Consider a long semiconductor bar which is doped uniformly with donor atoms, as shown in the Fig..1.

Fig.1 Light falls upon end of long n-type semiconductor bar |

The concentration n ≈ N

_{D}and is independent of position as bar is uniformly doped.
The end on which the light is incident is demoted as x = 0.

Due to photoexcitation, the covalent bonds are broken at the surface,
which is radiated. Due to this, the electron-hole pairs are generated
and are injected into the bar at x = 0.

The bar is n-type and free electrons are large in number. Thus
generated electron-hole pairs mainly affect the minority charge
concentration i.e. concentration of holes. Let us analyze the behaviour
of steady state minority carrier concentration p against the distance x
in the bar.

Let p' = concentration of injected minority charge carriers.

**Assumption**: It is assumed that the concentration p' is much smaller than the doping level i.e. p' <<n. This condition stating that minority concentration is much smaller than majority concentration and is called low level injection condiction.

It is known that J = (n μ

_{n}+ p μ_{p}) q E, thus the drift current can be neglected compared to electron drift current.
Now p = p' + p

_{o}<< n hence the hole drift current can be neglected compared to electron drift current.**Note**: It can be assumed that the hole current I

_{p}is entirely because of diffusion.

According yo continuity equation, the controlling differential equation for minority carrier concentration p is,

The diffusion length for the holes is L

_{p}and given by,
Hence the differential equation for the injected hole concentration p' = p - p

_{o}is given by,
Solution of this differential equation is given by,

where K

_{1}, K_{2}= Constants of integration
Now as x

**→**∞, the concentration can not become infinite. Thus the second term must be zero in the equation (4), for which K_{2}= 0.
At x = 0, the injected concentration is p'(0). Hence using in the equation (4),

p'(0) = K

_{1 }e^{o}i.e. K_{1 }= p'(0),
The equation (6) represents that the hoe concentration decreases
exponentially with the distance. This is shown in the Fig. 2.

Fig. 2 Exponential behavious of hole concentration in a n-type semiconductor bar |

__1.1 Diffusion Length (L__

_{p})
From the graph, the diffusion length L

_{p}represents the distance into the semiconductor at which the injected concentration drops to 1/e times its value at x = 0. Thus L_{p}is the diffusion length which is the distance x, at which the injected carrier concentration is 36.78 % of its value at x = 0.__1.2 Diffusion Currents__

The hole diffusion current is given by,

Where A = Area of cross-section of the bar

**J**

_{p }= Hole diffusion current density

From the diffusion current denstiy,

Using equations (8) and (9) in (7),

p(x) = p(0) (at x = 0) = p'(0) + p

_{o }.............. From Fig.1
This is the diffusion hole current which decreases exponentially with distance.

**Note**: This equation plays an important role in the derivation of diode current equation.

Let I

_{n }= Electron diffusion current = + A J_{n }_{ }According to equation of charge neutrality which is preserved under low level injection, n' = p' i.e.

Where no and p

_{o }are the thermal equilibrium concentrations which are independent of the position x. Differentiating the equation (14),
Thus the ratio of majority to minority diffusion current is given by -D

_{n }/D_{P}.
From the values of D

_{n }and D_{P}, the magnitude of majority to minority diffusion current is about 2.11 for germanium and is about 2.61 for silicon.__1.3 Drift Currents__

In the bar considered above, the external voltage applied is zero and
the bar is open circuited. Hence the net current through the bar must be
zero everywhere.

**.**Net current = Electron currents + Hole currents

^{.}.
0 = I

_{nd }+ I_{n }+ I_{p }
Where I

_{nd }= Electron drift current
I

_{n }= Electron diffusion current = -(D_{n }/D_{P}) I_{p }
I

_{p }= Hole diffusion current**.**I

^{.}._{nd }- (D

_{n}/D

_{P}) Ip+ Ip = 0

Thus as Ip decreases exponentially, the electron drift current also decreases exponentially with the distance.

To exist a drift current, there must exist an electric field E. Hence
internally an electric field E is developed in the bar, for electron
drift current to exist.

J

_{pd }= p q μ_{p }E = I_{pd }/A**Note**: All the equations derived are based on the assumption that hole drift current is zero.

But hole drift current I

_{pd }can be obtained as,
J

_{pd }= p q μ_{p }E = I_{pd}/A
But as p << n, I

_{pd << }I_{p }which justifies the assumption that hole drift current is negligibly small and the entire injected minority carrier current is a diffusion current.

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