Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 6 / a

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- Solution a

Solution

Count wavevectors in a large volume with periodic boundary conditions. One spin-degenerate spherical band supplies

\frac{N ( k )}{V} = \frac{2}{( 2 π ) ^{3}} \frac{4 π k ^{3}}{3} = \frac{k ^{3}}{3 π ^{2}}, \frac{1}{V} \frac{d N}{d k} = \frac{k ^{2}}{π ^{2}} .

For the conduction band,

k = 2 m_{e}^{*} (ϵ - ϵ_{c}) /ℏ

and

d k / d ϵ = m_{e}^{*} / (ℏ^{2} k)

. Transforming the state count to energy gives

g_{e} (ϵ) = \frac{1}{2 π ^{2}} (\frac{2 m _{e}^{*}}{ℏ ^{2}})^{3/2} ϵ - ϵ_{c}, ϵ \geq ϵ_{c} .

For the valence band use

∣ d k / d ϵ ∣ = m_{h}^{*} / (ℏ^{2} k)

and

k = 2 m_{h}^{*} (ϵ_{v} - ϵ) /ℏ

, obtaining

g_{h} (ϵ) = \frac{1}{2 π ^{2}} (\frac{2 m _{h}^{*}}{ℏ ^{2}})^{3/2} ϵ_{v} - ϵ, ϵ \leq ϵ_{v} .

Each density of states is zero beyond its indicated band edge. The factor two counts spin; an extra valley degeneracy was not assumed in these stipulated isotropic bands.

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