A crystal potential is periodic

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · 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

A crystal potential is periodic,

U(\mathbf r+\mathbf R)=U(\mathbf r)

for every Bravais lattice vector

\mathbf R

, so it has a Fourier expansion over the reciprocal lattice only,

U(\mathbf r) = \sum_{\mathbf G} U_{\mathbf G}\,e^{i\mathbf G\cdot\mathbf r}

, with

U_{\mathbf G}^{*}=U_{-\mathbf G}

because

U(\mathbf r)

is real. By Bloch's theorem a stationary state of wavevector

\mathbf k

can be expanded in plane waves carrying wavevectors

\mathbf k-\mathbf G

for every reciprocal lattice vector

\mathbf G

\psi_{\mathbf k}(\mathbf r) = \sum_{\mathbf G} c_{\mathbf k-\mathbf G}\, e^{i(\mathbf k-\mathbf G)\cdot\mathbf r}.

((a)) Substitute both expansions into

H\psi_{\mathbf k}=\varepsilon(\mathbf k)\psi_{\mathbf k}

with

H=-\dfrac{\hbar^2}{2m}\nabla^2+U(\mathbf r)

, and simplify the kinetic term acting on each plane wave. ((b)) Multiply the resulting equation by

e^{-i(\mathbf k-\mathbf G')\cdot \mathbf r}

for a fixed reciprocal lattice vector

\mathbf G'

and integrate over the crystal volume

V

, using

\frac{1}{V}\int_V e^{i(\mathbf G-\mathbf G')\cdot\mathbf r}\,d^3r=\delta_{\mathbf G\mathbf G'}

, to obtain the central equation. ((c)) State the two places where approximation could enter a practical calculation with this equation, given that its derivation used none.

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