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 Fermi Surface



1. Introduction

The Fermi surface defines a theoretical area of constant energy in reciprocal space (k-space). This surface is used to separate occupied electronic states from empty states in solid materials and defines the allowed energies of electrons. The Fermi surface was named after Italian physicist Enrico Fermi, who was one of the developers of the statistical theory of electrons.


2. Theoretical background
The Fermi surface exists due to the Pauli Exclusion Principle, which restricts the number of electrons allowed in a single quantum state (one particle per state). Therefore- in the lowest state of energy, the particles fill up all energy levels up to the Fermi energy -. Meaning, below the energy level there are exactly  states. We can now formally define the Fermi energy as the energy difference between the highest and lowest occupied single-particle states- in a quantum system of non-interacting fermions at absolute zero temperature.

Fermi surfaces are used to define the properties of different metals (such as thermal, electrical, magnetic, and optical). This is due to the fact that changes in the occupancy of states near the Fermi energy influence the currents.


3. Shape and Construction

The first existence of such a surface was determined in 1957 in an experiment on copper. The Fermi surface of copper was found to be distorted and not a perfect sphere due to the potential of the lattice.

Fermi surface of copper, as determined in 1957; two shapes were found to be consistent with the original data, and the other, slightly more deformed version turned out to be correct 

Figure 3.1: Fermi surface of copper, as determined in 1957

However, when no potential is applied, The Fermi Surface is a sphere in 3D (or a circle in 2D) in reciprocal space.

Figure 3.2: extended zone (see explanation below) scheme in 2D

The radius of the Fermi sphere is, as mentioned above, dependent on the value of the Fermi energy: .

The volume of a Fermi sphere in k-space is: . 

 Now, we can calculate the number of filled states () in a Fermi sphere:

From this we can define  as:  .

For a monovalent element, the volume of the Fermi surface is half of the Brillouin zone. In the software we consider a three-dimensional lattice, free electrons, which has two electrons per unit cell. The area of the Fermi sphere for the electrons then has a volume equal to the volume of the first Brillouin zone.

Furthermore, often in a metal the Fermi surface radius  is larger than the size of the first Brillouin zone, which results in a portion of the Fermi surface lying in the second (or higher) zones. This representation is called extended-zone and this is the representation which was chosen for the attached software.



[1]see Reciprocal Lattice at:


[2]see Periodic Table of the Fermi Surfaces of Elemental Solids at:


[3]More about the Construction of free-electron Fermi surfaces in 2D at:


[4]see Brillouin zone at: