Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section B / B6

In the Bloch wave sum, $N$ is the number of sites, $R_j=ja$ the equilibrium position of site $j$, $\varphi (r-R_j)$ a localized atomic orbital there, and $k$ the crystal wavevector. The phase factor gives the translation character of the Bloch state. The factor $1/\sqrt{N}$ normalizes it when the site orbitals are treated as orthonormal; appreciable intersite overlap requires an overlap-dependent normalization.

For a tight-binding model, choose localized orbitals and calculate the on-site and intersite Hamiltonian matrix elements. Form Bloch sums for each orbital type and solve $H(k)c=E(k)S(k)c$ at each $k$, where $S(k)$ is the orbital-overlap matrix. In the orthogonal tight-binding approximation, $S=I$. The eigenvalues give the energy bands and the eigenvectors their orbital composition; the nonorthogonal nearest-neighbour tight-binding band illustrates how retained overlap modifies this procedure.

For one orthonormal s orbital per site, keep on-site energy ${ϵ}_s$ and real nearest-neighbor hopping integral $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$. The two neighbors contribute phases $e^{\pm ika}$, giving $E_s(k)={ϵ}_s+2t\cos ka$. The band-curvature definition $1/m^{\ast }=E^{\prime \prime }(k)/{\hslash }^2$ givesWith the usual bonding convention $t<0$ for these s orbitals, the center is a minimum with positive electron effective mass and the edge a maximum with negative electron effective mass. Hole masses near a filled-band maximum have the opposite sign.

For p orbitals aligned along the chain, the orbital phase and the sign of one-dimensional hopping reverse the usual center-versus-edge ordering. Choose every $p_x$ orbital with its positive lobe on the right. Equal neighboring Bloch coefficients then bring opposite-sign lobes together between sites, giving an antibonding state at $k=0$. Alternating coefficients at $k=\pi /a$ make those facing lobes have the same sign, producing the lower bonding state. Thus a nearest-neighbor model has $E_p(k)={ϵ}_p+2t_{\sigma }\cos ka$ with the usual $t_{\sigma }>0$, maximum at zero and minima at the zone edges. This uses a fixed common orbital-phase convention at all sites.