Fermi level and Fermi-Dirac distribution

WHY do we even need this?

WHAT problem are we solving? We have billions of electrons and a continuum of allowed energy states (bands). We can't track each electron. We need a statistical rule: given a state at energy $E$, what's the chance an electron sits there?

WHY not just use classical Boltzmann statistics? Because electrons obey the Pauli exclusion principle: at most one electron per quantum state. Classical particles are distinguishable and can share states freely. That single constraint changes the whole distribution.

Deriving the Fermi–Dirac distribution from scratch

We derive $f(E)$ using the grand-canonical idea: a single state exchanges electrons with a big reservoir at temperature $T$ and chemical potential $\mu$.

Step 1 — A state can only hold 0 or 1 electron. Why this step? Pauli exclusion. Occupation number $n \in \{0, 1\}$ only.

Step 2 — Grand-canonical probability of each option. The probability of a state having energy $E$ and $n$ electrons is proportional to the Gibbs factor $e^{-(E n - \mu n)/k_BT}$. Why this step? $\mu$ (chemical potential) is the energy cost/gain of adding one particle from the reservoir; the $-\mu n$ term accounts for particle exchange.

• $n=0$: weight $= e^{0} = 1$
• $n=1$: weight $= e^{-(E-\mu)/k_BT}$

Step 3 — Normalize to get the average occupation.

Why this step? Average occupation = (sum of $n\times$weight) / (sum of weights).

Step 4 — Divide top and bottom by the numerator's exponential to get the clean form:

At $T=0$, $\mu \to E_F$, and even at finite $T$ in semiconductors we usually call $\mu \approx E_F$. So:

HOW $f(E)$ behaves — reading the curve

At $T=0$ K (the step function):

• $E < E_F$: $e^{(E-E_F)/k_BT} \to e^{-\infty} = 0 \Rightarrow f=1$ (all states below filled).
• $E > E_F$: $e^{+\infty}\to\infty \Rightarrow f=0$ (all states above empty).

At $T>0$: the sharp step smears over a width of a few $k_BT$. Some electrons get thermally kicked above $E_F$, leaving empty states (holes) below.

Boltzmann approximation (the 80/20 shortcut)

When $E - E_F \gg k_BT$ (states well above $E_F$, e.g. conduction band in a semiconductor), the $1$ in the denominator is negligible:

Why this matters: this is the classical Boltzmann tail — it's why we can compute carrier concentrations easily in non-degenerate semiconductors. This single approximation covers ~80% of device calculations.

Worked examples

Common mistakes (Steel-manned)

Active-recall flashcards

Connections

• Density of states — pair $f(E)$ with $g(E)$ to get carrier counts.
• Intrinsic and extrinsic semiconductors — doping shifts $E_F$ toward the bands.
• Carrier concentration n and p — $n=\int g_c(E)f(E)dE$, $p=\int g_v(E)(1-f)dE$.
• Pauli exclusion principle — the physical origin of the "1 +" in the denominator.
• Band gap and conduction/valence bands — where $E_F$ sits relative to bands.
• Chemical potential and equilibrium — flat $E_F$ = no net electron flow.

Concept Map

Hinglish (regional understanding)