Doped Semiconductors — Question 3

Question 3

A minute substitution fraction produces many mobile carriers

A crystal has host-site density $N_s=5.00\times10^{28}\,\mathrm{m^{-3}}$. One donor replaces one host atom per $10^7$ sites. Donors are fully ionized and intrinsic carriers are negligible.

1. Find donor density and mean donor spacing $\ell_D=N_D^{-1/3}$.

2. For a cube of side $10.0\,\mu\mathrm m$, find the mean donor number and its relative Poisson fluctuation $1/\sqrt{\bar N}$.

Question 3 - Solution

A minute substitution fraction produces many mobile carriers

(a) Method and calculation

$$\boxed{N_D=N_s/10^7=5.00\times10^{21}\,\mathrm{m^{-3}}}.$$$$\boxed{\ell_D=(5.00\times10^{21})^{-1/3}=58.48\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{\bar N=N_D(10^{-5})^3=5.00\times10^6}.$$$$\boxed{\sigma_N/\bar N=1/\sqrt{\bar N}=4.472\times10^{-4}}.$$

Physical check

The spacing is a density scale rather than a periodic donor lattice. Poisson fluctuations assume independent random substitution.