Phonon dispersion relation of SnSe at two different temperatures was calculated using a density functional perturbation theory [63] as implemented in Quantum ESPRESSO package [64]. PBE-GGA [65] ultrasoft pseudopotential [66] from the Standard Solid State Pseudopotentials (SSSP) library for all atomic species with a 35 RY energy cutoff for wavefunction and 350 Ry for charge density along with a 4 × 12 × 12 Monkhorst-Pack-type grid. 18 irreducible k-points were calculated from a 2 × 4 × 4 k-points mesh [67]. Figure 3.8 shows the calculated phonon dispersion relation of SnSe. The lattice parameters are based on the synchrotron XRD measurements at 65 K. Several noticeable low-lying optical modes can be found near Γ point, and the frequency of these branches are ∼ 1 T Hz, comparable to the value estimated from the fitted specific heat. Thus we may attribute the Einstein mode that appears in the specific heat data to those optical modes.
From the calculated phonon dispersion relation, we can also estimate the sound velocity near the zone center. Table 3.2 shows the velocities of different phonon branches. It can be seen that the sound velocity strongly depends on the mode of the phonon. The estimated Debye temperature, via θD= (4πV3N
0) 1 3hvm
kB , also exhibits similar behavior.
The calculated phonon density of states (PDOS) is plotted in Figure 3.9. A distinct phonon energy gap is found at ∼ 3 T Hz, corresponds to the gap at the same energy in phonon dispersion relation plot. Since the total energy U and specific heat at constant volume CV depends on PDOS,
via
U = Z
g (E) f (E) EdE =
Z g (E) E ekB TE − 1
Figure 3.8: Phonon dispersion relation of SnSe calculated using DFT, calculated by Drs. Ching-Ming Wei and Chen-Rong Hsing in Taiwan.
Figure 3.9: Calculated PDOS of SnSe at 65 K. The resemblance of the PDOS between 0 and 3 GHz and that above 3 GHz is due the softening of zone center optical mode, which mimics the dispersion of an acoustic mode. CV = ∂U ∂T V = Z g (E) E2ekB TE kBT2 ekB TE − 1 dE , (3.13) where g(E) denotes the PDOS, and f (E) is Bose-Einstein distribution function, we can calculate for CV and compares it with our experimental data.
The calculated specific heat are plotted in Figure 3.10. Some noteworthy remarks are: (1) The calculated Cv is close to the measured Cp over a wide range of temperatures. As
temperature goes higher, Cv approaches the Dulong-Petit limit.
(2) Theoretically, the Cp should be no less than that the Cv. However, this is not the case
below 125 K. One possible scenario is that the calculated Cv is based on PDOS at 65 K and we
assume that the PDOS doesn’t vary with temperature. Nonetheless, we found that at T > θD, the
Cp is noticeable greater than the Cv and also the D-P limit, and the difference between them gets
larger as temperature increases. This phenomenon is a clear signature of strong anharmonicity in SnSe.
Figure 3.10: (a-c) Comparison of measured (CP) and calculated (CV) specific heat at different
temperature ranges. (d) C/T vs. T2 plot for both measured and calculated data.
calculated data and using , we obtain a Debye temperature of ∼ 182 K, which is close to our experimental value 204 K (fitted with experimental Cp data).
For the vibration mode that the optical phonons represent, Li et al. [57] attribute it to T Oc mode, as illustrated in Figure 3.11. Li et al. suggested this atomic motion accounts for most
of the anharmonic effect in SnSe, greatly suppresses its thermal conductivity. Therefore, the low- lying optical phonons found in both Cp measurement and phonon dispersion relation calculations
represents the TOc mode, and the corresponding atomic motion is illustrated in Figure 3.11. Next, we will discuss how the presence of this low energy optical mode affects thermal conductivity, and also why this particular mode has such a low energy and strong anharmonicity. Above we have demonstrated that for SnSe single crystals, except for phonon-phonon interactions, all the other phonon scattering mechanisms are negligible. More specifically, the interaction is between T Ocmode
and acoustic phonons. The reason for them to interact originates from the fact that the interatomic force constant is not purely harmonic and contains higher order terms. To verify the existence of ph-ph coupling, further experiments such as inelastic neutron scattering is needed. Ideally, for strong anharmonic systems, the corresponding phonon modes should have a wide linewidth, as energy and
Figure 3.11: The displacement pattern for the lowest-energy T Oc mode (Pnma) in SnSe, taken from
Ref. [57].
time are conjugate to each other according to Heisenberg uncertainty principle. A broad linewidth indicates a short lifetime of that phonon mode.
In addition, avoided crossing of the coupled phonon modes may be observed in the measured phonon dispersion relation. Basically, avoided crossing indicates the presence of perturbation to the Hamiltonian which contains off-diagonal elements. Neumann and Wigner demonstrated that the two phonon branches would not cross if they belong to the same irreducible representation (i.e., symmetry) [68]. The avoided crossing is found to lower the group velocity where the phonon bands avoid crossing, thereby reducing the thermal conductivity [69]. In the case of SnSe, for example, the higher order terms in interatomic force constant can be treated as the perturbation to the harmonic Hamiltonian. This results in a split in the originally degenerate states at crossing point. Moreover, after avoided crossing, the new states are mixture of the original degenerate eigenstates. In fact, as can be seen in Figure 3.8, the avoided crossing between acoustic and low-lying optical modes in SnSe is found for some phonon states, for example, the phonon modes along Γ − T direction at around 1 THz. But the crossing in some directions (e.g., Γ − Y ) is not evidently avoided. Nonetheless, direct experimental measurements on these states are necessary to decisively determine the details.
We conclude that there exist low lying optical phonons near zone center in SnSe, which may be responsible for the low thermal conductivity. However, why SnSe possesses such a peculiar phonon mode and strong anharmonicity is unclear. In fact, it is believed that the anharmonicity in SnSe is closely related to the chemical bonding in the system. Hence, in next section, we need to look into the interatomic bonding in SnSe.