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.

In a direct optical transition, a photon promotes an electron between the valence band and conduction band at essentially the same crystal wavevector: optical photon momentum is small on the scale of the Brillouin zone. An indirect optical transition also needs a phonon or another momentum source. Near an indirect gap, phonon emission requires photon energy approximately $E_g+\hslash \Omega$, while phonon absorption permits $E_g-\hslash \Omega$ with a thermally occupied phonon.

Strong direct interband absorption needs both many allowed transitions and a nonzero optical matrix element. The joint density of states counts pairs with $E_c(\mathbf{k})-E_v(\mathbf{k})=\hslash \omega$; nearly parallel bands and critical points of this energy difference can enhance it. Dipole selection rules and polarization weight that count, so a large joint density of states alone does not guarantee a strong absorption peak.

The supplied infrared spectrum lies below the interband threshold and instead probes free-carrier response. In the Drude model, $ϵ(\omega )={ϵ}_{\infty }-{\omega }_p^2/[\omega (\omega +i/\tau )]$, with ${\omega }_p^2=n{e}^2/({ϵ}_0{m}^{\ast })$. At long wavelengths the mobile electrons screen the field and the reflectivity is high. As wavelength decreases through the screened plasma edge, the real dielectric function becomes positive and propagating light can enter. The reflectivity reaches its minimum near index matching, $ϵ\simeq 1$, then rises toward the bound-electron background reflectance.

For background ${ϵ}_{\infty }=16$, its refractive index is four and the normal-incidence Fresnel equations give $R_{\infty }=[(4-1)/(4+1){]}^2=9/25=0.36$. The short-wavelength side of the figure is approaching a reflectance of this order, consistent with that background; its shortest plotted wavelength still has a carrier correction. Here “infinity” means high frequency relative to the free-carrier scale while remaining below the remote interband resonances.

The Drude-Lorentz reflectivity minimum is near $ϵ=1$, not the plasma-edge zero $ϵ=0$. Reading the deep minimum as ${\lambda }_{\min }\simeq 22\mu \mathrm{m}$ and neglecting damping there gives ${\omega }_p^2=({ϵ}_{\infty }-1)(2\pi c/{\lambda }_{\min }{)}^2$. The carrier effective mass from a reflectivity minimum is thereforeThe graph supports only a rough read-off: shifting the minimum by $0.5\mu \mathrm{m}$ changes this mass by about five percent. The deep narrow minimum supports the weak-loss assumption $\omega \tau \gg 1$ in this region, although finite scattering explains why the long-wavelength reflectivity is below unity. The estimate also assumes one dominant carrier species, a nonmagnetic bulk sample and a nearly constant background dielectric function. If the plasma-edge zero is used instead, its denominator is ${ϵ}_{\infty }$ and its wavelength is slightly longer; the two wavelengths must not be interchanged.