Fermi Level Calculator

What Is the Fermi Level?

The Fermi level marks the highest occupied electron energy at absolute zero. In a free-electron gas model, it sets the boundary between filled and empty states and controls electrical conduction in metals.

Fermi Level Formulas

From electron number density $n$:

$$k_{\rm F} = \sqrt[3]{3\pi^2 n}$$ $$E_{\rm F} = \frac{\hbar^2 k_{\rm F}^2}{2 m_e}$$ $$v_{\rm F} = \frac{\hbar k_{\rm F}}{m_e}$$ $$T_{\rm F} = \frac{E_{\rm F}}{k_{\rm B}}$$

The Fermi-Dirac distribution gives the occupation probability at finite temperature:

$$f(E) = \frac{1}{e^{(E - E_{\rm F})/k_{\rm B}T} + 1}$$

Example: copper with $n = 8.47 \times 10^{28}\,\mathrm{m^{-3}}$ yields $E_{\rm F} \approx 7\,\mathrm{eV}$. Related tools: Logic Gate Calculator, Op-Amp Gain Calculator, and LM317 Calculator.

Frequently Asked Questions

What is Fermi energy used for?

It estimates conduction electron speeds, heat capacity contributions, and the energy scale where metals behave as a Fermi gas.

Why does copper have a higher Fermi energy than silver?

Copper has a larger electron number density, which increases $k_{\rm F}$ and therefore $E_{\rm F}$.

What is the Fermi function at E = E_F?

Regardless of temperature, $f(E_{\rm F}) = 0.5$ when the energy equals the Fermi level.

Does this apply to semiconductors?

The free-electron model fits metals well. Semiconductors need effective mass and band structure, but the same density-based formulas give order-of-magnitude estimates.

What units should number density use?

Use electrons per cubic meter. Preset metals in the tool use values in the $10^{28}\,\mathrm{m^{-3}}$ range.

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What Is the Fermi Level?

Fermi Level Formulas

Frequently Asked Questions

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