E (k) = E_{c} + ℏ^{2} k^{2} / (2 m_{e}^{*})

, with two spin states per wave vector. In the nondegenerate limit the Fermi-Dirac distribution reduces to its Boltzmann factor. The number per area is therefore

n = 2 \int \frac{d ^{2} k}{( 2 π ) ^{2}} exp [- β (E_{c} - μ + \frac{ℏ ^{2} k ^{2}}{2 m _{e}^{*}})] = \frac{e ^{- β (E_{c} - μ)}}{π} \int_{0}^{\infty} k e^{- β ℏ^{2} k^{2} / (2 m_{e}^{*})} d k = \frac{m _{e}^{*} k _{B} T}{π ℏ ^{2}} e^{- β (E_{c} - μ)} .

\int_{0}^{\infty} k e^{- a k^{2}} d k = 1/ (2 a)

. Thus

C = \frac{k _{B}}{π ℏ ^{2}}, a = 1, b = 1.

Equivalently, the two-dimensional density of states is constant,

D_{c} = m_{e}^{*} / (π ℏ^{2})

, and

n = N_{c} e^{- β (E_{c} - μ)}

with

N_{c} = D_{c} k_{B} T

. These expressions include twofold spin degeneracy and one valley. With a general total band degeneracy

g_{c}

, replace the factor

1/ π

g_{c} / (2 π)

; extra valleys or occupied confinement subbands would change the coefficient.

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