Temperature dependence of the non-local spin Seebeck effect in YIG/Pt nanostructures

Magnons, the collective excitations in magnetically ordered systems, represent an attractive option for information transfer and processing. Using the ferrimagnetic insulator Yttrium Iron Garnet (YIG) as a model system, magnon-based information processing schemes have been put forward on the basis of coherently excited spin waves.^1–4 Recent experiments in YIG/Pt heterostructures furthermore show that information can also be carried by incoherent non-equilibrium magnons^5,6 diffusing in YIG. This approach even allows for the implementation of logic operations within the magnetic system.⁷ The non-equilibrium magnons can be excited and detected electrically via spin scattering mechanisms at the YIG/Pt interface. In a scheme referred to as magnon-mediated magnetoresistance (MMR), magnons are generated by driving a dc charge current through an injector Pt strip and detected as a non-local voltage in a second strip. The MMR effect has been studied as a function of the distance d between injector and detector,⁵ temperature⁶ and magnetic field magnitude and orientation,⁸ allowing for the extraction of the length scales involved in the magnon diffusion process. In addition to the electrical injection, non-equilibrium magnons can also be generated thermally, via local Joule heating in the injector strip. The ensuing thermal non-local voltage is called non-local spin Seebeck effect in analogy to the well established local (longitudinal) spin Seebeck effect (SSE).⁹ While the microscopic mechanisms and in particular the relevant length scales for the SSE have been investigated in quite some detail,^10,11 the physics behind the non-local thermal signal is not well established. In this context, the non-local SSE has recently been studied at room temperature as a function of strip separation d and YIG thickness.¹² While at short distances d the local and the non-local SSE signals have the same sign, for larger distances the non-local SSE amplitude is inverted, which was attributed to the profile of the non-equilibrium magnon distribution in the YIG film.

In this article, we systematically study the non-local spin Seebeck effect in YIG/Pt nanostructures as a function of temperature and strip separation and find that the non-local SSE voltage changes sign at a characteristic strip separation d₀, which is strongly temperature dependent. We interpret our findings as evidence of a complex interplay between the temperature dependencies of the interfacial transparency, i.e. angular momentum transfer across the YIG/Pt interface, and the diffusive properties of the thermally excited non-equilibrium magnons.

We investigate the non-local spin Seebeck effect in YIG/Pt bilayers fabricated and nano-patterned at the Walther-Meißner-Institut (sample series A) and at Johannes Gutenberg-University Mainz (sample series B). Series A was fabricated starting from a commercially available 2 μm thick YIG film grown onto (111) oriented Gd₃Ga₅O₁₂ (GGG) via liquid phase epitaxy (LPE). After Piranha cleaning and annealing (see Ref. 13 for details) to improve the interface quality, 10 nm of Pt were deposited onto the YIG film using electron beam evaporation. A series of nanostructures consisting of 2 parallel Pt strips with length $l=100μm$⁠, width $w=500nm$ and an edge-to-edge separation of $20nm≤d≤10μm$ were patterned using electron beam lithography followed by Ar ion etching. Series B was fabricated using a 3.3 μm thick LPE-YIG film grown onto a GGG substrate as well. A series of strips with width $w=1μm$ and edge-to-edge distance $100nm≤d≤1.2μm$ were patterned using electron beam lithography followed by a lift-off process with a Pt thickness of 7.5 nm deposited by dc sputtering. An optical micrograph of one of these nanostructures is depicted in Fig. 1 (a). Figure 1 (b) is a schematic representation of the YIG/Pt bilayer (not to scale). The color coding in the YIG film qualitatively represents the magnon accumulation profile as proposed in Ref. 12, where red/blue corresponds to a magnon depletion/accumulation with respect to the local thermal equilibrium value. Two detectors are exemplarily shown for different injector/detector separations d, i.e. above an area of magnon depletion or accumulation for detector 1 and 2, respectively. The implications of the magnon accumulation profile for the detected non-local thermal signal are discussed in the following.

In order to study the devices in series A and B, the samples were mounted in the variable temperature insert of a superconducting magnet cryostat (⁠$10K≤T≤300K$⁠). For series A an external magnetic field $μ0H=1T$ was rotated in the thin film plane, while for series B the external magnetic field was applied along the y-direction and swept from $−250mT$ to $+250mT$ (⁠$α=90°$⁠, $270°$ in Fig. 1 (b)). For local longitudinal spin Seebeck effect measurements in one single (injector) strip, we used the current heating method described in Ref. 14: a charge current $Jc=100μA$ is applied to the Pt strip along the x direction using a Keithley 2400 source meter, inducing Joule heating in the normal metal. The ensuing temperature gradient across the Pt/YIG interface gives rise to the spin Seebeck effect and generates a spin current J_s flowing across the interface, with the spin current spin polarization s determined by the orientation of the magnetization M in YIG. This spin current is accompanied by a charge current J_ISHE flowing along the x-direction in the Pt, as shown in Fig. 1 (b). The voltage drop V_loc, which includes the local SSE and the resistive response of the injector strip is recorded in a 4-point measurement configuration using a Keithley 2182 nanovoltmeter. Since the spin Seebeck effect is of thermal origin, the SSE voltage is proportional to the Joule heating power in the Pt and therefore independent of the heating current direction. Using the switching scheme of Ref. 14, we extract V_therm,loc = (V_loc(+J_c) + V_loc(−J_c))/2, and thereby eliminate additional resistive effects such as the spin Hall magnetoresistance. Note that we do not consider contributions from the unidirectional SMR,¹⁵ since to our knowledge there is no experimental evidence of this effect in ferrimagnetic insulator/normal metal bilayers to date.

Figure 2 (a) shows V_therm,loc measured as a function of the magnetic field orientation $α$ with respect to the x-axis at 50 K for a device from series A. We observe the characteristic SSE dependence $Vtherm,loc∝sin(α)$ yielding a positive amplitude $ASSE,loc=Vtherm,loc(90°)−Vtherm,loc(270°)$ of the local SSE, as expected in YIG/Pt heterostructures for this field configuration.¹⁶ 

Using an additional nanovoltmeter, we simultaneously measure the voltage drop V_nl arising along the unbiased and electrically isolated second Pt strip. In analogy to the local thermal signal, the non-local thermal voltage is extracted as V_therm,nl = (V_nl(+J_c) + V_nl(−J_c))/2 in order to distinguish it from resistive non-local effects such as the magnon mediated magnetoresistance.⁶ V_therm,nl as a function of the external magnetic field orientation at 50 K is depicted in Fig. 2 (b) and (c) for strip separations of $d=200nm$ and $2μm$ (series A). In both devices we observe a sin$(α)$ dependence, with an amplitude A_SSE,nl about one order of magnitude smaller than for the local SSE. In the following we do not discuss the absolute amplitude of the non-local thermal signal in detail, but focus on the sign of A_SSE,nl as a function of separation between the Pt strips. While the signal is positive for the $d=200nm$ device, a negative A_SSE,nl is observed in the device with a larger injector-detector separation of $d=2μm$⁠. In order to confirm this sign change, we extract the amplitude of the non-local SSE measured at 50 K in different devices with strip separations ranging from $20nm$ to $10μm$⁠. The resulting data is shown in Fig. 2 (d) as green symbols. Indeed, a sign change is observed at a strip separation $d0≈560nm$⁠. Repeating these measurements as a function of temperature in the range between 10 K and 300 K yields the data compiled in Fig. 2 (d). For all temperatures, a sign change in A_SSE,nl is observed as a function of the strip separation. Invariably, for small gaps the local and non-local SSE are both positive, but for large gaps the non-local SSE becomes negative. The experimental data in Fig. 2 (d) show that the critical strip separation d₀, which is defined by A_SSE = 0, shifts to larger values as the temperature increases. The values d₀ extracted from Fig. 2 for different temperatures are shown in Fig. 3 as red symbols for sample series A. With increasing temperature d₀ increases monotonically and seems to saturate around $T=200K$⁠.

Similar experiments as a function of temperature and strip separation were performed on devices from series B and the critical strip separation extracted from these measurements is included in Fig. 3 as blue squares. While the temperature dependence is much steeper, a qualitatively similar increase of d₀ with temperature is observed in both series.

This characteristic sign change in the non-local SSE in YIG/Pt heterostructures above a particular separation d₀ has been previously observed by Shan et al.¹² at room temperature and was attributed to the spatial profile of the non-equilibrium magnon accumulation $μm$ in the YIG film, as shown schematically in Fig. 1 (b). Magnons are thermally excited in the ferrimagnet due to Joule heating in the injector strip and diffuse vertically towards the GGG/YIG interface as well as laterally to the sample edges. According to the model proposed by Shan et al.,¹² this leads to a depletion of magnons (⁠$μm<0$⁠, red in Fig. 1 (b)) compared to the thermal equilibrium population beneath the injector. On the other hand, diffusing magnons accumulate further away from the injector, giving rise to $μm>0$ (blue in Fig. 1 (b)). This model is applied to describe the increase of d₀ with increasing YIG thickness observed by Shan et al.:¹² in contrast to phonons, the magnons cannot cross the YIG/GGG interface and accumulate there. As a consequence, d₀ (which marks the sign change of $μm$⁠) shifts to smaller values for thinner YIG films. Since the overall profile of $μm$ is governed by diffusive magnon transport, the corresponding length scales can reach several μm.¹² Note that while the magnon accumulation profile is affected by changes in the magnon diffusion length $λm$⁠, d₀ is not directly proportional to $λm$ but exhibits a more complex behavior as discussed in the following. Furthermore, we expect d₀ to be independent of the Joule heating power to a first approximation, since in linear response the latter changes only the absolute number of non-equilibrium magnons which are thermally generated at the injector. The boundary conditions at the Pt/YIG and YIG/GGG interfaces, however, determine the qualitative profile of the magnon accumulation and remain independent of the heating power.

As shown in Fig. 1 (b), the sign of $μm$ determines the direction of the interfacial spin current J_s at the detector, i.e. towards (away from) the YIG for negative (positive) $μm$ at detector 1 (detector 2), and consequently governs the sign of the measured non-local ISHE voltage. Non-local SSE measurements as a function of the strip separation therefore allow us to map out the non-equilibrium magnon distribution in the YIG film. In particular, the characteristic length d₀ for the sign change of $μm$ can be determined.

In order to rationalize the measured temperature dependence of d₀, the parameters governing the angular momentum transfer across the YIG/Pt interface as well as the magnon diffusion process need to be analyzed. It has been shown that the transparency of the YIG/Pt interface, described by the effective spin-mixing conductance g_s, influences the magnon accumulation and hence the sign-reversal distance d₀.¹² For a fully opaque interface (obtained using an Al₂O₃ interlayer) which suppresses angular momentum backflow into the injector and therefore preserves a strong magnon depletion, an increase of the sign-reversal distance d₀ was observed.¹² Previous measurements of the MMR effect in a YIG/Pt heterostructure - prepared using exactly the same fabrication method as for sample series A - as a function of temperature have shown that the MMR signal decreases with decreasing temperature,⁶ consistent with $gs∝T3/2$ as predicted by theory.^17,18 A similar behavior was observed in a Pt/YIG/Pt heterostructure which is based on dc sputtered Pt patterned by a lift-off process.¹⁹ We therefore assume a qualitatively similar temperature dependence of the interfacial transparency for evaporated (series A) and sputtered Pt (series B). However, with this temperature dependence we expect a decreasing transparency of the YIG/Pt interface with decreasing temperature, leading to an increase of d₀ at low temperatures according to the model presented in Ref. 12. Since this is not consistent with our experimental observations depicted in Fig. 3, the interface properties alone are not sufficient to describe the temperature dependence of the non-local SSE.

In addition to the interfacial transparency, the magnon diffusion length $λm$ and the magnon spin conductivity $σm$ determine the spatial distribution of the non-equilibrium magnons in YIG. We extracted the magnon diffusion length from temperature dependent MMR measurements conducted in the sample series A and found an increase of $λm$ with decreasing temperature by about a factor of 3 between 300 K and 50 K, following a 1/T dependence. This is different from the temperature independent diffusion length reported by Cornelissen et al.,²⁰ who extracted $λm(T)=const.$ together with a magnon spin conductivity $σm$ vanishing at low temperatures. While the detailed evolution of $λm$ and $σm$ with T thus must be studied more systematically in future work, it is clear that these quantities depend on temperature. This implies that they can qualitatively impact the magnon diffusion process and consequently the non-local SSE. Indeed, the strong dependence of d₀ on the relative amplitudes of g_s, $λm$ and $σm$ at a fixed temperature has been demonstrated by Shan et al. using a one-dimensional analytical model for the spin Seebeck effect.¹² 

Based on the available experimental data, we can conclude that for a quantitative modeling of the non-local SSE the temperature dependence of both the angular momentum transfer across the YIG/Pt interface and the magnon diffusion in YIG must be taken into account. We furthermore note that additional effects due to phononic heat transport, known to be of importance for the local SSE, cannot fully be excluded at this point as a source of the non-local SSE-like signal. The local SSE originates from a finite difference of the effective temperature of the magnon and phonon subsystems in YIG close to the YIG/Pt interface, giving rise to a magnon spin current.²¹ While this difference may also influence the thermal signal measured at the non-local detector strip, it was shown that the thermalization of the magnon and phonon subsystems takes place on a length scale $λmp$ of the order of a few nm at room temperature,^22–24 due to a very efficient (magnon conserving) magnon-phonon scattering. The effective temperature model described in Ref. 21 is therefore applicable mainly in the local limit, i.e. close to the injector. The long-distance non-local limit, however, will be dominated by diffusing magnons and the magnon-phonon scattering which does not conserve the number of magnons and can reach a larger length scale $λm$ of the order of several $μ$m,¹⁸ as discussed above. Further contributions from phonons, i.e. an enhancement of the SSE signal due to phonon drag^25,26 might also play a role for the magnon accumulation profile (see supplementary material).

In summary, we have measured the non-local SSE in a YIG/Pt heterostructure as a function of injector-detector distance and temperature. The non-local SSE changes sign at a characteristic injector-detector separation d₀, consistent with previous observations put forward by Shan et al.¹² We furthermore observe a decrease of the characteristic separation d₀ with decreasing temperature, which was confirmed by the independent study of two sample series in different setups. Our results suggest a complex dependence of the non-local SSE on interfacial transparency, magnon diffusion properties as well as phonon heat transport.

Note added. During the review process of this paper, we have become aware that Zhou et al.²⁷ reported similar results on the temperature dependence of the non-local SSE sign change.