i.e. for small wavenumbers. However, this large computational cost in DFT can be avoided by calculating phonon dispersion relations using Density Functional Perturbation Theory, as described in [ 2 ]. 5.1.2 Molecular Dynamics Alternatively, the phonon frequencies can also be obtained from molecular dynamics runs [ 3 ]. The relative displacements u l j in Eq. 5.8 can be Fourier- transformed, leading to ± k j ð t Þ ¼ 1 N cell X j X l exp ð² i k R l Þ u l j / X m exp ð² i x ð k ; m Þ t Þ ; ð 5 : 12 Þ where N cell is the number of primitive cells in the supercell. Fourier-transforming equation 5.12 to frequency space gives ± k ð x Þ / X m d ð x ² x k ; m Þ : ð 5 : 13 Þ The spectral analysis of ± k ð x Þ ; i.e. finding sharp peaks in the power spectrum P k j ³ j ± k j ð x Þj 2 ð 5 : 14 Þ gives the phonon frequencies. The advantage of this method is that it can be used for more complicated systems, where explicit calculation of the full dynamical matrix would be extre- mely expensive. Furthermore, we can calculate the temperature dependence of the phonon spectrum by simply performing molecular dynamics simulations at dif- ferent temperatures. The temperature dependence of the phonon spectrum is due to anharmonic effects, i.e., at larger displacements when terms higher than second order contribute to the potential energy in Eq. 5.4 . 5.1.3 Thermodynamics The quantum mechanical solution of a system of harmonic oscillators [ 1 ] states that the allowed energies of a phonon mode labelled by k and m are E k m ¼ 1 2 þ n ³ ´ x ð k ; m Þ ; ð 5 : 15 Þ where ± h is the reduced Planck constant, and n is a non-negative integer. The canonical partition function of a system can be calculated as 5.1 Lattice Dynamics 53
Z ¼ X j exp ð² b E j Þ ; ð 5 : 16 Þ where E j is the energy of the j th state and b ¼ 1 k B T : Substituting 5.15 into this expression, we obtain Z vib : ¼ Y k ; m X 1 n k m ¼ 0 exp ² b 1 2 þ n k m ³ ´ ± h x ð k ; m Þ ³ ´ " # ð 5 : 17 Þ which can be simplified by using X 1 n ¼ 0 exp ð² nx Þ ¼ 1 1 ² exp ð² x Þ ð 5 : 18 Þ to Z vib : ¼ Y k ; m exp ð² b ± h x ð k ; m Þ = 2 Þ 1 ² exp ð² b ± h x ð k ; m ÞÞ : ð 5 : 19 Þ In the case of a crystal, the total partition function is Z ¼ exp ð² b/ 0 Þ Z vib : : ð 5 : 20 Þ The partition function can be used to obtain all thermodynamic quantities. For example, the free-energy can be obtained as F ¼ ² k B T ln Z ð 5 : 21 Þ ¼ / 0 þ k B T X k ; m ln 2 sinh ð b ± h x ð k ; m Þ = 2 Þ ½ ; ð 5 : 22 Þ and the internal energy is U ¼ 1 Z o Z o b ð 5 : 23 Þ ¼ / 0 þ X k ; m ± h x ð k ; m Þ 1 2 þ 1 exp ð² b ± h x ð k ; m ÞÞ ² 1 ³ ´ : ð 5 : 24 Þ This result leads us to a rather crude method for approximating the real temper- ature in the case of a classical molecular dynamics run [ 4 ]. We equate the kinetic energy E kin.
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