Band bending in semiconductors

Band bending is the curving of the energy bands ($E_c$, $E_v$, $E_{Fi}$) with position inside a semiconductor. It happens wherever there is an electric field, i.e. wherever the electrostatic potential $V(x)$ changes with position.

Why the bands bend

The energy of an electron at a point is its band energy plus its potential energy $-eV(x)$:

$$E_c(x) = E_{c0} - eV(x)$$

The field is $\mathcal{E} = -dV/dx$, so

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

• Where $\mathcal{E} = 0$ the bands are flat.
• Where a field exists, all three levels shift by the same amount, so the band gap $E_g$ stays constant but the bands tilt or curve.
• An electron "rolls down" the conduction band and a hole "floats up" the valence band.

Common cases

1. Applied voltage on a uniform bar: the field is uniform, so the bands tilt in a straight line. The Fermi level is no longer flat; current flows.
2. pn junction (equilibrium): electrons diffuse to the p side and holes to the n side, leaving fixed ion charges. These create a built-in field. Because the Fermi level must be the same throughout in equilibrium, the bands bend in the depletion region by $eV_0$.
3. Non-uniform doping, metal–semiconductor contact, surface charges: each creates a local space charge and field, so the bands bend near that region.

 p side            n side
 Ec ----\
         \__________ Ec
 EF ------------------- EF (flat)
 Ev ----\
         \__________ Ev
       |<-W->|  depletion region
   bending = eV0

Key rule

In equilibrium the Fermi level is flat; any difference in doping is accommodated by bending of $E_c$ and $E_v$. The amount of bending equals $e$ times the potential difference across the region.