Temperature dependence of carrier concentration: Foundations

This page assumes nothing. We build every letter, subscript, and picture the parent note leans on, in an order where each idea rests only on the ones before it.

0. The stage: energy, and a picture of "up"

Before any symbol, one picture. Imagine a vertical ladder where height = energy. Low rungs = low energy = comfortable, bound states. High rungs = high energy = free, mobile states. Every symbol below lives somewhere on this ladder.

Figure — Temperature dependence of carrier concentration

Why need it? Because the entire topic asks "can an electron afford to climb from here to there?" — and you can't ask that without a height axis.

1. The two shelves: valence band and conduction band

Look at figure s01. The ladder is not evenly filled — real crystals leave a forbidden gap with no rungs at all.

For silicon $E_{g} \approx 1.12$ eV. Remember this number; the parent page's worked examples orbit it. See Intrinsic vs extrinsic semiconductors for what "pure vs doped" does to this picture.

2. Heat, spelled out: $k_{B}$ , $T$ , and the budget $k_{B} T$

The parent page writes $k_{B} T$ everywhere. Here is what each letter is.

At $300$ K, $k_{B} T = 8.617 \times 1 0^{- 5} \times 300 \approx 0.02585$ eV. That is the $0.02585$ eV you see in the silicon example.

3. The exponential $e^{- E / k_{B} T}$ — why THIS function

The single most-used shape on the parent page is $e^{- E / k_{B} T}$ . Why an exponential and not, say, $1/ E$ ?

Look at figure s02: a modest change in $1/ T$ slides you along a curve whose steepness is set entirely by $E$ . Big $E$ (like $E_{g}$ ) → steep; small $E$ (like a donor's $E_{D}$ ) → gentle. That single fact is what makes the parent page's three-region plot have three different slopes.

4. Occupancy: the Fermi level $E_{F}$ and $f (E)$

We know the shelves; now: which rungs are actually occupied?

Read the pieces you now own: $E - E_{F}$ is "how far above the water line," and dividing by $k_{B} T$ compares that to the heat budget — the same ratio from §2, now inside the same exponential from §3.

5. Seats: density of states $g (E)$ and the constants $N_{C}, N_{V}$

Occupancy tells us the chance a rung is filled. But how many rungs are there at each height?

Counting electrons is then just seats × occupancy, summed over all heights — the whole of the parent's Step 4. Doing that sum collapses the messy $g (E)$ into a single tidy number per band:

Definition Effective density of states

$N_{C}$ and $N_{V}$ $N_{C} = 2 (\frac{2 π m _{e}^{*} k _{B} T}{h ^{2}})^{3/2}, N_{V} = 2 (\frac{2 π m _{h}^{*} k _{B} T}{h ^{2}})^{3/2}$ $N_{C}$ behaves as if all the conduction-band seats were bunched at the edge $E_{C}$ into one "effective" pile of size $N_{C}$ . Same idea for $N_{V}$ at $E_{V}$ .

The leading factor of 2 is spin degeneracy: every energy state can hold two electrons, one spin-up and one spin-down (a quantum rule). So each "seat" is really a bench for two — hence the ×2.
$m_{e}^{*}, m_{h}^{*}$ = effective masses — how "heavy" an electron/hole acts inside the crystal (not its true mass; the lattice fudges it).
$h$ = Planck's constant, the quantum bookkeeping factor that sets how densely states can pack.
Note the $T^{3/2}$ : warmer means more thermally-reachable seats, but this grows slowly compared to the exponential.

Result of seats × occupancy: $n = N_{C} e^{- (E_{C} - E_{F}) / k_{B} T}$ . Every symbol in it is now yours.

6. The carrier symbols: $n$ , $p$ , $n_{i}$ , and mass action

7. The doping symbols: donors and acceptors

Real devices aren't pure — we add dopants. See Doping and dopant ionization. There are two flavours, mirror images of each other.

8. The plot axes: $ln n$ and $1/ T$ — why these choices

The parent plots $ln n$ against $1/ T$ , not $n$ against $T$ . Why the strange axes?

That is the deep reason the parent's three regions show as three straight segments with three slopes.

Prerequisite map

Equipment checklist

Test yourself — you are ready for the parent page when you can answer each without peeking.

What is the band gap

E_{g}

in one sentence and a picture?

The empty energy height

E_{C} - E_{V}

an electron must jump to go from bound (valence) to free (conduction); the gap between the two shelves on the ladder.

What does

k_{B} T

physically represent?

The thermal energy budget per particle — heat converted into energy units via

k_{B} = 8.617 \times 1 0^{- 5}

eV/K.

Why does nature use

e^{- E / k_{B} T}

rather than a simple fraction?

Because raising an energy barrier multiplies the difficulty (odds), and exponentials encode multiplicative penalties.

What is the Fermi level

E_{F}

The energy at which a state is exactly 50% likely to be occupied — the "water line" of the electron sea.

When may we replace Fermi-Dirac

f (E)

with

e^{- (E - E_{F}) / k_{B} T}

When

E - E_{F} ≫ k_{B} T

(non-degenerate, far above the water line), so the

+ 1

in the denominator is negligible.

Why does

g (E)

rise as

E - E_{C}

at the band edge?

A parabolic band gives speed

\propto E - E_{C}

, and the number of momentum directions in 3-D grows with that speed — so seats pile up as

E - E_{C}

Why is there a leading factor of 2 in

N_{C}

and

N_{V}

Spin degeneracy — each energy state holds two electrons (spin-up and spin-down).

What do

N_{C}

and

N_{V}

summarise, and how do they scale with

T

The effective density of states bunched at each band edge; both scale as

T^{3/2}

What is a hole

p

A missing valence electron that behaves like a mobile positive charge.

State the law of mass action and why

E_{F}

vanishes from it.

n p = N_{C} N_{V} e^{- E_{g} / k_{B} T} = n_{i}^{2}

; multiplying

n

and

p

cancels the

E_{F}

terms, leaving a doping-independent product.

Why is

n_{i} \propto e^{- E_{g} /2 k_{B} T}

(factor of 2)?

Because

n_{i} = n p

, and taking the square root halves the exponent.

Difference between

N_{D}

and

N_{D}^{+}

(and their acceptor mirrors)?

N_{D}

is all donor atoms added;

N_{D}^{+}

is only those that have actually ionized. For acceptors,

N_{A}

is all added and

N_{A}^{-}

is those that captured an electron.

What is

ln x

and why plot

ln n

1/ T

ln

is the inverse of

e^{x}

, pulling the exponent out; it turns each exponential region into a straight line whose slope

- E / k_{B}

reveals the controlling energy.

Now that every symbol is earned, return to the parent note and read the derivations — nothing there will be unfamiliar.