The properties of a material is guided by the arrangement of atoms within the material, which reflects the shape of a Fermi surface, thus the Fermi surface has a direct correlation with properties of materials. In metals such as potassium, sodium and magnesium the shape of the Fermi surface is more or less spherical (a Fermi sphere), indicating that the behaviour of the electrons is analogous for any direction in momentum space. On the other hand, for materials such as lead and aluminium the Fermi surfaces retain more obscure shapes, typically with large bumps and depressions. No matter the shape of the Fermi surface, the behaviour of electrons residing near the surface is vital in the determination of the magnetic, electrical and all other properties and how they depend on direction within the crystal because at temperatures of absolute zero the electrons have enough energy to be raised above the Fermi surface and are free to move.
The free electron Fermi surfaces were developed from spheres of radius kF determined by the valence electron concentration. For an arbitrary electron concentration the Fermi surface is shown in Figure 1.
Figure 1 Figure 2
These are Brillouin zones[2] of a square lattice in two dimensions. The blue circle shown in Figure 1 illustrates a surface where free electrons have constant energy, in otherwords it will be the Fermi surface for a certain value of electron concentration. It is inconvenient to have sections of the Fermi surface that belong to the same Brillouin zone appear detached one from another. This can be fixed by a relating everything back to the first Brillouin zone. This is known as mapping the Fermi surface in the reduced Zone scheme. 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.
The black square shown Figure 1-4 is the first Brillouin zone, the blue circle shows a surface where free electrons have constant energy and the shaded area represents the already occupied electron states, so the firsts Brillouin Zone is entirely occupied.
In Figure 2 the Fermi surfaces of the second zone are represented by the blue lines and as before the shaded are shows the occupied electron states in that zone. In Figure 3 the blue lines are the Fermi surfaces for free electrons on the third zone, and the shaded area represents occupied electron states. In Figure 4, the blue lines are the Fermi surfaces for free electrons on the fourth zone, and the shaded area represents occupied electron states. The first Brillouin zone has been left in all the figures to show how each zone relates to the first zone.
Figure 3 Figure 4
Powerful experimental methods have been developed for the determination of Fermi surfaces. The easiest of which being the Haas-van Alphen (dHvA) effect because it exhibits the characteristic periodicity of the properties of metals within a uniform magnetic field very well without the need of detailed theoretical analysis.
The de Haas-van Alphen effect is the oscillation of the magnetic moment of a metal as a function of the static magnetic field intensity B. The effect can be observed in pure specimens at low temperatures in strong magnetic fields: we do not want quantization of the electron orbits to be blurred by collisions, and we do not want the population oscillations to be averaged out by thermal population of adjacent orbits.
The analysis of the dHvA[3] effect is usually given for absolute zero, were the electron spin is neglected. The treatments is given for a 2D system: in 3D we need only multiply the 2D wavefunction by plane wave factors exp(ikzz), where the magnetic field is parallel to the z axis. The area of an orbit in kx, ky space is quantized. The area between successive orbits is;
The area in k space occupied by a single orbital is (2π/L)2, neglecting spin, for a square specimen of side L. Using equation (above) we find that the number of free electron orbital’s that coalesce in single magnetic level is
where . Such a magnetic level is called a Landau level.
The magnetic moment µ of a system at absolute zero is given by. The moment here is an oscillatory function of 1/B. This oscillatory magnetic moment of the Fermi gas at low temperatures is the de Haas-van Alphen effect. The oscillations occur at equal intervals of 1/B such that
The momentum p of a free electron is related to the wavelength λ of the electronic wave by the equation below,
If the electron did not interact with the metallic lattice, the energy would not depend upon the direction of k, where k is the corresponding wave vector associated with the wavelength, and all constant-energy surfaces, including the Fermi surface, would be spherical. This is somewhat true for copper. The Fermi surface of copper was found to be distorted but was still a recognizable deformation of a sphere: eight necks make contact with the hexagonal faces of the first Brillouin zone of the face-centred cubic (fcc) lattice. The electron concentration in a monovalent metal with an fcc structure is n=4/a3; there are four electrons in a cube of volume a3.
The radius of a free electron Fermi sphere is;
The shortest distance across the Brillouin Zone, in otherwords the distance between the hexagonal faces, is (2π/a)(3)1/2 = 10.88/a, somewhat larger than the diameter of the free electron sphere. The sphere does not touch the zone boundary, but we know that presence of a zone boundary tends to lower the band energy near the boundary. Thus it is plausible that the Fermi surface should neck out to meet the closest (hexagonal) faces of the zone. The square faces of the zone are more distant, with separation 12.57/a, and the Fermi surface does not neck out to meet these faces.
References
[1] Sidney Perkowitz. Fermi surface. Available:
http://www.britannica.com/EBchecked/topic/204790/Fermi-surface. Last accessed 25th Mar 2012.
[2] Fermi Surface. Available: http://phycomp.technion.ac.il/~nika/fermi_surfaces.html. Last
Accessed 29th Mar 2012
[3] Charles Kittel. (1995). Fermi Surfaces and Metals. In: Introduction to Solid State Physics. 7th ed. Canada: John Wiley & Sons. p233-268
