Scaling relation between anomalous Nernst and Hall effect in ${\left[\mathrm{Pt}/\mathrm{Co}\right]}_n$ multilayers

The anomalous Nernst coefficient, anomalous Hall angle, and Seebeck coefficient have been measured in the same series of ${\left[\mathrm{Pt}/\mathrm{Co}\right]}_n$ superlattices in which spin-orbit coupling is dominated by interfaces. With an increase in the number of interfaces from 1 to 15, the anomalous Nernst coefficient and anomalous Hall angle simultaneously increase, by $350\%$ and $430\%$, respectively. Furthermore, they even scale linearly with each other, while the Seebeck coefficient and magnetization of the superlattices vary a little. Based on the linear scaling relation, a physical scenario behind the anomalous Nernst effect, as well as a formula relating the Nernst coefficient, the Hall angle, and the Seebeck coefficient, is proposed. The work not only demonstrates an effective way to enhance the anomalous Nernst effect in ferromagnetic conductors but also can bring about deeper insight into the anomalous Nernst effect.

Figure 1

Patterns for measuring (a) the resistance and Hall resistance and (b) the anomalous Nernst effect. Point E and pad F are along the easy axis and also along the $x$ axis. (c) Schematic of the experimental setup for measuring the Seebeck coefficient of a thin film. Tungsten probes with a spacing of $d$ are used to pick up the Seebeck voltage. The temperature gradient between point K and point Q in a silicon strip, after the geometry factor $\xi$ is taken into account, is used to estimate the counterpart in a sample on the top of the strip. A thin layer of thermal grease has to be used between the sample and the strip to improve the interfacial thermal conductivity and obtain a high signal-to-noise ratio of the Seebeck voltage.

Figure 2

(a) $H_z$ dependence of ${\rho }_{xy}$ of $\mathrm{SP}n$ at 300 K. Inset: Dependence of ${\rho }_{xx}$ on ($2n-1$) at 300 K. (b) Dependence of ${\theta }_H$ at 300 K on ($2n-1$) and its linear fitting as $n\le 5$. (c) Scaling law between ${\rho }_{xy}$ and ${\rho }_{xx}$ with varying $T$. Inset: Scaling exponent $\beta$. (d) $H_z$ dependence of the anisotropic magnetoresistance of Co and some superlattices.

Figure 3

(a) Field dependence of the $V_N$ of SP8 under opposite $\partial T/\partial x$ and $\omega =587\mathrm{mW}$. Inset: Positions of corresponding pads. (b) Field dependence of the $V_N$ of SP8 under different $\gamma$ at $\omega =250\mathrm{mW}$. Inset: Definition of the angle $\gamma$. (c) Field dependence of the $V_N$ of SP8 under elevated powers. Inset: Linear dependence of saturated $V_N$ on $\omega$. The solid red line shows the linear fitting result. (d) Dependence of ${\nu }_N=d{V}_N/d\omega$ (left axis) and ${\nu }_N/{\theta }_H$ (right axis) on the number of interfaces. Inset: Scaling relation between ${\nu }_N$ and ${\theta }_H$ of the superlattices. Solid red and blue lines show linear fitting results as $n\le 5$ and $n\ge 5$, respectively.

Figure 4

(a) $M-H$ curves of SP8 with $H$ along the in-plane easy and hard axes. (b) Dependence of the $M_0$ of the superlattices on ($2n-1$). (c) Dependence of the Seebeck voltage on the temperature difference. (d) Seebeck coefficient of the superlattices as well as some reference samples.

Figure 5

(a) $H_z$ dependence of $R_{xy}$ and (b, c, d) $H_z$ dependence of $V_N$ of $X$ (20 nm; Co, ${\mathrm{Co}}_{40}{\mathrm{Fe}}_{40}{\mathrm{B}}_{20}$, and ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$)/Pt(2 nm) at an elevated heating power $\omega$, respectively.