Bloch Band Model

From what we have learned above, particularly in Equation 16.36, a one-electron wave function in a crystal should be written as 

In calculations, the solution / of a one-electron SE is approximated by a linear combination of atomic orbitals, /n(r)  

The eigenvalues appear as energy bands. The energy levels are functions of the wave vector k  

The density of states is the number of states in a given energy interval. 

Improved calculational schemes were originally developed by the American physicists John C. Slater and Conyers Herring, using plane waves as basis sets. Slater used a muffin-tin potential, where the atomic wave functions inside the atomic spheres were fit to the plane waves. He was also the first person to use a viable local exchange approximation, which he called the Xa model. 

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The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

The appearance of one or more CT-excitons below the conduction band of the conjugated chain may not appear to be of major importance for the properties of the material. In fact the consequences of the occurrence of excitons are significant. Slater and Shockley (1936) demonstrated that the descriptions of the system by Bloch functions, i.e. the band model, and by localised excitations, i.e. excitons, were related to one another by a unitary transformation. They were also the first to consider the impact of the... 

Finally, we derived the BLOCH FUNCTION to show that these energy bands, in reciprocal space, do have some validity in quantum mechanics. It also gives Insight as to the nature of the Fermi level. We also illustrated band models in 5.3.9. What is Important to realize that the valence band there is drawn in two dimensions. Actually, it follows the Brlllouin Zone or k-space of the crystal lattice in three-dimensions. The Fermi level surface is also affected by both k-space and temperature. It is constrained by reciprocal space, just as the BrUlouin Zone is. We use the band model to illustrate certain aspects of each unique crystal. Otherwise, the required model would be quite complex, particularly those crystals with low symmetry. We usually illustrate some specific defect and the band model immediately adjoining it. For a phosphor, this would be the activator (impurity) center. Since we have already (Chapter 2) examined various point defects, let us now illustrate them within the periodic lattice as a function of the energy bands and the Band Model. 

Here, and N are the phonon polarization and the number of lattice molecules, respectively, while Vq is the Fourier transform of the particle-crystal interaction Vq, and Pq = /dr wo(r)pe i, with wo(r) the Wannier function of the lowest Bloch band [128], The validity of the single-band Hubbard model requires that J,V < A, and the temperature k T < A with A the energy gap to the first excited Bloch band. 

The general description of a metal is in terms of orbitals, which are eigenfunctions of energy and momentum. The total wave function is, in principle, a Slater determinant. The Swiss physicist Felix Bloch developed the general band model. 

Band theory is basically a one-electron theory. Electron-electron interactions are only included in the form of an average contribution to the effective electron-ion interaction potential. Thus, band theory should be most informative for modeling the electronic structure of liquids for which the MNM transition is of the Bloch-Wilson band-overlap variety. Fig. 2.13 illustrates some typical results for the electronic density of states of mercury in a series of structures with constant interatomic separation. With increasing density, the band-overlap transition is clearly evident as the gap closes between the lower, predominantly s-like band and the upper p-band. These results agree qualitatively with the observed electronic properties of expanded mercury although, as we shall see in chapter 4, the actual MNM transition occurs in a density range for which the band model still predicts a nonvanishing density of states at the Fermi level. 

More general expressions use a two-dimensional (2D) wavefunction and more than one band, i.e., for a (2/i -f-1) band model with Bloch wavefunctions in... 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

Wave propagation in periodic structures can be effieiently modeled using the eoncept of Bloeh (or Floquet-Bloch) modes . This approach is also applicable for the ealeulation of band diagrams of 1 -D and 2-D photonic crystals . Contrary to classical methods like the plane-wave expansion , the material dispersion ean be fully taken into aeeount without any additional effort. For brevity we present here only the basie prineiples of the method. 

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. 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

Pseudo ID crystals are often described by a tight binding Hubbard type model with matrix elements t j and tj. (t >>tj, ), an intramolecular Coulomb interaction, U, and an electron-phonon interaction parameter, X. The quasiparticle states of the electrons can be described by the usual Bloch model with wave vector, k, provided that U is sufficiently small compared with the band width, At. However, because of phase space considerations, particle-particle scattering is much more important in ID than in 3D systems. 

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