Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 2 / b

A linear combination of atomic orbitals expands a crystal wavefunction in localized orbitals on all sites,Here $\alpha$ labels the basis site, $\eta$ the retained atomic orbitals and $\mathbf{R}=m{\mathbf{a}}_1+n{\mathbf{a}}_2$. For the two-band model retain one relevant orbital on each B or N atom, usually the out-of-plane $p_z$ orbital, and neglect other orbital manifolds. Define normalized Bloch sums for $N_c$ cells,Then $\Psi ={\alpha }_B{\Phi }_B+{\alpha }_N{\Phi }_N$. Bloch periodicity puts the same two coefficients in every cell. We adopt the stated orthonormality, neglect longer-range hopping, and approximate the diagonal matrix elements by the supplied atomic energies ${ϵ}_B,{ϵ}_N$. More generally the crystal environment shifts these on-site energies, so equality with isolated-atom expectation values is an additional tight-binding approximation.

For equal real nearest-neighbour transfer integral $t_{\mathrm{nn}}$, this phase convention givesThe secular determinant is $({ϵ}_B-E)({ϵ}_N-E)-t_{\mathrm{nn}}^2|\gamma {|}^2=0$. ThusSince $({ϵ}_B-{ϵ}_N{)}^2/4=({ϵ}_B+{ϵ}_N{)}^2/4-{ϵ}_B{ϵ}_N$, this is the requested algebraic form. A hopping-sign convention or rephasing of the Bloch sums changes off-diagonal phases but not these eigenvalues. Explicitly,