The conductivity of a pure semiconductor is approximately proportional to T^ 3/2 e^ - E/2kT , where E represents the band gap. For germanium, the band gap is 0.74 eV at 4 K and 0.67 eV at 300 K . Determine the factor by which the conductivity of pure germanium increases when the temperature is elevated from 4 K to 300 K .

Step-by-step solution

The conductivity of a pure semiconductor is given by = C T^ 3/2 e^ - E/2kT . The ratio of conductivities at T₂ = 300 K and T₁ = 4 K is: ₂ ₁ = ( T₂ T₁ )^ 3/2 ( E₁ 2kT₁ - E₂ 2kT₂ ) Given T₁ = 4 K , E₁ = 0.74 eV and T₂ = 300 K , E₂ = 0.67 eV . Using Boltzmann's constant k 8.625 10⁻⁵ eV/K : E₁ 2kT₁ = 0.74 2 8.625 10⁻⁵ 4 1072.5 E₂ 2kT₂ = 0.67 2 8.625 10⁻⁵ 300 13.0 The difference in the exponent is 1072.5 - 13.0 = 1059.5 . The exponential term is e^ 1059.5 . Converting to base 10 using e 10^ 0.4343 : e^ 1059.5 = 10^ 1059