Consider a simple cubic crystal of lattice constant

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure

Solid State Physics, Ch Bloch's theorem and the nearly-free-electron method

Problem

Consider a simple cubic crystal of lattice constant aa, whose reciprocal lattice is simple cubic with primitive vectors of length 2π/a2\pi/a along each cubic axis. The crystal potential has only one pair of nonzero Fourier coefficients, UGU_{\mathbf G} and U−GU_{-\mathbf G}, for G=(2π/a)(1,0,0)\mathbf G=(2\pi/a)(1,0,0); take UGU_{\mathbf G} real (the crystal has an inversion centre at the origin), and every other Fourier coefficient zero. ((a)) Show that at the zone-boundary wavevector k0=(π/a,0,0)\mathbf k_0=(\pi/a,0,0), the two plane waves labelled k0\mathbf k_0 (i.e. G′=0\mathbf G'=\mathbf 0) and k0−G\mathbf k_0-\mathbf G are exactly degenerate in the free-electron energy ℏ2∣k0−G′∣2/2m\hbar^2|\mathbf k_0-\mathbf G'|^2/2m, and explain why truncating the central equation to just these two waves is a good approximation for this potential, though not an exact one: name the next plane waves the potential couples to them, and estimate from second-order perturbation theory the shift those waves produce in each level. ((b)) Solve the resulting 2×22\times2 secular determinant for the two energies ε±\varepsilon_\pm and give the two normalised real-space eigenstates. ((c)) Evaluate ε0=ℏ2(π/a)2/2m\varepsilon^0=\hbar^2(\pi/a)^2/2m, ε+\varepsilon_+ and ε−\varepsilon_- numerically for a=3.00 A˚a=3.00\,\mathrm{\AA} and ∣UG∣=1.10 eV|U_{\mathbf G}|=1.10\,\mathrm{eV}, together with the size of the shift estimated in (a).

Related theory

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