Semiconductor Physics and Carrier Transport

Excess carrier generation under illumination

When light whose photon energy exceeds the band gap is absorbed, it creates electron-hole pairs. The carriers added above their equilibrium values are called excess carriers, \delta n and \delta p. In a bulk region charge neutrality requires \delta n = \delta p, so the two excesses always rise and fall together. For a uniform generation rate G with no field or diffusion effects, steady state is set by the balance between generation and recombination. When the excess is small compared with the majority-carrier density (low-level injection), this gives \delta n = \delta p = G\tau, so the minority-carrier lifetime \tau controls the excess density while the doping sets the majority density. When the excess becomes comparable to the doping, this simplification fails and the total densities must be written as n = n_0 + \delta n and p = p_0 + \delta p. The product np then exceeds the equilibrium value n_i^2, because the excess pairs are added on top of both populations. For example, a p-type sample with p_0 = 10^{15} cm^{-3} under G = 10^{17} cm^{-3}s^{-1} with \tau = 1\ \mus gives \delta n \approx 10^{11} cm^{-3}, far below p_0, so low-level injection holds.

If generation occurs only at one face of a long bar, the excess minority carriers do not stay uniform. Diffusion carries them away from the illuminated surface while recombination removes them, so \delta n(x) = \delta n(0)\,e^{-x/L_n} with L_n = \sqrt{D_n\tau_n}. The electron diffusion length is therefore the distance over which the excess falls by a factor e, about 37 percent of its surface value. In a field-free bar the minority electron profile depends only on L_n, and the hole diffusion length does not enter. For example, if L_n = 5\ \mum, the excess at x = 5\ \mum is e^{-1} of its value at the surface.

The photo-generated electron current density in a zero-field region is J_n = qD_n\,d\delta n/dx. The gradient is fixed by the continuity equation D_n\,d^2\delta n/dx^2 - \delta n/\tau_n + G = 0 (with any extra recombination term included), so the current at a point equals the net generation integrated across the region between that point and the place where the excess vanishes. Current magnitude therefore cannot be read from the generation rate at one point alone.

Fermi level position in nondegenerate semiconductor

In a non-degenerate semiconductor Boltzmann statistics apply, and the Fermi level lies in the band gap several kT away from both band edges. For n-type material the electron density is n = N_C\exp[-(E_C - E_{Fn})/kT], so E_C - E_{Fn} = kT\ln(N_C/n). The separation depends only logarithmically on the density. Halving n therefore raises E_C - E_{Fn} by kT\ln 2, about 18 meV at 300 K, and the Fermi level moves toward mid-gap. Similarly, for holes E_{Fp} - E_V = kT\ln(N_V/p).

The hole density is p = N_V\exp[-(E_F - E_V)/kT]. When the temperature changes, two effects act together. The effective density of states N_V grows as T^{3/2}, and the exponent (E_F - E_V)/kT shrinks as kT grows. So for a Fermi level at a fixed energy above the valence band, p rises steeply with temperature. Calculations must use the temperature-scaled N_V and the kT at the required temperature, not the room-temperature value.

Donors are not always fully ionized. The fraction that remains neutral is N_D^0/N_D = 1/(1 + (1/g)\exp[(E_D - E_F)/kT]), where g is the donor degeneracy. A small neutral fraction means the donor level lies well above the Fermi level, so the donor is ionized. Because E_C - E_D = (E_C - E_F) - (E_D - E_F), the donor ionization energy follows directly once the neutral fraction and the Fermi level position are known. Shallow donors in silicon lie only tens of meV below E_C. For example, if E_C - E_F = 0.2 eV and E_D - E_F = 0.05 eV, then E_C - E_D = 0.15 eV.

Band bending in nonuniform doping

In equilibrium the Fermi level is flat throughout a sample. Where the doping varies, the band edges must bend to keep it flat, and this bending is the electrostatic potential energy. The band-edge energy and the electrostatic potential are related by E_V = \text{const} - q\phi, and the field is \mathcal{E} = -d\phi/dx. Combining these gives |\mathcal{E}| = \frac{1}{q}\left|\frac{dE_V}{dx}\right|. A linear band slope gives a uniform field, and if the valence band falls by \Delta over a length L, the field magnitude is \Delta/(qL). For example, a drop of 0.1 eV over 1 \mum gives a field of 0.1\ \text{V}/10^{-4}\ \text{cm} = 1000\ \text{V/cm}.

The direction of the currents follows from the electron concentration profile. Electron diffusion current flows toward higher electron concentration, because electrons diffuse down their own gradient and conventional current runs opposite to electron motion. In equilibrium the total electron current is zero, so J_n = qn\mu_n\mathcal{E} + qD_n\,dn/dx = 0, which gives \mathcal{E} = -(kT/q)(1/n)\,dn/dx. Conventional electron drift current flows parallel to the field. Where n decreases with x, the field points along +x, and the drift current along +x balances the diffusion current toward higher n in the -x direction.