Disentangling Spin, Anomalous, and Planar Hall Effects in Ferromagnet–Heavy-Metal Nanostructures

Ferromagnet (FM)℃heavy-metal (HM) nanostructures can be used for magnetic state readout in the proposed magnetoelectric spin-orbit logic by locally injecting a spin-polarized current and measuring the spin-to-charge conversion via the spin Hall effect. However, this local configuration is prone to spurious signals. Here, we address spurious Hall effects that can contaminate the spin Hall signal in these FM/HM T-shaped nanostructures. The most pronounced Hall effects in our ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}/\mathrm{Pt}$ nanostructures are the planar Hall effect and the anomalous Hall effect generated in the ferromagnetic nanowire. We find that the planar Hall effect, induced by misalignment between magnetization and current direction in the ferromagnetic wire, is manifested as a shift in the measured baseline resistance, but does not alter the spin Hall signal. In contrast, the anomalous Hall effect, arising from the charge-current distribution within the FM, adds to the spin Hall signal. However, the effect can be made insignificant by minimizing the shunting effect via proper design of the device. We conclude that local spin injection in FM/HM nanostructures is a suitable tool for measuring spin Hall signals and, therefore, a valid method for magnetic state readout in prospective spin-based logic.

Figure 1

(a),(b) False-colored top-view SEM images of a FM/HM nanostructure with ISHE and SHE measurement configurations, respectively, and orientation of the external magnetic field $H_x$. Blue indicates the CoFe electrode and yellow marks the T-shaped Pt nanostructure. (c),(d) Evolution of the transverse resistance ($R_T^{\mathrm{ISHE}}$ in ISHE configuration and $R_T^{\mathrm{SHE}}$ in SHE configuration, respectively) as a function of $H_x$ (trace and retrace) measured at $I_{\mathrm{bias}}=50\mu \mathrm{A}$ and 300 K. Difference between low- and high-resistance states (dashed black lines; gray shaded area represents associated error) is the experimentally obtained spin Hall signal $2\mathrm{\Delta }{R}_{(\mathrm{I})\mathrm{SHE}}^{\mathrm{expt}.}$.

Figure 2

(a) Representation of a FM electrode with magnetization m and the effect on m of misalignment angle ${\alpha }_0$ between the applied current bias $I_{\mathrm{bias}}$ and an external in-plane magnetic field H. Left panel: At small H, m aligns with the easy axis (i.e., longest dimension of the FM electrode), which is parallel to $I_{\mathrm{bias}}$, such that $\phi =0$ and the PHE contribution is zero. Right panel: At large H, m aligns with H, such that $\phi ={\alpha }_0$ and the PHE contribution is nonzero. (b) Transverse resistance $R_T^{\mathrm{ISHE}}(\alpha )$ as a function of the in-plane angle $\alpha$ [Fig. 1] at $I_{\mathrm{bias}}=50\mu \mathrm{A}$ and 300 K with a fixed magnetic field of 5 kOe. Black solid dots are measured data, whereas the red dashed line indicates fitting to Eq. (2). Fit can be separated into an ISHE component and a PHE component, as shown by blue and cyan curves, respectively. (c) Magnification of the ISHE component presented in (b) indicated by the orange square. Cyan dashed line presents the magnitude of the PHE signal at $\alpha =0^{\circ }$ and $\alpha ={180}^{\circ }(R_{\mathrm{PHE}}^{0^{\circ }\mathrm{\&}{180}^{\circ }})$. (d) Transverse resistance $R_T^{\mathrm{ISHE}}$ versus applied magnetic field in the x direction, as given in Fig. 1 after subtraction of the baseline obtained from the fit ($b=5.8001\mathrm{\Omega }$). High-resistance state corresponds to $\alpha =0^{\circ }$, whereas low-resistance state corresponds to $\alpha ={180}^{\circ }$. Shift of the transverse resistance by an amount of $R_{\mathrm{PHE}}^{0^{\circ }\mathrm{\&}{180}^{\circ }}$ (cyan dashed line) is observed at large magnetic fields, where m and $I_{\mathrm{bias}}$ are misaligned. Experimental spin Hall signal $2\mathrm{\Delta }{R}_{\mathrm{ISHE}}^{\mathrm{expt}.}$ is the same as the spin Hall signal obtained from fitting of the angle dependence of the transverse resistance $2\mathrm{\Delta }{R}_{\mathrm{ISHE}}^{\mathrm{fit}}$.

Figure 3

(a) Geometry, mesh, and electric potential map of 3D FEM model used for the FM/HM nanostructure to estimate the AHE in the ISHE configuration. 3D color map of electrical potential induced by a bias current $I_{\mathrm{bias}}$ (white arrow) forced from the $\mathrm{Co}\mathrm{Fe}$ electrode (port 1) to the $\mathrm{Pt}$ electrode (port 2). As all equations used for simulations are linear, voltage values correspond to an injected current arbitrarily set at 1 A. Inset: Side view of the device (x-z plane) picturing the charge-current density lines. It shows the existence of a z component in charge-current density, which gives rise to the AHE in the $\mathrm{Co}\mathrm{Fe}$ electrode. (b) Out-of-plane magnetic field dependence of transverse resistivity in a $\mathrm{Co}\mathrm{Fe}$ cross-shaped nanostructure. Anomalous Hall resistivity ${\rho }_{\mathrm{AHE}}$ extracted from this measurement is used to determine the strength of the AHE. Inset: AHE measurement setup with the $\mathrm{Co}\mathrm{Fe}$ Hall cross depicted in blue. (c) $\mathrm{Co}\mathrm{Fe}$ electrode-thickness dependence of the AHE signal obtained using 3D FEM simulations. Line is a guide to the eye. AHE can be eliminated by optimizing the thickness of the $\mathrm{Co}\mathrm{Fe}$ electrode. Optimum $\mathrm{Co}\mathrm{Fe}$ thickness at which the AHE contribution vanishes is about $13$ nm (red dashed line) in this device with a $\mathrm{Pt}$ electrode thickness of 8 nm and resistivities as presented in the text.

Figure 4

(a) False-colored top-view SEM image of the FM/HM nanostructure, where the FM ($\mathrm{Co}\mathrm{Fe}$) and HM ($\mathrm{Pt}$) electrodes are indicated by blue and yellow, respectively. The SHE measurement configuration is displayed. The in-plane rotation of the magnetic field is described by angle $\alpha$. (b) Transverse resistance $R_T^{\mathrm{SHE}}$ as a function of $\alpha$ at 300 K with a fixed magnetic field of 5 kOe. Black dots are measured data, and the red dashed line is a fit to Eq. (2). The fit is separated into an SHE component and a PHE component shown by blue and cyan curves, respectively. (c) Magnification of the ISHE component marked by the orange square in (b). The cyan dashed line presents the magnitude of the PHE signal at $\alpha =0^{\circ }$ and $\alpha ={180}^{\circ }(R_{\mathrm{PHE}}^{0^{\circ }\mathrm{\&}{180}^{\circ }})$. (d) Transverse resistance $R_T^{\mathrm{SHE}}$ versus applied magnetic field in the x direction, as given in Fig. 1 after subtraction of the baseline obtained from the fitting $(b=5.8121\mathrm{\Omega })$. The low-resistance state corresponds to $\alpha =0^{\circ }$, whereas the high-resistance state corresponds to $\alpha ={180}^{\circ }$. A shift of the transverse resistance by an amount $R_{\mathrm{PHE}}^{0^{\circ }\mathrm{\&}{180}^{\circ }}$ (cyan dashed line) is observed at large magnetic fields, where m and $I_{\mathrm{bias}}$ are misaligned. The experimental spin Hall signal $2\mathrm{\Delta }{R}_{\mathrm{ISHE}}^{\mathrm{expt.}}$ is the same as the spin Hall signal obtained from fitting the angle dependence to the transverse resistance $2\mathrm{\Delta }{R}_{\mathrm{ISHE}}^{\mathrm{fit}}$.

Figure 5

(a) Ordinary Hall measurement configuration at the $\mathrm{Pt}$ Hall cross fabricated next to the local spin-injection device. (b) Angular dependence of the Hall resistance $(R_{xy}^{\mathrm{OHE}})$ as a function of angle $\beta$ with a fixed magnetic field of 50 kOe. Fitting to a $\cos (\beta )$ function is show as a red line.

Figure 6

(a) Simulated mean magnetization components of $\mathrm{Co}\mathrm{Fe}$ in the area where the $\mathrm{Co}\mathrm{Fe}$ wire overlaps with the $\mathrm{Pt}$ wire as a function of a field, $H_x$, applied along the $\mathrm{Co}\mathrm{Fe}$ wire’s long axis. (b) Snapshots of the in-plane magnetization (see color wheel) at a height of z = 9 nm, which corresponds to the vertical middle of the $\mathrm{Co}\mathrm{Fe}$ electrode ($t_{\mathrm{Co}\mathrm{Fe}}=15\mathrm{nm}$, with 3 nm cell height), at specific fields during the switching of magnetization. Location of the snapshots in magnetic field are indicated by blue dots in panel (a). (c) Mean stray-field components generated in the $\mathrm{Pt}$ wire overlapping with the $\mathrm{Co}\mathrm{Fe}$ wire as a function of field $H_x$. The maximum value of $⟨B_{\mathrm{stray},z}⟩$ reaches about 70 mT and saturates to about 25 mT at high saturation fields. (d) Snapshots of the z component of the stray field in the $\mathrm{Pt}$ wire at z = 21 nm (i.e., 6 nm above the $\mathrm{Co}\mathrm{Fe}$ wire) evaluated at the same fields as $\mathrm{Co}\mathrm{Fe}$ magnetization in panel (b). Green dots in panel (c) indicate the magnetic field position of the snapshot.

Figure 7

Expected shape of transverse resistance $R_T^{\mathrm{ISHE}}(H)$ loop induced by the PHE contribution. (a),(b) Sketch of alignment of current $I_{\mathrm{bias}}$, magnetization $\mathbf{m}$, and external magnetic field $\mathbf{H}$ at high (green areas) and low magnetic fields (yellow areas) for negative and positive misalignment angles, ${\alpha }_0$, between $I_{\mathrm{bias}}$ and $\mathbf{m}$. When $I_{\mathrm{bias}}$ and $\mathbf{m}$ are parallel, the transverse resistance is equal to the ISHE resistance, i.e., $R_{\mathrm{ISHE}}$, if no other Hall effects are involved. If there is misalignment between $I_{\mathrm{bias}}$ and $\mathbf{H}$, misalignment is transferred to $\mathbf{m}$ above a certain threshold value of $\mathbf{H}$; therefore, the PHE contribution $R_{\mathrm{PHE}}$ appears only at magnetic fields above this threshold value. (c),(d) $R_T^{\mathrm{ISHE}}(H)$ loop as a function of applied magnetic field in the ISHE configuration, for negative and positive misalignment angles, respectively. Effect of the PHE on $R_T^{\mathrm{ISHE}}$ is twofold. First, PHE shifts the baseline signal down (negative misalignment angle) or up (positive misalignment angle). Second, PHE induces specific shapes (dips and upturns) close to the switching field of magnetization in the magnetic field dependence.