explain einstein relationeinstein relationthere

Question

Explain Einstein relation.

Einstein relation:

There exists a significant relation between the diffusion constant and the mobility. It is termed as the Einstein relation and may be deduced as given below:

Consider a semiconductor wherein there exists an electric field E_x and a concentration gradient as the current is zero. In these conditions the system is in the Boltzmann statistics and thermal equilibrium applies. Consider a potential V(x) producing at x an electric field E(x)= -dV/dx.

For the density of holes the Boltzmann expression as a function of x in thermal equilibrium is as

p(x) = Ce^-eV/KT

Here C is a constant.

Therefore the gradient of the hole density is given by

dp/dx = (-e/KT)p.(dv/dx)

= (e/KT)p.E

The hole current vanishes in thermal equilibrium. Therefore,

O=peu_p E_x-eD_p  dp/dx

=peup E_x-(e₂ /KT)D_p . E_x

Dp = (KT/e) Up (i.e. the Einstein relation).