Electronic and Semiconductor Properties

Conductivity of n-type semiconductor

An n-type semiconductor is made by doping a tetravalent element such as silicon with a pentavalent donor such as phosphorus or arsenic. Each donor atom contributes one loosely bound electron, which is easily ionized into the conduction band, so the donor level lies just below the conduction band edge E_c and nearly all donors are ionized at room temperature. When the donor density N_D is much larger than the intrinsic carrier density n_i, electrons are the majority carriers and n \approx N_D. The minority hole density follows from the mass-action law np = n_i^2, so p = n_i^2/N_D, which is usually negligible. Because the electron term dominates, the conductivity is approximately \sigma \approx n e \mu_n, and the full expression \sigma = e(n\mu_n + p\mu_p) is used only when the hole term is not negligible.

• donor — an impurity atom that supplies a free electron, e.g., phosphorus in silicon.
• majority carrier / minority carrier — the carrier type present in larger number (electrons in n-type) and the opposite type present in smaller number.
• mass-action law — the product np stays equal to n_i^2 in thermal equilibrium, which fixes the minority density.

Example: for N_D = 10^{15} cm^{-3} and n_i = 10^{10} cm^{-3}, p = 10^{5} cm^{-3}, so the holes are about 10^{-10} of the electrons and may be ignored.

Intrinsic semiconductor conductivity

An intrinsic semiconductor is a pure crystal with no deliberate doping, so its electrons and holes are created in pairs by thermal excitation across the band gap E_g. Therefore n = p = n_i, and the conductivity is \sigma = n_i e(\mu_n + \mu_p). The intrinsic carrier density depends on temperature approximately as n_i \propto T^{3/2}\exp(-E_g/2k_BT), so the conductivity rises steeply with temperature, which is the opposite of the behaviour of metals.

Pressure also matters. In common semiconductors such as Si and Ge the band gap generally narrows under hydrostatic compression, which increases n_i and hence the conductivity, so conductivity increases with both temperature and pressure in these materials.

The band gap can be extracted from a resistivity measurement. Since \rho = 1/\sigma, a measured resistivity gives n_i once the sum of the mobilities is known, and the Arrhenius form n_i = C\exp(-E_g/2k_BT) then yields E_g from the temperature dependence of n_i. Note that some problems use a simplified relation with \exp(-E_g/k_BT); always use the form stated in the problem.

• intrinsic — pure, undoped semiconductor where n = p.
• thermal excitation — promotion of electrons from valence to conduction band by thermal energy.

Example: if \sigma = 4\times10^{-4} S/m and \mu_n + \mu_p = 0.2 m^2/(V·s), then n_i = \sigma/(e \cdot 0.2) = 1.25\times10^{16} m^{-3}.

Electron and hole mobility conductivity

Mobility \mu is defined as the drift velocity per unit electric field, \mu = v_d/E, with units m^2 V^{-1} s^{-1}. In a semiconductor with both carrier types, the conductivity is the sum of the electron and hole contributions, \sigma = e(n\mu_e + p\mu_h). Because the two terms are weighted by their densities, a carrier type with a higher mobility does not automatically dominate; the product of density and mobility decides the result.

In silicon and gallium arsenide electrons usually have higher mobility than holes, so a given carrier density of electrons gives a larger current. When the ratio of mobilities and the two densities are known, the unknown mobility follows by substitution into the conductivity equation.

In a metal there is only one carrier type, and the conductivity is simply \sigma = n e \mu. Rearranging gives the carrier density n = \sigma/(e\mu) when the conductivity and electron mobility are measured, which is a standard way to estimate the free-electron density of a metal such as aluminium.

• mobility — drift velocity per unit field, \mu = v_d/E.
• carrier density — number of charge carriers per unit volume, in m^{-3} or cm^{-3}; convert units consistently with mobility (cm^2 vs m^2).

Example: with n = 2\times10^{21} m^{-3} and \mu = 0.01 m^2/(V·s), \sigma = 1.6\times10^{-19}\times2\times10^{21}\times0.01 = 3.2 S/m.

Fermi energy of free electron gas

In the free electron model, electrons move freely in a box of the metal with energy E = \hbar^2 k^2/2m. At T = 0 the states are filled up to the Fermi wavevector k_F, and the Fermi energy is E_F = \hbar^2 k_F^2/2m. The relation between k_F and the electron density depends on dimensionality, and spin degeneracy (factor 2) must be included. In three dimensions n = k_F^3/3\pi^2. In one dimension a wire with spin-degenerate states gives n = 2k_F/\pi, so k_F = \pi n/2 and E_F = \hbar^2\pi^2 n^2/8m. In two dimensions n = k_F^2/2\pi, giving E_F = \pi\hbar^2 n/m.

The quantum free electron theory treats valence electrons as delocalized over the whole crystal rather than bound to individual atoms. Electrons are fermions obeying the Pauli exclusion principle, so their occupancy follows Fermi-Dirac statistics, f(E) = 1/(\exp[(E-E_F)/k_BT]+1), and not Maxwell-Boltzmann statistics. The delocalized, degenerate electron gas explains the high electrical and thermal conductivity of metals.

• delocalized electron — an electron not attached to a single atom and free to move through the crystal.
• Pauli exclusion principle — no two electrons in the same state; this is why Fermi-Dirac statistics apply.

Example: for a 1D wire with n = 0.5 Å^{-1}, k_F = \pi n/2 \approx 0.785 Å^{-1}, and with \hbar^2/2m = 3.81 eV·Å^2, E_F = 3.81\times0.617 \approx 2.35 eV.