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|Title:||Phonon Hall Effect in Two-Dimensional Lattices||Authors:||ZHANG LIFA||Keywords:||phonon Hall effect, four-terminal junction, periodic crystal lattices, nonequilibrium Green's function, Green-Kubo formula, Berry phase||Issue Date:||6-Jun-2011||Citation:||ZHANG LIFA (2011-06-06). Phonon Hall Effect in Two-Dimensional Lattices. ScholarBank@NUS Repository.||Abstract:||Based on Raman spin-phonon interaction, we theoretically and numerically studied the phonon Hall effect (PHE) in the ballistic multiple-junction finite two-dimensional (2D) lattices by nonequilibrium Green's function (NEGF) method and and in the infinite 2D ballistic crystal lattices by Green-Kubo formula.
We first proposed a theory of the PHE in finite four-terminal paramagnetic dielectrics using the NEGF approach. We derived Green's functions for the four-terminal junctions with a spin-phonon interaction, by using which a formula of the relative Hall temperature difference was derived to denote the PHE in four-terminal junctions. Based on such proposed theory, our numerical calculation reproduced the essential experimental features of PHE, such as the magnitude and linear dependence on magnetic fields. The dependence on strong field and large-range temperatures was also studied, together with the size effect of the PHE. Applying this proposed theory to the ballistic thermal rectification, two necessary conditions for thermal rectification were found: one is phonon incoherence, another is asymmetry. Furthermore, we also found a universal phenomenon for the thermal transport, that is, the thermal rectification can change sign in a certain parameter range.
In the second part of the thesis, we investigated the PHE in infinite periodic systems by using Green-Kubo formula. We proposed topological theory of the PHE from two different theoretical derivations. The formula of phonon Hall conductivity in terms of Berry curvatures was derived. We found that there is no quantum phonon Hall effect because the phonon Hall conductivity is not directly proportional to the Chern number. However, it was found that the quantization effect, in the sense of discontinuous jumps in Chern numbers, manifests itself in the phonon Hall conductivity as singularity of the first derivative with respect to the magnetic field. The mechanism for the change of topology of band structures comes from the energy bands touching and splitting. For honeycomb lattices, there is one critical point. And for the kagome lattices there are three critical points correspond to the touching and splitting at three different symmetric center points in the wave-vector space.
From both the theories of PHE in four-terminal junctions and in infinite crystal systems, we found a nonmonotonic and even oscillatory behavior of PHE as a function of the magnetic field and temperatures. Both these two theories predicted a symmetry criterion for the PHE, that is, there is no PHE if the lattice satisfies a certain symmetry, which makes the dynamic matrix unchanged and the magnetic field reversed.
In conclusion, we confirmed the ballistic PHE from the proposed PHE theories in both finite and infinite systems, that is, nonlinearity is not necessary for the PHE. Together with the numerical finding of the various properties, this theoretical work on PHE can give sufficient guidance for the theoretical and experimental study on the thermal Hall effect in phonon or magnon systems for different materials. The topological nature and the associated phase transition of the PHE we found in this thesis provides a deep understanding of PHE and is also useful for uncovering intriguing Berry phase effects and topological properties in phonon transport and various phase transitions.
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