In this study, we systematically investigated the anomalous Nernst effect in perpendicularly magnetized amorphous TbFeCo thin films with various compositions. It was found that the magnitude of the off diagonal thermopower (ODT), which corresponds to the anomalous Nernst effect, can be uniformly explained with respect to the Tb content regardless of the concentration above or below the compensation composition. The maximum ODT of 1.3 $\mu $V/K and the thermoelectric conductivity of 1.59 A/mK at room temperature were obtained, which is more significant than other perpendicular magnetic anisotropy thin films to achieve a large Nernst voltage for roll-type thermoelectric devices. By considering the thermoelectric tensor, Mott’s equation, and the scaling law, it was shown both experimentally and theoretically that the magnitudes of the first and second terms contributing to the anomalous Nernst effect are comparable. It was also found that the ODT of TbFeCo thin films is twice or more significant than the product of the Seebeck coefficient and the Hall angle. Furthermore, amorphous metals and Mn-alloys with a large Berry curvature are located above the relation that the ODT is twice the product of the Seebeck coefficient and the Hall angle, which means that amorphous metals are expected to enhance the ANE.

## I. INTRODUCTION

Thermoelectric energy conversion is a promising approach for environmentally friendly energy generation. Traditionally, it has exploited the generation of an electric voltage along a thermal gradient, the Seebeck effect, and many researchers have devoted themselves to improving the efficiency of conventional thermoelectric materials by the enhancement of the Seebeck effect and reduction in the thermal conductivity. Recently, the research field dealing with a relationship between spin current and heat current has attracted much attention.^{1–4} The anomalous Nernst effect (ANE) is one of the promising approaches to thermoelectric energy conversion, which is the thermal counterpart of the anomalous Hall effect (AHE).^{5} Thin film materials cannot obtain a sufficient temperature difference along the thickness direction for practical heat flux density. To obtain a sufficient temperature difference, it is desirable to create a temperature difference within the film plane; thus, perpendicularly magnetic anisotropy (PMA) films are suitable for Nernst elements.^{6,7} The large Nernst voltage can be obtained with PMA when the temperature difference is applied along the in-plane direction across the width $w$ of the strip film since the Nernst voltage is proportional to the ratio $l/w$ where $l$ is the strip length. Moreover, the Nernst element with a single magnetic materials is a feasible way to easily generate voltage by the heat source or temperature gradient. The energy-conversion efficiency of Nernst elements could transcend that of Seebeck elements.^{8}

The ANE is experimentally observed as a transverse voltage generated in a magnetic material subjected to a thermal gradient, which was reported in various magnetic materials.^{9–16} The ANE in magnetic materials was considered to be proportional to magnetization,^{4,11} but recent studies suggest that Berry curvature effects^{17,18} also can play a dominant role. Moreover, Berry curvature effects in amorphous magnetic materials have been discussed recently.^{19–21} However, since the ANE is not large enough to use in practical application compared with the conventional Seebeck effect, further understanding of the ANE in magnetic materials is necessary to improve thermoelectric efficiency.

TbFeCo thin films are well-known materials to be amorphous transition metal (TM) and rare-earth (RE) metal alloys, to have perpendicularly magnetic anisotropy, which are suitable for systematically investigating the composition dependence of transport properties. In TbFeCo thin films, the AHE sign depends on the RE-composition, and the sign changes near compensation composition. In our previous work,^{15} it was found the ANE sign coincides with that of the AHE, and the absolute value of the ANE is proportional to the product of the Seebeck coefficient $ S x x$ and Hall angle $tan\u2061 \theta H$. However, the absolute value of the off diagonal thermopower (ODT) $ S y x$ corresponding to the ANE and the composition dependence of ODT in TbFeCo thin films have not been discussed yet in detail. In this study, we have carried out the systematic investigation of the composition dependence of ODT in TbFeCo thin films to clarify the key factor determining the absolute value of ODT.

## II. EXPERIMENTAL METHOD

The TbFeCo thin films were prepared by magnetron sputtering apparatus with a base pressure of $ 10 \u2212 4$ Pa. The samples of sub./AlN(25 nm)/TbFeCo(50 nm)/AlN(5 nm) were deposited onto glass substrate. The AlN layer was deposited by the AlN 4 in. target and the mixture gas of $ Ar$ and $ N 2$ by the sputtering power of 150 W and the Ar gas of 0.4 Pa, which is a protection layer against oxidization of the TbFeCo layer. The $ T b 30 F e 35 C o 35$ target was used as the sputtering target for the deposition of the TbFeCo layer, and the variation in the composition was done by the number and position of Fe and Co chips. The Ar gas pressure of 0.5 Pa and the power of 100 W were used for TbFeCo deposition. The Tb composition in each sample was determined by inductively coupled plasma and x-ray fluorescence. Each sample has the squareness $ M r/ M s$ is about unity and perpendicular anisotropy at room temperature (not shown), where $ M r$ and $ M s$ are the residual magnetization and the saturation magnetization, respectively. The electrical and thermoelectric properties were measured in each sample by applying the external magnetic field from $\u2212$1.2 to 1.2 T along the out-of-plane direction at ambient temperature. For the thermoelectric measurement, Peltier elements attached at the sample edges induced a uniform temperature difference of up to 5 K. The temperature difference along the sample was measured from the voltage between the attached constant strip. The length $L$ along the temperature gradient direction is shorter than the width $W$ perpendicular to temperature gradient, and the ratio $W/L$ was set to be $15/13$ as shown in Fig. 1(a). The point contacts were fabricated by a metal mask to evaluate intrinsic ODT voltage by removing the influence of geometrical contributions.^{7}

## III. RESULTS AND DISCUSSION

Figure 1 shows the composition dependence of transport properties. From Fig. 1(b), the resistivity in each sample was independent of the applied magnetic field, and the magnetoresistance was negligibly small as reported in our previous study.^{7} The longitudinal resistivity $ \rho x x$ monotonically increases as the Tb content increases since the conduction electron moves through TM-sites, and the RE-sites act the role of scatterer.^{22,23} On the other hands, the sign of anomalous Hall resistivity $ \rho y x$ changes beyond about $ 30 at . %$ corresponding to the compensation composition in these samples as shown in Fig. 1(c). Since the conduction electrons are polarized by the exchange interaction, the increase in the spin polarization of conduction electrons generates the intrinsic spin–orbit interaction(SOI). Thus, the sign change of anomalous Hall resistivity beyond the compensation composition is induced by a dominant spin direction. In the TM-rich region, the absolute value of anomalous Hall resistivity slightly increases as the Tb content increases while in the RE-rich region, the values were almost constant even if the Tb content changes. Figure 1(d) shows the composition dependence of Hall angle $tan\u2061 \theta H$, where the Hall angle $tan\u2061 \theta H$ is defined as $ \rho y x/ \rho x x$. The sign of Hall angle $tan\u2061 \theta H$ similarly changes near compensation composition, but the absolute value reaches its maximum near the compensation composition and slightly decreases in TM-rich sample. Karel *et al.* discussed that increasing transition metal fraction leads the large anomalous Hall angle in amorphous systems.^{20} The previous study also proposed introducing a rare-earth element or an element with large SOI to generate a larger nonzero Berry curvature in order to enhance anomalous Hall angle in amorphous systems. In this study, a TM-rich sample showed the large Hall angle $tan\u2061 \theta H$ of about 5%.

Figure 1(e) shows the composition dependence of Seebeck coefficient $ S x x$. The Seebeck coefficient monotonically increases as the Tb content decreases. General thermoelectric materials show that the Seebeck coefficient decreases as the electrical conductivity increases, while these samples have the opposite trend. Figure 1(f) also shows the composition dependence of ODT $ S y x$, which is the transverse voltage corresponding to the intrinsic Nernst voltage without the geometrical contribution.^{7} The ODT value is the transverse voltage at the external magnetic field $B=0$ corresponding to the remanent magnetic state. The sign of ODT coincided with that of anomalous Hall resistivity, and the absolute value of ODT $ S y x$ increases as the Tb content decreases. The maximum ODT value of about 1.3 $\mu $V/K was obtained in the TM-rich sample with the Tb content of 24 at.%. Although the ODT monotonically increases as the Tb content decreases, the samples with lower Tb content showed the deterioration of squareness in the hysteresis loops due to larger saturation magnetization. Moreover, the Tb content less than 21 at. % leads to the crystallization of FeCo thin films in this study. As the above-mentioned, regardless of whether the Tb content is in the TM-rich or RE-rich region, the absolute values of transport properties in various Tb contents can be explained by single trends.

^{15}since the Seebeck coefficient does not depend on the magnetic field in these samples, the second-term is negligible in Eq. (3). Thus, we can obtain the Seebeck coefficient $ S x x\u2245 \alpha x x \rho x x$. From the Eqs. (3) and (4), $ \rho x y=\u2212 \rho y x$, and $ \rho y x= \rho x xtan\u2061 \theta H$, we can obtain the elements of the thermoelectric conductivity tensor,

Figure 3 shows the composition dependence of two contributions in the ODT. From Fig. 3, the absolute value of first-term $| S y x ( 1 )|$ increases as the Tb content decreases, and the absolute value $| S y x ( 1 )|$ for the sample with the Tb content of 24 at. % is about 4.5 times larger than that of 40 at. %. On the other hand, the absolute value $| S y x ( 2 )|$ also increases as the Tb content decreases, which shows a similar trend as that of first-term $| S y x ( 1 )|$. The value $| S y x ( 2 )|$ for the sample with the Tb content of 24 at. % is about 1.8 times larger than that of 40 at. %. The contribution of the second-term is smaller than that of the first-term in the TM-rich region. By considering the absolute value of Hall angle $|tan\u2061 \theta H|$ is almost constant, the value $| S y x ( 2 )|$ depends only on the value of the Seebeck coefficient. In addition, because the resistivity $ \rho x x$ decreases as the Tb content corresponding to scatters decreases, it was confirmed that only the increase in the thermoelectric conductivity $ \alpha x x$ contributed to the increase in $| S x x|$. Thus, in the TbFeCo thin films, the relationship between the Seebeck coefficient and resistivity is different from those of general thermoelectric materials. Moreover, as shown in Fig. 3, it is confirmed that the ODT value is almost twice as the product of the Hall angle and the Seebeck coefficient.

The anomalous Hall effect follows the scaling relation of $ \rho y x=\lambda \rho x x n$ due to various natures, where $\lambda $ represents the strength of the SOI. This scaling relationship together with the observed large anomalous Hall conductivity $ \sigma y x$, the intrinsic mechanism from the large Berry curvature with the topological nature^{24}, and not scattering, such as skew scattering^{25} and side-jump.^{26} Figure 4 shows the scaling relation of $ \sigma x x$ and $| \sigma y x|$. Since the electrical conductivity $ \sigma x x$ is ranged from $4.7$ to $9.0\xd7 10 5 S / m$, the AHE in TbFeCo thin films is attributed among the impurity regime ( $n=0.4$) and the intrinsic regime $n=2$.^{27–29} It was confirmed that the electrical conductivities were positioned in the mid-regime. By fitting the data by the simple form of scaling relation, $ \sigma y x=\lambda \sigma x x 2 \u2212 n$, the scaling factor $n$ can be obtained to be almost $n=1.1$.

^{36}hence, these are given by

^{37}Moreover, the local magnetic moment in Fe–Co in these samples, which serves as the conduction path, is larger, further increasing the SOI.

^{4}It was confirmed that the experimental results show that the ODT is almost twice as the second-contribution $ S y x ( 2 )$ of the product $| S x xtan\u2061 \theta H|$, as shown in Fig. 5. Thus, the ODT can be enhanced by enlarging the dominant contribution of the Seebeck coefficient $ S x x$ and the Hall angle $ \theta H$.

Materials located above the relational expression $ S y x=2| S x xtan\u2061 \theta H|$ are Mn-based alloys and amorphous TM-RE alloys, which can be expected to have a large Berry curvature.^{17,21,38} This is related to the fact that first-term contribution $ S y x ( 1 )$ originates from the novel electronic structure. In recent years, it has been reported that the Nernst coefficient of amorphous Fe–Sn thin films are relatively large, and the local lattice structure may be involved in the increase in the anomalous Nernst coefficient,^{35} which means that amorphous metals are expected to enhance the ANE. Therefore, further studies on amorphous metals are expected to increase the anomalous Nernst effect, and this study gives a new perspective to enhance the anomalous Nernst effect.

Figure 6 shows the absolute values of the room-temperature $| S y x|$ for only the remanent state of PMA materials without an external magnetic field. The ODT of TbFeCo is relatively large among materials that include multilayer films and ordered alloys that exhibit perpendicular magnetic anisotropy, suggesting that a large ANE may be obtained by amorphous magnetic materials. The in-plane magnetized films of Heusler compounds show a larger ANE of up to 7 $\mu $V/K.^{42–44} However, the external magnetic field is necessary, or a sufficient temperature difference along the thickness direction cannot be obtained against practical heat flux density.

^{38,41,45–48}the variety of substrates, including flexible ones, is limited. On the other hand, amorphous TbFeCo thin films can obtain significant enough PMA without an annealing process, which shows the remanent out-of-plane magnetization without external magnetic field. It is desirable to create a temperature difference within the film plane to obtain a sufficient temperature difference without the need for special lithography. Thus, PMA films with high aspect ratio are suitable for Nernst elements.

^{6,7,41}The large effective Nernst voltage $\Delta V$ can be obtained by utilizing PMA materials when the temperature difference $\Delta T$ is applied along the in-plane direction across the width $w$ of the strip sample. Since the effective Nernst voltage is the sum of the Seebeck voltage and the effective Nernst voltage, which is proportional to the ratio $l/w$ where $l$ is the strip length when the flexible substrate with $l\u226bw$ is utilized, as shown in Fig. 7. This roll-type thermoelectric device with point contact electrodes

^{7}shows the thermoelectric voltage as

## IV. CONCLUSION

We systematically investigated the composition dependence of the transport properties of TbFeCo thin films and discussed in detail the off diagonal thermopower (ODT) corresponding to the Nernst effect. As a result, by estimating two contributions from each tensor based on the experimental results, it was found that the magnitudes of the first and second terms consisting of the anomalous Nernst effect are about the same. Furthermore, because the Hall angle is almost constant for the Tb composition and considering Mott’s relational formula and scaling law, the scaling factor $n$ is $1$, so the ODT is twice the product of the Seebeck coefficient and Hall angle. Therefore, when the scaling factor $n$ is $1$, both experiments and theory show that the contributions of the first and second terms to the anomalous Nernst effect of TbFeCo are comparable, and Mott’s relation and scaling law can determine the magnitude of the anomalous Nernst effect of TbFeCo. It was found that this can be explained systematically, regardless of the composition. The fact that the magnitude of ODT is determined only by the Seebeck coefficient and Hall angle without depending on the composition means that clear and simple guidelines for the material design have been obtained. This research proposes a new material development guideline to obtain a significant anomalous Nernst effect with PMA, and it is expected that further research on amorphous metal magnetic materials will enhance the anomalous Nernst effect.

## ACKNOWLEDGMENTS

This research was supported, in part, by a Grant-in-Aid for Scientific Research (B) (Grant Nos18H01698, 20H02196, 22H01805, and 23K17828), and a Fund for Fostering Joint International Research (B) (Grant No.18KK0132) from the Japan Society for the Promotion of Science.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Ryo Ando:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal). **Takashi Komine:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Thermoelectric and Thermomagnetic Effects and Applications*

*The Theory of the Properties of Metals and Alloys*