** 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

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.

**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:** http://phelafel.technion.ac.il/~sarikr/

[2]see **Periodic Table** of the Fermi Surfaces of Elemental Solids at: http://www.phys.ufl.edu/fermisurface/periodic_table.html

[3]More about the Construction of** free-electron Fermi
surfaces in 2D** at: http://phycomp.technion.ac.il/~nika/fermi_surfaces.html

[4]see **Brillouin zone at:** http://phelafel.technion.ac.il/~sarikr/