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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 Ex 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=peup Ex-eDp dp/dx
=peup Ex-(e2 /KT)Dp . Ex
Dp = (KT/e) Up (i.e. the Einstein relation).
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