Thickness dependence of spin Peltier effect visualized by thermal imaging technique

TL;DR

This study visualizes how the spin Peltier effect varies with YIG thickness using thermal imaging, providing insights into magnon propagation and validating models for magnon diffusion length estimation.

Contribution

It introduces a thermal imaging method to measure the thickness dependence of the spin Peltier effect in YIG, enabling precise analysis of magnon propagation.

Findings

Confirmed the t_YIG dependence of SPE with high resolution.

Validated phenomenological models for magnon diffusion length.

Demonstrated the effectiveness of lock-in thermography in spin caloritronics.

Abstract

Magnon propagation length in a ferrimagnetic insulator yttrium iron garnet (YIG) has been investigated by measuring and analyzing the YIG-thickness t_YIG dependence of the spin Peltier effect (SPE) in a Pt/YIG junction system. By means of the lock-in thermography technique, we measured the spatial distribution of the SPE-induced temperature modulation in the Pt/YIG system with the t_YIG gradation, allowing us to obtain the accurate t_YIG dependence of SPE with high t_YIG resolution. Based on the t_YIG dependence of SPE, we verified the applicability of several phenomenological models to estimate the magnon diffusion length in YIG.

Equations

$$\Delta T\propto 1-\exp \left(-t_{\mathrm{YIG}}/l_{\mathrm{m}}\right).$$

$$\Delta T\propto \frac{1}{S\coth \left(t_{\mathrm{YIG}}/l_{\mathrm{m}}\right)+1},$$

$$\Delta T\propto \frac{\cosh \left(t_{\mathrm{YIG}}/l_{\mathrm{m}}\right)-1}{\sinh \left(t_{\mathrm{YIG}}/l_{\mathrm{m}}\right)+r\cosh \left(t_{\mathrm{YIG}}/l_{\mathrm{m}}\right)},$$

Full text

Interconversion between spin and heat currents has been extensively studied in the field of spin caloritronics spincaloritronics1 ; spincaloritronics2 . One of the spin-caloritronic phenomena is the spin Seebeck effect (SSE) SSE3 , which generates a spin current as a result of a heat current in metal/magnetic-insulator junction systems. The Onsager reciprocal of SSE is the spin Peltier effect (SPE) SPE1 ; SPE2 ; SPE3 ; SPE6 ; SPEtheory3 ; SPE8 ; SPE9 ; Yamazaki . A typical system used for studying SPE and SSE is paramagnetic metal Pt/ferrimagnetic insulator yttrium iron garnet (YIG) junction systems SPE1 ; SPE2 ; SPE3 ; SSE3 ; SSE8 ; SSE20 ; SSE22 ; SPE4 , where the spin and heat currents are carried by electron spins in Pt and magnons in YIG SSEtheory5 ; SSEtheory11 ; SPEtheory2 ; SPEtheory4 .

SPE and SSE are characterized by their length scale including the magnon-spin diffusion length $l_{\rm{m}}$ and the magnon-phonon thermalization length $l_{\rm{mp}}$ in YIG Duine ; SSE36 ; SPEtheory2 . These length parameters have been investigated by measuring discrete YIG-thickness $t_{\rm{YIG}}$ dependence of SPE and SSE in several Pt/YIG junction systems with different $t_{\rm{YIG}}$ SSE20 ; SSE22 ; SSE26 ; SSE36 ; SPE3 . However, different junctions may have different magnetic properties, surface roughness, crystallinity, and interface condition. These variations make it hard to analyze the fine $t_{\rm{YIG}}$ dependence and obtain correct values of the length parameters.

In this letter, we measured the $t_{\rm{YIG}}$ dependence of SPE by using a single Pt/YIG system with a $t_{\rm{YIG}}$ gradient ($\nabla t_{\rm{YIG}}$). Since SPE induces temperature change reflecting the local $t_{\rm{YIG}}$ value, we can extract the $t_{\rm{YIG}}$ dependence of SPE from a temperature distribution. The spatial distribution of the SPE-induced temperature change was visualized by means of the lock-in thermography (LIT) SPE2 . The LIT method allows us to obtain the accurate $t_{\rm{YIG}}$ dependence of SPE with high $t_{\rm{YIG}}$ resolution. By means of the thermoelectric imaging technique based on laser heating Iguchi2019 , we also measured the $t_{\rm{YIG}}$ dependence of SSE in a single Pt/YIG system, which shows the same behavior as that of SPE. By analyzing the measured $t_{\rm{YIG}}$ dependence of SPE and using phenomenological models, we estimated $l_{\rm{m}}$ and determined the upper limit of $l_{\rm{mp}}$ for YIG.

The sample used for measuring SPE consists of a Pt film and an YIG film with $\nabla t_{\rm{YIG}}$ [Fig. 1(a),(d)]. $\nabla t_{\rm{YIG}}$ was introduced by obliquely polishing a single-crystalline YIG (111) film grown by a liquid phase epitaxy method on a single-crystalline Gd3Ga5O12 (GGG) (111) substrate. The obtained $\nabla t_{\rm{YIG}}$ is almost uniform in the measurement range, which was observed to be $9.2$ $\mu$m per a $1$ mm lateral length by a cross-sectional scanning electron microscopy [Fig. 1(f)]. After the polishing, a U-shaped Pt film with a thickness of 5 nm and width of 0.5 mm was sputtered on the surface of the YIG film. The longer lines of the U-shaped Pt film were along the $\nabla t_{\rm{YIG}}$ direction. In the microscope image of the sample in Fig. 1(d), the yellow (gray) area above (below) the white dotted line corresponds to the YIG film with $\nabla t_{\rm{YIG}}$ (GGG substrate).

SPE induces temperature modulation in the Pt/YIG/GGG sample in response to a charge current in the Pt film. When we apply a charge current ${\bf{J}}_{\rm{c}}$ to the Pt film as shown in Fig. 1(a), a spin current is generated by the spin Hall effect in Pt SHE5 ; SHE6 . The spin current induces a heat current across the Pt/YIG interface via SPE. The heat current results in a temperature change $\Delta T$ which satisfies the following relation SPE1 ; SPE2 : $\Delta T\propto\left({\bf{J}}_{\rm{c}}\times{\bf{M}}\right)\cdot{\bf{n}}$, where ${\bf{M}}$ and ${\bf{n}}$ are the magnetization vector of YIG and the normal vector of the Pt/YIG interface plane, respectively. Significantly, the SPE-induced temperature change reflects local $t_{\rm{YIG}}$ information because the temperature change induced by SPE is localized owing to the formation of dipolar heat sources SPE2 ; SPE3 . Based on the $\nabla t_{\rm{YIG}}$ value and the spatial resolution of our LIT system, we obtained the high $t_{\rm{YIG}}$ resolution of $92$ nm.

The procedure of the LIT-based SPE measurements are as follows SPE2 ; LIT1 ; LIT2 ; LIT3 ; LIT4 ; LIT5 ; AMPE . To excite SPE, we applied a rectangular a.c. charge current ${\bf{J}}_{\rm{c}}$ with the amplitude $J_{0}$, frequency $f$, and zero offset to the Pt film [Fig. 1(b)]. By extracting the first harmonic response of a temperature change $\Delta T_{1f}$ in this condition (SPE configuration), we can detect the pure SPE signal free from a Joule heating contribution SPE2 ; SPE3 . Here, $\Delta T_{1f}$ is defined as the temperature change oscillating in the same phase as ${\bf{J}}_{\rm{c}}$ because $\Delta T$ generated by SPE follows the ${\bf{J}}_{\rm{c}}$ oscillation [Fig. 1(b)] and the out-of-phase signal is negligibly small SPE2 ; SPE3 ; SPE5 ; SPE7 . In the LIT measurement, we detect the first harmonic component of the infrared light emission $\Delta I_{1f}$ caused by $\Delta T_{1f}$, where $\Delta I_{1f}=A\cos\phi$ with $A$ and $\phi$ respectively being the lock-in amplitude and phase. We converted $\Delta I_{1f}$ into $\Delta T_{1f}$ by considering spatial distribution of an infrared emissivity $\epsilon$ of the sample SPE2 ; SPE3 . All measurements of SPE were performed at room temperature and atmospheric pressure under a magnetic field with a magnitude of $20$ mT, where ${\bf{M}}$ aligns along the field direction.

Figure 2(a) shows the $\Delta I_{1f}$ image from the Pt/YIG/GGG sample in the SPE configuration with $J_{0}=8$ mA and $f=5$ Hz. Clear signals appear only on the Pt film with finite $t_{\rm{YIG}}$ but disappears on the Pt/GGG interface with $t_{\rm{YIG}}=0$ [compare Figs. 1(d) and 2(a)]. The sign of $\Delta I_{1f}$ is reversed when ${\bf{J}}_{\rm{c}}$ is reversed, which confirms that the observed infrared signal comes from SPE SPE1 ; SPE2 . The spatial profile of $\Delta I_{1f}$ along the $y$ direction is plotted in Fig. 2(c), where the $\Delta I_{1f}$ values are averaged along the $x$ direction in the area surrounded by the dotted line in Fig. 2(a). $\Delta I_{1f}$ gradually increases with small oscillation with increasing $t_{\rm{YIG}}$. The oscillation originates from the oscillation of $\epsilon$ of the sample due to multiple reflection and interference of the infrared light in the YIG film [see the infrared light image of the sample in Fig. 1(e)] SPE3 . To calibrate the $\epsilon$ oscillation in the $t_{\rm{YIG}}$ dependence of SPE, the infrared emission induced by the Joule heating was measured [Fig. 2(b)], where ${\bf{J}}_{\rm{c}}$ with the amplitude $\Delta J_{\rm{c}}=0.5$ mA, frequency $f=5$ Hz, and d.c. offset $J_{\rm{dc}}=8.0$ mA was used as an input for the LIT measurement [Fig. 1(c)] SPE2 . Since the temperature change induced by the Joule heating is uniform on the Pt film, the lock-in amplitude of the infrared emission $A$ on the Pt film depends only on the $\epsilon$ distribution. We found that the obtained $t_{\rm{YIG}}$ dependence of $\epsilon$ ($\propto A$ due to the Joule heating) and the $\Delta I_{1f}$ signal due to SPE exhibit the similar oscillating behavior [Fig. 2(c)]. By calibrating $\Delta I_{1f}$ by $\epsilon$, we obtained the $t_{\rm{YIG}}$ dependence of $\Delta T_{1f}$ induced by SPE [Fig. 2(d)]. $\Delta T_{1f}$ monotonically increases with increasing $t_{\rm{YIG}}$. The saturated $\Delta T_{1f}$ value for $t_{\rm{YIG}}>6$ $\mu$m was determined by using the same Pt/YIG/GGG sample coated with a black-ink infrared emission layer with high emissivity larger than 0.95.

The accurate $t_{\rm{YIG}}$ dependence of SPE with high $t_{\rm{YIG}}$ resolution allows us to verify the applicability of several phenomenological models used for discussing the behaviors of SPE. First of all, we found the obtained $t_{\rm{YIG}}$ dependence of the SPE-induced temperature modulation cannot be explained by a simple exponential approximation SSE20 ; SSE22 ; SSE26 ; SPE3 . Based on the assumption that the magnon diffuses in YIG with the magnon diffusion length $l_{\rm{m}}$, the simple exponential decay has been used for the analysis of the $t_{\rm{YIG}}$ dependence:

[TABLE]

However, in general, this expression cannot be used for the small thickness region since the exponential function should be modulated by the boundary conditions for the spin and heat currents. In fact, the fitting result using Eq. (1) significantly deviates from the experimental data when $t_{\rm{YIG}}<4$ $\mu$m (see the green curve in Fig. 3). The observed continuous $t_{\rm{YIG}}$ dependence of SPE thus requires advanced understanding of the spin-heat conversion phenomena.

Next we focus on two phenomenological models proposed in Refs. SPEtheory4, and SPEtheory1, . The model in Ref. SPEtheory4, is based on the linear Boltzmann’s theory for magnons and the $t_{\rm{YIG}}$ dependence of SPE is described as

[TABLE]

where $S$ is a $t_{\rm{YIG}}$-independent constant used as a fitting parameter in our analysis. The red curve in Fig. 3 shows the fitting result based on Eq. (2). We found that Eq. (2) shows the best agreement with the experimental result and $l_{\rm{m}}$ is estimated to be $3.9$ $\mu$m. We also analyzed the experimental results by the model based on the non-equilibrium thermodynamics SPEtheory1 :

[TABLE]

where $r$ is a $t_{\rm{YIG}}$-independent constant used as a fitting parameter in our analysis. In contrast to Eq. (2), Eq. (3) is less consistent with the experimental results in the whole thickness range (see the yellow curve in Fig. 3) and gives a shorter magnon diffusion length of $0.6$ $\mu$m. From the fitting results using Eqs. (1-3), we conclude that Eq. (2) is the best phenomenological model to explain the experimental results on SPE.

To check the reciprocity between SPE and SSE reciprocity , we also measured the $t_{\rm{YIG}}$ dependence of SSE by using a single Pt/YIG/GGG sample with $\nabla t_{\rm{YIG}}$. The YIG film used in the SSE measurement was obtained from the same YIG/GGG substrate and the SSE sample was prepared by the same method as that for the SPE sample. The $\nabla t_{\rm{YIG}}$ value of the SSE sample was determined to be $12.2$ $\mu$m per a $1$ mm lateral length.

To obtain the $t_{\rm{YIG}}$ dependence of SSE, the SSE signal was measured in the Pt/YIG/GGG sample by means of the thermoelectric imaging technique based on laser heating Iguchi2019 ; SSE4 ; SSE14 ; SSE29 . As shown in Fig. 4(a), the sample surface was irradiated by laser light with the wavelength of $1.3$ $\mu$m and the laser spot diameter of $5.2$ $\mu$m to generate a temperature gradient across the Pt/YIG interface. The local temperature gradient induces a spin current across the interface due to SSE. The spin current is then converted into a charge current via the inverse spin Hall effect in Pt SHE6 . To avoid the reduction of the spatial resolution of the SSE signal due to the thermal diffusion, we adopted a lock-in technique in the laser SSE measurement, where the laser intensity was modulated in a periodic square waveform with the frequency $f=5$ kHz and the thermopower signal $S_{1f}$ oscillating with the same frequency as that of the input laser was measured. The measurements at the high lock-in frequency realized high spatial resolution for the SSE signal because temperature broadening due to the heat diffusion is suppressed. Here, we defined the SSE signal $S_{\rm{SSE}}$ as $\left[S_{1f}(+50\ {\rm{mT}})-S_{1f}(-50\ {\rm{mT}})\right]/2$ to remove magnetic-field-independent background. By scanning the position of the laser spot on the sample, we visualized the spatial distribution of the SSE signal with high $t_{\rm{YIG}}$ resolution of $64$ nm.

Figure 4(c) shows the spatial distribution of $S_{\rm{SSE}}$ for the Pt/YIG/GGG sample. In response to the laser heating, the clear signal was observed to appear in the Pt film. The $t_{\rm{YIG}}$ dependence of the SSE signal is plotted in Fig. 4(d), where the $S_{\rm{SSE}}$ values were averaged along the $x$ direction in the area surrounded by the dotted line in Fig. 4(c). The SSE signal monotonically increases with increasing $t_{\rm{YIG}}$. Significantly, the $t_{\rm{YIG}}$ dependence of $S_{\rm{SSE}}$ shows the same behavior as that of the SPE-induced temperature modulation [Fig. 4(d)]. This result supports the reciprocity between SPE and SSE and strengthens our conclusion in the SPE measurement.

In the recent study on SSE in Ref. SSE36, , Parakash et al. reported non-monotonical increase of the SSE signal with $t_{\rm{YIG}}$. Since the SSE signal takes a local maximum at $t_{\rm{YIG}}\sim l_{\rm{mp}}$, they estimated $l_{\rm{mp}}$ as $250$ nm from the maximum point. However, in our Pt/YIG samples, the SSE and SPE signals monotonically increase with increasing $t_{\rm{YIG}}$. These results suggest that $l_{\rm{mp}}$ is shorter than the $t_{\rm{YIG}}$ resolution, $64$ nm, in our experiments. The conclusion is consistent with the theoretical expectation of $l_{\rm{mp}}\sim 1$ nm SPEtheory2 .

In conclusion, we have discussed the length scale of the spin and heat transport by magnons in YIG by measuring the $t_{\rm{YIG}}$ dependence of SPE in the Pt/YIG sample. This measurement was realized by using the YIG film with the $t_{\rm{YIG}}$ gradient and the LIT method, which allow us to obtain the continuous $t_{\rm{YIG}}$ dependence of SPE in the single Pt/YIG sample. The experimental result is well reproduced by the phenomenological model based on the linear Boltzmann’s theory for magnons referenced in Ref. SPEtheory4, and $l_{\rm{m}}$ is estimated to be $3.9$ $\mu$m for our YIG sample. We also measured the $t_{\rm{YIG}}$ dependence of SSE. The SPE and SSE signals show the same behavior in the $t_{\rm{YIG}}$ dependence and monotonically increase as $t_{\rm{YIG}}$ increases. The monotonic increase implies that $l_{\rm{mp}}$ is shorter than $64$ nm for our YIG sample. These results give crucial information to understand the microscopic origin of the spin-heat conversion phenomena.