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Mathematical Analysis

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(pDpt)1/2] exp[-x2 / 4Dpt]          (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  Dp is the diffusion coefficient of the minority carrier holes in an n-type material. Eq.(1) describes the diffusion profile as can be seen from the increasing Gaussian width, (2Dpt)1/2 with increasing time t.  Eq.(1) does not, however, describe the carrier drift motion because its peak remains fixed at x=0, independent of time t, nor it describes the recombination process because the total number of carriers is constant in time in eq.(1).

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(pDpt)1/2] exp[-(x - upEt)2 / 4Dpt]       (2)  [Diffusion and Drift]

where up is the mobility of the minority carrier hole in [cm2/Vs], and E is the electric field in [V/cm] given by the bias voltage divided by sample length.  For example, if  a +2 V bias is applied on a 0.2 cm long Ge sample bar, then E = 2 V/0.2 cm = 10 V/cm. Note that the drift velocity is upE.  The peak of the concentration profile moves by a distance upEt in time t. Note that the diffusion coefficient Dp and the mobility up is related through the Einstein's relation:

Dp = (kT / q) up     [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, tp.  Suppose that a time tp seconds elapses after the laser excitation, then the total number of excess carriers is reduced to N / e = N / 2.7182818.  Including the recombination loss of carriers,  eq.(2) becomes

p(x,t) = [N / 2(pDpt)1/2] exp(-t/tp) exp[-(x-upEt)2 / 4Dpt]     (3)  [Diffusion, Drift and Recombination]

If you integrate p(x,t) over the entire x range, then you should get N exp( -t / tp). This means that the total number of excess minority carriers, which was N initially, is reduced to a factor exp( -t / tp) at time t due to the net recombination loss.   Eq.(3) for p(x, t) is the final formula for the excess hole concentration, as a function of time t and position x. This equation is coded in the applet to simulate the excess minority carrier process.

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 = ( Nd / ni )2where Nd is the donor impurity concentration and ni is the intrinsic concentration (ni = 1010/cm3 in Si). For example, in an n-Si doped with As donor impurities to Nd = 1.0x1016 cm-3, n / p = ( 1.0E16 / 1.0E10 )2 = 1012.   Majority carrier concentrationis a trillion times the minority concentration.]

[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.