(i) A simple model of a crystal consists of an infinite linear array of sites equally spaced with separation $b$. The probability amplitude for an electron to be at the $n$-th site is $c_{n}, n=0, \pm 1, \pm 2, \ldots$. The Schrödinger equation for the $\left\{c_{n}\right\}$ is

$$E c_{n}=E_{0} c_{n}-A\left(c_{n-1}+c_{n+1}\right)$$

where $A$ is real and positive. Show that the allowed energies $E$ of the electron must lie in a band $\left|E-E_{0}\right| \leq 2 A$, and that the dispersion relation for $E$ written in terms of a certain parameter $k$ is given by

$$E=E_{0}-2 A \cos k b$$

What is the physical interpretation of $E_{0}, A$ and $k$ ?

(ii) Explain briefly the idea of group velocity and show that it is given by

$$v=\frac{1}{\hbar} \frac{d E(k)}{d k},$$

for an electron of momentum $\hbar k$ and energy $E(k)$.

An electron of charge $q$ confined to one dimension moves in a periodic potential under the influence of an electric field $\mathcal{E}$. Show that the equation of motion for the electron is

$$\dot{v}=\frac{q \mathcal{E}}{\hbar^{2}} \frac{d^{2} E}{d k^{2}},$$

where $v(t)$ is the group velocity of the electron at time $t$. Explain why

$$m^{*}=\hbar^{2}\left(\frac{d^{2} E}{d k^{2}}\right)^{-1}$$

can be interpreted as an effective mass.

Show briefly how the absence from a band of an electron of charge $q$ and effective mass $m^{*}<0$ can be interpreted as the presence of a 'hole' carrier of charge $-q$ and effective mass $-m^{*}$.

In the model of Part (i) show that

(a) for $k^{2} \ll 12 / b^{2}$ an electron behaves like a free particle of mass $\hbar^{2} /\left(2 A b^{2}\right)$;

(b) for $(\pi / b-k)^{2} \ll 12 / b^{2}$ a hole behaves like a free particle of mass $\hbar^{2} /\left(2 A b^{2}\right)$.