Free Electron Theory of Metals

Abstract

The most important characteristic of a metal is its high electrical conductivity. Around 1900, shortly after J.J. Thomson’s discovery of the electron, people became interested in understanding more about the mechanism of metallic conduction. The first work by E. Riecke in 1898 was quickly superseded by that of Drude in 1900. Drude (P. Drude, Annalen der Physik 1, 566 (1900); ibid., 3, 369 (1900); ibid., 7, 687 (1902)) proposed an exceedingly simple model that explained a well-known empirical law, the Wiedemann–Franz law (1853). This law states that at a given temperature the ratio of the thermal conductivity to the electrical conductivity is the same for all metals. The assumptions of the Drude model are:

• (i) a metal contains free electrons which form an electron gas.

• (ii) the electrons have some average thermal energy $\left\langle \frac{1}{2}mv_{\mathrm{T}}^2\right\rangle$, but they pursue random motions through the metal so that $\left\langle \mathbf v_{\mathrm{T}}\right\rangle =0$ even though $\left\langle v_{\mathrm{T}}^2 \right\rangle \ne 0$. The random motions result from collisions with the ions.

• (iii) because the ions have a very large mass, they are essentially immovable.

Appendices

Problems

3.1

A two-dimensional electron gas is contained within a square box of side L.

• (a) Apply periodic boundary conditions and determine $E(k_x, k_y)$ and $\varPsi _{\mathbf k} (x, y)$ for the free electron Hamiltonian $H=\frac{1}{2m}\left( p_x^2+p_y^2\right)$.

• (b) Determine the Fermi wave number $k_{\mathrm{F}}$ in terms of the density $n_0=\frac{N}{L^2}$.

• (c) Evaluate $G(\varepsilon )$ and $g(\varepsilon )$ for this system.

• (d) Use $n_0=\int _0^\infty d\varepsilon \, g(\varepsilon )f_0(\varepsilon )$ together with the Fermi function integration formula to determine how the chemical potential $\zeta$ depends on T.

• (e) Express the energy of the system in terms of the Fermi function integral and determine the specific heat of the electrons at low temperatures.

3.2

Consider an electron inside a metallic nanowire of a square cross section with sides $L_x=L_y=L$ lying along the z axis.

• (a) Show that the single particle eigenstates can be written as $\psi _{n_x, n_y}(k_z) = \sin {\frac{n_x\pi x}{L}}\sin {\frac{n_y\pi y}{L}}\mathrm{{e}}^{i k_z z}$ and $E_{n_x,n_y}(k_z) = \varepsilon (n_x, n_y) + \frac{\hbar ^2 k_z^2}{2m}$, where $(n_x, n_y)$, $k_z$, and $\varepsilon (n_x, n_y)$ are the quantum numbers describing the finite size effects of the cross section, the wave number along the wire, and the energy level of a particle in an infinite two-dimensional quantum well of dimension $L\times L$.

• (b) Show that the total density of states is given by $g(E) = \frac{2\sqrt{2m}}{\hbar } \frac{\varTheta (E-\varepsilon _{n_x,n_y})}{\sqrt{E-\varepsilon _{n_x, n_y}}}$, where $\varTheta (x)$ is the Heaviside function of unit step.

3.3

Consider the d dimensional system electrons or phonons for $d\ge 1$.

• (a) Show that density of states of the free electron gas scales as $g(E) \approx E^{d/2-1}$.

• (b) Determine the corresponding scaling law for the phonon density of states in the Debye model discussed in the previous chapter.

3.4

A metal is described by the conductivity tensor given by $\sigma _{xx} =\sigma _{yy} = \frac{\sigma _0 (1+i\omega \tau )}{(1+i\omega \tau )^2+(\omega _{\mathrm{c}}\tau )^2}$, $\sigma _{xy} =-\sigma _{yx} = \frac{\sigma _0 (-\omega _{\mathrm{c}} \tau )}{(1+i\omega \tau )^2+(\omega _{\mathrm{c}}\tau )^2}$, and $\sigma _{zz} = \frac{\sigma _0}{1+i\omega \tau }$ in the presence of a dc magnetic field $\mathbf B = B\hat{z}$.

• (a) Consider the propagation of an electromagnetic wave $E_{\pm } =\left( E_x \pm iE_y\right) \mathrm{{e}}^{i\omega t}$ parallel to the z-axis. Use Maxwell’s equations to obtain the wave equation, and show that $c^2k^2 = \omega ^2\epsilon _{\pm }(\omega )$, where ${\underline{\epsilon }}_\pm = {\mathbf 1} -\frac{4\pi i}{\omega }{\underline{\sigma }}_\pm (\omega )$.

• (b) Consider the cases $\omega _{\mathrm{c}}\tau \gg 1$ and $\omega _{\mathrm{c}} \gg \omega$ and show that $\omega = \frac{c^2k^2\omega _{\mathrm{c}}}{\omega _{\mathrm{p}}^2}$ for one circular polarization.

3.5

Let us consider the interface between a dielectric of dielectric constant $\epsilon _{\mathrm{D}}$ and a metal of dielectric function $\epsilon (\omega ) = 1-\frac{\omega _{\mathrm{p}}^2}{\omega ^2}$, where $\omega _{\mathrm{p}}^2 = \frac{4\pi n e^2}{m}$. It is illustrated in Fig. 3.6. If the normal to the surface is in the z direction and the wave vector $\mathbf q = (0, q_y, q_z)$, consider the region of $\omega -q_y$ space in which $q_z$ is imaginary (i.e. $q_z^2 <0$) both in the dielectric and in the metal. Impose the appropriate boundary conditions at $z=0$ and at $\mid z\mid \longrightarrow \infty$, and determine the dispersion relation ($\omega$ as a function of $q_y$) for these surface plasma modes.

Fig. 3.6

Interface between a dielectric and a metal

3.6

At a temperature T a semiconductor contains $n_{\mathrm{e}}$ electrons and $n_{\mathrm{h}}$ holes per unit volume in parabolic energy bands. The mass, charge, and collision time of the electrons and holes are $m_{\mathrm{e}}$, $-e$, $\tau _{\mathrm{e}}$ and $m_{\mathrm{h}}$, e, $\tau _{\mathrm{h}}$, respectively.

• (a) Use the equations of motion of charged particles in the presence of a dc magnetic field $\mathbf B = B\hat{z}$ and an ac electric field $\mathbf E={\mathbf E}_0 \mathrm{{e}}^{i\omega t}$ to determine $\underline{\sigma }_{\mathrm{e}}(\omega )$ and $\underline{\sigma }_{\mathrm{h}}(\omega )$, the electron and hole contributions to the frequency dependent magnetoconductivity tensor.

• (b) Consider $\omega _{\mathrm{ce}} =\frac{eB}{m_{\mathrm{e}}c}$ and $\omega _{\mathrm{ch}} =\frac{eB}{m_{\mathrm{h}}c}$ to be large compared to $\tau _{\mathrm{e}}^{-1}$ and $\tau _{\mathrm{h}}^{-1}$, respectively. Determine the Hall coefficient for $\omega =0$.

• (c) Under the conditions of part (b), determine the magnetoresistance.

Summary

In this chapter first we have briefly reviewed classical kinetic theories of an electron gas both by Drude and by Lorentz as simple models of metals. Then Sommerfeld’s elementary quantum mechanical theory of metals is discussed.

In the Drude model, the electrical conductivity $\sigma =\frac{n_0e^2\tau }{m}$ is determined by the Newton’s law of motion given by

$$ m\left( \frac{d\mathbf v_{\mathrm{D}}}{dt} +\frac{\mathbf v_{\mathrm{D}}}{\tau } \right) =-e\mathbf E. $$

Here $n_0=\frac{N}{V}$ and $-e$ are the electron concentration and the charge on an electron. The thermal conductivity is given by

$$ \kappa =\frac{w}{-\partial T/\partial x} =\frac{1}{3}n_0 v_{\mathrm{T}}^2\tau \frac{dE}{dT} =\frac{1}{3}v_{\mathrm{T}}^2 \tau C_{\mathrm{v}}, $$

where $C_{\mathrm{v}}=n_0 \frac{dE}{dT}$ is the electronic specific heat.

The electrical current density $\mathbf j$ and thermal current density $\mathbf w$ are given, in terms of distribution function $f$, by

$$ \mathbf j(\mathbf r,t) = \int (-e)\mathbf v\, f (\mathbf r,\mathbf v, t)\, d^3v \text{ and } \mathbf w(\mathbf r,t)=\int \varepsilon \mathbf v \, f (\mathbf r,\mathbf v, t)\, d^3v. $$

In the Sommerfeld model, states are labeled by $\{\mathbf k,\sigma \}=(k_x,k_y, k_z) \text{ and } \sigma$, where $\sigma$ is a spin index. The Fermi energy $\varepsilon _{\mathrm{F}} \left( \equiv \varTheta _{\mathrm{F}}\right) ,$ Fermi velocity $v_{\mathrm{F}}$, and Fermi temperature $T_{\mathrm{F}}\left( = \frac{\varTheta _{\mathrm{F}}}{k_{\mathrm{B}}}\right)$ are defined, respectively, by

$$ \varepsilon _{\mathrm{F}}=\frac{\hbar ^2 k_{\mathrm{F}}^2}{2m} =\frac{1}{2}mv_{\mathrm{F}}^2=\varTheta _{\mathrm{F}}, $$

where the Fermi wave number $k_{\mathrm{F}}$ is related to the carrier concentration $n_0$ by $k_{\mathrm{F}}^3 = 3\pi ^2 n_0$. The density of states of an electron gas is

$$ g(\varepsilon )=\frac{1}{2\pi ^2}\left( \frac{2m}{\hbar ^2}\right) ^{3/2}\varepsilon ^{1/2}. $$

For electrons moving in a periodic potential, $g(\varepsilon )$ does not have such a simple form. At a finite temperature, the chemical potential $\zeta$ is determined from

$$ N=V\int _0^\infty g(\varepsilon ) f _0(\varepsilon )d\varepsilon . $$

The internal energy U is given by

$$ \frac{U}{V} = u = \int _0^\infty d\varepsilon \, \varepsilon g(\varepsilon ) f _0(\varepsilon ). $$

These integrals are of the form $I = \int _0^\infty d\varepsilon \, f _0(\varepsilon )\frac{dF(\varepsilon )}{d\varepsilon }.$ At low temperatures, we have, to order $\varTheta ^2$,

$$ I=F(\zeta )+\frac{\pi ^2}{6}\varTheta ^2F^{{\prime \prime }}(\zeta ). $$

The electronic heat capacity $C_{\mathrm{v}} = \left( \frac{\partial U}{\partial T}\right) _V$ is given, at low temperature, by $C_{\mathrm{v}} = \gamma T,$ where $\gamma =\frac{\pi ^2 k_{\mathrm{B}}^2}{2\zeta _0} N$ for free electrons.

The electrical and thermal current densities $j_x$ and $w_x$ are, respectively, written as

$$ j_x = \left[ e^2E + e\varTheta \frac{\partial }{\partial x}\left( \frac{\zeta }{\varTheta }\right) \right] \mathcal {K}_1 +\frac{e}{\varTheta }\frac{\partial \varTheta }{\partial x} \mathcal {K}_2 $$

and

$$ w_x = -\left[ eE + \varTheta \frac{\partial }{\partial x} \left( \frac{\zeta }{\varTheta }\right) \right] \mathcal {K}_2 -\frac{1}{\varTheta }\frac{\partial \varTheta }{\partial x} \mathcal {K}_3. $$

where

$$ \mathcal {K}_n =\frac{n_0}{m\zeta _0^{3/2}} \int _0^\infty d\varepsilon \ \left( -\frac{\partial f _0}{\partial \varepsilon }\right) \varepsilon ^{n+1/2} \tau . $$

The function $\mathcal {K}_n$ is given by

$$ \mathcal {K}_n =\frac{n_0}{m\zeta _0^{3/2}} \left[ \zeta ^{n+1/2} \tau (\zeta ) + \frac{\pi ^2}{6}\varTheta ^2 \frac{d^2}{d\varepsilon ^2} \left( \varepsilon ^{n+1/2}\tau (\varepsilon )\right) \mid _{\varepsilon =\zeta }\right] . $$

The electrical and thermal conductivities are given, in terms of $\mathcal {K}_1$, by $\sigma =e\mathcal {K}_1$ and $\kappa _T= k_{\mathrm{B}} \frac{\mathcal {K}_3\mathcal {K}_1-\mathcal {K}_2^2}{\mathcal {K}_1\varTheta }.$ The Sommerfeld expression for $\kappa _T$ is $\kappa _T= \frac{\pi ^2}{3}k_{\mathrm{B}}^2 \frac{n_0\tau }{m}T.$

In the presence of an electric field $\mathbf E$ and a dc magnetic field $\mathbf B$, the magnetoconductivity tensor has nonzero components, for the case $\mathbf B$ along the z-axis, as follows: $\sigma _{xx}=\sigma _{yy} =\frac{\sigma _0(1+i\omega \tau )}{(1+i\omega \tau )^2+(\omega _{\mathrm{c}}\tau )^2}$, $\sigma _{xy}=-\sigma _{yx} =\frac{\sigma _0(-\omega _{\mathrm{c}}\tau )}{(1+i\omega \tau )^2+(\omega _{\mathrm{c}}\tau )^2}$, $\sigma _{zz} = \frac{\sigma _0}{1+i\omega \tau }$. Here $\omega _{\mathrm{c}} =\frac{eB}{mc}$ and $\sigma _0=\frac{n_0e^2\tau }{m}$ is just the Drude’s dc conductivity.

The electrical current density $\mathbf j$ can be thought of as the time rate of change of the polarization $\mathbf P$, that is, $\mathbf j=\dot{\mathbf P}=i\omega {\mathbf P}$, where $\mathbf P=\frac{\epsilon -1}{4\pi }\mathbf E$ and $\mathbf D$, $\mathbf P$, and $\mathbf E$ are assumed to vary as $\mathrm{{e}}^{i\omega t}$. Hence we have the relation

$$ \epsilon (\omega ) = 1 -\frac{4\pi i }{\omega }\sigma (\omega ). $$

$\epsilon (\omega )$ has real and imaginary parts, $\epsilon _1$ and $\epsilon _2$, respectively, and in the Drude model, we have $\epsilon _1(\omega ) = 1-\frac{\omega _p^2}{\omega ^2+1/\tau ^2}$ and $\epsilon _2(\omega ) = -\frac{\omega _p^2/\omega \tau }{\omega ^2+1/\tau ^2}.$ The two Maxwell equations $\mathbf \nabla \times \mathbf E = -\frac{1}{c}\, \dot{\mathbf B}$ and $\mathbf \nabla \times \mathbf B = \frac{1}{c}\, \underline{\epsilon }\cdot \dot{\mathbf E}$ can be combined to obtain the wave equation

$$ \mathbf q(\mathbf q\cdot \mathbf E)-q^2\mathbf E +\frac{\omega ^2}{c^2}{\underline{\epsilon }}\cdot \mathbf E =0. $$