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Introduction

As a function of the Fermi level, EF, the electron and hole concentrations are given, respectively, by:

Fermi - Dirac

n = Nc F1/2(hc), where hc = (EF - Ec)/kT, and

p = Nv F1/2(hv), where hv = (Ev - EF)/kT.

These are the exact formulae for electron and hole concentrations, according to the Fermi-Dirac statistics. The mysterious function, F1/2, is called the Fermi-Dirac integral of order 1/2 (in Dingle's notation [1]) where

Fn(h) = [1 / G(n+1)] xn / [ 1 + e(x - h) ] dx

People do not like to work with a black box function and thus they try to use an explicit and familiar function when they can. Fortunately, a simple approximation is available when h < -3. This is the so-called Maxwell-Boltzman approximation:

F1/2(h) » exp(h)   for h < -3.

Maxwell-Boltzman

n » Nc exp[ -(Ec - EF)/kT ],

p » Nv exp[ -(EF - Ev)/kT ].

Applies if  Ev + 3 kT < EF < Ec - 3 kT.

The following applet demonstrates the applicability of MB approximation vs. the Fermi level position. Here FD is the exact concentration [2] according to the Fermi-Dirac integral [3].


In the laboratory (i.e., cleanroom [footnote]), the electron concentration in the conduction band is determined by the doping level of Donor impurities and the temperature of the material.

In the classroom, the electron concentration, n, is expressed in terms of the Fermi level position, EF, and the temperature, T:

n depends on Ec - EF and kT.

The mathematical formula can take an approximate, but simple exponential Maxwell-Boltzman form if

Ec - EF > 3 kT (or, Nd < 0.05Nc)  --- [n-type]
Ef - Ev > 3 kT (or, Na < 0.05Nv)  --- [p-type]

The more complicated, accurate Fermi-Dirac formula must be used if

Ec - EF < 3 kT   --- [heavily-doped n-tyep]
Ef - Ev < 3 kT  --- [heavily-doped p-type]

The simple approximation is possible for lightly-to-moderately-doped semiconductor (called nondegenerate semiconductor ). Otherwise, the full formula must be used in heavily-doped semiconductor (also called, degenerate semiconductor because the fermi energy can be degenerate with, or 'the same' as, a band state.)

The applet demonstrates the discrepancy in electron (or hole in p-type) concentration between the approximate formula [Maxwell-Boltzman approximation] and the full, accurate formula [Fermi-Dirac formula] when the concentration is high.


footnote-1: In the cleanroom (or fab line), the majority carrier concentration is controlled by the precise doping level of impurity atoms.