Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 5

For a periodic single-electron Hamiltonian operator, Bloch's theorem givesTwo bands at the same $\mathbf{k}$ have the same translation eigenvalues but different cell-periodic eigenfunctions, generally different energies and different orbital/nodal character. They may be chosen orthogonal even at a band degeneracy. The basis choices are treated in the two scoped branches below.

For a slowly varying external potential $U(\mathbf{r})$, the semiclassical wavepacket Hamiltonian is ${ϵ}_b(\mathbf{k})+U(\mathbf{r})$. Hamilton's equation for the conjugate variables $\mathbf{r},\hslash \mathbf{k}$ gives $\hslash \dot{\mathbf{k}}=-\nabla U=\mathbf{F}$. Equivalently the electric perturbation changes the translation phase at the rate $q\mathbf{E}\cdot \mathbf{R}/\hslash$. This is the single-band acceleration theorem, requiring a slowly varying field and negligible interband tunnelling/scattering; it refers to crystal momentum, not free-space mechanical momentum.

Let $e>0$ and $K=e{E}_0/(\hslash \omega )$. Integrating from the specified initial condition $\mathbf{k}(0)=0$ gives, with the first field alone,With both components switched on,This is a circle centred at $(-K,0)$, not at the band minimum. Since $v_i=\hslash k_i/m_i^{\ast }$, integration givesThe switch-on drift in a circularly driven parabolic band accompanies unequal transverse scales: it is a stretched cycloid. In a frame moving with that drift, the oscillation is an ellipse with semiaxis ratio $m_y^{\ast }/m_x^{\ast }\gg 1$. A closed real-space ellipse would require different initial momentum. Indeed, after one period $x-x_0=-2\pi e{E}_0/(m_x^{\ast }{\omega }^2)\ne 0$, an explicit check against that interpretation.

Writing $\theta =\omega t$, the kinetic energy above the band minimum isFor $m_x^{\ast }<m_y^{\ast }$ it is largest at $\theta =\pi$, with value $2e^2{E}_0^2/(m_x^{\ast }{\omega }^2)$. Equating this to the optical-phonon energy gives the emission threshold estimateThis treats the optical branch as approximately dispersionless and assumes a momentum-allowed final state, a cold initial electron, and negligible phonon absorption. It is the amplitude of each Cartesian field component; the circular field has constant magnitude $E_0$. A steady periodic orbit centred at $\mathbf{k}=0$ would instead give twice this amplitude, but fails the stated switch-on condition.

The nearly-free electron model expands the Bloch state in plane waves,All basis functions have the same translation eigenvalues because $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. A weak periodic potential mixes them most strongly near Bragg degeneracies and opens band gaps. Delocalized $s,p$ conduction states of simple metals, such as sodium and aluminium, are typical applications; the approximation need not describe tightly bound core states.

The tight-binding model uses Bloch sums of localized atomic-like or Wannier orbitals,Including ${\mathbf{\tau }}_{\alpha }$ in the phase is an equivalent basis convention. Overlap and hopping between these localized orbitals create dispersive bands. Narrow transition-metal $d$ bands and the bonding/antibonding bands of covalent solids are natural examples; graphene's $p_z$ bands are another. The model can be accurate beyond weak hopping when enough orbitals and hopping integrals are retained.