Semiconductors — Question 8

Question 8

Equilibrium drift-diffusion balance yields the Einstein relation

Nondegenerate electrons have $n(x)\propto e^{e\phi(x)/(k_BT)}$, field $E=-d\phi/dx$, and conventional current $J_n=en\mu_nE+eD_n\,dn/dx$.

1. Use zero equilibrium current to derive the diffusion-mobility relation.

2. At thermal voltage $V_T=k_BT/e=.0250\,\mathrm V$ and $\mu_n=.120\,\mathrm{m^2/(V\,s)}$, evaluate $D_n$ and the field that balances a density profile $n=n_0e^{-x/L}$ with $L=10.0\,\mu\mathrm m$.

Question 8 - Solution

Equilibrium drift-diffusion balance yields the Einstein relation

(a) Method and calculation

$$\frac{dn}{dx}=\frac{en}{k_BT}\frac{d\phi}{dx}=-\frac{enE}{k_BT}.$$$$J_n=enE\left(\mu_n-\frac{eD_n}{k_BT}\right)=0\Rightarrow\boxed{D_n=\mu_nV_T}.$$

(b) Interpretation and further calculation

$$\boxed{D_n=.00300\,\mathrm{m^2/s}}.$$$$0=en(\mu_nE-D_n/L)\Rightarrow\boxed{E=V_T/L=2.50\times10^3\,\mathrm{V/m}}.$$

Physical check

Electron diffusion carries conventional current toward increasing electron density. The compensating drift current is toward +x for this decaying density profile.