Doped Semiconductors — Question 9

Question 9

A graded donor profile produces an equilibrium internal field

A slowly graded n-type region is locally neutral and fully ionized, with $n(x)\simeq N_D(x)=N_0e^{x/L}$. Electron current is $J_n=en\mu_nE+eD_n\,dn/dx$, and $D_n/\mu_n=V_T$.

1. Find the equilibrium field and electrostatic potential difference between x=0 and x=L.

2. For $V_T=.0250\,\mathrm V$ and $L=20.0\,\mu\mathrm m$, evaluate the field and electron band-edge change $E_c(L)-E_c(0)$.

Question 9 - Solution

A graded donor profile produces an equilibrium internal field

(a) Method and calculation

$$J_n=en(\mu_nE+D_n/L)=0\Rightarrow\boxed{E=-V_T/L}.$$$$\phi(L)-\phi(0)=-\int_0^LEdx=\boxed{V_T}.$$

(b) Interpretation and further calculation

$$\boxed{E=-1.25\times10^3\,\mathrm{V/m}}.$$$$E_c=-e\phi+\text{constant}\Rightarrow\boxed{\Delta E_c=-eV_T=-.0250\,\mathrm{eV}}.$$

Physical check

The field offsets diffusion, allowing a spatially constant equilibrium Fermi level. Local neutrality is approximate; a small space charge supports the field near boundaries.