Anomalous Hall Effect for the Phonon Heat Conductivity in Paramagnetic Dielectrics

Abstract

The theory of anomalous Hall effect for the heat transfer in a paramagnetic dielectric, discovered experimentally in [1], is developed. The appearance of the phonon heat flux normal to both the temperature gradient and the magnetic field is connected with the interaction of magnetic ions with the crystal field oscillations. In crystals with an arbitrary phonon spectrum this interaction creates the elliptical polarization of phonons. The kinetics related to phonon scattering induced by the spin-phonon interaction determines an origin of the off-diagonal phonon density matrix. The combination of the both factors is decisive for the phenomenon under consideration. PACS numbers: 66.70.+f, 72.15.Gd, 72.20.Pa A novel interesting phenomenon is found experimentally in recent paper [1]. The matter concerns an analog of the anomalous Hall effect (AHE) for the heat conductivity of an ionic paramagnetic dielectric. In fact, applying magnetic field B in the direction normal to heat flow j, the authors have discovered an appearance of the heat transfer in the direction normal to B and j. For the complete lack of free charged carriers and negligible role of the spin-spin coupling at the parameters concerned, the transverse flow is naturally associated with the evolution of the phonon system. However, the magnetic field does not directly act on phonons and only polarizes paramagnetic ions. It is the coupling between phonons and subsystem of isolated ions carrying magnetic moment M that determines the formation of the picture observed. Note in this aspect that the phenomenon concerned is an analog of the AHE in the paramagnetic phase of a ferromagnetic above the Curie point, discovered a long time ago [2]. An existence of the effect found in [1] has been confirmed in work [3]. In both [1] and [3] the choice of terbium gallium garnet (TbGG) was not accidental. In this compound for T < 10K the heat conductivity due to spin-phonon coupling (SPC) proves to be about two orders of the magnitude smaller as compared with gadolinium garnet [4] of the identical structure.As it is known, in most of ion dielectrics the spinphonon relaxation is determined by the two-phonon processes, see, e.g., [5] and [6]. Usually this supposes the presence of the Kramers doublet in the ground state and the transitions via virtual excitation of higher Kramers doublets. Trivalent ion T b 3+ has the even number of felectrons and the Kramers degeneration is absent. However, for rare earth ions with the even number of felectrons, an appearance of the quasi-doublet structure is typical for splitting the multiplet in the crystalline field with the level spacing ε 12 of about several Kelvins in the ground state. TbGG is a striking example of the similar compound, see, e.g., [7]. From the general symmetry reasons and the direct analysis [8, 9, 10] one can conclude that the SPC under conditions concerned reduces to a scalar production of magnetic moment M and the orbital moment related to the motion of surrounding ions. At low temperatures when the long wave acou...

Origin of the Phonon Hall Effect in Rare-Earth Garnets

“…An accurate evaluation of the integrals in Eq. (13) confirms that the primary contribution to κ xx comes from ω ≈ T = 5K. On the other hand the dominant contribution to κ xy comes from ω ∼ 30K -the phonon Hall effect is due to relatively "hot" phonons.…”

“…We stress that the PHE in TGG relies specifically upon scattering from superstoichiometric Tb 3+ ions, not just scattering from any impurities. This observation was not considered in all previous suggestions 12,13,15 for the mechanism behind the PHE. In this Letter, motivated by the above observation, we show that the PHE originates from the resonant skew scattering of phonons from the crystal field states of superstoichiometric Tb 3+ ions.…”

Energy Magnetization and the Thermal Hall Effect

“…Recently there has been surging experimental interest in studying the thermal Hall effect in various systems, such as the phonon Hall effect [2,3], magnon Hall effect [4], and so on. There are also many theoretical efforts to study these phenomena [5][6][7][8]. However, most of these theoretical studies face a fundamental issue: direct application of the Kubo formula, when done correctly without questionable ''tricks'' [9], always yields unphysical divergence at zero temperature [10,11].…”

Thermal Effects in Spintronics: Physics and Applications

“…Strohm et al observed the thermal Hall effect of phonons in a paramagnetic insulator of terbium gallium garnet [72]. The result was explained by the interaction of local magnetic ions with the local orbital angular momentum of oscillating surrounding ions [73,74]. The thermal Hall effect of magnons was also observed in an insulating ferromagnet LU 2 V 2 O 7 with pyrochlore structure [75], and the result was explained in terms of a Dzyaloshinskii-Moriya interaction.…”