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Mathematical Analysis

1. About the Energy-dependence of Carrier Concentration:

2. About the position-dependence of carrier concentration:
Minority carrier density decreases as it goes deep into the sample, at last it approaches to the log of the minority carrier density under zero bias and remains as a constant. The dependence of carrier density and position can be expressed as:
                              n(x) = n0 exp[qV/kT] exp[-x/L]
Where n0 is the carrier concentration at x=0 under zero bias. When under certain bias, we can obtain:
                              n(x, V0) = n0 exp[qV0/kT] exp[-x/L] =  n0' exp[-x/L]
Note n0' is a constant at a given bias. Also note that:
                             n(x0)= n(Ec, x0)exp[-(E-Ec)/kT]dE
L is called the diffusion length. It is the average distance a minority carrier travels before it recombines with a majority carrier, it is typically a few microns to a few millimeters. We can obtain the expression of L as:
                            L=sqrt[kT µ t / q]
where µ is the carrier mobility, is the carrier lifetime.

There have been many attempts to measure the minority-carrier lifetimes, mobilities, and diffusion lengths. For doping concentration greater than about 10E19cm-3, the experiments are quite difficult, since the minority-carrier concentrations are too small, and as a result there is quite a bit  of spread in the reported data. For pure purposed of device modeling, the following empirical equations have been proposed for minority-carrier electrons and minority-carrierr holes,


                          µn=232+1180/(1+(Na/(8*1016))0.9)
                          µp=130+370/(1+(Nd/(8*1017))1.25)
                         1/tn=3.45*10-12*Na+0.95*10-31*Na2
                         1/tp=7.8*10-13*Nd+1.8*10-31*Nd2


The data used in the applet is based on these equations. Here is a table of diffusion length of minority electrons and holes based on these equations:
 
 

Doping, p-side Ln(µm)  Doping, n-side Lp(µm) 
Na=1E15cm-3 1023  Nd=1E15cm-3 1290 
Na=1E16cm-3 307  Nd=1E16cm-3 407 
Na=1E17cm-3 76  Nd=1E17cm-3 124 
Na=1E18cm-3 16  Nd=1E18cm-3 28 
Na=1E19cm-3 3.8183  Nd=1E19cm-3 3.8210 

We can see from the upper table that the diffusion length decreases as the doping level increases. When under the same doping level under 1E19 cm-3, the diffusion length of the minority holes is greater than that of electrons. Note that in order to see the process of decrease of the minority density as it deep into the sample, the ratio of the diffusion length under different doping level is adjusted by some factor in the applet, not the exact number showed in the above table.

3. ABOUT THE TOTAL NUMBER OF INJECTED CARRIERS AT A GIVEN FORWARD BIAS, and the convenrsion between N and V.