Bloch wave vector

The Bom-von Karman contour condition demonstrates that the Bloch wave vector of free electrons in a cubic lattice is, according to Sommerfeld, constituted only by real components. The number of k values (k = p/h) admitted in a primitive cell of a reciprocal lattice is equal to the number of sites in the crystal. The linear momenta operator, p, is... 

Equation (1.4.42) for the gap width is very similar to the expression proposed by Phillips (1970) for Eg as recalled in Section 1.3. It has two contributions an ionicity gap (ec — fa) and a covalent gap, which, here, reads 2y/F. Since Fmin depends both upon the resonance integrals j8 and the Bloch wave vectors k at the gap edges, Fmin is a function of the nature and symmetry of the orbitals involved in the chemical bond, of their relative directions on neighbouring atoms and more generally of the crystal structure. 

For a given Bloch wave vector, the bulk modes m = Mexp(ikna)exp(i( t) and v = yexp(i/cna)exp(i(ot) satisfy (4.1.11). Their frequencies are equal to ... 

A surface mode is a singular solution of the bulk equations of motion, which obeys the surface boundary conditions. Its Bloch wave vector may be complex. In the case of the alternating linear chain, combining the two systems of equations leads to ... 

We assume that the Hartree-Fock orbitals of our problem have already been determined, based on a Fock matrix Fabik) whose subscripts a and b label Bloch states built from basis GTO s and whose argument fe is a Bloch-wave vector in units such that the Brillouin zone is of unit length. Our methods for the calculation of Fatik) and related quantities were reported in detail in our previous work [10]. 

Bloch s theorem states that in a periodic solid each electronic wave function can be expressed as the product of a wave-like component (with wave vector k) and a cell-periodic component/ (r) ... 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

Furthermore, the field operator is expanded in the Bloch waves with wave vector k in the band denoted by b as... 

Expressed formally, the wave must be matched in amplitude at the surface and in phase velocity parallel to the crystal strrface. This implies that the tangential components of D and H must be continuous across the strrface, and the components in the crystal strrface of the wavevectors inside and outside the strrface must be the same. If n is a rmit vector normal to the crystal strrface, whatever the values of k o or the resttlting Bloch wave inside the crystal, then... 

Let us consider a wave vector k of an occupied state close to kp and its equivalent k — 2kp as indicated in Fig. 1.32. The Bloch functions of both states will be given by Eq. (1.31) ... 

Here, gi is the length of the primitive reciprocal lattice vector. The constants a, p, and are determined either by considering leading Bloch waves or by fitting with first-principles calculations. The method of Harris and Liebsch is used extensively in the treatment of atom scattering data. With some modifications, the method of Harris and Liebsch is also applicable to calculate STM images. We will discuss it in detail in Chapter 5. 

If at the Fermi level, the only surface Bloch wave of the material is a sinusoidal function with Bloch vector q. 

The action of the space-group operator (R v) c G (with R c P), the point group of the space group G on the Bloch function kfr) gives the transformed function vJk(r). To find the transformed wave vector k we need the eigenvalue exp ( ik -t) of the translation operator ( t). 

Equation (3) shows that the space-group operator (R v) transforms a Bloch function with wave vector k BZ into one with wave vector R k, which either also lies in the BZ or is equivalent to ( ) a wave vector k in the first BZ. (The case Id = k is not excluded.) Therefore, as R runs over the whole R = P, the isogonal point group of G, it generates a basis ( 0kl for a representation of the space group G,... 

Let us first consider an infinite 3D lattice (2D and 1D infinite and truncated lattices are considered in Sections III and IV) with one dipole or many dipoles per unit cell. The matrix of R(z) in (1.33) is diagonalized by a transformation to Bloch states of wave vector K, defined, for instance, by the creation operator... 

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 ... 

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. 

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