Induced Electric Field

Consider a semiconductor that is non-uniformly doped with donor impurity atoms. The semiconductor is in thermal equilibrium, the Fermi energy level is through the crystal so the energy-band diagram may qualitatively look like that shown in Figure 1.1. The doping concentration decreases as x increases in this case. There will be a diffusion of majority carrier electrons from the region of high concentration to the region of low concentration, which is in the +x direction. The flow of negative electrons leaves behind positively charged donor ions. The separation of positive and negative charge induces an electric field that is in a direction to oppose the diffusion process. When equilibrium is reached, the mobile carrier concentration is not exactly equal to the fixed impurity concentration and the induced electric field prevents any further separation of charge. In most cases of interest, the space charge induced by this diffusion process is a small fraction of the impurity concentration thus the mobile carrier concentration is not too different from the impurity dopant density.

The electric potential ϕ is related to electron potential energy by the charge (- e), so we can write

(1)

The electric field for the one-dimensional situation is defined as

(2)

Figure 1.1 Energy-band diagram for a semiconductor in thermal equilibrium with a non-uniform donor impurity concentration.

If the intrinsic Fermi level changes as a function of distance through a semiconductor in thermal equilibrium, an electric field exists in the semiconductor.

If we assume a quasi-neutrality condition in which the electron concentration is almost equal to the donor impurity concentration, then we can still write

(3)

Solving for E_F - E_Fi , we obtain

(4)

The Fermi level is constant for thermal equilibrium so when we take the derivative with respect to x we obtain

(5)

The electric field can then be written, combining Equations (5) and (2), as

(6)

Since we have an electric field, there will be a potential difference through the semiconductor due to the non-uniform doping.