Fermi Energy Calculator

The Fermi Energy Calculator determines the Fermi energy (EF) of a system of electrons, which is a key concept in quantum mechanics and solid-state physics. Fermi energy represents the highest energy level occupied by electrons at absolute zero temperature in a system of non-interacting fermions. It is crucial in analyzing the electrical and thermal properties of metals, semiconductors, and degenerate electron gases. Scientists and engineers use this calculator to assess how electrons behave in a material, which helps in designing electronic components and conducting research in nanotechnology and condensed matter physics.

formula of Fermi Energy Calculator

Where:

EF = Fermi energy in joules
ℏ = reduced Planck's constant = 1.0545718 × 10⁻³⁴ J·s
m = mass of an electron = 9.10938356 × 10⁻³¹ kg
n = electron number density in electrons per cubic meter
π = pi ≈ 3.14159

To convert EF from joules to electronvolts:

EF (eV) = EF (J) / q

Where:
q = elementary charge = 1.602176634 × 10⁻¹⁹ C

This formula assumes a uniform electron gas and is valid for three-dimensional, non-relativistic, degenerate Fermi systems.

Table of Common Values for Materials

Material	Electron Density (n) (electrons/m³)	Fermi Energy (eV)
Copper (Cu)	8.5 × 10²⁸	~7.0
Silver (Ag)	5.86 × 10²⁸	~5.5
Sodium (Na)	2.65 × 10²⁸	~3.2
Gold (Au)	5.9 × 10²⁸	~5.5
Aluminum (Al)	18.1 × 10²⁸	~11.6

These values are typical approximations based on empirical data and can vary slightly with temperature and purity.

Example of Fermi Energy Calculator

Scenario:
Calculate the Fermi energy of copper with an electron density of 8.5 × 10²⁸ electrons/m³.

Step 1:
EF = (1.0545718 × 10⁻³⁴)² / (2 × 9.10938356 × 10⁻³¹) × (3π² × 8.5 × 10²⁸)^(2/3)

Step 2:
EF ≈ 1.118 × 10⁻¹⁸ joules

Step 3:
Convert to eV:
EF (eV) = 1.118 × 10⁻¹⁸ / 1.602176634 × 10⁻¹⁹ ≈ 6.98 eV

So, the Fermi energy for copper is approximately 7.0 eV, which matches typical reference values.

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