Anatomy of electrical signals and dc-voltage line shape in spin-torque ferromagnetic resonance

Yin Zhang, Q. Liu, B. F. Miao, H. F. Ding, and X. R. Wang

Phys. Rev. B 99, 064424 – Published 19 February 2019

Abstract

The electrical detection of spin-torque ferromagnetic resonance (st-FMR) is becoming a popular method for measuring the spin-Hall angle of heavy metals (HM). However, various sensible analysis on the same material with either the same or different experimental setups yielded different spin-Hall angles with large discrepancy, indicating some missing ingredients in our current understanding of st-FMR. Here we carry out a careful analysis of electrical signals of the st-FMR in a HM/ferromagnet (HM/FM) bilayer with an arbitrary magnetic anisotropy. The FM magnetization is driven by two radio-frequency (rf) forces: the rf Oersted field generated by an applied rf electric current and the so called rf spin-orbit torque from the spin current flowing perpendicularly from the HM to the FM due to the spin-Hall effect. By using the universal form of the dynamic susceptibility matrix of magnetic materials at the st-FMR, the electrical signals, originated from the anisotropic magnetoresistance, anomalous Hall effect, and inverse spin-Hall effect are analyzed and dc-voltage line shapes near the st-FMR are obtained. Angular dependence of dc voltage is given for two setups. A way of experimentally extracting the spin-Hall angle of a HM is proposed.

Received 30 October 2018
Revised 25 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.064424

Physics Subject Headings (PhySH)

Magnetism Spintronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yin Zhang^1,2, Q. Liu³, B. F. Miao^3,4, H. F. Ding^3,4,*, and X. R. Wang^1,2,†

¹Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
²HKUST Shenzhen Research Institute, Shenzhen 518057, China
³National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 22 Hankou Road, Nanjing 210093, China
⁴Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 22 Hankou Road, Nanjing 210093, China

^*Corresponding author: hfding@nju.edu.cn
^†Corresponding author: phxwan@ust.hk

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Vol. 99, Iss. 6 — 1 February 2019

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Images

Figure 1
Model system that mimics the experimental setups of st-FMR. The $x y z$ coordinate is fixed with respect to the sample. The HM/FM bilayer sample lies in the $x y$ plane. The $X Y Z$ is a moving coordinate with the $Z$ axis along $M_{0}$ and the $Y$ axis in the $x y$ plane. $θ$ and $ϕ$ are the polar and azimuthal angles of $M_{0}$ in the $x y z$ coordinate, i.e., $θ$ is the angle between the $Z$ and $z$ axes and $ϕ$ is the angle between the in-plane component of $M_{0}$ and the $x$ axis. $θ_{H}$ and $ϕ_{H}$ are the polar and azimuthal angles of the external static magnetic field $H$ in the $x y z$ coordinate. $j_{FM}$ and $j_{HM}$ are, respectively, the rf electric current in the FM and HM layers.
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Figure 2
Schematic illustration for two experimental configurations of st-FMR. A HM/FM bilayer lies in the $x y$ plane. The stable magnetization $M_{0}$ is in the sample plane by applying an in-plane static magnetic field $H$ . $ϕ$ is the angle between $M_{0}$ and the $x$ axis, and $ϕ_{H}$ is the angle between $H$ and the $x$ axis. The definitions of the $x y z$ and $X Y Z$ coordinates are the same as Fig. 1 with $θ = 90^{o}$ . The rf electric currents $j_{FM}$ and $j_{HM}$ are along the $x$ -direction, and the rf Oersted field $h$ is along the $y$ direction according to the Ampere's law. The displacement vector between two electrodes is along (a) the $x$ or (b) the $y$ directions.
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Figure 3
Comparison between the current theory (solid curves) and experimental results (squares). (a) The fitting of data from Ref. [21] for Pt(6)/Py(4) (red) and Pt(15)/Py(15) (green), respectively. The Pt/Py parameters are $M = 6.4 \times 10^{5}$ A/m for Pt(6)/Py(4) [or $M = 8.34 \times 10^{5}$ A/m for Pt(15)/Py(15)], $α = 0.028$ for Pt(6)/Py(4) [or $α = 0.0123$ for Pt(15)/Py(15)], $Δ ρ σ_{FM} = 2 %, g_{eff}^{↑ ↓} = 3.02 \times 10^{19} m^{- 2}$ , and $θ_{SH} = 0.05$ . $Δ ρ j_{HM} j_{FM} l_{x} = 1.536 \times 10^{8} V A m^{- 2}$ for Pt(6)/Py(4) and $Δ ρ j_{HM} j_{FM} l_{x} = 7.862 \times 10^{7} V A m^{- 2}$ for Pt(15)/Py(15) are used for the fittings. (b) The fitting of data from Ref. [22] for Pt(6)/Py(5.5). The Pt/Py parameters are $g_{eff}^{↑ ↓} = 3.96 \times 10^{19} m^{- 2}, M = 7.5 \times 10^{5}$ A/m, $α = 0.016$ , and $θ_{SH} = 0.045$ . (c) The fitting of data from Ref. [23] for Pt(4)/Py(2.5). The Pt/Py parameters are $g_{eff}^{↑ ↓} = 3.02 \times 10^{19} m^{- 2}, M = 5.8 \times 10^{5}$ A/m, $α = 0.043$ , and $θ_{SH} = 0.05$ . (d) The fitting of data from Ref. [24] for Pt(6)/Py(5). The Pt/Py parameters are $g_{eff}^{↑ ↓} = 3.02 \times 10^{19} m^{- 2}, M = 6.97 \times 10^{5}$ A/m, $α = 0.02$ , and $θ_{SH} = 0.061$ .
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Figure 4
Angular dependence of $A_{1}$ in units of $A_{0} = \frac{Δ ρ j_{HM} j_{FM} l_{x} a θ_{SH}}{2 α (2 H_{0} + M)} \frac{ℏ}{2 e}$ for the setup shown in Fig. 2. The model parameters are $ω = 2 π \times 9.0$ GHz, $M = 8.0 \times 10^{5}$ A/m, and easy axis anisotropy coefficient $K_{x} = 0.0$ (black curve) or $K_{x} = 0.05$ (blue curve). The black curve is plotted according to Eq. (42), and the blue curve is numerically calculated from Eq. (45).
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Figure 5
Schematics of new ISHE [(a) and (b)] and SHE [(c) and (d)] in magnetic materials. (a) For a spin current flowing along the spin polarization direction, a charge current can be generated along the magnetization perpendicular to spin flowing direction (as well as the spin polarization). (b) For a spin current flowing perpendicular to the spin polarization direction, a charge current flows along the spin flowing direction when the magnetization is along the spin polarization direction. (c) For a charge current flowing along the magnetization, the generated spin current flows perpendicular to the charge current and the spin polarization is along the spin flow direction. (d) For a charge current flowing perpendicular to the magnetization, the generated spin current is along the charge current and the spin polarization is along the magnetization.
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