Insulators — Question 8

Question 8

Subgap defect activation can dominate an insulating crystal

An insulator has two independent conductivity channels: $\sigma_b=C_be^{-E_b/(k_BT)}$ and $\sigma_d=C_de^{-E_d/(k_BT)}$. Let $E_b=1.00\,\mathrm{eV}$, $E_d=.20\,\mathrm{eV}$, $C_b/C_d=10^8$, and $k_B=8.617\times10^{-5}\,\mathrm{eV/K}$.

1. Derive an expression for the temperature where the conductivities are equal.

2. Evaluate the crossover temperature and identify the dominant channel below it.

Question 8 - Solution

Subgap defect activation can dominate an insulating crystal

(a) Method and calculation

$$\sigma_b/\sigma_d=(C_b/C_d)e^{-(E_b-E_d)/(k_BT)}.$$$$\sigma_b=\sigma_d\Rightarrow\boxed{T_\times=(E_b-E_d)/[k_B\ln(C_b/C_d)]}.$$

(b) Interpretation and further calculation

$$\boxed{T_\times=.80/[k_B\ln10^8]=504.0\,\mathrm K}.$$$$T<T_\times\Rightarrow\ln(\sigma_b/\sigma_d)<0\Rightarrow\boxed{\text{defect channel dominates}}.$$

Physical check

The small defect prefactor is overcome by its smaller activation barrier at low temperature. An apparent activation energy need not equal the intrinsic band-gap scale.