Origin of the Phonon Hall Effect in Rare-Earth Garnets

Abstract

The phonon Hall effect has been observed in the paramagnetic insulator, Tb3Gd5O12. A magnetic field applied perpendicularly to a heat current induces a temperature gradient that is perpendicular to both the field and the current. We show that this effect is due to resonant skew scattering of phonons from the crystal field states of superstoichiometric Tb 3+ ions. This scattering originates from the coupling between the quadrupole moment of Tb 3+ ions and the lattice strain. The estimated magnitude of the effect is consistent with experimental observations at T ∼ 5 K, and can be significantly enhanced by increasing temperature.PACS numbers: 72.20.Pa, 72.15.Gd When a linear magnetic field is applied perpendicularly to a heat current in a sample of terbium gallium garnet (TGG), Tb 3 Ga 5 O 12 , a transverse temperature gradient is induced in the third perpendicular direction 1,2 . This is the "phonon Hall effect (PHE)". The effect was observed in this insulator at low temperature (T ∼ 5 K), a situation in which there are no mobile charges such as electrons or holes 3 . The Neel temperature of TGG is 0.24 K 4 , so it is a paramagnet at T ∼ 5 K. Hence magnons do not contribute to the heat current and one does not expect a contribution from the magnon Hall effect 5-8 . Phonons are not charged and hence cannot be affected by the Lorentz force which gives rise to the usual classical Hall effect. Therefore the mechanism for the PHE must be related to the spin-orbit interaction. However, the spin-orbit interaction for phonons is not at all obvious, unlike in the anomalous Hall effect and spin Hall effect for electrons [9][10][11] . Thus, an understanding of the origin of the observed PHE is a fundamental problem.So far, there have been a few theoretical attempts to explain the PHE 12-15 . Refs. 12 and 13 assumed a Ramantype interaction between the spin of stoichiometric Tb 3+ ions and the phonon. This interaction results in "elliptically polarized" phonons. According to 12,13, the "elliptic polarization", in combination with scattering from impurities, leads to the PHE. In this scenario the type of impurity is unimportant and hence phonon -impurity scattering is considered in the leading Born approximation. This is an intrinsic-extrinsic scenario, i.e., the "elliptic polarization" is an intrinsic effect and the scattering from impurities is an extrinsic effect. The major problem with this scenario was realized in Ref. 14 -in spite of the "elliptic polarization" the Born approximation does not result in the PHE. Ref. 14 attempted to go beyond the leading Born approximation in impurity scattering. However, the problem has not been resolved yet. An intrinsic mechanism for the PHE, based on the Berry curvature of phonon bands, was suggested in Ref. 15. This is similar to the Berry curvature mechanism in the Hall effect for light 16 . The Berry curvature mechanism is certainly valid for materials with specially structured phonon bands, however, it is hard to see how the mechanism can be realized in TGG which has the simple...

Thermal Hall Effect in a Phonon-Glass Ba3CuSb2O9

“…Moreover, the calculation method used in refs. 6 and 7 has been criticized as inappropriate by several authors [8][9][10]. The works in the second group predict a T n dependence (n = −1 [8], 1, or 3 [9]), which does not agree with our findings.…”

“…In the third group, authors in ref. 10 show a nearly T 3 dependence of κ xy , which is different from that of BCSO. This discrepancy may be attributed to a difference in scattering mechanisms of phonons by the 4f Tb 3+ ions with a large quadrupole moment and that of the 3d Cu 2+ ions.…”

“…On the other hand, no Hall effect is expected in insulators because the Hall effect originates from conduction electrons. However, a Hall effect of charge-neutral excitations, which is observed as a thermal Hall effect (THE), has been predicted to occur in magnetic and non-magnetic insulators [2][3][4][5][6][7][8][9][10][11], thereby providing new insights into the research on quantum spin liquids and other frustrated materials. So far, three kinds of THEs have been reported in insulators: THEs of magnons in ordered magnets [12][13][14], spin excitations in disordered magnets [14][15][16], and phonon THEs [17,18].…”

“…The THEs are observed only in the Czochralski-grown crystals where κ xx is more strongly suppressed than that of flux-grown crystals, which leads the authors in ref. 10 to conclude that the phonon THE is caused by the excess phonon scattering by the superstoichiometric Tb 3+ ions. Therefore, it is tempting for us to associate the phonon THE in BCSO with the strong suppression of κ xx .…”

Approximately Quantized Thermal Hall Effect of Chiral Liquids Coupled to Phonons

“…Several mechanism can explain phonon Hall effects. In the literature a Raman-type interaction between phonons and large spins [40,41], effects of Berry curvature of phonon bands [42,43], and a resonant skew scattering of phonons [44] has been discussed.…”