Periodic wave functions, crystals

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. 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

In actual calculations on crystals, it is impractical to include all lO atoms and so we use the periodicity of the crystal. We know that the electron density and wave function for each unit cell is identical and so we form combinations of orbitals for the unit cell that reflect the periodicity of the crystal. Such combinations have patterns like the sine waves that we obtained from the particle-in-the-box calculation. For small molecules, the LCAO expression for molecular orbitals is... 

The electrical properties of many solids have been satisfactorily explained in terms of the band theory . Briefly, the motion of an electron detached from its parent atom but free to move in a periodically varying potential field, such as that existing between atoms on a crystal lattice, is expressed in terms of a wave function (Boch Function). This particular... 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

In a perfect crystal with periodic potentials, electron wave functions form delocalized Bloch waves [46]. Impurities and lattice defects in disordered... 

In crystalline semiconductors, it is relatively easy to understand the formation of gaps in energy states of electrons and hence of the valence and conduction bands using band theory (see Ziman, 1972). Band structure arises as a consequence of the translational periodicity in the crystalline materials. For a typical crystalline material which is a periodic array of atoms in three dimensions, the crystal hamiltonian is represented by a periodic array of potential wells, v(r), and therefore is of the form, 7/crystai = ip l2m) + v(r), where the first term p l2rri) represents the kinetic energy. It imposes the eondition that the electron wave functions, which are solutions to the hamiltonian equation, H V i = E, Y, are of the form... 

Because of the presence of the regularity associated with a crystal with periodicity, we may invoke Bloch s theorem which asserts that the wave function in one cell of the crystal differs from that in another by a phase factor. In particular, using the notation from earlier in this section, the total wave function for a periodic solid with one atom per unit cell is... 

We adopt the term Bloch functions for the cell-periodic part of the full electron wave-function in the crystal. 

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