The Fermi level is the energy at which the probability of finding an electron is exactly one-half at any temperature. It is the electrochemical potential of electrons in a solid and the universal reference for electronic behavior — from why metals conduct to how transistors switch. The closely related Fermi energy is the highest occupied electron energy at absolute zero. These two ideas underpin nearly all of solid-state physics and device engineering.
The Fermi-Dirac Distribution
Electrons are fermions, so they obey the Pauli exclusion principle: no two electrons can occupy the same quantum state. At thermal equilibrium, the probability that a state at energy \(E\) is occupied is given by the Fermi-Dirac distribution:
$$ f(E) = \frac{1}{1 + \exp\!\bigl((E - E_F)/(k_B T)\bigr)} $$Here \(E_F\) is the Fermi level (chemical potential), \(k_B\) is Boltzmann’s constant, and \(T\) is absolute temperature. This single equation contains the whole story:
- When \(E = E_F\), the exponential is 1 and \(f(E) = 0.5\). That half-occupation is the definition of the Fermi level at finite \(T\).
- When \(E\) is well below \(E_F\) (by more than a few \(k_B T\)), \(f(E)\) approaches 1: the state is almost certainly filled.
- When \(E\) is well above \(E_F\), \(f(E)\) approaches 0: the state is almost certainly empty.
At absolute zero the distribution becomes a sharp step: every state below \(E_F\) is filled, every state above is empty. As temperature rises, the step softens over roughly \(k_B T\) (about 26 meV at 300 K), exciting a small population of electrons above \(E_F\) and leaving holes below. Only carriers within a few \(k_B T\) of \(E_F\) participate in transport — which is why the DOS at the Fermi level, not the total valence-electron count, controls metallic conductivity and electronic heat capacity.
flowchart LR
A[Energy E] --> B{Compare E to E_F}
B -->|E much less than E_F| C["f(E) ≈ 1<br/>filled"]
B -->|E = E_F| D["f(E) = 0.5"]
B -->|E much greater than E_F| E["f(E) ≈ 0<br/>empty"]
C --> F[Transport and DOS<br/>near E_F]
D --> F
E --> FFermi Level vs Fermi Energy
The terms Fermi level and Fermi energy are often used interchangeably, but they are not identical.
| Quantity | Definition | Temperature dependence |
|---|---|---|
| Fermi energy (\(E_F\) at \(T=0\)) | Highest occupied state at absolute zero; measured from the band bottom in a free-electron picture | Defined only at \(T = 0\) |
| Fermi level (chemical potential \(\mu\)) | Energy where \(f(E) = 0.5\); the electrochemical potential | Varies with temperature and doping |
Strictly, the Fermi energy is the chemical potential at absolute zero; the Fermi level is that chemical potential at any temperature. In metals the two are numerically almost the same because \(E_F\) is large (several eV) compared to \(k_B T\). In semiconductors, where the Fermi level sits in a gap and shifts with temperature and doping, the distinction is real and important.
A free-electron estimate for a simple metal is
$$ E_F = \frac{\hbar^2}{2m}\,(3\pi^2 n)^{2/3} $$where \(n\) is the valence-electron density. Typical metals yield \(E_F \sim 2\)–\(10\,\mathrm{eV}\), so \(k_B T / E_F \ll 1\) even near melting. Semiconductors have no analogous “filled-to” energy inside a continuum of states; their chemical potential is fixed by charge neutrality in the gap.
Where the Fermi Level Sits: Metals, Semiconductors, Insulators
The position of the Fermi level relative to the energy bands distinguishes the great classes of solids.
Metals: the Fermi level lies inside a partially filled band. Empty states sit immediately above \(E_F\), so electrons move with negligible energy cost. In \(\mathbf{k}\)-space, points with energy \(E_F\) form the Fermi surface — a closed surface (or set of sheets) separating occupied from unoccupied states at \(T = 0\). Conductivity, quantum oscillations, and nesting-driven instabilities are properties of that surface. Copper’s is nearly spherical; transition and topological metals can have multi-sheet, open, or highly anisotropic surfaces.
Intrinsic semiconductors: the Fermi level lies in the middle of the band gap, roughly halfway between the valence-band maximum (VBM) and conduction-band minimum (CBM). No states exist exactly at \(E_F\); its mid-gap position means equal (and small) thermal populations of electrons and holes. There is no Fermi surface in the metallic sense — only a chemical potential floating in empty energy space.
Insulators: the Fermi level also sits in the gap, but the gap is so wide that thermal excitation is negligible at ordinary temperatures.
For an intrinsic semiconductor with effective densities of states \(N_c\) and \(N_v\),
$$ E_F^{\mathrm{(intrinsic)}} = \frac{E_c + E_v}{2} + \frac{1}{2} k_B T \ln\!\left(\frac{N_v}{N_c}\right) $$When \(N_c \approx N_v\), this reduces to mid-gap: \(E_F \approx E_v + E_g/2\). Silicon’s room-temperature gap is about \(1.12\,\mathrm{eV}\); germanium’s is about \(0.66\,\mathrm{eV}\); diamond’s is about \(5.5\,\mathrm{eV}\) — which is why diamond behaves as an insulator under ordinary conditions.
Temperature Dependence of the Fermi Level
In a metal, raising temperature barely moves \(E_F\). Charge neutrality and the large DOS at the Fermi surface force only a correction of order \((k_B T / E_F)^2\). Copper’s Fermi level at 300 K is indistinguishable from its \(T = 0\) Fermi energy for most purposes.
Semiconductors are different. The intrinsic carrier concentration grows exponentially:
$$ n_i(T) = \sqrt{N_c N_v}\,\exp\!\bigl(-E_g / 2k_B T\bigr) $$and \(E_F\) drifts as \(N_c(T)\) and \(N_v(T)\) change (both scale as \(T^{3/2}\)). In doped material three regimes appear:
- Freeze-out (low \(T\)): dopants are not fully ionized; \(E_F\) sits near the impurity level.
- Extrinsic plateau (device range): nearly all dopants are ionized; majority density is fixed by \(N_D\) or \(N_A\), and \(E_F\) sits a fixed distance from the majority band edge.
- Intrinsic takeover (high \(T\)): \(n_i(T)\) exceeds the dopant density; the material behaves as intrinsic and \(E_F\) returns toward mid-gap.
For silicon, freeze-out ends well below room temperature for shallow dopants (P, B ionization \(\sim 45\,\mathrm{meV}\)), the extrinsic regime covers ordinary device operation, and intrinsic takeover becomes serious only above roughly \(150\)–\(200\,^\circ\mathrm{C}\) depending on doping. Room-temperature Si electronics can therefore treat doping as setting a fixed \(E_F\).
How Doping Moves the Fermi Level
Semiconductors are useful because their Fermi level can be deliberately shifted by impurities — doping.
- n-type doping adds donors (phosphorus or arsenic on Si sites). After ionization, \(n \approx N_D\), and the Fermi level moves upward, toward the conduction band:
- p-type doping adds acceptors (boron or gallium). With \(p \approx N_A\), the Fermi level moves downward, toward the valence band:
Concrete silicon numbers at 300 K clarify the scale. Take \(N_c \approx 2.8 \times 10^{19}\,\mathrm{cm}^{-3}\), \(N_v \approx 1.0 \times 10^{19}\,\mathrm{cm}^{-3}\), and \(n_i \approx 1.0 \times 10^{10}\,\mathrm{cm}^{-3}\):
| Silicon sample | Dopant density | Approx. \(E_F\) position (300 K) | Majority carrier |
|---|---|---|---|
| Intrinsic | — | Mid-gap (\(\sim 0.56\,\mathrm{eV}\) above VBM) | \(n = p = n_i\) |
| Lightly n-type | \(N_D = 10^{15}\,\mathrm{cm}^{-3}\) | \(\sim 0.25\,\mathrm{eV}\) below CBM | Electrons |
| Heavily n-type | \(N_D = 10^{18}\,\mathrm{cm}^{-3}\) | \(\sim 0.09\,\mathrm{eV}\) below CBM | Electrons |
| Lightly p-type | \(N_A = 10^{15}\,\mathrm{cm}^{-3}\) | \(\sim 0.24\,\mathrm{eV}\) above VBM | Holes |
| Heavily p-type | \(N_A = 10^{18}\,\mathrm{cm}^{-3}\) | \(\sim 0.06\,\mathrm{eV}\) above VBM | Holes |
Values use the non-degenerate Boltzmann formulas above. At \(10^{19}\,\mathrm{cm}^{-3}\) and above, Fermi–Dirac integrals are required and \(E_F\) can enter the band — degenerate doping.
When n-type and p-type regions are joined, their Fermi levels must align at equilibrium — the electrochemical potential must be uniform. Band edges bend so \(E_F\) is flat across the junction; that bending is the built-in potential of a p-n junction:
$$ q V_{\mathrm{bi}} = E_g - (E_c - E_F)_n - (E_F - E_v)_p $$or \(V_{\mathrm{bi}} = (k_B T/q)\ln(N_A N_D / n_i^2)\). Depletion, diode rectification, solar-cell voltage, and bipolar injection all follow from this alignment.
| Doping type | Dopant example | Fermi level moves | Majority carrier |
|---|---|---|---|
| Intrinsic | None | Mid-gap | Balanced |
| n-type | Phosphorus, arsenic | Toward conduction band | Electrons |
| p-type | Boron, gallium | Toward valence band | Holes |
The Fermi Level in Devices: MOSFET Threshold
A MOSFET makes the picture concrete. In an n-channel device the substrate is p-type, so bulk \(E_F\) sits near the valence band. Positive gate voltage bends the bands at the Si/SiO\(_2\) interface. Threshold is reached when the surface potential \(\psi_s\) has bent far enough that the surface is as strongly n-type as the bulk is p-type — surface inversion:
$$ \psi_s(\mathrm{threshold}) = 2\psi_B,\qquad \psi_B = \frac{k_B T}{q}\ln\!\left(\frac{N_A}{n_i}\right) $$Here \(\psi_B\) is the bulk Fermi potential from mid-gap. For \(N_A = 10^{17}\,\mathrm{cm}^{-3}\) in Si at 300 K, \(\psi_B \approx 0.41\,\mathrm{V}\), so threshold bending is about \(0.82\,\mathrm{V}\) before oxide capacitance and interface charge set the gate voltage \(V_T\). Textbook \(V_T\) formulas translate “how far must we pull \(E_F\) at the surface to invert the channel.”
Under bias, separate electron and hole quasi-Fermi levels \(E_{Fn}\) and \(E_{Fp}\) describe non-equilibrium populations; their splitting \(E_{Fn} - E_{Fp} = qV\) drives current in diodes and solar cells.
The Work Function and the Fermi Level
The work function is the minimum energy needed to remove an electron from the Fermi level to a point just outside the surface (the vacuum level):
$$ \phi = E_{\mathrm{vacuum}} - E_F $$Because it is measured from \(E_F\), the work function is central to:
- Metal–semiconductor contacts, where the metal work function and semiconductor electron affinity decide ohmic versus Schottky behavior.
- Photoemission and thermionic emission, where electrons leave only if they overcome \(\phi\).
- Contact potentials between dissimilar metals, which arise from differing Fermi levels and surface dipoles.
In DFT slab calculations the work function comes from aligning the bulk Fermi level to the vacuum potential far from the surface — a direct computational route to an absolute \(E_F\) reference. Photoelectron spectroscopy measures the same alignment experimentally.
Density of States, DFT Output, and What Codes Print
How many electrons participate in a process depends on how many states exist near the Fermi level — the density of states (DOS) \(g(E)\). Conduction, electronic heat capacity, and Pauli paramagnetism are controlled by \(g(E_F)\):
- Electrical conduction needs empty states within \(\sim k_B T\) of \(E_F\).
- Electronic heat capacity of a metal scales as \(g(E_F)\, T\).
- Pauli susceptibility and many superconducting estimates scale with \(g(E_F)\).
Carrier densities integrate DOS against the Fermi–Dirac function:
$$ n = \int_{\mathrm{CB}} g_c(E)\, f(E)\, \mathrm{d}E,\qquad p = \int_{\mathrm{VB}} g_v(E)\,\bigl(1 - f(E)\bigr)\, \mathrm{d}E $$In a self-consistent DFT calculation the code finds \(E_F\) (or \(\mu\)) so that integrated occupations equal the electron count in the cell. Engines such as Quantum ESPRESSO print this after the SCF cycle. In a typical scf.out for a metal (with smearing):
the Fermi energy is 12.3456 ev
For insulators or semiconductors you more often see:
highest occupied, lowest unoccupied level (ev): -0.1234 0.9876
Band-structure and DOS plots are almost always shifted so that \(E_F = 0\) (metals) or the VBM is at zero (semiconductors). Reading the printed Fermi energy and subtracting it from the eigenvalues is standard post-processing — covered further in our blog posts on DOS and band structure, and on our computational engines page.
Smearing methods (Gaussian, Methfessel–Paxton, Marzari–Vanderbilt, Fermi–Dirac) broaden occupations near \(E_F\) so metallic SCF cycles converge; the printed “Fermi energy” is the chemical potential consistent with that smearing. Semiconductors usually use fixed occupations and a clear HOMO–LUMO gap.
Common Misconceptions
“The Fermi level is always the highest occupied state.” True at \(T = 0\) in a metal. At finite \(T\), and in any gapped semiconductor, \(E_F\) can sit where no state exists. Occupation would be 50% if a state were present there; the chemical potential remains well-defined either way.
“Metals and semiconductors both have a Fermi surface.” Only metals (and degenerately doped systems) have a Fermi surface in \(\mathbf{k}\)-space. In an intrinsic semiconductor, \(E_F\) is a number in the gap, not a constant-energy surface cutting through bands.
“Doping adds states at the Fermi level.” Shallow dopants add impurity levels near a band edge; they move \(E_F\) by changing the carrier density that charge neutrality must satisfy. Mid-gap DOS stays essentially zero until deep defects or disorder appear.
“The DFT Fermi energy is vacuum-referenced.” Plane-wave codes report \(E_F\) on an arbitrary internal scale set by the average cell potential. Absolute alignment (work functions, band offsets) needs a common reference — vacuum from a slab, a core level, or an interface calculation.
“\(E_F\) always sits at mid-gap when undoped.” Only when \(N_c \approx N_v\). Large mass asymmetry shifts the intrinsic level toward the heavier band’s edge.
“Raising temperature always moves \(E_F\) a lot.” In metals, almost not at all. In doped semiconductors within the extrinsic window, \(E_F\) moves only weakly through \(N_c(T)\) and \(N_v(T)\); the dramatic shift is the high-\(T\) return to mid-gap when intrinsic carriers dominate.
Run it on Simatra
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