Fermi Dirac-vs.-Maxwell Boltzman
(Carrier Concentration vs. Fermi Level)

The concentration of electrons, in a lightly-to-moderately doped semiconductor, with donor impurities for example, can be found using this simple formula if the Fermi level Ef is known: n = Nc exp[ -(Ec-Ef)/kT ], where Nc is a constant of material (density of states of conduction band), and Ec-Ef is the energy separation of the Conduction Band edge (Ec) from the Fermi level (Ef) and kT is the thermal energy (0.0259 eV at 300K). This simple formula, however, does not work if the semiconductor is heavily doped (a simple rule of thumb is n > 0.1 Nc for n-type or p > 0.1 Nv for p-type).

A heavily doped semiconductor material is found in the device regions such as:

The simple exponential concentration formula in the lightly-or-moderately doped semiconductor is called Maxwell-Boltzman (MB) approximation; where as the accurate experssion [required in a heavily-doped material] is called Fermi-Dirac (FD) formula. The FD is accurate and no approximation is involved; while the MB is an approximation, valid in not-heavily doped materials only. Use this applet to quickly find out the difference between these two formulas, particularly in the high carrier concentration regions. For heavy doping, look for: (i) n/Nc > 0.1, or p/Nv > 0.1; (ii) Ec - Ef < 3 kT, or Ef - Ev < 3 kT; or (iii) dn/n != 0, or dp/p != 0 where dn or dp is the difference between MB and FD.

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Copyright (c) C.R.Wie 1999-2000