Semiconductors — Question 6

Question 6

Carrier creation and mobility loss compete in the temperature coefficient

An intrinsic model has $n_i=C T^{3/2}e^{-E_g/(2k_BT)}$ and $\mu_n+\mu_p=M T^{-s}$, with constants C,M,Eg and s positive.

1. Derive the logarithmic slope of conductivity with respect to temperature.

2. For s=2, find where the model predicts a stationary conductivity and determine whether it is a maximum or minimum.

Question 6 - Solution

Carrier creation and mobility loss compete in the temperature coefficient

(a) Method and calculation

$$\sigma=eCM T^{3/2-s}e^{-E_g/(2k_BT)}.$$$$\frac{d\ln\sigma}{d\ln T}=\frac32-s+\frac{E_g}{2k_BT}.$$

(b) Interpretation and further calculation

$$s=2:\quad-\tfrac12+\frac{E_g}{2k_BT_*}=0\Rightarrow T_*=E_g/k_B.$$$$T<T_*:\ d\sigma/dT>0;\quad T>T_*:\ d\sigma/dT<0.$$Thus the stationary point is a maximum if the model remains valid that far.

Physical check

A formal extrapolated extremum need not fall within the temperature range where the assumed laws apply.