Abstract
The spin Hall effect and its inverse have key roles in spintronic devices as they allow conversion of charge currents to and from spin currents. The conversion efficiency strongly depends on material details, such as the electronic band structure and the nature of impurities. Here we show an anomaly in the inverse spin Hall effect in weak ferromagnetic NiPd alloys near their Curie temperatures with a shape independent of material details, such as Ni concentrations. By extending Kondo's model for the anomalous Hall effect, we explain the observed anomaly as originating from the secondorder nonlinear spin fluctuation of Ni moments. This brings to light an essential symmetry difference between the spin Hall effect and the anomalous Hall effect, which reflects the firstorder nonlinear fluctuations of local moments. Our finding opens up a new application of the spin Hall effect, by which a minuscule magnetic moment can be detected.
Introduction
Spintronics has provided an impetus for the intensive development of techniques to generate or detect spin currents, such as nonlocal spin injection^{1}, spin pumping^{2} and spin light emission devices^{3}. The spin Hall effect (SHE)^{4}, which converts a charge current to a spin current, and its inverse have been widely explored in semiconductors^{3,4,5} and nonmagnetic metals^{2,6,7} by using such techniques. Although most of the work has concentrated on the phenomenon and mechanism of the SHE itself, Uchida et al. have used the SHE as a detector of a spindependent chemical potential arising from the spin Seebeck effect^{8,9}. Further applications of the SHE are, however, still lacking. Recently, the doping of gold with individual iron impurities was considered as a mechanism to enhance the SHE through spin fluctuations of the iron moments^{10}. This raises the question whether the SHE might be sensitive to collective fluctuations, in particular, close to a magnetic phase transition where the fluctuations are strong. As for the anomalous Hall effect (AHE) in a pure ferromagnetic metal such as Fe^{11} and Ni^{11,12}, a sharp peak of the Hall resistivity (ρ_{H}) appears only below the Curie temperature (T_{C}). This can be understood as an effect of fluctuations of local magnetic moments^{13}.
In this report, we demonstrate experimentally, and provide a theoretical argument, that the inverse SHE (ISHE) also shows an anomaly, but that it is substantially different from that of the AHE. Figure 1 shows schematic images of the AHE and the ISHE of a weak ferromagnetic material near T_{C}. In the AHE, a charge current is injected into the ferromagnetic material, and the difference of the skew scattering probabilities for spinup and spindown electrons results in a Hall voltage. In the ISHE, on the other hand, a spin current is injected and both spinup and spindown electrons are deflected to the same side, which also results in a Hall voltage. Contrary to the AHE, however, we observe a dip and peak in the ISHE for weak ferromagnetic NiPd alloys below and above T_{C}, respectively. As the quantity of the involved Ni ions are very small, the present system could be used for sensitive magnetometry.
Results
AHE and ISHE of NiPd alloys
We measured the SHE of a weakly ferromagnetic NiPd nanowire with 7, 8 and 9% Ni using the nonlocal spin injection technique^{14,15,16} at 5–50 K. Figure 2a,b shows the schematic structure and scanning electron microscopic image of the lateral spin valve with the NiPd middle wire. The NiPd alloy is a weak ferromagnet when the Ni concentration is larger than 2.3% (ref. 17). Depending on the Ni concentration, its T_{C} varies from a few K to hundreds of K. To characterize the magnetic property of NiPd wires in our devices, AHE measurements were performed for the reference NiPd Hall bars prepared at the same time as the SHE devices. A representative AHE resistance R_{xy} for Ni_{0.08}Pd_{0.92}, which is proportional to the magnetization, is shown in Fig. 3a. We note that in the AHE measurements, the external field H was applied perpendicularly to the plane, which is different from the ISHE measurements (see Fig. 2b). At T=5 K, a hysteresis loop of the magnetization can be clearly seen. With increasing temperature, the magnetization continually decreases to zero and R_{xy} simply shows the ordinary Hall resistance above 30 K. The Curie temperature T_{C} for Ni_{0.08}Pd_{0.92} can be determined to be about 21 K by plotting the remnant magnetization M_{r} and coercive field H_{c} versus temperature (Fig. 3b), which is lower than the value for the bulk alloy^{18}. This is presumably due to the imperfect distribution of Ni in our alloy, prepared by the ion implantation technique, compared with bulk alloys (see Methods). We also performed the same measurements for the other two concentrations where T_{C} for Ni_{0.07}Pd_{0.93} and Ni_{0.09}Pd_{0.91} are 16 and 32 K, respectively.
The ISHE is based on the spin to charge current conversion, which follows the vector product as below: J_{C} J_{S} × s_{C}, where J_{C} and J_{S} are the charge and spin current density, s_{C} is the spin orientation of the conduction electron. Our device configuration is depicted in Fig. 2a. The charge current flows from one of two Ni_{0.81}Fe_{0.19} (hereafter Py) wires, that is, Py1 to the left hand side of the Cu wire. As the spin–orbit (SO) interaction of Cu is quite weak, it works as a transmitter of the generated spin current at the Cu/Py1 interface to the right hand side of the Cu wire. The spin current is preferentially absorbed into the NiPd middle wire below the Cu wire because of its strong SO interaction. This absorption is confirmed by a reduction of the nonlocal spin valve signal detected at the second Py electrode Py2, as detailed in refs 15 and 16. In the case of Ni_{0.08}Pd_{0.92}, the nonlocal spin valve signal is reduced to 0.25 compared with the value without the NiPd middle wire, and the spin diffusion length is estimated to be 10 nm. The absorbed spinup and spindown electrons with the orientation s_{C} parallel to the hard direction of the Py wires are scattered by Ni impurities in Pd, resulting in the Hall voltage V_{ISHE}. From the field variation, the overall change of ISHE resistance R_{ISHE} (equal to V_{ISHE} divided by the charge current) can be observed.
We show representative ISHE resistance R_{ISHE} curves for Ni_{0.08}Pd_{0.92} as a function of the magnetic field at various temperatures in Fig. 3c, where all curves are shifted for clarity. Generally, R_{ISHE} increases linearly with the magnetic field up to 2,000 Oe and then becomes flat. This is consistent with the saturation of the magnetization of Py1, indicated by the anisotropic magnetoresistance curve of Py1 shown in Fig. 3d. In the following, we focus on R_{ISHE} with saturated magnetization of Py1 (that is, H>2,000 Oe). For temperatures far below (10 K; black curve) and above (30 K; pink curve) T_{C}, R_{ISHE} is completely flat at high enough fields, which is as expected in the ISHE for nonmagnetic systems. The values of R_{ISHE} are much larger than those for pure Pd, where an intrinsic mechanism should apply^{16,19,20} and we will argue that this increase comes from the extrinsic effect of the Ni impurities.
For temperatures near T_{C}, in contrast, anomalous behaviour can be clearly seen. At 20 K just below T_{C} (≈21 K), R_{ISHE} decreases with increasing the magnetic field (H) up to 8,000 Oe and then flattens out (see the red curve in Fig. 3c). Just above T_{C}, at 22 K, on the other hand, R_{ISHE} has a positive slope versus magnetic field even above 2,000 Oe and is not yet saturated at our maximum field. By making many similar scans for a full temperature range from 10 to 30 K, we can display this behaviour more clearly seen in the threedimensional plot shown in Fig. 3e. The x and y axes are the magnetic field and temperature, respectively, whereas both the z axis and colour represent R_{ISHE}. When the temperature crosses T_{C}, a reduction first appears at 20 K as a light blue peak at negative fields or a lightyellow dip at positive fields (see red triangles in Fig. 3e); then an enhancement appears at 22 K as a dark blue dip at negative fields or a red peak at positive fields (see blue stars in Fig. 3e). Such a behaviour of R_{ISHE} is observed below and above but only in the vicinity of T_{C}. As already mentioned, R_{ISHE} for a nonmagnetic metal should be completely flat at any temperature. It is thus unambiguous that the observed change in R_{ISHE} of the NiPd alloys near T_{C} is correlated to the magnetic phase transition of the NiPd wire (see Fig. 3b). On the other hand, such an anomaly near T_{C} is not observed in the temperature dependence of the spin absorption rate or the spin diffusion length of NiPd. This is because the spin diffusion length depends simply on the resistivity and the SO interaction of the NiPd wire, whereas the ISHE signal is strongly affected by the type of scattering mechanisms, that is, skew scattering, sidejump or band structure.
Anomaly in ISHE of NiPd alloys near the Curie temperature
To better quantify the contribution to the ISHE intrinsic to the NiPd wire, we define the change in R_{ISHE} as ΔR_{ISHE}≡[R_{ISHE} (H_{max})−R_{ISHE} (−H_{max})]/2, as shown in Fig. 3c. As a maximum field H_{max}, we chose 8,800 Oe in this work. Certainly, R_{ISHE} at 22 K is not fully saturated even at 8,800 Oe, which is different from R_{ISHE} at 20 K. This is related to the fact that it is much more difficult to align the localized magnetic moments of Ni impurities above T_{C} compared with those below T_{C} even if high magnetic fields are applied. In Fig. 4a, we plot ΔR_{ISHE} of Ni_{0.08}Pd_{0.92} (shown by red dots) as a function of temperature. The anomaly of ΔR_{ISHE} near T_{C} is clearly seen. Below 18 K, ΔR_{ISHE} is almost constant with temperature. Only in the vicinity of T_{C}, however, it shows a dip and peak structure at 20 and 21.5 K, respectively. As ΔR_{ISHE} varies little with temperature except near T_{C}, it can be separated into the temperatureindependent background and the anomalous part δΔR_{ISHE}, as . The results for NiPd with 7% and 9% of Ni are plotted with black squares and blue triangles, respectively. The increase of with the doping concentration suggests the extrinsic contribution induced by the Ni impurities. Thus, comes from both the extrinsic SHE of Ni impurities and the intrinsic SHE of Pd^{16}. Despite the fact that depends strongly on the Ni concentrations, δΔR_{ISHE} does so primarily via a shift to be near the corresponding T_{C}. We emphasize that such critical behaviour was reproducible for each of at least three samples at each Ni concentration—thus our arguments are based on more than ten complete data sets, of which only one is shown in Fig. 3e.
For the three concentrations, δΔR_{ISHE} are plotted as a function of the reduced temperature (T−T_{C})/T_{C} in Fig. 4b. Note that each curve corresponds to one full data set: for example, the red line represents a reduction of all of Fig. 3e. Although the three devices have different T_{C} and , the anomalous behaviour appears to be universal near T_{C}: the three curves almost scale onto one. The amplitudes of the dip and peak are about 5 μΩ. The anomalous behaviour disappears when the reduced temperature differs from T_{C} by 10%. These results are obviously different from the case of the AHE^{13}, where the anomaly of the Hall resistivity ρ_{H} appears only below T_{C}.
Discussion
What is the origin of the anomaly of δΔR_{ISHE} observed only in the vicinity of T_{C}? Resistive anomalies near magnetic critical points T_{C} have been studied for a long time. The anomaly near T_{C} in the longitudinal resistivity appears within the first Born approximation from quadratic fluctuations (M_{i}−M_{i})(M_{j}−M_{j}) in the moments M_{i} and M_{j} of localized spins at sites i and j, where contributions come only when i and j are within a certain cutoff distance^{21}. The resistivity then depends on a linear local magnetic susceptibility , which is a summation of shortrange fluctuations up to this distance. To simplify notation we shall denote the sum of such shortrange correlations as ∑(M−M)^{2}, omitting the site subscripts, and we shall use similarly abbreviated notations for higherorder moments. As mentioned in the introduction, the anomaly of ρ_{H} in the AHE near T_{C} was explained by Kondo^{13}. Note that the quantities r_{1} and r_{2} in ref. 13 correspond to a firstorder nonlinear susceptibility and a secondorder nonlinear susceptibility , respectively. We change notation for simplicity, and also to emphasize that Kondo makes a strictly local approximation, whereas we extend the theory to include nearby correlations between atomic moments. The anomaly of ρ_{H} in the AHE appears from the firstorder fluctuation of local moments . The secondorder spin fluctuation is also defined in ref. 13, but made no contribution to the AHE. This will be explained below; we first note that the and terms appear in the sd Hamiltonian to the same order with respect to the SO coupling constant λ of the localized moment. The term in changes sign for spinup or down electrons, whereas does not. Thus, the scattering amplitudes for spinup and down conduction electrons are different and proportional to and , respectively^{13}. As illustrated in Fig. 1, in the case of the AHE configuration, spinup and down conduction electrons are scattered to opposite directions. As a result, ρ_{H} is proportional to the difference of scattering amplitudes between spinup and down electrons, that is, . On the other hand, in the ISHE configuration, both spinup and down electrons are scattered to the same side. Unlike the AHE, the obtained R_{ISHE} is proportional to the sum of scattering amplitudes between spinup and down electrons, that is, . The effect of such terms has been hidden for 50 years^{13}. However, it is naturally expected that the contributions related to can survive if the incident current is the spin current, which is exactly the case for the ISHE.
To make a comparison between theory and experiment, we should, in principle, calculate , which is integrated over the Fermi surface. This should be proportional to the experimental δΔR_{ISHE}. However, the integration depends essentially on the Fermi wave vector, which acts as a cutoff to any divergence. To avoid including materialdependent parameters, we compare with a quantity, the uniform secondorder nonlinear susceptibility , which is independent of cutoff. is the response of the moments on all the sites to a uniform magnetic field and is the sum of correlations over all distances without any cutoff (see Supplementary Discussion). In Fig, 4b we superimpose on the experimental data a meanfield calculation of . It diverges with opposites signs above and below T_{C}. We interpret the dip and peak observed in the experimental data as a smearing of such a divergence. In a full calculation the theoretical spin Hall resistance should also be smeared: the divergence is cutoff, just as for the longitudinal resistance^{21}. To confirm the above scenario, we have performed Monte Carlo simulations for where the contribution of nearestneighbour correlations is taken into account. As can be seen in Supplementary Fig. S1, the smearing of the divergence of at T_{C} is well reproduced. Thus, we argue that neither the full theory nor the experiments should diverge at T_{C}.
Finally, we note that the anomaly of ρ_{H} in the AHE in pure Fe^{11} and Ni^{11,12} may alternatively be explained by Karplus and Luttinger's theory, which is based on band effects^{22}. In this theory, ρ_{H} would be proportional to ρ_{xx}^{2}M where ρ_{xx} is the longitudinal resistivity and M is the averaged magnetization. As M is zero above T_{C}, the sign change of R_{ISHE} below and above T_{C} seems to preclude explanation by such band effects.
In conclusion, we have demonstrated a characteristic behaviour of the SHE at the magnetic phase transition of the weak ferromagnetic NiPd near its Curie temperature T_{C}. It can be explained by the fluctuation contributions to skew scattering via SO interactions, and appears as the secondorder nonlinear spin fluctuation . Considering the small volume of NiPd wire (10^{−16} cm^{3}) and the magnetic moment of NiPd, the total magnetic moment involved in the experiment is extremely small: <10^{−14} emu. Such a tiny moment is far below the sensitivity of magnetometer (for instance, 10^{−8} emu for SQUID). The successful extractions of the contribution of higherorder spin fluctuations in our experiments prove the extremely high sensitivity of our method. This technique could be very useful to understand more complex spin textures such as spinglasses and spinice. For those systems, the linear magnetic susceptibility has been well studied so far but there have been almost no experimental studies of the higherorder spin fluctuations, where some important physics might be still hidden. The present method constitutes a novel approach to detect highorder nonlinear fluctuations with an extremely high sensitivity, which is impossible to achieve by a direct measurement. This will broaden the application of the SHE from potential devices to a means of measuring higherorder spin fluctuations, particularly in the critical region.
Methods
Sample preparation
The spin Hall devices were fabricated on a thermally oxidized silicon substrate using electron beam lithography on poly(methyl methacrylate) resist and a subsequent liftoff process. Here we used a lateral spin valve structure, which consists of two Py wires (30nm thick and 100nm wide) with a separation distance of 1 μm and a NiPd middle wire (20nm thick and 100nm wide) bridged by a Cu wire (100nm thick and 100nm wide) as is detailed in Fig. 2 in the main text. A Pd wire was first deposited by electron beam evaporation using a 99.999% purity source. To form the NiPd alloy wire with a Ni content of 7, 8 and 9%, we implanted a suitable amount of Ni^{+} into the Pd wire using an Ion Implanter (NH20SR) with an acceleration voltage of 30 kV. As is mentioned in the main text, the prepared NiPd alloy is not perfectly homogenous and nickel is distributed over ~10 nm below the Cu/NiPd interface in 20 nm thick Pd wires. This nickel distribution would lower the T_{C} compared with bulk NiPd alloys. However, the spin diffusion length of NiPd of about 10 nm corresponds to the maximum distribution of Ni in Pd. That is why we could still observe the spin fluctuation of Ni moment. The two Py wires were deposited by electron beam evaporation, whereas the Cu bridge was fabricated by a Joule heating evaporator. Before the Cu evaporation, a careful Ar ion beam milling with a beam current of 12 mA and beam voltage of 600 V was performed for 30 s to clean the surfaces of Py and NiPd wires and to obtain highly transparent contacts.
Additional information
How to cite this article: Wei, D.H. et al. The spin Hall effect as a probe of nonlinear spin fluctuations. Nat. Commun. 3:1058 doi: 10.1038/ncomms2063 (2012).
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Acknowledgements
We thank Y. Iye and S. Katsumoto for the use of the lithography facilities. T.Z. also thanks the Advanced Science Research Center of the JAEA for support via the REIMEI program. This work was supported by KAKENHI and a GrantinAid for Scientific Research in Priority Area from MEXT.
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D.H.W. and Y.N. designed the experiments and fabricated devices. D.H.W. collected the data and carried out the analysis. B.G., T.Z. and S.M. developed the theoretical analysis. B.G. developed the detailed equations in Supplementary Information. Y.O. planned and supervised the project. All authors discussed the results and commented on the manuscript.
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Supplementary Figure S1, Supplementary Discussion and Supplementary References (PDF 68 kb)
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Wei, D., Niimi, Y., Gu, B. et al. The spin Hall effect as a probe of nonlinear spin fluctuations. Nat Commun 3, 1058 (2012). https://doi.org/10.1038/ncomms2063
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DOI: https://doi.org/10.1038/ncomms2063
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