Spin-fluctuation-induced sign reversal of the spin Hall angle in Pt100−x Co x alloys near the Curie temperature

The spin Hall effect (SHE), typically emerging in non-magnetic metals with strong spin-orbit couplings, has attracted significant attention for its ability to convert a charge current into a spin current, a key feature in power-efficient spintronic devices. Recently, an enhanced SHE has been detected in the magnetic alloys, where the spin Hall conductivity is strongly modified by the dynamical and thermal spin fluctuations. We find that the spin Hall angle ( θSH ) in Pt100−x Co x alloys dramatically changes at the Curie temperature, which is positive in the paramagnetic phase akin to Pt, while negative in the ferromagnetic phase. Such intriguing behavior of θSH stemming from individual and collective fluctuations in the magnetic moments is further substantiated with the full-fledged Monte Carlo simulations. Our work broadens insights into the SHE and highlights the importance of spin fluctuations for the spin-current generation near the ferromagnetic instability point of magnetic alloys.

The manipulation of magnetizations through the injection of electric current is a central issue in modern magnetism and is also an indispensable element for the low energy consumption in spintronic devices [1][2][3][4][5], where generating spin currents via the spin Hall effect (SHE) is crucial [6][7][8][9].Generally, the SHE stems from the spin orbit coupling (SOC), so that prior studies have focused predominately on the nonmagnetic heavy metals (HMs) with a strong SOC [10,11], such as Pt, W and Ta.However, these metals are far from fulfilling the demand of the energy saving in practical applications [12], and a high use efficiency of the SHE for spin currents, which is measured experimentally by the spin Hall angle θ SH = σ SH /σ C with σ SH(C) being the spin Hall (charge) conductivity, is desirable.A direct approach to improve the charge-spin conversion, which has been extensively adopted in experiments, is to enhance σ SH and suppress σ C simultaneously by alloying or doping HMs with nonmagnetic impurities [13][14][15][16][17][18].An example is the Au-Ta alloys with 10% Ta dopant, possessing a θ SH as high as 0.5 [18].Recently, an alternative pathway for the efficient generation of spin currents has been proposed through the introduction of magnetic dopants into HMs.This leads to prominent spin-dependent scattering, driven by the spin fluctuation of magnetic moments at finite temperatures [19][20][21].Essentially, the enhancements in the SHE originate from the dynamical spin fluctuations of individual magnetic moments [22,23], collective thermal magnetic fluctuations [24,25], and quantum spin fluctuations dominant at low temperatures.These results in a rather distinct temperature dependence of the SHE near the ferromagnetic instability.With the help of Monte Carlo simulations, Gu et al [22] have shown that the SOC in Au-Fe alloys is strongly renormalized by spin fluctuations.Based on this theory, the nontrivial temperature dependence of the SHE in Ni-Pd alloys [24] is attributed to the skew scattering via the collective spin fluctuations of Ni ions near the Curie temperature (T C ) conforming with the sharp peak of the SHE at T C in Pt-Fe alloys [25], which can be measured by the harmonic response spin orbit torque (SOT) techniques.Nevertheless, the strong SHE enhancements in the Ni-Pd and Fe-Pt alloys occur in a rather narrow temperature regime near T C restraining their possible applications.
In the present work, we have selected the Pt-Co alloys to avoid the complicated effects on the electronic and magnetic properties from the doped magnetic ions, as Co atoms are less susceptible to local environmental perturbations [26,27].In stark contrast to the dip-peak structure in Pt-Fe alloys [25], we found that the θ SH in Pt-Co alloys undergoes a dip structure below T C , arising from the nonlocal magnetic correlations, and then increases dramatically with temperature accompanied by a sign reversal to a saturated value, which persists in the paramagnetic phase instead of forming a peak.This asymmetry sign reversal of θ SH around T C arises from the contributions of individual Co moments and has not yet been reported to the best of our knowledge.
The samples were deposited by magnetron sputtering onto corning glass substrates at room temperature, with an argon pressure of 2.5 m Torr and a base pressure of 4 × 10 −9 Torr.To ensure the homogeneity of the thin films, the substrates were rotated at 20 rpm during sputtering.Pt 100−x Co x alloy films, where x represents the atomic proportion of Co, were prepared by co-sputtering the Pt and Co targets with diameters of 3 inches.The sputtering power of the Pt target is fixed at 100 W, and the atomic proportion of Co (x) was determined by x-ray photoelectron spectroscopy, which was adjusted by varying the sputtering power of the Co target.The single layers of Pt 100−x Co x reveal the basic properties of the alloys, such as the lattice constant, T C and the temperature-dependent resistivity.Meanwhile, the multilayers of Ta(3)/Pt 100−x Co x (4)/Pt(1)/Co(0.9)/Ta(4)(hereafter denoted with shorthand Ta/Pt 100−x Co x /Pt/Co/Ta, the numbers in the parenthesis denote the thicknesses of corresponding layers in units of nm) are prepared for the magneto-transport measurements.In these multilayers, the top and bottom Ta layers serve as anti-oxidation and seed layers, respectively.The Co layer, magnetized perpendicularly, detects the injected spin currents from the Pt-Co alloys, and the 1 nm thick Pt layer maintains a perpendicular magnetic anisotropy (PMA).The multilayers are patterned to the Hall bars with a dimension of 20 × 200 µm 2 by the standard photolithography and Ar-ion etching techniques.The damping-like and field-like SOTs are measured via the harmonic Hall voltage method using an Analog-Digital/Digital-Analog data acquisition card with a sinusoidal current at different temperatures.
Figure 1(a) illustrates that the x-ray diffraction patterns of the Pt 100−x Co x single layers exhibit a peak near 40 degrees, adjacent to the (111) peak of Pt in the face-centered cubic structure.This peak progressively shifts towards higher angles with increasing x, indicating the formation of the Pt 100−x Co x alloy during the co-sputtering process.As shown in figure 1(b), the lattice constant of Pt 100−x Co x alloys calculated from the Bragg equation drops as x grows reflecting a larger atomic radius of Pt compared with Co atoms.The resistivity of Pt 100−x Co x alloys as functions of x and temperature (figures 1(b) and (c)) increases remarkably after the doping, indicating that the Co atoms provoke significant scattering in the electronic transport.In order to obtain the T C of the Pt 100−x Co x , the temperature dependence of the magnetizations was measured under a 100 Oe in-plane field using VersaLab.Then, T C was obtained by finding the maximum derivativity of M(T) curve, as shown in the inset of figure 1(d).As expected, the obtained T C of the Pt 100−x Co x is found to be proportional to the concentration of Co atoms, as depicted in the inset.
In Ta/Pt 100−x Co x /Pt/Co/Ta multilayers, the Co layer has strong PMA due to the interfacial magnetic anisotropy at the Pt/Co interface with Pt having (111) texture.However, it remains uncertain whether the ferromagnetic phase Pt 100−x Co x alloy in these multilayers also possesses PMA.To investigate this, the hysteresis loops corresponding to the in-plane and out-of-plane magnetic fields were analyzed.Figure 2(b) reveals that the magnetic hysteresis loops feature a squared shape for the out-of-plane field and almost straight lines for the in-plane field, suggesting that the Pt 100−x Co x in Ta/Pt 100−x Co x /Pt/Co/Ta likely exhibits PMA.Furthermore, the temperature dependence of anomalous Hall resistance in the Ta/Pt 80 Co 20 /Pt/Co/Ta multilayer parallels the magnetization-temperature relationship in Pt 80 Co 20 , as shown in figure 2(c), further supporting the presence of PMA in the magnetic moment of the alloy layer.
The behaviors of the SHE in Ta/Pt 100−x Co x /Pt/Co/Ta are quantified by the SOT efficiency using the first (V 1ω ) and the second (V 2ω ) harmonic Hall voltages, subjected to a sinusoidal current (I x ) with a frequency of 133 Hz.The effective fields of the damping-like (H DL ) and field-like (H FL ) are obtained by sweeping the applied fields H x and H y , as defined in figure 2(a), and they can be calculated using the following equations [28,29]: where ∆H DL(FL) is the corrected damping-like (field-like) effective field after removing the contribution from the planar Hall effect.ξ = ∆R P /∆R A is the proportion of the planar Hall resistance ∆R P to the anomalous Hall resistance ∆R A .
Figures 2(d  To quantify this further, we calculate the θ SH corresponding to the damp-like field based on the following equation [30]: where e is the elementary charge, M s is the saturation magnetization, t FM is the thickness of the Co-layer, h is the reduced Planck constant.The renormalized temperature θ SH for various multilayers, as shown in figure 4(a), exhibits a dramatic change in the vicinity of T C , regardless of the composition of the alloys.Below T C , the negative θ SH decreases markedly, leading to a pronounced dip as the temperature approaches T C , and it then bounces back to a positive plateau as crossing T C .The θ SH of Pt-Fe paramagnetic phase has been reported, unlike our results, the θ SH of Pt-Fe is non-monotonic with temperature, and the peak of θ SH occurs when the temperature is close to T C due to strong spin fluctuations near the phase transition.The difference of θ SH above T C in Pt-Co and Pt-Fe maybe caused by the fact that the local field has an infinitesimal impact on the magnetic moment of cobalt in Pt 100−x Co x alloys [26,27], in contrast to Pt-Fe alloys [25] where the magnetic moment of Fe surrounded by Pt atoms is strongly altered.First-principles calculations suggest that the local magnetic moment of Co atoms is more stable compared to Fe atoms, as detailed in the supplementary materials figures S9 and S10.According to previous reports, the Rashba effect is temperature-independent [31,32].It is well known that the Pt-and Ta-layer exhibit positive and negative θ SH respectively, and, retain their signs regardless of temperature changes.However, the spin currents produced by Pt-and Ta-layer are relatively small compared to Pt 100−x Co x alloys, because the top Ta-layer is oxidized and the Pt-layer is only 1 nm thick.Therefore, the nontrivial behavior of θ SH in figure 4  reversal of θ SH is caused by this voltage, the strength of θ SH should gradually increase as the temperature decreases in the ferromagnetic phase of Pt 100−x Co x because the voltage is increasing.However, the θ SH obtained by the experiment does not increase monotonically with decreasing temperature but shows a peak in the ferromagnetic instability point of Pt 100−x Co x .In other words, the influence of the voltage caused by out-of-plane magnetic moment in Pt 100−x Co x is not the cause of the abnormal temperature dependence of θ SH .In addition, the polarized-neutron investigation has shown that the unpaired 3d electrons of Co lie dominantly on the t 2g -orbital independent of the magnetic order [33], giving rise to multi-orbital Kondo scattering with SOC around T C .Meanwhile, the small difference in the work function between Co and Pt lowers the charge transferring effect highlighting the role of thermal spin fluctuations in the SHE.Therefore, the sign reversal of θ SH of Pt-Co alloys near T C originates from the spin-orbit interactions between the conduction electrons and local t 2g -orbitals of Co atoms rather than the intrinsic SOC of the Pt element.
To explore the physical origin of the anomaly in spin Hall resistivity near T C , we performed systematic first-principles calculations as well as MC simulations for thermal fluctuations of magnetic moments (cf the details in Supporting Information).Within the Kondo physics [34,35], the behaviors of SHE are primarily determined by the d-electrons in the t 2g -orbital whose local density of states peak around 0.5 Ev regardless of the occupation of Co impurities, as shown in figures 4(b) and (c).We therefore expect that the orbital-dependent Kondo effects involving local and band electrons in t 2g -orbitals survive in Pt-Co alloys at high temperatures, and the SOC of t 2g -orbitals will give rise to the resonant skew scattering responsible for the dramatic changes at the critical points with a strong spin fluctuation.More quantitatively, the spin Hall resistivity can be written formally as: with K being a constant coefficient.B 2 is determined by the spin-spin correlations up to the second Born approximation as below )} sin 3 θdθdϕ (5) where F 0 , F 1 can be approximated as temperature independent parameters, θ and ϕ are the angles for spherical coordinates with respect to the longitudinal current,c 2 depends on the structure of the material, M n is the magnetic moment of the localized electron of the nth ion.k and k ′ are wave vectors of the conduction electrons, R n is the position of the nth ion.M n (⟨M n ⟩) is the (thermally averaged) magnetic moment of the localized electron in the nth ion.The temperature dependences of r 2a and r 2b are simulated based on the three-dimensional Heisenberg model with 10 3 lattice points.As shown in figures 4(d) and (e), the nearest-neighbor and on-site correlations diverge with a sign reversal at T C , in spite of distinct temperature characteristics.Near T C , the fourth-order correlations B 2 jumps from the negative extreme value to a positive saturation as the temperature increases, suggesting an unstable skew scattering during the magnetic phase transition.Theoretical analyzes unveil that this behavior is connected with the high-order thermal fluctuation of magnetic moments.Above T C , the third-order spin-spin correlations corresponding to the anomalous Hall effect vanish due to the symmetry, while the fourth-order correlations B 2 that engenders the SHE persists and is dominated by the on-site contribution without any momentum transfer.It appears that the dependence of B 2 on temperature closely resembles that of θ SH , hinting that the nontrivial dependence of θ SH on the temperature indeed springs from the on-site and off-site thermodynamic perturbations of the magnetic moment fluctuation.Furthermore, the monotonic increase of resistivity with the temperature near T C reinforces that the abrupt change in θ SH near T C owes predominantly to σ SH subjected to spin fluctuations rather than σ C .Moreover, according to our experimental results, the temperature at which the negative extreme value of θ SH appears diminishes continuously with the increase of Co content (cf figure 4(a)), consistent with the Monte-Carlo simulations, i.e. the larger magnetization results in a stronger contribution from the nearest neighbors.
In conclusion, the SHE in Pt 100−x Co x alloys has been measured by the harmonic Hall voltage measurement technique.The θ SH in Pt 100−x Co x alloys with x = 12, 16 and 20 is negative in the ferromagnetic phase, whereas positive in the paramagnetic phase accompanied with a sign reversal around T C , reflecting unstable spin Hall resistivity in the magnetic phase transition.The theoretical analyzes based on the first-principles calculations and MC simulations reveal that this feature can be attributed to Kondo scattering of d-electrons in the t 2g -orbital from the magnetic moment fluctuation.In particular, MC simulations well reproduce the nontrivial temperature dependence of θ SH , validating that the magnetic moment fluctuation plays a significant role to determine the behaviors of the SHE during the phase transition.Our findings deepen the understanding of SHE and highlight the importance of spin fluctuations for spin-current generation near the ferromagnetic instability.

Figure 1 .
Figure 1.(a) XRD patterns of Pt 100−x Cox alloys.(b) The lattice constant and resistivity at room temperature with the various atomic proportions of Co in Pt 100−x Cox alloys.(c) The temperature dependence of the resistivities of Pt88Co12, Pt84Co16and Pt80Co20 layers.(d) The temperature dependence of the magnetization in Pt 100−x Cox alloys with the magnetic field around 100 Oe.Inset: the Curie temperature TC as a function of x in Pt 100−x Cox alloys.

Figure 2 .
Figure 2. (a) A schematic of the Hall bar device.(b) Out-of-plane hysteresis loops of Ta/Pt80Co20/Pt/Co/Ta at different temperatures.The inset shows the in-plane hysteresis loops of Ta/Pt80Co20/Pt/Co/Ta. (c) Anomalous Hall resistance (red point) of Ta/Pt80Co20/Pt/Co/Ta and magnetization of Pt80Co20 (black point) as a function of temperature.The first harmonic Hall voltage (V 1ω ) (d) and second harmonic Hall voltage (V 2ω ) (e) as a function of the in-plane magnetic field (Hx) along the current direction.(f) The effective fields of the corrected damping-like SOT versus the current density.
) and (e) show the dependences of V 1ω and V 2ω on H x in the magnetization-up state for the Ta/Pt 80 Co 20 /Pt/Co/Ta multilayer, respectively.The slope of the linear function fitting the relation of V 2ω vs. H x declines gradually with the temperature, and a sign reversal arises when the temperature is increased from 240 K to 280 K with T C = 245 K for Pt 80 Co 20 .However, this behavior is absent in the Ta/Pt/Co/W samples in which θ SH is positively defined (see the supplementary materials S3).The current dependences of △H DL are displayed in figure2(f), and the slopes of the fitted curves define β DL = −1.55,−7.16, −0.553 and
(a) is most likely caused by the unusual SHE in Pt 100−x Co x layers.Although the out-of-plane magnetic moment in Pt 100−x Co x contributes to the Hall voltage in harmonic Hall voltage measurements, this voltage cannot account for the non-linear temperature dependence and sign reversal of θ SH .The magnetism of Pt 100−x Co x gradually increases as the temperature decreases, leading to a corresponding rise in the magnetic voltage.If the sign

Figure 4 .
Figure 4. (a) The normalized temperature (T/TC) dependence of θSH.First-principles calculations of density of states for magnetic elements of Co with SOC near the Fermi level in (b), (c) Co25Pt75.r2a and r 2b are obtained by the Monte Carlo simulations in the three-dimensional Heisenberg model under the impacts of the nearest neighbor term (d) and the on-site term (e), respectively.(f) Parameter B2, which is proportional to the SHE intensity, is obtained by considering the impacts of the on-site term and the nearest neighbor term from Monte Carlo simulations.