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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) = - (~~h~~^{2}/2m*)d^{2}/dx^{2}
y(x, t) = ~~h~~^{2}k^{2}/2m*
y(x, t).

Or,

E = ~~h~~^{2}k^{2}/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 = ~~h~~^{2}(k_{x}^{2}
+ k_{y}^{2} + k_{z}^{2})/2m*

= h^{2}(n_{x}^{2} + n_{y}^{2} +
n_{z}^{2})/2m*L^{2}

where n_{x}, n_{y}, n_{z} = ... -2, -1, 0, 1,
2, 3, ... Here, the point (n_{x}, n_{y}, n_{z})
in the 3-dimensional k_{x}, k_{y}, k_{z} space
corresponds to a quantum energy state (if the electron spin is considered,
two quantum states).

If you count the number of (n_{x}, n_{y}, n_{z})
states, per unit volume, over the energy range E ~ E+dE [1],
you find

g(E)dE = 4p (2m*/h^{2})^{3/2}
E^{1/2}.

II. Fermi Distribution Function, f(E)

See this page.