Fermi surface methods

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. 

As shown by Ruderman and Kittel (77) and Bloembergen and Rowland (78), Aij in a solid is dependent on the nature of the energy bands in the solid. For metals A is proportional to the product of the square of the electron density of Fermi surface electrons at the nucleus and the effective mass, and decreases as the inverse cube of the internuclear distance. Insulators have been treated by the energy band method (78) and by a molecular method (79) where each atom is considered to be bonded to its nearest neighbors. Unfortunately, both of these methods involve approximations in the evaluation of An which are quite crude at present. 

A very convenient method of investigating electronic effects in catalysis is to study the activities within a series of metal alloys. Changes in alloy composition will alter the energy density of electron levels at the Fermi surface. Moreover, in the particular case of transition metals, it has been seen that alloying with a metal of Group 16 decreases the number of holes in the d-band. These effects should profoundly alter the catalytic activities of the alloys. It is important to keep in mind that within such... 

For many metals, the "nearly free" electron description corresponds quite closely 10 the physical situation. The Fermi surface remains nearly spherical in shape. However, it may now he intersected by several Brillouin zone boundaries which break the surface into a number of separate sheets. It becomes useful to describe the Fermi surface in terms not only of zones or sheets filled with electrons, but also of zones or sheets of holes, that is. momentum space volumes which are empty of electrons. A conceptually simple method of constructing these successive sheets, often also referred lo as "first zone. "second zone." and so on was demonstrated by Harrison. An example of such construction is shown in Fig. 2. 

The electronic properties of organic conductors are discussed by physicists in terms of band structure and Fermi surface. The shape of the band structure is defined by the dispersion energy and characterizes the electronic properties of the material (semiconductor, semimetals, metals, etc.) the Fermi surface is the limit between empty and occupied electronic states, and its shape (open, closed, nested, etc.) characterizes the dimensionality of the electron gas. From band dispersion and filling one can easily deduce whether the studied material is a metal, a semiconductor, or an insulator (occurrence of a gap at the Fermi energy). The intra- and interchain band-widths can be estimated, for example, from normal-incidence polarized reflectance, and the densities of state at the Fermi level can be used in the modeling of physical observations. The Fermi surface topology is of importance to predict or explain the existence of instabilities of the electronic gas (nesting vector concept see Chapter 2 of this book). Fermi surfaces calculated from structural data can be compared to those observed by means of the Shubnikov-de Hass method in the case of two- or three-dimensional metals [152]. 

In the present contribution, we will examine the fundamentals of such an approach. We first describe some basic notions of the tight-binding method to build the COs of an infinite periodic solid. Then we consider how to analyze the nature of these COs from the viewpoint of orbital interaction by using some one-dimensional (ID) examples. We then introduce the notion of density of states (DOS) and its chemical analysis, which is especially valuable in understanding the structure of complex 3D sohds or in studying surface related phenomena. Later, we introduce the concept of Fermi surface needed to examine the transport properties of metallic systems and consider the different electronic instabilities of metals. Finally, a brief consideration of the more frequently used computational approaches to the electronic structure of solids is presented. 

The first accurate band structure calculations with inclusion of relativistic effects were published in the mid-sixties. Loucks published [64-67] his relativistic generalization of Slaters Augmented Plane Wave (APW) method. [68] Neither the first APW, nor its relativistic version (RAPW), were linearized, and calculations used ad hoc potentials based on Slaters s Xa scheme, [69] and were thus not strictly consistent with the density-functional theory. Nevertheless (or, maybe therefore ) good descriptions of the bands, Fermi surfaces etc. of heavy-element solids like W and Au were obtained.[3,65,70,71] With this background it was a rather simple matter to include [4,31,32,72] relativistic effects in the linear methods [30] when they (LMTO, LAPW) appeared in 1975. 

The relativistic LMTO and LAPW methods were used to calculate [77-80] the Fermi surface of UPta. This is a heavy fermion compound, and its physical properties axe strongly influenced the presence of the narrow U-/ bands at the Fermi level. The shape of the Fermi surface is then sensitive to relativistic effects, in particular the SO-coupling. The results of the calculations [78] were surprising since they showed that the topology of the Fermi surface was well described by these band structures although they were obtained within the LDA. A similar precision was not found for the effective cyclotron masses which were off by up to a factor of 30 when compared to experiments. The crystal potential enters in the LMTO via the potential parameters [30,73] for each (or each j in the relativistic version [4]), including the mass parameters fi (eq.(49)). A convenient way... 

Figure 13. UPts Fermi surface calculated within the LDA using the Dirac-relativistic LMTO method. The width of the hatched stripes is proportional to the U-/5/2 component. The U-/7/2 content is very low all-over the Fermi surface. Dotted stripes show the m, =l/2 contribution, right-hatched the [mj 1=3/2, and left-hatched the the mj =5/2 projections. Note that band 1 and 2 have regions with very low /-character. On these parts of the Fermi surface there is a strong hybridization with other states, mainly Pt-p and -d. (Ref. [80]).

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