Theory of spin-Hall magnetoresistance in the ac terahertz regime

In bilayers consisting of a normal metal (N) with spin-orbit coupling and a ferromagnet (F), the combination of the spin-Hall effect, the spin-transfer torque, and the inverse spin-Hall effect gives a small correction to the in-plane conductivity of N, which is referred to as spin-Hall magnetoresistance (SMR). We here present a theory of the SMR and the associated off-diagonal conductivity corrections for frequencies up to the terahertz regime. We show that the SMR signal has pronounced singularities at the spin-wave frequencies of F, which identifies it as a potential tool for all-electric spectroscopy of magnon modes. A systematic change of the magnitude of the SMR at lower frequencies is associated with the onset of a longitudinal magnonic contribution to spin transport across the F-N interface.

Figure 1

Explanation of the SMR in a ferromagnet$|$normal-metal ($\mathrm{F}|\mathrm{N}$) bilayer: Via the spin-Hall effect, an electric field applied to N creates a spin accumulation at its boundaries. The spin accumulation at the F-N boundary drives a spin current from N into F. This spin current is converted into a charge current in N via the inverse spin-Hall effect. For a time-dependent driving field, additionally magnons (spin waves) can be excited in F. The possibility to coherently excite magnons is what distinguishes the THz SMR from its DC counterpart.

Figure 2

$\mathrm{F}|\mathrm{N}$ bilayer with a normal metal of thickness $d_{\mathrm{N}}$ and a ferromagnet of thickness $d_{\mathrm{F}}$. Coordinates are chosen such that the $xy$ plane is the interface between the F and N layers. The electric field $\mathbf{E}$ is applied in the $x$ direction.

Figure 3

Equivalent AC magnetoelectronic circuit diagrams for $\delta {\sigma }^{xx}$. The left circuit shows the correction from the transverse spin accumulation at the F-N interface; the right one shows the correction from the longitudinal spin accumulation. The dashed lines in the upper part indicate the transport of charge, while the solid lines in the lower part indicate the transport of spin angular momentum. Spin currents and spin accumulations are related via (spin versions of) Ohm's law and Kirchoff's circuit laws, see Eqs. (11, 12, 13, 14, 15, 16, 17, 18, 19). The impedances with labels ${\sigma }_{\mathrm{SH}}$ and arrows indicate the conversion between charge and spin currents due to the SHE and ISHE, see Eqs. (8, 9, 10). The magnetization direction enters via the projection on longitudinal and transverse components, ${\sigma }_{\mathrm{SH}\parallel }={\theta }_{\mathrm{SH}}\hslash {\sigma }_{\mathrm{N}}{m}_y/2e$ and ${\sigma }_{\mathrm{SH}\perp }={\theta }_{\mathrm{SH}}\hslash {\sigma }_{\mathrm{N}}{(1-m_y^2)}^{1/2}/2e$. The spin currents $j_{\mathrm{SH}\parallel }^z$ and $j_{\mathrm{SH}\perp }^z$ generated via the SHE are given by $j_{\mathrm{SH}\parallel }^z={\theta }_{\text{SH}}\hslash {\sigma }_{\mathrm{N}}E{\mathbf{e}}_y·{\mathbf{e}}_{\parallel }/2e={\sigma }_{\mathrm{SH}\parallel }E$ and $j_{\mathrm{SH}\perp }^z={\theta }_{\text{SH}}\hslash {\sigma }_{\mathrm{N}}E{\mathbf{e}}_y·{\mathbf{e}}_{\perp }/2e={\sigma }_{\mathrm{SH}\perp }E$. The fact that spin is not a conserved quantity—neither in N due to flips of electron spins, nor in F due to Gilbert damping—is reflected by the presence of “spin sinks” in N and F, represented as dissipation channels to the “ground” in the circuit diagram.

Figure 4

Numerical estimates for the spin impedances of a $\mathrm{YIG}|\mathrm{Pt}$ bilayer as a function of the frequency $\omega /2\pi$. For the impedance $Z_{\mathrm{N}}$ the large-$d_{\mathrm{N}}$ limit is shown; for $|Z_{\mathrm{F}\perp }\left(\omega \right)|$ both the large-$d_{\mathrm{F}}$ limit (solid red line) and the finite-$d_{\mathrm{F}}$ case (dotted red line) are shown, as discussed in the main text. Parameter values are taken from Table 1. The inset shows the same impedances on a logarithmic scale in the GHz frequency regime. The horizontal lines indicate $Z_{\mathrm{N}}$ (blue dashes, main panel and inset), $Z_{\mathrm{FN}\perp }$ (green dot-dashes, inset), and $Z_{\mathrm{FN}\parallel }^{\mathrm{m}}$ (light-purple dots, inset).

Figure 5

Real and imaginary parts of the dimensionless characteristics ${\mathrm{\Delta }}_{\mathrm{SMR}}$ of the spin-Hall magnetoresistance (top) and ${\mathrm{\Delta }}_{\mathrm{AHE}}$ of the anomalous Hall effect (bottom) for a $\mathrm{YIG}|\mathrm{Pt}$ bilayer. The thick solid lines use the asymptotic value $Z_{\mathrm{F}\perp }^{\infty }\left(\omega \right)$ of Eq. (56); thin dashed lines use the full expression for $Z_{\mathrm{F}\perp }\left(\omega \right)$, which includes the effect of spin-wave resonances in the YIG layer. Material and device parameters are taken from Table 1.

Figure 6

Same as Fig. 5 but for an $\mathrm{Fe}|\mathrm{Au}$ bilayer. Parameter values are taken from Table 1.

Figure 7

Real part (top) and imaginary part (bottom) of the combined impedance $Z_{\parallel }^{\mathrm{m}}\left(\omega \right)$ from Eq. (A7) for a ballistic F layer (dashed and dot-dashed curve) and of the impedance sum $Z_{\mathrm{FN}\parallel }^{\mathrm{m}}+Z_{\mathrm{F}\parallel }^{\mathrm{m}}\left(\omega \right)$ (solid curves) of Secs. 3 and 4f, 4g. Material and device parameters are taken from Table 1.

Figure 8

Comparison of the characteristics ${\mathrm{\Delta }}_{\mathrm{SMR}}$ (dashed and dot-dashed curve) and ${\overline{\mathrm{\Delta }}}_{\mathrm{SMR}}$ (solid curves) with and without inclusion of the Oersted field. Thick (colored) curves are for the case with Oersted field; thin (black) curves are for the case without it. The left and right panels show real and imaginary parts, respectively. Parameter values are taken from Table 1.

Figure 9

Dimensionless characteristic ${\overline{\mathrm{\Delta }}}_{\mathrm{SMR}}$, see Eq. (69), of a $\mathrm{Fe}\left|\mathrm{Au}\right|\mathrm{Fe}$ trilayer (dashed blue line) and a $\mathrm{Fe}|\mathrm{Au}$ bilayer (solid red line) as function of the thickness of the Au layer. The $\mathrm{Fe}\left|\mathrm{Au}\right|\mathrm{Fe}$ trilayer has identical F layers with parallel magnetization directions and a thickness $d_{\mathrm{N}}$, while the thickness of the Au layer in the $\mathrm{Fe}|\mathrm{Au}$ bilayer is $d_{\mathrm{N}}/2$. (This ensures that the magnetic interface/volume ratio in both geometries is same.) Parameter values are taken from Tables 1 and 2. The two curves are for $\omega /2\pi =0$ Hz. For comparison, $d_{\mathrm{N}}^{\left(0\right)}=6\times {10}^{-8}$ m is the thickness of the N layer of Ref. [69], which is also the reference thickness used for the numerical estimates of Sec. 5.