 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2/a10/a/ii/solution

The equal supplied masses give equal effective density of states. At the specified temperature,

N_{c} = N_{v} = 2 [\frac{2 π ( 0.3 m _{e} ) k _{B} ( 400 K )}{h ^{2}}]^{3/2} ≃ 6.35 \times 1 0^{24} m^{- 3} .

Using the same energy units in the exponential,

n_{i} = N_{c} e^{- E_{g} / (2 k_{B} T)} ≃ (6.35 \times 1 0^{24}) e^{- 0.67/ [2 (8.61733 \times 1 0^{- 5}) (400)]} ≃ 3.82 \times 1 0^{20} m^{- 3} .

For equal electron and hole mobilities, both carriers contribute to the intrinsic electrical conductivity:

μ_{e} = μ_{h} = \frac{σ}{2 e n _{i}} ≃ \frac{48}{2 ( 1.60218 \times 1 0 ^{- 19} ) ( 3.82 \times 1 0 ^{20} )} ≃ 0.392 m^{2} V^{- 1} s^{- 1} .

The factor two is essential. A dimensional check gives

\frac{S m ^{- 1}}{C m ^{- 3}} = \frac{A V ^{- 1} m ^{2}}{C} = m^{2} V^{- 1} s^{- 1},

using

A = C s^{- 1}

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