Carrier distributions in a semiconductor

Move the Fermi level and temperature to see how the density of states and the occupancy factors combine into the electron and hole distributions. Panels share one energy axis, with E_v taken as zero.

Material

Temperature300 K

Fermi level

Let E_g follow temperature (Varshni) Overlay Boltzmann approximation Carrier distribution scale

Figure 1 Coordinated sketches of the energy band diagram, the density of states, the occupancy factors f(E) and 1 − f(E), and the resulting carrier distributions g_cf and g_v(1 − f). The energy axis spans the same 2.2 eV for every material, centered on that material's 300 K E_i, and the carrier axis spans a fixed 1E+00 to 1E+22 cm⁻³eV⁻¹ shared by both bands, so panels taken at different settings can be compared directly. After Pierret, Semiconductor Device Fundamentals, Fig. 2.16.

Band and statistics parameters
`kT`		Thermal energy
`E`_g		Bandgap at this temperature
`N`_C		Effective conduction band density of states
`N`_V		Effective valence band density of states
`E`_i − midgap		Intrinsic level offset, (3`kT`/4) ln(`m`_p/`m`_n)
`n`_i		Intrinsic carrier concentration

Carrier concentrations
`n`		Area under `g`_c`f`, integrated numerically
`p`		Area under `g`_v(1 − `f`)
`n` − `p`		Net ionized doping required for neutrality
`np`/`n`_i²		Unity while Boltzmann statistics hold
`n`/`n`_Boltz		Error incurred by the Boltzmann approximation
`p`/`p`_Boltz		Same test on the valence band side

Model, parameters, and what is left out

Energies are referenced to the valence band edge, so E_v = 0 and E_c = E_g. The three ingredients are exactly as specified:

f(E) = 1 / {1 + exp[(E − E_F)/kT]}

g_c(E) = m_n* √[2m_n*(E − E_c)] / (π²ℏ³), E ≥ E_c

g_v(E) = m_p* √[2m_p*(E_v − E)] / (π²ℏ³), E ≤ E_v

In practical units this is g(E) = 6.812×10²¹ (m*/m₀)^3/2 √(ΔE in eV) cm⁻³eV⁻¹, with the factor of two for spin already contained in the expression above. The same coefficient reproduces N_C = 2.51×10¹⁹(m*/m₀)^3/2 cm⁻³ at 300 K, which is the internal consistency check for the numbers in the tables.

Both axes are held fixed so that two settings can be compared by eye. The energy window runs from −0.35 eV to E_g(300 K) + 0.35 eV and depends on the material alone, never on temperature or Fermi level; the Fermi level slider stops 50 meV short of each end so its marker always stays on the plate. The carrier distribution axis is a fixed logarithmic decade scale from 1 to 10²² cm⁻³eV⁻¹, shared by both bands. Two consequences are worth pointing out in class: the Boltzmann tail becomes a straight line whose slope is 1/kT, and any part of a distribution falling below 1 cm⁻³eV⁻¹ is simply not drawn, which is why the minority carrier lobe disappears entirely once the material is strongly doped or cold. The printed peak values under the panel are exact regardless of what fits on the axis.

n and p are obtained by numerical quadrature of the plotted distributions, i.e. as Fermi–Dirac integrals of order 1/2, not from the Boltzmann approximation. The substitution u = √(E − E_c) removes the square-root singularity at the band edge; Simpson's rule on 2000 intervals then gives roughly six significant figures. The last two table rows compare the quadrature with n_Boltz = N_Cexp[−(E_c − E_F)/kT], so the onset of degeneracy is visible as a numerical ratio rather than only as a warning.

	Ge	Si	GaAs
`m`_n*/`m`₀	0.55	1.18	0.066
`m`_p*/`m`₀	0.37	0.81	0.52
`E`_g(300 K)	0.66 eV	1.12 eV	1.42 eV
`E`_g(0), α, β	0.7437, 4.774e-4, 235	1.170, 4.730e-4, 636	1.519, 5.405e-4, 204

The effective masses are the 300 K density-of-states values tabulated in Pierret, Table 2.1. Other compilations differ: Green's widely used values for silicon are 1.09 and 1.15, which change N_C and N_V by tens of percent while leaving every curve shape unaffected. If the numbers here are to be compared against a different table, that is the parameter to check first.

The optional temperature dependence uses the Varshni form E_g(T) = E_g(0) − αT²/(T + β), with the parameters listed above; it is off by default so that the tabulated 300 K gap is used at every temperature.

Deliberate simplifications, all of which matter somewhere in a real device:

Parabolic, isotropic bands with temperature-independent effective masses. Band non-parabolicity is what makes GaAs, with its very light electron, degenerate at modest doping.
No dopant levels, so no freeze-out and no incomplete ionization. The Fermi level is an independent input here; in a real sample it is fixed by the doping through charge neutrality, and the n − p row states which net ionized doping the chosen E_F would correspond to.
No bandgap narrowing, band tails, or impurity banding, all of which appear at the concentrations reached by the degenerate preset.
Only the single lowest conduction band minimum is represented through one density-of-states mass; the satellite valleys in GaAs are not included.