Many zinc-blende direct-gap semiconductors

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

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

Problem

Many zinc-blende direct-gap semiconductors, including GaAs, have two valence bands, heavy-hole and light-hole, that are degenerate at

\mathbf{k}=0

and each connect to the same conduction band by a direct, momentum-conserving transition at the same gap

E_g

. Each pair of bands contributes its own joint density of states of the form derived for two parabolic bands,

J_{cv}^{(i)}(\hbar\omega)=\frac{1}{2\pi^2}\left(\frac{2\mu_i}{\hbar^2}\right)^{3/2}(\hbar\omega-E_g)^{1/2},\qquad i=\mathrm{hh,\ lh},

with

1/\mu_i=1/m_e^*+1/m_i^*

, and the total interband

\varepsilon_2(\omega)

is the sum of the two independent golden-rule contributions, since the two hole bands do not interfere. ((a)) Show that the total

\varepsilon_2(\omega)

near threshold is still a single square-root edge, with a combined weight proportional to

\mu_{hh}^{3/2}+\mu_{lh}^{3/2}

; explain in a sentence why having two valence bands sharing the same

E_g

does not create two separate absorption edges, unlike the two thresholds produced by an assisting phonon at an indirect gap. ((b)) For GaAs at

300\,\mathrm{K}

m_e^*=0.067\,m_e

m_{hh}^*=0.45\,m_e

and

m_{lh}^*=0.082\,m_e

. Compute

\mu_{hh}

and

\mu_{lh}

in units of

m_e

, and hence the ratio of the light-hole to the heavy-hole contribution to

\varepsilon_2

near the edge. Take the polarisation-averaged

|p_{cv}|^2

to be the same for the heavy-hole and light-hole transitions.

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