First-Principles Calculation of Force Constants and Full Phonon Dispersions
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FIRST-PRINCIPLES CALCULATION OF FORCE CONSTANTS AND FULL PHONON DISPERSIONS SIQING WEI and M. Y. CHOU Georgia Institute of Technology, School of Physics, Atlanta, GA 30332-0430 ABSTRACT We calculated the real-space force constants and full phonon dispersion curves for elemental semiconductors (silicon and germanium) under the local-density approximation with the Hellmann-Feynman forces. The force constants are obtained through supercell calculations for planar displacements in three different symmetry directions. From these real-space force constants the dynamical matrix for an arbitrary wave vector in the Brillouin zone can be constructed. The procedure is simple in concept and requires no complicated computer programing. It is also possible in principle to handle the anharmonic effects.
INTRODUCTION The phonon dispersions are a vital information needed in the calculations of heat capacities, thermal expansion coefficients, and electron-phonon interactions. 1 Early efforts to study the dispersion spectrum have concentrated on empirical models such as the valence force fields and bond charge models 2,3 . Due to the empirical nature, these models obtain their parameters by fitting to the experimental results and, therefore, lack the predicting power. 4 ,5 With the development of the local-density approximation, two approaches have been used to calculate the phonon dispersions: (i) the "direct" approach, where a supercell is used to calculate the total energy or the Hellmann-Feynman 6 forces for the distorted system (hence, also known as the supercell approach); 7 ,8 (ii) the linear 10 response approach, where the atomic displacements are treated as perturbations.9, There are advantages and shortcomings of these two approaches.
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The size of the su-
percell restricts the former approach to high symmetry points or directions only, while the perturbation approximation handles only linear phenomena. METHODS In this paper, we are going to show a new method which avoids the drawbacks of the approaches mentioned above by calculating the real-space (interatomic) force constant matrices using the supercell approach. The method is based on the observation that the planar force constants are actually projected sums of the real-space force constants. Therefore, the real-space force constant matrices can be solved from a set of linear equations containing the planar force constants. (The completeness of the equations will Mat. Res. Soc. Symp. Proc. Vol. 291. @1993 Materials Research Society
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be discussed in detail elsewhere1").
The planar force constants can easily be calculated
for supercells along high symmetry directions. Once the real-space force constants are known, it will be straightforward to obtain the phonon spectrum. Thus we extend the supercell approach in the phonon spectrum calculation to arbitrary wave vectors. In this approach, the anharmonic effect can also be handled within the same framework. Other advantages of the method include the following: (i) it requires only standard total energy codes; an
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