Electronic properties of Solids

Charbel Tannous

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Charbel Tannous. Electronic properties of Solids. Master. Physique du Solide, Université de Brest, France. 2018, pp.25. hal-04169035

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# Electronic properties of Solids

C. Tannous

Université de Brest, Lab-STICC, CNRS-UMR 6285, F-29200 Brest, FRANCE

(Dated: July 23, 2023)

Electronic properties of Solids are explained from the Crystallographic, Thermodynamic and Statistical points of view. Properties and Applications of metals, semi-metals, semiconductors and Insulators are also reviewed. Band theory methods are briefly described and applied to Graphene.

PACS numbers: 71.15.-m, 71.20.-b, 78.67.De

Keywords: Electronic structure calculation methods, Band structure, Quantum wells

## Contents

I. Introduction 1

II. The free electron model 2

A. Drude classical description 2

1. Matthiessen Rule 4

2. Hall effect and cyclotron resonance 4

B. Quantum description 5

III. Statistical properties of a Fermion gas 5

A. Fermi-Dirac distribution 6

B. Dispersion relations 6

C. Density of states 7

D. The effective mass 8

E. Thermodynamic properties of the Fermi gas 9

IV. Band theory for a single electron in a Periodic potential 15

A. Derivation of the exact dispersion relation 16

V. Many-electron systems and Band structure calculations 18

A. Two-particle case 18

B. General case: the many body wave-function 19

C. Augmented Plane Wave method 20

D. Developments concerning the APW method 21

E. Tight-Binding method 22

F. Emergent and other Fermi varieties 22

A. Atomic units 23

B. Standard Metal properties 24

References 25

## I. INTRODUCTION

When particles are inside a particular environment such as condensed matter, their behavior is very different compared to the one in Vacuum or free open space where they are not interacting between themselves and their motion is not obstructed by various objects.

Particles inside solids become quasi-particles meaning they are dressed by their environment or by interactions among themselves. For instance, an electron in a solid is subject to a potential $V(r)$ created by atoms, ions, impurities, defects, vacancies, dislocations... present inside the solid.

The notion of quasi-particles originated in Landau theory of many-body Fermi liquids, which was originally created for basically understanding liquid Helium-3.
Landau (1; 2) hypothesized that the thermodynamic free energy *F* of the system should be described by a single phenomenological parameter, the spatial density *ρ*(*r*) of quasi-particles. *ρ*(*r*) is meant to preserve the basic characteristics of the Fermi-liquid such as charge, spin and (Fermi-Dirac) statistics.

Landau description implies we are, in fact, mapping the many-body system to an ensemble of weakly interacting fermions. This approach is equivalent to a first-order perturbation theory to a complex many-body problem.

Comparison between physical properties of particles and quasi-particles is summarized and commented in Table I.

<table><caption>TABLE I Fermi Particles (P) and Quasi-Particles (Q) with P and Q possessing mass and effective mass (*m* and *m*\*) respectively, spin and number of polarization states *N*<sub>*p*</sub>. If a P is massive and possesses a spin *S*, it has *N*<sub>*p*</sub> = 2*S* + 1 polarization states, whereas a zero-mass P with a spin *S* has polarization states (3) ±*S* i.e. their number is (*N*<sub>*p*</sub> = 2). Particles and quasi-particles are characterized by their dispersion relation *ϵ*(*k*) with *k* continuous for free particles, discrete in crystals for quasi-particles along with the corresponding dispersion relations *ϵ*<sub>*n*,σ*</sub>(*k*) with *n*, *σ* band index and spin that are proper to corresponding quasi-particles. When we have conservation of the number of particles or quasi-particles we should have a non-zero (4) chemical potential *μ*(*T*, *P*) depending on system temperature and pressure (*T*, *P*). In the non-conserved case, *μ*(*T*, *P*) = 0, ∀*T*, *P*. *n*, *σ* are band index and spin. For electrons and holes in crystals *n* ∈ N. Holes are encountered in semiconductors and correspond to absence of electron quasi-particles.</caption><thead><tr><th>Name</th><th>Nature</th><th>Mass</th><th>Charge</th><th>Spin</th><th><i>N</i><sub>p</sub></th><th>Statistics</th><th>Chemical Potential</th><th>Dispersion relation</th></tr></thead><tbody><tr><td colspan="9" style="text-align:center; font-weight:bold;">Free particles</td></tr><tr><td>Electron P</td><td></td><td><i>m</i><sub>e</sub></td><td>−<i>e</i></td><td>1/2</td><td>2</td><td>Fermi-Dirac</td><td>μ(<i>T</i>, <i>P</i>)</td><td><sup>ħ<sup>2</sup><i>k</i><sup>2</sup></sup>&frasl;<sub>2<i>m</i><sub>e</sub><i>g</i></sub></td></tr><tr><td>Fermion P</td><td></td><td><i>m</i></td><td><i>q</i></td><td><i>S</i></td><td>2<i>S</i> + 1</td><td>Fermi-Dirac</td><td>μ(<i>T</i>, <i>P</i>)</td><td><sup>ħ<sup>2</sup><i>k</i><sup>2</sup></sup>&frasl;<sub>2<i>m</i></sub></td></tr><tr><td colspan="9" style="text-align:center; font-weight:bold;">Quasi-Particles in a crystal</td></tr><tr><td>Electron Q</td><td></td><td><i>m</i><sup>*</sup><sub>e</sub></td><td>−<i>e</i></td><td>1/2</td><td>2</td><td>Fermi-Dirac</td><td>μ(<i>T</i>, <i>P</i>)</td><td>ε<sub>n,σ</sub>(<b>k</b>)</td></tr><tr><td>Hole Q</td><td></td><td><i>m</i><sup>*</sup><sub>h</sub></td><td>+<i>e</i></td><td>1/2</td><td>2</td><td>Fermi-Dirac</td><td>μ(<i>T</i>, <i>P</i>)</td><td>ε<sub>n,σ</sub>(<b>k</b>)</td></tr></tbody></table>

Landau (5) basic ansatz is that the free energy of the quasi-particle ensemble is a functional *F*[*ρ*(*r*)] and not a simple function, enabling him to define specific conditions that *F* should satisfy in order to qualify as the right free energy. This phenomenological idea explained many features of Fermi liquids and lead, later on, to band-structure calculations of many-body electronic systems with Density Functional Theory (DFT) and afterwards to Spin Density Functional Theory (SDFT) in order to describe magnetic systems.

Landau description, being some kind of first-order perturbation theory of weakly interacting fermions should not apply to strongly interacting fermion system such as 1D Luttinger-Tomonaga (6) model, 2D correlated fermion systems, superconducting systems in any dimension...

In order to set the stage for any quantitative description, solids can be classified in terms of their conductivity performance point of view. For instance, insulators have their conductivity *σ*<sub>*c*</sub> ∈ [10<sup>−18</sup>, 10<sup>−8</sup>] S/cm, whereas semi-conductors have *σ*<sub>*c*</sub> ∈ [10<sup>−8</sup>, 10<sup>+3</sup>] S/cm, semi-metals like graphite, As, Sb and Bi or metals possess *σ*<sub>*c*</sub> ∈ [10<sup>+3</sup>, 10<sup>+8</sup>] S/cm (7). S is Siemens or inverse Ohm Ω<sup>−1</sup> conductance unit.

## II. THE FREE ELECTRON MODEL

### A. Drude classical description

The behavior of metals was extremely puzzling to understand at the beginning of the 20th century despite centuries of metallurgy and metal studies.

One of the questions is the understanding of their heat capacity *C*<sub>*v*</sub> at high and low temperature:

* High temperature:

A 3D insulator possesses degrees of freedom given by the equipartition theorem (8) allocating 3/2 *k*<sub>*B*</sub>*T* energy per degree of freedom. This results in 3 *k*<sub>*B*</sub>*T* (*k*<sub>*B*</sub> being Boltzmann constant) resulting in a specific heat *C*<sub>*v*</sub> = 3 *k*<sub>*B*</sub>*N*<sub>*A*</sub> = 3*R* for *N*<sub>*A*</sub> (Avogadro number) ions since we have kinetic and vibrational degrees of freedom. This is Dulong-Petit law (9). *R* is gas constant.
In the 3D metal considered as a gas of non-interacting electrons the equipartition theorem (8) yields at high temperature a classical specific heat $C_v = \frac{3}{2}k_B N_e$ for $N_e$ electrons with only kinetic degrees of freedom. Taking $N_e = N_A$ we get $C_v = \frac{3}{2}R$ at high temperature.

* Low temperature:

For 3D insulators, the variation of $C_v \propto T^\alpha$ with exponent $\alpha \sim 3$ whereas 3D metals follow Wiedemann-Franz law with $C_v \propto T$ as displayed in Table. II.

P. Drude proposed a classical model for a free electron that predicts, thermal, electric, and optical properties of solids. Drude applied kinetic theory of gases to electrons to metals leading to description of DC and AC conductivity, Hall-effect, magneto-resistance phenomena and agreeing with Wiedemann-Franz law.

However, the specific heat of metals was overestimated and the issue has been tackled, later on with the advent of Quantum Mechanics. First of all, Ehrenfest theorem justified classical single electron treatment through building quantum packets obeying classical equations of motion and Sommerfeld replaced Maxwell-Boltzmann distribution with Fermi-Dirac statistics to account for Pauli Spin-Statistics theorem.

We assume that an electron (of charge $q = -e$ and mass $m_e$) is subjected to an electric field and its equation of motion is Newtonian written as:

$$m_e \ddot{\boldsymbol{r}} = -e \boldsymbol{E} \qquad (1)$$

Integrating once, we have: $\boldsymbol{v} = \dot{\boldsymbol{r}} = -e \boldsymbol{E} t / m_e$.

This implies the velocity increases linearly until a collision happens with other electrons impurities or limiting metal boundaries. Thus the velocity is reset to zero and re-increases linearly for a typical time $\tau_c$ until another collision event resets it. The average length traveled by the free electron is the mean free path $\ell$.

Thus we get a sawtooth type of motion of the electron with an average velocity between two successive collisions given by:

$$\bar{v} = \frac{1}{\tau_c} \int_0^{\tau_c} dt \, eE t / m_e = eE \tau_c / (2m_e) \qquad (2)$$

Drude suggested another mechanism which induces a smoother motion and that is a damping mechanism as a friction term proportional to velocity such that the equation of motion writes:

$$m_e \dot{\boldsymbol{v}} = -e \boldsymbol{E} - \frac{m_e \boldsymbol{v}}{\tau} \qquad (3)$$

$\tau$ is a relaxation time responsible for damping given by $\tau = \ell / \bar{v}$.

Integrating we have:

$$\boldsymbol{v} = \frac{e \boldsymbol{E} \tau}{m_e} + \boldsymbol{v}_0 \exp(-t/\tau) \qquad (4)$$

Thus, when $t \gg \tau$, $\boldsymbol{v} \to \boldsymbol{v}_\infty = \frac{e \boldsymbol{E} \tau}{m_e}$ resulting in a current density $\boldsymbol{J} = \frac{n_e e^2 \boldsymbol{E} \tau}{m_e}$ for an electron density $n_e$. The resulting Drude conductivity $\sigma_D$ as in $\boldsymbol{J} = \sigma_D \boldsymbol{E}$ is $\sigma_D = \frac{n_e e^2 \tau}{m_e}$.

It is interesting to note that if we equate Drude $\boldsymbol{v}_\infty = \frac{e \boldsymbol{E} \tau}{m_e}$ with the collision model velocity average $\bar{v} = eE \tau_c / (2m_e)$ we get the relation between the damping and collision times $\tau = \tau_c / 2$.

Moving on to a time-harmonic electric field $\boldsymbol{E} \propto \exp(i\omega t)$ we look for a solution of the form $\boldsymbol{v} \propto \exp(i\omega t)$ obtaining:

$$\boldsymbol{v} = - \frac{e \boldsymbol{E}}{m_e (i\omega + 1/\tau)} \qquad (5)$$
yielding a phase delay $\alpha$ between $v$ and $E$ given by

$$ \alpha = \pi/2 + \tan^{-1}(\omega\tau) \qquad (6) $$

and a frequency-dependent Drude conductivity given by:

$$ \sigma_D(\omega) = \frac{n_e \tau e^2}{m_e (1 + [\omega/\tau]^2)} \qquad (7) $$

## 1. Matthiessen Rule

Drude conductivity $\sigma_D = \frac{n_e e^2 \tau}{m_e}$ leads to resistivity $\rho_D = 1/\sigma_D = \frac{m_e}{n_e e^2 \tau}$ and a resistance $R_D \propto \rho_D$.

In the case of several damping processes (impurities, excitations, defects, vacancies, dislocations..) we add the corresponding resistances implying performing a harmonic sum of the different scattering times $\tau_i$ since the resistivity is inversely proportional to scattering time. Consequently we obtain a total inverse lifetime:

$$ \frac{1}{\tau} = \sum_{i=1}^{n_p} \frac{1}{\tau_i} \qquad (8) $$

where $n_p$ is the number of scattering processes: this is Matthiessen Rule.

The scattering times $\tau_i(T)$ generally depend on temperature $T$, thus all materials have a finite residual resistivity at $T \to 0$K with implications regarding superconductivity (9).

## 2. Hall effect and cyclotron resonance

The Hall-effect was discovered by Edwin Hall who was still a student in 1879 and can be explained with the Lorentz force acting on an electric charge $-e$ moving with velocity $\dot{r}$ in presence of $E$ and $B$ fields. Generalizing Drude approach, we write the equation of motion in $v$ as:

$$ m_e \dot{v} + \frac{m_e v}{\tau} = -e(E + v \times B) \qquad (9) $$

This is a linear system whose solution can be found from superposition of the transient response given by:

$$ m_e \dot{v}_T + \frac{m_e v_T}{\tau} = 0, \quad v_T = v_0 \exp(-t/\tau) \qquad (10) $$

and the stationary response given by:

$$ m_e \dot{v}_S = -e(E + v_S \times B) \qquad (11) $$

Using $v_S \sim \exp(i\omega t)$, $E = E_x x$, $B = B_z z$, we get the $x, y, z$ components as:

$$ m_e i \omega v_{Sx} = -e(E_x + v_{Sy} B_z), \quad m_e i \omega v_{Sy} = e v_{Sx} B_z, \quad v_{Sz} = 0 \qquad (12) $$

Solving the system we get: $v_{Sx} = \frac{eE_x im_e \omega}{m_e^2 \omega^2 - e^2 B_z^2}$.

The solution reveals existence of a resonance (cyclotron) given by $\omega_c = \frac{eB_z}{m_e}$.

Considering equilibrium when $v_{Sy} = 0$ we get appearance of electric fields:

$$ E_x = v_{Sx} B_z, \quad E_y = -\frac{e \tau E_x}{m_e} B_z = -\omega_c \tau E_x \qquad (13) $$
<figure><img src="image_2.png" /><figcaption>FIG. 1 Hall effect with magnetic induction $B = B_z z$ acting on electronic current $J = Jx$ resulting in transverse accumulation of charge due to Lorentz force leading to a transverse voltage along $y$ direction.</figcaption></figure>

The appearance of a transverse voltage originating from charge accumulation perpendicular to the original current direction along $x$ direction $J_x = \frac{n_e e^2 \tau}{m_e} E_x$ is the Hall effect (cf. fig. 1).

This leads to a Hall resistance given by $R_H = \frac{E_y}{J_x B_z} = -\frac{1}{en_e}$ indicating sign of carriers and being inversely proportional to their density.

## B. Quantum description

We start with 3D free electrons with mass $m_e$ in Vacuum with Hamiltonian:

$$ \hat{H} = -\frac{\hbar^2}{2m_e} \Delta \qquad (14) $$

The eigenfunctions are plane waves indexed by a continuous momentum $k$, $\psi_k(r) = \frac{1}{(\sqrt{2\pi})^3} \exp(ik \cdot r)$. They are orthonormal in the sense:

$$ \int dr \psi_k^*(r) \psi_{k'}(r) = \delta(k - k') \qquad (15) $$

The eigenvalues are given by:

$$ -\frac{\hbar^2}{2m_e} \Delta \psi_k(r) = \epsilon(k) \psi_k(r), \quad \epsilon(k) = \frac{\hbar^2 k^2}{2m_e} \qquad (16) $$

where $m_e$ is the free electron mass. Thus we have a parabolic energy dispersion relation (cf. Table. I) $\epsilon(k) = \frac{\hbar^2 k^2}{2m_e}$ (cf. fig. 2).

## III. STATISTICAL PROPERTIES OF A FERMION GAS

A gas is an ensemble of non-interacting particles and when we are aware of the fact that in condensed matter electrons are very close since atoms are distance by distances around several Angstroms, this might appear as extremely surprising.

Inside matter there are $N_A$ (Avogadro number) atoms, molecules and electrons per mole, thus it is inappropriate to talk about a gas behavior, however it was discovered that this hypothesis is true and later it was understood with the notion of screening and subsequently by Landau Fermi-liquid theory that explains the notion of a free single electron and how many electrons may interact together.
## A. Fermi-Dirac distribution

Particles and quasi-particles are characterized by their Mass, Charge, Spin, Polarizations, Statistical properties and dispersion relations.

They can be statistically distributed along Fermi-Dirac (8) depending when their spin is a half-integer.

In the Fermi-Dirac case, the occupation distribution is given by:

$$f(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \qquad (17)$$

with $\beta = 1/k_B T$ and $\mu$ the chemical potential that allows to determine the number of particles in the system. The chemical potential is a Lagrange multiplier corresponding to the conservation of particles when one determines the statistical distribution of a given system of particles (1). $\mu$ is positive, negative or zero whereas in the Boson case $\mu$ is negative and zero at condensation (4) temperature.

At zero temperature and according to Pauli principle, all Fermion states are filled while respecting the exclusion principle up to an energy $\epsilon_F$ called Fermi energy which is in fact the chemical potential at zero temperature: $\epsilon_F = \mu(T = 0)$.

## B. Dispersion relations

Metals contain free electrons and the first approach was to consider a metal as a container of free electrons bouncing off walls that are the metal sides with length $L$ located at $x = 0, L, y = 0, L, z = 0, L$.

This means we have a Hamiltonian:

$$\hat{H} = -\frac{\hbar^2}{2m_e}\Delta \qquad (18)$$

From a Quantum mechanical point of view we have Stationary Boundary Conditions (SBC) on metal walls:

$$\psi(x, y, z) = 0, \quad x = 0, L, \quad y = 0, L, \quad z = 0, L \qquad (19)$$

This means the electrons are subjected to a zero potential inside the containing metal and infinite on the container walls, i.e. an infinite 3D potential well.

The Schrödinger equation inside the metal as:

$$-\frac{\hbar^2}{2m_e}\Delta\psi(\mathbf{r}) = \epsilon(\mathbf{k})\psi_{\mathbf{k}}(\mathbf{r}) \qquad (20)$$

whose solutions are:

$$\psi_{\mathbf{k}}(\mathbf{r}) = A \sin\left(\frac{\pi n_x x}{L}\right) \sin\left(\frac{\pi n_y y}{L}\right) \sin\left(\frac{\pi n_z z}{L}\right) \qquad (21)$$

satisfy SBC conditions with $k_x = n_x\pi/L, k_y = n_y\pi/L, k_z = n_z\pi/L$ where $n_x, n_y, n_z \in \mathbb{Z}$.

In the case of a crystal we have to reconcile the notion of an limited structure made from a finite cell replicating itself ad infinitum. Born and Von Karman (BVK) introduced special boundary conditions to account for both these constraints.

In the BVK case, we differ from the SBC and write:
$$ \psi(x, y, z) = \psi(x + n_x L, y + n_y L, z + n_z L), \quad n_x, n_y, n_z \in \mathbb{Z} \qquad (22) $$

These periodic boundary conditions have several consequences:

* Quantization of momentum: Applying BVK yields $\psi_k(\mathbf{r}) \sim \exp(ik \cdot \mathbf{r})$ with $k_x, k_y, k_z = 0, \pm 2\pi/L, \pm 4\pi/L...$ contrasting with SBC case: $k_x, k_y, k_z = 0, \pm \pi/L, \pm 2\pi/L, \pm 3\pi/L, ...$

* Absence of walls meaning that we have an infinite domain, instead of the finite $L \times L \times L$ domain

* Existence of a unit cell and a reciprocal unit cell called the Brillouin Zone (BZ) leading to folding of the parabolic free electron bands $\epsilon(\mathbf{k}) = \frac{\hbar^2 k^2}{2m_e}$ back to the BZ as represented in fig. 2.

<figure><img src="image_3.png" /><figcaption>FIG. 2 (Color on line) Folding in of the free electron bands (full parabola) to the Brillouin zone [-π/a, π/a] as done with the blue lines above. a = L/N is the lattice parameter with L the length of the crystal and N the number of unit cells.</figcaption></figure>

* Existence of bands imply the existence of energy gaps, equivalent to absence of propagation for given sets of energy (filtering effect) and the existence of valence bands populated by absence of electrons (holes) and conduction bands populated by electrons.

* Existence of a current, since the wave-function is an imaginary plane wave $\psi_k(\mathbf{r}) \sim \exp(ik \cdot \mathbf{r})$ plane-wave function whereas in the SBC the wave-function $\psi_k(\mathbf{r}) = A \sin(\frac{\pi n_x x}{L}) \sin(\frac{\pi n_y y}{L}) \sin(\frac{\pi n_z z}{L})$ is real carrying zero current.

### C. Density of states

In a system with discrete energy eigenvalues $E_n$, the density of states is given by:

$$ D(E) = \sum_{n} \delta(E - E_n) \qquad (23) $$

In a continuous system the density of states $g(\epsilon)$ is defined such that the number of k states in the interval $[\epsilon, \epsilon+d\epsilon]$ is such that:

$$ g_{\sigma}(\epsilon)d\epsilon = \sum_{\substack{n,k \\ \epsilon \le \epsilon_{n,\sigma}(k) \le \epsilon + d\epsilon}} \qquad (24) $$

where $\epsilon_{n,\sigma}(k)$ is the dispersion relation of an electron band indexed with n, σ where n ∈ N and σ is spin component.
The density of states (DOS) is obtained from $\epsilon_{n,\sigma}(\mathbf{k})$ with formula (9):

$$D_{\sigma}(E) = \frac{V}{(2\pi)^d} \oint_{\epsilon_{n,\sigma}(\mathbf{k})=E} \frac{dS_E}{|\nabla_{\mathbf{k}}\epsilon_{n,\sigma}(\mathbf{k})|} \quad (25)$$

with $\nabla_{\mathbf{k}} = \frac{\partial}{\partial \mathbf{k}}$ and $V = L^3$ is a 3D normalization volume as in the plane wave case: $\frac{1}{\sqrt{V}} \exp(i\mathbf{k} \cdot \mathbf{r})$. The integration element $dS_E$ indicate the quadrature is performed over constant energy surfaces $\epsilon_{n,\sigma}(\mathbf{k}) = E$ for which the group velocity modulus is given by: $|v_g(E(n, \sigma))| = |\nabla_{\mathbf{k}}\epsilon_{n,\sigma}(\mathbf{k})|$.

Writing the density for a single spinless band as:

$$g(\epsilon) = 2 \left(\frac{L}{2\pi}\right)^d \int_{\epsilon(\mathbf{k})=E} \frac{dS_E}{|v_g(E)|} \quad (26)$$

where the factor 2 is spin degeneracy factor $2S + 1$ with $S = 1/2$.

The counting of states is based on the fact allowed $k$ values are distant by $2\pi/L$ along any direction, thus it suffices to consider a $k$ interval corresponding to an energy interval $dE$ and divide by $2\pi/L$ for every direction.

As an illustration, we get the following results in 1D,2D,3D and general dimension $d$:

* 1D: $g(\epsilon)d\epsilon = 2(\frac{L}{2\pi})2|dk| = 2(\frac{L}{2\pi})\frac{m}{\hbar}\frac{1}{2m\sqrt{\epsilon}}$, since we have an energy interval corresponding to $2|dk|$.

* 2D: $g(\epsilon)d\epsilon = 2(\frac{L}{2\pi})^22\pi k dk = \frac{L^2}{\pi}\frac{m}{\hbar^2}$, since we have an energy interval corresponding to a circular ring of surface $2\pi k dk$.

* 3D: $g(\epsilon)d\epsilon = 2(\frac{L}{2\pi})^34\pi k^2 dk = \frac{V}{2\pi^2}(\frac{2m}{\hbar^2})^{3/2}\epsilon^{1/2}$, since we have an energy interval corresponding to a spherical shell of volume $4\pi k^2 dk$.

* $d$ dimensions: We use the fact $g(\epsilon)d\epsilon \sim k^{d-1}dk$ yielding $g(\epsilon) \sim k^{d-1}|dk/d\epsilon|$. The free electron form $\epsilon \sim k^2$ finally results into: $g(\epsilon) \sim \epsilon^{\frac{d-2}{2}}$.

Note that for generality, we consider the electron to have a mass $m$ that could be either $m_e$ in Vacuum or another value in the quasi-particle case (inside a solid) as described below.

In a 3D free electron gas we can estimate, from the dispersion relation $\epsilon(\mathbf{k}) = \frac{\hbar^2 k^2}{2m_e}$ that $\epsilon_F = \frac{\hbar^2 k_F^2}{2m_e}$ where $k_F$ is the Fermi wave-vector. This means that at $T = 0$K, all electronic states lie within a sphere of radius $k_F$, thus we have:

$$2 \left(\frac{4\pi}{3}\right) k_F^3 \left(\frac{L}{2\pi}\right)^3 = N_e \quad (27)$$

yielding $k_F = (3\pi^2 N_e / V)^{1/3}$ where the volume $V = L^3$ (cf. Appendix IV).

Thus $k_F \sim \pi/a$ where $a$ is the lattice parameter and $\epsilon_F$ on the order of electron-volt energies.

## D. The effective mass

In the free electron dispersion relation $\epsilon(\mathbf{k}) = \frac{\hbar^2 k^2}{2m_e}$ it is possible to define the mass $m_e$ from the energy $\epsilon(\mathbf{k})$ by taking the second derivative with respect to momentum $k$ such that: $\frac{1}{m_e} = \frac{1}{\hbar^2} \frac{\partial^2 \epsilon(\mathbf{k})}{\partial \mathbf{k}^2}$.

Generalizing this approach to anisotropic metallic, semi-metallic, semi-conducting and insulating crystals, we define an anisotropic effective mass depending on band indexed by $\lambda$ as:

$$\frac{1}{m_{ij}^*(\lambda, \mathbf{k})} = \frac{1}{\hbar^2} \frac{\partial^2 \epsilon_{\lambda}(\mathbf{k})}{\partial k_i \partial k_j} \quad (28)$$

$\epsilon_{\lambda}(\mathbf{k})$ is a band specified with $\lambda = n, \sigma$, indicating band index and spin respectively.

This generalization of the notion of Newtonian mass to account for anisotropy would be extremely rewarding since $m_{ij}^*$ is not a scalar but a tensor depending on spatial coordinates, moreover it could be positive, negative and even zero (at Brillouin zone borders or in-band) leading to a potentially novel gravity theory endowed with gravitational screening akin to electromagnetic screening due to the existence of positive and negative mass (10).
E. Thermodynamic properties of the Fermi gas

As mentioned previously and for generality, we consider the electron to have a mass *m* that could be either *m*<sub>e</sub> as in Vacuum or an effective value *m*<sup>*</sup> as inside a solid.

Thermodynamic properties are based on integrals (11; 12) containing the Fermi distribution as for instance when we evaluate the number of electrons N<sub>e</sub>

$$
N_e = \sum_k \frac{1}{e^{\beta(\epsilon(\mathbf{k}) - \mu)} + 1} \qquad (29)
$$

where $\beta = 1/k_B T$, with $k_B$ the Boltzmann constant and $T$ the temperature.
Using the 3D density of states $g(\epsilon) = \frac{V}{2\pi^2} (\frac{2m}{\hbar^2})^{3/2} \epsilon^{1/2}$, we get:

$$
N_e = \frac{V}{\Gamma(3/2)} \left(\frac{m}{2\pi\hbar^2}\right)^{3/2} \int_0^\infty \frac{\epsilon^{1/2} d\epsilon}{e^{\beta(\epsilon-\mu)} + 1}, \quad (30)
$$

Thus we obtain integrals of the form:

$$
I = \int_{0}^{+\infty} \frac{G(\epsilon)d\epsilon}{e^{(\epsilon-\mu)/k_B T} + 1} \quad (31)
$$

with G(ϵ) an arbitrary function of energy ϵ.

Using change of variable $z = (\epsilon - \mu)/k_B T$ into $I$, we get:

$$
I = k_B T \int_{-\mu/k_B T}^{+\infty} \frac{G(\mu + k_B T z) dz}{e^z + 1} = k_B T \int_{-\mu/k_B T}^{0} \frac{G(\mu + k_B T z) dz}{e^z + 1} + k_B T \int_{0}^{+\infty} \frac{G(\mu + k_B T z) dz}{e^z + 1} \quad (32)
$$

Performing $z \to -z$ in the first integral and using the identity:

$$
\frac{1}{e^{-z} + 1} = 1 - \frac{1}{e^z + 1} \qquad (33)
$$

we get:

$$
I = \int_{0}^{\mu} G(z) dz - k_{B}T \int_{0}^{\mu/k_{B}T} \frac{G(\mu - k_{B}Tz) dz}{e^{z} + 1} + k_{B}T \int_{0}^{+\infty} \frac{G(\mu + k_{B}Tz) k_{B}T dz}{e^{z} + 1} \quad (34)
$$

Using the approximation $\mu/k_B T \approx \infty$, (since the Fermi energy is very large as explained later) we get:

$$
I = \int_{0}^{\mu} G(z) dz + k_{B}T \int_{0}^{\infty} \frac{[G(\mu + k_{B}Tz) - G(\mu - k_{B}Tz)] dz}{e^{z} + 1} \quad (35)
$$

Expanding G(z) around μ:

$$
G(\mu + x) - G(\mu - x) \approx G(\mu) + xG'(\mu) + \frac{x^2}{2!} G''(\mu) \dots \quad (36)
$$

where $G'$, $G''$, ... are the energy derivatives of $G$. Thus we get:

$$
I \approx \int_{0}^{\mu} G(z) dz + 2(k_{B}T)^{2} G'(\mu) \int_{0}^{\infty} \frac{z dz}{e^{z} + 1} \dots \quad (37)
$$
Using the identity $\int_0^\infty \frac{zdz}{e^z+1} = \frac{\pi^2}{12}$, we finally get the Sommerfeld expansion:

$$I \approx \int_0^\mu G(z) dz + \frac{\pi^2}{6} (k_B T)^2 G'(\mu) \dots \qquad (38)$$

Fermi integrals and Sommerfeld expansion allow us to evaluate the chemical potential $\mu(T)$, the energy of the Fermi gas $U$, the heat capacity at fixed volume $C_v$ and the thermal conductivity $\kappa_T$.

* Equation of state $\mu = \mu(N_e, V, T)$ for an ideal Fermi gas:

Let us introduce NIST (13) Fermi integrals $F_s(z)$ defined as:

$$F_s(x) = \frac{1}{\Gamma(s+1)} \int_0^\infty dt \frac{t^s}{e^{t-x}+1}, \quad s > -1 \qquad (39)$$

And the Polylogarithm function (13) $Li_s(z)$ as:

$$Li_s(z) = \frac{z}{\Gamma(s)} \int_0^\infty dx \frac{x^{s-1}}{e^x - z}. \qquad (40)$$

resulting into: $F_s(x) = -Li_{s+1}(-e^x)$.

Using particle density $N_e/V$ definition and $Li_s(z)$ we get:

$$\int_0^\infty \frac{\epsilon^{1/2} d\epsilon}{e^{\beta(\epsilon-\mu)} + 1} = \frac{1}{\beta^{3/2}} \int_0^\infty \frac{x^{1/2} dx}{\zeta e^x + 1} \qquad (41)$$

where $\zeta$ is defined as $\zeta = e^{-\beta\mu}$ yielding the result in a compact way.

For $s = 3/2$ and $\zeta \to \zeta^{-1}$ we have:

$$Li_{3/2}\left(\frac{1}{\zeta}\right) = \frac{1}{\Gamma(3/2)} \int_0^\infty \frac{t^{1/2} dt}{\zeta e^t - 1} \qquad (42)$$

which enable us to rewrite this expression as:

$$\frac{1}{\beta^{3/2}} \int_0^\infty \frac{x^{1/2} dx}{\zeta e^x - 1} = \frac{1}{\beta^{3/2}} Li_{3/2}(e^{\beta\mu}). \qquad (43)$$

Finally, we have:

$$\frac{N_e}{V} = - \left( \frac{mk_B T}{2\pi\hbar^2} \right)^{3/2} Li_{3/2}(-e^{\beta\mu}), \qquad (44)$$

Thus, the equation of state $\mu(N, V, T)$ is obtained with $\mu(N, V, T = 0)$ and Fermi energy $\epsilon_F = k_B T_F = \hbar^2 k_F^2 / 2m$ given by:

$$\epsilon_F = \frac{\hbar^2}{2m} \left( 6\pi^2 \frac{N_e}{V} \right)^{2/3}. \qquad (45)$$

Fermi-Dirac temperature $T_{FD}$ is given by:
$$ \frac{N_e}{V} = - \left( \frac{mk_B T_{FD}}{2\pi\hbar^2} \right)^{3/2} Li_{3/2}(-1) = \left( \frac{mk_B T_{FD}}{2\pi\hbar^2} \right)^{3/2} \qquad (46) $$

This enables us to check the temperature at which the chemical potential of a Fermi gas is zero (condensation forbidden by Pauli exclusion principle) as given by:

$$ T_{FD} = \frac{2\pi\hbar^2}{mk_B} \left(\frac{N_e}{V}\right)^{2/3} \qquad (47) $$

* Chemical potential $\mu(T)$

Using Sommerfeld expansion at zero and finite temperature with the chemical potential equal to $\epsilon_F$ the Fermi energy and $\mu(T)$ respectively at $T = 0$K and finite $T$ allow us to write:

$$ \begin{aligned} N_e(T = 0) &= \int_0^{\epsilon_F} d\epsilon g(\epsilon) \\ N_e(T) &= \int_0^{\mu} d\epsilon g(\epsilon) + \frac{\pi^2}{6} (k_B T)^2 g'(\mu) \end{aligned} \qquad (48) $$

Since both numbers $N_e(T = 0) = N_e(T)$ are equal by conservation of particle number, we have:

$$ \int_{\mu}^{\epsilon_F} d\epsilon g(\epsilon) = \frac{\pi^2}{6} (k_B T)^2 g'(\mu) \qquad (49) $$

Approximating $g'(\mu) \approx g'(\epsilon_F)$ and $\int_{\mu}^{\epsilon_F} d\epsilon g(\epsilon) \approx (\epsilon_F - \mu)g(\epsilon_F)$ results into:

$$ \mu(T) = \epsilon_F - \frac{\pi^2}{6} (k_B T)^2 \frac{g'(\epsilon_F)}{g(\epsilon_F)} \qquad (50) $$

* Total energy $U$

Proceeding exactly as in the Chemical potential determination, we use Sommerfeld expansion at zero and finite temperature for the energy to obtain:

$$ U(T) = \sum_{k} \frac{\epsilon(\mathbf{k})}{e^{\beta(\epsilon(\mathbf{k}) - \mu)} + 1} \qquad (51) $$

$\beta = 1/k_B T$, with $k_B$ the Boltzmann constant and $T$ the temperature.

We rewrite it explicitly as:

$$ U(T) = \int_{0}^{+\infty} \frac{\epsilon g(\epsilon) d\epsilon}{e^{(\epsilon - \mu)/k_B T} + 1} \qquad (52) $$

At $T = 0$K we have:

$$ U(0) = \int_{0}^{\epsilon_F} \epsilon g(\epsilon) d\epsilon \qquad (53) $$
Using the approximations: For $\epsilon = \mu$ the derivative $[\epsilon g(\epsilon)]' = \mu g'(\mu) + g(\mu) \approx \epsilon_F g'(\epsilon_F) + g(\epsilon_F)$ finally resulting into:

$$U(T) = U(0) + \frac{\pi^2}{6} (k_B T)^2 g(\epsilon_F) \qquad (54)$$

* Heat capacity $C_v$

$C_v = (\frac{\partial U}{\partial T})_v$ (8) is obtained immediately from eq. 54 as: $C_v = \frac{\pi^2}{3}(k_B^2 T)g(\epsilon_F)$.

Traditionally (9) this is written as $C_v = \gamma T$ with $\gamma = \frac{\pi^2}{3}k_B^2 g(\epsilon_F)$ the Grüneisen constant, a law that is valid in any dimension.

In this context, a question arises concerning the classical behavior of an electron gas. That means if we have $N_e$ electrons, there must be a classical crossover temperature $T_c$ such that $\gamma T_c = \frac{3}{2}k_B N_e$. Using $\gamma = \frac{\pi^2}{3}k_B^2 g(\epsilon_F)$ and the fact: $g(\epsilon_F)\epsilon_F = \frac{\pi^2}{3}N_e$, we get:

$$T_c = \frac{3}{\pi^2} \frac{\epsilon_F}{k_B} \qquad (55)$$

This is a very high temperature on the order of tens of thousands Kelvins. Thus the electron gas will be always likely to be observed as quantum explaining the difficulty of tackling the description of $C_v$ before the advent of Quantum Mechanics.

Considering a typical solid, we assume that other particles, quasi-particles, elementary excitations contribute to total energy and by extension to $C_v$.

For simplicity, if we consider lattice vibrations (phonons are the elementary excitations) $C_v$ contribution as $T^d$ in $d$ dimensions (9), then we have at low temperature:

$$C_v = aT + bT^d \qquad (56)$$

where $a, b$ are appropriate electronic and vibrational prefactors extractable from Table. II.

For example in a 3D metal we have at low temperature: $C_v = \gamma T + bT^3$ resulting in the experimental plot $C_v$ versus $T^2$ with $C_v/T = \gamma + bT^2$. The line $C_v$ versus $T^2$ provides the Grüneisen constant $\gamma$ as the $T = 0$K extrapolated intercept at $T \to 0$K whereas the line slope $b$ gives the Debye phonon coefficient as in fig. 3. Example room temperature Grüneisen constants for Cu and Al are respectively 0.695 and 1.35 mJ/mole.K$^2$

It is important to note that the electronic $C_v(T)$ allows to reveal the underlying mass of the excitations. Thus $\gamma = \frac{\pi^2}{3}k_B^2 g(\epsilon_F)$ and $g(\epsilon_F) \propto \frac{1}{\epsilon_F}$ we infer that $\gamma \propto m$ where $m = m_e, m^*....$

Consequently we have the ratio $\frac{\gamma_{th}}{\gamma_e} = \frac{m_{th}}{m_e}$, where $\gamma_{th}$ is the thermal Grüneisen coefficient measured with $C_v$ yielding a thermal mass $m_{th}$ whereas the free electron $\gamma_e$ corresponds to $m_e$.

Experimentally, materials such as UBe$_{13}$, CeAl$_3$, CeCuSi$_2$ have $\gamma_{th} \gg \gamma_e$ implying $m_{th} \gg m_e$. These materials are called Heavy Fermion materials since the mass ratio $\frac{m_{th}}{m_e} \sim 1000$ (cf subsection V.F).

In order to faithfully describe the $T$ dependence of any metallic solid, one has to account for electronic $C_v$ as well as vibrational $C_v$ deciphering fig. 4 (some metallic solids have additionally other excitations that should be accounted for with their appropriate $C_v$).
<figure><img src="image_4.png" /><figcaption>FIG. 3 Experimental graph of Potassium $C_v/T$ versus $T^2$ in order to determine a, b coefficient in $C_v(T)/T = a + bT^2$ agreeing with Kittel (9).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4 Temperature dependence of vibrational $C_v$ for several solids: Metallic Ag, Al, Semi-metallic Graphite, Insulator $Al_2O_3$ and Ionic insulator KCl. $C_v$ is normalized by $R/2$ (half the gas constant) whereas temperature is normalized by Debye temperature $\Theta_D$. At low temperature $T$, $C_v$ follows Debye power law in $T^3$ whereas at high temperature, it saturates to $3R$ after a break at a temperature close to $\Theta_D$. Adapted from F. Seitz, *The Modern Theory of Solids*, McGraw-Hill, New York (1940).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5 Temperature dependence of measured $C_v$ and $C_p$ for Copper (14). Note the high temperature behavior above the $3R$ classical limit. Recall that $C_p - C_v = 9\alpha_L^2VT/B$ where $\alpha_L$ is Copper linear expansion coefficient and $B$ is the Bulk modulus.</figcaption></figure>
<table><caption>TABLE II Electronic specific heat coefficient, Debye temperature and Z the number of electrons per atom in several metals. The total $C_v = \gamma_{exp}T + bT^3$ with Wiedemann-Franz law giving the electronic $C_v \propto T$ with coefficient $\gamma_{exp}$. The vibration $C_v = bT^3$ with $b = \frac{12\pi^4 N k_B}{56 D^3}$ (9). As an example, Potassium b coefficient can be evaluated as $b=2.58$ mJ/mol.K$^2$ with $N = N_A$ agreeing with Kittel (9) value.</caption><thead><tr><th>Element</th><th>Z</th><th>γ<sub>exp</sub> (mJ/mol.K<sup>2</sup>)</th><th>Θ<sub>D</sub> (K)</th></tr></thead><tbody><tr><td>Na</td><td>1</td><td>1.38</td><td>157</td></tr><tr><td>K</td><td>1</td><td>2.08</td><td>91</td></tr><tr><td>Cu</td><td>1</td><td>0.690</td><td>342</td></tr><tr><td>Ag</td><td>1</td><td>0.641</td><td>223</td></tr><tr><td>Au</td><td>1</td><td>0.725</td><td>163</td></tr><tr><td>Mg</td><td>2</td><td>1.30</td><td>396</td></tr><tr><td>Zn</td><td>2</td><td>0.638</td><td>319</td></tr><tr><td>Cd</td><td>2</td><td>0.688</td><td>209</td></tr><tr><td>Al</td><td>3</td><td>1.348</td><td>428</td></tr><tr><td>Pb</td><td>4</td><td>3.04</td><td>106</td></tr></tbody></table>

In fig. 4 vibrational $C_v$ for several solids: Metallic Ag, Al, Semi-metallic Graphite, Insulator Al$_2$O$_3$ and Ionic insulator KCl display a high temperature vibrational $C_v = 3R$.

The case of copper displayed in fig. 5 differs from our previous example displayed in fig. 4.

To summarize, in order to understand the behavior of $C_v(T)$ one should write for a simple solid with weakly interacting excitations:

$$C_v(T) = C_v(T)[\text{vibration}] + C_v(T)[\text{electronic}] + \sum_i C_v(T)[\text{excitation}]_i + \dots \quad (57)$$

For example, in the case of a non-simple solid such as a superconductor containing many-body interactions, there is a sizable coupling between vibrational and electronic excitations and the above expression is not valid.

### Thermal conductivity:

Thermal conductivity is given by (8): $\kappa_T = \frac{1}{3}C_v\bar{v}\ell$ where $C_v$ is the electronic heat capacity, $\bar{v}$ is an average electron velocity and $\ell$ its mean free path.

Considering that $\bar{v} \sim v_F$ it remains to describe the mean free path $\ell$ as a function of temperature. In general we expect $\ell$ to saturate at low $T$ and decrease at higher $T$ as $1/T^\alpha$ where $\alpha$ is an exponent depending on the metal and scattering processes followed by a saturation behavior. In fig. 6 we note an exponent $\alpha = 2$ as observed in many 3D metals.

For insulators we have $\kappa_T \sim T^3$ a 3D Debye behavior stemming from $C_v$ followed by a $1/T$ behavior originating from $\ell(T)$.

### Lorentz number and Wiedemann-Franz law:

For a metal one can estimate the mean free path $\ell(T)$ as $v_F\tau_R(T)$ where $\tau_R(T)$ is a relaxation time that saturates at low $T$ and decreases at higher $T$ as $1/T^\alpha$ $\alpha$ being an exponent depending on the metal and scattering processes.

Wiedemann-Franz law assumes a linear in $T$ behavior of $\kappa_T(T)$ as displayed in fig. 6.

Lorentz number $\mathcal{L}$ is given by the electric to thermal conductivity ratio at low temperatures: $\kappa/\sigma = \mathcal{L}T$. It is straightforward to derive this ratio from the expression $\kappa_T = \frac{1}{3}C_v v_F \ell$ and Drude conductivity $\sigma_D = n_e e^2 \ell / (m_e v_F)$ (cf. Table. IV).
<figure><img src="image_7.png" /><figcaption>FIG. 6 (Left) $\kappa_T$ for a 3D metal: At low $T$ we have the expected $T$ behavior originating from $C_v$. At high temperature we have a $1/T^2$ followed by saturation originating from $\ell(T)$ variation. (Right) $\kappa_T$ for a 3D insulator: At low $T$ we have the expected Debye $T^3$ behavior originating from $C_v$. At high temperature we have a $1/T$ originating from $\ell(T)$ variation. In general the electronic $\kappa_T$ is much larger than the phonon $\kappa_T$ with the exception of Diamond.</figcaption></figure>

Assuming both mean free path as same, and $C_v = \gamma T$ with $\gamma = \gamma_e$ the free electron value of Grüneisen constant, we get the value of the ratio as a universal number:

$$ \mathcal{L} = \frac{\pi^2 k_B^2}{3 e^2} \qquad (58) $$

Thus numerically we have: $\mathcal{L} = 2.45 \times 10^{-8}$ W. $\Omega/K$.

## IV. BAND THEORY FOR A SINGLE ELECTRON IN A PERIODIC POTENTIAL

The Kronig-Penney potential (15) is often used in the description of the electronic properties of crystals. It is based on a piecewise constant potential (see fig. 7). The eigenvalues are dispersive which means they depend on a wavevector reflecting the translational symmetry of the system.

In a perfect crystal the electrons are subjected to a periodic potential $\hat{V}_P$ and their physics is described by a single electron Schrödinger equation with Hamiltonian:

$$ \hat{H} = -\frac{\hbar^2}{2m_e} \Delta + \hat{V}_P(\mathbf{r}) \qquad (59) $$

Note that in this case we have $\hat{V}_P(\mathbf{r}) = 0$ in every crystal cell and that is different from a free electron in Vacuum where $\hat{V}(\mathbf{r}) = 0$ is non-periodic and zero everywhere.

In Vacuum there is no finite translational symmetry but continuous transformation symmetries. When an electron is in a crystal with a finite translational symmetry i.e.

$$ \hat{V}_P(\mathbf{r}) = \hat{V}_P(\mathbf{r} + \mathbf{T}), \quad \mathbf{T} = n_1\mathbf{a} + n_2\mathbf{b} + n_3\mathbf{c}, \quad n_1, n_2, n_3 \in \mathcal{Z} \qquad (60) $$

where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are lattice basis vectors.

Thus the Schrödinger equation solutions must respect Floquet theorem (16) imposing periodic boundary conditions on Partial differential equations containing periodic components and not ordinary boundary conditions as in the Vacuum case.

Introducing the finite translation operator (17–19) $\hat{T}_R = e^{i\mathbf{R} \cdot \mathbf{k}}$ we have: $\hat{T}_R\psi(\mathbf{r}) = \psi(\mathbf{r} + \mathbf{R})$

This leads to the commutation relations: $[\hat{T}_R, \Delta] = [\hat{T}_R, \hat{V}_P] = [\hat{T}_R, \hat{H}] = 0$
this implies the existence of a CSCO (Complete System of Commuting Observables):

$$ \hat{T}_R \psi_\lambda(\mathbf{r}) = \lambda \psi_\lambda(\mathbf{r}), \quad \hat{H} \psi_\lambda(\mathbf{r}) = E_\lambda \psi_\lambda(\mathbf{r}), \qquad (61) $$

where $\lambda$ indexes the common eigenfunctions for the $\hat{T}_R$, $\hat{H}$ operators.

This implies $\psi_\lambda(\mathbf{r} + \mathbf{r}') = \lambda \psi_\lambda(\mathbf{r})$ and from periodicity we infer that $|\psi_\lambda(\mathbf{r} + \mathbf{R})| = |\psi_\lambda(\mathbf{r})|$ leading to $|\lambda| = 1$.

Picking $\lambda = e^{i\alpha(\mathbf{R})}$ and performing two successive translations:

$$ \psi_{\lambda}(\mathbf{r} + \mathbf{R} + \mathbf{R}') = e^{i\alpha(\mathbf{R}+\mathbf{R}')} \psi_{\lambda}(\mathbf{r}) = e^{i\alpha(\mathbf{R})} e^{i\alpha(\mathbf{R}')} \psi_{\lambda}(\mathbf{r}) \qquad (62) $$

we deduce that $\alpha$ must be linear in $\mathbf{R}$ and we pick $\mathbf{k} \equiv \lambda$ to label eigenfunctions and eigenvalues.

Thus we infer Floquet (9; 16) equivalence with Bloch theorem yielding the solution wave function as:

$$ \psi_k(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_k(\mathbf{r}), \quad u_k(\mathbf{r} + \mathbf{R}) = u_k(\mathbf{r}) \qquad (63) $$

## A. Derivation of the exact dispersion relation

We consider an electron in the periodic piecewise constant potential $V(x)$ displayed in fig. 7.

<figure><img src="image_8.png" /><figcaption>FIG. 7 (Color on line) Ion core potentials (at left) and their simplified representation (at right) as periodic piecewise constant potential $V(x)$ displaying alternating regions of $V = 0$ and $V = V_0$ with periodicity $a_0 + b_0$. The energy bands are obtained for $E < V_0$. In the case we let $V_0 \to \infty$ and $b \to 0$ the barriers become delta functions sitting on a periodic lattice with parameter $a = a_0 + b_0$.</figcaption></figure>

Let $E > 0$ be an energy value as shown in fig. 7.

In region I, $-b_0 < x < 0$:

$$ -\frac{\hbar^2}{2m} \frac{\partial^2 \psi_I(x)}{\partial x^2} + V_0 \psi_I(x) = E \psi_I(x) $$

The solution of Schrödinger equation is: $\psi_I(x) = Ce^{Qx} + De^{-Qx}$, $Q^2 = \frac{2m(V_0-E)}{\hbar^2}$.

In region II: $0 < x < a_0$:

$$ -\frac{\hbar^2}{2m} \frac{\partial^2 \psi_{II}(x)}{\partial x^2} = E \psi_{II}(x) $$

The solution of Schrödinger equation is $\psi_{II}(x) = Ae^{i\kappa x} + Be^{-i\kappa x}$, $\kappa^2 = \frac{2mE}{\hbar^2}$.

In region III: $a_0 < x < (a_0 + b_0)$:

We use Bloch Theorem: $\psi_{III}(x) = \exp(ik[a_0 + b_0])\psi_I(x)$.

The boundary conditions at the boundaries 0 and a are:

* $\psi_I(0) = \psi_{II}(0)$ yielding $A + B = C + D$

* $\psi'_I(0) = \psi'_{II}(0)$ yielding $i\kappa(A - B) = Q(C - D)$

* $\psi_{II}(a_0) = \psi'_{III}(a_0)$ yields $Ae^{i\kappa a_0} + Be^{-i\kappa a_0} = (Ce^{-Qb_0} + De^{Qb_0}) \exp(ik[a_0 + b_0])$
$$ \bullet \psi'_{II}(a_0) = \psi'_{III}(a_0) \text{ yields } i\kappa(Ae^{i\kappa a_0} - Be^{-i\kappa a_0}) = -Q(Ce^{-Qb_0} - De^{Qb}) \exp(ik[a_0 + b_0]) $$

These conditions yield the following linear system:

$$ \begin{pmatrix} 1 & 1 & -1 & -1 \\ \kappa & -\kappa & -Q & Q \\ e^{i(\kappa-k)(a_0-b_0)} & e^{-i(\kappa+k)(a_0-b_0)} & -e^{-i(Q-k)b_0} & -e^{i(Q+k)b_0} \\ (\kappa-k)e^{i(\kappa-k)(a_0-b_0)} & -(\kappa+k)e^{-i(\alpha+k)(a_0-b_0)} & -(Q-k)e^{-i(Q-k)b_0} & (Q+k)e^{i(Q+k)b_0} \end{pmatrix} \begin{pmatrix} A \\ B \\ C \\ D \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \quad (64) $$

In order to have non-zero solutions for A, B, C, D, the determinant should be zero yielding the exact dispersion relation (9; 15) is given by:

$$ \frac{Q^2 - \kappa^2}{2Q\kappa} \sinh(Qb_0) \sin(\kappa a_0) + \cosh(Qb_0) \cos \kappa a_0 = \cos k(a_0 + b_0) \quad (65) $$

For simplicity, the dispersion relation (9) is transformed with the limiting process $V_0 \to \infty$ and $b_0 \to 0$ is (9):

$$ \frac{Q^2 b_0}{2\kappa} \sin(\kappa a_0) + \cos \kappa a_0 = \cos ka_0 \quad (66) $$

Starting with the wave function $y(x) = A \exp(i\kappa x) + B \exp(-i\kappa x)$, defined over the unit cell $x \in ]0, a[$ where the lattice parameter $a = a_0 + b_0$ and using both aforementioned conditions yields the dispersion relation:

$$ \frac{m_e g}{\kappa \hbar^2} \sin(\kappa a) + \cos \kappa a = \cos ka \quad (67) $$

Comparing both dispersion relations yields finally the value of the strength of the $\delta$ function potential as $g = \frac{Q^2 b_0 \hbar^2}{2m_e}$.

<figure><img src="image_9.png" /><figcaption>FIG. 8 (Color on line) Exact bands (in green) for the Kronig-Penney model obtained from the dispersion relation obtained numerically (20) from eq. 67. (a) is for a single band, (b),(c) and (d) are for 2, 4 and 8 bands respectively. In all cases, the strength of the potential $g = 0.5$ in atomic units. Gaps are region where electron propagation is forbidden.</figcaption></figure>

We may compare the free electron dispersion relation with the obtained bands by the process of folding them back to the BZ as displayed in fig. 2 with energy bands appearing where electron propagation is forbidden.

For all number of band values $n_B=1,2,4$ and 8 displayed in fig. 8. Equation. 67 allows the exact determination of the free wavevector $\kappa$ from a given Bloch wavevector $k$ and the exact band energy is obtained from $E = \frac{\hbar^2 \kappa^2}{2m_e}$.

The presence of gaps leads us to the main one separating occupied states (Valence band) and unoccupied states (Conduction band) akin to the HOMO (Highest Occupied Molecular Orbital) and LUMO (Lowest Unoccupied Molecular Orbital) in Quantum Chemistry. These levels result from Pauli principle ruling that filling of levels must proceed until all fermions are accounted for (8; 9).
The value of the main gap separating occupied (valence band) and unoccupied (conduction band) states allows us to classify metals as having a gap of zero value, a semi-metal with a negative gap, a semiconductor with a gap of a value $\epsilon \in [0.1 - 4]$ eV whereas an insulator has a gap larger $\ge 5$ eV.

Presently, band theory calculations are so advanced, they are able to help engineer new materials, predicting accurately their physical properties before they are actually mass produced for electronic applications. (section V discusses the versatility of band structure calculations).

# V. MANY-ELECTRON SYSTEMS AND BAND STRUCTURE CALCULATIONS

Band structure calculations are essentially based on solving the Schrödinger equation with the Born-Oppenheimer approximation to finally obtain either the wave-function $\Psi(r)$ or the Landau type electron density $\rho(r)$ for the many-electron system.

In an interacting electron ensemble, one has to account for Heisenberg exchange interactions originating from wave-functions overlap as well as Coulomb interactions between electrons. Moreover one has to account for Pauli exclusion principle originating from the spin-1/2 statistics of the electrons. Spin effects and Coulomb interactions combined are called correlations.

In order to treat many-electron systems (21) we start as an illustration of the potential issues with a two-particle system:

## A. Two-particle case

The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen wave functions of the individual particles:

$$ \Psi(\mathbf{x}_1, \mathbf{x}_2) = \psi_1(\mathbf{x}_1)\psi_2(\mathbf{x}_2). \qquad (68) $$

where the coordinates $\mathbf{x}$ mean that individual wave-functions: $\psi(\mathbf{x}) = \phi(\mathbf{r})|\sigma\rangle$ contain both spatial and spin coordinates.

This expression is the Hartree method as an approximation for the many-particle wave function. However, it is not satisfactory for fermions, such as electrons, because the wave function is not antisymmetric. An antisymmetric wave function can be analytically described as:

$$ \Psi(\mathbf{x}_1, \mathbf{x}_2) = -\Psi(\mathbf{x}_2, \mathbf{x}_1). \qquad (69) $$

Therefore the simple (Hartree) product does not satisfy Pauli principle. The remedy to the problem is to consider a linear combination of Hartree products:

$$ \Psi(\mathbf{x}_1, \mathbf{x}_2) = \frac{1}{\sqrt{2}} \{\psi_1(\mathbf{x}_1)\psi_2(\mathbf{x}_2) - \psi_1(\mathbf{x}_2)\psi_2(\mathbf{x}_1)\} = \frac{1}{\sqrt{2}} \begin{vmatrix} \psi_1(\mathbf{x}_1) & \psi_2(\mathbf{x}_1) \\ \psi_1(\mathbf{x}_2) & \psi_2(\mathbf{x}_2) \end{vmatrix} \qquad (70) $$

This normalized wave function is antisymmetric and no longer distinguishes between fermions. Moreover, it also goes to zero if any two wave functions or two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.

If we now take into account explicitly the spin degree of freedom $\psi(\mathbf{x}) = \phi(\mathbf{r})|\sigma\rangle$, we obtain:

$$ \Psi(\mathbf{x}_1, \mathbf{x}_2) = \frac{1}{\sqrt{2}} \{\phi_1(\mathbf{r}_1)|\sigma_1\rangle_1\phi_2(\mathbf{r}_2)|\sigma_2\rangle_2 - \phi_1(\mathbf{r}_2)|\sigma_1\rangle_1\phi_2(\mathbf{r}_1)|\sigma_2\rangle_2\} \qquad (71) $$
## B. General case: the many body wave-function

When treating a many-electron system (21) one has to account for the fact electrons are identical particles (17). This is seen in the Hamiltonian, exchanging indices and does not change the Hamiltonian, meaning the latter commutes with the particle exchange operator. This operator exchanges particle coordinates:

$$P_{ij} \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_i, \dots, \mathbf{x}_j, \dots, \mathbf{x}_N) = \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_j, \dots, \mathbf{x}_i, \dots, \mathbf{x}_N), \quad (72)$$

where the coordinates are both orbital and spin part.

The Hamiltonian commuting with the operator P, then its eigenvectors should simultaneously be eigenvectors of both operators. Since electrons are fermions, their many-body wave-function should be antisymmetric under exchange.

For the case of an independent particle Hamiltonian, which is the sum of one-electron Hamiltonians, we can write the solution to the Schrödinger equation as a product of one-particle wave-functions:

$$\Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N) = \psi_1(\mathbf{x}_1) \psi_2(\mathbf{x}_2) \cdots \psi_N(\mathbf{x}_N). \quad (73)$$

the one-particle states are eigenstates of the one-particle Hamiltonians. The probability density id then given by

$$\rho(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N) = |\psi_1(\mathbf{x}_1)|^2 |\psi_2(\mathbf{x}_2)|^2 \dots |\psi_N(\mathbf{x}_N)|^2, \quad (74)$$

which is just the product of one-particle densities implying that the corresponding probability density is uncorrelated.

Of course, a state with the coordinates permuted is also a solution, as are linear combinations of such states. But the requirement of antisymmetry implies that the wave function should have the form:

$$\Psi_{AS} = \frac{1}{N!} \sum_{\mathbf{P}} \mathrm{sgn}_{\mathbf{P}} \mathbf{P} \psi_1(\mathbf{x}_1) \psi_2(\mathbf{x}_2) \dots \psi_N(\mathbf{x}_N). \quad (75)$$

P is a permutation operator that permutes the coordinates, and not the wave-functions. The sign in the sum can be or according to whether the permutation can be written as a product of even or odd pair interchanges, respectively.

We can write this wave-function as a Slater determinant:

$$\Psi_{AS} = \begin{vmatrix} \psi_1(\mathbf{x}_1) & \psi_2(\mathbf{x}_1) & \cdots & \psi_N(\mathbf{x}_1) \\ \psi_1(\mathbf{x}_2) & \psi_2(\mathbf{x}_2) & \cdots & \psi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \psi_1(\mathbf{x}_N) & \psi_2(\mathbf{x}_N) & \cdots & \psi_N(\mathbf{x}_N) \end{vmatrix},$$

When the wave-function is transformed into its antisymmetric form, the electrons are generally correlated. In order to illustrate this, we write the probability density of finding two electrons with coordinates:

$$\rho(\mathbf{x}_1, \mathbf{x}_2) = \int dx_3 \cdots dx_N |\Psi_{AS}|^2 \quad (76)$$

$$= \frac{1}{N(N-1)} \sum_{k,l} \left[ |\psi_k(\mathbf{x}_1)|^2 |\psi_l(\mathbf{x}_2)|^2 - \psi_k^*(\mathbf{x}_1) \psi_k(\mathbf{x}_2) \psi_l^*(\mathbf{x}_2) \psi_l(\mathbf{x}_1) \right]. \quad (77)$$

The probability is evaluated after summing over spin variables:

$$\rho(\mathbf{r}_1, \mathbf{r}_2) = \sum_{s_1, s_2} \rho(\mathbf{x}_1, \mathbf{x}_2). \quad (78)$$

We infer that when spins are opposite, the second term vanishes yielding uncorrelated spin orbitals. For parallel spins, the two terms cancel when $\mathbf{r}_1 = \mathbf{r}_2$, thus electrons with equal spins cannot occupy the same space coordinates. Every electron is found surrounded by an exchange hole in which other electrons having similar spin are hardly
found. This is in fact Pauli's exclusion principle.

The different treatments of exchange and correlation effects are proper to the various Band structure methods detailed below.

This yields two distinct routes (21):

1. Wave-function $\Psi(r)$ route:

After making the orbital approximation, one may select the LCAO method, Roothaan, Ab-initio SCF, then treatment of correlations with later on MCSCF, CI, CC MPn/MBPT... or select the chemical Hückel method leading to EHT, EH-TB and after approximating the Hamiltonian indulge into semi-empirical methods such as: AM1, PM3, CNDO, INDO, MNDO...

2. Density $\rho(r)$ route:

After accounting of The Hohenberg-Kohn (21) theorems, one moves on to Density Functional Theory which is based on Landau Fermi Liquid Theory. Kohn-Sham work leads to KS-LDA (LSD) methods that might bifurcate into either APW, LAPW, LMTO, X-α, KKR, FPLO, FP-LAPW...

The treatment of exchange-correlations leads to KS NON-LDA (GGA) methods such as BP86, PBE... whereas the treatment of excited states leads to TD-DFT methods...

### C. Augmented Plane Wave method

Augmented Plane Wave method (21) starts by building a muffin-tin potential approximation as displayed in fig. 9. It consists of limiting the inter-core as well as the intra-core potentials by upper and lower bounds (cf. fig. 9). The upper bound allows us to write the wave-functions as plane waves whereas the lower bound gives us a hard core limit to the intra-core potential simplifying tremendously the solution wave-functions.

The wave-function is:

$$ \psi_k(\mathbf{r}) = \begin{cases} e^{i\mathbf{k}\cdot\mathbf{r}} & \text{for } |\mathbf{r}-\mathbf{R}| \ge r_0 \\ \text{Atomic function} & \text{for } |\mathbf{r}-\mathbf{R}| < r_0 \end{cases} \qquad (79) $$

<figure><img src="image_10.png" /><figcaption>FIG. 9 (Color on line) Muffin-tin potential approximation: The inter-core as well as the intra-core potentials are limited by upper and lower bound constants. The upper bound muffin-line approximation allows us to write the wave-functions as plane waves whereas the lower bound gives us a hard core limit to the intra-core potential.</figcaption></figure>

where $r_0$ is the core radius. Outside the core the function is a plane wave because the potential is constant. Inside the core the function is atom-like, and is found by solving the appropriate free-atom Schrödinger equation. Also, the atomic function is chosen such that it joins continuously to the plane wave at the surface of the sphere forming the core; this is the boundary condition.

Augmentation involves replacing the wave-function inside each sphere by some other functions, matching continuously and differentially the angular momentum components at the surface of the sphere.
It is the boundary conditions that determine the value of $\epsilon$ for a given $k$.

Thus APW leads to:

$$ \psi_k(\mathbf{r}) = \begin{cases} e^{i\mathbf{k} \cdot \mathbf{r}} & \text{when } |\mathbf{r} - \mathbf{R}| \ge r_0 \\ \sum_{lm} A_{lm} Y_{lm}(\theta, \phi) R_l(r, \epsilon). & \text{when } |\mathbf{r} - \mathbf{R}| < r_0 \end{cases} \qquad (80) $$

If an eigenfunction would be discontinuous, its kinetic energy would not be well-defined. In order to avoid such situation, we require that the plane wave outside the sphere matches the function inside the sphere over the complete surface of the sphere. Thus the plane wave is expanded in spherical harmonics:

$$ \exp(i\mathbf{k} \cdot \mathbf{r}) = 4\pi \sum_{l=0}^{\infty} \sum_{m=-l}^{l} i^l j_l(kr) Y_{lm}^*(\theta_k, \phi_k) Y_{lm}(\theta, \phi) \qquad (81) $$

where $r, \theta, \phi$ correspond to the polar representation. To keep the problem tractable, we limit all the expansions to a finite value of $l$. Requiring this at the sphere boundary means that all the coefficients of have to be equal for both parts of the function at the boundary. This condition fixes the and we obtain:

$$ \psi_{\mathbf{k}}^{\text{APW}}(\mathbf{r}) = 4\pi \sum_{lm} i^l \left[ \frac{j_l(kr_0)}{R_l(r_0)} \right] R_l(r) Y_{lm}^*(\theta_{\mathbf{k}}, \phi_{\mathbf{k}}) Y_{lm}(\theta, \phi) \qquad (82) $$

for the function inside the sphere.

The APW functions do not solve the Schrödinger equation, but they are convenient for approximating the true eigen-function by superposition of several APW functions having the same energy.

For any $\mathbf{k}$, the APW satisfies the Bloch condition, but in order to make the true eigen-function satisfy Bloch theorem we need to expand $\psi_{\mathbf{k}}(\mathbf{r})$ as:

$$ \psi_{\mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{K}} c_{\mathbf{K}} \psi_{\mathbf{k}+\mathbf{K}, \epsilon(\mathbf{k})}^{\text{APW}}(\mathbf{r}) \qquad (83) $$

where the sum is over all the reciprocal lattice vectors.

We still need a number of APW wave-functions to solve the full Schrödinger equation at the boundary and in the interstitial region. In practice, several hundreds can be used. In the process of doing this procedure, the energy does not change much, while more APW functions are added. Thus the process is continued until convergence is attained.

All the APW wave-functions have to be evaluated at the same energy. The coefficients are given by solving the generalized eigenvalue equation:

$$ Hc = ESc \qquad (84) $$

The most remarkable aspect of this eigenvalue equation is that its matrix elements depend on energy, thus we fix the energy, and look for values at which the secular expression is satisfied:

$$ (H - ES)c = 0 \qquad (85) $$

## D. Developments concerning the APW method

Note that the wave-function in general will have discontinuous derivatives on the boundary between the interstitial and atomic regions.
In the APW method the augmenting function corresponds to the exact muffin-tin potential eigenstate corresponding to the energy eigenvalue. Thus the eigenvalue problem will be non-linear in energy and has to be solved iteratively implying a costly procedure.

To overcome this problem, linearized versions of the APW method have been developed, with the energy set to a fixed value and the basis functions modified in order to gain extra flexibility through probing a larger energy region around the selected energy. These methods are the linearized APW method (LAPW).

## E. Tight-Binding method

Considering materials whose electrons can be described by a localized character as opposite to an extended wave-like character, thus we move to the case of tight-binding (TB) electronic states in a lattice of parameter a.

The Hamiltonian, in the site representation is given by:

$$ \hat{H} = \sum_{l} \epsilon_{l} |l\rangle \langle l| + \sum_{i,j} V_{i,j} |i\rangle \langle j| \quad (86) $$

with $\epsilon_l$ local energy at site $l$ (that reduces to a single value $\epsilon_0$ by translational symmetry) and $V_{i,j}$ overlap integral between neighboring sites $i$ and $j$ (that reduces to a single value $V_0$ between nearest neighbor sites). Note that a spatial FT links the site (direct lattice) representation to the wavevector (reciprocal lattice) representation:

$$ |k\rangle = \frac{1}{\sqrt{N}} \sum_{l} \exp(ik \cdot l) |l\rangle \quad (87) $$

with N the number of unit cells.

The tight-binding (TB) single-band electronic dispersion for the square $|a \times a|$ lattice of parameter a is:

$$ \epsilon(\mathbf{k}) = \epsilon_0 + 2V_0[\cos(k_x a) + \cos(k_y a)] \quad (88) $$

where $\epsilon_0$ is a mean energy, $V_0$ the neighboring sites overlap integral and $k_x, k_y$ wavevectors that belong to the Brillouin zone $[-\pi/a, \pi/a] \times [-\pi/a, \pi/a]$.

An example of Graphene 2D and 3D Brillouin zones with Linearized APW (LAPW) and Tight-Binding band structure calculations is provided in fig. 10.

## F. Emergent and other Fermi varieties

Some examples of emergent fermionic quasi-particles are:

* A Dirac fermion is a relativistic electron observed in materials such as Graphene.

* A Majorana fermion is a uncharged particle which is its own antiparticle, and can emerge as a quasi-particle in certain superconductors, or in a quantum spin liquid.

* A Weyl fermion is a massless fermion.

* A heavy fermion is a quasi-electron with an effective mass $m^* \gg m_e$
<figure><img src="image_11.png" /><figcaption>FIG. 10 (Color online) 3D Brillouin Zone (upper left) and 2D Graphene Brillouin zone (upper right) and band structure (lower left) for Graphene using LAPW method and a mixture of (nearly) free electron model and tight-binding (TB) methods. The non-TB bands originating from LAPW are displayed with red and blue dashed lines (lower right). Note that LAPW and TB band structures agree approximately if we ignore non-TB bands. Adapted from Kogan *et al.* (22)</figcaption></figure>

## APPENDIX A: Atomic units

In atomic and molecular physics (23), it is convenient to use the elementary charge $e$, as the charge unit, and the electron mass $m_e$ as the mass unit. Electrostatic forces and energies in atoms are proportional to $e^2/4\pi\epsilon_0$ and $\hbar$ that appears all over in quantum physics, so it is convenient to choose units of length and time such that $4\pi\epsilon_0 = 1$ and $\hbar = 1$.

Atomic units being based on typical dimensions of the hydrogen atom, length unit is the classical radius of the hydrogen electron orbital in 1s ground state, the Bohr radius $a_B$. Using dimensional analysis, $a_B = \frac{e^2}{m_e(e^2/4\pi\epsilon_0)}$, that is (0.529 Å). This stems from the Bohr model where the radius $r_s$ of the smallest orbit for an electron circling a fixed proton is $r_s = (1 + \frac{m_e}{m_p})a_B$, the reduced mass $\mu = \frac{m_p m_e}{m_p + m_e}$ and when $m_p \to \infty$ we obtain $r_s \to a_B$ and $\mu \to m_e$.

Similarly, we have another dimensionless parameter, the fine structure constant as $\alpha = \frac{e^2}{4\pi\epsilon_0 c\hbar} \sim \frac{1}{137}$ where c is velocity of light in vacuum.

The classical hydrogen ground state electron velocity $v_0$ is considered as unit of velocity. One may write $a_B/v_0 = \hbar/E_h$ with $E_h$ the unit of energy.

The unit of energy $E_h$ is the Hartree (27.2 eV), the Coulomb energy, in vacuum, of two electrons separated by the unit of length $a_B$, given by: $E_h = \frac{e^2}{4\pi\epsilon_0} \frac{1}{a_B} = (\frac{e^2}{4\pi\epsilon_0})^{\frac{2}{3}} \frac{m_e}{\hbar^2}$. The Hartree is twice the ground state energy of the Hydrogen atom $\frac{1}{2}(1 + \frac{m_e}{m_p})^{-1}E_h$ equal to the Rydberg (13.6 eV) when $m_p \to \infty$.
<table><caption>TABLE III Conversion table for energy expressed in J, eV, Hz and cm⁻¹.</caption><thead><tr><th></th><th>J</th><th>eV</th><th>Hz</th><th>cm<sup>-1</sup></th></tr></thead><tbody><tr><td>J</td><td>1</td><td>6.24151.10<sup>18</sup></td><td>1.50919.10<sup>33</sup></td><td>5.03411.10<sup>22</sup></td></tr><tr><td>eV</td><td>1.60219.10<sup>-19</sup></td><td>1</td><td>2.41797.10<sup>14</sup></td><td>8.06547.10<sup>3</sup></td></tr><tr><td>Hz</td><td>6.62619.10<sup>-34</sup></td><td>4.13570.10<sup>-15</sup></td><td>1</td><td>3.33564.10<sup>-11</sup></td></tr><tr><td>cm<sup>-1</sup></td><td>1.96648.10<sup>-23</sup></td><td>1.23935.10<sup>-4</sup></td><td>2.99792.10<sup>10</sup></td><td>1</td></tr></tbody></table>

## APPENDIX B: Standard Metal properties

For T = 0 K, we have the following values:

<table><caption>TABLE IV The data of the standard metal are taken from ref. (24) in order to compare with real metals. The data correspond to an idealized metal following the free electron model. In this model the electron concentration is assumed to be n<sub>e</sub> = N<sub>A</sub>/cm<sup>3</sup> (with N<sub>A</sub> the Avogadro number, this corresponds to an atomic volume of approximately 10 cm<sup>3</sup> for a monovalent metal).</caption><tr><td>Fermi energy</td><td>ε<sub>F</sub> = <sup>ħ<sup>2</sup></sup>&frasl;<sub>2m<sub>e</sub></sub>(3π<sup>2</sup>n<sub>e</sub>)<sup>2/3</sup> = 5.59 eV</td></tr><tr><td>Degeneracy temperature</td><td>T<sub>DG</sub> = 2ε<sub>F</sub>/5k<sub>B</sub> = 25900 K</td></tr><tr><td>Radius of the Fermi sphere</td><td>k<sub>F</sub> = (3π<sup>2</sup>n<sub>e</sub>)<sup>1/3</sup></td></tr><tr><td>Specific heat of electrons per unit volume</td><td><sup>2</sup>&frasl;<sub>3</sub>π<sup>2</sup>k<sub>B</sub><sup>2</sup>g(ε<sub>F</sub>)T = γT</td></tr><tr><td>Density of states at the Fermi level<br>(no spin degeneracy accounted for)</td><td>g(ε<sub>F</sub>) = <sup>m<sub>e</sub></sup>&frasl;<sub>2π<sup>2</sup>ħ<sup>2</sup></sub>(3π<sup>2</sup>n<sub>e</sub>)<sup>1/3</sup></td></tr><tr><td>Grüneisen coefficient γ</td><td>γ = <sup>2</sup>&frasl;<sub>3</sub>π<sup>2</sup>k<sub>B</sub><sup>2</sup>g(ε<sub>F</sub>)</td></tr><tr><td>Fermi velocity</td><td>v<sub>F</sub> = ħk<sub>F</sub>/m<sub>e</sub> = 1.4 × 10<sup>8</sup> cm/s</td></tr><tr><td>Surface of the Fermi sphere</td><td>4πk<sub>F</sub><sup>2</sup></td></tr><tr><td>Maximum cross-section of the Fermi sphere</td><td>πk<sub>F</sub><sup>2</sup></td></tr><tr><td>Cyclotron frequency of electrons in an induction field B</td><td>ω<sub>c</sub> = eB/m<sub>e</sub></td></tr><tr><td>Classical cyclotron radius for electrons<br>at the Fermi energy</td><td>v<sub>F</sub>/ω<sub>c</sub></td></tr><tr><td>Drude conductivity of Fermi level ε<sub>F</sub> electrons<br>with a mean free path ℓ</td><td>σ<sub>D</sub> = n<sub>e</sub>e<sup>2</sup>ℓ/(m<sub>e</sub>v<sub>F</sub>)</td></tr></table>
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