Fermi Surfaces |

*Introduction*

The Fermi surface is the surface of constant energy
in **k** space. The Fermi surface separates the unfilled orbitals from
the filled orbitals, at absolute zero. The electrical properties of the
metal are determined by the shape of the Fermi surface, because the current
is due to changes in the occupancy of states near Fermi surface. The free
electron Fermi surfaces were developed from spheres of radius
*k _{F}*
determined by the valence electron concentration.

*Construction of free-electron Fermi surfaces*

The free electron Fermi surface for the an arbitrary electron concentration is shown in Fig.1.

**Figure 1**

These are Brillouin zones of a square lattice in two dimensions. The
blue circle shown is a surface of constant energy for free electrons; it
will be the Fermi surface for some particular value of the electron concentration.

It is inconvenient to have sections of the Fermi
surface that belong to the same Brillouin zone appear detached one from
another. The detachment can be repaired by a transformation to the first
Brillouin zone. The procedure is known as mapping the Fermi surface in
the reduced zone scheme.

There is also another way to represent the Fermi
surface in the reduced and periodic zone scheme. Fermi surfaces for free
electrons are constructed by a procedure credited to Harrison, Fig.2.

**Figure 2**

The reciprocal lattice points of a square lattice are determined, and
free-electron sphere of radius appropriate to the electron concentration
is drawn around each point. Any point in **k** space that lies within
at least one sphere corresponds to an occupied state in the first zone.
Points within at least two spheres correspond to occupied states in the
second zone, and similarly for points in three or more spheres.

In Fig.3,

**Figure 3**

the black square shown is the first Brillouin zone, the blue circle
is the surface of constant energy for free electrons, and the shaded area
represents occupied electron states. As we can see, the first zone is entirely
occupied.

In Fig.4,

**Figure 4**

the black square shown is the first Brillouin zone, the blue lines are
the Fermi surfaces for free electrons on the second zone, and the shaded
area represents occupied electron states.

In Fig.5,

**Figure 5**

the black square shown is the first Brillouin zone, the blue lines are
the Fermi surfaces for free electrons on the third zone, and the shaded
area represents occupied electron states.

In Fig.6,

**Figure 6**

the black square shown is the first Brillouin zone, the blue lines are
the Fermi surfaces for free electrons on the fourth zone, and the shaded
area represents occupied electron states.

Thus, in Fig.7,

1st zone |
2nd zone |
3rd zone |
4th zone |

we show the free electron Fermi surface, as viewed in the reduced zone
scheme. The shaded areas represent occupied electron states. Parts of Fermi
surface (blue lines) fall in the second, third, and fourth zones. The first
zone is entirely occupied.

In Fig.8,

**Figure 8**

we show the Fermi surface for free electrons in the second zone as drawn
in the periodic scheme. The figure can be constructed by repeating the
second zone of Fig.7 or directly from Harrison construction.

In Fig.9,

**Figure 9**

we show the Fermi surface for free electrons in the third zone as drawn
in the periodic scheme. The figure can be constructed by repeating the
third zone of Fig.7 or directly from Harrison construction.

In Fig.10,

**Figure 10**

we show the Fermi surface for free electrons in the fourth zone as drawn
in the periodic scheme. The figure can be constructed by repeating the
fourth zone of Fig.7 or directly from the Harrison construction.

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