Carrier Statistics: Intrinsic and Extrinsic Semiconductors

The band picture sorts solids by gap width but leaves the carrier count qualitative. Making it quantitative is the task of this lesson: given the gap $E_{g}$ , the band-edge effective masses, the doping, and the temperature, how many electrons sit in the conduction band and how many holes in the valence band? The answer rests on two inputs — the density of states near each band edge and the Fermi-Dirac occupation that reaches into it — and produces the exponential temperature laws that govern every semiconductor device.

Carriers from the density of states

Near the bottom of the conduction band the energy is parabolic in the crystal momentum, $E (k) = E_{c} + ℏ^{2} k^{2} /2 m_{e}^{*}$ , with $m_{e}^{*}$ the conduction-band effective mass. A parabolic band has the same density of states as a free particle of that mass, shifted to start at $E_{c}$ :

g_{c} (E) = \frac{1}{2 π ^{2}} (\frac{2 m _{e}^{*}}{ℏ ^{2}})^{3/2} E - E_{c}, E \geq E_{c} .

Symmetrically, holes near the top of the valence band have $g_{v} (E) = \frac{1}{2 π ^{2}} (2 m_{h}^{*} / ℏ^{2})^{3/2} E_{v} - E$ for $E \leq E_{v}$ , with $m_{h}^{*}$ the hole effective mass. The electron concentration is the density of states weighted by the probability that a state is occupied:

n = E_{c} \int \infty g_{c} (E) f (E) d E, f (E) = \frac{1}{e ^{(E - E_{F}) / k_{B} T} + 1} .

In an undoped or lightly doped crystal the Fermi level sits deep in the gap, many $k_{B} T$ below $E_{c}$ , so throughout the conduction band $E - E_{F} ≫ k_{B} T$ and the Fermi-Dirac factor collapses to its Boltzmann tail, $f (E) \approx e^{- (E - E_{F}) / k_{B} T}$ . This is the nondegenerate limit, valid whenever the carriers are dilute. The integral is then a standard Gamma integral:

n = e^{- (E_{c} - E_{F}) / k_{B} T} \frac{1}{2 π ^{2}} (\frac{2 m _{e}^{*}}{ℏ ^{2}})^{3/2} E_{c} \int \infty E - E_{c} e^{- (E - E_{c}) / k_{B} T} d E .

With $x = (E - E_{c}) / k_{B} T$ the integral is $(k_{B} T)^{3/2} \int_{0}^{\infty} x^{1/2} e^{- x} d x = (k_{B} T)^{3/2} Γ (\frac{3}{2}) = (k_{B} T)^{3/2} \frac{π}{2}$ , and the concentration takes the compact form

n = N_{c} e^{- (E_{c} - E_{F}) / k_{B} T}, N_{c} = 2 (\frac{m _{e}^{*} k _{B} T}{2 π ℏ ^{2}})^{3/2} .

The prefactor $N_{c}$ is the effective density of states of the conduction band: the whole band acts as though $N_{c}$ states were collapsed onto the edge $E_{c}$ , a distance $E_{c} - E_{F}$ above the Fermi level. The identical argument for holes gives

p = N_{v} e^{- (E_{F} - E_{v}) / k_{B} T}, N_{v} = 2 (\frac{m _{h}^{*} k _{B} T}{2 π ℏ ^{2}})^{3/2} .

The electron concentration n is the overlap of the rising band-edge density of states g_c(E) with the falling Fermi-Dirac tail f(E); the shaded product peaks a few k_BT above E_c and gives n = N_c exp[-(E_c - E_F)/k_BT].

The intrinsic concentration and the law of mass action

The product $n p$ eliminates the Fermi level entirely:

n p = N_{c} N_{v} e^{- (E_{c} - E_{v}) / k_{B} T} = N_{c} N_{v} e^{- E_{g} / k_{B} T},

with $E_{g} = E_{c} - E_{v}$ the band gap. The right side depends only on the material and temperature, not on doping — this is the law of mass action. Defining the intrinsic carrier concentration $n_{i}$ by $n_{i}^{2} \equiv n p$ ,

n_{i} = N_{c} N_{v} e^{- E_{g} /2 k_{B} T} = 2 (\frac{k _{B} T}{2 π ℏ ^{2}})^{3/2} (m_{e}^{*} m_{h}^{*})^{3/4} e^{- E_{g} /2 k_{B} T} .

The exponent $e^{- E_{g} /2 k_{B} T}$ is the dominant factor. Because $E_{g} ≫ k_{B} T$ at room temperature (silicon: $E_{g} / k_{B} T \approx 43$ ), $n_{i}$ is astronomically smaller than the atomic density $\sim 5 \times 1 0^{22} cm^{- 3}$ , and it is exquisitely sensitive to both gap and temperature.

The mass-action balance is the reason a small dose of one carrier type strongly suppresses the other. In n-type silicon with $n = 1 0^{16} cm^{- 3}$ , the hole concentration is driven down to $p = n_{i}^{2} / n \approx 1 0^{4} cm^{- 3}$ , twelve orders of magnitude below the electrons. The doped carrier is the majority carrier; its mass-action partner is the minority carrier, and the minority population — small as it is — controls junction currents in the lessons that follow.

The law of mass action as a hyperbola np = n_i^2. Undoped material sits on the diagonal n = p = n_i; n-type doping slides the state down the hyperbola (n up, p down), p-type doping slides it the other way, always on the same curve.

The Fermi level in intrinsic material

Setting $n = p$ in the two Boltzmann forms fixes the intrinsic Fermi level $E_{i}$ . Equating $N_{c} e^{- (E_{c} - E_{i}) / k_{B} T} = N_{v} e^{- (E_{i} - E_{v}) / k_{B} T}$ and solving,

E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{1}{2} k_{B} T ln \frac{N _{v}}{N _{c}} = \frac{E _{c} + E _{v}}{2} + \frac{3}{4} k_{B} T ln \frac{m _{h}^{*}}{m _{e}^{*}} .

At $T = 0$ the Fermi level sits exactly at midgap. When the hole mass exceeds the electron mass (as in silicon, $m_{h}^{*} > m_{e}^{*}$ ) the second term is positive and $E_{i}$ drifts upward toward the conduction band as $T$ rises; the shift is only a few $k_{B} T ln (m_{h}^{*} / m_{e}^{*})$ , tens of millivolts at most, so $E_{i}$ stays close to midgap over the whole ordinary temperature range.

Doping and charge neutrality

A donor contributes a level $E_{d}$ a small energy $ε_{d} = E_{c} - E_{d}$ below the conduction band; an acceptor a level $E_{a}$ an energy $ε_{a} = E_{a} - E_{v}$ above the valence band. Both are shallow (silicon: $ε_{d} \approx 0.045 eV$ for phosphorus, $ε_{a} \approx 0.045 eV$ for boron), of order $k_{B} T$ at room temperature, so most dopants are ionized. A donor that has given up its electron is a fixed positive charge $N_{d}^{+}$ ; an acceptor that has captured one is a fixed negative charge $N_{a}^{-}$ . Global charge neutrality ties the mobile and fixed charges together:

n + N_{a}^{-} = p + N_{d}^{+} .

The ionized-donor fraction follows from the occupation of the donor level, which carries a factor $\frac{1}{2}$ from spin degeneracy of the bound state:

N_{d}^{+} = \frac{N _{d}}{1 + 2 e ^{(E_{F} - E_{d}) / k_{B} T}}, N_{a}^{-} = \frac{N _{a}}{1 + 4 e ^{(E_{a} - E_{F}) / k_{B} T}} .

Consider an n-type crystal with donor density $N_{d}$ and no acceptors. Neutrality reads $n = p + N_{d}^{+}$ . Together with $n p = n_{i}^{2}$ this is a closed system for $n$ and $p$ at each temperature. Three regimes emerge as temperature rises, and they are the defining behavior of a doped semiconductor.

The three doping regimes on a log-n versus 1/T plot for an n-type crystal. At low T carriers freeze onto donors (steep slope); an intermediate saturation plateau has all donors ionized so n = N_d; at high T intrinsic generation overwhelms the doping and n follows the steep n_i slope.

Freeze-out (low $T$ ). Thermal energy is too small to ionize all donors, so electrons condense back onto their donor atoms. With $n ≪ N_{d}$ and $p$ negligible, neutrality gives $n \approx N_{d}^{+}$ , and solving the occupation relation in the dilute limit yields

n \approx \frac{1}{2} N_{c} N_{d} e^{- ε_{d} /2 k_{B} T} .

The concentration falls off with a slope set by half the donor binding energy — the shallow analogue of the intrinsic slope, with $ε_{d}$ in place of $E_{g}$ .

Saturation / extrinsic ( $T$ intermediate). Once $k_{B} T ≳ ε_{d}$ every donor is ionized, $N_{d}^{+} = N_{d}$ , while intrinsic generation is still negligible ( $n_{i} ≪ N_{d}$ ). Neutrality collapses to $n \approx N_{d}$ : the electron concentration is pinned at the donor density and is essentially flat over a wide temperature window. This is the useful regime for devices, where the carrier count is set by doping and is nearly temperature-independent.

Intrinsic (high $T$ ). When $T$ climbs high enough that $n_{i} > N_{d}$ , thermal generation across the full gap dominates the dopants; $n \approx p \approx n_{i}$ and the doping becomes irrelevant. The concentration resumes the steep intrinsic slope $e^{- E_{g} /2 k_{B} T}$ . The onset temperature rises with doping and with gap, which is why wide-gap semiconductors like silicon carbide and gallium nitride keep their extrinsic behavior — and their device function — to much higher temperatures than germanium.

The temperature march of the Fermi level

Because $n = N_{c} e^{- (E_{c} - E_{F}) / k_{B} T}$ , the Fermi level position is a direct readout of the carrier count: $E_{c} - E_{F} = k_{B} T ln (N_{c} / n)$ . Following $n$ through the three regimes traces $E_{F}$ across the gap.

Near $T = 0$ : in n-type material $E_{F}$ lies between the donor level and the conduction band, because the highest occupied states are the donor electrons.
Saturation: with $n = N_{d}$ fixed, $E_{c} - E_{F} = k_{B} T ln (N_{c} / N_{d})$ grows linearly in $T$ — the Fermi level descends from near $E_{c}$ toward midgap as temperature rises.
Intrinsic: $E_{F} \to E_{i}$ near the middle of the gap, the same limit a p-type crystal approaches from below.

Fermi-level position versus temperature. In n-type material E_F starts just below the conduction band, descends through the saturation regime, and merges into the near-midgap intrinsic line E_i at high T; the p-type curve is the mirror image rising from just above the valence band.

The heavily doped limit deserves a caution. When $N_{d}$ approaches $N_{c}$ the Fermi level rises into or above the conduction band, the Boltzmann approximation fails, and the semiconductor becomes degenerate — the electron gas behaves like that of a metal, with the full Fermi-Dirac statistics and a Fermi surface. Degenerate doping is exactly what tunnel diodes and the ohmic contacts of every chip rely on. For the moderate doping of ordinary devices, however, the nondegenerate formulas above hold, and they set the stage for carrier transport: knowing how many carriers there are, the next question is how fast they move and how long they live.