Quantitative Temperature Dependence of Longitudinal Spin Seebeck Effect at High Temperatures

This article reports temperature-dependent measurements of longitudinal spin Seebeck effects (LSSEs) in Pt/Y$_3$Fe$_5$O$_{12}$ (YIG)/Pt systems in a high temperature range from room temperature to above the Curie temperature of YIG. The experimental results show that the magnitude of the LSSE voltage in the Pt/YIG/Pt systems rapidly decreases with increasing the temperature and disappears above the Curie temperature. The critical exponent of the LSSE voltage in the Pt/YIG/Pt systems at the Curie temperature was estimated to be 3, which is much greater than that for the magnetization curve of YIG. This difference highlights the fact that the mechanism of the LSSE cannot be explained in terms of simple static magnetic properties in YIG.


I. INTRODUCTION
The Seebeck effect converts a temperature difference into electric voltage in conductors [1]. Since the discovery of the Seebeck effect nearly 200 years ago, it has been studied intensively to realize simple and environmentallyfriendly energy-conversion technologies [2]. The Seebeck effect has been measured using various materials in a wide temperature range to investigate thermoelectric conversion performance and thermoelectric transport properties. Temperature-dependent measurements in a high temperature range are especially important in the investigation of the Seebeck effect, since thermoelectric devices are often used above room temperature [3,4].
The sample system for measuring the longitudinal SSE (LSSE) is a simple paramagnetic metal (PM)/ferrimagnetic insulator (FI) junction system. In many cases, Pt and Y 3 Fe 5 O 12 (YIG) are used as PM and FI, respectively. When a temperature gradient ∇T is applied to the PM/FI system perpendicular to the interface, the spin voltage is thermally generated and injects a spin current into the PM along the ∇T direction owing to thermal spin-pumping mechanism [38][39][40][41][42][43][44][45][46]. This thermally induced spin current is converted into an electric field E ISHE by the ISHE in the PM according to the relation where J s is the spatial direction of the thermally induced spin current and σ is the spin-polarization vector of electrons in the PM, which is parallel to the magnetization M of FI [see Fig. 1(a)]. By measuring E ISHE in the PM, one can detect the LSSE electrically.
In the experimental research on the SSE, temperaturedependent measurements also have been used for investigating various thermo-spin transport properties, such as phonon-mediated effects [15,16,39,47], correlation between the SSE and magnon excitation [29,48], and effects of metal-insulator phase transition [13]. However, all the experiments on the SSE to date have been performed around and below room temperature. In this article, we report quantitative temperature-dependent measurements of the LSSE in Pt/YIG systems in the high temperature range from room temperature to above the Curie temperature of YIG.

II. EXPERIMENTAL PROCEDURE
The sample system used in this study consists of a single-crystalline YIG slab covered with Pt films. One difference from conventional samples is that the Pt films  are put on both the top and bottom surfaces of the YIG slab [ Fig. 1(a)], while only the top surface of YIG is covered with a Pt film in conventional samples [18,23,27,32]. The lengths of the YIG slab along the x, y, and z directions are 3 mm, 7 mm, and 1 mm, respectively. 10-nm-thick Pt films were sputtered on the whole of the 3 × 7-mm 2 (111) surfaces of the YIG. The top and bottom Pt films are electrically insulated from each other because YIG is a very good insulator. Since YIG has the large charge gap of 2.7 eV [49], thermal excitation of charge carriers in YIG is vanishingly small even at the high temperatures.
To attach electrodes to both the Pt films symmetrically and to generate a uniform temperature gradient, we made the configuration shown in Fig. 1(b). Here, the Pt/YIG/Pt sample was sandwiched between two 0.5-mm-thick sapphire plates of which the surface is covered with two separated Au/Ti contacts. The distance between the two Au/Ti contacts is ∼ 6 mm. To extract voltage signals in the Pt films, both the ends of the Pt films are connected to the Au/Ti contacts via sintering metal paste, which can be used up to 900 • C, and thin Pt wires with the diameter of 0.1 mm were attached to the end of the contacts [see Fig. 1(b)]. The sapphire plates are thermally connected to heat baths of which the temperatures are controlled with the accuracy of < 0.6 K by using PID (proportional-integral-derivative) temperature controllers. We attached thermal paste to both the surfaces of the sapphire plates, except for the regions of the Au/Ti contacts, to improve the thermal contact [see Fig.  1(b)]. During the measurements of the LSSE, the temperature of the upper heat bath, T set H , is set to be higher than that of the lower one, T set L . According to the direction of the temperature gradient, hereafter, the top and bottom Pt films of the Pt/YIG/Pt sample are referred to as 'Pt H' and 'Pt L', respectively. In this setup, we can measure the electric voltage V H (V L ) and resistance R H (R L ) between the ends of Pt H (Pt L) without changing electrodes, wiring, and measurement equipment [see also the caption of Fig. 1]. An external magnetic field H (with the magnitude H) was applied along the x direction.
To investigate the temperature dependence of the LSSE quantitatively, it is important to estimate the temperature difference between the top and bottom of the YIG sample. However, in the conventional experiments, the temperatures of the heat baths, not the sample itself, were usually monitored [20]. Therefore, the measured temperature difference includes the contributions from the interfacial thermal resistance between the sample and heat baths and from small temperature gradients in the sample holders. To avoid this problem, in this study, we used the Pt films not only as spin-current detectors but also as temperature sensors [28,31]; we can know the temperatures of the Pt films from the temperature dependence of the resistance of the films, enabling the estimation of the temperature difference between the top and bottom of the sample during the LSSE measurements. Figure 1(c) shows the flow chart of the measurement processes. The measurement comprises the following three processes P1-P3, and the processes are repeated N times. Hereafter, T Hn and T Ln denote the set temperatures for each measurement step number n (= 1, 2, ..., N ) (see Table I, where the values of T Hn and T Ln for each n are shown). The process P1 is the measurement of the resistance of Pt H and Pt L, R H TABLE I: Set temperatures of the heat baths for each measurement step number n (≤ N ). Here, the THn and TLn values are increased by 10 K with every n increase, while THn − TLn is fixed at 8 K.
Step number n T Ln (K) T Hn (K)  1  290  298  2  300  308  3  310  318  4  320  328  5 330 338 · · · · · · · · · 31 590 598 32 600 608 33 610 618 34 (= N ) 620 628 and R L , in the isothermal condition, where the temperatures of the heat baths are set to the same temperature: In this isothermal condition, the temperature of the Pt/YIG/Pt sample is uniform and very close to the set temperature of the heat baths irrespective of the presence of the interfacial thermal resistance. Next, we apply a temperature gradient to the Pt/YIG/Pt sample by increasing the temperature of the upper heat bath, where T set H = T Hn and T set L = T Ln with T Hn > T Ln (see Table I). After waiting until the temperatures are stabilized, we measure the resistance of the Pt films under the temperature gradient: this is the process P2. The processes P1 and P2 are performed without applying H. Immediately after the process P2, we proceed to the process P3; the LSSE, i.e. the H dependence of V H and V L , is measured with keeping the magnitude of T set H − T set L constant. After finishing the LSSE measurements, we go on to the next step and increase n by 1, where the values of T Hn and T Ln are increased as shown in Table I. These measurement processes are summarized in Fig. 1(d).
The calibration method of the sample temperatures is as follows. From the process P1, we obtain the temperature (T Ln ) dependence of R H and R L under the isothermal condition. By comparing the resistance under the temperature gradient, obtained from the process P2, with the isothermal R H,L -T Ln curves, we can calibrate the temperature of the Pt films, T Pt H and T Pt L , under the temperature gradient, allowing us to estimate the average temperature T av (= (T Pt H + T Pt L )/2) and the temperature difference ∆T (= T Pt H − T Pt L ) in the YIG slab during the LSSE measurements. The T av and ∆T values are free from the contributions from the interfacial thermal resistance between the sample and heat baths and from the temperature gradients in the sapphire plates, enabling the quantitative evaluation of the LSSE at various temperatures. Although the experimental protocol proposed here cannot estimate the contributions of the interfacial thermal resistance at the Pt/YIG interfaces and temperature gradients in the Pt films, they are negligible compared with the temperature difference applied to the YIG slab [26,32]. observed to be the same as that in Pt L, a situation consistent with the scenario of the SSE [38,40,44,51]. Here we note that, although the large background voltage V BG appears due to unavoidable thermopower differences in the wires with increasing n [ Fig. 2(e)], the noise level in the V H and V L signals does not increase and the drift of V BG is small [Figs. 2(a) and 2(b)]. Therefore, we can extract the LSSE voltage simply by measuring the H dependence of V H and V L . We also found that the magnitude of the LSSE voltage in the Pt/YIG/Pt sample monotonically decreases with increasing the temperature.

III. RESULTS AND DISCUSSION
To quantitatively discuss the temperature dependence of the LSSE voltage in the Pt/YIG/Pt sample, we estimate T av and ∆T at each step number n by using the method explained in Sec. II. Figure 2(c) shows the T Ln dependence of R H and R L for the Pt films measured under the isothermal condition. By combining the isothermal R H,L -T Ln curves with the R H and R L data under the temperature gradient [the inset to Fig. 2(d)], we obtain the T av and ∆T values at each n [ Fig. 2(d)]. Importantly, the calibrated values of ∆T are dependent on n and always smaller than the temperature difference applied to the heat baths due to the interfacial thermal resistance and temperature gradients in the sapphire plates (note that T Hn − T Ln = 8 K for all the measurements as shown in Table I). We confirm again that the magnitude of V LSSE /∆T monotonically decreases with increasing the temperature and disappears above the Curie temperature T c of YIG, where T c of our YIG slab was experimentally estimated to be 553 K (see Appendix A). This behavior was observed not only in one sample but also in our different samples as exemplified in Fig. 3(c). Interestingly, the temperature dependence of V LSSE /∆T is significantly different from the magnetization (4πM ) curve of the YIG slab [compare Figs. 3(a) and 3(c)]; the magnitude of V LSSE /∆T rapidly decreases with a concaveup shape, while the magnetization curve of YIG exhibits a standard concave-down shape. We also checked that the strong temperature dependence of the LSSE voltage in the Pt/YIG/Pt sample cannot be explained by the weak temperature dependence of the thermal conductivity of YIG (see Appendix B). Similar difference in the temperature-dependent data between the ISHE voltage and magnetization was observed also in Pt/GaMnAs sys- tems in the measurement of the transverse SSEs [15,16]. The behavior of physical quantities near continuous phase transitions can be described by critical exponents in general. Here, we compare the critical exponents for the observed temperature dependences of the LSSE voltage in the Pt/YIG/Pt sample and the magnetization of YIG. First, we checked that the magnetization curve of YIG is well reproduced by a standard mean-field model [52]: where the critical exponent is fixed at 0.5 and A is an adjustable parameter [see Figs. 3(a) and 3(b)]. The critical exponent γ for the LSSE was estimated by fitting the experimental data in Fig. 3(c) with the following equation: where both S and γ are adjustable parameters and T av is regarded as T for the LSSE data. We found that the observed temperature dependence of V LSSE /∆T for the Pt/YIG/Pt sample is well fitted by Eq. (3) with γ = 3, which is much greater than the critical exponent for the magnetization curve [see also the double logarithmic plot in Fig. 3(d)]. This big difference in the critical exponents between the LSSE and magnetization emphasizes the fact that the LSSE is not attributed solely to static magnetic properties in YIG.
Here, we qualitatively discuss the origin of the temperature dependence of the LSSE voltage in the Pt/YIG/Pt sample. According to the thermal spin-pumping mechanism [38,44] and phenomenological calculation of the ISHE combined with short-circuit effects [53], the magnitude of the LSSE voltage is determined mainly by the following factors: the spin-mixing conductance [54][55][56][57] at the Pt/YIG interfaces, spin-diffusion length and spin-Hall angle of Pt, and difference between an effective magnon temperature in YIG and an effective electron temperature in Pt. Since the LSSE voltage is proportional to the spin-mixing conductance [38], it can contribute directly to the observed temperature dependence of the LSSE voltage. Recently, Ohnuma et al. formulated the relation between the spin-mixing conductance and interface s-d interaction at paramagnet/ferromagnet interfaces, and predicted that the spin-mixing conductance is proportional to (4πM s ) 2 of the ferromagnet [58].
By combining this prediction with Eq. (2), the spinmixing conductance is proportional to T c − T , of which the critical exponent (= 1) is greater than that for the magnetization curve. Although the temperature dependence of the spin-mixing conductance can explain the facts that the LSSE voltage monotonically decreases with increasing the temperature and disappears at T c , it is still much weaker than the observed (T c − T ) 3 dependence of the LSSE voltage. Furthermore, if the spin-diffusion length of Pt decreases with increasing the temperature [59], it can also contribute to reducing the LSSE voltage at high temperatures, while the spin-Hall angle of Pt was shown to exhibit weak temperature dependence [60] (note that the magnitude of the ISHE voltage monotonically decreases with decreasing the spin-diffusion length when the spin-Hall angle is constant [53]). The effective magnon-electron temperature difference could also be an important factor, but there is no clear framework to determine its temperature dependence at the present stage. Therefore, more elaborate investigations are necessary for the complete understanding of the temperature dependence of the LSSE voltage.

IV. CONCLUSION
In this study, we reported the longitudinal spin Seebeck effects (LSSEs) in Y 3 Fe 5 O 12 (YIG) slabs sandwiched by two Pt films in the high temperature range from room temperature to above the Curie temperature T c of YIG. To investigate the temperature dependence of the LSSE quantitatively, we used the Pt films not only as spin-current detectors but also as temperature sensors. The measurement processes used here enabled accurate estimation of the average temperature and temperature difference of the sample, being free from thermal artifacts. We found that the magnitude of the LSSE in the Pt/YIG/Pt sample rapidly decreases with increasing the temperature and disappears above T c of YIG; the observed LSSE voltage exhibits the (T c − T ) 3 dependence of which the critical exponent (= 3) is much greater than that of the magnetization of YIG (= 0.5). Although more detailed experimental and theoretical investigations are required to clarify the microscopic origin of this discrepancy, we anticipate that the quantitative temperaturedependent LSSE data at high temperatures will be helpful for obtaining full understanding of the mechanism of the LSSE.

ACKNOWLEDGMENTS
The authors thank S. Maekawa The Curie temperature T c of the YIG slab used in the present study was estimated from vibrating sample magnetometry and Arrott-plot analysis [52,61]. The inset to Fig. 4(a) shows the H dependence of the magnetization 4πM of the YIG slab for various values of T . From this result, we obtained the Arrott plots, i.e. H/4πM dependence of (4πM ) 2 , of the YIG slab [see Fig. 4(a)]. The saturation magnetization 4πM s of the YIG at each temperature was extracted by extrapolating the (4πM ) 2 data in the high-magnetic-field range to zero field [see red dotted lines in Fig. 4(a)]. As shown in Fig. 4(b), the T dependence of (4πM s ) 2 of the YIG slab is well fitted by a linear function; the horizontal intercept of the linear fit line corresponds to T c . The fitting result shows that the Curie temperature of our YIG slab is T c = 553 K, which is consistent with literature values [62,63]. The κ values were obtained by the combination of thermal diffusivity measured by a laser-flash method and specific heat C measured by a differential scanning calorimetry. Here, we measured the thermal diffusivity along the [111] direction of the single-crystalline YIG slab, which is parallel to the ∇T direction in the LSSE setup. As shown in Fig. 5(b), the measured C values are consistent with the Dulong-Petit (DP) law [1]; the difference of the C values from the DP specific heat of YIG, C DP = 0.676 J/gK, is less than 10 % of C DP for T > 350 K. The observed T dependence of κ is well fitted by κ ∝ T −1 , indicating that the thermal conductivity of the YIG is dominated by phonons in this temperature range [see also the inset to Fig. 5(a)]. The κ value at 300 K is consistent with literature values [64][65][66]. We also confirmed that the T dependence of κ is much weaker than that of the LSSE voltage in the Pt/YIG/Pt sample [compare Figs. 3(c) and 5(a)] and shows no anomaly around T c .