SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.6. CARRIER TRANSPORT BY DIFFUSION 121

whereDnis called the diffusion coefficient of the electron system and depends upon the scatter-
ing processes that control and theτsc.Sincethemeanfreepathisessentiallyvthτsc,wherevth
isthemeanthermalspeed,thediffusioncoefficientdependsuponthetemperatureaswell. In a
similar manner, the hole diffusion coefficient gives the hole flux due to a hole density gradient

φp(x, t)=−Dp

dp(x, t)

dx

(3.6.4)

The electron and hole flux causes current to flow in the structure This current is given by

Jtot(dif f)=Jn(dif f)+Jp(dif f)

= eDn

dn(x, t)

dx

−eDp

dp(x, t)

dx

(3.6.5)

Note that the electron charge is−ewhile the hole charge ise. While both electrons and holes
move in the direction of lower concentration of electrons and holes respectively, the currents they
carry are opposite, since electrons are negatively charged, while holes are positively charged.

3.6.1 Driftanddiffusiontransport:Einstein’srelation..............

In case both electric field and carrier concentration gradients are present, the current is given
by

Jn(x)=eμnn(x)E(x)+eDn

dn(x)

dx

Jp(x)=eμpp(x)E(x)−eDp

dp(x)

dx

(3.6.6)

The diffusion and drift processes are linked by scattering processes. We will now establish
an important relationship between mobility and diffusion coefficients. Consider a case where a
uniform electric field is applied, as shown in figure 3.17a. The potential energy associated with
the field is shown in figure 3.17b. There is a positive potential on the left-hand side in relation to
the right-hand side. For a uniform electric field the potential energy is

U(x)=U(0)−eEx (3.6.7)

The applied force is related to the potential energy by

Force =−∇U(x) (3.6.8)

Thus, since the electron charge−eis negative, the bands bend as shown in figure 3.17c according
to the relation
Ec(x)=Ec(0) +eEx (3.6.9)

Thus,ifapositivepotentialisappliedtotheleftofthematerialandanegativetotheright,the
energybandswillbelowerontheleft-handside,asshowninfigure3.17c.Theelectronsdrift
downhillintheenergybandpictureandthusoppositetothefield.