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Mathematical Analysis

1. About the Energy-dependence of Carrier Concentration:

In the applet, the color density within the blue rectangle in CB and red rectangle in the VB is proportional to the log of carrier concentration at the particular energy (the vertical scale in the Band diagram). This assumes a constant density of states (DOS) in the energy bands. This is discussed below:

The number of carriers (electrons or holes) in the energy range, E ~ E + dE, is given by the product of the number of energy states with the probability of occupation of a state by a carrier (electron or hole) :

n(E)dE = f(E) N(E)dE.       (1)

Where N(E)dE is the number of states per unit volume (N(E) = DOS) and f(E) is the probability of occupation (the Fermi-Dirac distribution function) :

f(E) = 1 / [ 1 + exp( (E-EF)/kT ) ]          (2)     Fermi distribution function

For electrons in the conduction band and for a not-so-heavily-doped material (say, donor concentration Nd << 1E18 cm-3), we may assume that E - EF >= Ec - EF >> kT, and therefore the exponetial term dominates over unity in eq.(2) :

f(E) ~= exp[ -(E-EF)/kT ]            (2')       Boltzman approximation.

The density of states for electrons in the conduction band is :

N(E) = p / 2( 2m/h2 )3/2 (E-Ec)1/2       (3)        Density of States

= a (E-Ec)1/2    where a = p / 2(2m/h2)3/2.

The energy dependence of electron concentration is therefore,

n(E) ~= a (E-Ec)1/2 exp[ -(E-EF)/kT ]    (4)     Energy dependence of concentration.

The function, x1/2 e-x/b, has a peak at x = b / 2.  The electron concentration is zero at the band edge, E = Ec. With increasing energy, n(E) increases until E = kT / 2  and then decreases (almost) exponentially for E > kT / 2.

Assumptions in the applet:

n(E) is approximated by

n(E) ~= a (E0 - Ec) exp[ -(E-EF)/kT ]    (4')    Concentration in the applet.

Concentration profile assumed in the applet is the eq.(4') as plotted in the following figure:

fig. for (4) & fig. for (4').

This is equivalent to a constant density of states :

N(E) ~= N(E0) = a (E0 - Ec)1/2     (3')     DOS in the applet.

If we set E0 such that the same total concentration of conduction electrons is obtained from (4') as the concentration from (4), then

E0 - Ec = 16 p kT ~= 50.27 kT.   (5)

The actual DOS and the assumed DOS are shown below:

fig. for (3) and (3').

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.