Semiconductors — Question 4

Question 4

The density-of-states prefactor modifies thermal activation

Over a specified range take $n_i(T)=C T^{3/2}e^{-E_g/(2k_BT)}$, with constant gap $E_g=1.00\,\mathrm{eV}$ and $k_B=8.617\times10^{-5}\,\mathrm{eV/K}$.

1. Derive the ratio $n_i(400\,\mathrm K)/n_i(300\,\mathrm K)$.

2. If both mobilities scale as $T^{-3/2}$, find the corresponding intrinsic conductivity ratio.

Question 4 - Solution

The density-of-states prefactor modifies thermal activation

(a) Method and calculation

$$\frac{n_i(T_2)}{n_i(T_1)}=\left(\frac{T_2}{T_1}\right)^{3/2}\exp\left[\frac{E_g}{2k_B}\left(\frac1{T_1}-\frac1{T_2}\right)\right].$$$$\boxed{n_i(400)/n_i(300)=(4/3)^{3/2}e^{4.8354}=193.8}.$$

(b) Interpretation and further calculation

$$\sigma_i=en_i(\mu_n+\mu_p)\propto e^{-E_g/(2k_BT)}.$$$$\boxed{\sigma_i(400)/\sigma_i(300)=e^{4.8354}=125.9}.$$

Physical check

The increasing carrier density outweighs the falling mobility in this model. Gap variation with temperature would require an additional correction.