State Bloch's theorem for a one dimensional lattice which is invariant under translations by $a$.

A simple model of a crystal consists of a one-dimensional linear array of identical sites with separation $a$. At the $n$th site the Hamiltonian, neglecting all other sites, is $H_{n}$ and an electron may occupy either of two states, $\phi_{n}(x)$ and $\chi_{n}(x)$, where

$$H_{n} \phi_{n}(x)=E_{0} \phi_{n}(x), \quad H_{n} \chi_{n}(x)=E_{1} \chi_{n}(x),$$

and $\phi_{n}$ and $\chi_{n}$ are orthonormal. How are $\phi_{n}(x)$ and $\chi_{n}(x)$ related to $\phi_{0}(x)$ and $\chi_{0}(x)$ ?

The full Hamiltonian is $H$ and is invariant under translations by $a$. Write trial wavefunctions $\psi(x)$ for the eigenstates of this model appropriate to a tight binding approximation if the electron has probability amplitudes $b_{n}$ and $c_{n}$ to be in the states $\phi_{n}$ and $\chi_{n}$ respectively.

Assume that the only non-zero matrix elements in this model are, for all $n$,

$$\begin{aligned} &\left(\phi_{n}, H_{n} \phi_{n}\right)=E_{0}, \quad\left(\chi_{n}, H_{n} \chi_{n}\right)=E_{1} \\ &\left(\phi_{n}, V \phi_{n \pm 1}\right)=\left(\chi_{n}, V \chi_{n \pm 1}\right)=\left(\phi_{n}, V \chi_{n \pm 1}\right)=\left(\chi_{n}, V \phi_{n \pm 1}\right)=-A, \end{aligned}$$

where $H=H_{n}+V$ and $A>0$. Show that the time-dependent Schrödinger equation governing the amplitudes becomes

$$\begin{aligned} i \hbar \dot{b}_{n} &=E_{0} b_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right) \\ i \hbar \dot{c}_{n} &=E_{1} c_{n}-A\left(c_{n+1}+c_{n-1}+b_{n+1}+b_{n-1}\right) \end{aligned}$$

By examining solutions of the form

$$\left(\begin{array}{l} b_{n} \\ c_{n} \end{array}\right)=\left(\begin{array}{l} B \\ C \end{array}\right) e^{i(k n a-E t / \hbar)}$$

show that the allowed energies of the electron are two bands given by

$$E=\frac{1}{2}\left(E_{0}+E_{1}-4 A \cos k a\right) \pm \frac{1}{2} \sqrt{\left(E_{0}-E_{1}\right)^{2}+16 A^{2} \cos ^{2} k a}$$

Define the Brillouin zone for this system and find the energies at the top and bottom of both bands. Hence, show that the energy gap between the bands is

$$\Delta E=-4 A+\sqrt{\left(E_{1}-E_{0}\right)^{2}+16 A^{2}}$$

Show that the wavefunctions $\psi(x)$ satisfy Bloch's theorem.

Describe briefly what are the crucial differences between insulators, conductors and semiconductors.