Count intrinsic and doped semiconductor carriers

Let $E_c$ and $E_v$ denote conduction and valence edges, with gap $E_g=E_c-E_v$. Electrons occupy a state of energy E with Fermi probability $f(E)=[1+e^{(E-E_F)/(k_BT)}]^{-1}$. The electron density integrates the conduction density of states against f; the hole density integrates the valence density against one minus f. A hole counts an unoccupied valence state, rather than a second electron species with an independently chosen equilibrium chemical potential.

Near an isotropic parabolic minimum, the three-dimensional density of states grows as the square root of energy above the edge. Its thermal integral produces the effective density $N_c$. For identical valleys, a factor $g_v$ counts equivalent minima and a separate factor two counts unresolved electron spin. If a tabulated density-of-states mass already incorporates valley multiplicity, adding that factor again is wrong. A conductivity effective mass describes acceleration and need not equal this state-counting mass.

In the nondegenerate equilibrium regime, holes obey the corresponding Boltzmann expression with effective valence density $N_v$. Multiplying the electron and hole formulas eliminates the common Fermi level and gives the mass-action law. Intrinsic material has equal electron and hole populations, although its Fermi level is not exactly at midgap unless the two effective densities match. Differences in band curvature move the intrinsic level without producing net charge in an otherwise ideal intrinsic crystal.

For uniformly doped neutral bulk, the positive charge of holes and ionized donors balances electrons and ionized acceptors. With complete ionization, the difference between donors and acceptors controls majority carriers, not their sum. Solving neutrality together with mass action gives a quadratic equation with one positive physical electron density. When the net donor density greatly exceeds the intrinsic density, that exact solution reduces to majority electrons approximately equal to the net donor count, with minority holes fixed by mass action.

At low temperature, donor electrons can remain bound to donor levels. Their ionized fraction depends on the Fermi level and impurity occupation statistics, so replacing $N_D^+$ by total $N_D$ fails. At strong doping, conduction occupation is no longer dilute and the full Fermi integral replaces the Boltzmann exponential. These are different failures: freeze-out concerns dopant ionization, whereas degeneracy concerns occupation of band states. Heavy doping may also change the bands themselves beyond this simple model.

A practical calculation should therefore check the inferred Fermi-level separation from both relevant edges and the assumed ionization range. Under illumination or injection, electrons and holes may require separate quasi-Fermi levels; their product need not equal the equilibrium intrinsic square. Charge neutrality remains useful in broad bulk regions, but cannot be imposed locally inside a depletion layer where space charge generates an electric field. Statistics and electrostatics complement each other rather than providing interchangeable constraints.