Nonlocal magnon-polaron transport in yttrium iron garnet

L.J. Cornelissen,1, ∗ K. Oyanagi,2, ∗ T. Kikkawa,2, 3 Z. Qiu,3 T.

Kuschel,1 G.E.W. Bauer,1, 2, 3, 4 B.J. van Wees,1 and E. Saitoh2, 3, 4, 5

1Physics of Nanodevices, Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands †2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan

5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan

The spin Seebeck effect (SSE) is observed in magnetic insulator|heavy metal bilayers as an inversespin Hall effect voltage under a temperature gradient. The SSE can be detected nonlocally aswell, viz. in terms of the voltage in a second metallic contact (detector) on the magnetic film,spatially separated from the first contact that is used to apply the temperature bias (injector).Magnon-polarons are hybridized lattice and spin waves in magnetic materials, generated by themagnetoelastic interaction. Kikkawa et al. [Phys. Rev. Lett. 117, 207203 (2016)] interpreted aresonant enhancement of the local SSE in yttrium iron garnet (YIG) as a function of the magneticfield in terms of magnon-polaron formation. Here we report the observation of magnon-polarons innonlocal magnon spin injection/detection devices for various injector-detector spacings and sampletemperatures. Unexpectedly, we find that the magnon-polaron resonances can suppress ratherthan enhance the nonlocal SSE. Using finite element modelling we explain our observations as acompetition between the SSE and spin diffusion in YIG. These results give unprecedented insightsinto the magnon-phonon interaction in a key magnetic material.

When sound travels through a magnet the local dis-tortions of the lattice exert torques on the magneticorder due to the magnetoelastic coupling1. By reci-procity, spin waves in a magnet affect the lattice dy-namics. The coupling between spin and lattice waves(magnons and phonons) has been intensively researchedin the last half century2,3. Yttrium iron garnet (YIG) hasbeen a singularly useful material here, because it can begrown with exceptional magnetic and acoustic quality2.Magnons and phonons hybridize at the (anti)crossing oftheir dispersion relations, a regime that has attractedrecent attention4–10. When the quasiparticle lifetime-broadening is smaller than the interaction strength, thestrong coupling regime is reached; the resulting fullymixed quasiparticles have been referred to as magnon-polarons6,7.

In spite of the long history and ubiquity of the magnon-phonon interaction, it still leads to surprises. Evidence ofa sizeable magnetoelastic coupling in YIG was recentlyfound in experiments on spin caloritronic effects, i.e. thespin Peltier11 and spin Seebeck effect12,13 (SPE and SSErespectively). Recently, Kikkawa et al. showed that thehybridization of magnons and phonons can lead to a res-onant enhancement of the local SSE in YIG9. Bozhkoet al. found that this hybridization can play a rolein the thermalization of parametrically excited magnonsusing Brillouin light scattering. They observed an ac-cumulation of magnon-polarons in the spectral regionnear the anticrossing between the magnon and trans-verse acoustic phonon modes14. However, these previousexperiments did not address the transport properties ofmagnon-polarons.

Nonlocal spin injection and detection experiments areof great importance in probing the transport of spin in

metals15, semiconductors16 and graphene17. Varying thedistance between the spin injection and detection con-tacts allows for the accurate determination of the trans-port properties of the spin information carriers in thechannel, such as the spin relaxation length18. Recently,it was shown that this kind of experiments are not limitedto (semi)conducting materials, but can also be performedon magnetic insulators19, where the spin information iscarried by magnons. Such nonlocal magnon spin trans-port experiments have provided additional insights in theproperties of magnons in YIG, for instance by studyingthe transport as a function of temperature20–23 or exter-nal magnetic field24. Finally, the nonlocal magnon spininjection/detection scheme can play a role in the develop-ment of efficient magnon spintronic devices, for examplemagnon based logic gates25,26. In this study, we makeuse of nonlocal magnon spin injection and detection de-vices to investigate the transport of magnon-polarons inYIG.

Magnons can be excited magnetically using the oscil-lating magnetic field generated by a microwave frequencyac current25, or electrically using a dc current in an ad-jacent material with a large spin Hall angle, such asplatinum19. Finally, they can be generated thermallyby the SSE27–30, in which a thermal gradient in the mag-netic insulator drives a magnon spin current parallel tothe induced heat current.

The generation of magnons via the SSE can be de-tected in several configurations: First, the heater-inducedconfiguration (hiSSE)31, which consists of a bilayerYIG|heavy metal sample that is subject to externalPeltier elements to apply a temperature gradient nor-mal to the plane of the sample. The SSE then gener-ates a voltage across the heavy metal film (explained in

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more detail below), which can be recorded. Second, thecurrent-induced configuration (ciSSE)28,32 in which theheavy metal detector used to detect the SSE voltage is si-multaneously used as a heater. A current is sent throughthe heavy metal film, creating a temperature gradient inthe YIG due to Joule heating. Due to this temperaturegradient, the SSE generates a voltage across the heavymetal film, which can again be recorded. Third, the non-local SSE (nlSSE)19,33, in which a current is sent througha narrow heavy metal strip to generate a thermal gradi-ent via Joule heating as well. However, the SSE signalresulting from this thermal gradient is detected in a sec-ond heavy metal strip, located some distance away fromthe injector.

In the nlSSE, the magnons responsible for generating asignal in the detector strip are generated in the injectorvicinity and then diffuse through the magnetic insula-tor to the detector. The temperature gradient under-neath a detector located several microns to tens of mi-crons from the injector does not contribute significantlyto the measured voltage23,34. In contrast, the hiSSE andciSSE always have a significant temperature gradient di-rectly underneath the detector. The hiSSE and ciSSE aretherefore local SSE configurations, contrary to the nlSSEwhich is nonlocal.

In all three configurations, the resulting voltage acrossthe heavy metal film is due to magnons which are ab-sorbed at the YIG|detector interface, causing spin-flipscattering of conduction electrons and generating a spincurrent and spin accumulation in the detector. Due tothe inverse spin Hall effect35, this spin accumulation isconverted into a charge voltage that is measured.

At specific values for the external magnetic field, thephonon dispersion is tangent to that of the magnons andthe magnon and phonon modes are strongly coupled overa relatively large region in momentum space. At theseresonant magnetic field values, the effect of the magne-toelastic coupling is at its strongest and magnon-polaronsare formed efficiently. If the acoustic quality of the YIGfilm is better than the magnetic one, magnon-polaronformation leads to an enhancement in the hiSSE signalat the resonant magnetic field9. This enhancement is at-tributed to an increase in the effective bulk spin Seebeckcoefficient ζ, which governs the generation of magnonspin current by a temperature gradient in the magnet.This was demonstrated experimentally by measuring thespin Seebeck voltage in the hiSSE configuration9, estab-lishing the role of magnon-polarons in the thermal gen-eration of magnon spin current.

Here we make use of the nlSSE configuration to di-rectly probe not only the generation, but also the trans-port of magnon-polarons. We show that in the YIG sam-ples under investigation not only ζ, but also the magnonspin conductivity σm is resonantly enhanced by the hy-bridization of magnons and phonons, which leads to sig-natures in the nonlocal magnon spin transport signalsclearly distinct from the hiSSE observations. Notably,resonant features in nonlocal transport experiments have

very recently been theoretically predicted by Flebus etal.10, who calculated the influence of magnon-polaronson the YIG transport parameters such as the magnonspin and heat conductivity and the magnon spin diffu-sion length.

I. RESULTS

A. Sample characteristics

Our nonlocal devices consist of multiple narrow, thinplatinum strips (typical dimensions are 100 µm × 100nm × 10 nm [l× w × t]) deposited on top of a YIG thinfilm and separated from each other by a centre-to-centredistance d. We have performed measurements of non-local devices on YIG films from Groningen and Sendai,both of which are grown by liquid phase epitaxy on aGGG substrate. The YIG film thickness is 210 nm (2.5µm) for YIG from Groningen (Sendai). In Sendai, fivebatches of devices where investigated (sample S1 to S4)on pieces cut from the same YIG wafer. In Groningen,two batches of devices where investigated (G1 and G2).The platinum strips are contacted using Ti/Au contacts(see Methods for fabrication details) Figure 1a shows anoptical microscope image of a typical device, with theelectrical connections indicated schematically. The cen-tral strip functions as a magnon injector while the twoouter strips are magnon detectors, measuring the nonlo-cal signal at different distances from the injector.

B. Experimental results

A nonlocal signal is generated in the devices by passinga low frequency ac current I(ω) (typically w/(2π) < 20Hz and Irms = 100 µA) through the injector. This leadsto both thermal and electrical generation of magnons,as outlined above. The voltage that is due to the ther-mally generated magnons is proportional to the excita-tion current squared, and hence can be directly detectedin the second harmonic detector response V (2ω) (i.e. thevoltage measured at twice the excitation frequency). Si-multaneously, the voltage due to electrically generatedmagnons can be measured in the first harmonic responseV (1ω)19. The sample is placed in an external magneticfield H, under an angle α = 90◦ to the injector/detectorstrips.

Figure 1b shows the results of two typical nonlo-cal measurements at different distances, in which µ0His varied from −3.0 to 3.0 T. Several distinct featurescan be seen in these results. As the magnetic field isswept through zero, the YIG magnetization and hencethe magnon spin polarization change direction, since amagnon always carries a spin opposite to the majorityspin in the magnet. This causes a reversal of the polar-ization of the spin current absorbed by the detector and

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consequently the voltage VnlSSE changes sign. Addition-ally, VnlSSE for short distance d (Figure 1b bottom pan-els) shows an opposite sign compared to VnlSSE for longdistance (Figure 1b top panels). This sign-reversal forshort distances is a characteristic feature of the nlSSE19

that has so far been observed to depend on both thethickness of the YIG film tYIG (roughly speaking, at roomtemperature when d < tYIG the sign will be opposite tothat for d > tYIG

33) as well as the sample temperature,where a lower temperature reduces the distance at whichthe sign-change occurs21,22.

The sign for short distances corresponds to the signone obtains when measuring the SSE in its local config-urations (hiSSE, indicated schematically in Figure 1c orciSSE). The results for a hiSSE measurement on sampleS3 as a function of H are shown in Figure 1d, and VhiSSEclearly shows the same sign as VnlSSE for short distance.We will discuss the origin of this sign-change in more de-tail later in this manuscript. The data shown in Figure 1are from samples with tYIG = 2.5 µm, hence the differentsigns for d = 2 µm and d = 6 µm.

Resonant features can be observed in the data for|µ0H| = µ0HTA ≈ 2.3 T, where the subscript TA sig-nifies that these features stems from the hybridization ofmagnons with phonons in the transverse acoustic mode,rather than the longitudinal acoustic mode (LA) which isexpected at larger magnetic fields. The rightmost panelsof Figure 1 show a close-up of the data around H = HTA.For small d the magnon-phonon hybridization causes aresonant enhancement (the absolute value is increased)of VnlSSE, while for large d a resonant suppression (theabsolute value is reduced) occurs.

Figure 2 shows the results of a magnetic field sweepfrom sample G1 for both electrically generated magnons(first harmonic) and thermally generated magnons (sec-ond harmonic). A feature at |H| = HTA can be resolvedboth in the first and second harmonic voltage. This sug-gests that magnon-phonon hybridization does not onlyaffect the YIG spin Seebeck coefficient, as the first har-monic signal is generated independent of ζ. It indicatesthat not only the generation, but also the transport ofmagnons is affected by the hybridization. In the secondharmonic, the signal is clearly suppressed at the resonantmagnetic field. Unfortunately, because the feature in thefirst harmonic is barely larger than the noise floor in themeasurements (see Fig. 2a and inset), we cannot concludewhether the signal due to electrical magnon generationis enhanced or suppressed at the resonance. Due to thefact that the effect in the first harmonic is so small, in theremainder of this paper we present a systematic study ofthe effect in the second harmonic, the nlSSE.

The resonant magnetic fields are different for the TAand LA modes (HTA and HLA, respectively). Due to thehigher sound velocity in the LA phonon mode, HTA <HLA, and the resonance due to magnons hybridizing withphonons in this mode can also be observed in our nonlo-cal experiments. In the Supplementary Material (sectionA) we show the results of a magnetic field scan over an

extended field range, and it can be seen that the res-onance at HLA also causes a suppression of the nlSSEsignal, similar to the HTA resonance. This is compara-ble to the case for the hiSSE configuration, in which theHLA and HTA resonances both show similar behaviour inthe sense that they both enhance the hiSSE signal. Forthe nlSSE case at distances larger than the sign-changedistance, both resonances suppress the signal.

We now focus on the resonance at HTA in the nlSSEdata and carried out nonlocal measurements as a functionof magnetic field for various temperatures and distances.Figure 3a (b) shows the distance (temperature) depen-dent results, obtained from sample S1 (S2). The regionswhere the sign of the nlSSE equals that of the hiSSE areshaded blue. From Figure 3a the sign-change in VnlSSEcan be clearly seen to occur between d = 2 and d = 5µm, as at d = 2 µm the nlSSE sign is equal to that ofthe hiSSE for any value of the magnetic field, whereas ford = 5 µm it is opposite. Additionally, when comparingthe VnlSSE−H curves for 300 K and 100 K in Figure 3b,the effect of the sample temperature on the sign-change isapparent: At 100 K, the nlSSE sign is opposite to that ofthe hiSSE over the whole curve. Furthermore, Figure 3bdemonstrates the influence of the magnetic field on thesign change, for instance in the curve for T = 160 K. Atlow magnetic fields, the nlSSE sign still agrees with thehiSSE sign (inside the blue shaded region), but around|µ0H| = 1.5 T the signal changes sign.

In addition, Figure 3a shows that the role of themagnon-polaron resonance changes as the nlSSE signalundergoes a sign change. For d ≤ 2 µm, magnon-phononhybridization enhances VnlSSE at H = HTA, whereas ford ≥ 5 µm VnlSSE is suppressed at the resonance mag-netic field. Similarly, from Figure 3b we observe that attemperatures T > 160 K, magnon-phonon hybridizationenhances the nlSSE signal at H = HTA, while at T ≤ 160K the nlSSE is suppressed at HTA. Since the thermallygenerated magnon spin current is related to the thermalgradient by jm ∝ −ζ∇T , a resonant enhancement in ζshould lead to an enhancement of the nlSSE signal at alldistances and temperatures, which is inconsistent withour observations. This is a further indication that notonly the generation, but also the transport of magnonsis influenced by magnon-polarons.

The temperature dependence of the low-field ampli-tude of the nlSSE V 0nlSSE and the magnitude of the reso-nance VTA (defined in Figure 1b) are shown in Figure 4aand 4b respectively. The curve for V 0nlSSE at d = 6 µmagrees well with an earlier reported temperature depen-dence of the nlSSE at distances which are larger than thefilm thickness23, while that at d = 2 µm qualitativelyagrees with earlier reports for distances shorter than theYIG film thickness21,22. Moreover, from the distance de-pendence of V 0nlSSE we have extracted the magnon spindiffusion length λm as a function of temperature, which isshown in the Supplementary Material (section B). λm(T )obtained from the Sendai YIG approximately agrees withthat for Groningen YIG23 for temperatures T > 30 K,

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but differs in the low temperature regime. For furtherdiscussion we refer to the Supplementary Material of thismanuscript. The temperature dependence of VTA is dif-ferent from that of V 0nlSSE, since first of all no change insign occurs here even for d = 2 µm and furthermore aclear minimum appears in the curve around T = 50 K.This indicates that the resonance has a different originthan the nlSSE signal itself, i.e. magnon-polarons areaffected differently by temperature than pure magnons.

The resonant magnetic field HTA decreases with in-creasing temperature, reducing from µ0HTA ≈ 2.5 T at3 K to µ0HTA ≈ 2.2 T at room temperature as shownin Figure 4c. In earlier work by some of us regardingthe magnetic field dependence of the nonlocal magnontransport signal at room temperature, structure in thedata at µ0H = 2.2 T was indeed observed

24, but notunderstood at that time. It is now clear that this struc-ture can be attributed to magnon-phonon hybridization.HTA depends on the following three parameters

9: TheYIG saturation magnetization Ms, the spin wave stiff-ness constant Dex and the TA-phonon sound velocitycTA. Dex is approximately constant for T < 300 K

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and both Ms and cTA decrease with temperature. Thereduction of HTA as temperature increases from 3 K to300 K can be explained by accounting for a 7 % decreaseof cTA in the same temperature interval, taking the tem-perature dependence of Ms into consideration

37. Theresults regarding the behaviour of the magnon-polaronresonance qualitatively agree for the Sendai and Gronin-gen YIG (see Supplementary Material (section C) for thetemperature dependent results for sample G2).

Moreover, we performed measurements of the nlSSEsignal as a function of the injector current, and foundthat the nlSSE scales linearly with the square of thecurrent at high temperatures, as expected. However, atlow temperatures (T < 10 K) and sufficiently high cur-rents (typically, I > 50 µA), this linear scaling breaksdown (see Supplementary Material (section D)). Thiscould be a consequence of the strong temperature depen-dence of the YIG and GGG heat conductivity at thesetemperatures38,39. The injector heating causes a small in-crease in the average sample temperature which increasesthe heat conductivities of the YIG and GGG, therebydriving the system out of the linear regime. However, itmight also be related to the bottleneck effect which is ob-served in parametrically excited YIG14. A more detailedinvestigation is needed in order to establish the origin ofthe nonlinearity.

Finally, we have investigated the ciSSE configuration,meaning that current heating of the Pt injector is used todrive the SSE and the (local) voltage across the injector ismeasured. The sign of the ciSSE voltage corresponds tothat obtained in the hiSSE configuration. However, noresonant features were observed in the ciSSE measure-ments, contrary to the hiSSE and nlSSE configurations.We believe that this is due to the low signal-to-noise ratioin the ciSSE configuration, which could cause the featureto be smaller than the noise level in our ciSSE measure-

ments. We refer to the Supplementary Material (sectionE) for further discussion.

C. Modelling

The physical picture underlying the thermal genera-tion of magnons has been a subject of debate in themagnon spintronics field recently. Previous theories ex-plain the SSE as being due to thermal spin pumping,caused by a temperature difference between magnons inthe YIG and electrons in the platinum13,40,41. However,the recent observations of nonlocal magnon spin trans-port and the nlSSE give evidence that not only the inter-face but also the bulk magnet actively contributes andeven dominates the spin current generation. At elevatedtemperatures the energy relaxation should be much moreefficient than the spin relaxation, which implies that themagnon chemical potential (and its gradient) is more im-portant as a non-equilibrium parameter than the temper-ature difference between magnons and phonons. A modelfor thermal generation of magnon spin currents basedon the bulk SSE42 which takes into account a non-zeromagnon chemical potential has been proposed in orderto explain the observations34.

This model has been reasonably successful in explain-ing the nonlocal signals (due to both thermal and electri-cal generation) in the long distance limit23,33, yet is notfully consistent with experiments in the short distancelimit for thermally generated magnons33. The model isexplained in detail in Refs. 33 and 34, and is describedconcisely in the Methods section of this manuscript. Thephysical picture captured by the model is explained inFigure 5a and b, where for this study we focus on thethermally generated magnons driving the nlSSE. In Fig-ure 5a a schematic side-view of the YIG|GGG samplewith a platinum injector strip on top is shown. A cur-rent is passed through the injector, causing it to heat upto temperature TH . The bottom of the GGG substrateis thermally anchored at T0. As a consequence of Jouleheating, a thermal gradient arises in the YIG, driving amagnon current JmQ = −ζ/T∇T parallel to the heat cur-rent, i.e. radially away from the injector. This reducesthe number of magnons in the region directly below theinjector (magnon depletion).

In Figure 5b the same schematic cross-section is shown,but now the colour coding refers to the magnon chemicalpotential µm. Directly below the injector contact µmis negative due to the magnon depletion in this region(µ−). At the YIG|GGG interface, magnons accumulatesince they are driven towards this interface by the SSEbut are reflected by the GGG, causing a positive magnonchemical potential µ+ to build up. Note that the µ−

and µ+ regions are not equal in size since part of themagnon depletion is replenished by the injector contact,which acts as a spin sink. Due to the gradient in magnonchemical potential, a diffuse magnon spin current Jmd nowarises in the YIG given by Jmd = −σm∇µm.

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The combination of these two processes leads to a typ-ical magnon chemical potential profile as shown in Fig-ure 5c, which is obtained from the finite element model(FEM) at room temperature. The sign change from µ−

to µ+ occurs at a distance of roughly dsc = 2.6 µm fromthe injector, comparable to the YIG film thickness.

Here we used the effective spin conductance of thePt|YIG interface gs as a free parameter in order to getapproximate agreement between the modelled and exper-imentally observed sign-change distance dsc (see Methodsfor the further details of the model). The value for gs isapproximately a factor 30 lower than what we calculatedfrom theory34 and used in our previous work23. Whenusing gs = 9.6×1012 S/m2 as in previous work, dsc ≈ 300nm which is much shorter than what we observe in theexperiments. This discrepancy between the models forelectrically and thermally generated magnon transportmight indicate that some of the material parameters suchas spin or heat conductivity and spin diffusion length (forboth YIG and platinum) we use are not fully accurate.However, it is also conceivable that the models are notcomplete and need to be refined further33, for instanceby including temperature difference at material interfaceswhich are currently neglected.

The value of dsc depends mainly on four parameters:The thickness of the YIG film tYIG, the transparencyof the platinum|YIG injector interface, parameterized inthe effective spin conductance gs, the magnon spin con-ductivity of the YIG σm and finally the magnon spindiffusion length λm. At high temperatures (i.e. closeto room temperature), the thermal conductivities κGGGand κYIG are similar in magnitude

43 and affect dsc onlyweakly, allowing us to focus here on the spin transport.

Increasing tYIG or σm increases dsc since this reducesthe spin resistance of the YIG film, allowing the depletedregion to spread further throughout the YIG. However,increasing gs or λm causes the opposite effect and re-duces dsc since this increases the amount of µ

− which isabsorbed by the injector contact compared to that whichrelaxes in the YIG. The precise dependency of dsc onthese parameters is nontrivial but can be explored usingour finite element model. Ganzhorn et al. and Zhou etal. in Refs. 21 and 22 observed that dsc becomes smallerwith lower temperatures. This indicates that the ratio ofthe effective spin resistance of YIG to that of the Pt con-tact increases, causing spins to relax preferentially intothe contact and thereby reducing the extend of µ−.

Flebus et al. developed a Boltzmann transport theoryfor magnon-polaron spin and heat transport in magneticinsulators10. Here we implement the salient features ofmagnon-polarons into our finite element model. We ob-serve that when the combination of gs, λm, σm, tYIGand d is such that the detector is probing the depletionregion, i.e. µ−, the magnon-polaron resonance causesenhancement of the nlSSE signal. Conversely, when thedetector is probing µ+ the resonance causes a suppres-sion of the signal. This cannot be explained by assumingthat the only effect of the magnon-polaron resonance is

the enhancement of ζ, as this would simply increase thethermally driven magnon spin current JmQ and hence en-

hance both µ− and µ+. To understand this behaviour,we have to account for the enhancement of σm by themagnon-polaron resonance as well.

A resonant increase in σm leads to an increased diffu-sive backflow current Jmd , which can lead to a reductionof the magnon spin current reaching the detector at largedistances. We model the effect of the magnon-phonon hy-bridization by assuming a field-dependent magnon spinconductivity σm(H) and bulk spin Seebeck coefficientζ(H), which are both enhanced at the resonant fieldHTA.Note that the field-dependence only includes the con-tribution from the magnon-polarons10, and does not in-clude the effect of magnons being frozen out by the mag-netic field24,44–46 since this is not the focus of this study.The parameter values used in the model are given in theMethods section of this paper. The model is used to cal-culate the spin current flowing into the detector contactas a function of magnetic field, from which we calculatethe voltage drop over the detector due to the inverse spinHall effect. We then vary the ratios of enhancement forσm and ζ, i.e. fσ = σm(HTA)/σ

0m and fζ = ζ(HTA)/ζ

0,where σ0m and ζ

0 are the zero field magnon spin conduc-tivity and spin Seebeck coefficient and σm(HTA), ζ(HTA)are these parameters at the resonant field. The ratio ofenhancement δ = fζ/fσ is crucial in obtaining agreementbetween the experimental and modelled data. To changedelta, we fix fζ = 1.09 and vary fσ. The value for fζis comparable to the enhancement in ζ calculated fromtheory for low temperatures10.

D. Comparison between model and experiment

Figure 6 shows a comparison between the distance de-pendence of V 0nlSSE and VTA obtained from experiments(Fig. 6a) and the finite element model (Fig. 6b and c) atroom temperature. In Figure 6a, V 0nlSSE shows a changein sign around d = 4µm, while VTA has a positive signover the whole distance range. Fig. 6b shows the modelresults for V 0nlSSE (red), and the voltage measured atH = HTA for δ = 2 (green) and δ = 0.5 (purple). Whilethe voltage obtained from the model is approximatelyone order of magnitude lower than in experiments, thequalitative behaviour of the experimental data is repro-duced. In particular, the modelled dsc approximatelyagrees with the experimentally observed distance.

For δ = 2, the modelled voltage at HTA is always en-hanced with respect to V 0nlSSE (for d < dsc, V (HTA) <V 0nlSSE and for d > dsc, V (HTA) > V

0nlSSE). This is

not consistent with the experiments as it leads to a signchange in VTA, which is defined as VTA = V

0nlSSE −

V (HTA), as can be seen from Fig. 6c.However, for δ = 0.5, V (HTA) is enhanced with respect

to V 0nlSSE for d < dsc but suppressed for d > dsc. Thisresults in a positive sign for VTA over the full distancerange, comparable to the experimental observations. The

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full magnetic field dependence obtained from the modelcan be found in the Supplementary Material (section F).As can be seen from the inset in Fig. 6c, δ = 0.5 results ina decay of VTA with distance which is comparable to theexperimentally observed VTA(d) (inset Fig. 6a). We fittedthe data for VTA obtained from both the experiments andthe simulations to VTA(d) = A exp−d/`TA, where A isthe amplitude and `TA the length scale over which VTAdecays. From the fits, we obtain `expTA = 6.3 ± 1.2 µmand `simTA = 10.6 ± 0.1 µm at room temperature, wherewe have fitted to the model results for δ = 0.5. Fromthe simulations, we find that `TA is influenced by thevalue used for δ, where a smaller δ leads to a longer `TA.This could indicate that δ has to be increased slightly toobtain better agreement between `expTA and `

simTA .

Therefore, in order to explain the observations, 0.5 <δ < 1, i.e. the relative enhancement due to magnon-phonon hybridization in σm has to be larger than that ofζ. `expTA is enhanced at low temperatures (see Supplemen-tary Material (section B) for the distance dependence ofVTA at low temperatures). This could indicate that δdecreases with decreasing temperatures. For further dis-cussion we refer to the Supplementary Material (sectionB).

The model results depend sensitively on gs. A larger gsreduces the dsc observed in the model, so that our modelno longer qualitatively fits the distance dependence ofVnlSSE obtained in experiments. As a consequence, theδ needed to model the resonant suppression of the signalat HTA for long distances decreases further, which wouldimply that the enhancement in σm is much stronger thanthat in ζ. Such a strong enhancement in σm should resultin a clear magnon-polaron resonance in the electricallygenerated magnon spin signal, whereas we observed onlya small effect here (see Fig. 2a). This is an indicationthat our choice of reducing gs compared to our previouswork is justified.

II. DISCUSSION

We report resonant features in the nlSSE as a functionof magnetic field, which we ascribe to the hybridizationof magnons and acoustic phonons. They occur at mag-netic fields that obey the “touch” condition at which themagnon frequency and group velocity agree with thatof the TA and LA phonons. The signals are enhanced(peaks) for short injector-detector distances and hightemperatures, but suppressed (dips) for long distancesand/or low temperatures. The temperature dependenceof the TA resonance differs from that of the low-fieldnlSSE voltage, indicating that different physical mechan-sims are involved (this in contrast to the local SSE con-figuration). The sign of the nlSSE signal correspondsto that of the signal in the hiSSE configuration for dis-tances below the sign-change distance. In this regime themagnon-polaron feature causes signal enhancement, sim-ilar to the hiSSE configuration. For distances longer than

the sign-change distance, the nlSSE signal is suppressedat the resonance magnetic field.

These results are consistent with a model in whichtransport is diffuse and carried by strongly coupledmagnons and phonons10 (magnon-polarons). Theorypredicts an enhancement of all transport coefficientswhen the acoustic quality of the crystal is better than themagnetic one. Simulations show that the dip observed inthe nlSSE is not caused by deteriorated acoustics, butby a competition between the thermally generated, SSEdriven magnon current and the diffuse backflow magnoncurrent which are both enhanced at the resonance. Moreexperiments including thermal transport as well as an ex-tension of the Boltzmann treatment presented in Ref. 10to 2D geometries are necessary to fully come to grips withheat and spin transport in YIG.

Additionally, we observed features in the electricallygenerated magnon spin signal at the resonance magneticfield. This is further evidence that not only the gener-ation of magnons via the SSE, but additional transportparameters such as the magnon spin conductivity are af-fected by magnon-polarons.

The nonlocal measurement scheme provides an excel-lent platform to study magnon transport phenomena andopens up new avenues for studying the magnetoelasticcoupling in magnetic insulators. Finally, these results arean important step towards a complete physical picture ofmagnon transport in magnetic insulators in its many as-pects, which is crucial for developing efficient magnonicdevices.

III. METHODS

Sample fabrication. The YIG films used in thisstudy were all grown on gadolinium gallium garnet(GGG) substrates by liquid phase epitaxy (LPE) in the[111] direction. The samples from the Sendai group havea thickness of 2.5 µm, the samples used in Groningen are210 nm thick. The Sendai samples were grown in-house,whereas the Groningen samples were obtained commer-cially from Matesy GmbH. In Sendai, five batches of de-vices where fabricated from the same YIG wafer (S1 toS4). The fabrication method and platinum strip geome-try are the same for all batches, but they were not fabri-cated at the same time, which might lead to variations infor instance the interface quality from batch to batch. InGroningen, two batches of devices were investigated (G1and G2). The nonlocal devices fabricated in Groningenare defined in three lithography steps: the first step wasused to define Ti/Au markers on top of the YIG film viae-beam evaporation, used to align the subsequent steps.In the second step, Pt injector and detector strips weredeposited using magnetron sputtering in an Ar+ plasma.In the final step, Ti/Au contacts were deposited by e-beam evaporation. Prior to the contact deposition, abrief Ar+ ion beam etching step was performed to removeany polymer residues from the Pt strip contact areas to

7

ensure optimal electrical contact to the devices. The non-local devices fabricated in Sendai were defined in a singlelithography step. Two parallel Pt strips and contact padswere patterned using e-beam lithography followed by alift-off process, in which 10-nm-thick Pt was depositedusing magnetron sputtering in an Ar+ plasma.Measurements. Electrical measurements were car-

ried out in Groningen and in Sendai, using a current-biased lock-in detection scheme. A low frequency accurrent of angular frequency ω (typical frequencies areω/(2π) < 20 Hz, and the typical amplitude is I = 100µArms) is sent through the injector strip, and the voltageon the detector strip is measured at both the frequenciesω (the first harmonic response) and 2ω (the second har-monic response). This allows us to separate processesthat are linear in the current, which govern the firstharmonic response, from processes that are quadratic inthe current which are measured in the second harmonicresponse19,28,47.

The measurements in Sendai were carried out in aQuantum Design Physical Properties Measurement Sys-tem (PPMS), using a superconducting solenoid to applythe external magnetic field (field range up to µ0H =±10.5 T). The measurements in Groningen were carriedout in a cryostat equipped with a Cryogenics Limitedvariable temperature insert (VTI) and superconductingsolenoid (magnetic field range up to µ0H = ±7.5 T).Electronic measurements in Groningen are carried out us-ing a home built current source and voltage pre-amplifier(gain 104) module galvanically isolated from the rest ofthe measurement electronics, resulting in a noise levelof approximately 3 nVr.m.s. at the output of the lockinamplifier for a time constant of τ = 3 s and a filterslope of 24 dB/octave. The electronic measurements inSendai were carried out by means of an ac and dc cur-rent source (Keithley model 6221) and a lockin amplifierusing a time constant of τ = 1 s and a filter slope of24 dB/octave. The data shown in Figure 1b and Fig-ure 3 is the asymmetric part of the measured voltagewith respect to the magnetic field. The antisymmetriza-tion procedure includes both the forward and backwardmagnetic field sweep, and the voltage shown in the fig-ures is given by VH+ = (Vbackward(H)−Vbackward(−H))/2and VH− = (Vforward(H) − Vforward(−H))/2, where VH+is the voltage at postive magnetic field values and VH−that at negative magnetic field values.

Simulations. The two-dimensional finite elementmodel is implemented in COMSOL MultiPhysics (v4.4).The linear response relation of heat and spin transportin the bulk of a magnetic insulator reads(

2e~ jmjQ

)= −

(σm ζ/T

~ζ/2e κ

)(∇µm∇T

), (1)

where jm is the magnon spin current, jQ the total(magnon and phonon) heat current, µm the magnonchemical potential, T the temperature (assumed to bethe same for magnons and phonons by efficient thermal-ization), σm the magnon spin conductivity, κ the total

(magnon and phonon) heat conductivity and ζ the spinSeebeck coefficient. We disregard temperature differ-ences arising from the Kapitza resistances at the Pt|YIGor YIG|GGG interfaces. −e is the electron charge and ~the reduced Planck constant. The diffusion equations forspin and heat read

∇2µm =µmλ2m

, (2)

∇2T = j2c

κσ, (3)

where jc is the charge current density in the injector con-tact, σ and κ the electrical and thermal conductivity andλm the magnon spin diffusion length. Eq. (3) representsthe Joule heating in the injector that drives the SSE.

In the simulations, tYIG = 2.5 µm and wYIG = 500 µmare the thickness and width of the YIG film, on top ofa GGG substrate that is 500 µm thick. wYIG is muchlarger than λm and finite size effects are absent. Theinjector has a thickness of tPt = 10 nm and a widthof wPt = 300 nm. The spin and heat currents normalto the YIG|vacuum, Pt|vacuum and GGG|vacuum inter-faces vanish. At the bottom of the GGG substrate theboundary condition T = T0 is used, i.e. the bottom ofthe sample is taken to be thermally anchored to the sam-ple probe. Furthermore, a spin current is not allowed toflow into the GGG. The spin current across the Pt|YIGinterface is given by jintm = gs (µs − µm), where gs is theeffective spin conductance of the interface, µs is the spinaccumulation on the metal side of the interface and µmis the magnon chemical potential on the YIG side of theinterface. The nonlocal voltage is then found by calculat-ing the average spin current density 〈js〉 flowing in thedetector, which is then converted to non-local voltageusing VnlSSE = θSHL〈js〉/σ, where θSH is the spin Hallangle in platinum and L is the length of the detectorstrip. The spin current in the platinum contact relaxesover the characteristic spin relaxation length λs.

The parameters we use for platinum in the model areθSH = 0.11, σ = 1.9 × 106 S/m, λs = 1.5 nm and κ =26 W/(m K). For YIG, we use σm = 3.7 × 105 S/m,λm = 9.4 µm which was obtained in our previous work

23.Furthermore, we use κ = 7 W/(m K), based on YIGthermal conductivity data from Ref. 39. For the bulkspin Seebeck coefficient at zero field we use ζ0 = 500A/m, based on our previous work in which we gave anestimate for ζ at room temperature33. For GGG, the spinconductivity and spin Seebeck coefficient are set to zero.For the GGG thermal conductivity we use κ = 9 W/(mK), based on data from Refs. 38 and 43. Finally, for theeffective spin conductance of the interface we use gs =3.4×1011 S/m2. We note that this is roughly a factor 30smaller than in our earlier work23. This variation of theinterface transparency in different experiments indicatesthe presence of physical processes that are not taken intoaccount in the modeling.

8

IV. ACKNOWLEDGEMENTS

We thank H. M. de Roosz, J.G. Holstein, H. Ademaand T.J. Schouten for technical assistance and R.A.Duine, B. Flebus and K. Shen for discussions. Thiswork is part of the research program of the Nether-lands Organization for Scientific Research (NWO) andsupported by NanoLab NL, EU FP7 ICT Grant No.612759 InSpin, the Zernike Institute for Advanced Mate-rials, Grant-in-Aid for Scientific Research on InnovativeArea ”Nano Spin Conversion Science” (Nos. JP26103005and JP26103006), Grant-in-Aid for Scientific Research(A) (No. JP25247056) and (S) (No. JP25220910) fromJSPS KAKENHI, Japan, and ERATO ”Spin QuantumRectification Project” (No. JPMJER1402) from JST,Japan. Further support by the DFG priority programSpin Caloric Transport (SPP 1538, KU3271/1-1) is grate-

fully acknowledged. K.O. acknowledges support fromGP-Spin at Tohoku University. T.Ki. is supported byJSPS through a research fellowship for young scientists(No. JP15J08026).

V. AUTHOR CONTRIBUTIONS

B.J.v.W., L.J.C., T.Ki. and E.S. conceived the ex-periments. Z.Q. fabricated the Sendai YIG films. K.O.and L.J.C. fabricated the nonlocal devices in Sendai andGroningen, respectively. K.O. and L.J.C. performedthe experiments. T.Ki. supervised the experiments inSendai. K.O., L.J.C., T.Ki., T.Ku., G.E.W.B. and E.S.analyzed and interpreted the data. L.J.C. performed thenumerical modelling. L.J.C., T.Ku. and G.E.W.B. in-terpreted the modelling results. L.J.C. wrote the paper,with the help of all co-authors.

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FIG. 1. Experimental geometries and main results. Figure a is an image of a typical device, with schematic currentand voltage connections. The three parallel lines are the Pt injector/detector strips, connected by Ti/Au contacts. α is theangle between the Pt strips and an applied magnetic field H (in b-d α = 90◦). b The nonlocal spin Seebeck (nlSSE) voltagefor an injector-detector distance d = 6 µm (top) and d = 2 µm (bottom) as a function of µ0H. At |µ0H| = |µ0HTA| ≈ 2.3T, a resonant structure is observed that we interpret in terms of magnon-polaron formation (indicated by blue triangles as aguide to the eye). The right column is a close-up of the anomalies for H > 0. The results can be summarized by the voltagesV 0nlSSE and VTA as indicated in the lower panels. c Schematic geometry of the local heater-induced hiSSE measurements. Herethe temperature gradient ∇T is applied by external Peltier elements on the top and bottom of the sample. d The hiSSEvoltage measured as a function of magnetic field. The close-up around the resonance field (right column) focusses on themagnon-polaron anomaly. All results were obtained at T = 200 K. The results for d = 6, d = 2 and d = 0 µm were obtainedfrom sample S1, S2, S3, respectively (see Methods for sample details).

11

FIG. 2. Nonlocal voltage due to electrically and thermally generated magnons as a function of magnetic field.Figure a shows the nonlocal voltage generated by magnons that are excited electrically (first harmonic response to an oscillatingcurrent in the injector contact). An anomaly is observed atH = |HTA| (the field that satisfies the touching condition for magnonsand transverse acoustic phonons). The inset shows a second set of data from the same sample, taken with a higher magneticfield resolution (µ0∆H = 15 mT), sweeping the magnetic field both in the forward (black) and backward (red) directions.Figure b shows the nlSSE voltage (second harmonic response) for the same device. VnlSSE is suppressed at H = |HTA|. Theinset shows the corresponding second harmonic data of the high resolution field sweep. The results were obtained on sampleG1 (thickness 210 nm) with d = 3.5µm and I = 150µAr.m.s., at room temperature. A constant background voltage Vbg = 575nV was subtracted from the data in Fig. a.

12

Temperature dependence for d = 2 μmDistance dependence for T = 300 Ka bnonlocal sign = local sign

FIG. 3. VnlSSE vs magnetic field as a function of distance and temperature. Figure a is a plot of VnlSSE vs H forvarious injector-detector separations at T = 300 K, while Figure b shows VnlSSE vs H for different temperatures and d = 2µm. The data in Figs. a and b are from sample S1 and S2, respectively. The magnon-polaron resonance is indicated by theblue arrows. The blue shading in the graphs indicates the region in which the sign of the nlSSE signal agrees with that of thehiSSE. The right column in both a and b shows close-ups of the data around the positive resonance field (blue triangles). Thedata in the close-ups has been antisymmetrized with respect to H, i.e. V = (V (+H)−V (−H))/2. Fig. a shows that when thecontacts are close (d ≤ 2 µm), the magnon-polaron resonance enhances VnlSSE, while for long distances VnlSSE is suppressed atthe resonance magnetic field. For very large distances (d ≥ 20 µm), the resonance cannot be observed anymore. Similarly inFig. b, for temperatures T ≥ 180 K, the magnon-polaron resonance enhances the nlSSE signal, while for lower temperaturesthe nlSSE signal is suppressed. The excitation current I = 100 µAr.m.s. for all measurements.

13

FIG. 4. Temperature dependence of V 0nlSSE, VTA and HTA. a displays the temperature dependence of the low-field V0nlSSE,

for d = 2 µm and d = 6 µm. For 2 µm, the signal changes sign around T = 143 K. The blue shading in the graph indicates theregime in which the sign agrees with that of the hiSSE. The temperature dependence of the magnon-polaron resonance VTA isshown in Figure b. Here, no sign change but a minimum around T = 50 K is observed, which is absent in Figure a. Figurec shows the temperature dependence of the resonance field HTA. Error bars in b and c reflect the peak-to-peak noise in thedata used to extract VTA and the step size in the magnetic field scans (µ0∆H = 20 mT), respectively.

FIG. 5. Physical concepts underlying the nlSSE signal and simulated magnon chemical potential profile. Figurea sketches the effects of Joule heating in the injector, heating it up to temperature TH , which leads to a thermal gradient in theYIG. The bulk SSE generates a magnon current JmQ antiparallel to the local temperature gradient, spreading into the film awayfrom the contact. When the spin conductance of the contact is sufficiently small, this leads to a depletion of magnons belowthe injector, indicated in Figure b as µ−. When the magnons are reflected at the GGG interface, JmQ accumulates magnons at

the YIG|GGG interface, shown in Figure b as µ+. The chemical potential gradient induces a backward and sideward diffusemagnon current Jmd . Both processes in Figure a and b are included in the finite element model (FEM). Its results are plottedin Figure c in terms of a typical magnon chemical potential profile. µm changes sign at some distance from the injector, also atthe YIG surface, where it can be detected by a second contact. The magnon-polaron resonance enhances both the spin Seebeckcoefficient ζ and the magnon spin conductivity σm. The increased backflow of magnons to the injector causes a suppression ofthe nonlocal signal at long distances (see Figure 6).

14

FIG. 6. Comparison of the experimental and simulated V 0nlSSE and VTA. Figure a shows the distance dependence ofV 0nlSSE and VTA (inset) measured at room temperature. The dashed line in the inset is an exponential fit to the data. V

0nlSSE

changes sign around d = 4µm, while VTA remains positive. Figure b is a plot of the calculated distance dependence of V0nlSSE

at zero magnetic field (red) and at the resonant field for δ = 2 (green) and δ = 0.5 (purple). Here δ is a parameter thatmeasures the relative enhancement of the spin Seebeck coefficient compared to the magnon spin conductivity, as explained inthe main text. The inset shows the signal decay at long distances on a logarithmic scale. Figure c shows the modelled distancedependence of VTA for various values of δ on a linear scale (inset for logarithmic scale). δ = 0.5 results in a positive sign forVTA over the full distance range with a slope that roughly agrees with experiments (cf. insets of Figure a and c). Reducing δfurther leads to a more gradual slope for VTA. In the simulations, the SSE enhancement is fζ = 1.09, while fσ is varied with δ.

Nonlocal magnon-polaron transport in yttrium iron garnetAbstractI ResultsA Sample characteristicsB Experimental resultsC ModellingD Comparison between model and experiment

II DiscussionIII MethodsIV AcknowledgementsV Author contributions References