Quantum Theory of Electrons in Metals

(``Free Electrons; Sommerfeld")

(AM, Ch 2)

Now let's put $QM$ into the model.

The most important effect of putting quantum mechanics into the theory of metals is the result of treating electrons as fermions with Fermi-Dirac statistics, rather than as a classical gas of particles obeying the Kinetic Theory of Gases. But let's start at the beginning.

When electrons are treated quantum mechanically, 2 things change: (a) possible states are quantized; (b) particles are indistinguishable.

=2.0 true in

Consider non-interacting electrons moving (freely) in a constant potential which we take to be zero. Schroedinger's equation in 1D is

$$\frac{\hbar^2}{2m}\frac{d^2\psi_n}{d x^2} = \varepsilon _n\psi_n$$

What boundary conditions do we use? (It doesn't matter all that much because the bulk properties aren't really affected by what goes on at the surface.)

(a)

Realistic, i.e., $\psi_n$ finite outside the metal but decaying exponentially $(\to 0$ as $x \to \infty$). This is practically never used for bulk calculations since $r_{decay} \sim$ atomic dimensions.

(b)

Box: Standing Wave Solutions $\qquad\qquad \psi_n = 0$ at walls
$\psi_n(x) = \sqrt{\frac{\textstyle 2}{\textstyle L}} \sin \left( \frac{\textst... ...arepsilon _n = \frac{\textstyle n^2\pi^2\hbar^2}{\textstyle 2mL^2} \quad n > 0$.
We're interested in transport - want traveling waves.

(c)

Periodic: $\psi_n(x+L) = \psi_n(x)$
Now traveling wave solutions are allowed.

$$\psi_n(x) = \sqrt{\frac{1}{L}} e^{ik_nx}$$

$$k_n = \frac{2n\pi}{L} \quad n \gtrless 0$$

$$E_n = \frac{\hbar^2k^2_n}{2m} = \frac{\hbar^2}{2m} \left( \frac{2n\pi}{L} \right)^2$$

It is easy to generalize this to 3D. Consider a box with sides $L_x, L_y, L_z$: Schroedinger's equation becomes

$$- \frac{\hbar^2}{2m} \nabla^2 \psi_n = - \frac{\hbar^2}{2m} \left... ...rtial y^2}+\frac{\partial ^2\psi_n}{\partial z^2}\right) = \varepsilon _n\psi_n$$

For periodic b.c.'s, the solution is a product wavefunction

$$\psi_n(x,y,z) = \frac{1}{\sqrt{V}} (\exp ik_xx) (\exp ik_yy)(\exp ik_zz)$$

$$= \frac{1}{\sqrt{V}} e^{i\vec{k}\cdot\vec{r}}$$

$$E_n = \frac{\hbar^2k^2}{2m}$$

with

$$k_x = \frac{2n_x\pi}{L_x} \qquad k_y = \frac{2n_y\pi}{L_y} \qquad k_z = \frac{2n_z\pi}{L_z}$$

The momentum carried in the plane wave state is $\vec{p} = \hbar\vec{k}$ and the velocity $\vec{v} = \frac{\textstyle \hbar\vec{k}}{\textstyle m}$. Each state is characterized by $\vec{k}$ and by the spin $\sigma$ along some chosen axis. There are 2 different states for each allowed value of $\vec{k}$. Note that the probability density $\vert\psi_n(\vec{r})\vert^2$ is uniform in space.

Density of States

Suppose we are interested in some property of the one-electron states, such as the total average number of electrons in them. We sum over the states

$$N = \sum\limits_{\vec{k}\sigma} n_{k\sigma}\qquad {\mbox{ sum ... ...{k}'s \atop \textstyle k_x = \frac{\textstyle 2n\pi}{\textstyle L}, {\rm etc.}}$$

where $n_{k\sigma}$ is the number of electrons in state $k$ with spin $\sigma$. Assume that $n_{k\sigma}$ is a rather smoothly varying function of $\vec{k}$. This allows us to transform the sum into an integral.

$$\sum\limits_{\vec{k}\sigma} n_{k\sigma} \to \sum\limits_{\sigma}\int d^3k\; n_{k\sigma} \varphi (\vec{k})$$

$\varphi (\vec{k})$ is the density of states per unit volume of $\vec{k}$-space. What is $\varphi (\vec{k})$? $k_x = \frac{\textstyle 2n\pi}{\textstyle L_x}$ implies that the density of allowed $k_x$ values along the $k_x$-axis is $\frac{1}{2\pi/L_x} = \frac{L_x}{2\pi}$. Similar arguments for $k_y$ and $k_z$ lead to

Note that this is independent of the ratio of $L_x, L_y$, and $L_z$. In fact, $\varphi (\vec{k})$ is independent of the sample shape in the limit $V \to \infty$, except for the lowest few states.

If spin is included, put in an extra factor of 2. Thus

$$\sum\limits_{\vec{k}\sigma} n_{k\sigma} \longrightarrow \sum\l... ... \int \frac{d^3k}{(2\pi )^3} n_{k\sigma} = \frac{2V}{(2\pi )^3} \int d^3k n_k$$

if $n_{k\sigma}$ is independent of $\sigma$. In general for any dimension $d$

$$\sum\limits_{\vec{k}\sigma} \longrightarrow \frac{2V_d}{(2\pi )^d} \int d^dk$$

where $V_d$ is the $d$-dimensional volume.