Germanium crystallises in the diamond structure: a face-centred cubic Bravais lattice

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

Solid State Physics, Ch Tight binding and the band structures of real materials

Problem

Germanium crystallises in the diamond structure: a face-centred cubic Bravais lattice, conventional cubic lattice constant

a=5.658\times10^{-10}\,\mathrm{m}

, with two atoms per primitive cell. For a face-centred cubic lattice the shortest reciprocal lattice vectors are of

\{111\}

type, with magnitude

|\mathbf G|=2\pi\sqrt3/a

. A pseudopotential calculation gives the Fourier component of the crystal potential at this

\mathbf G

|U_{\mathbf G}|=3.1\,\mathrm{eV}

. ((a)) Find the free-electron kinetic energy at the corresponding Brillouin zone face,

k=|\mathbf G|/2

. ((b)) Form the ratio

|U_{\mathbf G}|

to this zone-face energy, and use it, alongside the equivalent ratio for aluminium (

0.083

), to decide whether germanium is better started from free electrons or from atomic orbitals. State what a large ratio means for the nearly-free-electron method.

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