Phonons force-constant model

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

Isotope superlattices of nonpolar semiconductors gave an insight on how the coherent optical phonon wavepackets are created [49]. High-order coherent confined optical phonons were observed in 70Ge/74Ge isotope superlattices. Comparison with the calculated spectrum based on a planar force-constant model and a bond polarizability approach indicated that the coherent phonon amplitudes are determined solely by the degree of the atomic displacement, and that only the Raman active odd-number-order modes are observable. 

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

Figure 31. Surface phonon dispersion for Cu(lll). The open circles are from HAS experiments, and the open triangles are from EELS experiments. The surface modes shown as solid lines and bulk band boundaries are based on a simple force constant model. The X and Y designations indicate the polarizations of the corresponding modes as identified in the reduced zone diagram in the inset. (Reproduced from Fig. 3 in Ref. 99, with permission.)...

Figure 32. Surface phonon dispersion for Nb(OOl). The data are the solid points which were taken at 900 K. Panels a and b correspond to slab dynamics calculations with two different force constant models the calculation in panel b uses the force constants from the bulk phonon fits. The solid lines represent the surface phonons and resonances polarized mainly longitudinally (or parallel), the lines with long dashes represent phonons polarized mainly perpendicularly, and those with short dashes are shear horizontal. (Reproduced from Fig. 6 of Ref. 107, with permission.)...

We next apply the force-constant model to a simple example that includes all the essential features needed to demonstrate the behavior of phonons. Our example is based on a system that is periodic in one dimension (with lattice vector ai = (a/V5)[x -I- y]), but exists in a two-dimensional space, and has two atoms per unit cellatpositionsti = a/2 )k, ti = —(a/2V2)x. This type of atomic arrangement... 

The force-constant model Table 6.2. Phonon modes in Si. 

Use the Born force-constant model to calculate the phonon frequencies for silicon along the L - r - X directions, where L — (1,1, 1) and X = (1,0,0). Take the ratio of the bond stretching and bond bending force constants to be Kr/Kg = 16. Fit the value of Kr to reproduce the experimental value for the highest optical mode at r, which is 15.53 THz, and use this value to obtain the frequencies of the various modes at X compare these with the values given in Table 6.2. Determine the atomic displacements for the normal modes at F, the lowest acoustic branch at X and L, and the highest optical branch at X and L. 

In particular, the phonon dispersion relations and polarization vectors can be calculated with reasonable accuracy using force-constant models [59] or the embedded atom method [60-62], In recent calculations of Fe-ph and X for surface states, wave functions obtained from the one-electron model potential [63, 64] have been used. For the description of the deformation potential, the screened electron-ion potential as determined by the static dielectric function and the bare pseudopotential is used, Vq z) = f dz e (z,2/ gy)qy), where (jy is the modulus of the phonon momentum wave vector parallel to the surface, and bare Fourier transform parallel to the surface of the bare electron-ion... 

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

Aq is a constant except for the Cu close to Ni. The relation (26) is direct evidence for the attractive force being due to spin fluctuation, irrespective of any theoretical model. In the conventional superconductor, the isotope effect provides direct evidence for phonon-mediated superconductivity. The relation (26) is considered to correspond to the isotope effect in a phonon-mediated superconductivity model. 

The dynamical model employed in the theoretical calculations is the Shell Model, of which there are several variants [6, 9]. It is designed to approximate the physical situation of the ions in the crystalline environment more realistically than does the Bom-von Karman treatment with harmonic force constants between neighboring atoms as discussed in Section II, but its handling of the forces is not so very different. The Shell Model was developed for these materials to account for the bulk phonon dispersion that was measured by neutron scattering experiments as well as for their dielectric... 

DFT calculations of the static lattice were performed within the local density approximation (LDA) using planewave basis sets and ultrasoft pseudopotentials, and the results were used to construct a force constant matrix within a large superceU model. The phonon spectrum was then evaluated as a function of temperature using quasi-harmonic models that allowed us to constmct mode Gruneisen relationships [66] (Fig. 4.4). The results indicated a dip in the V(T) relation at 80 K, that was slightly smaller than that observed for diamond-structured Si (Fig. 4.5). 

---