Dephasing of transverse spin current in ferrimagnetic alloys

It has been predicted that transverse spin current can propagate coherently (without dephasing) over a long distance in antiferromagnetically ordered metals. Here, we estimate the dephasing length of transverse spin current in ferrimagnetic CoGd alloys by spin pumping measurements across the compensation point. A modified drift-diffusion model, which accounts for spin-current transmission through the ferrimagnet, reveals that the dephasing length is about 4–5 times longer in nearly compensated CoGd than in ferromagnetic metals. This finding suggests that antiferromagnetic order can mitigate spin dephasing—in a manner analogous to spin echo rephasing for nuclear and qubit spin systems—even in structurally disordered alloys at room temperature. We also find evidence that transverse spin current interacts more strongly with the Co sublattice than the Gd sublattice. Our results provide fundamental insights into the interplay between spin current and antiferromagnetic order, which are crucial for engineering spin torque effects in ferrimagnetic and antiferromagnetic metals.

Figure 1

Dephasing of a coherent transverse spin current excited by ferromagnetic resonance (FMR) in the spin source. The spin current carried by electrons is coherent in the normal metal (NM) spacer layer (indicated by the aligned black arrows), but enters the spin sink with different incident wave vectors (dashed gray lines). (a) In the ferromagnetic metal (FM) spin sink, the propagating spins accumulate different precessional phases in the ferromagnetic exchange field (red vertical arrows) and completely dephase within a short distance. (b) In the ideal antiferromagnetic metal (AFM) or ferrimagnetic metal (FIM), the spin current does not dephase completely in the alternating antiferromagnetic exchange field (blue and green vertical arrows), as any precession at one sublattice is compensated by the opposite precession at the other sublattice. In the case of a FIM that is an alloy of a transition metal (TM; such as Co) and rare-earth metal (RE; such as Gd), the TM constitutes one sublattice (e.g., blue vertical arrows) and the RE constitutes the other sublattice (e.g., green vertical arrows).

Figure 2

(a) Hysteresis loop of ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$(7)/Cu(4)/${\mathrm{Co}}_{75}{\mathrm{Gd}}_{25}$ (12) (unit: nm). The ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ and ${\mathrm{Co}}_{75}{\mathrm{Gd}}_{25}$ magnetizations switch separately at in-plane fields $\approx$ 0.2 mT and $\approx$ 13 mT, respectively. (b), (d), (f) Saturation magnetization $M_s$ and (c), (e), (g) coercivity ${\mu }_0{H}_c$ of ${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ with thicknesses of 40 nm [(b), (c)], 10 nm [(d), (e)], and 5 nm [(f), (g)].

Figure 3

(a) Half-width-at-half-maximum (HWHM) FMR linewidth versus frequency for stacks with a spin sink [NiFe/Cu/Co($d$: nm)], a stack without a spin sink (NiFe/Cu), and a stack with an insulating Ti-oxide spin blocker before the spin sink (NiFe/Cu/$\mathrm{Ti}{\mathrm{O}}_x$/Co). (b)–(d) FMR linewidth versus frequency for stacks with different thicknesses ($d$: nm) of ${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ spin sinks, where $x=23$ (b), 25 (c), and 28 (d). The slope [proportional to the damping parameter $\alpha$; see Eq. (2)] saturates at $d\approx 1$ nm for the FM Co spin sink (a), whereas the slope saturates at a much larger $d$ for the FIM ${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ spin sinks [(b)–(d)].

Figure 4

Nonlocal damping enhancement $\mathrm{\Delta }\alpha$ versus spin sink thickness $d$ for CoGd spin sinks with different compositions.

Figure 5

(a) Cartoon schematic of transverse spin current transport in the spin-source/spacer/spin-sink stack. The FMR-driven spin source pumps a coherent transverse spin current (polarized along the $x$ axis) through the Cu spacer. The spin current reflected at the Cu (spacer)/CoGd (spin sink) interface is parametrized by the reflected spin-mixing conductance ${\tilde{g}}_2^{\uparrow \downarrow }$, whereas the spin current transmitted through the CoGd spin sink is parametrized by the transmitted spin-mixing conductance $g_t^{\uparrow \downarrow }$. (b) Illustration of the dephasing of the transverse spin current propagating in the CoGd spin sink. The shrinking circles represent the decay of the transverse spin polarization due to dephasing while it precesses about the effective net exchange field $H_{\mathrm{ex}}^{\mathrm{net}}$. (c) Illustration of the oscillatory decay of the transverse spin current. The complex transmitted spin-mixing conductance $g_t^{\uparrow \downarrow }$ captures the transmitted transverse spin polarization components $s_x$ and $s_y$.

Figure 6

Left column: Spin sink thickness dependence of nonlocal damping enhancement $\mathrm{\Delta }\alpha$. The solid curve indicates the fit using the modified drift-diffusion model. Right column: The real and imaginary parts of the transmitted spin-mixing conductance $g_t^{\uparrow \downarrow }$ derived from the fit. The vertical dashed line indicates the spin dephasing length ${\lambda }_{\mathrm{dp}}$.

Figure 7

(a) Static magnetic properties (saturation magnetization $M_s$ and coercive field $H_c$) reproduced from Fig. 2, (b) spin dephasing length ${\lambda }_{\mathrm{dp}}$, and the (c) real and (d) imaginary parts of the interfacial contribution of the transmitted spin-mixing conductance $g_{t,0}^{\uparrow \downarrow }$ versus Gd content in the spin sink. The shaded region indicates the window of composition corresponding to magnetic compensation. The error bars in (b)–(d) show the 95% confidence interval.

Figure 8

Transmitted spin-mixing conductance $g_t^{\uparrow \downarrow }$ versus spin sink thickness $d$ near the magnetic compensation composition, derived from the modified drift-diffusion model fit (Fig. 6). Note the phase shift in $\mathrm{Im}\left[g_t^{\uparrow \downarrow }\right]$ for ${\mathrm{Co}}_{75}{\mathrm{Gd}}_{25}$ (b). The cartoons on the right illustrate the net precession direction of the transverse spin polarization in each CoGd spin sink. ${\mathrm{Co}}_{75}{\mathrm{Gd}}_{25}$ exhibits retrograde precession, opposite to the precession direction in the other spin sink compositions.

Figure 9

FMR spectrum of ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$(7)/Cu(4) at 6 GHz. The red curve represents the fit using Eq. (A1).

Figure 10

FMR spectra for ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$(7 nm)/Cu(4 nm)/${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$($d$ nm) with no spin sink (magenta) and with spin sink layer of pure Co (black), Gd 25% (red and green), and Gd 28% (blue and light blue). NiFe and Co exhibit well-separated FMR. The different thicknesses for Gd 25% and 28% was selected to show two different transport regimes, i.e., below/above the ${\lambda }_{\mathrm{dp}}$ found from Fig. 6. No FMR signal attributable to ${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ in our samples ($20\le x\le 30$) is detected.

Figure 11

Composition dependence of the renormalized reflected mixing conductance ${\tilde{g}}_r^{\uparrow \downarrow }$ at the Cu/${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ interface with the ${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$ spin sink thickness $d\gg {\lambda }_{\mathrm{dp}}$.

Figure 12

Dependence of the spin dephasing length ${\lambda }_{\mathrm{dp}}$ and interfacial transmitted spin-mixing conductance $g_{t,0}^{\uparrow \downarrow }$ of ${\mathrm{Co}}_{72}{\mathrm{Gd}}_{28}$ on the spin-flip length ${\lambda }_{\mathrm{sf}}$ in the modified drift-diffusion model.

Figure 13

Resistivity $\rho$ with varying Gd at. % for $\mathrm{Si}{\mathrm{O}}_x$(thermally oxidized)/${\mathrm{Co}}_{100-x}{\mathrm{Gd}}_x$(10)/$\mathrm{Ti}{\mathrm{O}}_x$(3) (unit: nm). The resistivities of intermediate compositions between $x=20$ and 30 are derived via linear interpolation.

Figure 14

Momentum-resolved $g_{t,0}^{\uparrow \downarrow }$ versus the in-plane momentum ${\kappa }^2$. $\mathrm{Re}\left[g_{t,0}^{\uparrow \downarrow }\right]$ in blue and $\mathrm{Im}\left[g_{t,0}^{\uparrow \downarrow }\right]$ in red.

Figure 15

Interfacial transmitted mixing conductance versus the exchange splitting $k_{\mathrm{\Delta }}$. $\mathrm{Re}\left[g_{t,0}^{\uparrow \downarrow }\right]$ in blue and $\mathrm{Im}\left[g_{t,0}^{\uparrow \downarrow }\right]$ in red.

Figure 16

Comparison of modeling results with (a) free $\mathrm{Re}\left[g_{t,0}^{\uparrow \downarrow }\right]$ and (b) fixed $\mathrm{Re}\left[g_{t,0}^{\uparrow \downarrow }\right]$. The shaded region indicates the window of composition corresponding to magnetic compensation.

Figure 17

Schematic of a ferrimagnetic trilayer as modeled below. The two magnetic sublattices are denoted by red and blue arrows pointing in opposite directions.

Figure 18

Dependence of the (a) reflected and (b) transmitted mixing conductance as a function of the ferrimagnetic layer thickness upon varying the exchange on sublattice B. The real (imaginary) part of the mixing conductance is reported in blue (red). The unit is in ($e^2/h$) and $\mathrm{per}\mathrm{n}{\mathrm{m}}^2$.

Figure 19

Inverse of the dephasing length as a function of the magnetic exchange.