Near the fundamental gap of a direct-gap semiconductor the conduction and valence…

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 the fundamental gap of a direct-gap semiconductor the conduction and valence bands are well approximated by parabolas centred at

\mathbf{k}=0

E_c(\mathbf{k})=E_g+\hbar^2k^2/2m_e^*

and

E_v(\mathbf{k})=-\hbar^2k^2/2m_h^*

. The interband contribution to the dielectric function derived from Fermi's golden rule is

\varepsilon_2(\omega)=\frac{\pi e^2|p_{cv}|^2}{\varepsilon_0 m_e^2\omega^2}\,J_{cv}(\hbar\omega),

where

|p_{cv}|^2

is the squared interband momentum matrix element, taken constant over the small energy range considered here, and

J_{cv}(\hbar\omega)

is the joint density of states per unit volume, counting both spin states. ((a)) By writing

E_c(\mathbf{k})-E_v(\mathbf{k})=E_g+\hbar^2k^2/2\mu_X

with

1/\mu_X=1/m_e^*+1/m_h^*

, and counting the

\mathbf{k}

-states inside a sphere of radius

k

, show that

J_{cv}(\hbar\omega)=\frac{1}{2\pi^2}\left(\frac{2\mu_X}{\hbar^2}\right)^{3/2}(\hbar\omega-E_g)^{1/2},\qquad \hbar\omega\ge E_g.

((b)) Hence show that near threshold

\varepsilon_2(\omega)\propto \mu_X^{3/2}(\hbar\omega-E_g)^{1/2}/\omega^2

. ((c)) CdTe is direct-gap, with

E_g=1.50\,\mathrm{eV}

300\,\mathrm{K}

m_e^*=0.11\,m_e

and

m_h^*=0.35\,m_e

. Compute the reduced mass

\mu_X

in units of

m_e

, and, treating

|p_{cv}|^2

as constant over this range, compute the ratio

\varepsilon_2(1.53\,\mathrm{eV})/\varepsilon_2(1.60\,\mathrm{eV})

Related theory

Quantum Mechanics

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