Impact of the out-of-plane conductivity on spin transport evaluation in a van der Waals material

Abstract

Layered materials are promising candidates for spintronic applications due to their unique electronic structures and spin transport properties. However, the strong anisotropic conductivity inherent in these materials complicates the quantitative evaluation of spin Hall conductivity and spin diffusion length. In this work, we present a comprehensive study of spin transport in a transition metal dichalcogenide PtTe$_2$ by combining a three-dimensional finite element model with nonlocal spin valve structures. We developed a theoretical model that treats an anisotropic spin diffusion in the same way as the conventional isotropic model, enabling the extraction of spin diffusion lengths along both the in-plane and out-of-plane directions. Our analysis revealed that the conventional isotropic assumption tends to overestimate some values, particularly for the out-of-plane spin diffusion length and spin Hall conductivity. These findings provide new insight into anisotropic spin diffusion and spin-charge conversions in layered materials and emphasize the importance of accounting for anisotropic conductivity in the design of spintronic devices.

Figures (11)

• Figure 1: (a) Schematic of the crystal structure of PtTe$_2$ (space group $P\overline{3}m1$, No. 164), illustrated using VESTA VESTA2011. Pt and Te atoms are shown as blue and yellow spheres, respectively. (b) False-colored SEM image of a typical device used for four-terminal measurements. (c) Cross-sectional TEM image of a PtTe$_2$ thin film along its longer direction.
• Figure 2: Temperature dependence of (a) $\rho_{\parallel}$ in a PtTe$_2$ thin film with thickness $t \approx 30$ nm and different widths, (b) $\rho_{\parallel}$ in a thin film with width $w \approx 100$ nm and different thicknesses, (c) $\rho_{\perp}$ in a bulk PtTe$_2$ (inset: optical microscope image of the sample), and (d) the anisotropy ratio $\rho_{\perp}/\rho_{\parallel}$, with $\rho_{\perp}$ from the bulk crystal and $\rho_{\parallel}$ from a thin film ($t = 30$ nm, $w = 1$$\mathrm{\mu m}$).
• Figure 3: (a) False-colored SEM image of a typical device used for ISHE measurements. The arrow indicates the positive direction of the magnetic field, and the $x$ and $y$ axes define the in-plane coordinate system. (b) Top panel: ISHE resistance $R_{\mathrm{ISHE}}$ of PtTe$_2$ measured at 50 K for Sample D. Red and blue circles represent forward and backward magnetic field sweeps, respectively. Background signals have been subtracted from the raw data. The ISHE resistance amplitude $\Delta R_{\mathrm{ISHE}}$ is defined as the difference between the average values of $R_{\mathrm{ISHE}}$ above $+3000$ Oe and below $-3000$ Oe, as indicated by the dashed lines. Bottom panel: Typical AMR signal of Py, showing the saturation of magnetization above 3000 Oe for $H_{x}$ applied along the hard axis. (c) Temperature dependence of the ISHE resistance amplitude $\Delta R_{\mathrm{ISHE}}$ for Sample D.
• Figure 4: (a) False-colored SEM image of a typical device used for NLSV signal measurements with a PtTe$_2$ nanowire. The arrow represents the positive direction of the magnetic field. (b) NLSV signals without (red) and with (blue) the PtTe$_2$ middle wire measured at $T = 50$ K for Sample A. (c) Temperature dependence of the spin signal amplitudes $R_{\mathrm{s}}^{\mathrm{without}}$ (red) and $R_{\mathrm{s}}^{\mathrm{with}}$ (blue). (d) Temperature dependence of the amplitude ratio $R_{\mathrm{s}}^{\mathrm{with}}/R_{\mathrm{s}}^{\mathrm{without}}$.
• Figure 5: (a) Typical 3D model with renormalized $z$-axis for a PtTe$_2$ nanowire and the calculated spin accumulation voltage in the NLSV geometry. (b) Spin diffusion length of PtTe$_2$ as a function of temperature. The black square shows the spin diffusion length $\lambda_{\mathrm{s}}^{\mathrm{iso}}$ obtained from the conventional isotropic analysis, assuming $\sigma = \sigma_{\parallel}$. The red circle indicates the in-plane spin diffusion length $\lambda_{\mathrm{s}}^{\parallel}$ obtained from the anisotropic analysis. The blue closed (open) circles indicate the out-of-plane spin diffusion length $\lambda_{\mathrm{s}}^{\perp}$ obtained by assuming the out-of-plane conductivity $\sigma_{\perp}=\sigma_{\perp}^{\mathrm{bulk}}$ ($\sigma_{\perp}^{\mathrm{bulk}}/10$). (c) Comparison between $\lambda_{\mathrm{s}}^{\mathrm{iso}}$ and $\lambda_{\mathrm{s}}^{\parallel}$. Through the renormalization in the anisotropic model, the conductivity is transformed to become isotropic for the in-plane and out-of-plane directions (i.e., $\sigma_{\parallel}' = \sigma_{\perp}' = \sigma_{\parallel}$) and the effective thickness is rescaled by a factor of $\sqrt{\sigma_{\parallel}/\sigma_{\perp}}$. This makes the thickness thicker compared to the isotropic case. In the present case, the actual thickness ($t = 26$ nm in Sample A) is much larger than the spin diffusion length. Thus, this renormalization has little impact on the total spin absorption.