Intrinsic Semiconductors and Carrier Concentration

The syllabus marks this one (Derivation), so the derivation is the content, not the decoration.

What an intrinsic semiconductor is

A pure semiconductor — silicon or germanium with no deliberate impurities. At absolute zero the valence band is full, the conduction band empty, and it is an insulator.

Warm it, and some electrons cross the gap. Each one that leaves the valence band leaves behind a vacancy. That vacancy is a hole, and it behaves as a mobile positive charge: a neighbouring electron slides into it, moving the vacancy the other way, and the bookkeeping is far simpler if you track the vacancy rather than the millions of electrons shuffling.

Because every conduction electron comes from the valence band,

in a pure sample. This equality is what "intrinsic" means, and it is the step the derivation has to reach.

Density of electrons in the conduction band

Two ingredients multiply together:

• $Z(E)\,dE$, the density of states — how many states exist between $E$ and $E+dE$;
• $f(E)$, the Fermi function from Module 2 — the probability each is occupied.

The density of states near the bottom of the conduction band, for a free electron of effective mass $m_e^*$ measured from the band edge $E_C$:

The Fermi function simplifies. $E_F$ sits near mid-gap, so for any state in the conduction band $E - E_F$ is at least half a gap — several hundred meV, against $k_BT = 26$ meV. The exponential is therefore enormous compared with 1:

This is the Boltzmann approximation, and it is what makes the integral doable. It is valid precisely because the semiconductor is non-degenerate — lightly doped, with $E_F$ well inside the gap. Dope it heavily enough that $E_F$ enters a band and this step fails.

Doing the integral:

Substitute $x = (E - E_C)/k_BT$ and pull out the constant part of the exponential:

The standard integral is $\Gamma(3/2) = \sqrt{\pi}/2$, giving

$N_C$ is the effective density of states in the conduction band. The whole band has been replaced by a single equivalent level at $E_C$ holding $N_C$ states — a considerable simplification, and legitimate because the Boltzmann factor falls so fast that only states within a few $k_BT$ of the edge matter.

Density of holes in the valence band

Identical argument, upside down. A hole is an unoccupied state, so its probability is $1 - f(E)$, and for $E$ well below $E_F$:

Integrating downwards from $E_V$ with effective mass $m_h^{*}$:

Intrinsic carrier concentration

Multiply the two. The $E_F$ terms cancel — which is the point:

The Fermi level has vanished. This is the law of mass action: $np$ depends only on the material and the temperature, not on doping. It holds for doped semiconductors too, and it is one of the most useful results in the subject.

For an intrinsic sample $n = p = n_i$, so

Note the $E_g/2$ — the square root halves the exponent. Forgetting it is the single most common error here.

Where the Fermi level sits

Setting $n = p$ and solving for $E_F$:

The first term is the exact middle of the gap. The second is a small correction — at 300 K it is a few meV — that vanishes if the effective masses are equal. For an intrinsic semiconductor the Fermi level lies essentially at mid-gap, and that is the answer an exam wants, with the correction quoted as the refinement.

Variation with temperature

Everything is dominated by the exponential. The $T^{3/2}$ in $N_C$ and $N_V$ is real but feeble by comparison.

The practical consequence is dramatic, as the second worked example shows: a hundred-degree rise multiplies the carrier concentration by hundreds. This is why semiconductor resistance falls steeply with temperature, and why power devices need heat sinks — a hot device conducts more, which heats it further, which makes it conduct more still. That loop is thermal runaway, and it is a real failure mode, not a theoretical curiosity.