The geometrical contribution to the anomalous Nernst effect in magnetic thin films was experimentally investigated by varying the aspect ratios and electrode configurations. The bar-type electrode configuration induces a short-circuit current near both edges of electrodes and decreases the effective Nernst voltage, while the point-contact (PC) electrode exploits the intrinsic Nernst voltage. In a sample with PC electrodes, as the sample width along the transverse direction of the thermal flow increases, the Nernst voltage increases monotonically. Thus, a much wider element with PC electrodes enables us to bring out a larger Nernst voltage by utilizing perpendicularly magnetized thin films.

## I. INTRODUCTION

Electric charge and spin transport under a temperature gradient in magnetic materials has recently attracted much attention, and thermoelectric effects in magnetic materials have been discussed.^{1–4} Sakuraba proposed the anomalous Nernst element by connecting two different materials with opposite signs, and reported an improvement in electric power generation by utilizing the anomalous Nernst effect (ANE) in magnetic materials.^{5} For practical applications, a further improvement in power generation is needed, which in turn requires the simultaneous improvement of two anomalous Nernst coefficients of magnetic materials in such elements. It is well known that transverse magnetic field effects, such as the Hall effect and Nernst effect, in thermoelectric elements depend not only on material properties but also the electrode shape and element shape.^{6–11} The well-known Corbino geometry enhances the magnetoresistance, which is proportional to the square of the magnetic field in semiconductors. However, the geometrical contribution to the thermoelectric effect in magnetic materials has not yet been reported. In this study, we experimentally investigated the geometrical contribution to the thermoelectric effect by varying the aspect ratios and the attached electrodes. Moreover, we performed numerical analysis of the electric potential in such elements, and discussed the geometrical contribution to the thermoelectric effect in magnetic thin film elements.

## II. EXPERIMENTAL

We previously reported that the perpendicularly magnetized TbFeCo thin film has a relatively large Hall angle, and the anomalous Nernst coefficient of the TbFeCo thin film is almost equal to the product of its intrinsic Seebeck coefficient and its Hall angle.^{4} Thus, in this study, TbFeCo thin films were prepared as thermoelectric materials by sputtering deposition. The thin film samples with different aspect ratios consisted of AlN(25nm)/TbFeCo(50nm)/AlN(5nm). ICP analysis revealed the TbFeCo composition to be rare earth (RE)-rich in comparison with the compensation composition. By varying the sample shape and the attached electrodes, the four types of samples shown in Fig. 1 were prepared. The aspect ratio is defined as *W*/*L*, where *W* is the sample width and *L* is the sample length. In order to investigate the effect of the electrode shape, point-contact (PC) electrodes and bar-type electrodes along the longitudinal direction were attached to each sample with different aspect ratios. Bar-type electrodes impose a short circuit (SC) or equipotential at both edges of the element, owing to the low resistivity of electrode materials, such as copper. Thus, bar-type electrodes hereafter denote SC-type electrodes. For each sample, the Hall voltage and Nernst voltage were measured under the temperature difference ranging from 2-9 K as shown in Fig. 1(a), and the effect of element shape was determined.

Under a uniform temperature difference *T*_{h} − *T*_{c}, the relationship between the Nernst voltage *V*_{ANE} and the off-diagonal thermoelectric coefficient *α*_{xy}(*M*) as a function of magnetization *M* can be expressed as

This equation indicates that the Nernst voltage is proportional to the aspect ratio of the sample. Thus, if the temperature difference is constant, the Nernst voltage is expected to increase as the aspect ratio increases. Figure 2 plots the magnetic field dependences of magnetoresistance and anomalous Hall resistance for each sample. The magnetoresistance does not change as a function of magnetic field in any of the samples. The resistances of the PC samples are lower than those of the SC sample because the electric current does not flow uniformly through the whole sample, *i.e.*, because the sample has a non-uniform current density. The potential differences between the four probe arms used to measure the resistance of PC samples are less than those for SC samples. On the other hand, the signs for anomalous Hall voltage in all four sample types are almost the same as that of the standard RE-rich sample.^{4} The coercivities of all samples are the same as well.

For PC electrodes, the anomalous Hall resistance *R*_{AHE} is given by

where *t* is the film thickness, *R*_{s} is the anomalous Hall coefficient and *M* is the out-of-plane component of the magnetization. According to the above equation, the Hall resistance is independent of the aspect ratio and depends only on the film thickness. However, as seen in Fig. 2, the Hall resistance is dependent on the electrode configuration, which is the geometrical contribution to Hall effect. The Hall effect in the samples with SC electrodes could be exactly solved by previous reports.^{12,13} The exact solution indicates that the SC electrode distorts the Hall angle in the sample, while the PC electrode preserves the intrinsic Hall angle in the sample. The distorted Hall angle at the center of sample goes to zero as the aspect ratio increases.

Figure 3 plots the magnetic field dependences of the Seebeck voltage and anomalous Nernst voltage per unit temperature difference for each sample. The Seebeck coefficients of all samples are independent of sample shape. The inset in Fig. 3(b) also shows the anomalous Nernst voltage as a function of temperature difference at a magnetic field of 0 T, which confirms that the anomalous Nernst voltage is proportional to the temperature difference. The signs of the anomalous Nernst voltages of the four sample types are the same. As seen in Fig. 3, the anomalous Nernst voltages are the same in narrower samples, while the wider PC sample shows a higher voltage than the wider SC samples, which is the geometrical contribution to the Nernst effect. Thus, the electrode configuration is important to know the geometrical contribution in these systems.

In order to investigate the influence of the electrodes on the output voltage in the Hall and Nernst effects, the phenomenological or generalized Ohm’s law including the thermoelectric effect was numerically analyzed. The generalized Ohm’s law including the thermoelectric effect under a magnetic field is expressible *via*

where ** J** is the electric current density,

*ϕ*the electrical potential, $\sigma \u0332$ the electrical conductivity tensor, and $\u03f5\u0332$ the thermoelectric tensor. The diagonal element of the conductivity tensor denotes the magnetoconductivity, and the diagonal element of the thermoelectric tensor corresponds to the Seebeck coefficient via $\alpha \u0332=\sigma \u0332\u22121\u03f5\u0332$. Each off-diagonal element includes a transverse effect, such as the Hall effect and the Nernst effect. Under the assumption of isotropic transport properties and through the equation of continuity for electric current density, div

**= 0, the two-dimensional Poisson equation under a magnetic field along the out-of-plane direction can be obtained as follows:**

*J*This equation was solved numerically through the finite element method by imposing a uniform temperature gradient and appropriate boundary conditions, which are governed by the electrode configuration. The material parameters in Table I are estimated from the experimental results. The Nernst voltages of narrower samples can be reproduced by these parameters.

ρ_{xx} [μΩm]
. | ρ_{xy} [μΩm]
. | α_{xx} [μV/K]
. | α_{xy} [μV/K]
. |
---|---|---|---|

3.6 | − 0.088 | − 6.9 | 0.4 |

ρ_{xx} [μΩm]
. | ρ_{xy} [μΩm]
. | α_{xx} [μV/K]
. | α_{xy} [μV/K]
. |
---|---|---|---|

3.6 | − 0.088 | − 6.9 | 0.4 |

Figure 4 shows the electropotential in each sample for a temperature difference of 1 K. In the simple case, the Nernst voltage can be described by equation (1). Thus, the Nernst voltage would be expected to be proportional to the aspect ratio *W*/*L*. However, as the aspect ratio *W*/*L* was increased, the Nernst voltage of the SC samples decreased while only the voltage of the wider PC sample increased. The experimental and calculation results are compared in Table II. The calculation results are in good agreement with the experimentally measured voltages.

Aspect ratio . | W/L = 15/13
. | W/L = 5/13
. | |||
---|---|---|---|---|---|

Electrode type . | SC . | PC . | SC . | PC . | |

V_{ANE} [μV/K] | Calc. | -0.28 | -0.60 | -0.18 | -0.19 |

Exp. | -0.38 | -0.60 | -0.23 | -0.27 | |

V_{SE} [μV/K] | Calc. | -6.7 | -6.8 | -6.7 | -6.8 |

Exp. | -6.0 | -6.7 | -6.7 | -6.7 |

Aspect ratio . | W/L = 15/13
. | W/L = 5/13
. | |||
---|---|---|---|---|---|

Electrode type . | SC . | PC . | SC . | PC . | |

V_{ANE} [μV/K] | Calc. | -0.28 | -0.60 | -0.18 | -0.19 |

Exp. | -0.38 | -0.60 | -0.23 | -0.27 | |

V_{SE} [μV/K] | Calc. | -6.7 | -6.8 | -6.7 | -6.8 |

Exp. | -6.0 | -6.7 | -6.7 | -6.7 |

Figure 4(c) illustrates the SC electrode contribution to the effective electric field. Since the SC electrodes impose an equipotential at both edges, the Nernst electric field falls as the aspect ratio increases. Thus, the PC electrode configuration is more appropriate than the SC electrode in terms of application of the anomalous Nernst voltage.

The Hall effect with SC electrodes was exactly solved by previous reports.^{12,13} Then, the Hall angle distribution, *θ*(*y*), along the perpendicular direction *y* to the heat flow at the center of sample can be expressed by:

where *θ*_{H} is the intrinsic Hall angle *ρ*_{xy}/*ρ*_{xx}, and the Nernst voltage can be evaluated by:

This function of eq. 5 means that the Hall angle are the intrinsic Hall angle at the both transverse sides and drastically decreases at the center of sample. Moreover, the specific aspect ratio *W*/*L* = 1 determines the Hall angle decay. In this study, since the thermoelectric field is $\u2212\u2207\varphi =\alpha \u0332\u2207T$, the angle of −∇*ϕ* is the rotation angle determined by the thermoelectric tensor $\alpha \u0332$ under uniform temperature gradient, which must be called the Nernst angle *α*_{xy}/*α*_{xx}.

In order to investigate aspect ratio dependence of Nernst voltage, the above theoretical value was numerically evaluated. Figure 5(a) shows the normalized Nernst angle at the center of sample with different aspect ratio *W*/*L*. These results show drastical decrease in the Nernst angle at the center of sample as the aspect ratio increases. Thus, even if the aspect ratio increases, the Nernst electric field at the center of sample gradually decreases. Figure 5 plots the calculated anomalous Nernst voltage in SC- and PC-type electrode configurations as a function of aspect ratio *W*/*L*. The results show a monotonic increase in Nernst voltage with increasing aspect ratio of PC samples. On the other hand, the Nernst voltage in SC sample are suppressed by the attached electrodes because of drastical decrease in Nernst angle at the center of sample. Therefore, a much wider element with PC electrodes enables us to exploit the intrinsic Nernst voltage, and is expected to lead to the implementation of a Nernst element through the use of perpendicularly magnetized thin films.

## III. CONCLUSION

We investigated the influence of element shape and electrode configuration on the thermoelectric effect in magnetic materials by varying the aspect ratio *W*/*L* and electrode configuration. Moreover, by calculating the electropotential in variously shaped elements under a temperature gradient through the finite element method, the influence of element shape on the Nernst voltage was investigated. It was found that the magnetoresistance of the point-contact (PC)-type element was lower than that of the short-circuit (SC)-type element because of the non-uniformity of the sense current through the element. The anomalous Nernst voltages of SC-type elements were almost constant and independent of *W*/*L*, while the Nernst voltages of PC-type elements increased with increasing *W*/*L*. Moreover, numerical analysis revealed that the anomalous Nernst voltages of PC-type elements were proportional to *W*/*L*, while those of SC-type elements barely changed. These calculation results are in quantitative agreement with the experimental results. Thus, the anomalous Nernst voltage can be improved by varying the aspect ratio in PC-type elements.

## ACKNOWLEDGMENTS

The authors would like to thank Prof. Hasegawa (Saitama University) for the measurement of thermoelectric properties and Dr. Katoh (Industrial Technology Institute of Ibaraki Prefecture) for sample preparation and chemical analysis of thin films. This research was supported in part by research grants from the Suzuki Science and Technology Foundation, the Research Foundation for the Electrotechnology of Chubu, and a Grant-in-Aid for Scientific Research (B) and (C), the Japan Society for the Promotion of Science.