The second principal mechanism of current flow in semiconductors is diffusion. If the concentration of free charge carriers at a particular location in a semiconductor is higher than at other surrounding locations, then the atomic forces which prevail cause the carriers to migrate away from the region of higher concentration towards those of lower concentration. This process is known as diffusion and gives rise to a current known as diffusion
current. This is illustrated in Figs. 2.1, 2.2 and 2.3, which shows the changing concentration of carriers with time after a quantity of carriers has been released at a location.
Note that the movement of carriers is always in the direction of decreasing concentration. The diffusion of carriers will continue as long as the concentration gradient is maintained. The resulting current will flow in a direction, which depends on the direction of the concentration gradient and the charge on the carrier.
The rate at which diffusion takes place determines the charge flux density due to diffusion. This depends on: dn dx
, dp dx
 q ,

the carrier concentration gradients, q
the charge on the carriers,
D n
, D p
the diffusion coefficients
The diffusion coefficients are a measure of the relative ease with which the carriers can move within the concentration gradients. The dimensions of these coefficients are cm 2 s -1 (area/unit time).
1

Fig. 2.1 The Process of Diffusion t = t
0 t
1
> t
0 t
2
> t
1 t
3
> t
2 t
4
> t
3
0 distance
Fig. 2.2 Changes in Carrier Concentration with Time During Diffusion
2

n t = t
1
Carrier flow n t = t
2
> t
1
Carrier flow x
Fig. 2.3 Change in Concentration Gradient with Diffusion
The charge flux density for diffusion is then:
J n diff
 -q .D
n
.
dn dx
 qD n dn dx for electrons x
J p diff
  q .D
p
.
dp dx
  qD p dp dx for holes
The negative signs in front of the concentration gradients above account for the fact that the carriers move in the direction of decreasing concentration. Note that if the concentration gradients for both holes and electrons are in the same direction, then both types of carrier will be travelling in the same direction and the charge flows will tend to cancel each other. If, on the other hand, the gradients are in opposite directions, then the flows of charge of opposite sign will tend to add together.
For a piece of homogeneous semiconductor of uniform crosssectional area, A, the total diffusion current due to both electrons and holes in the positive x direction is given as:
I diff
 J n diff
A  J p diff
A  Aq(D n dn dx
 D p dp dx
)
The polarity of the gradients will determine the direction of flow of individual carriers and, hence, the overall diffusion current.
3

The diffusion coefficient is related to the mobility of the carriers by the
Einstein Relation:
D
 n n
 kT q
 D n
  n kT q
  n
V
T for electrons
D p
 p
 kT q
 D p
  p kT q
  p
V
T for holes where
V
T
 kT q
= 26mV @ 300  K is known as the thermal voltage.
The Thermal Voltage is the potential difference in an electric field through which an electron with a charge of magnitude q must accelerate to gain the same amount of energy as it has due to temperature kT.
Consider a length of homogeneous p-type semiconductor of uniform cross-sectional area, A, as shown in Fig. 2.4. Under equilibrium conditions, the concentration of majority carriers is determined by the doping concentration. If excess minority charge carriers are injected into the semiconductor from an external source, this equilibrium is disturbed as the excess minority carriers injected begin to recombine with the majority carriers present. This gives rise to changes in the charge concentration within the material.
The rate of change of carrier concentration depends on the rate of injection of excess carriers and the rate of recombination in the volume.
Notation: for electrons
Total carrier concentration: n
Equilibrium carrier concentration: n
0
Excess Carrier Concentration: n’ so that:
Excess carrier concentration is: n'  n  n
0
4

excess minority carriers recombine with majority carriers inside volume of semiconductor
A uniform cross section
J n
(x) charge flux density becomes a function of distance into material excess minority carriers injected from external source
Fig 2.4 Excess Charge Carriers Injected into a Semiconductor
5

If there is no electric field across the semiconductor and the material is homogeneous and uniform, then the injected minority carriers will move only by diffusion in the x direction. The law governing this process is known as the Diffusion Equation which for one dimension is given as:
 n'

(x) t
 D n
 2 n' (x)
 x 2
 n' (x)
 n where  n
is the average carrier lifetime before recombination. If the supply of minority carriers into the material can be maintained by some external means and the majority carriers which recombine can also be replaced, then this will support the recombination process and eventually a steady-state condition will be reached and the concentration of charge within the volume will reach a stable value so that
 n' (x)
 t
 0
Under these conditions…
 2 n' (x)
 x 2
 n' (x)
D n
 n
This is a second-order differential equation which has a general solution of the form: n' (x)  Ae
-x
L n
 Be
 x
L n
Since recombination can only reduce the free carrier concentration in the semiconductor there is no growth term so that B = 0. Hence the solution is: n' (x)  n' (x  0)e
 x
L n where n’(x = 0) is the quantity of excess minority carriers injected into the material at x = 0.
6

The minority carriers will penetrate a distance, L n
, on average into the volume before recombining. This distance is equivalent to the depth of penetration which would take place if the concentrations at the boundary of the material were maintained. It is related to the average carrier lifetime through the diffusion coefficient so that:
D n
 n
 L n
2
The profile of the minority carrier concentration across the material into which they are injected therefore has an exponential form as shown in Fig. 2.5. Note also that if a thin slice of the semiconductor is taken, most of the injected carriers pass through the slice and emerge at the other side but some have recombined. The quantity of minority carriers which pass through the material depends on the thickness of the slice and the doping concentration of the material. n (x) n(x = 0)
Thin slice n’(x = 0) n
0
L n distance x
Fig. 2.5 Profile of Minority Carrier Concentration through the
Semiconductor
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