Evidence of orbital Hall current induced correlation in second harmonic response of longitudinal and transverse voltage in light metal-ferromagnet bilayers

In transverse harmonic Hall measurements, the first harmonic Hall resistance provides insight into the equilibrium direction of magnetization. Conversely, the second harmonic signal offers information regarding the tilting of magnetization induced by the spin current[5] and also the Nernst effect which arises from heating[22]. An AC I=$I_{0}sin(\omega t)$ is applied to the bilayer devices along the x-axis and a transverse resistance is measured along the y-axis, where $I_{0}$ is the current amplitude and $\omega$=2$\pi\times 13s^{-1}$ is the angular frequency. The current is distributed between the light metal(LM) and ferromagnetic material(FM) based on their respective resistivities. To maintain uniform current density in the LM, we applied a parallel-circuit model to the bilayer configuration which is written as,
$J_{lm}$=$\frac{I_{total}}{(w_{lm}\times t_{lm})[1+(\frac{\rho_{lm}}{\rho_{fm}})\times(\frac{t_{fm}}{t_{lm}})]}$ and $J_{fm}$=$\frac{I_{total}\times(\frac{\rho_{lm}}{\rho_{fm}})\times(\frac{t_{fm}}{t_{lm}})}{(w_{fm}\times t_{fm})[1+(\frac{\rho_{lm}}{\rho_{fm}})\times(\frac{t_{fm}}{t_{lm}})]}$.
In light metal, the application of current induces a nonequilibrium state, leading to a transverse flow of orbital angular momentum and the emergence of a finite orbital current $J_{OH}$[10]. This orbital current does not directly interact with the static magnetization of the ferromagnetic metal; hence, it needs to undergo conversion into spin current for interaction. Recent findings suggest that certain ferromagnetic materials can facilitate the conversion of orbital current into spin current through the utilization of the spin-orbit coupling (SOC) inherent in the material. However, recent predictions highlight Nickel(Ni) as having superior efficiency in converting orbital current to spin current compared to other materials, which exhibit insufficient conversion capabilities[13, 14, 17, 19]. Current-induced effective fields ($B_{SL}$ and $B_{FL}$) induce periodic oscillations on the magnetization around its equilibrium position. The first harmonic Hall resistance measurement is equivalent to the DC resistance measurement (external magnetic field, $B_{ext}$ or time-independent) but the higher harmonic signals are strongly dependent on $B_{ext}$. For small oscillations of the magnetization, the Hall resistance $R_{xy}(t)$ can be expanded up to the first order[22]. This expansion allows for the characterization of the sample’s magnetic properties and the determination of the strength and behavior of $B_{SL}$ and $B_{FL}$. By analyzing the first harmonic and second harmonic responses, valuable information about the SOT effects can be obtained. The angular dependence of $R_{xy}^{2\omega}$ is given by[23, 24, 25]

Where, $R_{AHE}$ is the anomalous Hall resistance, $B_{Oe}$ is the Oersted field, $B_{eff}^{k}$ is the effective anisotropy field, $\phi$ is the angle between magnetization and current, $\alpha$ is the ordinary Nernst (ONE) co-efficient, $R_{{\Delta}T}^{0}$ is the anomalous Nernst (ANE) co-efficient, and $R_{PNE}$ is the planar Nernst coefficient.

To analyze the SOT effects, the values of $R_{AHE}$ and $B_{k}^{eff}$ are essential parameters. The values of $R_{AHE}$ and $B_{k}^{eff}$ were obtained through an out-of-plane magnetic field ($B_{z}$) sweep spanning a range from +1.1 T to -1.1T(see supplementary). To extract any effects resulting from field misalignment, the data was antisymmetrized. By performing a linear fit to the high-field region, the value of $R_{AHE}$ was extracted. Simultaneously, the interception of low-field region and high-field region linear fit provided the value of $B_{k}^{eff}$. we have shown $2\omega$ signal for two different (30mT and 200mT) magnetic fields in FIG2. and from there we can say that the oscillations of $2\omega$ signal suppress by the increase of the value of $B_{ext}$. By fitting the graph between the Coefficient of $cos(\Phi)and\frac{1}{B_{ext}+B^{k}_{eff}}$with the equation shown above, we were able to extract $B_{SL}$ from the graph and calculated the torque efficiency per unit applied current density as[26]

Where e, $\hbar$, $M_{s}$, $B_{SL}$, $t_{FM}$, and $J_{NM}$ are the electronic charge, reduced Planck’s constant, saturation magnetization, induced SL(FL) fields, the thickness of the FM, and current density flowing through the heavy metal layer respectively. The torque efficiency of all the devices is shown in Fig. Fig(5). We also calculated the field-like effective field and their corresponding Oersted field based on Eq.(1), as shown in the supplementary information. By conducting control tests on a single ferromagnet layer without any other layer above it, we were able to rule out the possibility of self-induced torque caused by spin-polarized current flowing in that layer in our samples.

Analogous to the physics underlying spin unidirectional magnetoresistance (UMR), the magnetoresistance (MR) contribution in this system depends on the current direction. Specifically, the resistance of the bilayer increases when the majority spin direction in the ferromagnetic layer aligns parallel to the spin accumulation and decreases when they are antiparallel.[6, 7, 8, 9]. We simultaneously performed second-harmonic magnetoresistance (MR) measurements with torque measurements to investigate unidirectional magnetoresistance (UMR). Our findings in this work demonstrate that orbital UMR which depends on the current or magnetization direction, can be observed in ferromagnets and light metals, even when heavy metals with large soc are not taken into consideration.The angular dependence of $R_{xx}^{2\omega}$ is given by[27, 28]

Where $\Delta R_{xx}^{1\omega}$ is the change in the first harmonic resistance. $R^{*}=gR_{\Delta T}+R_{UMR}$, The term $R^{*}$ refers to the longitudinal magnetoresistance, which takes into account the combined effects of thermal voltage$\sim(\Delta T\times m).x$ and UMR$\sim J_{x}m_{y}$ influences.g = l/w is the geometric factor, with l and w representing the length and width of the hall bar, respectively. The value is consistently 4($\approx\frac{\Delta R^{1\omega}_{xx}}{\Delta R^{1\omega}_{xy}}$) for all devices, and $R_{\Delta T}$ is determined from the field dependence of the previously mentioned second harmonic Hall signal.

We focused on the first term in the equation3 by measuring the angular dependency of $R^{2\omega}_{xx}$, as illustrated in figure 4, which aligns well with the equation as mentioned earlier, yielding R*=-0.0023,-0.0021,-0.001 and -0.0008 for Ni/Ti, NiFe/Ti, Ni/Nb and NiFe/Nb, respectively. Accordingly, the UMR is attributed to the higher magnetoresistance in all of the devices shown in Fig.(5). An amplified unidirectional magnetoresistance (UMR) is observed in Ni/Ti and Ni/Nb bilayers compared to NiFe/Ti and NiFe/Nb. This enhancement, which we term "orbital UMR," is attributed to scattering between orbital angular momentum and the substantial spin polarization associated with the ferromagnetic magnetization in Ni. The increased orbital contribution in Ni-based systems, relative to NiFe-based systems, suggests that orbital-to-spin scattering interactions play a significant role in enhancing the UMR in these structures. A striking difference between orbital and spin UMR is the absence of low-field enhancement of UMR in LM/FM bilayers compared to HM/FM bilayers. This discrepancy arises because, in HM/FM systems, spin current can excite or annihilate magnons, leading to electron-magnon scattering that enhances UMR at low fields. In contrast, this mechanism is less active in LM/FM bilayers, resulting in a reduced low-field response. In the case of Ni/Ti bilayers, no low-field enhancement of UMR is observed. However, in Ni/Nb bilayers, a low-field enhancement is present, which we attribute to the spin current generated by Nb. This observation is further verified with NiFe/Nb devices, as shown in Fig.5(d). The absence of orbital-magnon coupling can be attributed to the nature of magnons as bosonic spin excitations. Since magnons represent collective spin oscillations, they do not directly couple to orbital angular momentum, leading to a decoupling between orbital effects and magnonic spin dynamics. However, an orbital current can directly influence the electrical conductivity in a ferromagnet through orbital-selective s→d transitions. This interaction enhances the UMR due to increased scattering arising from orbital contributions, leading to the observed amplification in systems where orbital scattering plays a significant role. This orbital scattering effect, which enhances UMR, is absent in single ferromagnetic (FM) layer devices. The absence of additional layers restricts the generation of orbital currents and their associated s→d transitions, resulting in no such enhancement in the UMR for single FM layers. The second term in Eq. (3), representing the field-like (FL) effective field with the Oersted field, was also calculated and closely matches the FL term obtained from the Hall measurements(see Supplementary Material).