Born-von Karman

The Born-von Karman boundary conditions then restrict the allowed electronic states to those in the graphene Brillouin zone that satisfy... 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

One way of seeing this explicitly is to consider the Schrodinger equation modified for a periodic lattice with Born-von Karman periodic boundary conditions assuming a wavefunction ij/(r) = Eqcq exp(iq r) and a potential U(r) which has the periodicity of the lattice U(r) = 2,GUG exp(/G r), where the Fourier58 coefficients UG are given by UG = JCeiiG(r) exp (—zG r) dr, the Schrodinger equation is rewritten as... 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Up to this point, the analysis is rigorously correct and general it is simply a restatement of the Born-Von Karman expansion of the energy in terms of relative displacements—.see Eq. (8-17). However, we shall now make a major approximation in taking the force constants from the very simple valence force field that we described in Chapter 8. This will give us a clear and correct qualitative description of the vibration spectra and will even give semiquantitativc estimates of the frequencies. Afterward, we shall consider the influence of the many terms that are omitted in this simple model. 

Fig. 3.1. Phonon dispersion curves of diamond along the main symmetry directions calculated from a Born-von Karman model fitted to neutron scattering experimental data (after [50]). The frequencies are expressed in wavenumber v = uj/2itc. Along the [110] directions (E), the modes are neither purely longitudinal nor transverse, and three branches exist for each category. Copyright 1992 by the American Physical Society...

Let us consider the ID lattice (1) with repeat distance a, where each unit cell contains Just one atomic orbital (AO) /. The simplest way to describe the energy levels of this system is to impose Born-von Karman boundary conditions. Essentially, this means that we are bending our system as shown in (2), that is, we are transforming the chain into a loop. However, since the number N) of sites is very... 

The occupied valence bands consist of two low-lying a bands and a double degenerate tt band just below the Fermi level. Also the lowest unoccupied band is of tt symmetry. Thus, without the DC field the four energetically lowest valence bands are double occupied and all other bands are empty. We shall use this information below in quantifying the effects of the external DC field in different approximations, i.e., we shall analyze the occupation of the different bands as a function of band index. Moreover, in order to quantify the electronic distribution, we shall use the number of electrons inside the muffin-tin spheres (with radii of 1.1 a.u.) for the 24 atoms per Born von Karman zone. 

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

Dispersion Curves and the Bom-von Karman Constants 1179 Table 12.1 The Born-von Karman force constants for sodium at temperature 90 K (after [9]). 

The coefficients 0ap in (12.14), which represent the second derivatives of the potential energy with respect to the atomic displacements determined at the equilibrium points, are called atomic force constants. By definition, they have an ex-pUcit physical meaning. The coefficient 4>afi(lk I k ) is equal to the minus force which acts on the atom (Ik) in the direction a, when the other atom (Vk ) deviates per unit distance in the direction /3. The Born-von Karman model implies that all other atoms stay at their equihbrium positions. 

Table 12.2 The Born-von Karman force constants for niobiur...

This approximation has the benefit of allowing us to introduce the Born-von Karman boundary conditions given by Equation (11.20), where N is the number of atoms in the ring ... 

The Hamiltonian H we were talking about represents an effective one-electron Hamiltonian. From Chapter 8, we know that it may be taken as the Fock operator. A crystal represents nothing but a huge (quasi-intinite) molecule, and assuming the Born-von Karman condition, a huge tyclic molecule. This is how we will get the Hartree-Fock solution for the crystal - by preparing the Hartree-Fock solution for a cyclic molecule and then letting the number of unit cells N go to infinity. 

Fig. 31. Phonon dispersion curves for CcPds al room temperature. Solid symbols are longitudinal modes, open symbols transverse modes. In the [110] direction A, transverse polarized in (110) plane O, transverse polarized in (100) plane. Results of Born-von Karman fit including breathing term are shown as solid and dashed lines (the latter for A data points). Dotted lines if breathing term is neglected (Severing el al. 1988).

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