Carrier concentration equations (n, p, ni): Visual walkthrough

We are going to answer one question: how many mobile electrons live in the conduction band?

The whole plan, in one diagram:

Step 1 — Draw the two energy bands and the gap

WHAT. Energy $E$ runs upward on a vertical axis (like altitude). Two shaded regions matter:

the valence band, everything at or below an energy we name $E_{v}$ (the "main floor", packed with electrons),
the conduction band, everything at or above an energy we name $E_{c}$ (the "balcony", where mobile electrons live).

Between them is an empty stripe — no allowed energies at all — of height $E_{g} = E_{c} - E_{v} .$

Here each symbol is doing a job:

$E$ ::: the energy of one electron state (vertical position on our picture).
$E_{c}$ ::: the bottom edge of the conduction band (lowest energy a conducting electron can have).
$E_{v}$ ::: the top edge of the valence band.
$E_{g}$ ::: the band gap, the forbidden height an electron must leap to conduct.

WHY. An electron only conducts if it sits above $E_{c}$ . So the count $n$ will only ever integrate over the balcony — this picture fixes the integration region before we write a single integral. (See Band gap Eg and Intrinsic vs extrinsic semiconductors.)

PICTURE.

Figure — Carrier concentration equations (n, p, ni)

Step 2 — Count the seats: the density of states $g_{c} (E)$

WHAT. Not every energy in the balcony offers the same number of "seats". The number of available states per unit energy per unit volume is the density of states, written $g_{c} (E)$ . Near the edge it grows as a square root: $g_{c} (E) = a constant \frac{1}{2 π ^{2}} (\frac{2 m _{n}^{*}}{ℏ ^{2}})^{3/2} shape E - E_{c} .$

Term by term:

$E - E_{c}$ ::: distance you have climbed above the edge; zero right at $E_{c}$ , bigger higher up.
$m_{n}^{*}$ ::: the electron's Effective mass — heavier electrons pack more states in.
$ℏ$ ::: the reduced Planck constant — Planck's constant $h$ divided by $2 π$ , so $ℏ = h / (2 π)$ ; it sets the size of one quantum "seat". We will meet $h$ itself again in Step 7.
the whole prefactor ::: just a fixed number that scales the curve.

WHY a square root? In three dimensions the number of quantum states with energy up to $E$ grows like the volume of a sphere in wave-number space. Differentiate a cube-root-radius sphere's volume with respect to energy and out falls $E - E_{c}$ . So: more room higher up. (Full construction lives in Density of states.)

PICTURE. The orange curve starts flat at $E_{c}$ and fans outward as we climb.

Step 3 — Ask who actually sits: the occupancy $f (E)$

WHAT. Seats existing is not the same as seats filled. The probability that a state at energy $E$ is occupied is the Fermi-Dirac distribution: $f (E) = \frac{1}{1 + e ^{(E - E_{F}) / k T}} .$

Term by term:

$E_{F}$ ::: the Fermi level — the energy at which a seat is filled exactly half the time ( $f = \frac{1}{2}$ ).
$E - E_{F}$ ::: how far the state sits above the Fermi level.
$k$ ::: Boltzmann's constant (converts temperature into energy).
$T$ ::: absolute temperature; $k T$ is the "thermal fuzziness" energy ( $\approx 0.0259$ eV at 300 K).

WHY this shape? Electrons obey the exclusion rule: one per state, and warmth smears the cutoff. $f$ is $\approx 1$ far below $E_{F}$ (seats full), drops through $\frac{1}{2}$ at $E_{F}$ , and $\approx 0$ far above (seats empty). The steepness of the drop is set by $k T$ .

PICTURE. Blue S-curve falling from 1 to 0, half-way exactly at $E_{F}$ .

Step 4 — Only the tail matters: the Boltzmann approximation

WHAT. In a lightly doped ("non-degenerate") crystal, $E_{F}$ sits deep in the gap, far below $E_{c}$ . So for every conducting state ( $E \geq E_{c}$ ) we have $E - E_{F} ≫ k T$ , the $e^{(E - E_{F}) / k T}$ term dwarfs the "+1", and $f (E) \approx e^{- (E - E_{F}) / k T} .$

Term by term:

the "+1" ::: dropped because $e^{(E - E_{F}) / k T}$ is already huge in the balcony.
$e^{- (E - E_{F}) / k T}$ ::: a plain exponential decay — the far tail of the S-curve.

WHY do this? A bare exponential is integrable in closed form; the full Fermi–Dirac is not. This one approximation is what turns the whole problem into a clean boxed formula.

PICTURE. Zoom into the region above $E_{c}$ : the blue exact curve and the green Boltzmann tail lie on top of each other.

Step 5 — Multiply the two curves: seats × occupancy

WHAT. The number of filled seats at energy $E$ is (seats there) × (chance each is filled): $g_{c} (E) f (E) \approx \frac{1}{2 π ^{2}} (\frac{2 m _{n}^{*}}{ℏ ^{2}})^{3/2} E - E_{c} e^{- (E - E_{F}) / k T} .$

Two opposing tendencies collide:

$E - E_{c}$ ::: rising — more seats higher up.
$e^{- (E - E_{F}) / k T}$ ::: falling — less likely to be filled higher up.

WHY it makes a bump. Rising × falling = a hump that peaks a little above $E_{c}$ and dies out fast. Almost all conduction electrons live within a few $k T$ of the band edge — that is what the hump shows.

PICTURE. Orange (rising) and green (falling) drawn faint; their product is the solid purple hump.

Step 6 — Add it all up: the integral becomes an area

WHAT. To get the total electron count $n$ we sum the hump over all balcony energies — that is exactly an integral: $n = \int_{E_{c}}^{\infty} g_{c} (E) f (E) d E = shaded area under the purple hump .$

Now substitute $x = (E - E_{c}) / k T$ (measure energy in units of $k T$ above the edge). The whole prefactor pops out and the leftover integral is a pure number: $\int_{0}^{\infty} x e^{- x} d x = \frac{π}{2} .$

$x$ ::: dimensionless height above $E_{c}$ , in thermal units.
$\frac{π}{2}$ ::: the fixed area of the "standard hump" — a universal number, the same for every material.

WHY substitute? It separates the shape (a universal number) from the scale ( $N_{c}$ and the exponential). This is the algebra that produces the boxed answer.

PICTURE. The purple hump, filled in, with its area labelled $n$ .

Step 7 — The result: $n = N_{c} e^{- (E_{c} - E_{F}) / k T}$

WHAT. Multiplying the prefactor from Step 5 by $\frac{π}{2}$ from Step 6 and folding in the $(k T)^{3/2}$ that came out of the substitution gives: $n = N_{c} e^{- (E_{c} - E_{F}) / k T}, N_{c} = 2 (\frac{2 π m _{n}^{*} k T}{h ^{2}})^{3/2} .$

Term by term:

$N_{c}$ ::: the effective density of states — pretend the entire smeared band collapsed into one single level at $E_{c}$ holding $N_{c}$ seats. ( $\approx 2.8 \times 1 0^{19} cm^{- 3}$ for Si at 300 K.)
$h$ ::: Planck's constant — the same constant as before, since $h = 2 π ℏ$ . We swap $ℏ$ for $h$ only to make the final box tidy; nothing new is introduced.
$e^{- (E_{c} - E_{F}) / k T}$ ::: the Boltzmann factor measuring how far $E_{F}$ sits below the edge; the deeper $E_{F}$ , the exponentially fewer electrons.
$E_{c} - E_{F}$ ::: measured downward from the band edge — "electrons fall from the top."

WHY it's beautiful. A messy integral over infinitely many states becomes: (one effective seat count) × (one occupancy factor) — the recurring idea of this whole chapter.

By the identical picture flipped upside-down (empty seats $1 - f$ below $E_{v}$ ), holes give $p = N_{v} e^{- (E_{F} - E_{v}) / k T}, N_{v} = 2 (\frac{2 π m _{p}^{*} k T}{h ^{2}})^{3/2} .$

$N_{v}$ ::: the effective density of states in the valence band — the mirror of $N_{c}$ , as if the whole valence band collapsed into one level at $E_{v}$ holding $N_{v}$ seats. It uses the hole effective mass $m_{p}^{*}$ . ( $\approx 1.04 \times 1 0^{19} cm^{- 3}$ for Si at 300 K.)

PICTURE. The real band's smeared electrons redrawn as one sharp spike of height $N_{c}$ at $E_{c}$ , scaled by the exponential.

Step 8 — The degenerate edge case: when Step 4 breaks

WHAT. Push doping until $E_{F}$ reaches $E_{c}$ . Now near the edge $E - E_{F} \approx 0$ , so $f \approx \frac{1}{2}$ , not the small exponential. Boltzmann predicted a decaying tail; the truth is a nearly-full band edge that cannot exceed $f = 1$ .

WHY it matters. The boxed formula would keep growing without bound as $E_{F}$ rises, but reality saturates. The correct count needs the Fermi–Dirac integral $F_{1/2}$ , not $e^{- (E_{c} - E_{F}) / k T}$ .

Rule of thumb: trust the box only while $E_{c} - E_{F} ≳ 3 k T .$

PICTURE. Two occupancy curves overlaid: non-degenerate (green tail, formula valid) vs degenerate (blue, $f$ flattening near 1, formula fails).

The one-picture summary

Everything at once: the two rising/falling curves, their product hump, the shaded area $n$ , and the collapse into a single effective level $N_{c}$ scaled by the Boltzmann factor.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a theater. The balcony is the conduction band; a seat's height is its energy. Step 1: I mark where the balcony floor starts ( $E_{c}$ ) and how far below the packed main floor sits ( $E_{g}$ down at $E_{v}$ ). Step 2: I count the seats — there are more of them the higher you climb, growing like a square root. Step 3: But existing isn't sitting. The chance a seat is filled is the Fermi–Dirac S-curve — full low down, empty high up, exactly half full at the Fermi level $E_{F}$ . Step 4: Since $E_{F}$ is way below the balcony, the chance up there is just a fading exponential tail. Step 5: Multiply "many seats" (rising) by "small chance" (falling) and you get a little hump hugging the balcony floor — almost everyone sits near the edge. Step 6: Sum the hump — that shaded area is the electron count $n$ . Step 7: The sum tidies up into one clean idea: pretend all those seats squished into one level at $E_{c}$ holding $N_{c}$ of them, times how far $E_{F}$ sits below. Step 8: The one caution — if you dope so hard that $E_{F}$ reaches the balcony, seats can't be more than 100% full, the exponential lie breaks, and you need the honest Fermi–Dirac count.

Related: Doping and charge neutrality · Fermi level position · Effective mass · Density of states · Fermi-Dirac distribution · Band gap Eg