SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.6. CARRIER TRANSPORT BY DIFFUSION 123

individually zero and we have from equation 3.6.6 for the electrons

E(x)=−

Dn

μn

1

n(x)

dn(x)

dx

(3.6.10)

To obtain the derivative of carrier concentration, we writen(x)in terms of the intrinsic Fermi
level,EFi, which serves as a reference level, and the Fermi level in the semiconductor,EF(x).
If we assume that the electron distribution is given by the Boltzmann distribution we have

n(x)=niexp

{

−

(

EFi−EF(x)

kBT

)}

(3.6.11)

This gives
dn(x)
dx

=

n(x)

kBT

(

−

dEFi

dx

+

dEF

dx

)

(3.6.12)

Atequilibrium,theFermilevel cannotvaryspatially,otherwisetheprobabilityoffinding
electronsalongaconstantenergypositionwillvaryalongthesemiconductor. This would cause
electrons at a given energy in a region where the probability is low to move to the same energy
in a region where the probability is high. Since this is not allowed by definition of equilibrium
conditions, i.e. no current is flowing, the Fermi level has to be uniform in space at equilibrium,
or
dEF
dx

=0 (3.6.13)

We then have from equation 3.6.10 and equation 3.6.12

Dn

μn

=

kBT

e

using

E(x)=

1

e

dEFi

dx

This relation is known as the Einstein relation with an analogous relation for the holes. As we can
see from table 3.4 which lists the mobilities and diffusion coefficients for a few semiconductors
at room temperature, the Einstein relation is quite accurate.

Example 3.10Use the velocity–field relations for electrons in silicon to obtain the

diffusion coefficient at an electric field of 1 kV/cm and 10 kV/cm at 300 K.

According to thev-Erelations given in figure 3.10, the velocity of electrons in silicon is∼

1.4× 106 cm/s and∼ 7 × 106 cm/s at 1 kV/cm and 10 kV/cm. Using the Einstein

relation, we have for the diffusion coefficient

D=

μkBT

e

=

vkBT

eE