Semiconductors — Question 4

Question 4

Increasing doping can eventually reduce conductivity in a supplied mobility model

In a specified high-doping model, all donors contribute one electron and holes are negligible: $n=N_D$. Mobility is $\mu(N_D)=\mu_0/[1+(N_D/N_*)^2]$.

1. Find conductivity and the donor density that maximizes it.

2. Find the maximum conductivity and compare the low- and high-doping limits.

Question 4 - Solution

Increasing doping can eventually reduce conductivity in a supplied mobility model

(a) Method and calculation

$$\sigma=e\mu_0N_*\frac{x}{1+x^2},\quad x=N_D/N_*.$$$$\frac{d\sigma}{dx}=e\mu_0N_*\frac{1-x^2}{(1+x^2)^2}=0\Rightarrow N_D=N_*.$$

(b) Interpretation and further calculation

$$\sigma_{\max}=e\mu_0N_*/2.$$$$x\ll1:\ \sigma\simeq e\mu_0N_D;\qquad x\gg1:\ \sigma\simeq e\mu_0N_*^2/N_D.$$

Physical check

The optimum follows the supplied mobility law; it is not a universal claim about every doped semiconductor.