Detection and quantification of inverse spin Hall effect from spin pumping in permalloy/normal metal bilayers

Spin pumping is a mechanism that generates spin currents from ferromagnetic resonance over macroscopic interfacial areas, thereby enabling sensitive detection of the inverse spin Hall effect that transforms spin into charge currents in nonmagnetic conductors. Here we study the spin-pumping-induced voltages due to the inverse spin Hall effect in permalloy/normal metal bilayers integrated into coplanar waveguides for different normal metals and as a function of angle of the applied magnetic field direction, as well as microwave frequency and power. We find good agreement between experimental data and a theoretical model that includes contributions from anisotropic magnetoresistance and inverse spin Hall effect. The analysis provides consistent results over a wide range of experimental conditions as long as the precise magnetization trajectory is taken into account. The spin Hall angles for Pt, Pd, Au, and Mo were determined with high precision to be $0.013\pm 0.002$, $0.0064\pm 0.001$, $0.0035\pm 0.0003$, and $-0.0005\pm 0.0001$, respectively.

Figure 1

(Color online) Schematic model of spin pumping in Py/N bilayer. $\vec{s}$ shows that the polarization of the spin current is oscillating in time with the frequency of the magnetization precession. The polarization of the spin current is perpendicular to the instantaneous magnetization direction $\vec{m}$ and the rate of magnetization change $\partial \vec{m}/\partial t$.

Figure 2

(Color online) Experimental setup: (a) Optical image of the Py/Pt bilayer integrated into the coplanar waveguide. (b) Contacts are added at the end of the bilayer to measure the voltage along the waveguide direction.

Figure 3

(Color online) Experimental data (symbols) for FMR peak positions and FMR linewidths of a Py/Pt bilayer are shown as a function of rf frequency in (a) and (b), respectively. The solid line in (a) represents the fit to Eq. (11) and results in $M_s=852\text{G}$. The solid line in (b) represents a linear fit to the linewidth vs frequency dependence.

Figure 4

(Color online) Derivative FMR spectra at 4 GHz for (a) Py and (b) Py/Pt. Solid lines are Lorentzian line-shape fits.

Figure 5

(Color online) Voltage measured along the samples vs field $H_{\text{dc}}$ for Py and Py/Pt at 4 GHz is shown with symbols in (a) and (b), respectively. Only the AMR contribution is present in the Py sample. Solid line in (a) shows a fit to Eq. (12). Both AMR and ISHE effects are observed in the Py/Pt. Dotted and dashed-dotted lines show the AMR and ISHE contributions, respectively, which are extracted from fitting the data to Eqs. (12, 18); the solid line in (b) shows the combined fit for the Py/Pt sample.

Figure 6

(Color online) Splitting of rf current due to capacitive coupling is schematically shown in (a) together with the directions of the applied dc magnetic field ${\vec{H}}_{\text{dc}}$ and the rf driving field ${\vec{h}}_{\text{rf}}$ with respect to the bilayer and waveguide. (b) Schematic of $\vec{m}$ precessing in Py. $\vec{m}$ precesses around its equilibrium direction given by ${\vec{H}}_{\text{dc}}$ at the driving frequency ${\omega }_f$ and with a phase delay ${\phi }_0$ with respect to $h_{\text{rf}}$. $\alpha$ is the angle between ${\vec{H}}_{\text{dc}}$ and the waveguide axis (along $y$), $\theta$ is the cone angle described by $\vec{m}$ and $\psi$ is the angle between $\vec{m}$ and the waveguide axis. Due to the strong capacitive coupling part of $I_{\text{rf}}$ flows through the Py given by $I_{\text{rf},S}$. (c) Geometry of the dc component of the pumped spin current with polarization direction $⟨\vec{s}⟩$ along the equilibrium magnetization direction ${\vec{m}}_{\parallel }$. The charge current due to ISHE ${\vec{j}}_c^{\text{ISH}}$ is orthogonal to the spin current direction (normal to the interface) and $⟨\vec{s}⟩$. The voltage due to the ISHE is measured along $y$ (waveguide axis). Solid arrows indicate the spin accumulation inside N, which decays with the spin-diffusion length ${\lambda }_{sd}$.

Figure 7

Elliptical precession trajectory results in a time-dependent cone angle of magnetization precession that modifies the dc component of pumped spin current. The ellipticity correction factor $P$ for the dc component of spin current is calculated as a function of microwave frequency according to Eq. (9).

Figure 8

(Color online) (a) and (b) show voltages measured at 4 and 11 GHz. Voltages measured at 11 GHz are smaller due to a decreased precession cone angle. Note that the ratio between the ISHE and AMR contributions change due to a faster decrease in the AMR voltage at high frequencies. (c) Spin Hall angle in Pt as a function of frequency. The slightly decreased values at lower frequencies may be due to nonlinear effects and a concomitant decrease in spin pumping at large angles of magnetization precession.

Figure 9

(Color online) (a) Voltage measured at 4 GHz as a function of angle $\alpha$ of the external magnetic field with respect to the coplanar waveguide axis. Experimental data and fits are shown with symbols and solid lines, respectively. (b) Spin Hall angle extracted from the fits. The theoretical model correctly takes into account the angular dependence for the ISHE and AMR contributions.

Figure 10

(Color online) Power dependence of the symmetric ISHE voltage contribution measured at 4 GHz. The inset shows that the maximum of the ISHE signal is linear vs rf excitation power. The highest power (150 mW) deviates from the linear behavior due to excitation of nonuniform modes. This deviation is also observed in the corresponding FMR spectra.

Figure 11

(Color online) Voltages measured at 11 GHz for (a) Py/Pd, (b) Py/Au, and (c) Py/Mo. Shown are data (symbols), combined fits (black lines), and individual AMR and ISHE contributions, with dotted green and dashed-dotted blue lines, respectively. Note the opposite sign of the ISHE contribution for Mo compared to those for Au and Pd.