Fermi Energy Calculator
Find the Fermi energy of a metal's conduction electrons using E_F = (ħ²/2m)(3π²n)^(2/3).
🧊 What is the Fermi Energy Calculator?
The Fermi energy calculator finds the highest occupied electron energy in a metal's conduction band at absolute zero, using the free electron gas model. Pick a metal preset or enter a custom electron density, and it returns the Fermi energy in electronvolts, plus the Fermi velocity and Fermi temperature.
Because electrons are fermions, the Pauli exclusion principle forbids more than one electron (of each spin) from occupying the same quantum state. As a result, even at absolute zero, a metal's conduction electrons cannot all pile into the lowest energy level, they fill up a "sea" of states from the bottom, and the Fermi energy marks the top of that filled sea.
The key relationship, E_F = (ħ²/2m)(3π²n)^(2/3), shows that Fermi energy depends only on the electron density n, growing with its two-thirds power. Real metals have astonishingly high conduction electron densities, around 10^28 to 10^29 per cubic metre, which is why typical Fermi energies land in the few-electronvolt range and Fermi temperatures reach tens of thousands of kelvin, far beyond any metal's melting point.
This calculator is useful for solid-state physics and materials science students studying electrical conductivity, specific heat, and the quantum-statistical behaviour of metals, since it handles the physical constants and unit conversions automatically.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Copper
Example 2 - Sodium
Example 3 - Aluminum
❓ Frequently Asked Questions
🔗 Related Calculators
What is Fermi energy?
The Fermi energy is the energy of the highest occupied electron state in a metal at absolute zero temperature, in the free electron gas model. Because electrons are fermions and obey the Pauli exclusion principle, they cannot all sit in the lowest energy state, they fill up states from the bottom, and the Fermi energy marks the 'water line' of that filled sea of electrons.
What is the formula for Fermi energy?
E_F = (ħ² / 2m) × (3π²n)^(2/3), where ħ is the reduced Planck constant, m is the electron mass, and n is the number of conduction electrons per unit volume. Fermi energy grows with the two-thirds power of electron density, so denser electron gases have higher Fermi energies.
Why do metals have Fermi energies of several electronvolts?
Typical metals have roughly 10^28 to 10^29 conduction electrons per cubic metre, extremely dense by everyday standards. Plugging this density into the Fermi energy formula gives values of a few electronvolts, copper is about 7.0 eV, sodium about 3.2 eV, and aluminum about 11.7 eV, reflecting how tightly packed the electrons are.
What is Fermi temperature?
Fermi temperature is TF = EF / kB, the temperature at which the thermal energy kBT would equal the Fermi energy. For typical metals TF is tens of thousands of kelvin, far above any metal's melting point, meaning the conduction electrons remain 'quantum degenerate' (governed by Fermi statistics, not classical statistics) at any temperature a metal can survive.
What is Fermi velocity?
Fermi velocity is the speed of an electron at the Fermi energy, vF = √(2EF/m). For typical metals this is on the order of 10^6 metres per second, over a thousand times faster than random thermal motion in a classical gas at room temperature would suggest, another sign of how different a degenerate electron gas is from a classical one.
Does Fermi energy depend on temperature?
In the simplest free electron model, Fermi energy depends only on electron density, not temperature. At finite temperature the sharp cutoff at EF smooths out slightly (some electrons are thermally excited just above it and states just below it empty out), but for metals at any achievable temperature this correction is tiny because EF is so much larger than kBT.
How does Fermi energy relate to electrical and thermal conductivity?
Only electrons near the Fermi energy can be excited into nearby empty states by an electric field or a temperature gradient, since states deep below EF are already full and Pauli exclusion blocks them from moving. This is why metals conduct electricity and heat so well, a large population of mobile electrons sits right at the Fermi surface, ready to respond to small perturbations.
Does the free electron model work for all materials?
It works well for simple metals with weakly-bound, delocalized conduction electrons, such as the alkali metals (sodium, potassium) and to a good approximation the noble metals (copper, silver, gold). It works poorly for semiconductors, insulators, and materials with strong electron-electron interactions or complex band structures, which need more sophisticated band-theory treatments.
How is electron density n determined for a metal?
For a simple metal, n is estimated as the number of valence electrons per atom (1 for copper, silver, gold and the alkali metals; 3 for aluminum) multiplied by the atom density, which comes from the metal's mass density, molar mass, and Avogadro's number. This straightforward counting is why the free electron model works best for metals with one loosely bound valence electron per atom.
Why does aluminum have a higher Fermi energy than copper despite having a similar atom density?
Aluminum contributes three valence electrons per atom to the conduction band, compared to copper's one, so its conduction electron density n is more than three times higher even though the two metals have comparable atomic densities. Because Fermi energy scales as n^(2/3), this higher density translates into a substantially larger Fermi energy for aluminum.