential generated by the (ionised) atom on this site, while the other sites can be viewed as a relatively weak perturbation. We suppose that we already know the atomic states $\phi_a(\mathbf{r} - \mathbf{R}_j - \mathbf{\delta}_m)$ for an electron bound to its atom situated at the position $\mathbf{R}_j + \mathbf{\delta}_m$, it is essentially an atomic orbital of an isolated atom that we do not calculate here but that we simply consider to be known from an atomic-physics calculation. Quite generally, there are many atomic orbitals per lattice node, but we are only interested in those that host an electron in the vicinity of the Fermi level. For the moment, we consider that there is only one orbital per lattice node, but the situation is generalised in a straightforward manner if we formally multiply the number of lattice nodes by the number of orbitals that we take into account. Formally, $N_{\text{basis}}$ is then the number of nodes times the number of orbitals per atom that we take into account, and some $m$ thus share the same displacement vector $\mathbf{\delta}_m$ in the unit cell.
### 2.2.1 Bloch's theorem for a Bravais lattice

For the moment, we discard the above-mentioned complications and consider a Bravais lattice with a single node and a single atomic orbital per unit cell that we place at the Bravais-lattice nodes for convenience, $\phi_a(\mathbf{r} - \mathbf{R}_j)$. Based on pure inspection and intuition obtained from plane waves describing free particles, Bloch proposed a *fully delocalised* superposition of these *localised* atomic wave functions by adding a plane wave factor $\exp(i\mathbf{k} \cdot \mathbf{R}_j)$ to each of the orbital functions. This yields the *Bloch wave function*

$$ \psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{j} e^{i\mathbf{k} \cdot \mathbf{R}_j} \phi_a(\mathbf{r} - \mathbf{R}_j), \qquad (2.11) $$

where $N$ denotes the number of lattice nodes or unit cells, which goes to infinity in the thermodynamic limit, similarly to the factor $V$ in the prefactor of the plane waves of free particles [see Eq. (2.6)]. One can directly show, with the help of Eq. (2.5) and its action on the atomic wave functions depending on $\mathbf{r}$, that Bloch's proposal (2.11) is actually the good one. It is indeed an eigenstate of the discrete translation operator (2.9) since it satisfies the eigenvalue equation

$$ \mathcal{T}_{\mathbf{R}_j} \psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{R}_j} \psi_{\mathbf{k}}(\mathbf{r}). \qquad (2.12) $$

We notice immediately that, apart from a factor $\hbar$, the wave vector $\mathbf{k}$ is nothing other than the *lattice* or *quasi-momentum* $\hbar\mathbf{k}$ mentioned in the introduction, similarly to the case of free particles. In the following, we will mostly speak about the wave vector rather than lattice momentum, but keep in mind that it is the same quantity.

Because the translation operator commutes with the periodic Hamiltonian, the Bloch wave function is an eigenstate of the latter, and we have already done the major part of the diagonalisation of the Hamiltonian and the calculation of its spectrum. We will see below that this is also the case in the more complex situation of a general lattice with a basis of several lattice sites. In order to appreciate this point, we consider the so-called *tight-binding model* in which electrons are allowed to pass to a site $\mathbf{R}_j$ from another one $\mathbf{R}_j + \mathbf{a}_l$ situated at a distance $\mathbf{a}_l$ by the usual quantum mechanical tunnel effect with the characteristic tunneling amplitude $t_l$. If we represent the quantum state centred at the lattice site $\mathbf{R}_j$ by the Hilbert-
<figure><img src="image_16.png" /><figcaption>Figure 2.2: Band spectra for the square lattice (left panel) and the triangular lattice (right panel). We have chosen $t_1 = t_2 = t_3 = -t$ for illustration.</figcaption></figure>

space vector $|\mathbf{R}_j\rangle$,<fnref n="4" /> the above-mentioned tunneling event can be represented by the operator $t_l|\mathbf{R}_j\rangle\langle\mathbf{R}_j + \mathbf{a}_l|$. The overall Hamiltonian, which takes into account all different tunnel events between nearest neighbouring sites can therefore be written as a superposition of all these operators,

$$ \hat{H} = \sum_{j,l} t_l |\mathbf{R}_j\rangle\langle\mathbf{R}_j + \mathbf{a}_l|, \qquad (2.13) $$

where the different $\mathbf{a}_l$ are the vectors connecting an arbitrary site to its nearest neighbours, as it is depicted in the insets of figure 2.2 for a square lattice (left panel) and the triangular lattice (right panel). In the case of a Bravais lattice, we can even write the Hamiltonian (2.13) in terms of the discrete translation operators because $\mathcal{T}_{\mathbf{a}_l}|\mathbf{R}_j\rangle = |\mathbf{R}_j - \mathbf{a}_l\rangle$.<fnref n="5" /> Indeed the discrete translation operators can also be written as

$$ \mathcal{T}_{\mathbf{a}_l} = \sum_j |\mathbf{R}_j\rangle\langle\mathbf{R}_j + \mathbf{a}_l|, \qquad (2.14) $$

<footnote n="4">⁴We use these states $|\mathbf{R}_j\rangle$ as a basis of our Hilbert space that is thus of dimension $N$. Strictly speaking, these states are only orthogonal if we can neglect the overlap $\langle\mathbf{R}_j|\mathbf{R}_{j'}\rangle = \int d^d r \phi_a^*(\mathbf{r} - \mathbf{R}_j)\phi_a(\mathbf{r} - \mathbf{R}_{j'})$ between different sites $j \neq j'$ on the lattice, which we consider to be itself $d$-dimensional. In the following, we make this assumption not because it is generally valid but to simplify the discussions. The critical reader may be reassured that this is a good approximation in many situations and that in most cases overlap corrections can be taken into account without doing harm to the overall structure of the tight-binding model, which we use here.</footnote>

<footnote n="5">⁵This is just the Dirac “bra and ket” notation of the equation $\mathcal{T}_{\mathbf{a}_l}\phi_a(\mathbf{r} - \mathbf{R}_j) = \phi_a(\mathbf{r} - (\mathbf{R}_j - \mathbf{a}_l))$ in terms of the wave functions $\phi_a(\mathbf{r} - \mathbf{R}_j) = \langle\mathbf{r}|\mathbf{R}_j\rangle$.</footnote>
so that the tight-binding Hamiltonian (2.13) may be easily expressed in terms of the latter as

$$ \hat{H} = \sum_{l} t_{l} \mathcal{T}_{a_{l}}. \qquad (2.15) $$

This allows us also to obtain immediately, without calculations but simply by appealing to Eq. (2.12), the spectrum of the Hamiltonian

$$ E(\mathbf{k}) = \sum_{l} t_{l} e^{i\mathbf{k} \cdot \mathbf{a}_{l}}, \qquad (2.16) $$

which is nothing other than an *electronic band*. It is shown in figure 2.2 for a square lattice (left panel), with $\mathbf{a}_1 = a\mathbf{u}_x$ and $\mathbf{a}_2 = a\mathbf{u}_y$, and for a triangular lattice (right panel), with $\mathbf{a}_1 = a\mathbf{u}_x$, $\mathbf{a}_2 = (a/2)(\mathbf{u}_x + \sqrt{3}\mathbf{u}_y)$ and $\mathbf{a}_3 = (a/2)(-\mathbf{u}_x + \sqrt{3}\mathbf{u}_y)$. For illustration reasons, we have chosen all $t_l = -t$ to be equal in these plots. Notice that if $\mathbf{a}_l$ is a primitive lattice vector relating two nearest neighbour sites, then so is $-\mathbf{a}_l$, and it is associated with the same tunneling amplitude in order to maintain the symmetry of the Bravais lattice.<fnref n="6" /> The tunneling amplitude will henceforth be called *hopping parameter*. The phase factors $\exp(i\mathbf{k} \cdot \mathbf{a}_l)$ and $\exp(-i\mathbf{k} \cdot \mathbf{a}_l)$ of the spectrum (2.16) thus come along with the same hopping parameter and thus simply yield a cosine, $2 \cos(\mathbf{k} \cdot \mathbf{a}_l)$, and the spectrum thus becomes

$$ E(\mathbf{k}) = 2 [t_1 \cos(k_x a) + t_2 \cos(k_y a)] \qquad (2.17) $$

for the square lattice and

$$ E(\mathbf{k}) = 2 \left[ t_1 \cos(k_x a) + t_2 \cos \left( \frac{k_x + \sqrt{3}k_y}{2} a \right) + t_3 \cos \left( \frac{-k_x + \sqrt{3}k_y}{2} a \right) \right] \qquad (2.18) $$

for the triangular lattice.

When we take a closer look at the bands in figure 2.2, we notice that they are periodic in the space of wave vectors (or lattice momenta). This space is called *reciprocal space*, and its periodicity is a very important property of lattice systems that is absent for particles moving in a space with a continuous symmetry. Indeed, the dispersion $E(\mathbf{k}) = \hbar^2\mathbf{k}^2/2m$ of a free non-relativistic

<footnote n="6">⁶If you think about it, you will notice that if the hopping parameters were not equal for $\mathbf{a}_l$ and $-\mathbf{a}_l$, nearest neighbours related by these vectors are no longer inequivalent, and one would need to redefine a larger Bravais lattice with these two neighbours as the basis.</footnote>
particle, described in terms of the Schrödinger equation, is evidently not periodic in the wave vector but monotonically growing. The periodicity of physical quantities in reciprocal space is actually a consequence of the periodicity in the direct space of the Bravais lattice. In order to understand this point in more detail, let us introduce the reciprocal lattice, first in a formal manner – its physical significance should become clearer in the following paragraphs. In the same manner as the direct lattice may be constructed from the primitive lattice vectors $\mathbf{a}_i$, $i = 1, ..., d$ (for a $d$-dimensional lattice, by $\mathbf{R}_j = j_1\mathbf{a}_1 + ... + j_d\mathbf{a}_d$, the reciprocal lattice is spanned by the primitive vectors $\mathbf{a}_i^*$ and has nodes at the positions $\mathbf{G}_\mu = \mu_1\mathbf{a}_1^* + ... + \mu_d\mathbf{a}_d^*$, where $\mu = (\mu_1, ..., \mu_d)$ is a set of $d$ positive or negative integers. The basis vectors of the two lattices, direct and reciprocal, are related by the “orthogonality” relations

$$ \mathbf{a}_i \cdot \mathbf{a}_{i'}^* = 2\pi\delta_{i,i'}. \qquad (2.19) $$

While this might yet seem awfully formal, consider a displacement of the Bloch wave function in reciprocal space by a reciprocal-lattice vector, i.e. we “translate” the wave vector $\mathbf{k}$ by the quantity $\mathbf{G}_\mu$, $\mathbf{k} \rightarrow \mathbf{k} + \mathbf{G}_\mu$. You may immediately see from the definition of the Bloch wave function (2.11), the relation (2.19) and the fact that the product of two integers $j_i$ and $\mu_i$ is itself an integer, that the eigenstate is unaltered, $\psi_\mathbf{k} = \psi_{\mathbf{k}+\mathbf{G}}$. This periodicity in reciprocal space of the Bloch wave function is then naturally transmitted to all of its associated physical quantities such as its energy. This is the origin of the periodicity of the Bloch bands in the wave vectors $E(\mathbf{k}) = E(\mathbf{k} + \mathbf{G})$.

The above-mentioned periodicity in reciprocal space, which is a consequence of the original lattice periodicity, is a fundamental property of band theory, and it also turns out to be essential in the understanding of the topological aspects of Bloch bands that are discussed in Chap. 6. In contrast to free particles in a continuous space, not all wave vectors or momenta are therefore inequivalent – this is the basic difference between a lattice momentum and the usual momentum associated with a displacement in a continuous space – but they are defined modulo a reciprocal-lattice vector. We are therefore allowed (or even obliged, in order to avoid multiple counting of the same states) to restrict states in reciprocal space to those wave vectors that are inequivalent. The ensemble of these inequivalent wave vectors defines the first Brillouin zone, and there are various manners to define it. The conventional one, which we limit ourselves to, is the Wigner-Seitz construction represented in figure 2.3. Take a site of the reciprocal lattice – any one will do the job
<figure><img src="image_17.png" /><figcaption>Figure 2.3: Wigner-Seitz construction of the first Brillouin zone of the square lattice (left panel) and the triangular lattice (right panel). The blue vectors $a_1$ and $a_2$ indicate the primitive lattice vectors of the direct lattice, while $a_1^*$ and $a_2^*$ are those of the reciprocal lattice. The grey lines, which are perpendicular to the reciprocal-lattice vectors and cut the latter at half-way, delimit the first Brillouin zone.</figcaption></figure>

since they are all equivalent as we know by now – and call it the $\Gamma$-point, which is the center of the Brillouin zone. Then take all vectors relating the $\Gamma$-point to its (nearest) neighbours on the reciprocal lattice and cut them in the middle by a line perpendicular to the respective vectors. These lines delimit precisely the first Brillouin zone, as it is shown in figure 2.3 for the square (left panel) and the triangular lattice (right panel).

## Illustration of the physical significance of the Brillouin zone

The equivalence between states that differ in their wave vector by a reciprocal-lattice vector can be illustrated nicely in the following example of an excitation in a one-dimensional crystal. It is noteworthy that the concepts of reciprocal lattice and Brillouin zone are not restricted to the theoretical description of the electronic motion in a crystal. Indeed, these concepts apply to all wave-like excitations in a periodic structure, and the most natural one is certainly the vibration of the lattice itself. Consider a one-dimensional chain of evenly spaced atoms, with a lattice spacing $a$, and a harmonic transverse displacement of the atoms in the direction perpendicular to the chain. This is depicted in panel (a) of figure 2.4. In order to describe this harmonic displacement, a rational person – for example a child that is asked to draw a line connecting the red point of the figure – would naturally propose an interpolation such as that shown in panel (b), described by the formula
<figure><img src="image_18.png" /><figcaption>Figure 2.4: Illustration of the equivalence of oscillations in a one-dimensional lattice with wave vectors that differ by a reciprocal-lattice vector. Panel (a) represents a physical excitation of the lattice with a lattice spacing *a* of fixed wavelength that we arbitrarily choose to be $\lambda = 4\pi a$. Here, it is a transverse oscillation in which the atoms are displaced in a direction perpendicular to the lattice direction *x*. In the absence of the excitation, the atoms are placed at the discrete positions *x* = *ja*, where *j* are integers. (b) Interpolation with a continuous function $f_1(x) = -2 \cos(k_x x) = -2 \cos(k_x j a)$. (c) Interpolation with a continuous function $f_2(x) = -2 \cos[(k_x + 2\pi/a)x] = -2 \cos[(k_x a + 2\pi)j a]$. (d) The superposition of the two functions shows that both describe the same physical excitation.</figcaption></figure>
$f_1(x) = -2 \cos(k_x x) = -2 \cos(k_x j a)$. Notice that this formula is now continuous and it describes, since it is a good interpolation, also positions between the discrete positions of the atoms $x = j a$ in the $x$-direction, where $j$ is an integer.

In addition to this very rational proposal, a mathematician might also suggest an alternative, more complicated interpolation formula depicted in panel (c). Instead of connecting two neighbouring atoms by a straight line, the mathematician's proposal has an additional oscillation in the space between the two atoms. This interpolation is given by the formula $f_2(x) = -2 \cos[(k_x + 2\pi/a)x] = -2 \cos[(k_x a + 2\pi)j)$, and even if this proposal seems unnatural, it described the same elongation of the atoms in the excitation. As screwed as it may seem, it is therefore a formula that is as legitimate as the first one $f_1(x)$ since only the discrete positions of the atoms matter and not the space in between them, as it is shown in panel (d), where we superpose the two functions with the discrete lattice excitation. As reasonable physicists, we therefore need to identify the two periodic functions that are different in their wave vector: while $f_1$ has a wave vector $k_x$, that of $f_2$ is $k_x + 2\pi/a$. However, $a^* = 2\pi/a$ is precisely the reciprocal lattice vector of a periodic one-dimensional lattice with lattice spacing $a$, satisfying Eq. (2.19) for a one-dimensional lattice. Similarly, any function $f_{\mu+1}(x) = 2 \cos[(k_x + 2\pi\mu/a)x)$ with an integer value of $\mu$ is as eligible as or equivalent to $f_1$ or $f_2$. In order to avoid redundancies, we therefore need to identify all these functions, which describe the same lattice excitation, and this is precisely done by restricting the wave vectors to the first Brillouin zone, which in our one-dimensional example is given by the interval $[-a^*/2, a^*/2] = [-\pi/a, \pi/a]$.

Continuum limit and band mass

Most physics students are confronted, in undergraduate classes on electronic transport, with the somewhat cryptic statement by their professors that electrons in a conductor, i.e. in a crystalline structure such as those discussed above, behave as free electrons albeit with another mass, called *band mass*, that bluntly falls from the sky. This is the case in the Drude model, which has been introduced in the introduction and that provides a microscopic approach to Ohm's law. We are now armed to understand better this enigmatic band mass. Indeed, the energy bands of electrons, which we have just calculated for the square or the triangular lattice, are not parabolic but given by the periodic functions (2.17) and (2.18). However, consider for the moment
that these bands are only sparsely filled by electrons. In this case, the electrons occupy only states in the vicinity of the Γ-point where the bands have their minimum in the examples that we have chosen. The position of this minimum is the origin of the wave vectors (at the Γ-point, we have $k = 0$). In this limit, we only need to consider wave vectors in the vicinity of the Fermi wave vector $k_F a \ll 1$ so that we can make a Taylor expansion of the bands in the vicinity of the Γ-point. This limit is also sometimes called the continuum limit because the wavelengths are much larger than the lattice spacing such that the atomic positions are coarse-grained.<fnref n="7" />

The Taylor expansion of the band dispersion (2.17) for the square lattice then yields

$$E(\mathbf{k}) \approx 2(t_1 + t_2) - t_1 k_x^2 a^2 - t_2 k_y^2 a^2 = E_0 + \frac{\hbar^2 k_x^2}{2m_x} + \frac{\hbar^2 k_y^2}{2m_y} \quad (2.20)$$

around the Γ-point, where $E_0 = 2(t_1 + t_2)$ is an irrelevant constant and the band masses are defined as

$$m_x = -\frac{\hbar^2}{2t_1 a^2} \quad \text{et} \quad m_y = -\frac{\hbar^2}{2t_2 a^2}. \qquad (2.21)$$

In the following, we consider the hopping parameters $t_1$ and $t_2$ to be negative so that $k = 0$ represents the band minimum. We need to point out two observations. First, the band mass generally depends on the direction of propagation, in contrast to the usual bare mass of free electrons, as a consequence of the isotropy of free space. However, if the crystal has a square symmetry ($t_1 = t_2$), the band mass becomes isotropic with $m_x = m_y$. Second, the band mass has nothing (!) to do with the bare mass $m_0$ of electrons in free space, as one clearly sees from the last equation: it is given entirely in terms of the lattice parameters – the lattice spacing and the hopping parameters. The origin of the formal equivalence of an electron at the band bottom and a free electron is simply due to the fact that a Taylor expansion around

<footnote n="7">⁷This naming is a little bit unfortunate since the term “continuum limit” is also frequently used to denote the limit of a crystal of size $L$ with periodic boundary conditions. As you have learnt in your statistical-physics classes (e.g PHY433), the wave vector in the same direction is then given by $k = 2\pi p/L$, in terms of the integer $p$, and the continuum limit is defined by $L \to \infty$ in this case. In the present lecture notes, however, we always consider this limit, and we use the term “continuum limit” to denote wave vectors the modulus of which is small with respect to the inverse lattice spacing, $|k|a \ll 1$.</footnote>
a minimum yields a parabolic function to lowest non-trivial order, and the curvature can be identified with a mass on a purely dimensional basis.

Similarly to the square lattice, a Taylor expansion of the band dispersion (2.18) around the Γ-point for the triangular lattice yields a slightly more complicated expression

$$E(\mathbf{k}) \approx E_0 - \left(t_1 + \frac{t_2+t_3}{4}\right) k_x^2 a^2 - \frac{3}{4}(t_2+t_3)k_y^2 a^2 - \frac{\sqrt{3}(t_2-t_3)}{2}k_x k_y, \quad (2.22)$$

with the unimportant constant $E_0 = 2(t_1 + t_2 + t_3)$. Even if it is more complicated than the expression for the square lattice for pure geometric reasons, it keeps nevertheless a quadratic form in the wave vectors that can be rendered diagonal by a spatial rotation, $E(\mathbf{k}) \approx E_0 + (\hbar^2/2)(k_+^2/m_+ + k_-^2/m_-)$. However, we are not interested, here, in this rotation, but we stick to the simpler case $t_2 = t_3$, for which the dispersion relation can be written in the same manner as for the square lattice,

$$E(\mathbf{k}) \approx E_0 + \frac{\hbar^2 k_x^2}{2m_x} + \frac{\hbar^2 k_y^2}{2m_y}, \quad (2.23)$$

in terms of the band masses

$$m_x = -\frac{\hbar^2}{2(t_1 + 2t_2)a^2} \quad \text{and} \quad m_y = -\frac{\hbar^2}{6t_2a^2} \quad (2.24)$$

in the x- and y-directions. Similarly to the square lattice the band becomes isotropic in the vicinity of the Γ-point if all hopping parameters are equal, $t_1 = t_2 = t_3$.

### 2.2.2 Bloch's theorem for an arbitrary lattice

We now turn to the more complicated situation of Bloch's theorem for a general lattice that is not a Bravais lattice. As we have discussed above, such a lattice can always be decomposed into a basis of $N_{\text{basis}}$ sites or orbitals<fnref n="8" /> and a periodic Bravais lattice obtained by translating the basis by the lattice vectors $R_j$. Consider for the moment that there are no tunneling events (hoppings) between sites of different sublattices, i.e. the sublattices

<footnote n="8">⁸In the following, we use the term sites generically to denote sites and orbitals, to keep the discussion as simple as possible.</footnote>
<figure><img src="image_19.png" /><figcaption>Figure 2.5: Non-Bravais lattice composed of two square sublattices A and B. The sublattice B is displaced by a vector $\delta_m$ with respect to sublattice A. Bloch wave function are constructed on each of the sublattices that serve as a basis, and a Bloch wave function for the whole lattice may be obtained by a quantum-mechanical superposition. The prefactors in this superposition determine the relative weight of the wave functions on the two sublattices.</figcaption></figure>

are perfectly decoupled. In this case, we can describe the electronic motion on each of the sublattices simply by a Bloch wave function

$$ \psi_{\mathbf{k}}^{(m)}(\mathbf{r}) = \frac{1}{\sqrt{N_{\text{u.c.}}}} \sum_{j} e^{i\mathbf{k} \cdot \mathbf{R}_j} \phi_m(\mathbf{r} - \mathbf{R}_j - \delta_m) \quad (2.25) $$

on each of the sublattices $m = 1, \dots, N_{\text{basis}}$, where $N_{\text{u.c.}}$ denotes the number of primitive unit cells that is the number of sites per sublattice. Quantum mechanics tells us that each superposition of these Bloch wave functions is an equally valid state on the full lattice,

$$ \psi_{\mathbf{k}}(\mathbf{r}) = \sum_{m=1}^{N_{\text{basis}}} u^{(m)}(\mathbf{k}) \psi_{\mathbf{k}}^{(m)}(\mathbf{r}), \quad (2.26) $$

since it still satisfies the lattice symmetries. If we now consider couplings between the sublattices, the amplitudes $u^{(m)}(\mathbf{k})$ are no longer freely choosable but fixed by the diagonalisation of the Hamiltonian in the subspace of the Bloch wave functions (2.25) at each of the wave vectors $\mathbf{k}$.

Before discussing this diagonalisation, which allows us to obtain the spectrum of $N_{\text{basis}}$ electronic bands, let us investigate in more detail the structure of the Hilbert space that we are confronting here. If we consider a lattice with $N$ sites (atomic orbitals), the full Hilbert space must naturally be of dimension $N$. This number is decomposed into the number of unit cells $N_{\text{u.c.}}$ and that of sites per unit cell $N_{\text{basis}}$, $N = N_{\text{u.c.}} \times N_{\text{basis}}$. As mentioned above,
Bloch's theorem applies to the underlying periodic Bravais lattice and thus allows us to diagonalise this periodic part in terms of the wave vectors in the first Brillouin zone in reciprocal space. If we consider periodic boundary conditions, the number of (discrete) wave vectors therefore equals that of unit cells $N_{\text{u.c.}}$ (remember your statistical-physics classes, e.g. PHY433). Bloch's theorem therefore allows us to perform an essential part of the diagonalisation, but we are still confronted with a Hamiltonian that is yet not diagonal in the subspace of a given wave vector $\mathbf{k}$. In order to retrieve the correct number of quantum states, each of these subspaces must be of dimension $N_{\text{basis}}$, i.e. the subspace Hamiltonian is given by an $N_{\text{basis}} \times N_{\text{basis}}$ matrix that we will study in detail below. Put differently, this matrix has a spectrum of $N_{\text{basis}}$ eigenvalues that evolve in reciprocal space to the $N_{\text{basis}}$ energy bands that are functions precisely of the wave vector, $E_n(\mathbf{k})$.

Let us now take a closer look at the Hamiltonian in the $N_{\text{basis}}$-dimensional subspace, which is nothing other than the *Bloch Hamiltonian* $H_{\mathbf{k}}$, which is characterised by its matrix elements

$$H_{\mathbf{k}}^{m,n} = \langle \psi_{\mathbf{k}}^{(m)} | \hat{H} | \psi_{\mathbf{k}}^{(n)} \rangle \qquad (2.27)$$

the matrix elements of which are given in terms of the sublattice Bloch wave functions (2.25). In order to appreciate this point, consider the Schrödinger equation $H|\psi_{\mathbf{k}}\rangle = E|\psi_{\mathbf{k}}\rangle$ for the full Bloch wave function (2.26), $\psi_{\mathbf{k}}(\mathbf{r}) = \langle \mathbf{r} | \psi_{\mathbf{k}} \rangle$, and multiply it from the left with the *bra* $\langle \psi_{\mathbf{k}}|$,

$$\langle \psi_{\mathbf{k}} | H | \psi_{\mathbf{k}} \rangle = E \langle \psi_{\mathbf{k}} | \psi_{\mathbf{k}} \rangle. \qquad (2.28)$$

The scalar product $\langle \psi_{\mathbf{k}} | \psi_{\mathbf{k}} \rangle$ accounts for the fact that the Bloch wave function is not necessarily normalised. With the use of the decomposition (2.26), this equation becomes

$$\sum_{m,m'} u^{(m)*}(\mathbf{k}) \left[ \langle \psi_{\mathbf{k}}^{(m)} | \hat{H} | \psi_{\mathbf{k}}^{(m')} \rangle - E \langle \psi_{\mathbf{k}}^{(m)} | \psi_{\mathbf{k}}^{(m')} \rangle \right] u^{(m')}(\mathbf{k}) = 0, \qquad (2.29)$$

which implies the *secular equation*

$$\det[H_{\mathbf{k}} - E S_{\mathbf{k}}] = 0, \qquad (2.30)$$

the solution of which<fnref n="9" /> yields the $N_{\text{basis}}$ energy bands $E_n(\mathbf{k})$ we are looking for. In addition to the Bloch Hamiltonian, this equation appeals to the *overlap*

<footnote n="9">⁹The secular equation is a polynomial of order $N_{\text{basis}}$ in the energy $E$.</footnote>
matrix $S_k = \langle \psi_k^{(m)} | \psi_k^{(m')} \rangle$. Consider this overlap matrix to be simply the one matrix $S_k = \delta_{m,m'}$, i.e. we neglect the (exponentially suppressed) overlap between atoms on different lattice sites.<fnref n="10" /> In this case, the secular equation is the solution of a new static Schrödinger equation

$$H_k |u_n(\mathbf{k})\rangle = E_n(\mathbf{k}) |u_n(\mathbf{k})\rangle, \qquad (2.31)$$

where the periodic Hamiltonian $H$ is replaced by the Bloch Hamiltonian $H_k$ and the Bloch wave functions by the *Bloch states* $|u_n(\mathbf{k})\rangle$. These states are precisely formed by the components $u_n^{(m)}(\mathbf{k})$,

$$|u_n(\mathbf{k})\rangle \sim \begin{pmatrix} u_n^{(1)}(\mathbf{k}) \\ \vdots \\ u_n^{(m)}(\mathbf{k}) \\ \vdots \\ u_n^{(N_{\text{basis}})}(\mathbf{k}) \end{pmatrix}. \qquad (2.32)$$

The reader should be warned about two things here! First, there is a non-negligible danger to get lost in the indices. The subscript $n$ indicates the $n$-th eigenstate, which is associated with the band $E_n(\mathbf{k})$, while the superscript $(m)$ denotes the sublattice component. Second, the nomenclature is somewhat unusual: contrary to the usual quantum-mechanical relation between Hilbert-space state and wave function, the Bloch wave function is *not* the real-space representation of the Bloch state $|u_n(\mathbf{k})\rangle$ but of $|\psi_n(\mathbf{k})\rangle$. As it is discussed in more detail below in section 2.2.3, the Bloch state $|u_n(\mathbf{k})\rangle$ represents rather an envelope function $u_n(\mathbf{k}, \mathbf{r})$ that is itself periodic in the underlying Bravais lattice. If we wanted to be precise, we should call $|u_n(\mathbf{k})\rangle$ the “state corresponding to the periodic part of the Bloch wave function”. However, this expression is rather cumbersome in view of the use we make of $|u_n(\mathbf{k})\rangle$ in these classes so that we stick to the term “Bloch states”.

Before a more exhaustive comparison between the two bases, associated with the periodic Hamiltonian and the Bloch Hamiltonian, respectively, we finish this section with a receipe to diagonalise the Bloch Hamiltonian and to obtain the different bands in terms of the atomic energies $t_{m,m'}(\mathbf{R}_j) =$

<footnote n="10">¹⁰In many cases, this is a good approximation, and corrections due to the non-zero overlap between the sublattice wave functions can be taken into account in modified effective Bloch Hamiltonians. However, as in many approximate schemes, there exist subtle counter-examples that we exclude here in a first discussion of Bloch states.</footnote>
$\int d^Dr \phi^{(m)*}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m) \Delta \hat{H} \phi^{(m')}(\mathbf{r} - \boldsymbol{\delta}_{m'})$, which are matrix elements of the Hamiltonian sandwiched between two atomic wave functions. The atomic wave function $\phi^{(m')}(\mathbf{r} - \boldsymbol{\delta}_{m'})$ is that at the $m'$-th site in the unit cell that defines the origin of the Bravais lattice, while $\phi^{(m)}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m)$ is that at the site $m$ in the unit cell at the (Bravais) lattice vector $\mathbf{R}_j$. The attentive reader may have noticed that it is actually not the full periodic Hamiltonian $\hat{H}$ that enters in this matrix element but only a part $\Delta \hat{H}$ of it, which can be understood in by the following reasoning. The potential that enters the full Hamiltonian is the periodic lattice potential $V(\mathbf{r})$, which is formed by all the atoms of the lattice. However, the atomic wave function $\phi^{(m)}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m)$ is itself an eigenstate of the atomic Hamiltonian

$$ \hat{H}_a \phi^{(m)}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m) = E_a^{(m)} \phi^{(m)}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m), \quad (2.33) $$

which already takes into account the potential formed by the atom at the lattice site $\mathbf{R}_j + \boldsymbol{\delta}_m$. In order to avoid double counting, the contribution from this atom to the potential $V(\mathbf{r})$ must therefore be omitted, and the potential formed by the other atoms is therefore perturbatively taken into account by the part $\Delta \hat{H}$ of the Hamiltonian, which formally reads

$$ \hat{H} = \hat{H}_a + \Delta \hat{H}. \quad (2.34) $$

With this in mind, the matrix element (2.27) of the Bloch Hamiltonian can be calculated, in $d$ dimensions, as

$$ \begin{aligned} & \langle \psi_{\mathbf{k}}^{(m)} | H | \psi_{\mathbf{k}}^{(m')} \rangle \\ &= \sum_{j,j'} \frac{e^{i\mathbf{k} \cdot (\mathbf{R}_{j'} - \mathbf{R}_j)}}{N_{\text{u.c.}}} \int d^d\mathbf{r} \, \phi^{(m)*}(\mathbf{r} - \mathbf{R}_j - \boldsymbol{\delta}_m) \hat{H} \phi^{(m')}(\mathbf{r} - \mathbf{R}_{j'} - \boldsymbol{\delta}_{m'}) \\ &= \sum_l e^{i\mathbf{k} \cdot \mathbf{R}_l} \int d^d\mathbf{r}' \, \phi^{(m)*}(\mathbf{r}' + \mathbf{R}_l - \boldsymbol{\delta}_{m-m'}) (\hat{H}_a + \Delta \hat{H}) \phi^{(m')}(\mathbf{r}') \\ &= [E_a^{(m')}(\mathbf{k}) + T_{m,m'}(\mathbf{k})], \end{aligned} \quad (2.35) $$

where we have performed diverse variable changes that allow us to omit one sum over the lattice vectors. It simply yields a factor of $N_{\text{u.c.}}$ which cancels that in the denominator in the second line. Furthermore, we have introduced the relative distance $\boldsymbol{\delta}_{m-m'} = \boldsymbol{\delta}_m - \boldsymbol{\delta}_{m'}$ between two different sites inside the same unit cell.<fnref n="11" /> As mentioned above, we neglect the off-diagonal elements

<footnote n="11">¹¹This distance should not be mixed up with the Kronecker delta, $\delta_{m,m'}$!</footnote>
of the overlap matrix, which we consider to be<fnref n="12" /> $S_{m,m'}(\mathbf{k}) \approx \delta_{m,m'}$, and we have defined the hopping matrix

$$T_{m,m'}(\mathbf{k}) = \sum_{j} e^{i\mathbf{k} \cdot \mathbf{R}_{j}} t_{m,m'}(\mathbf{R}_{j}). \qquad (2.36)$$

It is given by the atomic energy

$$t_{m,m'}(\mathbf{R}_j) = \int d^d r \phi^{(m)*}(\mathbf{r} - \mathbf{R}_j - \delta_{m-m'}) \Delta \hat{H} \phi^{(m')}(\mathbf{r}), \qquad (2.37)$$

which we have introduced above and that takes into account the hopping (tunnel effect) between atomic orbitals, and it is weighted by the plane wave states $\exp(i\mathbf{k} \cdot \mathbf{R}_j)$ inherited from the Bloch wave function. We thus obtain the same structure as in the case of the tight-binding model on a Bravais lattice, where the hoppings were also weighted by these plane-wave factors, but we now need to distinguish between the different sites inside the unit cell.

In conclusion, here is the simplified receipe for writing down the tight-binding model on a general lattice. Fix the origin of the Bravais lattice on an arbitrary lattice point. A tunnel event to a neighbouring site is then given by the hopping parameter $t_{m,m'}(\mathbf{R}_j)$. Usually these hopping parameters depend on the relative distance between the origin and the site where the electron may hop to, and often the hopping parameter is then the same for all nearest neighbours.<fnref n="13" /> Check whether the site, which is the target in the hopping process, is inside or outside the unit cell which contains the origin. If it is inside, we have $\mathbf{R}_j = 0$ and the phase factor that needs to be multiplied with the hopping parameter is simply one. If it is outside this unit cell, one needs to multiply the hopping parameter with the phase factor $\exp(i\mathbf{k} \cdot \mathbf{R}_j)$ where $\mathbf{R}_j$ is the vector that connects the unit cell of the origin with that of the target site. Please pay attention here: it is not the vector that relates the origin, which resides, e.g, on the sublattice A, to the target site, which may belong to a different sublattice. Instead, the vectors $\mathbf{R}_j$ connect unit cells, i.e. it connects the origin on the sublattice A to the site of type A inside

<footnote n="12">¹²This time, it is well the Kronecker delta, $\delta_{m,m'}$.</footnote>

<footnote n="13">¹³This is often used in a more general framework where all hopping parameters connecting sites separated by the same distance are set equal. However, this need not necessarily be true in all situations, e.g. if the orbitals are oriented in different directions with respect to the vector connecting the sites. Here, we omit this difficulty since there is no specific orientational distinction in the cases we consider in these classes.</footnote>
the unit cell that contains the target site. The relative distances $\delta_m$ do thus not occur in these phase factors but only the (Bravais) lattice vectors $R_j$, as one clearly sees from Eq. (2.36). The off-diagonal elements of the Bloch Hamiltonian are then directly given by these weighted hopping parameters (2.36), while the diagonal elements have two contributions. The first contribution simply stems from the atomic energies $E_a^{(m)}$, which are naturally equal if the sites are occupied by the same atoms in a chemical sense (such as in silicon, graphite, copper,...), but they may be different in lattices with different chemical components (such as the semiconductors GaAs, InSb,... or the alloys $Bi_xSb_{1-x}$, ...). This receipe will be illustrated in section 2.3, where we calculate and discuss the band structure of 2D graphene. There may be a second contribution to the diagonal elements if we consider hopping between sites on the same sublattice. However, these sites are generally no longer nearest neighbours (but next-nearest neighbours) for which we neglect the hopping in simplified tight-binding models, while nearest neighbours reside on different sublattices.

### 2.2.3 Periodic Hamiltonian and Bloch Hamiltonian

Before studying the concrete example of the electronic band structure in graphene, let us briefly discuss a complementary formulation of Bloch's theorem that is also of interest for the understanding of the topological version of band theory (Chap. 6) and that yields another perspective on the Bloch Hamiltonian and the Bloch states. Indeed, the Bloch wave function (for each component $m$) can also be written as

$$ \psi_{\mathbf{k}}^{(m)}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u^{(m)}(\mathbf{k}, \mathbf{r}), \qquad (2.38) $$

i.e. as the product of a plane wave $\exp(i\mathbf{k}\cdot\mathbf{r})$ and the function $u^{(m)}(\mathbf{k}, \mathbf{r})$. Substituting this expression into Eq. (2.12) shows that the function $u^{(m)}(\mathbf{k}, \mathbf{r}) = u^{(m)}(\mathbf{k}, \mathbf{r} - R_j)$ is itself periodic in the lattice vectors $R_j$. It is precisely the envelope function associated with the Bloch state $|u(\mathbf{k})\rangle$ or, more precisely, its $m$-th component. Notice the following orthogonality property, which will turn out to be extremely important in the geometric and topological characterisation of the Bloch bands: while the orthogonality of the Bloch wave functions at different wave vectors is insured by the plane-wave factors, $\langle\psi_{\mathbf{k}}|\psi_{\mathbf{k}'}\rangle = \delta_{\mathbf{k},\mathbf{k}'}$, this is no longer the case for the Bloch states $\langle u(\mathbf{k})|u(\mathbf{k}')\rangle \neq \delta_{\mathbf{k},\mathbf{k}'}$. This non-orthogonality eventually allows us to define
an evolution of the Bloch states in a neighbourhood defined in its associated Hilbert space when one explores the vicinity of a specific wave vector in reciprocal space, while this is impossible for the orthogonal Bloch wave functions.

In terms of the eigenstates of the periodic Hamiltonian $\hat{H}$, Eq. (2.38) reads

$$|\psi_{n,\mathbf{k}}\rangle = e^{i\mathbf{k}\cdot\hat{\mathbf{r}}}|u_n(\mathbf{k})\rangle. \qquad (2.39)$$

This allows us to understand the relation between these different states from yet another perspective. Indeed, the Bloch wave function is obtained from the Bloch state by a translation by the wave vector $\mathbf{k}$ in reciprocal space, since the position operator $\hat{\mathbf{r}}$ is the generator of translations or reciprocal space. The unitary operator $\exp(i\mathbf{k}\cdot\hat{\mathbf{r}})$ thus allows us to connect also the periodic Hamiltonian $\hat{H}$ to the Bloch Hamiltonian by the very same unitary transformation

$$H_{\mathbf{k}} = e^{-i\mathbf{k}\cdot\hat{\mathbf{r}}} \hat{H} e^{i\mathbf{k}\cdot\hat{\mathbf{r}}}. \qquad (2.40)$$

Applying this unitary transformation to the original periodic Hamiltonian [see Eq. (2.8)], one obtains

$$H_{\mathbf{k}} = \frac{(\mathbf{p} + \hbar\mathbf{k})^2}{2m} + V(\mathbf{r}), \qquad (2.41)$$

since $\exp(-i\mathbf{k} \cdot \hat{\mathbf{r}})\mathbf{p} \exp(i\mathbf{k} \cdot \hat{\mathbf{r}}) = \mathbf{p} + \hbar\mathbf{k}$, as it follows directly from the canonical commutation relations for the position and momentum operators, and the position operator naturally commutes with the (position-dependent) potential.

### Operators in the Bloch basis – the case of the velocity operator

The expression (2.41) of the Bloch Hamiltonian allows us to represent also other operators $\hat{O}$ in the Bloch basis, via the same unitary transformation,

$$\mathcal{O}(\mathbf{k}) = e^{-i\mathbf{k}\cdot\hat{\mathbf{r}}} \hat{O} e^{i\mathbf{k}\cdot\hat{\mathbf{r}}}. \qquad (2.42)$$

Let us illustrate this with the velocity operator, which happens to be of strong relevance in the calculation of both transport and optical properties of crystalline systems. In the original basis of the periodic Hamiltonian, the $\mu$-th component ($\mu = x, y, z$ in three dimensions) of the velocity operator is simply given by the Heisenberg equation of motion

$$v_{\mu} = \dot{x}_{\mu} = \frac{1}{i\hbar}[x_{\mu}, \hat{H}], \qquad (2.43)$$
<table><caption>Table 2.1: Comparison between the original basis of the periodoc Hamiltonian and the Bloch basis.</caption><thead><tr><th>physical object</th><th>original basis<br>(periodic Hamiltonian)</th><th>Bloch basis</th></tr></thead><tbody><tr><td>operators</td><td><i>O</i></td><td><i>O</i>(<b>k</b>) = e<sup>-i<b>k</b>&middot;<i>r</i></sup><i>O</i>e<sup>i<b>k</b>&middot;<i>r</i></sup></td></tr><tr><td>states</td><td>|<i>&psi;</i><sub><b>k</b></sub>&rangle;</td><td>|<i>u</i>(<b>k</b>)&rangle; = e<sup>-i<b>k</b>&middot;<i>r</i></sup>|<i>&psi;</i><sub><b>k</b></sub>&rangle;</td></tr><tr><td>explicit Hamiltonian</td><td><i>H</i> = <sup><i>p</i><sup>2</sup></sup>&frasl;<sub>2<i>m</i></sub> + <i>V</i>(<b>r</b>)</td><td><i>H</i><sub><b>k</b></sub><sup><i>m</i>,<i>m</i>'</sup> = &lt;<i>&psi;</i><sub><b>k</b></sub><sup>(<i>m</i>)</sup>|<i>H</i>|<i>&psi;</i><sub><b>k</b></sub><sup>(<i>m</i>')</sup>&gt;<br><i>H</i><sub><b>k</b></sub> = (<sup><b>p</b> + <i>h</i><b>k</b></sup>&frasl;<sub>2<i>m</i></sub>)<sup>2</sup> + <i>V</i>(<b>r</b>)</td></tr><tr><td>explicit eigenstates</td><td>Bloch wave functions<br>|<i>&psi;</i><sub><i>n</i>,<b>k</b></sub>&rangle;</td><td>Bloch states<br>|<i>u</i><sub><i>n</i></sub>(<b>k</b>)&rangle;</td></tr><tr><td>Schrödinger equation</td><td><i>H</i>|<i>&psi;</i><sub><i>n</i>,<b>k</b></sub>&rangle; = <i>E</i><sub><i>n</i></sub>(<b>k</b>)|<i>&psi;</i><sub><i>n</i>,<b>k</b></sub>&rangle;</td><td><i>H</i><sub><b>k</b></sub>|<i>u</i><sub><i>n</i></sub>(<b>k</b>)&rangle; = <i>E</i><sub><i>n</i></sub>(<b>k</b>)|<i>u</i><sub><i>n</i></sub>(<b>k</b>)&rangle;</td></tr><tr><td>velocity operator</td><td><i>v</i><sub><i>&mu;</i></sub> = <sup>1</sup>&frasl;<sub><i>i</i><i>h</i></sub>[<i>x</i><sub><i>&mu;</i></sub>, <i>H</i>]</td><td><i>v</i><sub><i>&mu;</i></sub>(<b>k</b>) = <sup>1</sup>&frasl;<sub><i>h</i></sub> <sup>&part;<i>H</i><sub><b>k</b></sub></sup>&frasl;<sub>&part;<i>k</i><sub><i>&mu;</i></sub></sub></td></tr></tbody></table>

which becomes, in the Bloch basis,

$$v_{\mu}(\mathbf{k}) = \frac{1}{i\hbar} e^{-i\mathbf{k}\cdot\hat{\mathbf{r}}} [x_{\mu}, \hat{H}] e^{i\mathbf{k}\cdot\hat{\mathbf{r}}} = \frac{1}{i\hbar} [x_{\mu}, H_{\mathbf{k}}], \quad (2.44)$$

where we have made use of the commutativity between the unitary transformation and the components of the position operator $x_{\mu}$ in the last step. If we now use the expression (2.41) for the Bloch Hamiltonian, we may also write the velocity operator in the Bloch basis as

$$v_{\mu}(\mathbf{k}) = \frac{p_{\mu} + \hbar k_{\mu}}{m} = \frac{1}{\hbar} \frac{\partial H_{\mathbf{k}}}{\partial k_{\mu}}, \quad (2.45)$$

i.e. the velocity operator in the Bloch basis is simply the **k**-space gradient of the Bloch Hamiltonian (devided by a factor of $\hbar$). While the last step may seem quite formal, it turns out to be extremely practical if the Bloch Hamiltonian is written explicitly in its explicit matrix form as in Eq. (2.27) instead of Eq. (2.41).

The different bases, their relation and some relevant quantities are summarised in table 2.1.

## Bloch's theorem in the presence of a constant electric field and Bloch oscillations

A very important fact about Bloch's theorem is that it remains valid in the presence of a constant electric field **E**. This is remarkable because the
Hamiltonian

$$ \hat{H} = \frac{\mathbf{p}^2}{2m} + V(\mathbf{r}) + e\mathbf{E} \cdot \mathbf{r}, \qquad (2.46) $$

which takes into account the energy of the charged electron in the presence of the field $\mathbf{E}$, is no longer periodic in $\mathbf{r}$ even if the potential $V(\mathbf{r})$ is so. In order to understand why Bloch's theorem remains valid nevertheless, we may perform a gauge transformation that allows us to get rid of the last term, which is not periodic. You all know about gauge transformations in classical electromagnetism, where the electric and magnetic fields remain unchanged even if the underlying potentials $\Phi(\mathbf{r})$ and $\mathbf{A}(\mathbf{r})$ are modified by a position- and time-dependent function $\Lambda(\mathbf{r}, t)$, according to

$$ \Phi(\mathbf{r}) \rightarrow \Phi(\mathbf{r}) - \frac{\partial \Lambda}{\partial t} \quad \text{and} \quad \mathbf{A}(\mathbf{r}) \rightarrow \mathbf{A}(\mathbf{r}) + \nabla \Lambda. \qquad (2.47) $$

In quantum mechanics, a gauge transformation is reminiscent of a symmetry operation: it is also represented by a unitary transformation acting on the Hilbert-space vectors, but it is *local* in space and time. The generator of the gauge transformation is precisely given by the operator $(e/\hbar)\Lambda(\mathbf{r}, t)$, where the prefactor $e/\hbar$ has been added for pure dimensional reasons so that the generator is dimensionless. The unitary transformation is then given by

$$ U = e^{i \frac{e}{\hbar} \Lambda(\mathbf{r}, t)} \quad \text{with} \quad |\psi'\rangle = U |\psi\rangle \quad \text{and} \quad \mathcal{O}' = U \mathcal{O} U^{\dagger}, \qquad (2.48) $$

where $|\psi\rangle$ is an arbitrary Hilbert-space state and $\mathcal{O}$ an arbitrary operator, and the primes indicate the transformed quantities. According to the gauge transformation (2.48), the momentum operator $\mathbf{p}$ transforms as

$$ \mathbf{p}' = e^{i \frac{e}{\hbar} \Lambda(\mathbf{r}, t)} \mathbf{p} e^{-i \frac{e}{\hbar} \Lambda(\mathbf{r}, t)} = \mathbf{p} - e \nabla \Lambda(\mathbf{r}, t). \qquad (2.49) $$

Let us now see how the gauge transformation affects the Schrödinger equation $\hat{H}|\psi\rangle = i\hbar(\partial/\partial t)|\psi\rangle$, which should maintain its structure once transformed

$$ \hat{H}' |\psi'\rangle = i\hbar \frac{\partial}{\partial t} |\psi'\rangle. \qquad (2.50) $$

Notice that we need to take into account the time derivative explicitly in the Schrödinger equation, and we cannot use the stationary version of it directly since the gauge transformation depends itself explicitly on time. Indeed, through this gauge transformation, the Schrödinger equation reads

$$ U H U^{\dagger} |\psi'\rangle = U \left( i \hbar \frac{\partial}{\partial t} \right) U^{\dagger} |\psi'\rangle = \left( e \hbar \frac{\partial \Lambda}{\partial t} + i \hbar \frac{\partial}{\partial t} \right) |\psi'\rangle. \qquad (2.51) $$
The transformed Hamiltonian therefore needs to be modified by the extra term due to the time derivative on the right-hand side, and one finds

$$ \hat{H}' = \frac{(\mathbf{p} - e\nabla\Lambda)^2}{2m} + V(\mathbf{r}) + e\mathbf{E} \cdot \mathbf{r} - e\frac{\partial\Lambda}{\partial t}. \quad (2.52) $$

We may now use the last term to compensate the second-last term, which contains the electric field that we consider to be static, by fixing the gauge function

$$ \Lambda(\mathbf{r}, t) = \mathbf{E} \cdot \mathbf{r} t \quad \text{and} \quad U = e^{i\frac{e}{\hbar}\mathbf{E} \cdot \mathbf{r}t}. \quad (2.53) $$

The transformed momentum thus reads

$$ \mathbf{p}' = \mathbf{p} - e\mathbf{E}t, \quad (2.54) $$

and the Hamiltonian may finally be written as

$$ \hat{H}' = \frac{(\mathbf{p} - e\mathbf{E}t)^2}{2m} + V(\mathbf{r}), \quad (2.55) $$

which is evidently periodic in $\mathbf{r}$ so that we can save Bloch's theorem.

This has also important practical consequences for the Bloch Hamiltonian that may be obtained from the periodic Hamiltonian $\hat{H}'$ by the same transformation (2.40) as that in the absence of an electric field. Formally, this is ensured by the commutativity between the gauge transformation (2.48) and the unitary transformation $\exp(i\mathbf{k} \cdot \hat{\mathbf{r}})$, which links the periodic to the Bloch Hamiltonian, and the latter now reads

$$ H'_{\mathbf{k}} = \frac{[\mathbf{p} + (\hbar\mathbf{k} - e\mathbf{E}t)]^2}{2m} + V(\mathbf{r}) \Leftrightarrow H'_{\mathbf{k}} = H_{\mathbf{k}-(e/\hbar)\mathbf{E}t}. \quad (2.56) $$

This form of the Hamiltonian indicates us that we may continue to use the Bloch Hamiltonian, which we have e.g. obtained within the tight-binding model, in the presence of a constant electric field. Its effect is simply to provide a dynamical character to the wave vector $\mathbf{k}$ by

$$ \mathbf{k}(t) = \mathbf{k} - \frac{e}{\hbar}\mathbf{E}t, \quad (2.57) $$

which grows linearly in time.

Hold on! We have already seen above that the wave vector cannot grow constantly, but it is restricted to the first Brillouin zone. Indeed, once the
wave vector reaches the border of the first Brillouin zone, it must be folded back, i.e. it reappears at the equivalent point on the opposite side of the first Brillouin zone. This backfolding of the wave vector is at the origin of the so-called *Bloch oscillation*. To see that this implies also an oscillation in position space, we may investigate the velocity operator (2.45) in the Bloch basis averaged over the Bloch state associated with an isolated band $n$,

$$ \mathbf{v}_n(\mathbf{k}) = \langle u_n(\mathbf{k}) | \mathbf{v} \left( \mathbf{k} - \frac{e}{\hbar} \mathbf{E} t \right) | u_n(\mathbf{k}) \rangle. \quad (2.58) $$

One notices from this expression that the average velocity changes in time when the wave vector explores a path in the first Brillouin zone. Most saliently, we have seen that the physical quantities associate with a Bloch state are periodic in reciprocal space, and so is this average velocity. As a consequence it changes periodically its sign in time, e.g. when approaching the Brillouin-zone border, where the dispersion becomes flat (and the velocity thus vanishes). This periodic change in the velocity is at the origin of a real-space oscillation of an electron in a Bloch band. As we discuss in section 3.2 of Chap. 3, this oscillation is however strongly suppressed by scattering events at time scales $\tau$ that are usually much smaller than the period of the Bloch oscillation $T \sim \hbar/e|\mathbf{E}|a$, where $a$ is the characteristic lattice spacing.

## 2.3 Band structure of graphene and boron nitride

After these general considerations about Bloch's theorem, we are now armed to confront an explicit and practical example with the calculation of the band structure of graphene. We do not pretend to provide a full band structure calculation, but we use a simplified tight-binding model, which reproduces the relevant electronic bands in the vicinity of the Fermi level that are responsible for the very particular electronic properties of this amazing material. Furthermore, we discuss on equal footing the band structure of two-dimensional (2D) boron nitride (BN), which has the same lattice structure as graphene so that the tight-binding model is almost the same. Both materials are 2D crystals that consist of a single layer of atoms arranged in the honeycomb lattice shown in figure 2.6(a). This particular crystalline structure can be understood on the basis of the chemical properties of the carbon atoms that
<figure><img src="image_20.png" /><figcaption>Figure 2.6: (a) Crystal structure of graphene and boron nitride. The unit cell is depicted by the blue area and spanned by the primitive lattice vectors **a**₁ and **a**₂. The vectors **e**₁, **e**₂ and **e**₃ connect the nearest-neighbour sites which belong to different sublattices. The honeycomb lattice consists of two triangular sublattices A and B that are both occupied by carbon atoms in the case of graphene and by different atoms (boron and nitrogen) for boron nitride. (b) First Brillouin zone of the underlying triangular lattice. The center is the Γ-point, while the corners of the hexagonal first Brillouin zone represent two inequivalent points K and K'. The other corners of the hexagon are connected to one of the two points by the reciprocal lattice vectors **a**<sub>1</sub>* and **a**<sub>2</sub>* (or combinations of them). The centers of the borders of the hexagon are the M-points, three of which are inequivalent.</figcaption></figure>

constitute the graphene lattice, namely on the basis of the electronic occu-
pation of the atomic orbitals. In a carbon atom, there are four electrons that
occupy the 2s and 2p orbitals, three of which take part in the so-called sp²
hybridisation. These hybridised orbitals form coplanar lobes with mutual
120° angles, which are at the origin of extremely strong covalent bonds<fnref n="14" />
that form the 2D lattice. The orbitals are thus occupied by three electrons
per cabon atom, while one electron still occupies the unhybridised orbital
p_z responsible for the electronic properties of graphene in the vicinity of the
Fermi level. Notice already that this orbital is thus “half occupied” since it
can host up to two electrons due to their twofold spin degeneracy.

The electrons in the *p*<sub>*z*</sub> orbitals, which occupy the nodes of the honeycomb
lattice, are precisely the ones we take into account within our tight-binding
model. In a first step we characterise the lattice geometrically as well as its
reciprocal lattice. Each lattice site has three nearest neighbours, which are
connected by the vectors *e*₁, *e*₂ and *e*₃ in figure 2.6(a) and that reside on a

<footnote n="14">¹⁴Chemically, these bonds are as strong as those in diamond, making graphene one of the most resistant materials even if it is extremely thin (one atom thick).</footnote>
different sublattice than the original site. Indeed, the honeycomb lattice is not a Bravais lattice, but it consists of two triangular sublattices displaced by one of the vectors $\mathbf{e}_m$ of length $a \simeq 0.14$ nm. The triangular Bravais lattice is spanned by two primitive lattice vectors that we can choose to be

$$ \mathbf{a}_1 = \frac{\sqrt{3}}{2} a(\mathbf{u}_x + \sqrt{3}\mathbf{u}_y) \quad \text{and} \quad \mathbf{a}_2 = \frac{\sqrt{3}}{2} a(-\mathbf{u}_x + \sqrt{3}\mathbf{u}_y), \quad (2.59) $$

where $\mathbf{u}_x$ and $\mathbf{u}_y$ are the unit vectors in the directions $x$ and $y$, respectively [see figure 2.6(a)]. Similarly, with the help of Eq. (2.19), one readily finds the primitive vectors

$$ \mathbf{a}_1^* = \frac{2\pi}{3a}(\sqrt{3}\mathbf{u}_x + \mathbf{u}_y) \quad \text{and} \quad \mathbf{a}_2^* = \frac{2\pi}{3a}(-\sqrt{3}\mathbf{u}_x + \mathbf{u}_y) \quad (2.60) $$

of the reciprocal lattice, which is itself a triangular lattice, as we have already discussed above. The Wigner-Seitz construction yields the first Brillouin zone shown in figure 2.6(b), which is again that of the triangular lattice.<fnref n="15" /> Anticipating their role in the band structure calculated below, let us concentrate first on the six corners of the first Brillouin zone, which can be put into two classes of points $K$ and $K'$. In order to appreciate this classification consider the utmost right and left corners, which are given by the wave vectors

$$ \mathbf{K} = \frac{4\pi}{3\sqrt{3}a}\mathbf{u}_x, \quad \text{and} \quad \mathbf{K}' = -\mathbf{K} = -\frac{4\pi}{3\sqrt{3}a}\mathbf{u}_x, \quad (2.61) $$

for the $K$ and $K'$ points, respectively. All other four corners can be connected to one of them by the primitive reciprocal-lattice vectors $\pm \mathbf{a}_1^*$ and $\pm \mathbf{a}_2^*$ and are therefore equivalent to $\pm \mathbf{K}$ – as mentioned in our illustration of the reciprocal lattice, they represent the same excitation on the discrete lattice.

We have already mentioned that we take only into account electronic hopping between the $p_z$ orbitals in the tight-binding model. Furthermore, we make the simplification that only the hopping amplitudes<fnref n="16" /> $t = 3$ eV between nearest neighbours are considered in the model.<fnref n="17" /> In order to obtain the corresponding Bloch Hamiltonian, let us apply the receipe mentioned at the

<footnote n="15">¹⁵Remember that the reciprocal lattice is defined with respect to the periodic Bravais lattice in real space, which underlies the general lattice (here, the honeycomb lattice).</footnote>

<footnote n="16">¹⁶In order to obtain this value, quantum-chemical calculations need to be performed that do not interest us here.</footnote>

<footnote n="17">¹⁷This is actually not the case in graphene, where the hopping amplitude between next-nearest neighbours is roughly one order of magnitude smaller than $t$. However, in this first</footnote>
end of section 2.2.2. Neglecting the overlap matrix, the Bloch Hamiltonian is a 2 × 2 matrix

$$H_k = \begin{pmatrix} E_a^A & T_{A,B}(k) \\ T_{B,A}(k) & E_a^B \end{pmatrix}, \qquad (2.62)$$

where we consider first the diagonal terms, which are given by the atomic energies. As mentioned, we have $E_a^A = E_a^B$ in graphene because both sublattices are occupied by chemically identical carbon atoms, and we set this energy to zero henceforth, anticipating that it coincides with the Fermi level. However, in boron nitride, the A sublattice is occupied by, say, boron and the B sublattice by nitrogen atoms so that $E_a^A \neq E_a^B$. If we fix the zero in energy at their average value, we can set $E_a^A = \Delta$ and $E_a^B = -\Delta$. Furthermore, there are no hopping contributions in our model because a “diagonal” tunneling event connects sites on the same sublattice that are at best next-nearest neighbours for which we set $t_{nnn} = 0$, here.

Let us now investigate the off-diagonal terms that are related by complex conjugation, $T_{B,A}(k) = T_{A,B}^*(k)$, in order to have a Hermitian Hamiltonian, and let us apply the general receipe. The term $T_{B,A}(k)$ describes hopping from a site on the B sublattice to its nearest neighbours on the A sublattice represented by the green vectors $e_m$, with the amplitude $-t$.<fnref n="18" /> The vector $e_3$ connects the B site to an A site in the same basis if we define the basis as in figure 2.6(a). According to our receipe, the contribution to the element $T_{B,A}(k)$ of the hopping matrix is therefore simply $-t$ with no phase factor. In contrast, if we consider the A site that is reached by the vector $e_1$, it is not in the same unit cell but in a unit cell that is displaced by the vector $a_1$ with respect to the original one, and its hopping contribution comes therefore along with a phase factor $\exp(i\mathbf{k} \cdot \mathbf{a}_1)$. By the same token, the A site at the vector $e_2$ is situated in the unit cell displaced by the vector $a_2$, and the corresponding phase factor is $\exp(i\mathbf{k} \cdot \mathbf{a}_2)$. One thus obtains for the off-diagonal elements of the matrix (2.62)

$$T_{B,A}(k) = T_{A,B}^*(k) = -t\gamma_k = -t(1 + e^{i\mathbf{k}\cdot\mathbf{a}_1} + e^{i\mathbf{k}\cdot\mathbf{a}_2}) \quad (2.63)$$

approach, we stick to this simplified model, which captures indeed the relevant electronic behaviour in graphene around the Fermi level. Next-nearest-neighbour hopping will be taken into account in a later chapter (Chap. 7), where we discuss a variant of graphene in the framework of the so-called Haldane model. It will become clear that the inclusion of further hoppings is technically straight-forward even if the expressions become a bit more cumbersome.

<footnote n="18">¹⁸The sign is of no importance here, but quantum-chemistry calculations happen to provide a negative sign, and we use $t > 0$ for pure convenience.</footnote>
<figure><img src="image_21.png" /><figcaption>Figure 2.7: (a) Band structure of graphene. The valence band touches the conduction band at zero energy at the points K et K', at the corners of the first Brillouin zone. Each of the bands has a width of 3t. The inset shows a zoom on the band structure in the vicinity of the band contact points, where the bands disperse linearly in the wave vector q, which is defined with respect to the points K and K'. This is reminiscent of massless ultra-relativistic particles (electrons in the conduction band) and anti-particles (holes in the valence band). (b) Band structure of boron nitride. The conduction and valence bands are well separated in energy, but the gap is smallest at the points K and K', where it is given by 2Δ.</figcaption></figure>

so that our tight-binding Hamiltonian finally reads

$$H_k = \begin{pmatrix} \Delta & -t\gamma_k^* \\ -t\gamma_k & -\Delta \end{pmatrix}. \qquad (2.64)$$

As mentioned above, we have $\Delta = 0$ in the case of graphene. This Hamiltonian is readily diagonalised, and one finds the pair of bands

$$E_{\lambda}(\mathbf{k}) = \lambda \sqrt{\Delta^2 + t^2 |\gamma_{\mathbf{k}}|^2}, \qquad (2.65)$$

where $\lambda = \pm$ is the band index. Remember that these bands stem from the $p_z$ orbitals that host each one single electron. This “half filling” of the orbitals is then transferred to the full band structure, which is itself half-filled. Therefore the band with $\lambda = -$, which is called the *valence band* is completely filled, while the *conduction band* with $\lambda = +$ is empty in undoped graphene. This means that the Fermi level resides precisely at the band contact point, which we have set to zero, in graphene, while it is situated in the band gap in the case of boron nitride.

The electronic bands of graphene, with $\Delta = 0$ in Eq. (2.65), are shown in figure 2.7(a). One notices already from the figure that the conduction
band touches the valence band at isolated points that happen to be the corners $K$ and $K'$ of the first Brillouin zone, as we calculate explicitly below. Moreover, the bands are very special in the vicinity of these band contact points, where one obtains a linear dispersion of the energy as a function of the wave vector $q$ that is now defined with respect to $\pm K$ (i.e. the points $K$ and $K'$, respectively). This is shown in the inset of figure 2.7(a). At this stage, one already notices that it is not possible to describe the bands in terms of a band mass in the low-energy window around the Fermi level, i.e. in terms of non-relativistic electrons such as in the Schrödinger equation. Indeed, the linearity of the bands is rather reminiscent of the dispersion of *massless ultra-relativistic* particles, $\epsilon = \pm \hbar v |q|$. As much as the band mass, which we have discussed in the framework of the square and the triangular lattice in section 2.2.1, is completely unrelated to the bare electron mass, the velocity $v$, which we calculate below, is not the true speed of light but given in terms of the parameters of the tight-binding model. In graphene, it happens to be 300 times smaller than the speed of light.

The band structure of boron nitride is shown in figure 2.7(b). As a consequence of $\Delta \neq 0$, the bands are now well separated in energy with a minimal gap of $2\Delta$ at the $K$ and $K'$ points. While the bands are approximately parabolic there and may thus be described in terms of a band mass, they are more appropriately given by the dispersion $\epsilon = \pm \sqrt{\Delta^2 + \hbar^2 v^2 |q|^2}$, which is that of *massive relativistic* particles, where $\Delta = mv^2$ plays the role of the rest mass, which may thus be identified with the band mass here, $m = \Delta/v^2$. Since the Fermi level resides in the gap between the valence and the conduction bands, boron nitride is an *insulator* from an electronic point of view, while graphene is a *zero-gap semimetal*. Indeed, graphene is somewhat in between a semiconductor and a metal: it is neither a true semiconductor because the band gap vanishes nor a true metal since there are only two isolated states at the Fermi level (one at the $K$ point and one at the $K'$ point) instead of a macroscopic number of states.

### 2.3.1 Low-energy model of graphene and boron nitride – the emergence of 2D Dirac fermions

In order to understand the relation of electrons in graphene and boron nitride with relativistic particles more quantitatively, we make a Taylor expansion around the band contact points $\pm K$, which are obtained by setting the sum
of phase factors in Eq. (2.63) to zero,

$$ \gamma_K = 0 = 1 + e^{i\mathbf{K} \cdot \mathbf{a}_1} + e^{i\mathbf{K} \cdot \mathbf{a}_2}. \qquad (2.66) $$

The phases must therefore be the third roots of unity, with

$$ \mathbf{K} \cdot \mathbf{a}_1 = \frac{2\pi}{3} = -\mathbf{K} \cdot \mathbf{a}_2 \quad \text{or} \quad \mathbf{K}' \cdot \mathbf{a}_1 = -\frac{2\pi}{3} = \mathbf{K}' \cdot \mathbf{a}_2. \qquad (2.67) $$

A possible choice is then

$$ \mathbf{K} = \frac{4\pi}{3\sqrt{3}a} \mathbf{u}_x \quad \text{and} \quad \mathbf{K}' = -\mathbf{K} = -\frac{4\pi}{3\sqrt{3}a} \mathbf{u}_x, \qquad (2.68) $$

which, as we have already mentioned, represent the $K$ and $K'$ points, respectively. Remember that the other four corners of the first Brillouin zone do not represent different wave vectors since they are connected to $K$ and $K'$ by reciprocal-lattice vectors. The low-energy properties of graphene (and boron nitride) are therefore governed by excitations around these two inequivalent points, which are also called the *valleys* and that we denote henceforth with the *valley index* $\xi = \pm$ ($\xi = +$ for the $K$ valley at $K$ and $\xi = -$ for the $K'$ valley at $-K$).

Let us now investigate the bands in the vicinity of the $K$ and $K'$ points via the linear expansion in the wave vector. This is indeed justified since deviations from linearity are expected at energies that are a substantial fraction of the hopping amplitude $t$, i.e. in the 1 eV range away from the Fermi level, which corresponds to temperatures on the order of 10 000 K. We thus concentrate on energies $E \ll t$, where the linear approximation is justified since $E \sim t|\mathbf{q}|a$, i.e. we consider small wave vectors with respect to the inverse lattice spacing, $|\mathbf{q}|a \ll 1$. Let us first expand the sum of the phase factors $\gamma_k$ around the $K$ point to linear order

$$ \gamma_{K+\mathbf{q}} \approx \gamma_K + i\mathbf{q} \cdot \mathbf{a}_1 e^{i2\pi/3} + i\mathbf{q} \cdot \mathbf{a}_2 e^{-i2\pi/3} \approx \frac{3a}{2}(-q_x - iq_y), \qquad (2.69) $$

where we have made use of the explicit expressions (2.59) for the primitive lattice vectors $\mathbf{a}_1$ and $\mathbf{a}_2$ and the values $\sin(2\pi/3) = \sqrt{3}/2$ and $\cos(2\pi/3) = -1/2$, to separate the phases into their real and imaginary parts. Furthermore, by definition of the band contacts, we have $\gamma_{\xi K} = 0$. In the same manner, one obtains for the linear expansion of $\gamma_k$ around the $K'$ point

$$ \gamma_{-K+\mathbf{q}} \approx \gamma_{-K} + i\mathbf{q} \cdot \mathbf{a}_1 e^{-i2\pi/3} + i\mathbf{q} \cdot \mathbf{a}_2 e^{i2\pi/3} \approx \frac{3a}{2}(q_x - iq_y). \qquad (2.70) $$
Here and in the remainder, we use the following notational convention, which we have already implicitly made: we use $\mathbf{k}$ to denote the wave vectors in the full Brillouin zone, and their origin is naturally the $\Gamma$-point, while $\mathbf{q}$ is used for the wave vector in the continuum limit $|\mathbf{q}|a \ll 1$. In contrast to $\mathbf{k}$, its origin is given by the low-energy expansion of the energy bands and can be multiple valleys, such as the $K$ and $K'$ points in the present example.

Because of these two valleys, we obtain two low-energy Hamiltonians when we use the above small-wave vector expansions for the sum of the phase factors in Eq. (2.64),

$$ H_{\xi}(\mathbf{q}) = \begin{pmatrix} \Delta & \hbar v(\xi q_x - i q_y) \\ \hbar v(\xi q_x + i q_y) & -\Delta \end{pmatrix} = \hbar v(\xi q_x \sigma_x + q_y \sigma_y) + \Delta \sigma_z, \qquad (2.71) $$

where we have used the Pauli matrices

$$ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \text{et} \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \quad (2.72) $$

and defined the characteristic velocity

$$ v = \frac{3ta}{2\hbar}. \qquad (2.73) $$

As already anticipated in the qualitative discussion of the energy bands in graphene, this velocity plays the role of the speed of light, here, but the analogy is only formal: this velocity is given in terms of the tight-binding parameters $t$ and $a$, as it is also the case for the band masses, and is by no means related to the true speed of light. However, not only the low-energy parts of the bands

$$ E_{\xi,\lambda}(\mathbf{q}) = \lambda \sqrt{m^2 v^4 + \hbar^2 q^2 v^2}, \qquad (2.74) $$

but also the Hamiltonians (2.71) are the same as those that describe the quantum-mechanical properties of relativistic fermions. They are the 2D version of the Dirac Hamiltonian, which is more frequently used to describe the relativistic quantum mechanics of high-energy particles.<fnref n="19" /> Indeed, it came to many physicists as a surprise that this relativistic equation eventually finds

<footnote n="19">¹⁹A more detailed account of the Dirac equation and its role in condensed-matter physics may be found in Chap. 8, in the framework of 3D topological insulators and Weyl semimetals.</footnote>
an application in low-energy condensed-matter physics. However, it has been a source of many experimentally measured phenomena, originally discussed in the framework of high-energy physics, ever since the advent of graphene. Because the Dirac Hamiltonian in high-energy physics describes relativistic fermionic particles, the low-energy electrons in graphene are also regularly called Dirac fermions. Notice, however, that this is somewhat abusive. While electrons are indeed fermions, regardless of whether we consider low-energy electrons in graphene or electrons promoted to the high-energy parts, we do not bother about the statistical properties of the electrons here in the calculation of the energy spectra. The tight-binding model is a one-particle model for particles, be they fermions or bosons, moving in a periodic structure, and the fermionic nature of the electrons only indicates the position of the Fermi level and thus the relevant parts of their energy bands.

As it may be seen from Eq. (2.74) for the low-energy band dispersion in graphene and boron nitride, they do not depend on the valley index $\xi$. This may be compared to the presence of the physical spin of the electrons $s = \uparrow, \downarrow$. In most situations (e.g. in the absence of the Zeeman effect), the electronic energy does not depend on this internal degree of freedom, and the electronic energy is said to be two-fold degenerate. This is also the case in the present situation, where the electronic energy is two-fold *valley-degenerate*. Even if the low-energy parts of the bands are situated at different positions in the first Brillouin zone, this piece of information is lost at low energies, for example if we consider some thermodynamic quantities in the vicinity of the Fermi level, such as the conductivity which is discussed in the following chapter. This two-fold valley degeneracy, or more generally the emergence of two Dirac fermions, described by the two Hamiltonians in Eq. (2.71), is actually an illustration of the *fermion-doubling theorem* first described by Nielssen and Ninomiya in 1983. At that time, high-energy physicists took the opposite way as we have just taken: instead of describing the low-energy physics in terms of Dirac Hamiltonians, via a Taylor expansion, they started from the Dirac Hamiltonian, which they tried to mimic as the low-energy limit of some lattice model. This is sometimes useful for numerical implementations. However, they were confronted with the seemingly bizarre problem that instead of describing only one (Dirac) fermion on the lattice, they obtained a second one that was unwanted. In contrast to this situation in high-energy physics, the fermion doubling is a natural effect in electronic systems on a lattice and physically relevant in many thermodynamic situations. Furthermore, we show in Chap. 6 that this doubling of fermions is a necessary condition
for lattice electrons due to the periodicity of the first Brillouin zone, which
can alternatively be viewed as a compact manifold, and it is related to the
topological properties of several materials.

**Notions and concepts to retain**

• Bravais lattice *vs*. general lattice

• basis (motive), sublattice and unit cell

• direct and reciprocal lattice, lattice vectors

• Bloch's theorem, periodic Hamiltonian *vs*. Bloch Hamiltonian, Bloch wave function |ψ<sub>n, k</sub>⟩ *vs*. Bloch state |u<sub>n</sub>(k)⟩

• continuum limit, valleys

• Dirac Hamiltonian, Dirac fermion
# Chapter 3

## Conduction and conductivity:
Ohm's law and its limits

In the introduction, we have already encountered a first theoretical approach of electronic transport in materials within the Drude model. It has also been argued that there are severe limitations that are related to the particular form of the band structure in the vicinity of the Fermi level, even in the classical diffusive limit where the system size exceeds both mean free path and phase-coherence length, $l_e \ll L_φ \ll L$. In the present chapter, we first give an account of this classical transport within a model based on statistical physics. Even if classical electronic transport arises naturally in three-dimensional metals, it is immediately discussed in different dimensions. As we have seen in section 1.5 in the introduction, two-dimensional (2D) systems are readily fabricated via MBE at the interface between different semiconductors, *e.g.* in GaAs/AlGaAs heterostructures. These 2D conducting systems are also the basis, *e.g* in the form of semiconductor wafers, for electric nano-circuits that are then defined lithographically. These nano-circuits can be viewed as a network of 1D nanowires the width of which is on the order of some 10 nm that is also the order of magnitude of the Fermi wavelength in 2D electron gases. The 1D character is then obtained simply by the lateral quantum confinement.

The first part of this chapter is devoted to a reminder of the density of states (section 3.1), which, as you should remember from your past statistical-physics classes, is an essential quantity in the calculation of thermodynamic or average quantities. As we remind below, it allows us to replace the often cumbersome discrete sum over the quantum numbers of the orbital degrees of
freedom (here, the wave vectors) by an integral over energy. This approach is then used to calculate electric conductivity in Bloch bands (section 3.2), where we make a connection between the conductivity and diffusion processes via the Einstein relation and retrieve the Drude model for parabolic bands. In section (3.3), we revisit the relation between conductivity and conductance and argue that the latter is more fundamental from an experimental point of view and generally depends on the manner how a conductor is connected to the exterior world via electric contacts. We then provide a detailed investigation of quantum transport through a 1D conductor, both in the ballistic and the diffusive elastic regime. The Landauer formula for the conductance is derived within this framework, and we shall see that the conductance can be viewed as a transmission coefficient times the quantum of conductance. The chapter finishes with a generalisation of the Landauer formula to the multi-channel case, where we lift the constraint of having a purely 1D system. This allows us to make a connection with Ohm's law and the Drude model in the appropriate limits.

## 3.1 Density of states

Quite generally, the average of a physical quantity in statistical physics can be written as

$$A = \sum_{\mathbf{k},s} A_{\mathbf{k},s} f[\epsilon(\mathbf{k})] = \frac{L^d}{(2\pi)^d} \sum_s \int d^d\mathbf{k} A_{\mathbf{k},s} f[\epsilon(\mathbf{k})] = g \frac{L^d}{(2\pi)^d} \int d^d\mathbf{k} A_{\mathbf{k}} f[\epsilon(\mathbf{k})], \quad (3.1)$$

where $A_{\mathbf{k},s} = \langle \mathbf{k}, s | \hat{A} | \mathbf{k}, s \rangle$ is the value the quantity takes in the quantum state $| \mathbf{k}, s \rangle$. Here, the quantum numbers are separated into two classes: the external or orbital degrees of freedom, which we consider to be the wave vectors $\mathbf{k}$ in our case, and internal degrees of freedom $s$, such as the spin of the electrons or emergent ones such as the valley index in the case of graphene. We suppose that neither the eigenenergies $\epsilon(\mathbf{k})$ nor the microscopic quantity $A_{\mathbf{k},s} = A_{\mathbf{k}}$ associated with the state $| \mathbf{k}, s \rangle$ depends on the internal quantum number $s$ so that the latter is simply taken into account by the degeneracy factor $g$ in the last expression. In the case of the spin, this is a natural assumption if the system is not submitted to a magnetic field that lifts the spin degeneracy by the Zeeman effect. Furthermore, we have considered the system of volume $L^d$ in $d$ spatial dimensions to be sufficiently large so that we
may take the wave vectors as continuous variables so that the sum becomes an integral. As it is common in statistical physics, the average quantity A is obtained by the weighted sum of the microscopic quantity $A_k$, and this weight – in quantum statistics – is given by the Fermi-Dirac occupation factor, since our electrons follow fermionic statistics,

$$f[\epsilon(\mathbf{k}), \mu] = \frac{1}{e^{\beta[\epsilon(\mathbf{k}) - \mu]} + 1}, \qquad (3.2)$$

where $\beta = 1/k_B T$ is the inverse temperature multiplied by the Boltzmann constant $k_B$, and $\mu$ is the chemical potential, which coincides with the Fermi level at zero temperature.

Instead of calculating the multi-dimensional integral in Eq. (3.1), it is much simpler to perform an integral over the energy $\epsilon$. This approach is justified since the Fermi-Dirac occupation factor (3.2) takes into account only the energy of the states, and this is also very often the case for the quantity $A_k = A(\epsilon)$ that we want to average. This can be formally achieved with the help of

$$A = g \frac{L^d}{(2\pi)^d} \int d^d k A_k f[\epsilon(\mathbf{k}), \mu] = L^2 \int d\epsilon \rho(\epsilon) A(\epsilon) f(\epsilon, \mu), \qquad (3.3)$$

where the quantity $\rho(\epsilon)$, which is called the density of states (per unit volume),<fnref n="1" /> is some sort of dictionary that translates between the discrete sum over the wave vectors and the integral over energy. It counts how many states contribute to the average within a certain energy window $[\epsilon, \epsilon + d\epsilon]$. The density of states may formally be written as

$$\rho(\epsilon) = \frac{1}{L^d} \sum_{\mathbf{k}} \delta(\epsilon - \epsilon(\mathbf{k})), \qquad (3.4)$$

in terms of Dirac $\delta$-functions. While this expression corroborates the above-mentioned state-counting picture, it is itself not very practical in analytical calculations.

One may, however, obtain analytical expressions for the density of states in the case of an isotropic and monotonic dispersion $\epsilon(\mathbf{k}) \sim k^\alpha$. While we have

<footnote n="1">¹Sometimes one defines also $D(\epsilon) = L^d \rho(\epsilon)$ as the density of states, but we stick to the other convention here and use the density of states per unit volume throughout these classes.</footnote>
seen that this is generally not the case in electronic bands in crystals, it is a good approximation in the continuum limit and in the vicinity of the Fermi level, where we saw in the previous chapter that $\epsilon(\mathbf{k}) \simeq \hbar^2|\mathbf{k}|^2/2m$ in the case of approximately parabolic bands, $\epsilon(\mathbf{k}) \simeq \hbar v|\mathbf{k}|$ for linearly dispersing bands, such as in graphene, or still $\epsilon(\mathbf{k}) = \sqrt{\Delta^2 + \hbar^2 v^2}|\mathbf{k}|^2$ for massive Dirac fermions. In this case, the appropriate coordinates for the calculation of the integral over the wave vectors in Eq. (3.1) are (generalised) spherical coordinates, in which case the integral may be written as

$$ \sum_{\mathbf{k}} \dots = \frac{L^d}{(2\pi)^d} \int d^d\mathbf{k} \dots = \frac{L^d}{(2\pi)^d} S_d \int dk k^{d-1} \dots, \quad (3.5) $$

where $S_d$ is the surface of the unit sphere in $d$ dimensions. The integral over energy is then obtained by inverting the dispersion relations, $k = k(\epsilon)$ and by a change of integration variables $dk = (d\epsilon/dk)^{-1}d\epsilon = (1/\hbar v)d\epsilon$, where we have made use of the expression $v = (1/\hbar)d\epsilon/dk$ for the group velocity in the radial direction. The density of states therefore reads

$$ \rho(\epsilon) = \frac{g}{(2\pi)^d} S_d \left(\frac{dk}{d\epsilon}\right) k^{d-1}(\epsilon) = \frac{g}{(2\pi)^d} S_d \left(\frac{1}{\hbar v}\right) k^{d-1}(\epsilon), \quad (3.6) $$

with

$$ S_{d=3} = 4\pi, \quad S_{d=2} = 2\pi, \quad S_{d=1} = 2 \quad (3.7) $$

for the surfaces of the unit sphere in the dimensions $d = 1, 2, 3$ that interest us here.

### Density of states for a parabolic dispersion

In a first step, we investigate an isotropic parabolic dispersion relation

$$ \epsilon = \frac{\hbar^2 k^2}{2m} \Leftrightarrow k(\epsilon) = \frac{\sqrt{2m\epsilon}}{\hbar}, \quad (3.8) $$

for which we find $dk/d\epsilon = \sqrt{m/2\hbar^2\epsilon}$, and the density of states therefore reads

$$ \rho(\epsilon) = \frac{g}{(2\pi)^d} S_d \left(\frac{m}{\hbar^2}\right)^{d/2} (2\epsilon)^{d/2-1} \quad (3.9) $$

in $d$ dimensions.

The expressions for the density of states associate with a parabolic dispersion are listed in table 3.1, for the dimensions $d = 1, 2$ and $3$.
<table><caption>Table 3.1: Density of states for a parabolic dispersion relation $\epsilon = \hbar^2|\mathbf{k}|^2/2m$ in different dimensions.</caption><thead><tr><th>dimension</th><th>d = 1</th><th>d = 2</th><th>d = 3</th></tr></thead><tbody><tr><td>density of states (per unit volume)</td><td><sup>g &radic;2m</sup>&frasl;<sub>2&pi;</sub> <sup>1</sup>&frasl;<sub>&radic;&epsilon;</sub></td><td><sup>g m</sup>&frasl;<sub>2&pi; &hbar;<sup>2</sup></sub></td><td><sup>g &radic;2m<sup>3</sup></sup>&frasl;<sub>2&pi;<sup>2</sup> &hbar;<sup>3</sup></sub> &radic;&epsilon;</td></tr></tbody></table>

<table><caption>Table 3.2: Density of states for a parabolic dispersion relation $\epsilon = \hbar v|\mathbf{k}|$ in different dimensions.</caption><thead><tr><th>dimension</th><th>d = 1</th><th>d = 2</th><th>d = 3</th></tr></thead><tbody><tr><td>density of states (per unit volume)</td><td><sup>g</sup>&frasl;<sub>2&pi; &hbar;v</sub></td><td><sup>g</sup>&frasl;<sub>2&pi; &hbar;<sup>2</sup>v<sup>2</sup></sub></td><td><sup>g</sup>&frasl;<sub>2&pi;<sup>2</sup> &hbar;<sup>3</sup>v<sup>3</sup></sub></td></tr></tbody></table>

### Density of states for a linear dispersion

In the case of an isotropic linear dispersion relation

$$ \epsilon = \hbar v |\mathbf{k}| \quad \Leftrightarrow \quad k(\epsilon) = \frac{\epsilon}{\hbar v}, \qquad (3.10) $$

we simply have $dk/d\epsilon = 1/\hbar v$ – indeed, $v$ is also the group velocity here – and the $d$-dimensional density of states reads

$$ \rho(\epsilon) = \frac{g}{(2\pi)^d} S_d \frac{\epsilon^{d-1}}{(\hbar v)^d}. \qquad (3.11) $$

As in the case of the parabolic dispersion relation, the density of states is listed in the table 3.2, for the dimensions $d = 1, 2$ and $3$.

### Density of states for a general algebraic dispersion $k^\alpha$

For a general dispersion relation $\epsilon = Ck^\alpha$, where $C$ is just a constant that ensures the correct physical dimension, we naturally have $k(\epsilon) = (\epsilon/C)^{1/\alpha}$ and $dk/d\epsilon = (1/\alpha)\epsilon^{1/(\alpha-1)}/C^{1/\alpha}$ so that the density of states becomes

$$ \rho(\epsilon) = \frac{g}{(2\pi)^d} \frac{S_d}{C^{d/\alpha}} \epsilon^{\frac{d}{\alpha}-1}. \qquad (3.12) $$

Comparing this result with those obtained above, we see that the density of states for a parabolic and a linear dispersion relation follow well this general behaviour, for $\alpha = 2$ and $\alpha = 1$, respectively.

From a general point of view, the important point to retain from the formulas of the density of states is the algebraic dependence on energy, $\rho(\epsilon) \sim$
$\epsilon^{d/\alpha-1}$ rather than the prefactors. While they may be important in the full expressions of certain physical quantities, they often cancel out if we express some thermodynamic quantities in terms of others. As an example we may consider the internal energy of a Fermi gas, which is the average energy

$$U = \sum_{\mathbf{k},s} \epsilon_{\mathbf{k}} f[\epsilon(\mathbf{k}), \mu] = L^d \int d\epsilon \rho(\epsilon) \epsilon f(\epsilon, \mu), \quad (3.13)$$

as well as the average number of fermions

$$N = \sum_{\mathbf{k},s} f[\epsilon(\mathbf{k}), \mu] = L^d \int d\epsilon \rho(\epsilon) f(\epsilon, \mu). \quad (3.14)$$

Just by looking at the expressions, one notices that they look similar, and we may evaluate them in the zero-temperature limit $T \ll T_F$, i.e. at temperatures that are low with respect to the Fermi temperature $T_F = \epsilon_F/k_B$ in terms of the Fermi energy $\epsilon_F$. In this limit, one may replace the Fermi-Dirac occupation factor by a simple Heaviside step function $f(\epsilon) \approx \Theta(\epsilon_F - \epsilon)^2$ so that the two integrals are easily evaluated,

$$U = \frac{\alpha}{d + \alpha} \frac{g}{(2\pi)^d} \frac{S_d L^d}{C^{d/\alpha}} \epsilon_F^{\frac{d}{\alpha}+1} \quad (3.15)$$

and

$$N = \frac{\alpha}{d} \frac{g}{(2\pi)^d} \frac{S_d L^d}{C^{d/\alpha}} \epsilon_F^{\frac{d}{\alpha}} = \frac{\alpha}{d} L^d \rho(\epsilon_F) \epsilon_F \quad (3.16)$$

While these expressions are quite cumbersome due to the various prefactors, one directly sees that they are both related by the rather simple expression

$$U = \frac{d}{d + \alpha} N \epsilon_F, \quad (3.17)$$

where we have got rid of all these prefactors. The expression reproduces the internal energies

$$U_{\text{non-rel}} = \frac{3}{5} N \epsilon_F \quad \text{and} \quad U_{\text{rel}} = \frac{3}{4} N \epsilon_F, \quad (3.18)$$

for non-relativistic and relativistic 3D Fermi gases, respectively, as you should remember from your basic statistical-physics classes.
<figure><img src="image_22.png" /><figcaption>Figure 3.1: (a) Family of 2D bands obtained by dimensional reduction due to confinement of the electronic motion in the z-direction. (b) Associated density of states. Each 2D band yields a constant contribution to the density of states provided that the energy is larger than that of the band bottom.</figcaption></figure>

## Dimensional reduction and evolution of the density of states

In the above discussion of the density of states, we have encountered dispersion relations in several dimensions even if we naturally live in a three-dimensional (3D) world. The discussion of the different spatial dimensions is indeed not just a stylistic exercise, but one naturally encounters it when we artificially reduce the spatial dimensions of a system. We have already mentioned, in the introduction, that this naturally occurs in the formation of two-dimensional (2D) electron systems at the interface between two different semiconductors. If we consider an originally 3D parabolic band, the kinetic energy must be accompanied by a confining potential in one of the direction, which is usually the z-direction. In the case of MBE grown modulation-doped heterostructures, this confinement potential $V(z)$ has a triangular shape, but the precise shape is of minor importance here. The overall Hamiltonian therefore reads

$$H = \frac{\hbar^2(k_x^2 + k_y^2)}{2m} + H_z, \quad \text{with} \quad H_z = \frac{\hbar^2 k_z^2}{2m} + V(z). \qquad (3.19)$$

Instead of a full solution, which requires knowledge of the precise form of the confinement potential, let us just inspect the structure of the energy spectrum. It is clear that the wave-vector components $k_x$ and $k_y$ remain good quantum numbers, i.e. the motion in the plane is that of free particles, while the component $k_z$ does not commute with the confinement potential.

<footnote n="2">²Remember that $\Theta(x) = 1$ for $x \le 1$ and $\Theta(x) = 0$ for $x > 0$.</footnote>
The motion in the z-direction is therefore quantized, and the effective one-dimensional (1D) quantum-mechanical problem is described by the Hamiltonian $H_z$. The latter yields a discrete spectrum $\epsilon_n$ via the solution of the Schrödinger equation $H_z|n\rangle_z = \epsilon_n|n\rangle_z$. The energy spectrum thus consists of a family of 2D bands

$$ \epsilon_n(k_x, k_y) = \frac{\hbar^2(k_x^2 + k_y^2)}{2m} + \epsilon_n \quad (3.20) $$

that are schematically shown in figure 3.1(a), where $k_\parallel = \sqrt{k_x^2 + k_y^2}$ is the inplane component of the wave vector. What about the density of states in this situation? We are now confronted with 2D bands, and we therefore need to take the expression for the 2D density of states, which is constant in energy for parabolic bands. The full density of states is therefore the sum of the contributions from all bands. However, in order to contribute to the full density of states at a given energy $\epsilon$, only bands whose band bottom is lower in energy, $\epsilon_n < \epsilon$, must be taken into account,

$$ \rho(\epsilon) = \frac{m}{\pi\hbar^2} \sum_{n=0}^{\infty} \Theta(\epsilon - \epsilon_n), \quad (3.21) $$

in terms of the Heaviside step function.

The dimension of the system can still be reduced to effective 1D bands if we remember that a 2D electron gas, e.g. in an AlGaAs/GaAs heterostructure, can be constricted into channels by specially patterned gate electrodes. If we consider a square potential in the y-direction of width $W$, defined electrostatically by such gates, the electronic motion in the y-direction is equally confined and we obtain, in the same manner as above, a family of 1D bands

$$ \epsilon_{n,j}(k_x) = \frac{\hbar^2 k_x^2}{2m} + \tilde{\epsilon}_j + \epsilon_n, \quad (3.22) $$

where $\tilde{\epsilon}_j = \hbar^2\pi^2j^2/2mW^2$ for the square potential. The associated density of states is now that of 1D bands with their characteristic $1/\sqrt{\epsilon}$ decrease. Again only bands with minimal energy $\epsilon_n + \tilde{\epsilon}_j < \epsilon$ contribute to the density of states at the energy $\epsilon$,

$$ \rho(\epsilon) = \frac{\sqrt{2m}}{\pi\hbar} \sum_{n=0}^{\infty} \sum_{j=1}^{\infty} \frac{\Theta(\epsilon - \epsilon_n - \hbar^2\pi^2j^2/2mW^2)}{\sqrt{\epsilon - \epsilon_n - \hbar^2\pi^2j^2/2mW^2}}. \quad (3.23) $$
## 3.2 Statistical model for conductivity

The above reminder of quantum statistics allows us to approach the conduction properties of electrons in diffusive conductors, where we start from the current density. The current density is given in terms of the average velocity of the electrons

$$ \mathbf{j} = -\frac{e}{L^d} \sum_{n;s,\mathbf{k}} \mathbf{v}_{n,\mathbf{k}} f[\epsilon_n(\mathbf{k}), \mu], \quad (3.24) $$

where the index $n$ denotes the different electronic bands that may contribute to the electronic transport.

Let us first consider this expression in the absence of an electric field. Naively, we would expect that there is then no current flowing through the conductor, and our intuition turns out to be correct. In order to see this formally, we suppose that the system has time-reversal symmetry. Time-reversal symmetry, which we discuss in more detail in Chap. 7, tells us that the fundamental equations of motion – in the case of quantum mechanics, the Schrödinger equation – remain invariant if we change the sense of the time arrow in a thought experiment. It is eventually broken in the presence of a magnetic field that changes itself its orientation upon the change of the time arrow. In the absence of a magnetic field, time-reversal invariance tells us that, if we have an electronic state at a wave vector $\mathbf{k}$ and energy $\epsilon_n(\mathbf{k})$, there is also a state at $-\mathbf{k}$ with the same energy, $\epsilon_n(-\mathbf{k}) = \epsilon_n(\mathbf{k})$.

In order to show that there is no current flowing in the absence of an electric field, consider the velocity (2.58) for $\mathbf{E} = 0$

$$ \mathbf{v}_n(\mathbf{k}) = \langle u_n(\mathbf{k}) | \mathbf{v}(\mathbf{k}) | u_n(\mathbf{k}) \rangle = \frac{1}{\hbar} \langle u_n(\mathbf{k}) | \nabla_{\mathbf{k}} H_{\mathbf{k}} | u_n(\mathbf{k}) \rangle, \quad (3.25) $$

where we have made use of the expression (2.45) for the velocity operator $\mathbf{v}_\mathbf{k}$ in the Bloch basis. A straight-forward calculation indicates that the band
velocity is then nothing other than the group velocity<fnref n="3" />

$$v_n(\mathbf{k}) = \frac{1}{\hbar} \nabla_{\mathbf{k}} \epsilon_n(\mathbf{k}), \qquad (3.26)$$

and time-reversal symmetry then imposes

$$v_n(\mathbf{k}) = -v_n(-\mathbf{k}). \qquad (3.27)$$

Because the states at $\mathbf{k}$ and $-\mathbf{k}$ have the same energy, they are either both below or both above the Fermi level. In the latter case, they do not contribute to the electric transport anyway, while in the former case, these two contributions mutually annihilate each other due to the opposite sign in their velocity.

In order to overcome this mutual cancellation of the contributions at opposite wave vectors, we now apply an electric field. We have already seen how the average velocity changes in the presence of a homogeneous electric field which does not spoil the periodicity of the underlying lattice upon a gauge transformation, but it changes the wave vector according to Eq. (2.57) by the quantity $\Delta\mathbf{k} = -e\mathbf{E}t/\hbar$. As mentioned in section 2.2.3 of the preceding chapter, this leads to Bloch oscillations that are usually cut off by scattering events at the characteristic scattering time $\tau$, which replaces the time $t$ in the expression. The shift

$$\Delta \mathbf{k} = -\frac{e}{\hbar} \mathbf{E} \tau \qquad (3.28)$$

thus becomes static – it can alternatively be viewed as due to the stationary solution of the Newton equation – and we may furthermore transfer, in Eq.

$$\begin{align*} \nabla_k \epsilon_n(\mathbf{k}) &= \nabla_k \langle u_n(\mathbf{k}) | H_k | u_n(\mathbf{k}) \rangle \\ &= \langle \nabla_k u_n(\mathbf{k}) | H_k | u_n(\mathbf{k}) \rangle + \langle u_n(\mathbf{k}) | \nabla_k H_k | u_n(\mathbf{k}) \rangle + \langle u_n(\mathbf{k}) | H_k | \nabla_k u_n(\mathbf{k}) \rangle \\ &= \langle \nabla_k u_n(\mathbf{k}) | u_n(\mathbf{k}) \rangle \epsilon_n(\mathbf{k}) + \langle u_n(\mathbf{k}) | \nabla_k H_k | u_n(\mathbf{k}) \rangle + \epsilon_n(\mathbf{k}) \langle u_n(\mathbf{k}) | \nabla_k u_n(\mathbf{k}) \rangle \end{align*}$$

Notice that the first and the last term in the last line cancel each other because the gradient $\nabla_k$ is an anti-Hermitian operator so that $\langle u_n(\mathbf{k})|\nabla_k u_n(\mathbf{k})\rangle = -\langle \nabla_k u_n(\mathbf{k})|u_n(\mathbf{k})\rangle$, and we obtain the announced result.

<footnote n="3">³While this expression seems natural, one may derive it explicitly from the Schrödinger equation $H_k|u_n(\mathbf{k})\rangle = \epsilon_n(\mathbf{k})|u_n(\mathbf{k})\rangle$, which we multiply from the left with the bra $\langle u_n(\mathbf{k})|$ before taking the $k$-space gradient $\nabla_k$,</footnote>
<figure><img src="image_23.png" /><figcaption>Figure 3.2: (a) Full band in the absence of an electric field. The states at the wave vectors $\mathbf{k}$ and $-\mathbf{k}$ have the same energy but opposite velocities $\mathbf{v}_n(-\mathbf{k}) = -\mathbf{v}_n(\mathbf{k})$ so that there is no net current flow due to the pairwise cancellation of the velocities. (b) In the presence of an electric field, all states are shifted by $\Delta\mathbf{k}$ so that some states are now situated outside the first Brillouin zone. However, these states have equivalent ones inside the first Brillouin zone via a translation of a reciprocal-lattice vector $\mathbf{a}^*$ (in red).</figcaption></figure>

(3.24) the shift to the occupation factor

$$ 
\begin{aligned}
\mathbf{j} &= -\frac{e}{L^d} \sum_{n;s,\mathbf{k}} \mathbf{v}_n(\mathbf{k} + \Delta\mathbf{k}) f[\epsilon_n(\mathbf{k}), \mu] \\
&= -\frac{e}{L^d} \sum_{n;s,\mathbf{k}} \mathbf{v}_n(\mathbf{k}) f[\epsilon_n(\mathbf{k} - \Delta\mathbf{k}), \mu] = \sum_n \mathbf{j}_n
\end{aligned} 
\quad (3.29) 
$$

by a simple change of the integration variable. In the last expression, we have explicitly written that the current density has contributions $\mathbf{j}_n$ from all bands $n$.

We are now armed to confirm a central fact of electric transport in lattices: The contribution $\mathbf{j}_n$ of any fully occupied band to the current density is zero! This can immediately be seen from the equation for $\mathbf{j}_n$ and the fact the occupation number is one for all states, i.e. for all wave vectors in the first Brillouin zone,

$$ \mathbf{j}_n = -\frac{e}{L^d} \sum_{s,\mathbf{k}} \mathbf{v}_n(\mathbf{k}) = 0. \quad (3.30) $$

As in the case of no electric field, which may here simply be omitted by a simple change of integration variable, all contributions at $\mathbf{k}$ and $-\mathbf{k}$ cancel pairwise. The vanishing contribution $\mathbf{j}_n$ of a filled band can also be understood from the periodicity of the electronic bands in reciprocal space, as it is shown in figure 3.2. If all states are shifted by $\Delta\mathbf{k}$, some states, which were initially occupied at the (left) border of the first Brillouin zone sketched
in figure 3.2(b), are now empty while some others (at the right border) are shifted outside the first Brillouin zone. The latter need to be folded back into the first Brillouin zone, with the help of a reciprocal-lattice vector, and one notices that these states now occupy the emptied states (on the left border). This is simply due to the fact that a shift of a full periodicity of a periodic function remains a full periodicity, and the situation is therefore equivalent to the original one in the absence of an electric field [see figure 3.2(a)].

In order to calculate the current density for a partially filled band $n$, we simplify the above expression (3.29) and suppose that the electric field is oriented in the $x$-direction, $\mathbf{E} = E\mathbf{u}_x$, which is also the direction of the wave vector shift then $\Delta k_x = -eE\tau/\hbar$. If we consider the electric field to be sufficiently small, that is $\Delta k_x \ll k_F$ where $k_F$ is the Fermi wave vector in the band, we can use linear-response theory and expand the Fermi-Dirac occupation factor in $\Delta k_x$,

$$f[\epsilon_n(\mathbf{k} - \Delta\mathbf{k}), \mu] \approx f[\epsilon(\mathbf{k}), \mu] - \frac{\partial f}{\partial \epsilon} \frac{\partial \epsilon}{\partial k_x} \Delta k_x = f[\epsilon(\mathbf{k}), \mu] + \hbar v_x \left(-\frac{\partial f}{\partial \epsilon}\right) \Delta k_x, \quad (3.31)$$

where we have used $\partial\epsilon/\partial k_x = \hbar v_x$ in the last step. The linearised Fermi-Dirac occupation factor is then substituted into the expression of the current density. While the first term reproduces the zero result in the absence of an electric field, the relevant $x$-component of the current density in the $n$-th band becomes

$$j_{x,n} = -\frac{\hbar e}{L^d} \sum_{n;s,\mathbf{k}} v_{n,x}^2(\mathbf{k}) \left(-\frac{\partial f}{\partial \epsilon}\right) \Delta k_x = \frac{e^2 \tau}{L^d} \sum_{n;s,\mathbf{k}} v_{n,x}^2(\mathbf{k}) \left(-\frac{\partial f}{\partial \epsilon}\right) E, \quad (3.32)$$

i.e. we obtain the searched-for linear relation between current density and electric field (in the same direction). We can thus immediately extract the conductivity, which is precisely the proportionality constant between the two, and one has

$$\sigma = \frac{e^2 \tau}{L^d} \sum_{n;s,\mathbf{k}} v_{n,x}^2(\mathbf{k}) \left(-\frac{\partial f}{\partial \epsilon}\right) = e^2 \tau \sum_n \int d\epsilon \rho_n(\epsilon) v_{n,x}^2(\epsilon) \left(-\frac{\partial f}{\partial \epsilon}\right), \quad (3.33)$$

where we have made use of the density of states $\rho_n(\epsilon)$ for the $n$-th band, in line with the discussion of the previous section.

In the last expressions, we have used the (negative) derivative of the Fermi-Dirac occupation factor with respect to energy, which is indeed suggestive: it is a distribution of width $\sim k_B T$ centered around the Fermi level.
It reduces to the Dirac $\delta$-function in the zero-temperature limit

$$ -\frac{\partial f}{\partial \epsilon} \approx \delta(\epsilon - \epsilon_F) \qquad (3.34) $$

so that we obtain the zero-temperature result, valid for $T \ll T_F$,

$$ \sigma \approx e^2 \tau \sum_n \rho_n(\epsilon_F) v_{n,x}^2(\epsilon_F). \qquad (3.35) $$

We already see from this expression that the conductivity is an additive quantity in two respects. First, if we have different bands, one may associate a conductivity $\sigma_n$ with each band, and the total conductivity is $\sigma = \sum_n \sigma_n$. From an electric-transport point of view, we may see that the conductivities of the different bands, or more precisely their associate conductances, are combined as conductivities (conductances) in parallel. The second point is more subtle and appeals to the different types of scattering processes, described by different relaxation times, $\tau_1, \tau_2, ...$. In this case we need to add the inverses of the scattering times to obtain the total scattering time (Matthiesen's rule), $1/\tau = 1/\tau_1 + 1/\tau_2 + ...$, in agreement with the sum of the resistances in series.

### 3.2.1 Einstein relation and link with diffusion processes

The expression (3.35) for the zero-temperature conductivity actually makes a connection with diffusive scattering processes. Let us first notice that the $x$-component of the average band velocity $v_{n,x}(\epsilon_F)$ at the Fermi level in the $n$-th band is the same as that in the other directions if we consider an isotropic band dispersion. In this case, the average square velocity $\langle v_n^2 \rangle = \sum_{\mu=1}^d v_{n\mu}^2(\epsilon_F) = dv_{n,x}(\epsilon_F)$ for a $d$-dimensional system, which is nothing other than the Fermi velocity in the $n$-th band, $\langle v_n^2 \rangle = v_{F,n}^2$. We may thus write the conductivity in terms of the diffusion constant

$$ D_n = \tau v_{n,x}^2(\epsilon_F) = \tau \frac{v_{F,n}^2}{d} = \frac{v_{F,n} l_{e,n}}{d}, \qquad (3.36) $$

where $l_{e,n} = v_{F,n}\tau$ is the mean free path for an electron in the $n$-th band. We finally obtain the so-called *Einstein relation*

$$ \sigma = e^2 \sum_n D_n \rho_n(\epsilon_F), \qquad (3.37) $$
which relates the electric conductivity to a diffusion process, such as a random walk.

Notice that the above expression for the Einstein relation (3.37) is that for several bands that possibly cross the Fermi level. This is a more complex situation than that of the single-band model, which is perhaps more often used in the literature. One may then omit the sum as well as the band indices, and the single-band Einstein relation simply reads

$$ \sigma = e^2 D \rho(\epsilon_F). \qquad (3.38) $$

Furthermore, we notice that this relation for the conductivity is quite general and does not make any assumption about the energy bands, be they parabolic, linear or following an arbitrary algebraic law. The Einstein relation is therefore more general than the expression obtained within the Drude model, which we retrieve for a single parabolic band, as we show below.

### 3.2.2 Back to the Drude model

In order to find again the expression (1.6) of the conductivity of the Drude model, we need to consider a parabolic band with a band mass $m$. As we have discussed in the previous section, the density of states, associated with a general algebraic dependence of the energy on the wave vector $\epsilon = Ck^\alpha$, scales as $\rho(\epsilon) = \tilde{C}\epsilon^{d/\alpha-1}$, where $\tilde{C}$ is an unimportant constant that contains the microscopic parameters. The electronic density is then simply obtained by integrating the density of states in energy up to the Fermi level, since we still consider the zero-temperature limit,

$$ n_{\text{el}} = \int_{0}^{\epsilon_F} d\epsilon \rho(\epsilon) = \frac{2}{d} \tilde{C} \epsilon_F^{d/\alpha} = \frac{\alpha}{d} \rho(\epsilon_F) \epsilon_F, \qquad (3.39) $$

in line with Eq. (3.16). We may thus substitute the density of states $\rho(\epsilon_F) = (d/\alpha)n_{\text{el}}/\epsilon_F$ into the single-band conductivity (3.38), and one obtains

$$ \sigma = \frac{e^2 v_F^2 n_{\text{el}} \tau}{\alpha \epsilon_F}. \qquad (3.40) $$

In the case of a parabolic band with $\alpha = 2$, the Drude conductivity

$$ \sigma = \frac{e^2 n_{\text{el}} \tau}{m} \qquad (3.41) $$
is then finally obtained if we make use of the expression $\epsilon_F = (1/2)mv_F^2$ for the Fermi energy in terms of the Fermi velocity and the band mass.

In the case of graphene with its linear dispersion relation, $\alpha = 1$, we have $\rho(\epsilon) = C\epsilon^{d-1}$, we obtain the conductivity

$$ \sigma = \frac{e^2 v_F^2 n_{el} \tau}{\epsilon_F} = \frac{e^2 v \tau}{\hbar k_F}, \qquad (3.42) $$

where we have used $\epsilon_F = \hbar vk_F$ in the last step as well as $v_F = v$.

## 3.3 Conductivity and conductance

In the previous section, we have obtained the conductivity in the classical diffusive regime, i.e. in the regime where the system size exceeds both mean free path and phase-coherence length, $l_e \ll L_\phi \ll L$. In this case, the conductance of a conductor of length $L$ and cross section $S = W^{d-1}$ (in terms of the width $W$) is simply obtained from the conductivity, in a single-band model,<fnref n="4" /> by

$$ G = \sigma \frac{W^{d-1}}{L} = \frac{e^2 v_F^2 \tau}{d (2\pi)^d} S_d \frac{k_F^{d-1}}{\hbar v_F} \frac{W^{d-1}}{L}, \qquad (3.43) $$

where we have used the explicit expressions for the diffusion constant (3.36) and the $d$-dimensional density of states (3.6). This yet complicated expression can be simplified if we introduce the Fermi wavelength $\lambda_F = 2\pi/k_F$, the mean free path $l_e = v_F\tau$ and the Planck constant $h = 2\pi\hbar$. One can then get rid of the $d$ factors $2\pi$

$$ G = g \frac{e^2}{\hbar} A_d \left(\frac{W}{\lambda_F}\right)^{d-1} \left(\frac{l_e}{L}\right), \qquad (3.44) $$

where we have introduced the volume $A_d = S_d/d$ of the unit sphere in $d$ dimensions. One notices that, apart from the degeneracy and this geometric prefactor, the conductance is proportional to the cross section $W^{d-1}$, where the width is measured in units of the Fermi wavelength, and inversely proportional to length measured in units of the mean free path. These are dimensionless length scales, and the conductance itself is given in units of

<footnote n="4">⁴The multi-band case is obtained by direct generalisation.</footnote>