Fermi-Dirac Distribution

In one sentence: For electrons and similar particles, it gives the chance that an energy level is occupied, from 0 (always empty) to 1 (always full).

$f(E) = \frac{1}{e^{(E - \mu)/(k_B T)} + 1}$

Read it out loud: “f of E equals 1 divided by the quantity e to the power of E minus mu, divided by k B times T, all plus 1. f of E is the chance that a state with energy E is occupied. E is the energy of that state. Mu, said mew, is the chemical potential, the energy at which the chance is one half. e is Euler's number, about 2.718. k B is the Boltzmann constant, which links temperature to energy. T is the absolute temperature in kelvin.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Powers of ten, like $10^{-23}$ and $10^{-19}$: A negative exponent means divide by ten that many times. $10^{-23}$ is a decimal point followed by 22 zeros and then a 1, so the 1 sits in the 23rd place after the decimal point. The Boltzmann constant, $1.380649 \times 10^{-23}$ J/K, is a very small number for this reason. An electron volt is $1.602176634 \times 10^{-19}$ J. When you divide powers of ten, subtract the exponents: $10^{-21} \div 10^{-19} = 10^{-2}$ .
Euler’s number $e$: The letter $e$ stands for Euler’s number, about 2.71828. It is a fixed number, like $\pi$ , and it appears whenever something grows or shrinks at a rate proportional to its current size. $e^{x}$ means $e$ raised to the power $x$ . A negative power such as $e^{-x}$ equals 1 divided by $e^{x}$ , which is a small number when $x$ is large and positive. Calculators have a key labeled $e^{x}$ .
Probability, from 0 to 1: A probability is a number from 0 to 1 that measures how likely something is. A probability of 0 means impossible, and 1 means certain. A probability of one half means a fair coin toss is equally likely to come up either way. The value $f(E)$ is a probability for one energy level, not a count of electrons.
Fermions and the exclusion principle: A fermion is a particle with half-integer spin, such as an electron, a proton or a neutron. Spin is a built-in quantum property, like a tiny fixed amount of spinning, and electrons have spin one half. The Pauli exclusion principle says two identical fermions cannot be in the same quantum state, which is why electrons spread across energy levels instead of piling up in the lowest one.
Energy levels and quantum states: In an atom or a metal, an electron can have only certain allowed energies. Each allowed way for an electron to exist, with one definite energy, is called a quantum state, or an energy level. Think of rungs on a ladder: a particle sits on one rung at a time, and the rungs are at fixed heights.
The chemical potential, $\mu$ (mu): The Greek letter $\mu$ is said mew. The chemical potential is an energy that works as a dividing line. States with energy below $\mu$ are likely to be full, and states above $\mu$ are likely to be empty. At exactly $E = \mu$ the chance is one half, whatever the temperature.
Boltzmann constant and thermal energy, $k_B T$: The Boltzmann constant $k_B = 1.380649 \times 10^{-23}$ J/K converts a temperature into an energy. The product $k_B T$ is the typical thermal energy at temperature $T$ . At 300 K it is $0.025852$ eV, about 0.026 eV. An energy level that sits several of these thermal units above $\mu$ is rarely occupied.

What each symbol means

Symbol	Meaning
$f(E)$	The occupation probability of one state with energy $E$ . It is a number from 0 (always empty) to 1 (always full), with no unit. A value of 0.02 means the state is filled about 2 times in 100 at the moment described.
$E$	The energy of the state, in joules (J) or electron volts (eV). Electrons in metals have energies of a few electron volts, so eV is the natural unit for this formula.
$\mu$	The chemical potential, an energy in joules or electron volts. It is the energy at which the chance is one half at any temperature. In a metal at absolute zero, it equals the Fermi energy.
$e$	Euler’s number, about 2.71828. It is a fixed number with no unit, and $e^{x}$ means it raised to the power $x$ .
$k_B$	The Boltzmann constant, $1.380649 \times 10^{-23}$ J/K. Its unit is joules per kelvin, which means energy per degree of temperature. It converts temperature into energy.
$T$	The absolute temperature in kelvin (K), counted from absolute zero. Room temperature is about 300 K. The formula needs $T$ above zero.

The formula piece by piece

$E - \mu$: The energy of the state minus the chemical potential. It tells us how far above or below the dividing line the state sits. A negative result means the state is below the line, and a positive result means it is above.
$k_B T$: The Boltzmann constant times the temperature, which is the typical thermal energy at that temperature. It is the scale against which $E - \mu$ is measured.
$\frac{E - \mu}{k_B T}$: The distance from the dividing line measured in thermal units. Dividing by $k_B T$ makes the answer a plain number with no unit, so it can be used as a power of $e$ .
$e^{(E - \mu)/(k_B T)}$: Euler’s number raised to that power. When the state is well below $\mu$ , this number is small, and when the state is well above $\mu$ , it is very large.
$e^{(E - \mu)/(k_B T)} + 1$: Add one to the previous number. The plus one keeps the bottom of the fraction bigger than 1, which keeps the final chance below 1, and it reflects the exclusion rule.
$f(E) = \frac{1}{e^{(E - \mu)/(k_B T)} + 1}$: The whole formula in one sentence: the chance that a state is occupied equals one divided by Euler’s number raised to the distance above the dividing line in thermal units, plus one. It is always between 0 and 1.

What the formula is doing, and why

What goes in and what comes out. The inputs are the energy $E$ of one state, the chemical potential $\mu$ , the temperature $T$ , and the Boltzmann constant $k_B$ . The output $f(E)$ is a number from 0 to 1, the chance that the state is occupied by an electron. The formula does not say which electron sits in that state. It gives only the average chance for the whole group of electrons.

Why this shape? The exponent $\frac{E - \mu}{k_B T}$ measures how far the state sits above or below the dividing energy $\mu$ , compared with the typical thermal energy $k_B T$ . Dividing by $k_B T$ makes it a plain number. Then $e$ raised to that power shows how fast the chance falls off. When the state is far below $\mu$ , the power is a large negative number and $f$ is close to 1. When it is far above, the power is a large positive number and $f$ is close to 0. At $E = \mu$ the power is zero, and $f = \frac{1}{2}$ .

Why the plus sign? The Pauli exclusion principle says two identical fermions cannot share a state. The $+1$ in the bottom expresses that rule. Because the bottom is always bigger than 1, the chance can never go above 1, so a state is never more than fully occupied. Bosons, which can share a state, use a minus sign instead. That is the difference from the Bose-Einstein formula.

Sharp at low temperature, soft at high. At absolute zero the step is sharp: every state below $\mu$ is full and every state above $\mu$ is empty. At higher temperature, electrons near $\mu$ gain thermal energy, so the step blurs over a range of a few $k_B T$ . At room temperature $k_B T$ is only about 0.026 eV, so the blur is small on the scale of metal energies. The formula describes electrons in metals and semiconductors, where they are in thermal balance.

An everyday picture

Think of a theater where each seat holds at most one person. The seats near the stage fill first, and the back rows stay empty, with a boundary between the full and the empty seats. The chemical potential is where that boundary sits, and the temperature decides how blurry the boundary is, because a more energetic crowd shuffles around more.

Worked example: An electron state at room temperature

Take $T = 300$ K, which is about room temperature, and a state with energy $E = \mu + 0.1$ eV, which is 0.1 eV above the chemical potential. Use $k_B = 1.380649 \times 10^{-23}$ J/K and $1$ eV $= 1.602176634 \times 10^{-19}$ J to express the thermal energy in electron volts.

Find the thermal energy in electron volts. The formula needs $k_B T$ in the same unit as $E - \mu$ , which is eV. Multiply $k_B$ by 300: $1.380649 \times 10^{-23} \times 300 = 4.141947 \times 10^{-21}$ J. Then divide by $1.602176634 \times 10^{-19}$ J per eV. The powers of ten subtract, $10^{-21} \div 10^{-19} = 10^{-2}$ , and the ordinary numbers give $4.141947 / 1.602177 \approx 2.5852$ . So $k_B T \approx 0.025852$ eV.
Compute the exponent. The exponent is $\frac{E - \mu}{k_B T} = \frac{0.1}{0.025852} \approx 3.868$ . This number says the state sits about 3.9 thermal units above the dividing line. Dividing by $k_B T$ makes the exponent a pure number with no unit, so it can be used as a power of $e$ .
Raise e to that power. We need $e^{3.868}$ . A calculator gives $e^{3.868} \approx 47.85$ . The number is large because the exponent is positive, which means the state lies well above the dividing line.
Add one and take the reciprocal. The bottom of the fraction is $47.85 + 1 = 48.85$ . The chance is $f = \frac{1}{48.85} \approx 0.02047$ . Written as a percentage, that is about 2 percent, so the state is usually empty.
Read the answer in plain words. The ratio of empty chance to full chance is $e^{3.868} \approx 47.85$ . So this state is about 48 times more likely to be empty than full, which means it is almost always empty.

Answer: A state 0.1 eV above $\mu$ is occupied with probability about 0.0205, or about 2 percent, at 300 K. It is about 48 times more likely to be empty than full, so it is almost always empty.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Reversing the sign of the exponent. Above $\mu$ the occupation is small, and below $\mu$ it is close to 1.
Mixing joules and electron volts. Convert $k_B T$ into eV, or convert $E$ and $\mu$ into joules, before dividing. Mixing them gives an answer that is wrong by a factor of about $6 \times 10^{18}$ .
Reading $f$ as a number of electrons. It is a probability for one state, between 0 and 1. Multiply it by the number of states to get the expected number of electrons.
Using this formula for bosons. Electrons follow Fermi-Dirac statistics, while photons follow Bose-Einstein statistics, which has a minus sign in place of the plus sign.

Where you will meet it

Metals and semiconductors, where the occupation near the Fermi energy sets their electrical and thermal properties.
White dwarf stars, where electrons are packed so tightly that their pressure, governed by these statistics, holds the star up against gravity.
Nuclear physics, where protons and neutrons fill energy levels inside a nucleus following the same exclusion rule.

Questions people ask

What is the chemical potential?

It is the energy at which a state has a 50 percent chance of being occupied, at any temperature. For electrons in a metal at absolute zero, it equals the Fermi energy, which is the top of the filled energy levels.

What happens at absolute zero?

The chance drops in a sharp step. States below $\mu$ are all full, and states above $\mu$ are all empty. At any temperature above zero the step is a little blurred.

Why can two electrons not share one state?

The Pauli exclusion principle forbids two identical fermions from occupying the same quantum state. That rule is why the chance can never go above 1. A state here includes spin, so one spatial orbit can hold two electrons with opposite spins.

Fermi Energy: In a metal, electrons fill energy levels from the bottom up, and the Fermi energy is the energy of the highest level that gets filled.
Bose-Einstein Distribution: For bosons such as photons, it gives the average number of particles that share one energy level, and that number can be more than one.
Electron Shell Capacity: Each electron shell can hold a fixed maximum number of electrons: shell 1 holds 2, shell 2 holds 8, and shell 3 holds 18.
Spin One-Half: An electron has a built-in spin that, measured along any direction, gives only one of two values, called up or down.