Doped Semiconductors — Question 7

Question 7

Impurity scattering makes conductivity gains saturate

For a fully ionized donor-dominated semiconductor, use $n=N_D$ and the empirical mobility law $\mu(N_D)=\mu_0/(1+N_D/N_*)$. Neglect holes.

1. Derive conductivity and its high-density limit within this model.

2. Find the donor density at which conductivity reaches 90 percent of that limit, and the mobility there.

Question 7 - Solution

Impurity scattering makes conductivity gains saturate

(a) Method and calculation

$$\sigma=eN_D\mu_0/(1+N_D/N_*)=e\mu_0N_*\frac{x}{1+x},\quad x=N_D/N_*.$$$$\boxed{\sigma_{\infty}=e\mu_0N_*}.$$

(b) Interpretation and further calculation

$$x/(1+x)=.90\Rightarrow\boxed{x=9,\quad N_D=9N_*}.$$$$\boxed{\mu(9N_*)=\mu_0/10,\quad\sigma=.9e\mu_0N_*}.$$

Physical check

The asymptote is a property of the stated mobility law, not a universal bound. Increasing dopant density raises carrier count while also increasing scattering.