Frozen phonon

Figure 1. Representation of unit cells for (a) FeaNi and (b) CuZn. Corresponding to a tetragonal symmetry in the case of FeaNi (Ni atoms are marked black) and to the LI2 (CuaAu) structure in the case of c/a = 1. CuZn shows also tetragonal symmetry, whereby c/a = 1 corresponds to the B2 structure (black circles represent Cu atoms). In (b) a frozen phonon in [001] direction is indicated for the Zn atom.

FegNi. Frozen phonon calculations combined with the determination of the electron-phonon matrix in the framework of the theory of Varma and Weber have been carried out for the ferrous alloy. The resulting phonon dispersion for the bet phase was already presented elsewhere . As expected, no softening or anomalous curvatures have been detected. This confirms the existence of a bet ground state for FesNi. 

In the cuprates the phonon modes that modulate the Cu-0 bondlength induce similar charge transfer. It was found10, however, that the direction of the electronic polarization was opposite of the ionic polarization when the cuprates are doped with holes, and the electronic polarization is strongly dependent on the wavevector q. Fig. 1 shows the effective charge of oxygen as a function of q, calculated for the two-band Hubbard model with a frozen phonon,... 

Thermally-induced network vibrations broaden the absorption edge and shift the band gap of semiconductors. The thermal disorder couples to the optical transition through the deformation potential, which describes how the electronic energy varies with the displacement of the atoms. The bond strain in an amorphous material is also a displacement of atoms from their ideal position, and can be described by a similar approach. The description of static disorder in terms of frozen phonons is a helpful concept which goes back 20 years. Amorphous materials, of course, also have the additional disordering of the real phonon vibrations. 

According to the theory of the Urbach absorption edge in crystals, the slope E is proportional to the thermal displacement of atoms r(7). The frozen phonon model assumes that an amorphous semiconductor has an additional temperature independent term, r , representing the displacements which originate from the static disorder, so that... 

More difficult to calculate are the properties which depend on the response of the solid to an outside influence (stress, electric field, magnetic field, radiation). Elastic constants are obtained by considering the response of the crystal to deformation. Interatomic potential methods often provide good values for these and indeed experimental elastic constants are often used in fitting the potential parameters. Force constants for lattice vibrations (phonons) can be calculated from the energy as a function of atomic coordinates. In the frozen phonon approach, the energy is obtained explicitly as a function of the atom coordinates. Alternatively the deriva-tive, 5 - can be calculated at the equilibrium geometry. 

The term M p,is the eph coupling constant, and ba is the annihilation operator of the mode a, whose frequency and normal mode coordinate are represented by Q,a and Qp, respectively. The sites for electrons i( T) coupled with phonons are restricted to the C region or a subpart of C. The focused modes should be sufficiently localized on the molecule in term of their definition. Practically, these internal modes can be calculated by means of a frozen-phonon approximation, where displaced atoms are atoms in the c region (or its subpart) denoted as a vibrational box though a check for convergence to the size of the vibrational box is necessary [90]. 

Although the preferred method of calculating a spectrum is to perform an ab initio calculation on an extended solid, extracting frequencies and displacements across the Brillouin zone, on a fine A-grid, this approach can be computationally very expensive. In plane wave codes like CASTEP [18], CPMD [19], TWSCF [20], VASP [21], ABINIT [22], and some others, the number of plane waves that are taken into consideration, the selected correlation fimctional and the choice of pseudopotential will all have an impact on the quality of the calculations. Some codes (e.g. ABINIT) alleviate the problem by permitting frozen phonon calculations at the symmetry zone boundary, i.e. (0,0,0), (l/2,0,0), (l/2,l/2,0) and (l/2,l/2,l/2) and so determine the dynamical matrix at these points. The code then interpolates values of the d3mamical matrix for all the points within the Brillouin zone and uses these to calculate the solution to the vibrational problem inside the zone. 

An advanced approach to lattice dynamics involves the precise determination of the crystalline total energy as a function of the lattice displacement associated with a particular phonon. This method is commonly referred to as the frozen-phonon method. It utilizes first-principles band-structure techniques to obtain the total energy for each frozen-in position of the lattice. The phonon frequency can then be obtained from the resultant potential-energy curve. 

The two current ways of carrying out the "direct" ab initio calculations of energies or forces have come to be known as the "frozen phonon" method and the force constant method. In the former approach, the atoms are displaced in a given phonon mode and the phonon frequency is calculated either form the energy E = (i/2)kx2 or the force F= -kx. In this case both E and F provide exactly equivalent information and Frozen phonon calculations have prooved very successful and have been applied to many modes in Ge, GaAs, ... 

The above work demonstrated that the LDF approximation is precise enough to be useful in "direct" approaches to lattice dynamics, and that its accuracy is sufficient - in fact extreme. Among all the recent applications of DF to different substances in various situations (atoms, molecules, crystals, surfaces. Interfaces, etc.) the frozen phonon calculations have so far been the most stringent test for the validity of the DF method itself. 

The fundamentals ofj 2 he DF method are discussed in detail elsewhere in this volume the present lecture notes start where those of R. M. Martin ended the method provides us with 1) energy of the unit cell 2) forces on atoms and 3) stress over a unit volume. Only those details of the method that are specific to our present applications are summarized in Section 2. The successive steps leading to dynamical properties - static equilibrium, frozen phonon method - are then explained in Sections 3. and 4 the topic of frozen phonons is treated only briefly in these notes, because an adequate text already exists detailed material completing Section 4 is to be found in Ref. 13. 

Having checked the calculated static equilibrium (Section 3), we can start displacing the atoms and evaluate the energy-differences between distorted and undlstorted structures. The frequency of the "frozen phonon" defined by the selected displacement pattern is related to its energy by... 

We are encountering a feature that is inherent in all "direct" approaches to lattice dynamics as the total energy can only be calculated with finite displacements u, the harmonic terms appear intertwined with the enharmonic contributions, and we have to treat all the expansion terms simultaneously from the very beginning. In addition to the frozen phonon frequencies, we then obtain detailed Information on the anharmonicity of the mode in question, data which are difficult to find by other means, both theoretical and experimental. In some cases, it can be verified that the displacement u is small enough and does not give rise to any noticeable enharmonic effects. With most displacement patterns, however, the total energy has to be evaluated for several magnitudes of the displacement (typically 5 to 25 values of ) ... 

The most important aspects of the frozen phonon approach have been thoroughly explained already in Ref. 13, and the calculated frequencies of principal frozen phonons in GaAs are given, for illustration, in Tab. 4.1 the reader is recommended to look for all details at pp. 75 - 93 of Ref. 13. The following Section is a summary of the most important conclusions ... 

Large relative errors in the calculated TA(X) frequency (see Tab. 4.1) come from nearly complete cancellation between the ion-ion and electronic terms here we have approached the limits of reliability of the approximations presently used (viz. local potentials and Slater exchange) - but not those of the DF- or frozen phonon method using the more precise norm-conserving potentials, TA(X) have been obtained with excellent precision on Si, Ge in Ref. 5,37,38. Note that the sign of the Grunelsen parameter for TA(X) is correctly predicted (Tab. 4.1), even if the absolute value is off TA(X) is a soft mode, its frequency decreases with pressure. 

The L0(F) mode cannot be treated in the same manner as the T0(F) frozen phonon, because of the presence of a macroscopic electric field. The method proposed in Ref.6 is the ab initio evaluation of effective charges that determine the LO-TO splitting and will be discussed in detail in Section 6. 

Several calculations applying the "frozen phonon" concept to different substances h ve accumulated in the past 3-4 years. Be d l the worJcQon Si already mentioned in Section 1, phonons in Ge and A1 were studied as well, with the aid of the norm-conserving pseudopotentials. Ref. 41 demonstrates the use of forces, instead of energies, in frozen phonon calculations. (This will still be explained in Section 5.1.) In the context of static calculations (pha g diagrams), S. Froyen evaluated also the TO(r) frequency in GaAs. (7.84 THz), and energies of the frozen phonons at r and X in NaCl. ... 

FORCES AND PLANAR FORCE CONSTANTS 5.1. Forces and Frozen Phonons... 

We have seen already, in the context of frozen phonons, that anharmonic terms appear in the "direct" approach simultaneously... 

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