Introduction

As demand for magnetic-field sensing increases in a recently developing internet-of-things (IoT) society, the various types of magnetometers have been developed and immensely contributed to a broad range of applications1,2,3,4. The most commonly used two types of magnetometers are semiconductor Hall sensors5,6 and ferromagnetic magnetoresistive sensors7,8, which generate the output signal against the scalar quantity of the magnetic field along a principal axis. However, future automation in the automotive industry and robotics requires detecting the motion of objects via determination of the magnetic field (H) vector in three dimensions, enabling the control of precise position, angle, and rotation of fundamental components in an electric appliance. For magnetic-field sensing in three dimensions, individual detection of the three-dimensional (3D) components of the magnetic field is necessary. The classical way to detect 3D components of H includes placing three Hall sensors5,9,10,11 or magnetoresistive sensors orthogonally (along x-, y-, and z-axis)12,13,14 and attaching magnetic flux guides in the planar devices15,16. However, these techniques face the requirements of three power sources and 3D space arrangement. To expand the 3D sensing of the magnetic field into versatile electric appliances, it is essential to develop the 3D magnetometer based on a single planar-type device with low power consumption and a simple measurement unit. The appealing phenomena in ferromagnetic films such as spin–orbit torque17 and unidirectional magnetoresistance (UMR)18 are beneficial to trigger the development of a planar 3D magnetometer beyond the conventional Hall devices.

The 3D sensing of the magnetic-field vector corresponds to the function for the detection of the magnetic field amplitude and direction with polar coordinates. Figure 1a represents the magnetic-field vector H with its amplitude H, polar angle \({\theta }_{{{{{{\rm{H}}}}}}}\), and azimuthal angle \({\phi }_{{{{{{\rm{H}}}}}}}\). The 3D magnetometer has to uniquely determine these polar coordinates via electrical measurements. Typically, the Hall effect is available to measure the amplitude H and the polar angle \({\theta }_{{{{{{\rm{H}}}}}}}\). Conventional semiconductors, such as Si, GaAs, and InAs, have been well applied to practical Hall magnetometers owing to good controllability of sensitivity based on high-mobility charge carriers5. Recently, in addition to the conventional semiconductor devices driven by the ordinary Hall effect, ferromagnetic heterostructures with high Curie temperatures have been recognized as a good candidate for the room-temperature Hall magnetometer19,20,21, driven by anomalous Hall effect (AHE)22. In particular, the sizable anomalous Hall response for the detection of H is enabled by the intrinsic mechanism owing to the specific band features of the topologically nontrivial ferromagnetic materials22,23. Moreover, the determination of azimuthal angle \({\phi }_{{{{{{\rm{H}}}}}}}\) is critically important to develop compact planar-type 3D magnetometers since it is rather difficult for a single semiconductor Hall device. By using magnetic materials, anisotropic magnetoresistance (AMR) is applicable to measure the \({\phi }_{{{{{{\rm{H}}}}}}}\) because its amplitude reflects the relative angle between in-plane directions of electric current (I) and magnetization (M)7. When the direction of M follows to H, the AMR provides the information of the direction of H. However, the AMR in one component provides four identical values owing to the 180° period, which is insufficient to uniquely determine the \({\phi }_{{{{{{\rm{H}}}}}}}\).

Fig. 1: Detection concept for magnetic-field vector by a Fe–Sn heterostructure device.
figure 1

a Definition of the polar (\({\theta }_{{{{{{\rm{H}}}}}}}\)) and azimuthal (\({\phi }_{{{{{{\rm{H}}}}}}}\)) angles of the magnetic-field vector H. b Schematic of layer structure consisting of a SiOx cap and 4-nm-thick Fe-Sn layer grown on Al2O3 substrate. Magnetization of the Fe-Sn layer follows the magnetic field, providing resistance changes with respect to strength and direction of the field. c Top-view photograph of the planar Hall device and measurement setup. d Summary of the signal output from AHE, AMR, and UMR.

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In this study, we apply two AMR probes and UMR effect18,24,25,26,27,28 to solve this problem. Here, we report on the demonstration of magnetic-field vector detection by implementing the AHE, AMR, and UMR effects into a single planar-type 3D magnetometer based on the ferromagnetic Fe–Sn heterostructure. The device overcomes the limitation of conventional semiconductor devices and is highly sensitive at room temperature.

Results and discussion

Concept of device structure and measurement setup

A SiOx capped 4-nm-thick Fe-Sn heterostructure (Fig. 1b) was patterned into a device consisting of two Hall-bar channels oriented along x-axis (channel 1) and −45o from x-axis (channel 2) in series, as shown in Fig. 1c. To disentangle the AMR and UMR signals, harmonics measurements of the sheet and Hall resistance were employed using lock-in amplifiers under the application of the ac current with modulation frequency f = 2πω24. The first harmonic Hall resistance \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) and the first and second harmonic sheet resistance \({R}_{1}^{\omega }\) and \({R}_{1}^{2\omega }\) for channel 1, and the first harmonic sheet resistance \({R}_{2}^{\omega }\) for channel 2 were measured simultaneously. Figure 1d summarizes the concept for the determination of the polar coordinates \({\theta }_{{{{{{\rm{H}}}}}}}\) by \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) and \({\phi }_{{{{{{\rm{H}}}}}}}\) by \({R}_{1}^{\omega }\), \({R}_{2}^{\omega }\), and \({R}_{1}^{2\omega }\). The polar angle θH (out-of-plane field direction) is monitored by \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) via AHE. In usual polar coordinates, the θH should be only defined in 0o \(\le\)  θH \(\le\) 180o. The azimuthal angle ϕH (in-plane field direction) is measured by the values of \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) and the sign of UMR in \({R}_{1}^{2\omega }\). Judging from the combination of two AMR values \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\), four identical values are reduced to two. The UMR signal follows the sign of \({{{{{\bf{M}}}}}}\times {{{{{\bf{I}}}}}}\) under broken inversion symmetry along out-of-plane direction18, resulting in a 360° period against ϕH. As a consequence, the combinations of positive and negative values of θH and ϕH listed in Fig. 1d resolve the unique direction of a magnetic-field vector by simultaneous measurements of AHE, AMR, and UMR in the ferromagnetic heterostructure device.

Vector rotation in xz plane for verification of θ H detection

The AHE was evaluated to examine the correspondence between the \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) and the H direction as shown in Fig. 2. Under the perpendicular H condition in Fig. 2a, the \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) increases linearly up to μ0H = 0.6 T with no hysteresis behavior, being suitable for detection of out-of-plane component of the magnetic field. The measurable amplitude of H is limited to the saturation field of M, which can be modulated by the magnetic shape anisotropy or the interfacial magnetic anisotropy. The extracted values of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) at μ0H = 0.3 and 1 T are plotted as a function of modulation frequency f (Fig. 2b) and current amplitude Iac (Fig. 2c). The Hall resistance \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) is stably maintained against f and Iac, meaning that the Ohmic behavior persists in high frequency up to 300 Hz. The large Hall voltage under ac operation is an essential prerequisite for sensor application in terms of detectivity29 and low-energy consumption5. Figure 2d, e represent the polar angle θH dependence of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) measured at 1 T. The \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) is initially positive when μ0H = 1 T points along +z (θH = 0o). With sweeping θH from 0o to 180o in the xz plane, the \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) monotonically decreases with sign reversal at θH = 90o. Note that the \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\) did not follow the sinusoidal function of θH (black dashed line), implying that the magnetization was not saturated along the direction of μ0H = 1 T due to the in-plane magnetic anisotropy (Supplementary Note 3 and Supplementary Fig. 3). While the amplitude of H should be evaluated from two values of different component \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) (Supplementary Note 4), the normalized function of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\) guarantees that the angle of θH is uniquely determined in the range of 0o \(\le\) \({\theta }_{{{{{{\rm{H}}}}}}}\) \(\le\) 180o. Note that \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\) can be a good reference curve for identifying \({\theta }_{{{{{{\rm{H}}}}}}}\) even if the magnetization is not saturated along H.

Fig. 2: Hall effect measurements for determination of θH.
figure 2

a Hall resistance (\({R}_{{{{{{\rm{H}}}}}}}^{\omega }\)) as a function of out-of-plane magnetic field (\({\mu }_{0}H\)) measured at 0.2 mA with various modulation frequency from 3 Hz to 1 kHz. The data below f = 300 Hz are merged in a single curve. b, c The values of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) at \({\mu }_{0}H\) = 0.3 and 1 T plotted as a function of the modulation frequency f (b) and amplitude of current (c). d Schematic of \({\theta }_{{{{{{\rm{H}}}}}}}\) rotation of the magnetic field (H). \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) produces a positive value for 0 < \({\theta }_{{{{{{\rm{H}}}}}}}\) < 90o (red region) while it produces a negative value for 90o < \({\theta }_{{{{{{\rm{H}}}}}}}\) < 180o (green). e \({\theta }_{{{{{{\rm{H}}}}}}}\) dependence of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) when \({\mu }_{0}H\) = 1 T. The H was rotated in the xz plane (inset).

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Vector rotation in xy plane for verification of ϕ H detection

We here refer to the determination of the azimuthal angle ϕH using AMR and UMR in the magnetic Fe–Sn Hall-bar device. The primary and secondary harmonic components of \({R}_{1}^{\omega }\), \({R}_{2}^{\omega }\), and \({R}_{1}^{2\omega }\) as shown in Fig. 3a were measured during magnetic-field rotation in the xy plane. By contrast to the well-studied 180° period of AMR7, the UMR signal expectedly exhibits 360° period as depicted in Fig. 3b24,25. Figure 3c presents typical AMR, the variations of \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) (denoted as \(\Delta {R}_{1}^{\omega }\) and \({\Delta R}_{2}^{\omega }\)) as a function of ϕH under μ0H = 1 T. The periodic functions of \(\Delta {R}_{1}^{\omega }\) and \(\Delta {R}_{2}^{\omega }\) with 180° indicate that the resistance variation becomes positively large when the field is parallel to the current whereas it is negatively large when the field is perpendicular to the current. The AMR signal obeys sinusoidal function of cos2ϕH (black dashed lines), implying that the direction of in-plane M follows that of H. This agreement can be ascribed to the isotropic feature of M in the sample plane. The AMR amplitude \({R}_{1}^{{{{{{\rm{AMR}}}}}}}\) and \({R}_{2}^{{{{{{\rm{AMR}}}}}}}\), defined by a half value of the peak-to-peak amplitude of \(\Delta {R}_{1}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\) and \(\Delta {R}_{2}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\) (vertical black arrows in Fig. 3c) corresponds to about 0.1% of the base resistance of about 800 Ω, which is a typical value of ferromagnetic metals30. The phase shift \(\Delta {\phi }_{{{{{{\rm{H}}}}}}}\) between \(\Delta {R}_{1}^{\omega }\) and \(\Delta {R}_{2}^{\omega }\) defined in Fig. 3c and AMR signal amplitude are plotted in Fig. 3d as a function of ac injection current Iac. The uniform distribution of electric current along the series channel 1 and 2 is maintained in the three orders of magnitude in Iac. With the almost constant phase shift of 45.2 ± 0.75°, the simultaneous measurement of \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) allows the two possible angles of ϕH in either 0 < ϕH < 180° or −180 < ϕH < 0°. To uniquely determine ϕH in the range of −180° < ϕH < 180°, the UMR plays a significant role although a signal amplitude at room temperature is rather small compared to that of AMR.

Fig. 3: AMR and UMR measurements for determination of θH.
figure 3

a Top view of measurement setup. b Schematic of \({\phi }_{{{{{{\rm{H}}}}}}}\) rotation of the magnetic field. The UMR signal measured in \({R}_{1}^{2\omega }\) produces a positive value for 0° < \({\phi }_{{{{{{\rm{H}}}}}}}\) < 180° (red region) while it produces a negative value for −180° < \({\phi }_{{{{{{\rm{H}}}}}}}\) < 0° (green). c \({\phi }_{{{{{{\rm{H}}}}}}}\) dependences of \(\Delta {R}_{1}^{\omega }\) (green circles) and \({\Delta R}_{2}^{\omega }\) (orange circles) when \({\mu }_{0}H\) = 1 T was rotated in the xy plane. The measurement was performed at Iac = 0.2 mA and f = 13 Hz. d Current amplitude (Iac) dependence of the phase shift of \({\phi }_{{{{{{\rm{H}}}}}}}\) (\(\Delta {\phi }_{{{{{{\rm{H}}}}}}}\)) between the AMR signals, defined by the horizontal arrow in (c), and AMR signal \({R}_{1}^{{{{{{\rm{AMR}}}}}}}\) (green circles) and \({R}_{2}^{{{{{{\rm{AMR}}}}}}}\) (orange circles), defined by the vertical arrows in (c). e \({\phi }_{{{{{{\rm{H}}}}}}}\) dependence of \({\Delta R}_{1}^{2\omega }\) when \({\mu }_{0}H\) = 1 T was rotated in the xy plane. f Iac dependence of UMR signal \({R}_{1}^{{{{{{\rm{UMR}}}}}}}\)defined as vertical arrows in (e). Error bars denote the interval with the 95% reliability.

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Figure 3e shows the UMR of \({R}_{1}^{2\omega }\) measured with Iac of 0.2 mA, which is 360° period with its sign being positive in 0° < ϕH < 180° and negative for −180° < ϕH < 0° (see also Supplementary Note 7 and Supplementary Fig. 6). This valid judgment enables us to define ϕH uniquely. The amplitude of UMR signal characterized by \({R}_{1}^{{{{{{\rm{UMR}}}}}}}=\left[\Delta {R}_{1}^{2\omega }\left({\phi }_{H}={90}^{{{{{{\rm{o}}}}}}}\right)-\Delta {R}_{1}^{2\omega }({\phi }_{H}={-90}^{{{{{{\rm{o}}}}}}})\right]/2\) in Fig. 3f linearly increases with increasing the amplitude Iac, which is consistent to the origin of UMR effect at the broken inversion symmetry at the interface with Rashba spin-orbit interaction18 or magnetothermal effect25,31. The relative amplitude of UMR to the channel resistance per current density \(\frac{{R}_{1}^{{{{{{\rm{UMR}}}}}}}/{R}_{1}^{{{{{{\rm{\omega }}}}}}}}{{I}_{{{{{{\rm{ac}}}}}}}/{tw}}\) = 4.14 × 10−16 (m2 A−1) with \({R}_{1}^{{{{{{\rm{UMR}}}}}}}/{I}_{{{{{{\rm{ac}}}}}}}\) = 8.42 ΩA−1 being a linear slope in Fig. 3f, t = 4 nm Fe–Sn film thickness, w = 10 μm the channel width, and the \({R}_{1}^{\omega }\) = 813 Ω. This value is five times larger than the reported value in heavy metal/ferromagnet bilayer heterostructures25 (Supplementary Note 5 and Supplementary Table 2). In the trilayer structure of SiOx/4-nm-thick Fe-Sn/Al2O3 substrate, the band offsets at the top and bottom interfaces may not be perfectly canceled, leading to the broken inversion symmetry along the out-of-plane direction. The amplitude of UMR may be further enhanced by optimizing the layer structure31 or by introducing a superlattice structure. Considering the large AHE of the Fe–Sn nanocrystalline thin films fabricated by sputtering technique, which is probably ascribed to the intrinsic mechanism21,32, the spin-orbit interaction and specific band feature cooperatively contributes to yield the large UMR effect at room temperature. Combining the AMR originated from M and the UMR from broken inversion symmetry and specific band structure, the ϕH is uniquely determined in a planar-type Hall device based on ferromagnetic Fe–Sn heterostructure.

Experimental demonstration of 3D magnetic-field sensing

Finally, we demonstrated the detection of the magnetic-field vector consisting of θH and ϕH with a constant magnetic-field strength of 1 T along the flow chart depicted in Fig. 4a. The determination of θH and ϕH was performed by the evaluation of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) using AHE, and \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) using AMR and UMR, subsequently. Here, \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\) at \({\mu }_{0}H\) = 1 T (Fig. 2e) was used for the determination of \({\theta }_{{{{{{\rm{H}}}}}}}\). The beforehand determination of θH is a reasonable route to explore the magnetic-field direction since the Hall effect is a simple and independent measurement. Following the determination of θH, the AMR curve defines two possible in-plane directions ϕH or ϕH −180o. Note that because the amplitude of AMR depends on θH, the comparison with the normalized reference curves (Supplementary Note 6 and Supplementary Fig. 5) is necessary. Then, the sign of UMR in \({R}_{1}^{2\omega }\) is employed to uniquely determine ϕH. The reference curves of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\), \({R}_{1}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\), and \({R}_{2}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\) depend on the scalar quantity of the magnetic field H. Therefore, for determination of \({\phi }_{{{{{{\rm{H}}}}}}}\) and \({\theta }_{{{{{{\rm{H}}}}}}}\), the scalar quantity of H should be first evaluated by the independent analysis of \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) (Supplementary Note 4 and Supplementary Fig. 4) and the cross-checking of the projected value of H along z-direction with θH. As demonstration for the determination of magnetic-field vector H, the intentionally controlled H shown in Fig. 4b was measured with a planar Fe–Sn Hall device in a superconducting vector magnet. In the regions (i)–(iii), the sensing operation was examined along the out-of-plane rotation, in-plane rotation, and diagonal rotation. Figure 4c shows the time evolution of the experimentally detected θH and ϕH. The θH rotation (blue) at xz plane in the region (i) is systematically detected by \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\), which is consistent to controlled θH from 90° to 45°. Then, the ϕH rotation (green) under θH = 45° in the region (ii) is evaluated by the AMR curves of \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\) and the sign of UMR in \({R}_{1}^{2\omega }\). For the diagonal rotation examination, the subsequent scan of θH and ϕH was performed in the region (iii). At each step of θH rotation, the flow chart of Fig. 4a was applied to determine the angles. Overall, the excellent agreement between controlled and experimentally detected magnetic-field angles demonstrates that the planar-type single Fe–Sn device is capable of detecting the magnetic-field vector in three dimensions. Although the present demonstration was performed at \({\mu }_{0}H\) = 1 T, the flow chart of Fig. 4a is valid by selecting the proper reference curve of \({R}_{{{{{{\rm{H}}}}}}}^{\omega }({\theta }_{{{{{{\rm{H}}}}}}})\), \({R}_{1}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\), and \({R}_{2}^{\omega }({\phi }_{{{{{{\rm{H}}}}}}})\) depending on the field strength (Supplementary Note 4). Along this detection process, the device is capable of detecting the magnetic-field vector in the range of 0.1–1.0 T with a sensitivity of 0.4 V A−1degree−1 for \({\theta }_{{{{{{\rm{H}}}}}}}\), 0.04 VA−1degree−1 for \({\phi }_{{{{{{\rm{H}}}}}}}\) and 51.3 VA−1T−1 for the field amplitude (see Supplementary Note 8 and Supplementary Table 3). The detection functionality may be further extended by evaluating the scalar quantity of the field (Supplementary Fig. 4), such as detecting the position of the moving object.

Fig. 4: Demonstration of detection of the magnetic field vector.
figure 4

a Flow chart for determination of the magnetic-field vector by the Fe–Sn magnetometer. The polar angle \({\theta }_{{{{{{\rm{H}}}}}}}\) is determined by \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) through AHE whereas the azimuthal angle \({\phi }_{{{{{{\rm{H}}}}}}}\) is determined by \({R}_{1}^{\omega }\), \({R}_{2}^{\omega }\), and the sign of \({R}_{1}^{2\omega }\) through AMR and UMR, respectively. b Time evolution of \({\theta }_{{{{{{\rm{H}}}}}}}\) (blue) and \({\phi }_{{{{{{\rm{H}}}}}}}\) (green) of the controlled magnetic field H. Schematics for the rotation of H vector is depicted in insets. c Experimentally detected \({\theta }_{{{{{{\rm{H}}}}}}}\) and \({\phi }_{{{{{{\rm{H}}}}}}}\) by the Fe–Sn device. The error bar of θH and ϕH was defined by the contribution of AMR to \({R}_{{{{{{\rm{H}}}}}}}^{\omega }\) and the difference in detected angles by \({R}_{1}^{\omega }\) and \({R}_{2}^{\omega }\), respectively.

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Conclusion

By employing AHE, AMR, and UMR, we have demonstrated a feasible detection of the magnetic field vector with a Fe–Sn heterostructure-based Hall device. Compared with conventional semiconductor Hall sensors and magnetoresistance sensors, the advantages of our device are a compact device in a planar geometry with a single current source and a room-temperature fabrication process with abundant non-toxic elements. The simple configuration of the 3D magnetometer enables us to expand the possibility of implementing the various conventional sensors in smart electronics. The aspect of the low-cost sputtering process of the Fe–Sn-based magnetometers is an advantage in integration into the conventional silicon-based circuits or flexible substrate. The observation of the substantial signal from UMR in a SiOx/Fe–Sn/Al2O3 structure implies the importance of a specific band structure with the spin–orbit interaction at the heterointerface. The device performance will be further improved by applying extensive knowledge of topological material science owing to large AHE and specific UMR effect originating from the electronic band topology and spin–orbit interaction22,32,33,34.

Methods

Sample preparation

The heterostructure consisting of 4-nm-thick Fe0.62Sn0.38 ferromagnetic layer and SiOx cap layer was grown by co-sputtering technique at room temperature on Al2O3(0001) single crystal substrate. The thickness was measured by X-ray reflectivity (Supplementary Note 1 and Supplementary Fig. 1). The Fe/Sn composition ratio was calibrated using electron energy dispersive X-ray spectroscopy in a 48-nm-thick film deposited at the identical condition with the 4-nm-thick sample. We examined the uniformity of the Fe composition in as wide as a few mm2 in the film (Supplementary Note 1 and Supplementary Table 1). The device was fabricated with the SiOx/Fe–Sn/Al2O3 heterostructure by photolithography and Ar ion milling. The Ti/Pt electrodes were defined by photolithography and deposited by ion beam sputtering at room temperature after in situ Ar ion milling.

Electrical measurements

The device was placed in a variable temperature insert equipped with a superconducting vector magnet. Details of the measurement setup are shown in Supplementary Note 2 and Supplementary Figure 2. The measurement was performed by a lock-in technique at 300 K. The primary component of the channel resistance and Hall resistance were measured at the same frequency 13 Hz of the current by lock-in amplifiers. The secondary harmonic component of the resistance was measured by a lock-in amplifier with a second harmonic mode.