Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 2 / A10 / b / i / Solution

Near the zone centre, $\cos (k_{i}a)\simeq 1-k_i^2{a}^2/2$. The tight-binding model becomesOnly $|\mathbf{k}{|}^2$ appears at quadratic order, so the band effective mass tensor is isotropic. Higher-order lattice terms need not be isotropic.

For the conduction band, ${\beta }_c<0$, so the curvature is positive andHere $a=5.42\times 10^{-10}$ m and both beta parameters must be converted from eV to joules. The band edges at the zone centre areso the model has a direct gap $\boxed{E_g=1.240\mathrm{eV}}$. The mass values describe curvature at these extrema; they are not constant masses throughout the full tight-binding bands.