Competing Ordinary and Hanle Magnetoresistance in Pt and Ti Thin Films

TL;DR

The paper investigates magnetoresistance in Pt and Ti thin films to disentangle ordinary magnetoresistance (OMR) from Hanle magnetoresistance (HMR) arising from spin and orbital Hall effects. By measuring Pt across a wide thickness range and comparing crystalline versus amorphous samples, and by extending the analysis to Ti, it shows that OMR often dominates in crystalline films while HMR can emerge in less crystalline or amorphous samples, particularly in Ti where orbital effects are implicated. A key finding is that the diffusion coefficient D, linked to crystallinity, strongly influences HMR amplitude, explaining discrepancies across literature and suggesting coexistence of OMR and HMR in materials with spin or orbital Hall conductivities. The work provides practical criteria to distinguish MR origins and emphasizes that crystallinity and scattering govern which MR mechanism prevails, with implications for designing spintronic devices and interpreting spin/orbital transport experiments.

Abstract

One of the key elements in spintronics research is the spin Hall effect, allowing to generate spin currents from charge currents. A large spin Hall effect is observed in materials with strong spin orbit coupling, e.g., Pt. Recent research suggests the existence of an orbital Hall effect, the orbital analogue to the spin Hall effect, which also arises in weakly spin orbit coupled materials like Ti, Mn or Cr. In Pt both effects are predicted to coexist. In any of these materials, a magnetic field perpendicular to the spin or orbital accumulation leads to additional Hanle dephasing and thereby the Hanle magnetoresistance (MR). To reveal the MR behavior of a material with both spin and orbital Hall effect, we thus study the MR of Pt thin films over a wide range of thicknesses. Careful evaluation shows that the MR of our textured samples is dominated by the ordinary MR rather than by the Hanle effect. We analyze the intrinsic properties of Pt films deposited by different groups and show that next to the resistivity also the structural properties of the film influence which MR dominates. We further show that this correlation can be found in both spin Hall active materials like Pt and orbital Hall active materials, like Ti. For both materials, the crystalline samples shows a MR attributed to the ordinary MR, whereas we find a large Hanle MR for the samples without apparent structural order. We then provide a set of rules to distinguish between the ordinary and the Hanle MR. We conclude that in all materials with a spin or orbital Hall effect the Hanle MR and the ordinary MR coexist and the purity and crystallinity of the thin film determine the dominating effect.

Figures (11)

• Figure 1: (a) Experimental setup and coordinate system utilized in this paper. After deposition of a Pt film with thickness $t_{\mathrm{Pt}}$, a Hallbar with a width $w$ and length between contacts $L$ is defined. A constant current $I$ is applied and the voltage drop $V$ is measured to determine the longitudinal magnetoresistance. Simultaneously, the transversal voltage V$_\mathrm{H}$ is detected. Along each of the coordinate axes in (a), the longitudinal resistivity $\rho (\textbf{B})$ and transversal resistivity $\rho_\mathrm{H} (\textbf{B})$ is calculated by taking the geometry of the Hall bar into consideration. (b) Change in longitudinal resistivity $\Delta \rho = \rho (\textbf{B}) - \rho_0(B\,=\,$0T) normalized by $\rho_0$ of the exemplary MR for 5nm Pt as reported in Sala et al. sala_orbital_2023, depicting a MR compatible with HMR theory, i.e., a finite MR for B$\perp$t and no MR for B||t. (c) Direction dependent MR for our 8nm thick Pt film as a function of the external field strength $B$, which cannot be explained by HMR theory. (d),(e) Transversal resistivity for the same samples as in (b) and (c). While a transversal HMR effect can be observed in the sample reported by Sala et al. sala_orbital_2023 (d), no extra signal can be found in our Pt (e). After subtraction of the linear ordinary Hall contribution along B||n (see Appendix C), no change in transversal resistivity is found within the resolution limit.
• Figure 2: Normalized MR for the field sweeps along j, t, n as a function of Pt thickness $t_{\mathrm{Pt}}$. The amplitude $\Delta \rho$ at 6T is obtained from a quadratic fit to the data. In all cases, a non-trivial dependency on the thickness can be observed. The thickness dependence expected from the HMR theory with $\theta_\mathrm{s}$ = 4%, $\lambda_{\mathrm{s}}$ = 2.4nm (as calculated from Sagasta et al. sagasta_SHE) and $D =$ 3.1e-5m^2/s (as discussed later [Fig. \ref{['Fig_5_params']}(b)]) is shown in amber color. For thin films below 20nm, a reasonable agreement with the data for B||j is visible. Above that thickness, however, the expected HMR and the data deviate substantially. Furthermore, the changes in the MR are visible along all directions. From HMR theory no MR is expected for B||t, while similar amplitudes are expected for B||j and B||n.
• Figure 3: (a) Amplitude of the MR at 6T ($\Delta \rho$) normalized by the resistivity at 0T ($\rho_0$) along all directions plotted versus $\rho^{-1}_0$ to highlight the scaling. (b) Kohler-plot of the MR for magnetic fields applied along t, where no HMR contribution is expected. The MR $\Delta \rho / \rho_0$ is plotted over the external field divided by $\rho_0$. A color code from black (4nm) to light gray (120nm) is utilized to distinguish between the different thicknesses, which all show a similar behavior.
• Figure 4: (a) Resistivity at 0T $\rho_0$, (b) amplitude $A_\mathrm{i}$ as well as (c) exponent $n$ of the respective Kohler fit versus the sample thickness. The resistivity for the bulk value is derived using an extended Fuchs-Sondheimer model fuchs_conductivity_1938sondheimer_mean_1952althammer_quantitative_2013. The parameters obtained from fitting Eq. \ref{['Eq_Kohler']} to each individual sample along j, t, n, are almost identical for all samples and independent of the thickness. The mean values for $A_\mathrm{i}$ are 3.35, 3.58 and 3.75aΩ m T and 1.67, 1.72 and 1.76 for $n$ along j, t, n, respectively.
• Figure 5: (a) HMR amplitude at 6T, (b) spin relaxation time $\tau$ for works from Tab. \ref{['Tab_MR']}, (c) diffusion coefficient $D$ and (d) spin relaxation time $\tau$ for selected points on a smaller scale all versus the conductivity. It becomes apparent that for the HMR not only the resistivity but also the $D$ (or $\tau$) plays an important role. Since $\tau$ = $\lambda^2 / D$ and $\lambda \propto \sigma$sagasta_SHEliu_direct_2015, a scaling with the conductivity is expected. From the Drude model we assume $\tau \propto \sigma$ and therefore also $D \propto \sigma$. The open symbols in (c) and (d) are the values as detailed in the respective work. While the linear fit is in reasonable agreement in (c), the data points in (d) deviate. For a better comparison, we recalculate all HMR parameters by taking the (intrinsic) scaling from Sagasta et al. sagasta_SHE for $\sigma_\mathrm{s}$ and $\lambda_{\mathrm{s}}$ into account, which leaves only $D$ as the fit parameter. The so extracted $D$ is given by the full symbols. From this linear dependency, the diffusion coefficient used for HMR calculations in Fig. \ref{['Fig_2_HMR_all']} and Fig. \ref{['Fig_A3_data_Hall']} is calculated. The linear behavior shows, that the Pt from one group can consistently be described using the Drude assumption however, to explain the discrepancies between the different works, an additional contribution besides the conductivity (resistivity) is needed.