Foundations — Fermi level and Fermi-Dirac distribution

This page assumes nothing. Every letter, every symbol, every idea the parent note leans on is built here from scratch, in the order they depend on each other. If a piece of the parent note ever felt like it appeared out of nowhere, this is where it was quietly assumed — and now we make it explicit. Because the symbols must be earned before use, the finished formula only appears at the very end (§9), after every letter inside it has been built.

1. Energy $E$ — the vertical axis of everything

Plain words. Energy is a number we attach to a state that says "how much punch a particle here carries." For electrons in a solid we measure it in electron-volts (eV): the energy one electron gains falling through one volt. It is tiny — about $1.6\times 10^{-19}$ joules — so eV is the comfortable unit.

The picture. Draw a single vertical line. Down is low energy, up is high energy. Every diagram in this whole topic uses this same vertical axis. A "state" is a little shelf at some height $E$ on that line.

Why the topic needs it. The Fermi–Dirac curve (built in §9) is a function of $E$. Without a clear "up = more energy" axis, phrases like "states above the water line" or "the band gap" have no meaning. This axis is the stage.

2. The system and its reservoir — who is exchanging what

Plain words. Our system is the tiny thing we care about — in the extreme case, a single state (one shelf). Around it sits a huge reservoir: the rest of the solid, an enormous bath of electrons and heat so large that giving or taking a few electrons or a bit of energy doesn't change its temperature at all.

The picture. A small cup (the state) floating in a vast ocean (the reservoir). The ocean can hand the cup an electron, or take one back, freely — but the ocean itself stays at the same fixed temperature $T$ the whole time.

Why the topic needs it. The Fermi–Dirac curve is derived by asking "how often does the reservoir put an electron into this one state?" We can only answer that if we first name the two players and say they swap electrons and heat. Every time this page later says "reservoir," it means exactly this surrounding ocean.

3. Temperature $T$ and the thermal budget $k_{B}T$

Plain words. ==Temperature $T$ (measured in kelvin, K==, where $0$ K is the coldest possible, absolute zero) is a measure of how much random jiggling energy the reservoir hands out. Hotter means more jiggle.

To turn a temperature into an energy we multiply by ==Boltzmann's constant $k_B$==, a fixed conversion factor: $k_B=8.617\times 10^{-5}\text{eV/K}.$

The product ==$k_{B}T$== is therefore an energy — the typical thermal "kick" available at temperature $T$.

The picture. Imagine a short ruler of length $k_{B}T$ laid against the energy axis. Every energy gap in the theory gets compared to this ruler. If a gap is many rulers tall, thermal jiggle can't bridge it; if a gap is a fraction of a ruler, jiggle crosses it easily.

Why this tool and not another? Every formula in this topic compares an energy difference to the available thermal energy. Bundling them as a single ratio (an energy divided by $k_{B}T$, a pure number with no units) is what makes the maths clean — you will see this exact ratio inside every exponential in §4.

4. The exponential $e^x$ — the shape of "how likely"

Plain words. ==$e$== is a fixed number, $e\approx 2.718$. Raising it to a power, ==$e^x$== ("$e$ to the $x$"), is the exponential function. Its whole personality:

• $x=0\Rightarrow e^0=1$ (the pivot).
• $x$ large positive $\Rightarrow e^x$ shoots to huge.
• $x$ large negative $\Rightarrow e^x$ shrinks toward zero (never quite reaching it).

The picture. A curve hugging the floor on the left, passing through height $1$ at $x=0$, then rocketing upward to the right.

Why this tool and not another? Nature's rule for "how much more likely is a higher-energy arrangement" is that likelihood falls off exponentially with energy cost, per thermal budget: $e^{-(\text{energy cost})/k_{B}T}$. This single expression is called the Boltzmann factor: it is the weight the reservoir (§2) attaches to an arrangement, and it shrinks fast as the arrangement costs more energy. (When particle exchange with the reservoir is added — the $\mu$ term of §6 — the same expression is often called the Gibbs factor, but you can safely think of it as "the Boltzmann factor, now bookkeeping electrons too.") We use the exponential because it is the unique function whose rate of change is proportional to itself — the mathematical signature of "each extra unit of energy multiplies the odds by the same fixed factor." That is exactly how the state-versus-reservoir counting works in the parent note's derivation.

5. The natural logarithm $\ln$ — the exponential run backwards

Plain words. ==$\ln$== (natural log) is the button that undoes $e^x$. It answers: "$e$ to what power gives this number?" $\ln \left(e^x\right)=x,e^{\ln y}=y.$ Examples the parent note uses: $\ln 9=2.197$, and $\ln \left(\frac{1-f}{f}\right)$ appears in the inversion formula.

The picture. Take the exponential curve of §4 and flip it across the diagonal line $y=x$. That mirror image is $\ln$.

Why the topic needs it. Once you have the occupation curve and want to go the other way — "for this occupation value, how far above or below the water line am I?" — you must undo the exponential. That is precisely what the parent's worked example 3 does: $E-E_F=k_{B}T\ln \left(\frac{1-f}{f}\right).$ (The symbols $E_F$ and $f$ here are earned in §6 and §8.) Without $\ln$ we could plot the curve but never solve it backwards.

6. Chemical potential $\mu$ and the Fermi level $E_F$

Plain words. The ==chemical potential $\mu$== is the energy cost (or gain) of adding one more electron to the system from the reservoir (§2). Think of it as the current "water line" of the electron sea. In this topic, for electrons at the temperatures we care about, $\mu$ is what we call the ==Fermi level $E_F$== (the parent note uses $\mu \to E_F$). So $E_F$ is the special energy on our axis that marks the water surface.

The picture. Water filling a tank. The surface of the water is $E_F$. Below the surface: full (electrons). Above: empty. Add heat and the surface gets fuzzy — a few droplets splash above the line, leaving gaps below.

Why the topic needs it. $E_F$ is the reference point for the whole curve — everything only ever cares about $E-E_F$, the height relative to the water line. And in equilibrium this line is flat across a connected system, which anchors everything later about junctions and doping in Intrinsic and extrinsic semiconductors.

7. Pauli exclusion and the occupation number $n$

Plain words. The Pauli exclusion principle says at most one electron per quantum state. Two electrons refuse to occupy the identical state.

The picture. A stack of shelves, each shelf marked "MAX 1." Electrons queue and fill from the bottom; once a shelf holds one, the next electron must go higher.

Why this tool and not another? Classical particles (marbles) can pile arbitrarily many into the lowest energy slot. If electrons behaved that way we would use plain Boltzmann statistics and the whole story would collapse to a simple decaying exponential. The one-per-state cap forces the occupation of a state to saturate at 1, never more — and that ceiling is exactly why the final formula (§9) has that "$1+\dots$" in the denominator. See Pauli exclusion principle for the deeper reason (electron spin/antisymmetry).

8. The occupation probability

Plain words. The occupation probability — which we will name $f(E)$ in §9 — is a probability, a pure number between $0$ and $1$ (no units). Value $1$ means "certainly occupied," $0$ means "certainly empty," $\frac{1}{2}$ means "fifty-fifty."

The picture. For each height $E$ on the energy axis, this probability is a dial reading between $0$ and $1$. Stack those dials and you trace out the Fermi–Dirac curve: pinned at $1$ low down, sliding through $\frac{1}{2}$ at $E_F$, falling to $0$ high up.

Why "probability" and not "number of electrons"? Because a single state can only ever hold $0$ or $1$ electron (Pauli, §7), its average occupation is automatically a fraction between $0$ and $1$ — i.e. exactly a probability. Calling it a count is the parent note's third steel-manned mistake.

9. Assembling the Fermi–Dirac distribution

Now every symbol is earned — energy $E$ (§1), the reservoir (§2), the ruler $k_{B}T$ (§3), the exponential (§4), the water line $E_F$ (§6), the Pauli cap and average occupation (§7–8). Put them together and the occupation probability, written ==$f(E)$==, is:

This boxed formula is the whole topic in one line; the parent note reads and applies it. Everything on this page was the vocabulary needed to earn every one of its symbols.

10. Bands, the gap, and density of states $g(E)$ — where states even exist

Plain words. In a solid the allowed energies bunch into bands (continuous ranges of allowed states) separated by a band gap (a forbidden zone with no states). The ==Density of states $g(E)$== tells you how many states sit in a thin slice of energy near $E$.

The picture. Along the energy axis, states are dense inside bands and absent in the gap; $g(E)$ is the thickness of that band shading at each height. The figure below shows the two bands, the empty gap, and how multiplying $g(E)$ by $f(E)$ picks out the actually-occupied electrons.

Why the topic needs it. $f(E)$ only gives a probability; it is silent about whether a state even exists at $E$. To count real electrons you must multiply "chance a shelf is occupied" by "how many shelves there are": $n=\int g(E)f(E)dE.$ Here $n$ is the total carrier count (the second meaning of $n$ flagged in §7), not the single-state occupation. This is why the parent note insists $f(E)$ is not an electron count — that job needs $g(E)$ (Density of states, Carrier concentration n and p).

How these foundations feed the topic

The chain of dependencies, in words (each link means "is needed to build the next"):

1. Energy axis $E$ (§1) + reservoir (§2) set the stage.
2. Temperature $T$ (§3) becomes the energy ruler $k_{B}T$.
3. An energy difference divided by $k_{B}T$ is a pure ratio, fed into the exponential $e^x$ (§4).
4. The Fermi level $E_F$ (§6) provides the reference the difference $E-E_F$ is measured from.
5. Pauli exclusion (§7) supplies the "$1+$" cap; the average occupation (§8) is the probability.
6. Steps 3–5 assemble into the Fermi–Dirac curve $f(E)$ (§9).
7. Multiply by the density of states $g(E)$ (§10) to get the total carrier count $n$.
8. The natural log $\ln$ (§5) lets you run $f(E)$ backwards to find $E-E_F$.

Equipment checklist

Test yourself — cover the right side. If any answer is shaky, re-read that section before the parent note.

Connections

• Parent: Fermi level and Fermi–Dirac distribution — this page is its from-zero foundation.
• Pauli exclusion principle · Chemical potential and equilibrium · Density of states — the three pillars.
• Band gap and conduction/valence bands · Intrinsic and extrinsic semiconductors · Carrier concentration n and p — where these tools get used next.