 Solution

At cryogenic temperatures, the intrinsic carrier concentration is exponentially small, while a low phonon population permits relatively sharp band-edge and exciton spectral features. Near room temperature, thermal electron–hole generation increases strongly:This generally raises intrinsic conductivity, although increased phonon scattering tends to lower the individual carrier mobilities. With a gap of $1.24$ eV, the material remains far from a degenerate metal at room temperature.

For optical structure, compare $k_{\mathrm{B}}T\simeq 25.9$ meV at 300 K with the exciton binding energies. The $n=2$ binding of $9.56$ meV is substantially smaller, so excited bound states are readily thermally disrupted and broadened. The ground binding of $38.24$ meV is larger but only about $1.48k_{\mathrm{B}}T$: the ground exciton need not disappear completely, yet strong thermal ionization and broadening can weaken its sharp line. A comparison with $k_{\mathrm{B}}T$ is a useful scale argument, not an exact bound-pair fraction independent of density or lifetime.

Thermal lattice expansion and electron–phonon interactions can also shift a real material's gap and optical edge. The given fixed tight-binding parameters contain no temperature dependence, so they do not determine the magnitude or even a universal sign of that extra gap shift. Within the stated model, the dependable trends are more thermally generated carriers and less distinct excitonic structure on warming.