4.14 Sspd_chapter_2.2.9-2.2.10-2.2.11:diffusion,life-time and  (Page 3/3)

Life-Time varies from ns to ms depending upon the impurty concentration and crystal imperfections.If impurity concentration and density of imperfection is low then we have lifetime of ‘ms’ interval otherwise we have ‘ns’ interval life-time.

2.2.11.Continuity Equation in Silicon Bar optically excited by a continuous beam of light under steady state condition.

In Figure 2.2.44. N-Type Bar (N _D =10 ¹⁴ /cc) is shown which is illuminated by continuous light which has energy packet of 2eV on the left end of the sample.In due course of time the concentration profile of the majority and minority carriers become time-invariant and we say that thermal equilibrium condition has been achieved.

Here we have assumed minority carrier hole life-time=2μs.

Let Optical Generation Rate = G number of EHPs are being opticall generated per second.

The photon of the incident light has an energy packet=2eV=hν.

Suppose the rate of incidence of energy=10mW = 10mJ/s.

Since photon is 2eV which exceeds the energy band-gap=1.12eV hence it is powerful enough to cause photo-excitation. Hence all photons are absorbed by the left surface and equal number of EHPs are optically generated.

The physics of photo-excitation of EHP and recombination of EHP is shown in Figure 2.2.45.

Assuming 1 optically generated EHP for each absorbed incident photon, the optical generation rate G is as follows:

Therefore under optical excitation, the left surface experiences a photon boosted generation rate of EHPs = 3.125×10 ¹⁶ EHPs/sec.

Let the photons be absorbed within a volume=∆Vcc.

Therefore optical generation rate -= g _op =10 ¹³ EHP/(cc-μs)= 10 ¹⁹ EHP/(cc-s).

In DARK condition, through out the bar, thermal generation rate decides the equilibrium concentrations.

In Left End illumination, the photo-generation + thermal generation together decide the equilibrium concentration values on the left end of the Si-Bar.

Let new equilibrium concentration on the left end be p _n0 and n _n0 .

Generation-Recombination equation under optical-excitation condition is:

Now Law of Mass Action does not hold good because of external illumination.

Therefore n _n0 ×p _n0 ≠ n _i ² ;

Some time after illumination, carrier concentration assumes a steady state value through out the Si-Bar along the z-axis.

This steady state concentration profile has to be determined.

Let us consider holes only.

Refer to Figure 2.2.46.

(2.2.11.6)

The typical diffusion length in this particular example is:

L _p ² =τ _p D _p =2×10 ^-6 ×12.5cm ² /s = 25×10 ^-6 cm ² ;

Therefore diffusion length=L _p =5×10 ^-3 cm=0.5μm.

In Figure 2.2.47. steady state carrier concentration profiles are given with one end of the Si-Bar photo-excited and for the case Length of the Bar ‘L’>>L _p . This is a wide-bulk case.

In Si-Bar, when we have L ~ Lp or L<<L _p then we have narrow-bulk case and steady state carrier concentration profile is linear as shown in Figure 2.2.48.

The linearity of the concentration profile is due to the fact that in a narrow bulk thee is hardly any recombination hence diffusion current is constant along the z-axis and hence carrier concentration of the carrier is also constant. This leads to a linear slope within the bulk for both majority and minoreity carriers.

In our example, in case of narrow-bulk Si-Bar, the far end steady state concentration values depend upon surface recombination velocity. In our case we have assumed infinite surface recombination velocity hence:

The case for finite surface recombination velocity will be dealt in the Appendix of this Chapter.