Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 34B / Solution

The Bloch theorem says that for a translation-invariant lattice Hamiltonian, eigenstates can be chosen as ${\psi }_k(x+a)=e^{ika}{\psi }_k(x)$, or ${\psi }_k(x)=e^{ikx}{u}_k(x)$ with $u_k$ periodic of period $a$. The localized site orbitals are translates, ${\varphi }_n(x)={\varphi }_0(x-na)$ and ${\chi }_n(x)={\chi }_0(x-na)$. In the tight-binding model, neglect intersite overlap and useProjecting the Schrödinger equation onto each localized orbital gives its onsite energy times its own coefficient, plus the four stated neighbor matrix elements. ThusFor the Bloch amplitudes, set $z=\cos ka$. The coefficient eigenproblem isso the two energy bands areA first Brillouin zone is $-\pi /a\le k<\pi /a$; wavevectors differing by $2\pi /a$ are equivalent. Put $\Sigma =E_0+E_1$, $D=\sqrt{(E_1-E_0{)}^2+16A^2}$. Each band is nonincreasing as a function of $z\in [-1,1]$, because $d{E}_{\pm }/dz=-2A\pm 8A^{2}z/\sqrt{(E_1-E_0{)}^2+16A^2{z}^2}\le 0$. HenceThe extrema occur at $k=0$ and at the zone edge; they may be nonunique when $E_0=E_1$, in which case the gap vanishes. In the trial wavefunction, replacing $x$ by $x+a$ and reindexing $n$ produces exactly the factor $e^{ika}$, verifying Bloch theorem directly.

An electrical conductor has available nearby unoccupied states in a partially filled band or overlapping bands. A band insulator has filled lower bands separated from empty higher bands by a large gap. A semiconductor has the same filled-band structure with a smaller gap, so thermal excitation or doping supplies mobile electrons and holes. Band filling and carrier availability, not just the existence of two bands, determine which description applies.