Near an indirect gap

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

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

Problem

Near an indirect gap, absorption of a photon of energy ℏω\hbar\omega close to EgE_g requires the simultaneous absorption or emission of a phonon of energy ℏωph\hbar\omega_{ph} to supply the crystal momentum. Combining the second-order golden-rule rate through one intermediate state, a standard result of second-order perturbation theory quoted without re-derivation, with the two parabolic bands' densities of states, and using ∫0ΔxΔ−x dx=π8Δ2,\int_0^\Delta \sqrt{x}\sqrt{\Delta-x}\,dx=\frac{\pi}{8}\Delta^2, the absorption coefficient near an indirect edge assisted by a single phonon branch is α(ℏω)=A[(nph+1)(ℏω−Eg−ℏωph)2 Θ(ℏω−Eg−ℏωph)+nph(ℏω−Eg+ℏωph)2 Θ(ℏω−Eg+ℏωph)],\begin{aligned} \alpha(\hbar\omega)=A\Big[&(n_{ph}+1)(\hbar\omega-E_g-\hbar\omega_{ph})^2\,\Theta(\hbar\omega-E_g-\hbar\omega_{ph}) \\ &+ n_{ph}(\hbar\omega-E_g+\hbar\omega_{ph})^2\,\Theta(\hbar\omega-E_g+\hbar\omega_{ph})\Big], \end{aligned} where AA is a constant, Θ\Theta is the unit step function, and nph=[exp⁡(ℏωph/kBT)−1]−1n_{ph}=\left[\exp(\hbar\omega_{ph}/k_BT)-1\right]^{-1} is the equilibrium occupation of the assisting phonon mode. ((a)) State in one or two sentences why the phonon-emission term needs no minimum phonon population to proceed while the phonon-absorption term does, and why each contributes a squared rather than a square-root threshold. ((b)) Show that the ratio of the two branches' statistical weights, (nph+1)/nph(n_{ph}+1)/n_{ph}, is exactly exp⁡(ℏωph/kBT)\exp(\hbar\omega_{ph}/k_BT). ((c)) A silicon crystal at an unknown temperature gives a plot of α\sqrt\alpha against ℏω\hbar\omega whose lower straight section extrapolates to zero at 1.1066 eV1.1066\,\mathrm{eV}. Subtracting that section's extrapolation from the data above the upper threshold leaves the phonon-emission branch alone, whose α\sqrt{\alpha} is also a straight line; it extrapolates to zero at 1.1434 eV1.1434\,\mathrm{eV} and its slope is 1.501.50 times that of the lower section. Find EgE_g, the assisting phonon's energy ℏωph\hbar\omega_{ph} and the crystal's temperature, using kB=8.617×10−5 eV K−1k_B=8.617\times10^{-5}\,\mathrm{eV\,K^{-1}}.

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