Fermi Energy Calculator - Calculator Academy

Calculate Fermi energy or electron number density from the other using the free-electron model, with results in joules or electron volts.

Fermi Energy Formula

The following equation (free-electron model, 3D electron gas at 0 K) is commonly used to calculate the Fermi Energy.

E_F = (²) / (2mₑ)(3π² n)⁽2 / 3)

Variables:

Where E_F is the Fermi Energy (J)
n is the electron number density (m⁻³)
ħ is the reduced Planck constant (ħ = h/2π = 1.054571817×10⁻³⁴ J·s, with h = 6.62607015×10⁻³⁴ J·s)
m_e is the electron mass (9.1093837015×10⁻³¹ kg)

How to Calculate Fermi Energy?

The following example problem outlines the steps and information needed in order to calculate Fermi Energy.

Example Problem #1:

First, determine the electron number density (electrons per unit volume). In this example, n = 8.5×10²⁸ m⁻³.
Finally, calculate the Fermi Energy using the formula above.

E_F = (ħ²/(2m_e)) × (3π²n)^2/3

Inserting the values from above and solving the equation with the input values gives:

E_F = ( (1.054571817e-34)² / (2×9.1093837015e-31) ) × ( 3×π²×8.5e28 )^2/3 ≈ 1.13e-18 J (≈ 7.06 eV)

FAQ

What is Fermi Energy?

Fermi Energy is a concept in quantum mechanics that describes the highest energy level that electrons occupy at absolute zero temperature. It’s a crucial property in understanding the behavior of electrons in solids, particularly metals and semiconductors.

Why is the Fermi Energy important in physics and engineering?

The Fermi energy plays a vital role in determining the electrical, thermal, and optical properties of materials. It helps in understanding how materials conduct electricity, their electron distribution, and overall behavior under various conditions, making it essential for designing electronic devices.

How does temperature affect Fermi Energy?

While the Fermi energy is defined at absolute zero temperature, increasing temperature can excite electrons to higher energy states. In many materials (especially metals), the chemical potential (often called the “Fermi level” at finite temperature) shifts only slightly with temperature, but the occupancy of states around the Fermi level changes according to the Fermi-Dirac distribution, which can affect conductivity and other properties.