Here we consider excess minority carriers (holes) in an n-type semiconductor as simulated in this applet.

The incident laser pulse, very narrow, produces excess minority carrier
holes whose concentration profile initially assimilates a delta function,
when viewed as a function of position along the sample length. This
excess minority carrier pulse, p(x,t), quickly disperses to become a *Gaussian*
profile.

`p(x,t) = [N / 2(`p`D _{p}t)^{1/2}]
exp[-x^{2 }/ 4D_{p}t]
(1) [Diffusion only]`

where *N *is the total number of excess minority carriers initially
generated, per unit cross-sectional area of the sample bar, by the incident
laser pulse, and *D _{p}* is the diffusion coefficient
of the minority carrier holes in an n-type material. Eq.(1) describes the

In order to account for the carrier **drift **process, we make the
*Gaussian* peak of eq.(1) move at a speed set by the electric field,
*E*:

`p(x,t) = [N / 2(`p`D _{p}t)^{1/2}]
exp[-(x - u_{p}Et)^{2 }/ 4D_{p}t]
(2) [Diffusion and Drift]`

where *u _{p}* is the mobility of the minority carrier hole
in [cm

`D _{p} = (kT / q) u_{p} [Einstein
relation]`

Eq.(2) does not, however, adequately describe the recombination loss
of excess carriers because, according to eq.(2), the total number of minority
carrier holes remains constant at *N*, independent of time* t*.
You can check this by integrating *p(x,t)* over the entire length
of x, - infinity to + infinity. The result of this integration should be
equal to *N*. Thus, eq.(2) needs to be modified in order to
describe the recombination loss of excess carriers.

Electron-hole pairs recomine all the time. The recombination eliminates
the electons and holes in pairs. Under thermal equilibrium, *i.e.*,
in the absence of excess carriers or external stimuli, the rate of e-h
recombination is matched exactly by the rate of thermal e-h generation,
so that their concentrations remain constant, independent of time.
However, if excess carriers are present, then the rate of recombination
should be greater than the rate of *thermal* generation. This net
recombination rate reduces the excess concentration to zero over time,
eventually returning the concentration
back to the thermal
equilibrium value.*

The net **recombination **is characterized by the **minority carrier
lifetime**, *t _{p}*. Suppose
that a time

`p(x,t) = [N / 2(`p`D _{p}t)^{1/2}]
exp(-t/`t

If you integrate *p(x,t)* over the entire x range, then you should
get
*N exp( -t / t _{p})*. This
means that the total number of excess minority carriers, which was N initially,
is reduced to a factor

**Two questions** come to mind: Why did we talk about only the excess
minority carriers, and not the excess majority carriers ? Also,
why did we express the excess concentration of minority carriers as if
it was equal to the total minority concention in Eqs. (1), (2), and (3)
?

The **excess **carrier concentration is usually** much smaller**
than the equilibrium **majority **carrier concentration, and **much
larger** than the equilibrium **minority **concentration. Therefore,
the total majority concentration is little affected by the excess concentration;
but the total minority concentration is dominated by the excess concentration.
This last statement answers the second question as well: that why did we
express the excess minority concentration as if it is the total minority
concentration ? *total minority conc. = equil. conc. + excess conc.
*»*
excess conc.*

[Note that in most extrinsic semiconductors at thermal equlibirum, the
**majority
**carrier
concentration is overwhelmingly larger than the
**minority
**carrier
concentration. In an n-type semiconductor, * n / p = ( N _{d}
/ n_{i} )^{2}*where

[*Use this applet to
familiarize yourself with the majority and minority carrier densities in
Si, GaAs, and Ge.*]

Example: Calculate the excess hole concentration when a constant laser beam shines on the Si surface.

* To maintain a constant amount of excess carriers,
a steady amount of external stimulation needs to be present, such as a
constant laser beam or a DC voltage bias. These excess carriers are the
signal currents in a photodetector, a pn junction diode, or the bipolar
junction transistor.