Observation of the spin Peltier effect
J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, B. J. van Wees
OObservation of the spin Peltier effect
J. Flipse, ∗ F. K. Dejene, D. Wagenaar, G. E. W. Bauer,
2, 3
J. Ben Youssef, and B. J. van Wees Physics of Nanodevices, Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai, Japan Universit´e de Bretagne Occidentale, Laboratoire de Magn´etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, France. (Dated: September 28, 2018)We report the observation of the spin Peltier effect (SPE) in the ferrimagnetic insulator YttriumIron Garnet (YIG), i.e. a heat current generated by a spin current flowing through a Platinum(Pt) | YIG interface. The effect can be explained by the spin torque that transforms the spin current inthe Pt into a magnon current in the YIG. Via magnon-phonon interactions the magnetic fluctuationsmodulate the phonon temperature that is detected by a thermopile close to the interface. By finite-element modelling we verify the reciprocity between the spin Peltier and spin Seebeck effect. Theobserved strong coupling between thermal magnons and phonons in YIG is attractive for nanoscalecooling techniques.
The discovery of the spin Seebeck effect (SSE) inYIG | Pt bilayers [1] opened up a new research direction inthe field of spin caloritronics. In the SSE a temperaturedifference between the magnons in the magnetic insulatorand the electrons in the metal contact leads to thermalpumping of a spin current [2–4]. In a suitable metal suchas Pt this spin current is transformed into an observabletransverse voltage by the inverse spin Hall effect [5]. Nu-merical simulations of the phonon, magnon and electrontemperatures show good agreement with experiments [6].In this Letter we report the observation of the spin Peltiereffect (SPE), which is the Onsager reciprocal [7] of theSSE.The SPE is the generation of a magnon heat currentin the magnetic insulator by a spin current through theinterface with the metal contact. The latter can be gen-erated by a charge current in the Pt film that by the spinHall effect generates a transverse spin current normal tothe interface. The spin Peltier heat current generatesa temperature difference between magnons and phononsin the YIG that when relaxing leads to a change in thelattice temperature. We confirm this scenario experi-mentally by picking up such temperature changes viaproximity thermocouples. According to our modellingthe experimental results are consistent with Onsager reci-procity between the SPE and the SSE, which we measureseparately (see supplementary IV). Our results confirmrecent indications for a strong magnon-phonon interac-tion in YIG at room temperature [6, 8, 9].A charge current through a Pt strip generates a trans-verse spin current induced by the spin Hall effect thatleads to a spin accumulation V s at the boundaries. Atthe interface to YIG the spin current is absorbed as aspin transfer torque proportional to the spin mixing con-ductance [10, 11], as depicted in Fig. 1(a). When themagnetic moment of the spin accumulation ( µ s ) at thePt | YIG interface is parallel (antiparallel) to the averagemagnetization direction, the spin torque transfers mag- netic momentum and energy from the electrons in thePt to the magnons in the YIG (or vice versa). Magnonsare thereby annihilated (excited) (see Fig. 1(b)) lead-ing to cooling (heating) of the magnetic order parame-ter (see Fig. 1(c)). Since thermal magnons equilibratewith the lattice by magnon-phonon scattering, the non-equilibrium magnons affect the lattice temperature (seeFig. 1(b) and (c)) depending on the magnetization di-rection.In the SSE [2] the spin current density ( J s ) pumpedfrom the YIG into the non magnetic metal is proportionalto the temperature difference between the magnons andelectrons at the interface ( T m - e = T m − T e ) and the in-terface spin Seebeck coefficient L S , J s = L S T m - e . In or-der to arrive at a symmetric linear response matrix thatreflects Onsager symmetry, the sum of the products ofcurrents and driving forces should be proportional to thedissipation [12], leading to (see supplementary I) (cid:18) J s Q m - e (cid:19) = (cid:18) g S L S TL S T κ IS T (cid:19) (cid:18) V s / T m - e /T (cid:19) (1)Here we used the Onsager Kelvin relation Π S = S S T = L S / ( g S T ) , where the spin Seebeck S S = ( dV s / dT ) J s =0 and spin Peltier Π S = ( dQ m - e /dJ s ) ∂T m - e =0 coefficientshave been defined. g S is the average spin conductanceper unit area when spin accumulation and magnetiza-tion are collinear, i.e. the V s at the YIG | Pt interface iseither parallel or antiparallel to the average YIG magne-tization. g S ≈ . g r at room temperature [13], where g r is the real part of the spin-mixing conductance perunit area. κ IS is the magnetic contribution to the inter-face heat conductance per unit area [6]. The SPE heatcurrent density we set out to discover is therefore Q m − e = L S T V s . (2)The devices designed for observing the SPE are fab-ricated on top of a 200 nm thick single-crystal (111) a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov J c M T p T m M μ s T e T T T p T m M T T e (a) (b) (c) T μ s FIG. 1. (color online). Schematic figure of the spin Peltier effect at a Pt | YIG interface. (a) A charge current through the Ptcreates a transverse spin current induced by the spin Hall effect that generates a spin accumulation V s at the boundaries. (b)When the spin magnetic moment µ s is antiparallel to M the spin torque transfers angular momentum and energy from theelectrons in the Pt to the magnons in the YIG thereby cooling the electrons and heating the magnons, effectively raising themagnon temperature T m with respect to the electron temperature T e . (c) When µ s is parallel to M the spin torque transfersangular momentum and energy from the magnons in the YIG to the electrons in the Pt thereby cooling the magnons, effectivelylowering T m with respect to T e . Y Fe O (YIG) film grown on a 500 µ m thick (111)Gd Ga O (GGG) substrate by liquid-phase-epitaxy.Two temperature sensors are fabricated in close prox-imity to the Pt | YIG interface. The optical microscopeimage in Fig. 2(b) shows the 20 × µ m and 5 nmthick Pt injector film. The thermopile sensors consistof five 40 nm thick Pt-Constantan (Ni Cu ) thermo-couples in series that are very sensitive because of thelarge difference in the Seebeck coefficient of these met-als. In the thermopile on the right of the Pt injector thePt | Ni Cu order is reversed for additional cross checkmeasurements. The two thermopiles and the Pt injectorare connected to 5 |
100 nm thick titanium | gold contacts,providing good thermal anchoring and electrical contactto bonding pads 30 µ m away. All structures are patternedby electron beam lithography. The Pt injector and theNi Cu are deposited by DC sputtering while electronbeam evaporation has been used to make the Au contactsand Pt thermocouple components.An AC current is sent through the Pt injector, from I + to I − (Fig. 2(b)), to create V s . The voltage over the ther-mopile (V + and V − ) is simultaneously recorded. Usinga standard lock-in detection technique the first harmonicresponse ( V ∝ I ) is extracted from the measured voltage.A low excitation frequency of 17 Hz was used to ensurea thermal steady-state condition. All measurements arecarried out at room temperature.In Fig. 2(a) the first harmonic voltage over the ther-mopile is shown as a function of an applied in-planemagnetic field ( B ) for a root-mean-square current of 3mA through the Pt injector. A clear switch is observedjust after the applied field becomes positive, in line withthe magnetization reversal of YIG at very small coer-cive fields [14]. The signal switches back to its original value when reversing the field with a small hysteresis.We measure a SPE signal of 33 nV on top of a back-ground voltage of 0 . µ V. We observe linear scalingof the SPE signal for currents between 1 and 4 mA inthe Pt ( I Pt injector )(see Fig. 2(c)). Results for four dif-ferent samples (from two different batches) match thesignal presented here within 15 %. The measurementswere repeated with B rotated 90 ◦ . No SPE signal wasobserved in this configuration while the background re-mained the same (see supplementary II), which confirmsour interpretation.In order to obtain quantitative information we carryout 3D finite element modelling of our devices [15]. Asdiscussed above, the SPE heat current ( Q m - e ) flows be-tween the electron and magnon systems through thePt | YIG interface. Q m - e is calculated using Eq. (2) and V s = θJ c · η · tanh (cid:18) t λ (cid:19) (3)where θ is the spin Hall angle, t the Pt film thickness, J c the charge current density through the Pt injector, ρ the Pt resistivity, λ the spin-flip diffusion length and η = 2 λρ · [1 + g S ρλ coth (cid:0) tλ (cid:1) ] − a backflow correctionfactor. The heat charge current densities in Pt are mod-elled by a three reservoir model of thermalized phonons,magnons and electrons at temperatures T ph , T m and T e ,respectively [6]. In linear response the charge ( J c ) andheat ( Q ) current densities in the bulk of the materialsare related to their driving forces, i.e. gradients of ( V , T ph , T m and T e ) as (cid:126)Q x = κ x (cid:126) ∇ T x and (cid:18) (cid:126)J c (cid:126)Q e (cid:19) = − (cid:18) σ σSσST κ e (cid:19) (cid:18) (cid:126) ∇ V(cid:126) ∇ T e (cid:19) (4)where x is ph or m , σ is the electrical conductivity, −10 −5 0 5 100.450.460.470.48 V t h e r m o p i l e ( μ V ) B (mT) I Pt injector = 3mA ∆ V s p i n P e l t i e r ( n V ) I Pt injector (mA) Pt NiCu
NiCu Pt V+V- I+I- B (a)(b) (c) FIG. 2. (color online). (a) First harmonic voltage acrossthe thermopile as a function of applied magnetic field. Thedifference between the voltage at positive and negative fieldsis the spin Peltier signal. (b) Optical microscope picture ofthe device. (c) The spin Peltier signal (∆ V spin Peltier ) as afunction of the charge current through the Pt injector. S the Seebeck coefficient and κ ph , κ m and κ e are thephonon, magnon and electron thermal conductivities, re-spectively. The interaction between the magnon andphonon subsystems in YIG and between the phonon andelectron subsystems in Pt are taken into account by us-ing thermal relaxation lengths, λ m − ph and λ e − ph , respec-tively (see supplementary III), ∇ T m - ph = T m - ph λ m - ph and ∇ T e - ph = T e - ph λ e - ph . (5)The phonon interface heat conductance ( κ Iph ) and heatexchange between magnons and electrons across the in-terface ( κ IS ) are treated as boundary conditions [6] (seeSupplementary III).This model is evaluated for the material parameterslisted in table I. Additionally, we adopt g r = 7 × Ω − m − [16], θ = 0 .
11 [6], λ = 1 . L S =7 . × A/(m K) [2, 6]. The magnon heat conductivityof YIG ( κ m ) at room temperature is not well known sowe used a κ m of 10 − and 10 − W/(mK) in order tocover the range of estimated values [6, 17]. For Pt | YIGa κ Iph of 2 . × W/(m K) obtained from the acousticmismatch model was adopted [6]. Since this model tendsto overestimate the heat conductance [18], we also used
TABLE I. Material parameters used in the model. Both σ andS are measured in separate devices [19] except for σ of the Ptinjector, which is extracted from the SPE devices directly. κ ph κ e is adopted from Ref. 6 and the total κ = κ ph + κ e iscalculated using κ = σσ bulk κ bulk . σ S κ ph κ e (S/m) ( µ V/K) (W/(m · K)) (W/(m · K))YIG - - 6 -GGG - - 8 -Au 2.7 · · -5 3 23Pt thermocouple 4.2 · -5 4 28Ni Cu · -30 1 9 × W/(m K). In figure 3(a) the results are shown as afunction of λ m - ph . The semi transparent blue horizontalbar indicates the range of measured SPE signals that arebest fitted by a λ m - ph of 0.1 to 0.2 nm for the ranges of κ m and κ Iph discussed above.SSE samples were fabricated and simulated by thesame model and parameters used above (see Supplemen-tary IV). In Fig. 3(b) the results are plotted and bestfitted by λ m - ph between 0.2 and 0.5 nm, which is con-sistent with the values found for the SPE, as is indeedrequired by Onsager reciprocity. This implies that ourmodel captures the essential physics of the interactingelectron, magnon and phonon systems.The observed SPE signal in Fig. 2(a) corresponds toa phonon temperature difference of 0.25 mK at the ther-mopile, which according to the model is 39 % of thephonon temperature difference directly at the Pt | YIGinterface. By engineering devices in which the phononheat loss through the substrate is minimized by thinneror etched YIG films could therefore significantly enhancethe measured signal. Altering the Pt injector coupling tothe heat sink or placing the thermocouple on top of thePt injector might also help.The λ m - ph found here is smaller than the one adoptedby Ref. 6 ( ≈ et al .’s simulations might agree bet-ter with their measurements for smaller values as well. λ m - ph extracted from Fig. 3 is quite sensitive to smallvariations in the modelling, which implies a large uncer-tainty. Nevertheless even when accepting a large errorbar from 0.1 to 6 nm for λ m − ph we may conclude thatthermal magnons and phonons interact strongly [8].The background signal in the SPE data is a factor 20higher than we would expect from conventional chargePeltier heating and cooling at the Au | Pt injector inter-faces. Reference measurements on the second thermopileon the other side of the Pt injector excludes charge cur-rent leakage to the thermopile. For an identical configu-ration, V + on the same side as I + , we find an oppositesign of the measured voltage, as expected for a thermal s p i n S ee b e c k s i g n a l ( μ V ) s p i n P e l t i e r s i g n a l ( n V ) λ m–ph (nm) λ m–ph (nm) (a) (b) κ m = 1e-3 and κ Iph = 2e8κ m = 1e-2 and κ Iph = 2.79e8κ m = 1e-2 and κ Iph = 2e8κ m = 1e-3 and κ Iph = 2.79e8κ m = 1e-3 and κ Iph = 2e8κ m = 1e-2 and κ Iph = 2.79e8κ m = 1e-2 and κ Iph = 2e8κ m = 1e-3 and κ Iph = 2.79e8
FIG. 3. (color online). The modeled SPE (a) and SSE (b) signal versus λ m - ph for two different values of κ m (W/(m K)) andtwo different values of κ Iph (W/(m K)). The semi transparent blue bar indicates the range of measured SPE and SSE effectsignals. signal since the Pt | Ni Cu thermopile sequence is in-verted. A current leak would not change sign and cantherefore be excluded. The background in the secondharmonic voltage is likely to be caused by the thermo-voltage across the thermopile due to Joule heating in thePt injector, since its value agrees within 17% with themodeled one. Additional measurements of frequency de-pendent properties (see Supplementary V) rule out pick-ups due to capacitive or inductive couplings.We checked that the sign of the experimentally ob-served SPE and SSE signals obey reciprocity. Further-more the voltage measured across the Pt detector in aRF spin pumping measurement matches the sign of theSSE voltage for the same geometry when heating the YIGrelative to the Pt, as previously reported [20, 21]. How-ever, the absolute sign of these three effects is still underinvestigation.In conclusion, we report experimental proof that aspin accumulation at a Pt | YIG interface induces heat ex-change between electrons and magnons on both sides.Using thermal modelling to knit the theory of inter-face transport to the observables we demonstrate thatthe SPE is the Onsager reciprocal of the SSE and con-firm a strong interaction between thermal magnons andphonons in YIG, as reported earlier [8]. We hope thatthese results can contribute to a better understandingof coupling between thermomagnetic and thermoelectricproperties. Our proof of principle opens new strategiesfor nanoscale cooling applications.We would like to acknowledge B. Wolfs, M. de Rooszand J. G. Holstein for technical assistance. This workis part of the research program of the Foundation forFundamental Research on Matter (FOM) and supportedby NanoLab NL, Marie Curie ITN Spinicur, DFG Pri-ority Programme 1538 ”Spin-Caloric Transport”, Grant- in-Aid for Scientific Research A (Kakenhi) 25247056 andthe Zernike Institute for Advanced Materials. ∗ [email protected][1] K. Uchida, J. Xiao, H. Adachi, J. I. Ohe, S. Takahashi,J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,G. E. W. Bauer, S. Maekawa, and E. Saitoh, NatureMaterials , 894 (2010).[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh,S. Maekawa, Phys. Rev. B , 214418 (2010).[3] H. Adachi, J. I. Ohe, S. Takahashi, S. Maekawa, Phys.Rev. B , 094410 (2011).[4] S. Hoffman, K. Sato, Y. Tserkovnyak, Phys. Rev. B ,064408 (2013).[5] A. Hoffmann, IEEE Trans. Magnetics , 5172 (2013).[6] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E.W. Bauer, R. Gross, S. T. B. Goennenwein, Phys. Rev.B , 094410 (2013).[7] L. Onsager, Phys. Rev. , 405426 (1931).[8] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A.D. Karenowska, G. A. Melkov, B. Hillebrands, Phys.Rev. Lett. , 107204 (2013).[9] N. Roschewsky, M. Schreier, A. Kamra, F. Schade,K. Ganzhorn, S. Meyer, H. Huebl, S. Gepr¨ags, R. Gross,S. T. B. Goennenwein, arXiv 1309.3986v1 (2013).[10] A. Brataas, G. E. W. Bauer, P. J. Kelly, Phys. Rep. , 157 (2006).[11] Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, A. Slavin,Phys. Rev. Lett. , 146602 (2011).[12] H. B. Callen, Phys. Rev. , 1349 (1948).[13] H. J. Jiao, J. Xiao, G. E. W. Bauer, in preparation.[14] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, J. BenYoussef, Phys. Rev. B , 184421 (2013).[15] COMSOL Multiphysics ® , 032401(2013). [17] R. L. Douglass, Phys. Rev. , 1132 (1963).[18] E. T. Swartz, R. O. Pohl, Rev. Mod. Phys. , 605(1989).[19] F. L. Bakker, J. Flipse, B. J. van Wees, J. Appl. Phys. , 084306 (2012).[20] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A.A. Serga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh,B. Hillebrands, Phys. Rev. Lett. , 216601 (2011).[21] M. Weiler, M. Althammer, M. Schreier, J. Lotze,M. Pernpeintner, S. Meyer, H. Huebl, R. Gross,A. Kamra, J. Xiao, Y. T. Chen, H. J. Jiao, G. E.W. Bauer, S. T. B. Goennenwein, Phys. Rev. Lett. , 176601 (2013).[22] W. Wang, D. G. Cahill, Phys. Rev. Lett. , 175503(2012).[23] K. Uchida, H. Adachi, T. Ota, H. Nakayama,S. Maekawa, E. Saitoh, Appl. Phys. Lett. , 172505(2010).[24] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,M. S. Wagner, M. Opel, I. Imort, G. Reiss, A. Thomas,R. Gross, S. T. B. Goennenwein, Phys. Rev. Lett. ,106602 (2012). SUPPLEMENTARY INFORMATIONI. Onsager reciprocity for the spin Seebeck and spin Peltier effect
The linear response matrix of thermoelectrics reflects Onsager reciprocity when the sum of the products of currentstimes driving forces equals the dissipation [12]. When I c and Q are the charge and heat currents driven by voltageand temperature differences ∆ V and ∆ T , ˙ F = I c ∆ V + Q ∆ T /T equals the dissipation and we obtain the symmetricresponse matrix [12]: (cid:18) I c Q (cid:19) = − (cid:18) G GSTGST KT (cid:19) (cid:18) ∆ V ∆ T /T (cid:19) (6)where G is the electrical conductance, S the Seebeck coefficient, and K the heat conductance. Here the Onsager-Thomson relation for the Peltier coefficient Π = ST has already been implemented.In the case of the spin Seebeck effect (SSE) and the spin Peltier effect (SPE) for magnetic insulators, the spinaccumulation at the interface drives a spin current. To ensure reciprocity, we have to compute the Joule heatingcaused by the spin currents: ˙ F = I ↑ ∆ V ↑ + I ↓ ∆ V ↓ = G ↑ ∆ V ↑ + G ↓ ∆ V ↓ (7)where the subscripts denote the up and down spin contribution. Comparing this with the product of I s with ∆ V s : I s ∆ V s = ( G ↑ ∆ V ↑ − G ↓ ∆ V ↓ )(∆ V ↑ − ∆ V ↓ ) = 2( G ↑ ∆ V ↑ + G ↓ ∆ V ↓ ) (8)we conclude that ∆ V s / densities at the interface between a normal metal and a ferromagnet then reads (Eq. (1) of the main text, where all variablesand parameters are introduced): (cid:18) J s Q m − e (cid:19) = (cid:18) g S L S TL S T κ IS T (cid:19) (cid:18) V s / T m - e /T (cid:19) (9)and we omitted the charge sector because we are dealing with a ferromagnetic insulator. II. Measurement with B rotated 90 ◦ We repeated the measurements in the main text after rotating the magnetic field by 90 ◦ (see Fig. 4). Themagnetization direction is here parallel to the current in the Pt film such that the current-induced spin accumulationis normal to the magnetization. The background voltage and noise level remain unmodified; we do not detect aheating or cooling of the ferromagnet. This confirms our interpretation of the experiments in the main text. The spintorque normal to the magnetization is not expected to affect the magnon temperature and the SPE should vanish, asobserved. III. The 3D finite element model
We adopt the three reservoir model of thermalized electron, magnon, and phonon systems. Charges are transportedby the electron system only, while heat currents flow in all subsystems. We take into account spin angular momentumcurrents in the electron and magnon system, but disregard the phonon angular momentum current. The bulk chargeand heat currents in linear response are given by Eq. (4) in the main text. The charge and energy conservationrelations read: (cid:126) ∇ · (cid:126)J c (cid:126)Q ph (cid:126)Q m (cid:126)Q e = − P m - ph ) ( κ m + κ ph )4 λ m - ph ( T m − T ph ) + ( − P e - ph ) ( κ e + κ ph )4 λ e - ph ( T e − T ph ) − ( − P m - ph ) ( κ m + κ ph )4 λ m - ph ( T m − T ph ) J c σ − ( − P e - ph ) ( κ e + κ ph )4 λ e - ph ( T e − T ph ) (10) −10 −5 0 5 100.450.460.47 PtNiCu NiCuPt
V+V- I+I- B I Pt injector = 3mA
B (mT) V t h e r m o p i l e ( μ V ) FIG. 4. (a) First harmonic voltage across the thermopile as a function of applied magnetic field parallel to the Pt injector. (b)Optical microscope picture of the measured device and measurement geometry. with P m − ph = ( κ m − κ ph ) / ( κ m + κ ph ), P e − ph = ( κ e − κ ph ) / ( κ e + κ ph ) and J c /σ accounts for Joule heating. Theother terms describe the heat exchange between phonons, magnons and electrons, as indicated by the subscripts.All parameters have been defined in the main text, except for the thermal relaxation lengths λ m − ph and λ e − ph . Theformer is used as an adjustable parameter while the latter λ e − ph = (cid:112) κ e /g is calculated from the electron-phononcoupling parameter ( g ) for Au and Pt [22]. This leads to a λ P te − ph of 4.5 nm and a λ Aue − ph of 80 nm at room temperature.The boundary conditions at the metal | YIG interfaces are governed by the phonon ( κ Iph ) and magnetic ( κ IS ) interfaceheat conductances. We adopt the values for κ Iph from Ref. 6, while that for Ni Cu | YIG is assumed to be equal tothat of Pt | YIG. The interface magnetic heat conductance is calculated using [6] κ IS = he k B T (cid:126) µ B k B g r πM s V m (cid:18) g S ρλ coth (cid:18) tλ (cid:19)(cid:19) − (11)where V m is the (spin pumping) magnetic coherence volume. The currents through the interface read J Ic Q Iph Q Im Q Ie = κ Iph ( T aph − T bph ) − κ IS ( T m − T e ) − L S T V s κ IS ( T m − T e ) + L S T V s (12)where T Pt/YIG ph are the phonon temperatures on the Pt/YIG side of the interface. Q Im and Q Ie are the interfacemagnetic heat currents. The first terms on the r.h.s represent the heat current driven by temperature differences,while the second term is the heat current associated with the magnon injection by an applied V s (see Eq. (2) in themain text) that is responsible for the SPE.Eq. (4) in the main text is solved for the SPE and SSE configurations, taking into account energy conservation(Eq. (10)) and boundary conditions (Eq. (12)). The results are plotted in Figs. 3 (a) and (b) of the main text. IV. Spin Seebeck effect
To verify reciprocity we fabricated samples nominally identical to the Pt | YIG heterostructures used for the SPEexperiments in order to measure the longitudinal SSE [23, 24]. Fig. 5(b) gives a picture of such a device consisting −20 −15 −10 −5 0 5 10 15 20−30−20−100102030 V P t d e t e c t o r ( μ V ) B (mT) I heater = 5mA BI + I - V + V -
100 μm (a) (b)
FIG. 5. (a) Second harmonic voltage across the Pt detector as a function of applied magnetic field. (b) Optical microscopepicture of the measured device and the measurement geometry used. of a 5 nm thick sputtered Pt detector (250 x 10 µ m ) on top of a GGG | YIG substrate as used for the study of theSPE effect. The Pt detector is covered with a ∼
70 nm aluminium oxide (Al O ) layer that electrically isolates thePt detector from a 40 nm Pt film heater evaporated on top. Both the detector and heater are contacted by a 100 nmthick Au layer to large bonding pads.By Joule heating, a charge current through the heater creates a thermal gradient over the Pt | YIG interface. Thehot electrons transfer energy to the cold magnons, which is associated with a spin current in Pt that is converted toan observable charge current by the inverse spin Hall effect. This is the SSE.In Fig. 5(a) the second harmonic voltage versus magnetic field is a measure of the Joule heating. A clear SSE signalis observed, which changes sign when the magnetization is reversed, as expected. The signal scales quadratically withthe current and for B parallel to the Pt detector no SSE signal is detected, which confirms that the voltage is due tothe inverse spin Hall effect.The model discussed in the main text and the previous section can be applied to this measurement geometry tofind the T m − e at the interface and the transverse voltage over the Pt detector [6]: V SSE = 2 · g r γ (cid:126) k B πM s V m T m − e · πe θρl · λt tanh (cid:0) t λ (cid:1) g S ρλ coth ( t/λ ) (13)where l is the length of the Pt detector. The results obtained from the SSE modeling are shown in Fig. 3(b) of themain text. V t h e r m o c o u p l e X - c o m p o n e n t ( μ V ) Measurement frequency (Hz) V t h e r m o c o u p l e Y - c o m p o n e n t ( μ V ) Measurement frequency (Hz)(a) (b)I
Pt injector = 2 mA I
Pt injector = 2 mA
FIG. 6. (a) First harmonic voltage across the thermopile in phase with the current ( X -component). (b) First harmonic voltageacross the thermopile out-of-phase with the current ( Y -component). −10 −5 0 5 100.70.720.74 V t h e r m o p i l e ( μ V ) B (mT) I Pt injector = 4 mA
FIG. 7. First harmonic voltage across the thermopile as a function of applied magnetic field for a 4 mA current through thePt injector with frequency of 3 Hz.
V. Frequency dependent measurements