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Mathematical Analysis

I. Density of States (DOS) for electrons in a semiconductor crystal.

Electrons in a semiconductor crystal, under the influence of the periodic, crystal-lattice potential, are modeled as free particles (i.e., zero net applied force), but with a different mass from its mass in vacuum. This new mass of this 'free' electron is called effective mass, denoted m*.

Mathematically (in quantum mechanics or wave mechanics), a free particle moving in 1-dimension is expressed as

y(x, t) = A exp[ i (kx + w t)]. ----- (wavefunction of a free particle)

This is a travelling sinusoidal wave and is called the wavefunction of a free electron.

The electron energy is given by

Ey(x, t) = - (h2/2m*)d2/dx2 y(x, t) = h2k2/2m* y(x, t).

Or,

E = h2k2/2m*.

For a piece of semiconductor, we require that the wavefunction is exactly the same at the opposing surface (periodic boundary condition). If we consider a semiconductor dimension of L x L x L = V, then

y(x+L, t) = y(x, t). ----- (periodic boundary condition)

This yields,

exp[i kL] = 1.

Or,

kL = n 2p/ L.    n = ... -2, -1, 0, 1, 2, 3, ...

For an electron moving around in a bulk semiconductor (a 3-dimensional object), its energy is then

E = h2(kx2 + ky2 + kz2)/2m*
= h2(nx2 + ny2 + nz2)/2m*L2

where nx, ny, nz = ... -2, -1, 0, 1, 2, 3, ... Here, the point (nx, ny, nz) in the 3-dimensional kx, ky, kz space corresponds to a quantum energy state (if the electron spin is considered, two quantum states).

If you count the number of (nx, ny, nz) states, per unit volume, over the energy range E ~ E+dE [1], you find

g(E)dE = 4p (2m*/h2)3/2 E1/2.

II. Fermi Distribution Function, f(E)