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1 Long-range spin Seebeck effect and acoustic spin pumping K. Uchida 1,, H. Adachi,3, T. An 1,, T. Ota 1,, M. Toda 4, B. Hillebrands 5, S. Maekawa,3 and E. Saitoh 1,,3,6 1 Institute for Materials Research, Tohoku University, Sendai , Japan CREST, Japan Science and Technology Agency, Sanbancho, Tokyo , Japan 3 Advanced Science Research Centre, Japan Atomic Energy Agency, Tokai , Japan 4 Graduate School of Engineering, Tohoku University, Sendai , Japan 5 Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern, Germany 6 PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo , Japan saitoheiji@imr.tohoku.ac.jp. SA. Influence of voltage-probe contact Here we demonstrate that the contact of voltage probes does not affect the temperature distribution of the sapphire/[ni 81 Fe 19 /Pt-wire] samples used in the experiments shown in Figs To check the influence of the voltage-probe contact, we measured temperature T images of the sapphire/[two Ni 81 Fe 19 /Pt-wires] sample under a temperature gradient T before and after attaching the voltage probes to the ends of the Ni 81 Fe 19 /Pt wires (Fig. S1a). The difference images between them confirm that the temperature of the sapphire/[two Ni 81 Fe 19 /Pt-wires] sample does not change at every position by attaching the voltage probes (see also the T diff profiles along the x ( T ) direction in Figs. S1b and S1c, where T diff denotes the temperature variation of the sample due to the voltage-probe contact). SB. Linear-response theory of acoustic spin Seebeck effect In this section, we formulate a linear-response theory of the acoustic spin Seebeck effect (SSE), demonstrated in Figs. 1-3 in the main text, and show that the experimental observation can be explained as a consequence of the energy transfer from non-equilibrium phonons to magnons. We consider a model shown in Fig. Sa, and investigate the spin injection from a ferromagnetic metal (FM, in the experiments Ni 81 Fe 19 ) into an attached paramagnetic metal (PM, in the experiments Pt) through the s-d exchange interaction acting at the FM/PM interface: NATURE MATERIALS 1

2 a Higher T Lower T before attaching voltage probes sapphire Ni 81 Fe 19 /Pt T b (K) T b (K) after attaching voltage probes voltage probe T a (K) T a (K) x x difference (after - before) -3 3 T diff (K) -3 3 T diff (K) Tdiff (K) Tdiff (K) b c difference (Lower T) difference (Higher T) x (mm) Figure S1 Temperature variation due to voltage-probe contact. a, Temperature images of the sapphire/[two Ni 81 Fe 19 /Pt-wires] sample, measured with an infrared camera (NEC-Avio TH9100MR). We measured the images before and after attaching the voltage probes (tungsten needles) to the ends of the Ni 81 Fe 19 /Pt wires with applying the temperature difference of 0 K to the sapphire substrate. Temperatures on the metallic wires cannot be measured due to very low infrared emittance. The images in the right column represent the difference between those in the left and centre columns; T diff = T a T b, where T b(a) denotes the temperature before (after) attaching the voltage probes to the sample. b, c, T diff profiles along the x direction, measured when the voltage probes are attached to the ends of the Ni 81 Fe 19 /Pt wires placed near the lower-temperature (b) and higher-temperature (c) ends of the substrate. H sd = J sd r 0 interface s(r 0 ) S(r 0 ), (1) where s is the conduction-electrons spin density in PM, S is the spin density in FM, and J sd is the strength of the interface s-d exchange coupling. In the spin-wave region, the spin density S can be expanded by the magnon creation (annihilation) operator a q (a q ) with the relation S x S (r i )= 0 N F q (a q +a q )e iq r i, S y S (r i )= i 0 N F q (a q a q )e iq r i, and S z (r i )= S N F q,q a qa q+q e iq ri, where S 0 = S and N F is the number of lattice sites in FM. The spin current J s injected into PM can be calculated as a rate of change of the spin density in PM as J s = r N ts(r,t), where denotes the statistical average at a given time t. Assuming that the spin-orbit interaction is weak enough in the neighborhoods of the interface, the Heisenberg equation of motion for s yields 7 J s = q,k J k q sd S0 NF N P ReC k,q < (ω), () NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION where J k q sd is the Fourier transform of J sd (r) =J sd r 0 interface a3 F δ(r r 0) with the effective block spin volume a 3 F. Here, C< k,q (ω) is the Fourier transform of the interface correlation C k,q < (t, t )= i a + q (t )s k (t) between the magnons and the spin density s k = 1 N P r i [s x (r i ) is y (r i )]e ik r i with N P being the number of lattice sites in PM (note that the definition is different from Ref. 7 by a factor S 0 ). Using equation (), it has been shown that, when both the spin density in FM and the conduction electrons in PM are in local thermal equilibrium, no spin current is injected into PM because of the balance between thermal fluctuations in PM and those in FM 7. Conversely, when the spin density in FM deviates from local thermal equilibrium, a finite spin current is injected into PM. One of such processes was already considered in Ref. 7 where magnons in FM are deviated from local thermal equilibrium by the lateral correlation of magnons feeling the temperature gradient inside FM. Another thermal spin injection process is non-equilibrium phonons which drive non-equilibrium dynamics of magnons 6. To discuss such a phonon-mediated spin injection process, we consider the exchange Hamiltonian describing the dynamics of the spin density S in FM: H ex = R i,r j J ex (R i R j ) S(R i ) S(R j ), (3) where J ex (R i R j ) is the strength of the exchange coupling between the ions at R i and R j. The instantaneous position of the ion is written as R i = r i + u(r i ) where the lattice displacement u(r i ) is separated from the equilibrium position r i. In the following, we consider a situation where the polarization index ζ is not mixed with each other, hence ζ is hereafter omitted. Note that in the case of the acoustic spin pumping, demonstrated in Fig. 4 in the main text, the dominant contribution comes from the longitudinal component because of the selection rule in the magnon-phonon interaction. Up to the linear order in the displacement, the exchange Hamiltonian (3) can be written in the form H ex = q ω q a qa q + H mag ph, (4) where ω q =S 0 δ J ex(δ) q [ 1 cos(q δ) ] is the magnon frequency with the lattice vector δ and H mag ph = r i,δ (g δ/a S ) [u(r i ) u(r i +δ) ] S(r i )S(r i +δ) is the magnon-phonon interaction 31 with the magnon-phonon coupling g given by J ex (δ) =(g/ δ ) δ. The displacement u can be expressed in terms of a phonon field as 3 u(r i )= K (iê ( K) ν K M bk ionn F e ik ri b ) K e ik ri, where M ion is the ion mass and b K (b K) is the phonon creation (annihilation) operator with wavevector K, the polarization vector ê K, and the phonon frequency ν K. These equations yield NATURE MATERIALS 3

4 the following magnon-phonon interaction: H mag ph = 1 Γ K,q B K a q+k a q, (5) NF q,k where B K = b K + b K is the phonon field operator and the magnon-phonon vertex is given by Γ K,q =S 0 g K δ M ion v ( δ ê p K )( δ K) [ ] 1 cos(q δ) with the sound velocity v p. When the anisotropy in the magnon-phonon coupling is neglected, we obtain an approximate expression K Γ K,q g( ω q ) M ion vp, (6) where g = g δ /J ex is the dimensionless magnon-phonon coupling constant. With these apparatus, let us consider the phonon-mediated spin injection process shown in Fig. Sa, where non-equilibrium acoustic phonons driven by the temperature gradient in the substrate (sapphire) disturb the distribution function of magnons in FM (Ni 81 Fe 19 ), which then results in a finite spin injection into PM (Pt). The spin current injected into PM in this process is given by (J J s = sd S 0 )N int ν p N P N F NS (Λ/b) (Γ K,q) Ω K K k,q,k,k,k dν π A k,q(ν) DK R (ν) DK(ν) R ImDK R (ν)[ coth( k B T ) coth( k B T 1 ) ], (7) where N S is the number of lattice sites in the substrate, N int is the number of localized spins at the FM/PM interface, ν p is the characteristic frequency corresponding to the phonon high-energy cutoff, Ω K K represents the phonon interaction between the substrate and FM, D R K (ν) = (ν ν K +i/τ p ) 1 (ν + ν K +i/τ p ) 1 is the retarded component of the phonon propagator in the substrate with its mode frequency ν K and phonon lifetime τ p, b/λ is the lattice constant of the substrate divided by the dimension of the substrate along the temperature gradient. In the above equation, the quantity A k,q (ν) is given by A k,q (ν) = Imχ R k (ω)imxq K(ω R ν) Xq R (ω) [coth( (ω ν) k B T 1 ) coth( ω k B T 1 )], (8) ω where χ R k (ω) =χ P/(1 + λ sf k iωτ sf ) is the retarded component of the spin-density propagator in PM with χ P, λ sf, and τ sf being respectively the paramagnetic susceptibility, the spin diffusion length, and the spin relaxation time and X R q (ω) =(ω ω q +iαω) 1 is the retarded component of the magnon propagator with ω q = γh 0 +ω q and α being respectively the magnon frequency and the Gilbert damping constant. In equation (7), the prime in Γ K,q = g ( ω q ) K /(M ion vp) means that the phonon-magnon interaction is limited to the neighborhoods of the FM/substrate interface due to the heat balance condition, since FM in the present situation is not in contact with heat bath. Making use of A k,q (ν) (χ P ω q τ sf ) coth( ω q T 1 )[ 1 ω q Im χ k(ω q ) χ P ] and coth( k B T ) 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION ( ) coth( k BT 1 ) (k BT ) / sinh ( k ) BT T by setting T 1 = T,T T 1 = T ( T ) as well as performing a bit lengthy calculation, equation (7) is transformed into J s = ( kb ) τ p Γ effrb T, (9) ( ) where the dimensionless constant Γ eff is given by Γ eff = eg eν p M ion vp K, which should ν K +1/τp be interpreted as a phenomenological parameter representing an effective magnon-phonon coupling near the substrate/fm interface ( eg eν p M ion v ) and an effective phonon transmission amplitude p between the substrate and FM ( Ω K K ). Note that the latter factor is proportional ν K +1/τp to T S F =4Z S Z F /(Z S + Z F ), where Z S(F) is the characteristic acoustic impedance 7 of the substrate (the FM layer); the factor T S F ( 1) is maximized when Z S = Z F. In equation (9), R = 0.1 J sd S 0N int χ P π (λ sf /a) 3 (Λ/b) is a factor measuring the strength of the magnetic coupling at the FM/PM interface and B = (T/e T m) 9/ 0π 5 Ω K ( k BT e m τ sf ) 3 T e m/t v dv 7/ 0 tanh(v/) is a function of thermally excited magnons with T m being the characteristic temperature corresponding to the magnon high-energy cutoff. Combining equation (9) with equation (1) in the main text, we obtain equation () in the main text. Using λ sf = 7 nm, a = b =0.3 nm, θ SH =0.01, ρ =0.9 µωm, χ P = cm 3 /g, τ sf = 1 ps, S 0 = 1, J sd = 1 ev, T m = 850 K, and Z S(F) = 41 (48) 10 6 kg/m s (the resultant T S F is 1 at the sapphire/ni 81 Fe 19 interface), the dimensionless magnon-phonon coupling constant g times the phonon lifetime τ p in the sapphire substrate is roughly estimated as g τ p s to reproduce the SSE voltage V/ T 0.0 µv/k observed in the experiments (see Figs. d and e in the main text). a Acoustic spin Seebeck effect b Acoustic spin pumping J s J s Pt J s Ni 81 Fe 19 sapphire substrate silicone Pt YIG T = T 1 T = T T = T 3 T piezoelectric actuator Figure S Linear-response calculation. a, Feynman diagrams for calculating spin currents in the Pt layers in the sapphire/[ni 81 Fe 19 /Pt-wire] sample used in the experiments on the acoustic spin Seebeck effect (Figs. 1-3 in the main text). Here, the system is divided into three domains with their temperatures T 1, T, and T 3. J s denotes the spatial direction NATURE MATERIALS 5

6 of the spin current. b, Feynman diagram for calculating a spin current in the Pt layer in the Pt/Y 3Fe 5O 1 (YIG) sample used in the experiments on the acoustic spin pumping (Fig. 4 in the main text). In this case, the interaction between phonons and magnons is given by equation (5) with the thermal phonon operator being replaced by the external sound-wave operator. The thin solid lines with arrows, bold lines, and dotted lines represent electron, magnon, and phonon (or external-sound-wave) propagators, respectively. SC. Temperature dependence of acoustic spin Seebeck effect To further buttress the acoustic mechanism of the SSE, demonstrated in Figs. 1-3 in the main text, we measured the SSE voltage in the sapphire/[ni 81 Fe 19 /Pt-wire] sample at various temperatures (30 90 K). Figure S3 shows V/ T between the ends of the Pt layer as a function of the sample temperature T, measured when the Ni 81 Fe 19 /Pt wire was placed near the highertemperature end of the sapphire substrate. Here, the magnetic field of 300 Oe and the uniform temperature gradient were applied along the x direction. In all the temperature range, the clear SSE signals appear. Notable is that, around T = 40 K, the magnitude of the SSE voltage V/ T is strongly enhanced. This V enhancement provides a crucial evidence that the SSE signal in the sapphire/[ni 81 Fe 19 /Pt-wire] sample is dominated by the phonon-mediated process through the sapphire substrate, since the V -peak structure at low temperatures corresponds to the increase of the phonon lifetime in the sapphire substrate 6,8,33. V/ T (10-7 V/K) sapphire/[ni 81 Fe 19 /Pt-wire] H = 300 Oe Higher T Ni 81 Fe 19 /Pt wire y z x H LO sapphire substrate V HI T T (K) 300 Figure S3 Temperature dependence. Dependence of V/ T on the temperature T in the sapphire/[ni 81 Fe 19 /Pt-wire] sample at the magnetic field H = 300 Oe, measured when the Ni 81 Fe 19 /Pt wire was placed at the distance of 1.9 mm from the higher-temperature end of the sapphire substrate. The temperatures of the lower- and higher-temperature ends of the substrate were stabilized to T and T + T, respectively. 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION SD. Sign difference between acoustic spin pumping and heating effect Here we discuss the sign of the V signals observed in the Pt/Y 3 Fe 5 O 1 (YIG) sample, where the signal coming from the sound-wave-driven spin injection is opposite in sign to that coming from the heating effect by the piezoelectric actuator (see Fig. 4d in the main text). In the Pt/YIG sample, there are two contributions to the spin injection process; the excitation of magnons in the YIG slab injects a spin current into the Pt film (Fig. Sb), while the excitation of conduction electrons in the Pt film ejects a spin current from itself 7. Then, in the case of the sound-wavedriven spin injection, the sound wave interacts with magnons in the YIG slab efficiently, while it does not interact with conduction electrons in the Pt film because the thickness of the Pt film (15 nm) is too small for conduction electrons in the Pt to feel the sound wave (f MHz) which has a wavelength of the order of millimetres. This means that the sound wave does excite magnons in YIG slab efficiently while it does not excite conduction electrons in the Pt film. That is, a spin current in this case is injected into the Pt film (Fig. Sb). On the other hand, in the case of heating by the piezoelectric actuator, spin densities in the YIG slab and the Pt film are excited by a heat current, i.e., a flow of thermal phonons. Since the thermal de Broglie length of phonons is much shorter than a millimetre, the thermal phonons are able to excite conduction electrons in the Pt film. Moreover, since the silicone rubber in the present set-up acts as a heat sink due to heat conservation, the heat current carried by thermal phonons is forced to penetrate into the Pt film. Under such circumstances, conduction electrons in the Pt film are excited much stronger than magnons in the YIG slab as the electron-phonon interaction in Pt is much stronger than the magnon-phonon interaction in YIG. This means that, in the latter case, a spin current is ejected from the Pt film as was demonstrated in recent experiments 5. These considerations explain the difference in the sign of the signal for both processes as seen in Fig. 4d. Additional References 31. Stern, H. Thermal conductivity at the magnetic transition. J. Phys. Chem. Solids 6, (1965). 3. Mahan, G. Many-Particle Physics (Kluwer Academic, 1981). 33. Touloukian, Y. S. (ed) Thermophysical Properties of Matter (IFI/Plenum, 1970). NATURE MATERIALS 7