Theory of Phonon Hall Effect in Paramagnetic Dielectrics. L. Sheng, D. N. Sheng and C. S. Ting

Transcription

1 Journal Club by Oleg Chalaev Theory of Phonon Hall Effect in Paramagnetic Dielectrics L. Sheng, D. N. Sheng and C. S. Ting cond-mat/ will probably be published soon... best viewed with Acrobat Reader r version 4

2 Thermo-magneto-electric effects see [Abr87]6.2 el. field = E = σ [ ] j + R H j + Q T [ ] + N H T heat flux = q = Π [ ] j + NT H j χ T [ ] + L H T

3 Thermo-magneto-electric effects see [Abr87]6.2 el. field = E = σ [ ] j + R H j T = 0, R= Hall resistivity

4 Thermo-magneto-electric effects see [Abr87]6.2 el. field = E = σ [ ] j + R H j + Q T [ ] 0 = NT H j χ T q = 0, adiabatic Hall effect: E y = (R + QNT/χ)Hj x

5 Thermo-magneto-electric effects see [Abr87]6.2 heat flux = q = Π [ ] j + NT H j χ T [ ] + L H T q y = 0, j y = 0 = T y = NT χ Hj x Ettingshausen effect

6 Thermo-magneto-electric effects see [Abr87]6.2 heat flux = q = Π [ ] j + NT H j χ T [ ] + L H T j x = 0, j y = 0, q y = 0 = T = L H T, Leduc-Righi effect y χ x

7 Thermo-magneto-electric effects see [Abr87]6.2 el. field = E = σ [ ] j + R H j + Q T [ ] + N H T heat flux = q = Π [ ] j + NT H j χ T [ ] + L H T In insulators there is no Leduc-Righi effect (due to electrons). What about phonons? Phonons do not interact with magnetic field directly = we need unusual terms in the Hamiltonian

8 Experiment: Leduc-Righi effect for insulators See [SRW05]. material: Tb 3 Ga 5 O 12 dielectric, cubic, paramagnetic.

9 Experiment: Leduc-Righi effect for insulators See [SRW05]. material: Tb 3 Ga 5 O 12 dielectric, cubic, paramagnetic. immersed in 4 He-bath: T = 5.45K= there is NO optical phonons. producing temperature gradient via electrical heater thermometers: thermally activated hoping. eliminating thermometers misaligment and even in H magnetoresistance effect: T = 1 2s [ R(+B, T +) R( B, T )] phonon transport is diffusive, i.e., mean free path sample size. thermomagnetoresistance = magn. field affects phonon scattering

10 Experimental results

11 Spin-Phonon interactions: (pseudo)spin of an ion interacts with phonon Phonons do not interact with magnetic field directly Phonon excitations total angular momentum of an atom external magnetic field. Here is the requested unusual term in the Hamiltonian Raman interaction (mean field): V = K m M Ω m m=atom number. After quantisation: V = 1 ωqσ (a qσ + a 2 ω qσ)(a qσ a qσ ) qσ q,σ,σ qσσ where qσσ = i KM (ê qσ ê qσ )

12 Thermal current operator J E = 1 2V (R m R n )Φ αβ (R m R n )u α mv β n, mnαβ where u α m and v α m with α = x, y and z are the α-components of the center-of-mass displacement u m and velocity v m of the m-th unit cell, respectively, and Φ αβ (R m R n ) are the stiffness matrix elements of the lattice with R m the equilibrium position of the unit cell. Serious question: with Raman interaction included, the continuity equation for the heat flux [Har63] may not hold any more and then we have problems in how to define heat flux just like in case of spin current (which is not conserved in systems with SOI). The authors did not mention this problem, and did not explain their derivation of the heat flux. May be they just wanted to hide this problem from the referee... Did they cheat?

13 Thermal current operator continued Modified due to spin-phonon interactions. J E = J (0) E + J (1) E, where J (0) E = 1 2V J (1) E = 1 2V with q,σ,σ j qσσ q,σ,σ,σ j qσσ ωqσ ω qσ (a qσ + a qσ) (a qσ a qσ ), ( qσ σ ω qσ ω qσ ) (a qσ +a qσ)(a qσ +a qσ ), j qσσ = ω qσ δ σσ q ω qσ + 4 (ω2 qσ ω 2 qσ ) α [ ( q ê α qσ)ê α qσ ê α qσ( q ê α qσ )]. Here, J (1) E comes from the Raman interaction.

14 Linear responce thermal current Note: the derivation is not similar to the charge conductivity! Does any one know simple and clear derivations? A long (but clear) derivation is available in [Ф. М. Куни81]. (see also Kubo s handwaving in ([KTH85]4.6.48)) κ xy = V T /kb T dλ 0 0 dt J x E ( iλ)j y E(t) where J x E is the x component of the energy flux operator J E of the phonons, and J E (t) = e iht/ J E e iht/.

15 Result κ xy = γk BKM 2π 2 c s ( ) kb T ΘD/T 0 x dx, (1) e x 1 where γ = (5 δ)(1 + δ) 4 /[4δ 2 (9 + 18δ 3 ) 1/3 ] with δ = c L /c T, c s is the average sound speed defined by 3/c 3 s = (1/c 3 L + 2/c 3 T) and Θ D = ω D /k B = (6π 2 /ν 0 ) 1/3 c s /k B is the Debye temperature with ν 0 the volume of a unit cell. comparison with experiment:

16 Conclusions theory fits experiment well = everyone is happy this document is available here.

17 Alexey Alexeevich Abrikosov. Fundamentals of the Theory of Metals. Nauka, см. DVD 5(по-русски). R. J. Hardy. Energy-flux operator for a lattice. Phys. Rev., 132: , см. DVD 5. A. S. Ioselevich and H. Capellmann. Strongly correlated spin-phonon systems: A scenario for heavy fermions. Phys. Rev. B, 51: , May см. DVD 5. R. Kubo, M. Toda, and N. Hashitsume. Statistical Physics II: Nonequilibrium statistical mechanics. Springer series in solid state physics. Springer, 1985.

18 см. DVD 5. C. Strohm, G. L. J. A. Rikken, and P. Wyder. Phenomenological evidence for the phonon hall effect. Phys. Rev. Lett., 95:155901, October J. M. Ziman. Electrons and Phonons. The Theory of Transport Phenomena in Solids. The int. series of monographs on physics. Изд. ин. лит-ры., см. DVD 5 Дж. Займан «Электроны и фононы. Явления переноса в твёрдыx телаx». Ф. М. Куни. Статистическая физика и термодинамика. Наука, Москва, см. DVD 5.

19 Help desk Non-standart thermoconductivity in dielectrics: [Zim62]8.8. Raman interaction=coupling between phonons and localized spins, see [IC95]