SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMEARY IORMAIO A. Scattering theory Here we propoe a mechanim for the pin Seebeck effect (SSE) oberved in a magnetic inulator in term of the effective pin-wave (magnon) temperature magnetic film (, in the experiment LaY 2 e 5 O 12 ) and the electron temperature in the in the normal metal contact (, in the experiment Pt). he thermal fluctuation of the magnetization m at an / interface are excited by a thermal random magnetic field ( j) h= j h (j = 0, 1), which atifie the fluctuation-diipation theorem by the equal-poition time-correlation function j () ( j) ( j) h t h t' = 2 k α γmv δ δ δ( t t'), where ( j) ( ') i i' B a jj ' ii' ( 0) α i the bulk Gilbert damping parameter, k B the Boltzmann contant, γ the gyromagnetic ratio, M the aturation magnetization, 0 =, () 1 =, and V a the magnetic coherence volume in the layer, which depend on temperature and pin-wave tiffne contant D. ( 1) α = γ g 4πMV i the r a damping enhancement due to the pin pumping, where g r i the real part of the pin-mixing conductance of the interface 24. he net thermal pin current acro the / interface i given by the um of the fluctuating thermal pin-pumping current Jp from to proportional to and the Johnon-yquit pin-current noie J fl from to proportional to (ref ): MV a α γ () 1 () 1 J = J p + Jfl = + γ m m m h (S1) he DC component along the magnetization equilibrium direction (x direction) reduce to J J = 2 α k ( ) (S2) (1) x B he remaining tak i the evaluation of the patial profile of induced by a global temperature bia Δ over the two end of the layer. Here we aume that the electron nature material 1

2 upplementary information temperature of the contact, the temperature difference, equal the lattice temperature in the layer. In thi ituation, i induced at the boundarie to the heat bath, where the phonon are trongly coupled while the magnon are (almot) thermally inulated 29. he olution of a imple heat-rate equation of the coupled magnon-phonon ytem yield 1 = ηδ inh ( / λ ), where η ( L/ λ ) coth ( L/ 2λ ) x m, x i the poition along the m temperature-gradient direction (x = 0 at the centre of the layer), and L i the length of the layer along the direction. he (quared) magnon relaxation length i decribed a λ 3.9 Dk τ τ, where τ mm (τ mp ) i the magnon-magnon (magnon-phonon) 2 2 m B mm mp relaxation time. In the preent LaY 2 e 5 O 12 /Pt ample, the pin current hown in equation (S2) i converted into electric voltage in the Pt layer due to the invere pin Hall effect (ISHE) leading to: ISHE ( λ ) V = ξδ inh x/ (S3) m m where ξ ηθ e k ρlg γ ( π M V A) =, θ SH SH B r a i the pin-hall angle ( for Pt 11 ), e the able S1 Parameter for LaY 2 e 5 O 12 /Pt. γ electron charge, ρ the reitivity of the Pt 4πM A/m layer, l the length of the Pt layer, and A the contact area. By uing the parameter in able S1, we find ξ = 0.37 μv/k and λ m = 4.8 mm for LaY 2 e 5 O 12 at = 300 K. hee calculated value roughly agree with the experimental ξ = 0.16 μv/k and λ m = 6.7 mm, which are obtained by fitting of the data hown in ig. 3 uing equation (S3) (ee c (Curie temp.) K D (ref. 21) Jm 2 α τ mm τ mp (ref. 31, 32) g r /A m -2 V a 1/3 L l ρ 5.43 nm 8 mm 4 mm 0.91 μωm 2 nature MAERIALS

3 upplementary information dahed line in ig. 3). he hyperbolic ine ditribution of the ISHE voltage in equation (S3) i confirmed by the temperature dependent meaurement. igure S1a how V/Δ a a function of x Pt in the LaY 2 e 5 O 12 /Pt ytem for variou value of the ample temperature. When > 200 K, V varie almot linearly with repect to x Pt. In contrat, below 150 K, the x Pt dependence of V deviate from the linear function; the magnitude of V decay within everal millimeter from both the end of the LaY 2 e 5 O 12 layer. In all the temperature range, the oberved V ditribution can be fitted by equation (S3), where ξ and λ m are fitting parameter (ee olid curve in ig. S1a). hee reult upport our model, although the precie temperature dependence of the parameter remain to be undertood. igure S1 emperature dependence. a, Dependence of V/Δ on x Pt, the diplacement of the Pt wire from the centre of the LaY 2 e 5 O 12 layer along the x direction, in the LaY 2 e 5 O 12 /Pt ample for variou value of at H = 100 Oe (olid circle). he olid curve are the fitting reult uing a hyperbolic-ine function ξinh(x/λ m ), where ξ and λ m are nature material 3

4 upplementary information adjutable parameter. b, A chematic illutration of the LaY 2 e 5 O 12 /Pt ample and the patial ditribution of the electric voltage induced by the ISHE V ISHE (ee equation (S3)). c, dependence of λ m. Inet to c how the dependence of ξ. B. Linear-repone theory In thi ection, we how within the linear-repone formalim that we can explain the magnitude and patial dependence of the pin current J in the Pt wire induced by the thermal motion of localized magnetic moment in the LaY 2 e 5 O 12 layer. he qualitative picture given in the main text i confirmed by thi linear-repone approach 33 a well. We tart from the Landau-Lifhitz-Gilbert equation decribing the thermal motion of the localized magnetic moment in the LaY 2 e 5 O 12 : mrt = γ H + hrt + D mrt mrt + αmrt mr t (S4) t 2 (,) [ eff (,) (,)] (,) (,) t (,) where m= M / M i the unit magnetization normalized by the aturation magnetization M, α the Gilbert damping contant, H eff the effective magnetic field, and h expree the effect of thermal noie atifying h( r, th ) ( r', t') = 2 k( r) α γm δ δ( r r ') δ( t t') through the i i' B ii' fluctuation-diipation theorem. he thermal motion of the localized magnetic moment in the LaY 2 e 5 O 12 affect the dynamic of the conduction-electron pin in the Pt wire through the following -d interaction at the LaY 2 e 5 O 12 /Pt interface 34, H = Ω S r () mr () (S5) d d 0 r interface where Ω d i the -d interaction trength and S = Ma γ i the ize of the localized pin 3 0 with the effective block pin volume 3 a. hi interaction then induce a pin current J x 2 t into the Pt wire acro the LaY 2 e 5 O 12 /Pt interface (note that the pin quantizing direction i along the x direction). Uing the Heienberg equation of motion for x, 4 nature MAERIALS

5 upplementary information the reultant pin current J can be calculated a an interface correlation 35 between and m: Ω S = d 0 + J m t t 2 r interface Im (,) r (,) r (S6) where m ± = m y ± im z and ± = y ± i z. hi interface correlation can be evaluated perturbatively in term of the -d interaction trength Ω d. When neglecting the influence of the temperature gradient inide of the LaY 2 e 5 O 12 layer (proce P 1 in ig. S2), we find that the teady-tate pin current J in the Pt wire vanihe becaue of the fluctuation-diipation theorem. In contrat, when taking account of the nonzero temperature variation (proce P 1 and P 3 ), we obtain a nonzero pin current a J = L[ ( x) ( x )] (S7) 0 where (x) denote the temperature at poition x (x 0 i the centre of the ample: ( x0) = 2 in ig. S2), and L S Ω ( a/ λ ) χτ d int f = kb π α( Λ / a ) with λ, χ, τ f, a, int, and Λ being the pin diffuion length, pin uceptibility, pin relaxation time, lattice contant in the Pt wire, the number of local magnetic moment at the LaY 2 e 5 O 12 /Pt interface, and the ize of a temperature domain along the temperature gradient, repectively. ote that the pin current at the centre of the ample vanihe becaue the two relevant procee (P 2 and P 2 ) cancel out. he pin current dicued here ha the ame linear profile a that oberved in our experiment (ee ig. 3 in the main part of the article), on the bai that the invere pin-hall voltage due to the SSE i proportional to the pin current induced in the Pt wire, i.e., ( ρ ) VISHE = θshj 2 e w with w being the width of the Pt wire. ote that the oberved millimeter-cale patial ditribution i conitent with the recent experiment on Y 3 e 5 O 12 (ref. 30). Uing λ = 7 nm, a = 0.2 nm, θ SH = , ρ = 0.91 μωm, χ = cm 3 /g, τ f = 1 p, a = 1.23 nm, and the interface -d coupling extracted from the independent experiment, Ω d ~ 10 mev (ref. 21), the pin current derived in equation (S7) explain the magnitude of the voltage ignal V ISHE /Δ ~ 0.1 μv/k oberved in our experiment. nature material 5

6 upplementary information inally, we point out that the concept to explain the preent experiment i quite different from that invoked for the previou i 81 e 19 -baed SSE experiment. All previou model dealt with the effect of tatic magnetic moment in local equilibrium with the conduction electron 36. In contrat, here we demontrate that the SSE in a magnetic inulator can be undertood only by conidering dynamical magnetic moment, i.e., pin wave or magnon. he finding of thi unconventional mechanim ugget that magnon play a key role alo in the SSE of metallic magnet. herefore, the obervation of the magnon-driven SSE in the inulator LaY 2 e 5 O 12 and it theoretical formulation have poibly larger ramification for the phyic of metallic magnet. igure S2 Linear-repone calculation. eynman diagram for calculating the pin current flowing acro the interface between a magnetic inulator (, in the experiment LaY 2 e 5 O 12 ) and nomal metal (, in the experiment Pt). Here, the ytem i divided into three temperature domain ( 1 / 1, 2 / 2, 3 / 3 ) with their temperature 1, 2, 3. he thin olid line with arrow (bold line without arrow) repreent electron propagator (magnon propagator). Additional Reference 31. Kittel, C. & Abraham, E. Relaxation proce in ferromagnetim. Rev. Mod. Phy. 25, (1953). 32. Demokritov, S. O. et al. Boe Eintein condenation of quai-equilibrium magnon at 6 nature MAERIALS

7 upplementary information room temperature under pumping. ature 443, (2006). 33. Šimánek, E. & Heinrich, B. Gilbert damping in magnetic multilayer. Phy. Rev. B 67, (2003). 34. Zhang, S. & Li, Z. Role of nonequilibrium conduction electron on the magnetization dynamic of ferromagnet. Phy. Rev. Lett. 93, (2004). 35. Caroli, C., Compecot, R., oziere, P. & Saint-Jame, D. Direct calculation of the tunneling current. J. Phy. C: Solid State Phy. 4, (1971). 36. Hatami, M., Bauer, G. E. W., akahahi, S. & Maekawa, S. hermoelectric pin diffuion in a ferromagnetic metal. Solid State Commun. 150, (2010). nature material 7