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 kF 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
Figure 7

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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