**Branch :**Computer Science and Engineering

**Subject :**Fundamental of Electronic Devices

## The Einstein Relation

In discussing the motion of carriers in an electric field, we should indicate the influence of the field on the energies of electrons in the band diagrams.

Assuming an electric field ξ(x) in the x-direction, we can draw the energy bands as in Fig. below, to include the change in potential energy of electrons in the field. Since electrons drift in a direction opposite to the field, we expect the potential energy for electrons to increase in the direction of the field, as in Fig. The electrostatic potential v(x) varies in the opposite direction, since it is defined in terms of positive charges and is therefore related to the electron potential energy E(x) displayed in the figure by

From the definition of electric field,

we can relate ξ(x) to the electron potential energy in the band diagram by choosing some reference in the band for the electrostatic potential. We are interested only in the spatial variation v(x) for Eq above. Choosing E_{i} as a convenient reference, we can relate the electric field to this reference by

Therefore, the variation of band energies with ξ(x) as drawn in Fig. is correct. The direction of the slope in the bands relative to ξ is simple to remember: Since the diagram indicates electron energies, we know the slope in the bands must be such that electrons drift "downhill" in the field. Therefore, ξ points "uphill" in the band diagram. At equilibrium, no net current flows in a semiconductor. Thus any fluctuation which would begin a diffusion current also sets up an electric field which redistributes carriers by drift. An examination of the requirements for equilibrium indicates that the diffusion coefficient and mobility must be related.

Setting Eq.of the drift & diffusion component equal to zero for equilibrium, we have

put the value of the P(x)

Then we get

The equilibrium Fermi level does not vary with x, and the derivative of E_{r} Thus Eq. above reduces to