The Fermi Surface

From the above discussion of the simple band model (no electron interactions are included), it follows that there is a surface in phase space at T = 0°K across which there is a discontinuity in the occupa- 

In conclusion, the principal features of collective electrons that appear to have physical significance are the Brillouin zone and band-structure concepts that follow from the lattice periodicity and the concept of a Fermi surface. 

For the calculation of the dielectric matrix to second order in the pseudopotential, we will need the Fermi surface to the same order. For this 

The Fermi energy to second order in the pseudopotential is to be found from the corresponding expression for the energy  

The equation for the magnitude of the Fermi wave vector then becomes  

a factor of 2 is included to account for the spin degeneracy. Because 

Only the angular integration is left over. Inserting the expression for F in this equation, one obtains  

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The Fermi energy is the energy of the highest-energy filled orbital, analogous to a HOMO energy. If the orbital is half-filled, its energy will be found at a collection of points in /c-space, called the Fermi surface. 

Figure 5.1 The parabolic distribution in energy, N(E), as function of energy, E, for free electrons. The Fermi surface represents the upper limit of electron energy at the absolute zero of temperature, but at higher temperatures a small fraction of the electrons can be excited to higher energy levels...

In order to discuss electron transport properties we need to know about the electronic distribution. This means that, for the case of metals and semimetals, we must have a model for the Fermi surface and for the phonon spectrum. The electronic structure is discussed in Chap. 5. We also need to estimate or determine some characteristic lengths. 

A. R. Mackintosh and O. K. Andersen, Chapter 5.3 in Electrons at the Fermi Surface, edited by M. Springford (Cambridge University Press, Cambridge, England, 1980). 

Figure 9 Majority Fermi surfaces for Co, Cu, and Co5Cu4. For values of kj greater than approximately 0.6 there are no allowed values of in Co. An electron in a copper layer with a value of k. greater than this value cannot scatter into the cobalt layers while conserving k,. The Fermi surface of Co5Cu4 shows two modes that are localized on the copper layers.

Momentum conservation implies that the wave vectors of the phonons, interacting with the electrons close to the Fermi surface, are either small (forward scattering) or close to 2kp=7i/a (backward scattering). In Eq. (3.10) forward scattering is neglected, as the electron interaction with the acoustic phonons is weak. Neglecting also the weak (/-dependence of the optical phonon frequency, the lattice energy reads ... 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

In this equation v is a phonon frequency, such that hv is approximately k, with the Debye characteristic temperature of the metal. The quantity p is the product of the density of electrons in energy at the Fermi surface, N(0), and the electron-phonon interaction energy, V. 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

In the case of Cu, the effects of the SIC on the band structure are summarized as follows [21], The width of the s-type band is not affected. The relative position of the d-bands with respect to the Fermi energy is lowered by 2 eV, and the width of the d-band is reduced by 15%. As a result, the electrons in the d-bands are more localized. The s-d hybridization near the Fermi energy is reduced. Consequently, I have got somewhat controversial results on the geometry of the Fermi surface. As reference,... 

Rajput, S.S., Prasad, R., Singru, R.M., Triftshauser, W., Eckert, A., Kogel, G., Kaprzyk, S. and Bansil, A. (1993) A study of the Fermi surface of lithium and disordered lithium-magnesium alloy theory and experiment, J. Phys. Condens. Matter, 5,6419-6432. 

Sakurai, Y., Tanaka, Y., Bansil, A., Kaprzyk, S., Stewart, A.T. Nagashima, Y., Hyodo, T., Nanao, S., Kawata, H. and Shiotani, N. (1995) High-resolution Compton scattering study of Li asphericity of the Fermi surface and electronic correlation effects, Phys. Rev. Lett., 74, 2252-2255. 

In this chapter we will have a closer look at the methods of the reconstruction of the momentum densities and the occupation number densities for the case of CuAl alloys. An analogous reconstruction was successfully performed for LiMg alloys by Stutz etal. in 1995 [3], It was found that the shape of the Fermi surface changed and its included volume grew with Mg concentration. Finally the Fermi surface came into contact with the boundary of the first Brillouin zone in the [110] direction. Similar changes of the shape and the included volume of the Fermi surface can be expected for CuAl [4], although the higher atomic number of Cu compared to that of Li leads to problems with the reconstruction, which will be examined. 

One can clearly see the large positive anisotropy in the [111] direction near the boundary of the first Brillouin zone (BZB). It is caused by the [111] high momentum component, which produces a continuous distribution of the momentum density across the BZB, as the Fermi surface has contact with the BZB in this direction. In the other directions, especially in [100], calculations show a steep decrease of the momentum density at the Fermi momentum and therefore a negative deviation from the spherical mean value. 

Figure 8. Occupation number density of Cu in the repeated zone scheme the solid line marks the boundary of the first BZB the bold contour line marks the Fermi surface.

The Fermi surfaces exhibit the well-known neck structure in the (111) directions. Their radii also increase with A1 concentration. 

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