Band structures and Bloch function

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 existence of surface states is a consequence of the atomic structure of solids. In an infinite and uniform periodic potential, Bloch functions exist, which explains the band structures of different solids (Kittel, 1986). On solid surfaces, surface states exist at energy levels in the gap of the energy band (Tamm, 1932 Shockley, 1939 Heine, 1963). 

Cf. the introduction to band-structure calculations with Bloch functions i/r = J2n ,knaXn by R. Hoffmann, Angew. Chem., 99, 871 (1987) Angew. Chem., Int. Ed. Engl., 26, 846 (1987) and references cited therein. 

Band Structure for the Free-Electron Case. If the electron is free, then the Bloch functions are simple plane waves, because the wavef unctions nk(r) used for the expansion Eq. (8.4.2) are themselves plane waves. For an electron gas with no lattice and no imposed symmetry, Fermi-Dirac statistics apply At 0 K all electrons pair up (spin-up and spin-down), with an occupancy of 2 for every k value from k 0 to the Fermi wavevector kF =1.92/rs = 3.63 a0/rs, and from zero energy up to the Fermi energy eF = h2kF2 /rn 50.1 eV rs/a0) 2, where rs is the radius per conduction electron and a0 is the Bohr radius, and the energy levels are spherically symmetric in k-space. The Fermi surface is a sphere of radius kF. Note that the ratio (rs/a0) varies from 0.2 to 1.0 nm for metals (Table 8.3). 

Each of these component band structures could be understood in further detail.77 Take the S 3p substructure at F. The unit cell contains two S atoms, redrawn in a two-dimensional slice of the lattice in 105 to emphasize the inversion symmety. Diagrams 106-108 are representative x, y, and z combinations of one S two-dimensional hexagonal layer at T. Obviously, x and y are degenerate, and the x, y combination should be above z—the former is locally o antibonding, the latter x bonding. Now combine two layers. The x, y layer Bloch functions will interact less (x overlap) than the z functions (o antibonding for the V point, 109). These qualitative considerations (x, y above z, the z bands split more than the x, y bands) are clearly visible in the positioning of bands 3-8 in Fig. 37a and b. 

First, the irreducible part of the Brillouin zone now varies from k = 0 to k = Tr/d = tt/2d. Indeed, doubling the parameter of the unit cell in real space halves the size of the Brillouin zone (or the reciprocal-space unit cell). Second, recall that orbital interactions are additive and that the final MO diagram (or band structure) is just the result of the sum of all the orbital interactions. Within each individual H2 unit the interactions simply correspond to the bonding (a) and antibonding (a ) MOs of each individual H2 unit. There are three types of interactions involving the MOs of different H2 units interactions between all the a orbitals interactions between all the a orbitals and interactions between the a and the a orbitals. Since all the an orbitals are equivalent by translational symmetry, their interaction is described by the Bloch function ... 

The replacement of main-group atoms in clusters by transition-metal atoms generates a richer structural chemistry superimposed on the cluster basics illustrated by the p-block systems. A logical question arises here. What would a one-dimensional material containing a transition metal look like Well, the d AOs will generate bands in a similar manner as the s and p orbitals. The major novelty will be the introduction of orbitals of 8 symmetry. Let s look at a hypothetical chain composed of equidistant Ni atoms (d = 2.5 A). The computed band structure, DOS and COOP are illustrated in Figure 6.14. The COs at k = 0 and ir/d arc drawn below. As a review of the previous section, we will reconstruct it starting from the Bloch functions associated with the nine Ni AOs. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

The Band Concept. The eigenfunctions for the delocalization of Si a electrons along the skeleton are described by Bloch functions (22-23). A good quantum number is not a space coordinate but a wave vector K. An example of a band structure is shown in Figure 7. The band gap energy (Eg) is the difference between the edges of the conduction and valence bands. 

Formally the energy-band structure for an infinite crystal is defined to be the eigenvalues Ej(k) of the one-electron Schrodinger equation (1.4) obtained as functions of the Bloch vector k. Physically, this definition is of course not very illuminating and I shall therefore now give the simplest possible derivation of a condition for the formation of energy bands, which has a very appealing physical interpretation. 

The fundamental difference between an organic and an inorganic semiconductor is in their bonding. The covalent bonding that prevails in inorganic semiconductors results in a large electronic wave function overlap, which is the basis for forming the Bloch waves and the band structure. In OSCs,... 

The important term electronic band structure is identical with the course of the energy of an extended wave function as a function of k, and we seek for E ip k,r)), the crystal s equivalent to a molecular orbital diagram. As stated before, the fc-dependent wave function ip k,r) is called a crystal orbital, and there may be many one-electron wave functions per k, just as there may be several molecular orbitals per molecule. Due to the existence of these periodic wave functions, there results stationary states in which the electrons are travelling from atom to atom the Bloch theorem thereby explains why the periodic potential is compatible with the fact that the conduction electrons do not bounce against the ionic cores. 

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