The conductivity of a pure semiconductor is roughly proportional to $T^{3 / 2} e^{- Δ E / 2 K T}$ where $Δ E$ is the band gap. The band gap for germanium is 0.74 eV at 4 K and 0.67 eV at 300 K. By what factor does the conductivity of pure germanium increase as the temperature is raised from 4 K to 300 K ?

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Solution

Here $δ = T^{3 / 2} e^{- Δ E / 2 K T}$

$∴ \frac{σ_{2}}{σ_{1}} = {(\frac{T_{2}}{T_{1}})}^{3 / 2} \frac{e^{- Δ E_{2} / 2 K T_{2}}}{e^{- Δ E_{1} / 2 K T_{1}}}$

$= {(\frac{300}{4})}^{3 / 2} \frac{e^{- 0.67 / (2 \times 8.62 \times 10^{- 5} \times 300)}}{e^{- 0.74 / (2 \times 8.62 \times 10^{- 5} \times 4)}}$

$\approx 10^{463}$

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The conductivity of an intrinsic semiconductor depends on temperature as $σ = σ_{0} e^{- Δ E / 2 K T},$ where $σ_{0}$ is a constant. Find the temperature at which the conductivity of an intrinsic germanium semiconductor will be double of its value at T = 300 K. Assume that the gap for germanium is 0.650 eV and remains constant as the temperature is increased.

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(Take k = 8.62 \times 10^{- 5} {eVK}^{- 1}, e^{0.82} = 2.27)

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The band gap for silicon is 1.1 eV. (a) Find the ratio of the band gap to kT for silicon at room temperature 300 K. (b) At what temperature does this ratio become one tenth of the value at 300 K ? (Silicon will not retain its structure at these high temperatures.)

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The conductivity of a pure semiconductor is roughly proportional to T3/2 e−ΔE/2KT where ΔE is the band gap. The band gap for germanium is 0.74 eV at 4 K and 0.67 eV at 300 K. By what factor does the conductivity of pure germanium increase as the temperature is raised from 4 K to 300 K ?

Here δ=T3/2 e−ΔE/2KT ∴ σ2σ1=(T2T1)3/2 e−ΔE2/2KT2e−ΔE1/2KT1 =(3004)3/2 e−0.67/(2×8.62×10−5×300)e−0.74/(2×8.62×10−5×4) ≈ 10463