Pure semiconductors possess conductivities too small for practical electronics. By introducing minute concentrations of substitutional impurity atoms—doping—electrical conductivity can be modulated across many orders of magnitude. The quantum electrostatics of shallow donors and acceptors is governed by a hydrogenic model renormalized by dielectric screening and effective mass.
1. The Hydrogenic Model of Shallow Impurity States
Consider a Group V donor atom (e.g., Phosphorus, Arsenic) substituting for a tetravalent Silicon atom in the diamond lattice. Four valence electrons form covalent (sp^3) bonds with adjacent silicon atoms, leaving the fifth electron bound to the excess positive nuclear charge (+e) of the donor core.
Because the extra electron travels through the host crystal, its Coulomb interaction with the ion core is screened by the relative dielectric permittivity (psilon_r) ((psilon_r approx 11.7) for Si, (13.1) for GaAs), and its kinetic energy is scaled by the conduction band effective mass (m_e^*). Under the effective mass approximation, the Hamiltonian for the donor electron is hydrogenic:
[
left[ -rac{hbar^2}{2m_e^*}
abla^2 – rac{e^2}{4pi psilon_0 psilon_r r}
ight] psi_D(mathbf{r}) = E_D psi_D(mathbf{r})
]
The ground-state ionization energy (E_d) (binding energy below (E_c)) and effective Bohr radius (a_B^*) scale directly from Bohr’s atomic theory:
[
E_d = E_c – E_D = rac{m_e^*}{m_0} rac{1}{psilon_r^2} R_y, quad a_B^* = psilon_r left( rac{m_0}{m_e^*}
ight) a_0
]
where (R_y = 13.6 ext{ eV}) and (a_0 = 0.529 ext{ AA}). In Silicon ((m_e^* approx 0.26 m_0, psilon_r = 11.7)):
[
E_d approx 13.6 imes rac{0.26}{(11.7)^2} approx 25.8 ext{ meV}, quad a_B^* approx 0.529 imes rac{11.7}{0.26} approx 24 ext{ AA}
]
Because (E_d approx k_B T) at room temperature ((k_B T approx 25.9 ext{ meV}) at (300 ext{ K})), thermal lattice vibrations readily ionize almost all donor atoms, releasing electrons into the conduction band.
2. Complete Charge Neutrality and Fermi Level Positioning
In a uniformly doped semiconductor containing donor density (N_D) and acceptor density (N_A), global electrostatic neutrality requires:
[
n + N_A^- = p + N_D^+
]
where the ionized donor and acceptor densities are governed by Fermi-Dirac occupancy with spin degeneracy factors ((g_D = 2, g_A = 4)):
[
N_D^+ = rac{N_D}{1 + g_D e^{(E_F – E_D)/k_B T}}, quad N_A^- = rac{N_A}{1 + g_A e^{(E_A – E_F)/k_B T}}
]
For an n-type semiconductor where (N_D gg N_A) and complete ionization is achieved ((N_D^+ approx N_D, N_A^- approx 0)), the neutrality condition simplifies to (n = p + N_D). Substituting (p = n_i^2 / n) yields the quadratic equation:
[
n^2 – N_D n – n_i^2 = 0 implies n = rac{N_D}{2} + sqrt{left(rac{N_D}{2}
ight)^2 + n_i^2}
]
When (N_D gg n_i), the electron density is clamped to the donor concentration: (n approx N_D), and the hole density becomes negligible: (p approx n_i^2 / N_D ll n). The corresponding Fermi level lies close to the conduction band edge:
[
E_F = E_c – k_B T lnleft( rac{N_c}{N_D}
ight)
]
3. Temperature Regimes: Freeze-Out, Extrinsic, and Intrinsic
The carrier concentration exhibits three distinct operational temperature regimes:
- Freeze-Out Regime ((T o 0 ext{ K})): Thermal energy is insufficient to ionize dopants ((k_B T ll E_d)). Electrons collapse into donor ground states ((n o 0)), causing carrier freeze-out.
- Extrinsic Saturation Regime ((sim 100 ext{ K} – 500 ext{ K})): All donors are fully ionized ((n approx N_D)), while intrinsic generation is negligible ((N_D gg n_i(T))). This wide temperature window is the operational domain of standard solid-state microelectronics.
- Intrinsic Regime ((T > 500 ext{ K})): Intrinsic carrier generation over the bandgap overwhelms the dopant density ((n_i(T) gg N_D)). The semiconductor loses its n-type or p-type identity and behaves as an intrinsic material: (n approx p approx n_i(T)).
