Symmetries of the band structure

In three dimensions there are 14 different types of Bravais lattices, 32 different types of point groups, and a total of 230 different space groups, of which 73 are symmorphic and 157 are non-symmorphic. We will give a more detailed account of symmetry groups in three dimensions after we have considered a simple example in two dimensions. 

In the following we will need to apply symmetry operations to functions of the space variable r, so we define a new set of operators whose effect is to change r  

In this definition of 0 u t], the action is always on the vector r itself. We will prove that the group formed by the operators 0 u t is isomorphic to the group of 

Having defined the basic formalism for taking advantage of the crystal synune-tries, we will now apply it to simplify the description of the eigenfunctions and eigenvalues of the single-particle hamiltonian. We will prove that for any element of the space group 1/ t e S, we have 

To show this we need to prove first, that ir r) and (9 [/ t V t r) are eigenstates of the hamiltonian with the same eigenvalue and second, that  

---

The band structure and Bloch functions of metals have been extensively published. In particular, the results are compiled as standard tables. The book Calculated Electronic Properties of Metals by Moruzzi, Janak, and Williams (1978) is still a standard source, and a revised edition is to be published soon. Papaconstantopoulos s Handbook of the Band Structure of Elemental Solids (1986) listed the band structure and related information for 53 elements. In Fig. 4.14, the electronic structure of Pt is reproduced from Papaconstantopoulos s book. Near the Fermi level, the DOS of s and p states are much less than 1%. The d states are listed according to their symmetry properties in the cubic lattice (see Kittel, 1963). Type 2 includes atomic orbitals with basis functions xy, yz, xz], and type e, includes 3z - r-), (x - y ). The DOS from d orbitals comprises 98% of the total DOS at the Fermi level. 

The observed evolution of the shape of the band-structure upon doping satisfies the Luttinger sum rule [9], It should be noted that only in the Anderson lattice-like limit of the Emery model it is possible to obtain the observed evolution of the FS upon doping. In all other cases, the oxygen symmetry of the FS can be attributed to the (non-renormalized) oxygen band and therefore the strong doping dependence of the band structure cannot be expected. 

The situation for (TMTSF)2N03 is somewhat different since the triangular N03 anion adopts a (2, 0, 0) periodicity below TAO = 45 K [74]. This new lattice symmetry implies a folding of the band structure about planes kx = Tt/2a. Consequently, the anion ordering triggers a transition between a quasi-one-dimensional conductor at high temperature and a two-... 

Indirect but important data on molecular symmetry are provided by transport and magnetic measurements in solids. These properties reflect the collective behavior of electrons in the system, and are indicative of the band structure. 

Solids may be structurally disordered or crystalline. Perfect crystals with completely periodic structures do not exist in Nature. However, most of the discussion here will be based on such idealiized models, and the electronic structure is described in terms of band structures, dispersion relations between formal one-electron energies, s, and wavevectors, k e(k). First, in Section 2, we illustrate how the relativistic shifts (mass-velocity and Darwin) of parts of the band structure with respect to each other may affect the physical properties, including the crystal structure. The second subject (Section 3) treated in this chapter concerns the simultaneous influence of the crystal symmetry and the SO-coupling on e(fe), spin splitting effects, i.e. effects which are without atomic counterparts. 

The Hamiltonian of the electron-photon interaction will be used in a very simplified form taking into account only the simplest band structure of a semiconductor with parabolic electron and hole bands without complications related to heavy and light holes, spin-orbit splitted hole band or with the Dirac model of the band structure in the case of small band gap semiconductors. In the case of simple parabolic band after their size quantization in a spherical symmetry quantum dots the electrons and holes are characterized by envelope wave functions with the quantum numbers I, n, m. An essential simplification of the future calculations is the fact that in the selected simple model the band-to-band transitions under the influence of the electron-photon interaction Hamiltonian take place with the creation of an e-h pair with exactly the same quantum numbers for electron and for hole as follows e l,n,m), h l,n,m). ... 

Clearly, the treatment of a solid involving of the order of 10 3 electrons is even a more complicated matter than that of an isolated molecule or complex in spite of the simplifications introduced by symmetry, and the use of effective potentials, and thus of a band theoretical approach, is probably not adequate in the discussion of wave function sensitive parameters such as spin distributions. But many important properties of solids reflect the electronic energy levels, rather than the finer details of the electronic distributions, and in spite of the fact that band calculations are rarely carried through to self consistency, band structures and energies of simple compounds may be determined sufficiently well to provide a good comparison with experimental data. The main effort has been directed to metals, where the valence electrons are weakly bound, and to simple compounds of high symmetry with the sodium chloride or diamond-like structure. In the latter case this effort also reflects the importance of these compounds in solid state physics and electronics and the elucidation of the band structure was essential for an understanding of many of the important properties of these materials. 

The bands have been calculated by many groups of researchers with similar results. Figure 9.6 shows the comparison of the band structures of MgB2 and graphite. The linearized augmented plane waves method was used for calculations. The symmetry point notations are those for the simple hexagonal Brillouin zone, presented in Figure 4.15. 

---