Fermi-Dirac Distribution Simulator — Quantum Statistics of Electrons
Visualize the electron occupation probability f(E) from Fermi level, temperature, and observed energy. Thermal energy kT and the ratio to a neighbouring energy update in real time, and the temperature-dependent transition band is highlighted.
Parameters
Fermi energy E_F
eV
Observed energy E
eV
Temperature T
K
Energy spacing ΔE
eV
Typical Fermi levels: Cu E_F ≈ 7.0 eV, Ag 5.5 eV, Au 5.5 eV, Al 11.7 eV. The intrinsic Fermi level of a semiconductor sits near the middle of the band gap.
Results
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f(E) occupation
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f(E+ΔE)
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Thermal energy kT
—
f(E)/f(E+ΔE)
Fermi-Dirac distribution f(E)
x = energy E (eV) / y = occupation f(E) / blue dashed = E_F (f = 0.5), green band = E_F ± 2 kT transition zone / yellow lines = observed E and E+ΔE
Temperature comparison (T/2, T, 2T)
Same E_F at three temperatures. Orange = T/2, blue = current T (highlighted), red = 2T. Higher T widens the transition band around E_F.
Theory & Key Formulas
The probability that an electron (a fermion) occupies a state of energy $E$ in thermal equilibrium follows the Fermi-Dirac distribution.
Occupation probability ($E_F$ is the Fermi level, $k$ Boltzmann's constant, $T$ absolute temperature):
At $E = E_F$ the value is always $f = 0.5$ regardless of temperature. As $T \to 0$ the curve becomes a step (1 below $E_F$, 0 above).
What is the Fermi-Dirac Simulator
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I have heard of the "Fermi-Dirac distribution", but what does the formula actually describe?
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It tells you the probability that a fermion — an electron, for example — occupies a state of energy E at temperature T. The form is $f(E) = 1/(1+\exp((E-E_F)/kT))$, where E_F is the Fermi level, the energy at which the occupation is exactly 0.5. Fermi and Dirac derived it independently in 1926. It shows up everywhere quantum statistics matters: metallic conduction, semiconductor carrier densities, even the support pressure of a white dwarf.
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The default output shows f(E) = 0.0205 at E = 5.10 eV with E_F = 5.00 eV and T = 300 K. Why is it under 5 % from just 0.10 eV above E_F?
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That is exactly the signature of Fermi statistics. At room temperature kT is only 25.85 meV, so 100 meV is about 3.87 kT. The exponent exp(3.87) ≈ 47.85 dominates the denominator, and f(E) = 1/48.85 ≈ 0.0205. Move just 0.05 eV further (E + ΔE = 5.15 eV) and exp(5.80) ≈ 331 drops the occupation to 0.00301; the ratio f(E)/f(E+ΔE) = 6.80. Near the Fermi level a few kT shift can change the occupation by an order of magnitude.
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When I press "Sweep temperature" the curve at E_F gets fuzzier — the transition band widens.
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Right. At T → 0 the distribution is a perfect step: every state below E_F is filled and every state above is empty. As T rises, a band of width ~ 4 kT opens up around E_F, thermally promoting electrons to states above the Fermi level. The orange-blue-red trio in the comparison plot makes this widening visible. That broadening of the band is precisely what makes semiconductor carrier density depend so strongly on temperature.
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The FAQ mentions "reduces to Boltzmann at high energy". When does that approximation start to be safe?
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Empirically, once $E - E_F$ exceeds about 3 kT — roughly 80 meV at room temperature — the form $f(E) \approx \exp(-(E-E_F)/kT)$ is correct to within a few percent. That is what the standard non-degenerate semiconductor formula $n = N_c \exp(-(E_c - E_F)/kT)$ relies on. Closer to E_F the approximation breaks down completely and you need full Fermi-Dirac integrals; that is the regime of degenerate semiconductors and metals.
FAQ
Thermodynamically, the Fermi level is the chemical potential μ for adding one electron — the energy required to add a particle to the system. It coincides with the energy at which f = 0.5. In metals it is the maximum occupied state at T = 0. In semiconductors it moves through the band gap with doping, temperature, and bias, controlling threshold voltage, carrier density, and depletion-layer width. Designing electronic devices is largely an exercise in placing the Fermi level where you want it.
Classical statistics treat particles as distinguishable and allow any number per state. Fermi-Dirac enforces the Pauli exclusion principle: each state holds at most one fermion (two with spin). Therefore f(E) → 1 saturates for E ≪ E_F, and matches Boltzmann only for E ≫ E_F. The quantum statistics dominate at low temperature and high density, explaining the linear-in-T electronic specific heat of metals, the support pressure of white dwarfs, and Coulomb-blockade staircases in quantum dots.
The conduction-band electron density is the density of states g(E) times f(E), integrated from the band edge: $n = \int g_c(E) f(E)\,dE$. When E_c − E_F > 3 kT (non-degenerate regime) this simplifies to $n \approx N_c \exp(-(E_c - E_F)/kT)$, with N_c the effective density of states. Holes follow $p \approx N_v \exp(-(E_F - E_v)/kT)$. This simulator focuses on f(E) alone, which is the building block for those carrier-density calculations.
The transition band — the region where f drops from 1 to 0 — has a width of roughly 4 kT. At room temperature (300 K) that is about 0.1 eV; at 77 K (liquid nitrogen) about 26 meV; at 4.2 K (liquid helium) only 1.4 meV, which is extremely sharp. This is why scanning tunneling spectroscopy (STS) and angle-resolved photoemission (ARPES) experiments aiming for high energy resolution are performed at cryogenic temperatures.
Real-world applications
Semiconductor device design: Almost every semiconductor device — MOSFETs, bipolar transistors, diodes, LEDs, solar cells — starts from carrier-density calculations using the Fermi-Dirac distribution. SPICE-class circuit simulators rely on E_F position to fix boundary conditions of the drift-diffusion equations and predict thresholds, including the famous 60 mV/decade subthreshold slope. Sliding the Fermi level toward the conduction band edge in this simulator quickly shows where the occupation changes most rapidly.
Electronic specific heat and conductivity in metals: The electronic specific heat of a metal is not the classical $(3/2)R$ but linear in T, $C_e = \gamma T$, with the Sommerfeld coefficient γ proportional to the density of states at E_F. Only the small fraction of electrons within kT of E_F can be thermally excited. The same mechanism keeps the resistivity of metals only weakly temperature-dependent at room temperature (phonon scattering dominates) — both consequences of Fermi-Dirac statistics.
STM and ARPES spectroscopy: Tunneling current in STS is $I \propto \int [f(E) - f(E + eV)] \rho(E) T(E)\,dE$, so the Fermi-Dirac difference directly sets the temperature-limited energy resolution. ARPES measurements of the Fermi surface and band structure require cryogenic temperatures because kT broadens the spectral function — the ~ 4 kT transition band shown here is precisely the experimental constraint.
White dwarfs and neutron stars: A white dwarf is supported against gravitational collapse by the degeneracy pressure of electrons, the pressure of an essentially T = 0 Fermi-Dirac distribution stacked under the Pauli exclusion principle. The Chandrasekhar mass limit (~1.4 M_⊙) follows from the same statistics; in neutron stars neutron degeneracy plays the same role. Linking white-dwarf physics to room-temperature semiconductor design through one statistical formula is part of the beauty of quantum statistics.
Common misconceptions and caveats
The most common misconception is to treat the Fermi level as the highest occupied state. That definition is strictly valid only at absolute zero. At finite temperature, the Fermi level is the energy where f = 0.5. In semiconductors it sits in the band gap, where there are no electronic states at all, yet it still controls carrier densities through the Boltzmann factor $\exp(-(E-E_F)/kT)$. The fact that "no electrons live there but the chemical potential is well defined" is the conceptually subtle part.
A second pitfall is to assume the Boltzmann approximation always holds. Near the Fermi level (in degenerate semiconductors and metals), $f(E)$ and $\exp(-(E-E_F)/kT)$ disagree strongly, and the full Fermi-Dirac distribution must be used. Try moving E close to E_F in this simulator and compare the curve with the exponential approximation; you can see precisely where the simple form breaks down. Always verify $E - E_F > 3 kT$ before applying the Boltzmann form in design formulas.
Finally, do not conflate $f(E)$ with a number of particles. $f(E)$ is the probability that a single quantum state is occupied, not a density. To obtain the actual carrier concentration you must multiply by the density of states $g(E)$ and integrate: $n = \int g(E) f(E)\,dE$. The form of $g(E)$ depends strongly on dimensionality (bulk, quantum well, wire, dot). This simulator isolates the behaviour of $f(E)$; in real-device modelling it always combines with $g(E)$.