The square root of the absorption coefficient,

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Dielectric and optical properties of solids · Dielectric and optical properties of solids

Solid State Physics, Ch Optical properties: interband absorption, excitons and luminescence

Problem

The square root of the absorption coefficient, α\sqrt{\alpha}, of an indirect-gap crystal is measured near its threshold and forms two straight lines when plotted against photon energy ℏω\hbar\omega, consistent with α∝(ℏω−Eg∓ℏωph)\sqrt{\alpha}\propto(\hbar\omega-E_g\mp\hbar\omega_{\rm ph}). Two points on the lower-energy line are (ℏω,α)=(0.640 eV, 2.0 (cm−1)1/2)(\hbar\omega,\sqrt{\alpha})=(0.640\,\mathrm{eV},\,2.0\,\mathrm{(cm^{-1})^{1/2}}) and (0.660 eV, 6.0 (cm−1)1/2)(0.660\,\mathrm{eV},\,6.0\,\mathrm{(cm^{-1})^{1/2}}). Two points on the higher-energy line are (0.700 eV, 2.0 (cm−1)1/2)(0.700\,\mathrm{eV},\,2.0\,\mathrm{(cm^{-1})^{1/2}}) and (0.720 eV, 6.0 (cm−1)1/2)(0.720\,\mathrm{eV},\,6.0\,\mathrm{(cm^{-1})^{1/2}}). ((a)) Find the photon-energy intercept of each line. ((b)) Find the indirect gap EgE_g and the energy of the assisting phonon ℏωph\hbar\omega_{\rm ph}. ((c)) State which of the two thresholds involves absorbing a phonon and which involves emitting one, and predict which threshold disappears on cooling to low temperature.

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