Magnetic field dependence of the nonlocal spin Seebeck effect in Pt/YIG/Pt systems at low temperatures

Spin caloritronics¹ is an emerging field to study the interconversion between spin and heat currents. The spin Seebeck effect (SSE) is one of the fundamental phenomena in this field, referring to the spin-current generation from a heat current. The SSE is well studied in a longitudinal configuration,^2,3 which consists of a heavy metal(HM)/ferromagnet(FM) bilayer system, typically a Pt/Y₃Fe₅O₁₂(YIG) junction. When a thermal gradient is applied perpendicular to the interface, a magnon spin current is generated in the FM and converted into a conduction-electron spin current in the HM via the interfacial exchange interaction,⁴ which is subsequently detected as a transverse electric voltage via the inverse spin Hall effect (ISHE).^5,6 Recent studies of the longitudinal SSE (LSSE)^7–9 suggest that magnon transport in the FM plays a key role in SSEs.

A nonlocal experiment is a powerful tool to investigate the transport of spin currents in various magnetic insulators.^10–21 In such experiments, when a spin current is excited via a thermal gradient, it is called a nonlocal SSE (nlSSE).^10,11 A typical nonlocal device consists of two HM wires on top of a magnetic insulator, which are electrically separated with the distance d. In nlSSE measurements, one of the HM wires is used as a heater; the Joule heating of an applied charge current (I) drives magnon spin currents in the magnetic insulator. Some of the magnons reach the other HM wire and inject a spin current, which is converted into a voltage via the ISHE. By changing the injector-detector separation distance d, we can address the transport property of magnon spin currents.

In this paper, we report the high magnetic field (B) dependence of the nlSSE in lateral Pt/YIG/Pt systems at various temperatures from T = 300 K to 3 K and up to |B| = 10 T. We observed that the nlSSE signal V_2ω decreases with increasing B, but the feature turns out to depend on the amplitude of the applied I. In a linear regime (V_2ω ∝ I²), substantial B-induced suppression of the nlSSE was observed below 10 K, which is consistent with the previous LSSE results in Pt/YIG.^8,22 In a nonlinear regime (V_2ω $̸∝$ I²), however, the nlSSE signal remains almost unchanged under high B at low Ts. By measuring the d dependence, we estimate the magnon diffusion length λ as functions of T and B.

We prepared a series of nonlocal Pt/YIG/Pt devices, schematically shown in Fig. 1(a). A 2.5-μm-thick YIG film was grown by liquid phase epitaxy on a Gd₃Ga₅O₁₂ (111) substrate.²³ On top of the YIG film, we fabricated two Pt wires using e-beam lithography and the lift-off process.^16,20 The dimensions of the Pt wires are 200 μm length, 100 nm width, and 10 nm thickness. The Pt wires were deposited by magnetron sputtering in Ar⁺ atmosphere. We investigated four batches of samples (S1–S4) cut from the same YIG wafer. The d dependence was studied in S1 (d = 5, 6, 8, 13, 15, and 20 μm) and S2 (d = 9 μm), while the I dependence at low Ts in S3 (d = 2 μm) and S4 (d = 8 μm). We measured a nonlocal voltage using a lock-in detection technique; we applied an ac charge current, I, of 13.423 Hz in frequency to the injector Pt wire and measured a second harmonic nonlocal voltage V_2ω across the detector Pt wire.¹¹ 

First, we confirmed that the obtained nonlocal voltage satisfies the features of the nlSSE at room temperature. Figure 1(b) shows typical V_2ω as a function of in-plane B (θ = 90°) at 300 K in the d = 9 μm sample. A clear V_2ω appears, whose sign changes with respect to the B direction. V_2ω disappears when B is applied perpendicular to the plane (θ = 0). This symmetry is consistent with that of the SSE.³ We define the low-field amplitude of the voltage signal as V_nlSSE = [V_2ω(0.18 T) − V_2ω(−0.18 T)]/2, at which the magnetization of YIG is fully saturated along B. As shown in Fig. 1(c), V_nlSSE is proportional to I², indicating that V_nlSSE appears due to Joule heating. With increasing B, V_2ω gradually decreases, and at around ±2.2 T, sharp dip structures show up, which are induced by magnon polarons due to magnon–TA-phonon hybridization.^16,24,25

By changing the injector-detector separation distance d, we estimate the length scale of the magnon spin current.²⁶ As shown in Fig. 1(d), V_nlSSE decreases with increasing d. A one-dimensional spin diffusion model^11,27 describes the decay, which reads

where λ is the magnon spin diffusion length and C is the d-independent constant. We fit Eq. (1) to the d dependence of V_nlSSE and obtain λ = 6.76 ± 0.16 μm at 300 K. Similar values are reported in previous studies in both thin (200 nm)¹¹ and thick (50 μm)²⁸ YIG films.

Next, we measured the T dependence of V_2ω with I = 100 μA. As shown in Fig. 2(a), at 300 K, negative voltages are observed for the d = 0.5 and 1.5 μm samples, while the positive ones show up for the d = 8 and 15 μm samples. With decreasing T, the d = 8 and 15 μm samples exhibit a monotonic increase in V_nlSSE. On the other hand, with decreasing T, the negative voltages observed for the d = 0.5 and 1.5 μm samples at 300 K change their sign at several tens of kelvin. The sign change of V_nlSSE with changing d and T has been observed in previous nlSSE experiments and explained as a result of a spatial profile of the magnon chemical potential μ_m that governs the sign and amplitude of V_nlSSE; a negative μ_m created beneath the Pt injector exponentially decays apart from the injector and above a certain distance a positive one manifests due to the presence of YIG/GGG interface. The overall μ_m profile varies with T.^{14–17,26–28} Furthermore, we found a second sign change for the d = 0.5 μm sample at 3 K, which is unclear at this moment.

We now focus on the magnetic field B dependent features of V_2ω. Figure 2(b) shows the B dependence of λ at various Ts obtained by fitting Eq. (1) to V_nlSSE(d). At 300 K, λ decreases with increasing B by 30% up to 3 T [from λ = 6.8 μm at B = 0.18 T to 4 μm at 3 T, see blue filled circles in Fig. 2(b)]. A similar field-induced decrease in λ has been observed in the time-resolved LSSE,²⁹ nlSSE,³⁰ and electrically excited magnon transport experiment³⁰ at room temperature. On the other hand, at lower Ts, λ was found to be less sensitive to B [see Fig. 2(b)].

To further investigate the effect of high B on the nlSSE, we applied larger magnetic fields up to 10 T. Figure 2(c) shows a typical V_2ω–B result for |B| < 10 T in the d = 1.5 μm sample with I = 100 μA at 300 K. High B-induced suppression of V_2ω is clearly observed. In Fig. 2(d), we plot the degree of B-induced V_2ω suppression up to 8 T, defined as

as a function of T for the d = 0.5, 1.5, 8.0, and 15 μm samples. At 300 K, all the samples show the substantial high B-induced V_2ω reduction: 65% < δ < 75% for the d = 1.5, 8.0, and 15 μm samples and δ = 39% for the d = 0.5 μm sample. For the d = 8.0 and 15 μm samples, with decreasing T, δ gradually decreases in the range of 20 K < T < 300 K and slightly increases below 20 K. For the d = 0.5 and 1.5 μm samples, more complicated T dependences were observed, which may be related to the nonmonotonic T responses of V_2ω as shown in Fig. 2(a). The T–δ behavior above 20 K for the d = 8.0 and 15 μm samples qualitatively agrees with the previous LSSE result in Pt/YIG-bulk systems.^8,31 However, below 20 K, the present nlSSE and previous LSSE results are totally different; δ of the LSSE becomes more outstanding with decreasing T and reaches δ ∼ 100% at ∼3 K,²² much greater than the present nlSSE results.

Significantly, we found that the disagreement at low temperatures is relevant to the applied current intensity I. So far, the nlSSE experiments were carried out with I = 100 μA. Below 20 K, however, V_2ω turned out to deviate from the I² scaling in this I range. To see this, we introduce the normalization factor,

If V_2ω is proportional to I², S keeps a constant with I, which was indeed confirmed above 20 K for I < 100 μA. Figure 3(a) shows the I dependence of S at 3 K at the low B of 0.18 T for the d = 2, 8, and 9 μm samples. S takes almost the same value for I ≲ 5 μA [see the gray colored area in Fig. 3(a)], but for I ≳ 5 μA, S decreases with increasing I. We refer the former region to the linear regime (V_2ω ∝ I²), while the latter refers to the nonlinear regime (V_2ω$̸∝$I²). In Fig. 3(b), we plot the T dependence of S in the linear and nonlinear regimes at B = 0.5 T for the d = 8 μm sample. The difference in S between the linear and nonlinear regimes becomes significant with decreasing T, and at 3 K, S in the linear regime is about 4 times greater than that in the nonlinear regime. Importantly, the B dependence of V_2ω and δ also varies between the linear and nonlinear regimes. In Fig. 3(c), we show representative results on V_2ω vs B with several I values at 3 K for the d = 2.0 μm sample. In the linear regime (for I = 3 μA), clear B-induced V_2ω suppression was observed (δ = 78%). By increasing I and entering into the nonlinear regime, however, the B-induced V_2ω reduction becomes less visible and, when I = 100 μA, V_2ω is almost flat against B (δ = −0.1%). In Fig. 3(d), we summarize the δ values as a function of T obtained in the linear (red filled circles) and nonlinear (blue filled circles) regimes for the d = 8.0 μm sample and compare them to the previous LSSE result (gray filled triangles).²² Interestingly, the T dependence of δ for the nlSSE agrees well with that for the LSSE.

The matching of the T–δ results in the low-T range between the nlSSE in the linear regime and the LSSE indicates that the same mechanism governs the B-induced suppression. In Refs. 8 and 22, the T dependence of δ for the LSSE at low Ts was well reproduced based on a conventional LSSE theory in which the effect of the Zeeman-gap opening in a magnon dispersion (∝gμ_BB, where g is the g-factor and μ_B is the Bohr magneton) was taken into account; the competition between thermal occupation of the magnon mode relevant for the LSSE (whose energy is of the order of k_BT) and the Zeeman gap (gμ_BB) dominates the B-induced LSSE reduction. When k_BT ≪ gμ_BB (≈10 K at 8 T), magnons cannot be thermally excited, leading to the suppression of the LSSE [see Fig. 3(d)]. Our results indicate that the same scenario is valid also for the nlSSE in the linear regime.

Finally, we discuss the nonlinear feature of the nlSSE. Both the S and δ values of the nlSSE in the nonlinear regime gradually increase with decreasing T [see Figs. 3(b) and 3(d)]. However, their increasing rates are much smaller than those for the linear regime; both S and δ at 3 K in the nonlinear regime are ∼4 times smaller than those at the same T for the linear regime and also comparable to those at 12 K for the linear regime. These results suggest that the energy scale of magnons driving the nlSSE in the nonlinear regime at 3 K may be much higher than the thermal energy k_BT at 3 K and the Zeeman energy gμ_BB at 8 T. We note that, in the nonlinear regime, the system temperature at least remains unchanged during the measurements, indicating that the temperature rise due to Joule heating is negligible. Furthermore, we found that, in the nonlinear regime of I = 100 μA, the intensity of magnon-polaron dips at 3 K at B = 2.5 T (9.2 T) is smaller (larger) than that in the linear regime of I = 3 μA at the same T [see the dip structures marked by blue (red) triangles in Fig. 3(c)]. Here, the dip at the low B (high B) originates from the spin currents carried by hybridized magnon–TA-phonon (magnon–LA-phonon) modes with the fixed energy of E_MTA ≈ 6 K (E_MLA ≈ 26 K). The dip intensity should thereby be maximized when the magnon mode at the energy of E_MTA (E_MLA) is most significantly occupied under the condition of k_BT ≈ 6 K (26 K), and apart from this temperature, the intensity of the magnon-polaron dip decreases. Therefore, the small (large) magnon-polaron dip at B = 2.5 T (9.2 T) at 3 K in the nonlinear regime also indicates the over occupation of high-energy magnons than that expected from the thermal energy k_BT at 3 K, as with the S and δ results discussed above. Future work should address the origin of such high-nonequilibrium state realized in this regime.

In summary, we systematically investigated the nonlocal spin Seebeck effect (nlSSE) in the lateral Pt/YIG/Pt systems as functions of separation distance (d), magnetic field (B), temperature (T), and excitation current (I). We found that below 20 K, the nlSSE voltage V_2ω deviates from the conventional I² scaling for I ≳ 5 μA. In this nonlinear regime, the amplitudes of V_2ω and B-induced signal reduction δ become smaller than those in the linear regime, where V_2ω ∝ I² and I < 5 μA. In the linear regime, the T dependence of δ of the nlSSE agrees well with that of the longitudinal SSE (LSSE), which can be attributed to the suppression of magnon excitation by the Zeeman effect. Our results provide an important clue in unraveling the B-induced suppression of the nlSSE and useful information on the nonlinear effect in nonlocal spin transport at low temperatures.

We are thankful to G. E. W. Bauer, B. J. van Wees, L. J. Cornelissen, J. Shan, T. Kuschel, F. Casanova, J. M. Gomez-Perez, S. Takahashi, Z. Qiu, Y. Chen, and R. Yahiro for fruitful discussions and K. Nagase for technical help. This work was a part of the research program of ERATO Spin Quantum Rectification Project (No. JPMJER1402) from JST, the Grant-in-Aid for Scientific Research on Innovative Area Nano Spin Conversion Science (Grant No. JP26103005), the Grant-in-Aid for Scientific Research (S) (Grant No. JP19H05600), and the Grant-in-Aid for Research Activity Start-up (Grant No. JP19K21031) from JSPS KAKENHI, JSPS Core-to-Core Program, the International Research Center for New-Concept Spintronics Devices, World Premier International Research Center Initiative (WPI) from MEXT, Japan. K.O. acknowledges support from GP-Spin at Tohoku University.