Hanle lineshapes and spin-rotation signatures from in-plane anisotropic spin relaxation in heterogeneous spin devices

Abstract

Spin precession experiments in lateral spin devices are a powerful tool for probing the spin transport properties of materials. These experiments can be quantitatively described using the Bloch diffusion equation, which offers a practical framework for modeling spin-related phenomena. In this work, we present calculations of the spin density across heterogeneous, diffusive spintronic devices. The modeled devices feature spin transport channels that include both isotropic and in-plane anisotropic spin relaxation regions. We analyze how different geometric configurations and spin transport parameters influence the lineshape of spin precession signals under magnetic fields applied in different orientations and compare with experimental observations. Our results introduce a theoretical framework for interpreting spin transport measurements in lateral graphene spin devices. The framework is especially relevant when the graphene is partially proximitized by other two-dimensional materials, where proximity-induced spin-orbit coupling leads to anisotropic spin relaxation.

Figures (9)

• Figure 1: (a) Schematics of a heterogeneous lateral spin device comprising one anisotropic and four isotropic regions (Reg. I-V). Region III denotes the anisotropic region, where spins oriented along the three spatial directions can exhibit different lifetimes. A pure spin current flows between the injector (FM1) and detector (FM2) ferromagnetic electrodes. The injected spins, that then diffuse along $x$, are initially aligned with $y$, the easy magnetization axis of FM1 and FM2. The red arrows, labeled $x'$ and $y'$, denote the principal axes of spin relaxation, which do not need to coincide with $x$ and $y$; they are rotated by an angle $\phi$ with respect to the device axes. An external magnetic field $B$ is used to induce spin precession and vary the orientation of the spins in region III. (b) Spin-dependent electrochemical potentials $\mu_x$, $\mu_y$, $\mu_z$ across the device for $\phi=40^\circ$, with $B_y=0$ (solid curves) and $B_y=20 \ $mT (dashed curves). Although spins are injected along $y$, anisotropic relaxation yields a nonzero $\mu_x$ even at $B_y=0$. Applying $B_y$ induces spin precession, as spins are no longer parallel to $y$, which leads to the generation of an additional $\mu_z$ component (dashed blue curve). Inset: Schematic representation of the direction of dominant SOF (large red arrow) and the spin orientation at the injection point and after diffusing through the anisotropic region (brown arrows).
• Figure 2: Spin rotation in the anisotropic region. (a) $\delta$ vs. $\phi$ for different $\tau_{x'}$. Inset: definition of $\delta$. In the limit where the $y'$ spin component is fully suppressed ($\tau_{y'}<\tau_{x'}$ by definition), spins align with $x'$, yielding $\delta = 90\degree-\phi$. (b) $\delta$ vs. $\zeta_{x'y'}$ for different $w_\mathrm{H}$ at $\phi=25\degree$. In (a) and (b) $\zeta_{x'y'}$ is modulated by varying $\tau_{x'}$ and keeping $\tau_{y'}$ fixed at 20 ps. (c) $\delta$ vs. $\phi$ for different $w_\mathrm{H}$. As $w_\mathrm{H}$ increases, the $y'$ spin component decays exponentially faster than the $x'$ component, and $\delta$ approaches $90\degree-\phi$. (d) $\delta$ vs. $\phi$ for different $\tau_{y'}$. (e)$\, \delta \,$vs. $\zeta_{x'y'}$ for different $w_\mathrm{H}$. In (d) and (e) $\zeta_{x'y'}$ is modulated by varying $\tau_{y'}$ and keeping $\tau_{x'}$ fixed at 600 ps. (f) $\delta$ vs. $\zeta_{x'y'}$ for different $\phi$. The values of all the other parameters are listed in Table$\ $\ref{['Box:Par1']}.
• Figure 3: Dependence of the nonlocal resistance $R_\mathrm{nl}$ lineshape on $\phi$ and $\zeta_{x'y'}$. (a) $R_\mathrm{nl}$ vs. $B_y$ at different $\phi$ when $\zeta_{x'y'}=20$. Curves are normalized to the value of $R_\mathrm{nl}$ when $\phi=90 \degree$. (b)$\ R_\mathrm{nl}$ vs. $B_y$ for different $\zeta_{x'y'}$ when $\phi= 40 \degree$. $\zeta_{x'y'}$ is modulated by changing $\tau_{y'}$ and keeping $\tau_{x'}$ constant at 600$\ $ps. Curves are normalized to the value of $R_\mathrm{nl}$ when $\zeta_{x'y'}=1$. The values of all the other parameters are listed in Table$\ $\ref{['Box:Par1']}.
• Figure 4: Dependence of the in-plane spin precession lineshape on the device geometry. (a) $R_\mathrm{nl}$ vs. $B_z$ precession curves for different $l_\mathrm{d}$ and constant $l= 3 \, \micro$m. (b) $R_\mathrm{nl}$ vs. $B_z$ precession curves for different $l$ and constant $l_\mathrm{d}= 3 \, \micro$m. Notice that the curves in (a) are a mirror image of the curves in (b). All the curves are individually normalized to the value of $R_\mathrm{nl}$ at $B=0$. Spin transport parameters and $w_\mathrm{H}$ are listed in Table$\ $\ref{['Box:Par1']}.
• Figure 5: Spin precession lineshape under an out-of-plane magnetic field $B_z$ as a function of $\phi$. (a) $R_\mathrm{nl}$ vs. $B_z$ precession curves for selected $\phi$ in a symmetric device ($l=l_\mathrm{d}= 5 \ \micro$m). The curves remain symmetric about $B_z=0$ for all $\phi$. (b, c) $R_\mathrm{nl}$ vs. $B_z$ for different $\phi$ in an asymmetric device ($l=3 \ \micro$m, $l_\mathrm{d}= 7 \ \micro$m) with $\phi \in [0\degree,90\degree]$ in (b) and $\phi \in [90\degree,180\degree]$ in (c). The curves in (b) are a mirror image of the curves in (c) with respect to $B_z=0$. Even in an asymmetric device, the curves are symmetric when $\phi=\pi/2 \ n$ (dashed lines). Spin transport parameters and $w_\mathrm{H}$ are listed in Table$\ $\ref{['Box:Par1']}.