Einstein Relation [for Reference Purpose]

EINSTEIN RELATION [For Reference Purpose]

Even though the drift and diffusion transport are independent phenomenon, they both can be directly related.

Einstein showed that the mobility (μ) and the diffusion coefficient (D) can be related as shown in Fig. 2.25.

This graph shows that the diffusion always occurs from higher density region to lower density region, in order to achieve uniform distribution.

Thus, under thermal equilibrium, we can write

Drift current = Diffusion current

Equating equations (3) and (9), we can write

 …….(1)

If F=eE is the force required to restore the excess charge carriers to equilibrium and to achieve uniform distribution of charge carriers, then, the force due to the electrons (Δn_e) can be written as

F=Δ n_e eE…..(2)

From Equations (1) and (2) We can write

 …….(3)

The force corresponding to pressure gradient is

 ………(4)

Eqns. (3) and (4) we get

Diffusion coefficient for electrons is

…….(5)

 ………(6)

Similarly for holes we can write the diffusion coefficient as

……...(7)

 ……..(8)

Equations (5) and (7) are called Einstein's relation which represents the relationship between diffusion coefficient and mobility of charge carriers.

From equations (6) and (8) we can write

……….(9)

Equation (9) gives the relationship between diffusion coefficient in respect of their mobilities.