Transport de chaleur via les spins

Transcription

1 Transport de chaleur via les spins Joseph P. Heremans Ohio State University Department of Mechanical and Aerospace Engineering Department of Physics Reference: Stephen R. Boona, Roberto C. Myers and Joseph P. Heremans, Spin Caloritronics, Energy Environ. Sci (014) DOI: /C3EE4399H Trans.Spins-1

2 Types of magnetism Diamagnetism Independent atomic moments: Paramagnetism Uncompensated spin/angular momentum Permanent atomic moments T Ferromagnetism Cooperating atomic moments Ferrimagnetism Free electrons Pauli spin Paramagnetism Landau orbital Diamagnetism Band anti / ferromagnetism Antiferro magnetism ME 8194 Ch 7 Magnetism

3 Transport Electrons with or without spin Elemental excitations E Charge g g L Heat Spin Caloritronics M Magn. Moment g Phonons? Magnons Magnon Drag Trans.Spins-3

4 Structure of this lecture 1. Longitudinal spin Seebeck. Spin waves in the Ferromagnet: Magnon thermal conductivity Thermally driven spin/moment flux 3. Spin polarized electrons and spin/orbit interactions: Spin Hall effect Inverse spin Hall effect 4. At the interface: spin mixing conductance 5. Transverse Spin Seebeck effect Non local version of longitudinal spin Seebeck effect GaMnAs: the role of phonons 6. Phonon magnon electron drag induced spin Seebeck effect: GaMnAs 7. The giant spin Seebeck effect (InSb): Pure phonon electron drag, no magnons 8. Phonon diamagnetism 9. Applications Trans.Spins-4

5 1. Longitudinal spin Seebeck Heat Spin Caloritronics Magn. Moment Trans.Spins-5

6 Spin-Seebeck effect definition Measurements rely on inverse spin-hall effect: E ISHE D ISHE J S σ Spin Seebeck Coef. S xy x T x Grossly exaggerated for effect Trans.Spins-6

7 Transverse Spin-Seebeck Effect (TSSE) Longitudinal Spin-Seebeck Effect (LSSE) z y Pt V 1 V V3 V 4 x V E ihse HOT j s COLD j s j s E ihse s Ferromagnet: T, j q H z Metal: Pu (Uchida & al, Nature, 008) Semiconductor: GaMnAs (Jaworski & al, Nat. Mater, 010) Insulator: YIG (Uchida & al, Nat. Mater, 010) Ferromagnet: Can only be an Insulator: YIG (Uchida & al, APL010) Trans.Spins-7

8 ISHE medium Platinum Theoretical understanding of the SSE 1. T drives the magnons out of thermal equilibrium: 1.1 magnon thermal conductivity 1. phonon drag J s. Spin torque results from magnons striving back to equilibrium T Ferromagnet 3. Net spin flux diffuses into Pt 4. Spin orbit interactions in Pt convert J S => E ISHE Sinova, Finkelstein, Nat. Commun. 013 Tserkovnyak, PRB, 014 H. Adachi, J.-I. Ohe, S. Takahashi, and S. Maekawa, PRB 83, (011). Trans.Spins-8

9 . In the Ferromagnet: Magnon thermal conductivity Thermally driven spin/moment flux Heat M Magn. Moment Magnons Trans.Spins-9

10 Ferromagnetic spin chain: magnons Magnetic moments localized on core electrons on atoms (f or d levels) wavevector k π wavelength Slide by Burkard Hillebrands, U. Kaiserslautern Trans.Spins-10 Burkard Hillebrands 1th Joint MMM/Intermag Conference Chicago, January 14, 013

11 U Spin waves, simple Heisenberg ferromagnet The ground state has all spins parallel. Exchange integral N J S S p1 p S p p1 = angular momentum of spin at site pl Treat spins as classical vectors of magnitude S => The ground state exchange energy is The excited state has one spin reversed. This increases the excited state energy to U 1 U0 8JS Too expensive U0 NJS The excited state has one spin reversed. Much lower energy state is all atoms share a little bit of the spin reversal, and precess. Trans.Spins-

12 Magnon dispersion, ferromagnets (THz) k B gh z JSa k Yttrium Iron Garnet (YIG) Y 3 Fe (FeO 4 ) 3, or Y 3 Fe 5 O 1 T CURIE = 550 K, T DEBYE = 538 & 567 K Tb-YIG, 95K & 83K k 110 v MAGNON 8500 ms -1 k 100 Plant, J. Phys. C (1983) Trans.Spins-

13 Yttrium Iron Garnet (YIG) Y 3 Fe (FeO 4 ) 3, or Y 3 Fe 5 O 1 Types of magnons in reality Thermal magnons at room temperature Dispersion? You judge. Optical like Magnons at 3 4 THz (150 K) Dispersion becomes cmplicated 1 THz = 49 K Magnons below 700 GHz ~ 30 K Parabolic dispersion (Kittel model) Thermal magnons at T < 40 K Subthermal magnons at T> 40 K Magnons excited by microwaves: 6 GHz Energies much below thermal magnons Very high population density by FMR J. S. Plant, J. Phys. C: Solid State Phys.. 16 (1983) Trans.Spins-13

14 Magnon density of states D(), ferromagnets k B gh z JSa k Magnons have only a single polarization for each value of k Number of magnon states of frequency between and +d 0 D( ) 1 4 JSa 3/ gh B, gh Magnetic fields can freeze out magnons, 10 4 ev/t or 1.3 K / T, B z z / / D gh B z Trans.Spins-14

15 U ( T ) C MAGNON 0 D( ) ( T ) du dt f 0 ( ) d kbt JSa 3/ E exp k Magnon specific heat Assuming phonons and magnons are independent, one can assume additivity Phonons + magnons gm B T Mostly phonon specific heat phonons only C(0T) C(7T) C(0T) C(7T) Approximate: magnons and phonons hybridize Trans.Spins-15

16 Magnon specific heat in YIG Mostly phonon specific heat C MAGNON ( T ) du dt kbt JSa 3/ exp E k gm B T C TOTAL C MAGNON T 1.1 C(0T) C(7T) C PHONON T 3 Trans.Spins-16

17 Magnon thermal conductivity in YIG MAGNON 1 C 3 MAGNON v MAGNON MAGNON Same idea: freeze out magnons by applying magnetic field TOTAL PHONON MAGNON along [100] axis Trs.Spins-17

18 Magnon and phonon mean free paths (YIG) PHONON MAGNON MAGNON PHONON 1 C 3 1 C 3 MAGNON PHONON v v 8500 ms 1 MAGNON PHONON 5000 ms 1 MAGNON PHONON crystal thickness In YIG at low temperature the magnon mean free path 5 to 50 m Magnons and phonons have comparable mfp s MFP MAGNON MFP PHONON Tr.Spins-18

19 Heat current implies spin current Heat current of spin-wave implies a spin current => A temperature gradient implies a magnetization gradient. j j j j Q Q S M k MAGNON B. T. j g. MAGNON B j MAGNON j T MAGNON Trans.Spins-19

20 Pure spin current without charge current? YES Charge current j e : conjugate force = voltage difference x electron charge j e > 0 - p p p p Power dissipated as heat (phonons) T increases Spin current j M : Spin polarized charge carriers No net charge transport Net spin transport Need a spin flip transformation at some interface j e = Conjugate force? Magnetization gradient Trans.Spins-0

21 Heat M Treatment of spin flux S Magn. Moment g Magnons For magnetic systems, the most important thermodynamic variable is magnetization itself, whose conjugate force is just the Landau Lifshitz effective field H eff. The (M,H eff ) pair enters Onsager symmetry on par with other thermoelectric quantities. In fact, any well defined thermodynamic quantity (along with its conjugate force) based on a physical observable would obey Onsager reciprocity. In general spin flux is not conserved. This does not create any obstacles for invoking Onsager reciprocity for the magnetic dynamics (couple to charge currents, energy currents, mechanical motion etc.). Tserkovnyak et al, PRB DOI: /PhysRevB In some special cases, when the spin relaxation is weak, one could approximate ferromagnetic metals by a two fluid model: spin up and spin down electrons, and use spin up and spin down densities as well defined thermodynamic quantities (which could be approximately conserved) that enter Onsager reciprocity relations. In general, spin orbit interaction makes this impossible. Trans.Spins-1

22 Spin Onsager relations j S j e j j q M ej j F / T n L L L EE TE ME L L L ET TT MT Entropy production L L L EM TM MM e T H 0 LF Reciprocity holds Neither spin nor heat are conserved Discussion in Bauer, Tserkovnyak et al., PRB DOI: /PhysRevB Trans.Spins-

23 Caution 1: Magnon and phonon temperatures are not equal Magnons don t couple directly to heat reservoirs => they couple through the phonons => difference between phonon temperature and magnon temperature during heat conduction j Q Ax TM x T sinh( ) ( ) 0 x M P Acosh( AL ) CPCM P M 1 D. J. Sanders and D. Walton, Phys. Rev. B A C (1977) P CM P M MP Trans.Spins-3

24 Caution : Sometimes magnons hybridize with phonons YMnO 3 S. Pailhès et al., Phys. Rev. B 79, (009) Optical phonon Magnon + phonon hybridized T < T N Transverse Acoustic phonon (TA 100 ) T > T N The whole discussion above is oversimplified: it treated magnons phonons as different and independent particles. Trans.Spins-4

25 3. Spin polarized electrons and spin/orbit interactions: Spin Hall effect Inverse spin Hall effect Electrons with or without spin Initial theoretical suggestion: M. I. Dyakonov and V. I. Perel,; Perel' Sov. Phys. JETP Lett. 13: 467 (1971). Observation: Y. Kato; R. C. Myers, A. C. Gossard, D. D. Awschalom Science 306 (5703): (004). Inverse Spin Hall effect: S.O. Valenzuela; M. Tinkham Nature 44 (7099): (006) E. Saitoh; M Ueda, H. Miyajima, and G. Tatara Applied Physics Letters 88 (18): (006). Charge g Magn. Moment Trans.Spins-5

26 Spin polarized band electrons: Stoner model = E F = E F Trans.Elec-6

27 Spin Hall and inverse spin Hall Sadamichi Maekawa, Nat. Mater. VOL 8 p777, OCTOBER 009 Trans.Spins-7

28 Spin Hall effect Spin Hall Effect (SHE) : the appearance of spin accumulation on the lateral surfaces of an electric current carrying sample. Result of spin orbit interaction, does NOT require magnetization The signs of the spin directions is opposite on the opposing boundaries. In a cylindrical wire, the current induced surface spins will wind around the wire. When the current direction is reversed, the directions of spin orientation is also reversed. awsch web.physics.ucsb.edu Experiment: Application electric field Measured Kerr rotation as a function of magnetic field at two positions: left and right edges Observe Lorentzian curves=>out of plane spin polarization (Hanle effect). Trans.Spins-8

29 Spin Hall effect in GaAs Trans.Spins-9

30 Spin Hall effect in Au/Fe T. Seki et al., Nature Mater. 7, 15 (008). Guang-Yu Guo, Sadamichi Maekawa, and Naoto Nagaosa Phys. Rev. Lett. 10, (009) Charge injection Charge consists of spin-up and spin-down polarized electrons Scattering depends on spin-polarization Trans.Spins-30

31 Inverse Spin Hall effect Inverse Spin Hall Effect: an electrical current is induced by a spin flow. Due to a space-dependent spin polarization [7] The existence of both direct and inverse effects is demonstrated in metals and semiconductors. [ E ISHE D ISHE J S σ Trans.Spins-31

32 Electrons with or without spin Charge g g 4. Spin transfer at the interface Magn. Moment g Magnons Trans.Spins-3

33 Spin transfer torque and spin pumping Spin Polarized matter Normal Metal Normal Metal Spin Polarized matter M M x x The extreme case of the study of the effect of M Trans.Spins-33

34 Spin transfer across interfaces e Magnon transport in YIG (FM insulator) by heat current or FMR pumping Spin transfer YIGPt at the interface: exchange interaction between free electrons in metal ( s ) and d electrons on Fe atoms in YIG YIG Pt Reversible effect follows Onsager reciprocity Spin accumulation in Pt (e.g. by spin Hall effect) Spin transfer PtYIG, generating magnons in YIG Exponential decay of magnon current in YIG Efficient spin transfer across interface is essential: FM insulator with very low damping High interfacial spin mixing conductance g YIG with Excellent structural and magnetic uniformity in bulk and at surface Clean, atomically sharp YIG/Pt interface: strong interfacial exchange coupling Trans.Spins-34

35 Electrical spin pumping: 1. Injection M. Johnson and R. H. Silsbee, Phys. Rev (1988) Fully spin Polarized matter Normal Metal Partially spin Polarized matter Normal Metal Polarization ratio: n n n Trans.Spins-35

36 Current injection from ferromagnet to normal metal FM PM Trans.Spins-36

37 Fully spin Polarized V 0 Normal Metal Current injection Apply electrical potential difference V 0 Drive electrical current j e across interface Magnetization is injected at a rate proportional to the electric current. The transport of magnetization is proportional to the magnetization of each electron, B For non-fully polarized FM s: j M B e The injected magnetization current is then: j e j M B e j e M. Johnson and R. H. Silsbee, Phys. Rev (1988) Trans.Spins-37

38 Define interface conductance per unit area Weak coupling between & subbands => Interface conductance Do this for spin up and spin-down electrons separately G je e D( EF ) v V 0 DOS G G transmission probability x T Average velocity across interface e D e D ( E ( E F F ) v ) v In effect, an electrical current is related to a difference in magnetization potential across the interface M H EFF H Thermodynamic formalism: M. Johnson and R. H. Silsbee, Phys. Rev (1987) Trans.Spins-38 x x T T

39 Onsager formalism for spin dependent transconductance S C B M q G G G G G G G G e j j G G j j G G G G G G j j e j j j j C C B M q 1 charge current magnetization current charge chemical potential spin accumulation potential charge conductance spin mixing conductance Ferromagnet Normal metal G G e j j B e M Spin injection efficiency Trans.Spins-39

40 Spins decay in the normal metal n M n B n n Spin Polarized matter M, n Normal Metal n 0 ( x) n exp( x / L ) x Spin diffusion length L E F x E F spin lifetime L L D S Electron diffusion constant Trans.Spins-40

41 Inverse injection Normal Metal with spinpolarized electrons Spin Polarized matter If a normal metal in which there is an imbalance between spin up and spin down electron is put in contact with a metal in which there is a difference between the spinup and spin down DOS, apply the reverse argument for G separately. There is a current across the interface proportional to the non equilibrium polarized electron population in the normal metal. The difference in number of non equilibrium spins drives an electron current. Trans.Spins-41

42 Voltage bias due to inverse injection Normal Metal with spinpolarized electrons Spin Polarized matter Unbalence in electrons: F D Solve for V D : n M / B D ( E) D ( E) V d PAULI BM e PAULI D( E B F ) E ev E F f ( E) de Trans.Spins-4

43 Electrical spin pumping: Experiment M. Johnson and R. H. Silsbee, Phys. Rev (1988) Permalloy Aluminum Permalloy Trans.Spins-43

44 Transfer of spin flux across the YIG/normal metal interface YIG (Pt) Spin transfer across Pt/YIG interface through the electron magnon exchange interaction at the interface. Loosely speaking an "s d interaction with "s" referring to electrons in Pt and "d" to local Fe moments in YIG Spin sink j s1 = 0 j s = j s, FERROMAGNET Use inverse spin Hall in Pt for detection Silas Homan, Koji Sato, and Yaroslav Tserkovnyak, Landau-Lifshitz theory of the spin Seebeck effect, arxiv: (013) Trans.Spins-44

45 Spin mixing conductance formalism applies to LSSE YIG/Pt Our experimental results support present, exclusively spin current based, theoretical models using a single set of plausible parameters for spin mixing conductance, spin Hall angle and spin diffusion length Note: thermal excitation gives highest flux M. Weiler,, S. Goennenwein, arxiv: v1 (013) Trans.Spins-45

46 Thermal and electrical pumping T T L L L L L L L L L L L L L q j j J S C TT ST ET TS TE S q 1 Generalize: Spin is not conserved, but neither is heat Spin Seebeck coefficient is function of L TS / (L +L ), Spin transport is dissipative, because F J Q. T T L L L L L L L q j j TT ET ET TE TE 0 0 Spin Dependent Thermoelectric coefficients: L TE Bart van Wees, Ron Janssens Or generalize differently: Trans.Spins-46

47 Longitudinal Spin Seebeck Effect YIG/Pt Inverse spin Hall effect in the Pt E ISHE D ISHE j S σ S K.Uchida & al. Appl. Phys. Lett. 97, (010) Trans.Spins-47

48 Longitudinal Spin Seebeck Effect YIG/Pt -x x y z K.Uchida & al. Appl. Phys. Lett. 97, (010) Trans.Spins-48

49 LSSE YIG/Pt Film versus single crystal Spin Seebeck Coefficient (nv/k) T (K) H. Jin, et al., to be pblished GGG YIG Pt (0.5mm GGG + 4μm YIG + 10nm Pt) No low temperature enhancement More similar to polycrystalline YIG Uchida, et al., JAP 111, (01) Bulk YIG (1 mm YIG + 15nm Pt) Peak in SSE at 50K Does not match with phonon peak Normalized results from our LSSE thin film measurements Trans.Spins-49

50 SSE ratio in YIG (from Uchida, et al.) LSSE YIG/Pt Relation to thermal conductivity? Thermal conductivity in monocrystalline YIG slab (Mostly phonons) SSE(T)/SSE(90K) for monocrystalline YIG slab (Diffusive + phonon drag) SSE in 4μm YIG film (Diffusive only) Temperature (K) Temperature (K) Trans.Spins-50 Thermal conductivity (W/m K) Spin Seebeck coefficient (nv/k)

51 Caution 3: What can go wrong experimentally? Thermomagnetic & Galvanomagnetic Effects Slide from Gerrit Bauer Trans.Spins-51

52 Longitudinal Spin Seebeck Effect ONLY on YIG/Pt The planar Nernst effect has exactly the same symmetry as the longitudinal spin Seebeck effect LSSE measurements impossible on electrically conducting ferromagnets Trans.Spins-5

53 5. Transverse Spin Seebeck Effect Local versus non local spin injection LSSE = local TSSE = non local GaMnAs Phonon Drag (phonon magnon, phononelectron) Trans.Spins-53

54 Local vs non local spin injection Non local electrical spin injection: spin polarized charge current is driven by an applied electric field. spin current parallel to a charge current => spin current diffuses from the ferromagnet into the normal metal. Equivalent for electrical spin injection: j C, j S, j Q j C, j S, j Q F. J. Jedema, A. T. Filip & B. J. van Wees, Nature (001) Trans.Spins-54

55 CASE 1: In plane or cubic Mn content 7-18% 40 [-110] [100] [010] [110] [-1,1,0] & [100] are easy axes M (emu/cm 3 ) 0 [-110] 0-0 [110] K B (Oe) [100] 16% Mn Two possible experiments: Case 1a: B//T//easy axes Case 1b: B//T at 45 o from easy axes Trans.Spins-55

56 CASE 1a: B // T // easy axes Trans.Spins-56

57 Dependences on temperature gradient 1. Dependence on temperature gradient is linear => can assign a "Seebeck coefficient" to the slope. Dependence on strip position is totally unusual for transport coefficient Contrast with charge Seebeck between strips S XY XX E X E Y X T X T VY T X VX T X L w Trans.Spins-57

58 Position dependence of Spin Seebeck S xy A skewed and offcenter sinh(x) function, with a characteristic length scale of 4-6 mm S XY L ( x) sinh[ ( x x0 )] Remember D. J. Sanders and D. Walton, Phys. Rev. B (1977)? Trans.Spins-58

59 Temperature dependence and position dependence T and x dependence normalize Trans.Spins-59

60 Scratch the sample => spin current is vertical! => What causes the long range of the effect? 0.3 mm wide scratches z Scratch Scratch 1 x Pt4 Pt3 Pt1 Pt Scratch 1 Scratch y If spin current were horizontal scratching the sample in half would result in independent samples, There would be a negative signal above the scratch and positive signal below the scratch. Trans.Spins-60

61 Spin Seebeck is NOT due to spin current along Scratch me twice! Crack Scratch # Scratch #1 V y S xy V y /T x (V/K) Both Scratches 1 Scratch Intact B // [110] T avg (K) Cold end L 0 No change in signal Spin-Seebeck does not result from a macroscopic spin-current. The substrate, not the film, carries the mm-range information Substrate has ONLY PHONONS L Hot end Trans.Spins-61

62 6. Phonon Drag Electrons with or without spin Charge g L Heat Spin Caloritronics Magn. Moment g Phonons Magnons Magnon Drag Trans.Spins-6

63 Extended T dependence in GaMnAs Jaworski et al., Phys. Rev. Lett (011) (a) 6 S xy (V K -1 ) (b) xx (mv K -1 ) T (K) Thermal conductivity of substrate High (GaAs) (W cm -1 K -1 ) M (emu cm -3 ) Low Spin Seebeck Big Small Phonon -Drag Thermopower Big Small (a) (GaAs) (W cm -1 K -1 ) (b) S xy (V K -1 ) (c) xx (V K -1 ) a+bt 3 bt -1 a+bt 1.5 C p (J kg -1 K -1 ) a+bt T (K) b(t c -T) - b(t c -T) T (K) M (emu cm -3 ) 63

64 E SHE + y T (K) L z Hot T P T M _ x Magnetization GaMnAs GaAs x (a) 0 _ E SHE + L Cold Understanding so far Two mechanisms can push magnon fluxes: 1. Magnon thermal conductivity. Phonon-magnon drag Hot end: Tmagnon < Tphonon => T M >0 Drag heats magnons Drag decreases average M x => M x <0 T M (K) x (b) Cold end: inverse, T M <0, Drag cools magnons Drag increases average Mx=> M x >0 M x M x M x M x (c) x CHANGE in M x => CHANGE in spinpolarization of whatever carries the spin Inverse spin-hall => (d) V y V y V y V y S xy V y M x T M H x H x H x Trans.Spins-64

65 Adachi et al. Applied Physics Letters (010) Trans.Spins-65

66 A = oil droplets B = air molecules A wall collisions dominate A B collisions dominate Phonon Drag A = phonons, B = magnons A = phonons B = electrons When: 1. A B collisions dominate both A scattering and B scattering. A particles have drift velocity Then: 1. A particles impel B particles with momentum IN ONE DIRECTION. Out of thermal equilibrium 3. Very intense (a) S xy (V K -1 ) (b) xx (mv K -1 ) T (K) (GaAs) (W cm -1 K -1 ) M (emu cm -3 )? How do phonondrag curves get to have a maximum in temperature? Trans.Spins-66

67 The MacDonald Formalism, low temperature Phonons exert pressure on electrons feel a pressure because of electron phonon collisions. U(T) = phonon internal energy density Phonon Pressure is then given by Apply T gradient to a sample: More phonons on hot side than on cold side => pressure gradient => force per unit volume F x dp dx Assume that phonons interact ONLY with electrons F x pushes electrons toward cold side of sample. Electrostatic force nee x balances: Phonon electron drag thermopower 1 p U(T ) du( T ) dx 1 3 nee E x du( T ) dt x PED F T x dt dx C Fx CV dt ne 3ne dx Ex CV dt 3ne dx V dt dx Trans.Spins-67

68 Electrons with or without spin Charge 7. InSb No exchange coupling: spin polarization from Landau levels No magnon conductivity: phonon drag Giant spin Seebeck like effect g L Heat Spin Caloritronics Magn. Moment Phonons Trans.Spins-68

69 InSb and its Landau levels Ferromagnetism not necessary Spin Polarization necessary InSb is a very narrow gap semiconductor E g 0. ev Very strong spin/orbit interactions, effective Landé factor g* 50 Separate energy levels into Landau levels Landau diamagnetism and Pauli paramagnetism for free electrons Landau 1 3 Pauli R. Peierls, Quantum Theory of Solids, pp (Oxford 1955) Trans.Spins-69

70 InSb and its Landau levels Landau level Orbital and Zeeman splitting k x E ( E) * mc eh x eh x c * m m * c n 1 C sg * H B 0.1 (4,) n=.8x cm -3 x (3,) (,) (,) Polarization (%) K B (T) E (ev) (1,) (1,) (0,) E F (0,) From this field on up, most electrons are on the last Landau level (ultra quantum limit), spin polarized by Zeeman splitting H (T) Trans.Spins-70

71 InSb Spin Seebeck data 1. Signal is very large, 8 mv/k S xy (V K -1 ) 000 V K K. Even symmetric and small oddsymmetric portions as function of magnetic field 3. Even in field part: 1. large. ultra quantum field region B x (T) H x (T) H x // x T x Jaworski et al., Nature (01) Trans.Spins-71

72 T dependent Spin Seebeck Data.75K 6.3K 5.1K S xy (V K -1 ) S xy (V K -1 ) 500 V K -1 S xy (V K -1 ) 000 V K V K V K K 1000 V K K B x (T) S xy ~e -b*t+a S xy (V K -1 ) S xy (V K -1 ) S xy,max ( V K -1 ) B x (T) B x (T) Blue = hotter end; red = colder end (experimentally some variation) Signal many mv/k below 10 K, much larger than any parasitic T (K) Trans.Spins-7

73 Temperature dependence of amplitude signature of Zeeman splitting S xy max (V K -1 ) R T T (K) R T decays slower than SdH oscillations in resistivity Spin Seebeck effect exists even when orbital quantization is no longer resolved R T kbt g B B kbt sinh( g B Ratio between thermal energy (k B T) and Zeeman energy (g B B) for electrons on helical orbits Only adjustable parameter = amplitude Amplitude (Arb. Units) T (K) B ) Shoenberg D. Magnetic Oscillations in Metals, Cambridge 1984 Trans.Spins-73

74 The physics: 1. Temperature gradient creates phonon flux Change in phonon momenta: q k BT c. Strong phonon-drag impels additional momentum k to electrons: k x q 3. Strong spin-orbit interactions transform k into a change in Zeeman splitting energy: k k x We actually can estimate from published electronconcentration dependence of g factor => no adjustable parameters, for T=5K, T=40 mk: 10 ev 5% of k k B T Trans.Spins-74

75 Polarization is odd in field 100 Explanation based on the two-momentum model (electrons versus phonons) H (T) 5K Phonons are colder on cold side => slow down electrons more Phonon drag DIFFERENCE vis a vis Middle plane Phonons are warmer on hot side => accelerate electrons more k x -q x x T k x +q x x H x // x T Trans.Spins-75

76 Phonon-drag driven, spin-orbit induced spin splitting k = -q x 0 +q x H x // x T S +x <0 =0 >0 E E = g B H q x g B H g B H+q x x Spin polarized electron population is not at equilibrium spin pumping into Pt Non uniformly across sample (spatial dependence) Trans.Spins-76

77 Phonon-drag driven, spin-orbit induced spin splitting k = -q x 0 +q x S +x <0 =0 >0 -H x // x T E E = g B H q x g B H g B H+q x In both field directions, S +x <0 =0 >0 x Therefore, S +x is even symmetric with magnetic field. Trans.Spins-77

78 8. Phonon Diamagnetism L Phonons Heat Spin Caloritronics Magn. Moment new effect Trans.Spins-78

79 Anharmonicity & Grüneisen U x 0 ( k B T) x Anharmonic Potential d U dx k( x) Harmonic Potential F k( x x ) U k df dx 0 x Fdx k d U Cst dx k B T Anharmonicity is characterized by Apply to a solid The phonon frequencies with anharmonic bonds Grüneisen parameter ( x k Anharmonicity governs the phonon phonon interaction Hamiltonian 0 k( x, x, k, m 0 ) d ln( ) d atoms ) ln( V ) Trans.Spins-79

80 Thermal Conductivity (W cm -1 K -1 ) Anharmonicity and lattice thermal conductivity 100 Phonons scattered by crystal boundaries Morelli, Heremans & Slack Physical Review B 66, (00) Phonons scattered by defects (isotopes, ) Silicon model data Temperature (K) Phonons scattered by other phonons Average Mass of Atoms L A Smatterin g of Constants Debye Temp Volume per atom M n V 3 1 / 3 atom 3 T Grüneisen Parameter # atoms in cell Trans.Spins-80

81 Experimental setup: tuning fork geometry 1.4x10 15 cm 3 doped n type InSb single crystal Cernox Q L Q // H // [100] Q S Q 30 mm T L T diff T S L Geometry from T. H. Geballe and G. W. Hull, La physique des basses temperatures, p. 460 (1955) H H w L Size effect on boundary scattering phonons (Ballistic phonons in small arm) Difference in thermal conductivity Small arm serves as reference for phonon scattering in large arm w S th w L (3mm) ~ 3w S (1mm) A L ~ 3A S Trans.Spins-81

82 T diff (mk) Principle of measurement: thermal potentiometer no heat heater on heater off T = 30K H = 0T Constant Q L Step Q S Time (s) When T diff = 0 (T S = T L ): A Q L S L Q L S A L QS T L T diff T S Q S Typical random error in T diff ~ 10µK T diff (mk) T = 30K H = 0T Q L = 44.5mW A Q A Q L S L S L S Q (mw) S Advantages: 1. T diff = 0, heat flux only measurement. Eliminates calibration errors on thermometers 3. Eliminates magnetic field sensitivity of thermometers 4. Minimizes heat losses between arms 5. Enables accuracy of 1:10 4 or even 10 5 Trans.Spins-8

83 Temperature & Magnetic field dependence Magnetic field dependent lattice thermal conductivity Electronic < 10 5 times smaller than lattice conductivity No d or f electrons in system Effect is even in field L / S T = 4.4K T = 3K H (T) Trans.Spins-83

84 Physical meaning of L / S 1. Effect occurs where the transport is still ballistic, but starts picking up a phonon scattering component 1 L S 1 3 C V 1 B C C v V V v v C BL BS BL BS V v. Thermal potentiometer eliminates specific heat and sound velocities from the physics of the problem. 3. If we consider the limiting case where 4. For long wavelength phonons (T<< D ) Grüneisen a=3 for cubic crystals (1) 1 a A a= for trigonal crystals () T Proportionality constant S (W / m K) T (K) BS L BS BS BL S BL L S 1 BS 1 Klemens model 1 Trans.Spins-84 T S

85 What is magnetic in here? What is most likely to depend on magnetic field? 1. Electronic thermal conductivity? No: L 10 3 W m 1 K 1 ; E 10 W m 1 K L. Electron phonon scattering? No: would go the other way. Electrons freeze out at low T and high H 3. Specific heat? No: we checked experimentally 4. Phonon phonon scattering? Yes: T -3 => phonon phonon interactions. L S 1 BS 1 T 3 Trans.Spins-85

86 Concept: frozen phonon Theory of phonon diamagnetism Magnetization around frozen phonon 0 Frozen phonon [101] [101] [010] In Sb M ( B /Å 3 ) Sb In [101] Valence band structure frozen phonon 5x10 6 F M Sb [010] H=7 T // [010] Local moments weak Gradients very sharp Magnetic force on atoms: ( r) M ( r) H H ( r) Magnetic force is anharmonic alters thermal conductivity Trans.Spins-86

87 B /Å 3 Origin of the diamagnetic moment Orbital magnetism in valence band Logarithmic mesh, 0.18 Å In displacement Sb In Diamagnetic susceptibility of the phonon A B /Å 3 Of course, net magnetic moment integrated over (a) the solid (b) time, is zero Trans.Spins-87

88 Diamagnetic Moment vs. Displacement 0 Material has bandgap > 0: What type of diamagnetism? Susceptibility (m B /T) Orbital diamagnetism (semiconductor) Landau diamagnetism (metal) Displacement (Å) Diamagnetic χ function of displacement => Induced χ affects atom displacement Trans.Spins-88

89 Orbital phonon diamagnetism (insulators): Localized electrons in valence band Magnetic field (H APPL ) induces electron precession generating magnetic moments General expression for magnetic moment: r m Ze B 4m e r = mean distance from all electrons from atom Phonon induced Larmor like behavior of the valence electrons Langevin diamagnetsm Trans.Spins-89

90 H APPL Local displacement coordinate Grüneisen parameter Field induced anharmonicity: how it works x M LOCAL Phonon Magnetic force F M F MAGN LOCAL MAGN ( M H) F ( x) H ( x) H k F APPL k H APPL F k1 ( x x0) H APPL... 1 APPL Interatomic harmonic force Total: V V Diamagnetic anharmonicity causes Phonon frequency: ( H 1 L ( H APPL 1 ( H ( H L ) APPL ) APPL APPL ) ) 0 k( x x0 ) Cst k k1 H APPL m m Need to make a mode and frequency average Trans.Spins-90

91 Comparison theory experiment (no adjustable parameter) S 1 BS 1 1 L => L S H APPL L 7T S L H APPL S 0T H APPL 0T H APPL 7T H APPL H 0T APPL 0T 0 0 No adjustable parameters - / - (%) Theory Experiment ( L / S ) / ( L / S ) (%) T (K) Jin, Restrepo, Antonin, Boona, Windl, Myers, Heremans, under review Trans.Spins-91

92 9. Applications Heat dissipation by spin flux Thermal spin pumping efficiency Solid state heat engines Kovalev / Tserkovnyak design Kirihara / LSSE design Trans.Spins-9

93 Dissipation from spin currents? Joule heating J c > 0 - p p p p Power dissipated as heat (phonons) T increases Pure spin current Theory and assorted statements in literature: pure spin currents allow for dissipationless information transfer. If J c = 0, then Joule Heat = 0 J c = Phonons are still produced Phonon spin interactions important in spin Seebeck Trans.Spins-93

94 Spin Caloritronic engines based on domain wall motion Alexey A. Kovalev and Yaroslav Tserkovnyak Solid State Communications (010) Mechanisation: heat pushes domain wall Connect two of those with different pinning energy Heat pump/cooler Power generator Efficiency equations similar to thermoelectrics T. Spin-94

95 E based spin caloritronic energy recovering cloth, NEC corp. Akihiro Kirihara & al., Nat. Mater (01) Trans.Spins-95

96 ZT of LSSE process z CARNOT S T Ferromagnet Efficiency lz LZ l Z T z STh 1 z STh LZ Th XY S YIG Pt T Involves two materials Ferromagnet Inverse spin Hall S l L SInSb BiTe3 Pt BiTe3 10nm 100m H. Adachi, Spin Caloritronic V, Ohio State, 013 Z Trans.Spins-96 Z 8

97 Physics in its infancy Spin Seebeck effect is getting fairly well understood Spin mixing conductance good concept for spin fluxes in the linear transport regime Phonons induce diamagnetic moments Conclusions: spin caloritronics Spin Caloritronics Heat Spin Thermoelectrics Spintronics Charge Electrons Magnons Phonons Heat kt kt kt Magnetism Spin, Orbit B nd order Landau/Orbital Charge e Trans.Spins-97