The electrical conductivity of semiconductors is dictated by the thermodynamic availability of free charge carriers. Governed by Pauli exclusion and Fermi-Dirac statistics, the distribution of electrons in the conduction band and holes in the valence band leads to the exact derivation of intrinsic carrier concentrations and the mass-action law.
1. Density of States (DOS) in 3D Parabolic Bands
Near the conduction band minimum (E_c) and valence band maximum (E_v), the energy dispersion can be approximated quadratically in terms of scalar density-of-states effective masses (m_d^*) and (m_{dh}^*):
[
E(mathbf{k}) approx E_c + rac{hbar^2 k^2}{2m_{de}^*}, quad E(mathbf{k}) approx E_v – rac{hbar^2 k^2}{2m_{dh}^*}
]
Integrating in spherical (mathbf{k})-space with a factor of 2 for electron spin degeneracy yields the canonical square-root 3D density of states per unit volume:
[
g_c(E) = rac{1}{2pi^2} left( rac{2m_{de}^*}{hbar^2}
ight)^{3/2} sqrt{E – E_c}, quad (E ge E_c)
]
[
g_v(E) = rac{1}{2pi^2} left( rac{2m_{dh}^*}{hbar^2}
ight)^{3/2} sqrt{E_v – E}, quad (E le E_v)
]
2. Fermi-Dirac Statistics and Carrier Integrals
The probability that an electronic state at energy (E) is occupied in thermodynamic equilibrium at temperature (T) is given by the Fermi-Dirac distribution:
[
f(E) = rac{1}{1 + e^{(E – E_F)/k_B T}}
]
where (E_F) is the Fermi level (chemical potential (mu)). The total concentration of conduction electrons (n) is obtained by integrating the product of the DOS and the occupation probability:
[
n = int_{E_c}^infty g_c(E) f(E) , dE = N_c mathcal{F}_{1/2}left( rac{E_F – E_c}{k_B T}
ight)
]
where (mathcal{F}_{1/2}(ta) = rac{2}{sqrt{pi}} int_0^infty rac{xi^{1/2}}{1 + e^{xi – ta}} , dxi) is the complete Fermi-Dirac integral of order (1/2), and (N_c) is the effective density of states for the conduction band:
[
N_c = 2 left( rac{2pi m_{de}^* k_B T}{h^2}
ight)^{3/2}
]
Symmetrically, the hole density in the valence band is governed by the probability of an unoccupied state (1 – f(E)):
[
p = int_{-infty}^{E_v} g_v(E) [1 – f(E)] , dE = N_v mathcal{F}_{1/2}left( rac{E_v – E_F}{k_B T}
ight), quad N_v = 2 left( rac{2pi m_{dh}^* k_B T}{h^2}
ight)^{3/2}
]
3. The Non-Degenerate Maxwell-Boltzmann Approximation
When the Fermi level lies within the bandgap at least (3 k_B T) away from both band edges ((E_c – E_F ge 3 k_B T)), the exponential dominates: (e^{(E – E_F)/k_B T} gg 1). The Fermi-Dirac distribution reduces asymptotically to the classical Maxwell-Boltzmann factor:
[
f(E) approx e^{-(E – E_F)/k_B T}
]
Evaluating the carrier integrals analytically yields the closed-form carrier density formulas:
[
n = N_c e^{-(E_c – E_F)/k_B T}, quad p = N_v e^{-(E_F – E_v)/k_B T}
]
4. Intrinsic Carrier Concentration and the Mass-Action Law
In an undoped, perfectly pure semiconductor, every electron excited into the conduction band leaves behind exactly one hole in the valence band: (n = p = n_i), where (n_i) is the intrinsic carrier concentration. Multiplying (n) and (p) causes the Fermi level (E_F) to cancel identically, establishing the fundamental Mass-Action Law:
[
n cdot p = n_i^2 = N_c N_v e^{-(E_c – E_v)/k_B T} = N_c N_v e^{-E_g / k_B T}
]
[
n_i = sqrt{N_c N_v} e^{-E_g / (2 k_B T)}
]
For Silicon at (T = 300 ext{ K}) ((E_g approx 1.12 ext{ eV})), (n_i approx 1.0 imes 10^{10} ext{ cm}^{-3}), while in GaAs ((E_g approx 1.42 ext{ eV})), (n_i approx 2.1 imes 10^6 ext{ cm}^{-3}).
Equating (n = p = n_i) reveals the precise position of the intrinsic Fermi level (E_i):
[
E_i = rac{E_c + E_v}{2} + rac{3}{4} k_B T lnleft( rac{m_{dh}^*}{m_{de}^*}
ight)
]
The intrinsic Fermi level lies almost precisely at mid-gap, shifted only slightly by the natural logarithm of the effective mass asymmetry between electrons and holes.
