Bravais sublattice

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

Fig. 3.1. A planar lattice consisting of n Bravais sublattices of identical oriented molecules.

In passing from the first to the second problem, a feature of importance should be borne in mind. The periods of the orientational structure (3.1.9) can exceed those of the basic Bravais sublattice, Ai, A2. If this is the case, the unit cell A, A2 should be enlarged so that conditions (3.1.10) can be met and translations onto the new vectors R can reproduce the orientations of adsorbed molecules. Then the excitation Hamiltonian (3.1.3) can be represented in the Fourier form with respect to the wave-vector K as... 

The cubic structure is found in many crystals, but with different arrangements of the atoms. The simplest ones are the NaCl and the CsCl structures. The NaCl structure is the superposition of two identical fee Bravais sublattices shifted by 1/2 of the edges of their unit cell one Na+ (Cl ) ion has 6 Cl (Na+) nns along <100> directions. The CsCl structure is the superposition of two identical simple cubic (sc) sublattices translated by 1 /2 of the diagonal of their unit cell one Cs+ (C/ ) ion has 8 Cl (Cs+) nns of the other sublattice along < 111 > directions. The symmorphic space group of CsCl is Oh1 (Pm3m). 

The diamond structure and the cubic ZnS (sphalerite or zinc-blende) structure can be seen as the superposition of two identical fee Bravais sublattices translated by one quarter of the diagonal of their unit cell. In these structures,... 

All sites involved in the diffusion process are crystallographi-cally and thus also energetically equivalent (Bravais sublattice). 

Like the diamond stracmre discussed earlier, the honeycomb stracture is not itself a Bravais lattice. If the lattice is translated by one nearest-neighbor distance, the lattice does not go into itself. There are two nonequivalent, or distinct types of sites per unit cell, atoms a and b, separated by a distance Uq, as shown later in Figure 4.6. However, a Bravais lattice can be created by taking this pair of distinct atoms to serve as the basis. Doing so, shows that the vectors of the two-dimensional hexagonal lattice, a and U2, are primitive translation vectors. A given site on one sublattice with coordinates (0, 0), has three nearest neighbors on the other sublattice. They are located at (0, U2), (fli, 0), and (- , 0). 

Substitutional defects are either impurities or antisites. Impurities , as one would expect, refers to atoms which are not constituents of the semiconductor host. They are not intrinsic to the nature of a solid, but result from its incomplete purification or intentional contamination. Thus, they are referred to as extrinsic defects. Antisites, as with vacancies, are intrinsic to compounds. Antisite defects are found only in crystals with more than one sublattice and having different atoms on each. Si has two sublattices, both fee Bravais lattices, translated by 1/4, 1/4, 1/4 with respect to each other. However, because the atoms on both are the same, moving a Si atom from one sublattice to the other has no effect. By contrast, GaAs has two sublattices, one on which only Ga atoms reside, the other contains only As. Thus, there are two antisite defects, a Ga on an As site (GaAs) or an As on a Ga site (Asoa)- Antisite defects may, and often do, have multiple charge states in the energy gap. 

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