Band bending and Einstein relationship

Band bending

Band bending is the change of the energy band edges ($E_c$, $E_v$) with position inside a semiconductor. It happens whenever there is an electric field or a space charge, for example in a non-uniformly doped sample, at a pn junction, or at a metal-semiconductor contact or surface.

How it occurs:

1. In thermal equilibrium the Fermi level is constant throughout the material.
2. If doping varies with position (say $N_d$ falls with $x$), the gap $E_c - E_F$ must be smaller where doping is high and larger where it is low, because $n = N_c\exp[-(E_c - E_F)/kT]$.
3. Electrons diffuse from high to low concentration, leaving positive donor ions. This space charge sets up an internal field $E$ and a potential $V(x)$.
4. The electron potential energy is $-eV(x)$, so all bands shift by $-eV(x)$: they bend. The slope gives the field:

$$E = \frac{1}{e}\frac{dE_c}{dx}$$

 high Nd                    low Nd
                   _________  Ec
          ________/
 Ec  ____/
 EF ------------------------  (flat)
                   _________  Ev
          ________/
 Ev  ____/
      ---> e- diffuse
      <--- e- drift (built-in E)

Electrons roll "downhill" in energy; holes float "uphill". At a pn junction the total bending equals $eV_0$.

Einstein relation

In the sample above, in equilibrium, the electron current is zero:

$$J_e = en\mu_eE + eD_e\frac{dn}{dx} = 0 \quad (1)$$

With constant $E_F$:

$$\frac{dn}{dx} = -\frac{n}{kT}\frac{dE_c}{dx} = -\frac{neE}{kT} \quad (2)$$

Substituting (2) in (1):

$$\begin{aligned} en\mu_eE - eD_e\frac{neE}{kT} &= 0 \\ \frac{D_e}{\mu_e} &= \frac{kT}{e} \end{aligned}$$

Similarly for holes, $D_h/\mu_h = kT/e$. This is the Einstein relation. At 300 K, $kT/e = 25.9$ mV, so for Si electrons ($\mu_e = 1350$ cm$^2$/Vs), $D_e \approx 35$ cm$^2$/s.

It shows that drift and diffusion are linked: both depend on the same random thermal motion and scattering of carriers.