Observation of the crossover between metallic and insulating regimes of the spin Hall effect

Abstract

The physics of the anomalous and spin Hall effects is one of the most intriguing aspects of condensed matter physics. An important finding from a large collection of experimental and theoretical results is the universal scaling of the anomalous or spin Hall conductivity with the electric conductivity. This scaling has been successfully described by the intrinsic Berry curvature and extrinsic scattering mechanisms for metallic systems, revealing the topological nature of these effects. In contrast, the underlying physics in the opposite limit, the disordered insulating regime, is still unclear. In particular, it remains a major challenge, both experimentally and theoretically, to explore the spin Hall effect in the insulating regime. Here, we report the observation of the crossover between the metallic and insulating regimes of the spin Hall effect. The result demonstrates a direct correspondence between the spin and anomalous Hall effects, which will advance the fundamental understanding of spin transport.

Introduction

The family of Hall effects has played a central role in the development of condensed matter physics^1,2,3,4,5,6. An important member of this family is the spin Hall effect (SHE), which was theoretically predicted about half a century ago⁷. The SHE has attracted extensive attention for its fascinating topological, relativistic, and quantum mechanical nature, as well as spintronics applications^3,8. This effect enables electric generation and detection of spin currents in solid-state devices, providing an avenue for discovering a variety of spintronics phenomena³.

The SHE in metallic systems arises from intrinsic and extrinsic contributions, the mechanisms that are also responsible for the anomalous Hall effect (AHE) in ferromagnets². The intrinsic contribution can be explained by the Berry curvature associated with the Fermi surface and the band structure of the materials. On the other hand, the extrinsic contribution is caused by spin-dependent scattering on structural defects or impurities. For the AHE, a large collection of experimental results on a wide class of ferromagnets has revealed that there are three different regimes², characterized by the power-law relation, ${\sigma }_{xy}\propto {\sigma }_{xx}^{\gamma }$, between the anomalous Hall conductivity σ_xy and the longitudinal conductivity σ_xx; (i) the superclean regime, where σ_xy ∝ σ_xx due to the dominant contribution from the skew-scattering, (ii) the moderately dirty regime, where the intrinsic mechanism is dominant, and σ_xy is roughly insensitive to σ_xx, and (iii) the dirty regime with the scaling exponent γ generically larger than unity. The discovery of the crossover of the AHE has provided important information for the fundamental understanding of the physics of spin transport. Recently, the crossover between these regimes has also been confirmed for the SHE^9,10,11, demonstrating an important correspondence between the AHE and SHE in the metallic regime.

Although the existing theories have been successful in describing the AHE and SHE in metallic systems based on the intrinsic Berry curvature or the extrinsic scattering mechanisms, it remains a major challenge to understand the full-range phase diagram of these phenomena. The last important step is to explore the AHE and SHE in disordered insulating systems, where the carrier transport is dominated by hopping. In the insulating regime, the scaling relation is nontrivial because the mechanism of the AHE and SHE in hopping systems is clearly different from that in metallic systems; in the insulating regime, the AHE is attributed to a phase that a carrier gains when hopping around closed-loop paths in the presence of spin-orbit coupling (SOC) and background magnetization of the localized moments^12,13. Nevertheless, experimental observations have demonstrated that the scaling of the anomalous Hall conductivity with the electric conductivity prevails not only in the dirty metallic regime but also deep into the disordered insulating hopping regime². An analogous trend is expected for the SHE because of the similarity in the underlying physics. However, in contrast to the large collection of experimental results for the AHE, it remains a major experimental challenge to explore the SHE in the disordered insulating regime.

In this work, we report the observation of the crossover between the metallic and insulating regimes of the SHE in Pt, a model system for the study of the SHE. To explore the SHE in the insulating regime, the electric conductivity of Pt films is varied by three orders of magnitude by incorporating oxygen. We find the scaling behavior of the spin Hall conductivity, which prevails not only in the dirty metallic regime but also in the disordered insulating regime, despite the distinct difference in the transport mechanism. The observed variation of the spin Hall conductivity over the wide range of electric conductivity is reminiscent of the scaling of the anomalous Hall conductivity, illustrating a direct correspondence between the SHE and AHE in both the metallic and insulating regimes.

Results

Transport measurement

We use the spin-torque ferromagnetic resonance (ST-FMR)^14,15,16 to study the SHE of PtO_x films with various oxidation levels. Figure 1a shows a schematic illustration of the device. The device structure is SiO₂(4 nm)/Ni₈₁Fe₁₉(d_F)/TiN(2 nm)/PtO_x(10 nm)/SiO₂-substrate, where the numbers in parentheses represent the thickness (for details, see “Methods”). The separation between the Ni₈₁Fe₁₉ and PtO_x layers by the insertion layer minimizes the interfacial SOC effects, including the spin memory loss and interfacial spin-orbit torques^17,18. To minimize possible oxidation effects on the insertion layer due to its proximity to the PtO_x layer, we chose TiN, which is less susceptible to oxidation¹⁹, instead of other light metals, such as Cu and Ti. We can neglect the spin relaxation in the insertion layer because of the long spin diffusion length of around 40 nm in TiN due to the weak SOC²⁰. In the Ni₈₁Fe₁₉/TiN/PtO_x device, to control the oxidation level of the PtO_x layer, argon and oxygen gases were introduced into the chamber during the sputtering of the PtO_x film, and the amount of oxygen gas in the reactive mixture, Q, was varied between 0 and 10%. As shown in Fig. 1b, the electric resistivity ρ_N of the PtO_x film increases with Q, indicating that the oxidation level of the PtO_x layer is controlled by tuning Q. The change in the oxidation level induced by tuning Q is supported by X-ray photoelectron spectroscopy (XPS) spectra, shown in Fig. 1c.

Fig. 1: Device structure.

a Schematic illustration of the SiO₂/Ni₈₁Fe₁₉/TiN/PtO_x/SiO₂-substrate device used for the spin-torque ferromagnetic resonance (ST-FMR) measurement. θ is the angle between the direction of the radio frequency (RF) current and the in-plane applied magnetic field H. M denotes the magnetization, and the letters S and V in the illustration denote the RF signal generator and the nanovoltmeter, respectively. b Oxygen-gas flow ratio Q dependence of the electric resistivity ρ_N, measured by the standard four-probe method, of the PtO_x film. c X-ray photoelectron spectroscopy (XPS) spectra for the PtO_x films with Q = 1, 5, and 10%. The gray curve is the experimental data (exp.), and the red curve is the fitting result. The binding energies of the Pt 4f_7/2 peak for Pt, PtO, and PtO₂ are around 71.3, 72.3, 74.0 eV, respectively⁴⁸.

To characterize the carrier transport in the PtO_x films with different oxidation levels, we measured temperature T dependence of the sheet resistance R_s for the PtO_x films. Figure 2a shows the T dependence of R_s for the PtO_x film with Q = 1%. This result shows that R_s decreases monotonically with decreasing T: dR_s/dT > 0, showing a typical metallic behavior. By increasing the oxidation level, the T dependence of R_s is clearly changed. As shown in Fig. 2b and c, R_s increases with decreasing T for the PtO_x film with Q = 5% and Q = 10%: dR_s/dT < 0.

Fig. 2: Transport mechanism.

Temperature T dependence of the sheet resistance R_s for the PtO_x films with a Q = 1%, b Q = 5%, and c Q = 10%. We also compare the T dependence of R_s for the films with different Q in the inset to (a). The insets in b, c show T^−1/4 dependence of ${{{{{{\mathrm{log}}}}}}}\,{R}_{{{{{{{{\rm{s}}}}}}}}}$, where 200 K ≤ T ≤ 300 K. The solid circles are the experimental data and the solid lines are the linear fitting results.

To clarify the transport mechanism in the PtO_x films around room temperature, we plot ${{{{{{\mathrm{log}}}}}}}\,{R}_{{{{{{{{\rm{s}}}}}}}}}$ as a function of T^−1/4 (see the inset to Fig. 2b and c). This result shows that, for 200 K ≤ T ≤ 300 K, the T dependence of R_s is well described by the Mott variable range hopping (Mott-VRH) mechanism²¹:

$${R}_{{{{{{{{\rm{s}}}}}}}}}={R}_{{{{{{{{\rm{Mott}}}}}}}}}\exp \left[{\left(\frac{{T}_{{{{{{{{\rm{Mott}}}}}}}}}}{T}\right)}^{1/4}\right],$$

(1)

where R_Mott and T_Mott are the resistance parameter and the Mott characteristic temperature, respectively. This result indicates that the PtO_x film with Q = 5% is near the crossover between the band and hopping transport regimes, and the transport in the PtO_x film with Q = 10% is dominated by the Mott-VRH.

Spin-torque ferromagnetic resonance

Figure 3a shows the ST-FMR spectra for the Ni₈₁Fe₁₉(8 nm)/TiN(2 nm)/PtO_x(10 nm) trilayers with Q = 0, 6, and 10%, measured at room temperature. For the measurement, a radio frequency (RF) charge current was applied along the longitudinal direction of the device, and an in-plane external magnetic field H was applied with an angle of θ = 45° from the longitudinal direction (see also Fig. 1a). In the trilayer, the RF current generates damping-like (DL) and field-like (FL) spin-orbit torques, as well as an Oersted field, which drive magnetization precession in the Ni₈₁Fe₁₉ layer under the FMR condition. The magnetization precession induces an oscillation of the resistance due to the anisotropic magnetoresistance, resulting in the generation of a direct current (DC) voltage V_DC through the mixing of the RF charge current and oscillating resistance^14,15:

$${V}_{{{{{{{{\rm{DC}}}}}}}}}={V}_{{{{{{{{\rm{sym}}}}}}}}}\frac{{W}^{2}}{{({\mu }_{0}H-{\mu }_{0}{H}_{{{{{{{{\rm{res}}}}}}}}})}^{2}+{W}^{2}}+{V}_{{{{{{{{\rm{antisym}}}}}}}}}\frac{W({\mu }_{0}H-{\mu }_{0}{H}_{{{{{{{{\rm{res}}}}}}}}})}{{({\mu }_{0}H-{\mu }_{0}{H}_{{{{{{{{\rm{res}}}}}}}}})}^{2}+{W}^{2}},$$

(2)

where W is the linewidth and ${H}_{{{{{{{{\rm{res}}}}}}}}}$ is the FMR field. Here, V_sym and V_antisym are the magnitude of the symmetric and antisymmetric components, respectively; V_sym is proportional to the out-of-plane effective field H_⊥, which is dominated by the DL spin-orbit effective field H_DL, while V_antisym is proportional to the in-plane effective field H_∥, which is the sum of the Oersted field H_Oe and FL spin-orbit effective field H_FL. As shown in Fig. 3a, the V_DC spectrum varies systematically by changing the RF frequency f. We have confirmed that the variation of the resonance field ${H}_{{{{{{{{\rm{res}}}}}}}}}$ is consistent with the Kittel formula: $(2\pi f/\gamma )=\sqrt{{\mu }_{0}{H}_{{{\mbox{res}}}}({\mu }_{0}{H}_{{{\mbox{res}}}}+{\mu }_{0}{M}_{{{\mbox{eff}}}})}$, where M_eff is the effective demagnetization field and γ is the gyromagnetic ratio.

Fig. 3: Spin-torque ferromagnetic resonance.

a Magnetic field H dependence of the direct current (DC) voltage V_DC for the Ni₈₁Fe₁₉/TiN/PtO_x films with Q = 0, 6, and 10% at the frequencies from 5.0 to 8.0 GHz with an applied radio frequency (RF) power of 20 dBm. The Ni₈₁Fe₁₉-layer thickness is d_F = 8 nm. b In-plane magnetic field angle θ dependence of the V_sym (red) and V_antisym (blue) components of the spin-torque ferromagnetic resonance (ST-FMR) signal at f = 6.5 GHz for the Ni₈₁Fe₁₉/TiN/PtO_x film with Q = 10%. Here, θ is defined as the angle between the direction of the RF current and the in-plane applied magnetic field. The circles are the experimental data. Error bars, which represent the standard deviation of the fitting procedure, are smaller than the symbols. The solid curves are the fitting result using a function proportional to $\sin 2\theta \cos \theta$. c H dependence of V_DC at f = 7.5 GHz with an applied RF power of 20 dBm for the Ni₈₁Fe₁₉/TiN film, where the PtO_x layer is absent. d Q dependence of M_s/M_s(Q = 0), where M_s(Q = 0) is the saturation magnetization of the Ni₈₁Fe₁₉/TiN/PtO_x film with Q = 0%. The black line is a guide to the eye. e Frequency f dependence of ${\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$ for the Ni₈₁Fe₁₉/TiN/PtO_x film with Q = 10% and d_F = 12 nm. The black line is a guide to the eye.

The ST-FMR signals observed for the Ni₈₁Fe₁₉/TiN/PtO_x films originate from the RF current flowing in the PtO_x layer. In Fig. 3b, we show magnetic field angle θ dependence of the symmetric V_sym and antisymmetric V_antisym components, extracted by fitting the measured V_DC using Eq. (2), where θ is defined as the angle between the direction of the RF current and the in-plane applied magnetic field H (see Fig. 1a). Figure 3b shows that V_antisym is proportional to $\sin 2\theta \cos \theta$. This result indicates that the in-plane effective field, H_∥ = H_FL + H_Oe, is independent of θ, which is consistent with that H_FL and H_Oe are independent of the magnetization direction (see also Supplementary Note 1)^15,22,23. We note that V_sym is also proportional to $\sin 2\theta \cos \theta$. This result indicates that the out-of-plane effective field, H_⊥, is proportional to $\cos \theta$, (see also Supplementary Note 1)^15,22,23. The angular dependence of H_⊥ is consistent with the prediction of the out-of-plane effective field generated by the SHE: ${H}_{\perp }={H}_{{{{{{{{\rm{DL}}}}}}}}}| {{{{{{{\bf{m}}}}}}}}\times {{{{{{{\boldsymbol{\sigma }}}}}}}}| ={H}_{{{{{{{{\rm{DL}}}}}}}}}\cos \theta$, where m and σ are the unit vectors of the magnetization in the Ni₈₁Fe₁₉ layer and the spin polarization direction of the spin current generated by the SHE in the PtO_x layer, respectively²². We also note that the V_DC signal disappears in a Ni₈₁Fe₁₉(8 nm)/TiN(2 nm) film, where the PtO_x layer is absent, as shown in Fig. 3c. This result shows that the anomalous spin-orbit torque in the Ni₈₁Fe₁₉ layer, as well as the spin-orbit torques generated by the TiN layer is negligible in the Ni₈₁Fe₁₉/TiN/PtO_x films. The interfacial spin-orbit torques originating at the TiN/PtO_x interface also play a minor role in the Ni₈₁Fe₁₉/TiN/PtO_x film because interfacial SOC effects are notable only when PtO_x is directly contacted with Ni₈₁Fe₁₉¹⁸. These results indicate that the sizable symmetric voltage V_DC observed for the Ni₈₁Fe₁₉/TiN/PtO_x films originates from the DL torque generated by the SHE in the PtO_x layer (see also Supplementary Note 2).

The ST-FMR for the Ni₈₁Fe₁₉/TiN/PtO_x films allows us to quantify the DL-torque efficiency due to the SHE in the PtO_x layer. For the trilayer, we define the effective FMR spin-torque efficiency as ref. ²⁴

$${\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}=\frac{{V}_{{{\mbox{sym}}}}}{{V}_{{{\mbox{antisym}}}}}\frac{e{\mu }_{0}{M}_{{{{{{{{\rm{s}}}}}}}}}{d}_{{{{{{{{\rm{F}}}}}}}}}{d}_{{{{{{{{\rm{N}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}}{\hslash }\sqrt{1+\frac{{\mu }_{0}{M}_{{{{{{{{\rm{eff}}}}}}}}}}{{\mu }_{0}{H}_{{{{{{{{\rm{res}}}}}}}}}}},$$

(3)

where ${d}_{{{{{{{{\rm{N}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}={d}_{{{{{{{{\rm{N}}}}}}}}}+({\rho }_{{{{{{{{\rm{N}}}}}}}}}/{\rho }_{{{{{{{{\rm{I}}}}}}}}}){d}_{{{{{{{{\rm{I}}}}}}}}}$ is the effective thickness of the nonmagnetic layer. In Eq. (3), d_N = 10 nm is the thickness of the PtO_x layer; d_I = 2 nm and ρ_I = 462 μΩcm are the thickness and resistivity of the TiN layer, respectively. Here, M_s is the saturation magnetization, which is independent of Q as shown in Fig. 3d. The negligible change in M_s with Q shows that the Ni₈₁Fe₁₉ layer is not affected by the change of the oxidation level of the PtO_x layer, supporting that the Ni₈₁Fe₁₉ and PtO_x layers are well separated by the TiN insertion layer. We have also confirmed that ${\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$, extracted by fitting the V_DC spectra, is independent of the frequency f of the applied RF current, as shown in Fig. 3e. The negligible change in ${\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$ with f shows that the observed voltage is dominated by the ST-FMR, and possible spin pumping and thermoelectric contributions are negligible in the ST-FMR spectra (see also Supplementary Note 1)²⁵. From the effective FMR spin-torque efficiency ${\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$, the DL(FL) torque efficiencies per unit applied electric field E, ${\xi }_{{{{{{{{\rm{DL(FL)}}}}}}}}}^{E}=(2e/\hslash ){\mu }_{0}{M}_{{{{{{{{\rm{s}}}}}}}}}{d}_{{{{{{{{\rm{F}}}}}}}}}{H}_{{{{{{{{\rm{DL(FL)}}}}}}}}}/E$, can be determined using refs. ^24,26

$$\frac{1}{{\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}}=\frac{1}{{\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}}\left(\frac{1}{{\rho }_{{{{{{{{\rm{N}}}}}}}}}}+\frac{\hslash }{e}\frac{{\xi }_{{{{{{{{\rm{FL}}}}}}}}}^{E}}{{\mu }_{0}{M}_{{{{{{{{\rm{s}}}}}}}}}{d}_{{{{{{{{\rm{F}}}}}}}}}{d}_{{{{{{{{\rm{N}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}}\right).$$

(4)

To determine ${\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ using Eq. (4), we measured the ST-FMR for the Ni₈₁Fe₁₉/TiN/PtO_x films with different Ni₈₁Fe₁₉ layer thicknesses d_F, as shown in Fig. 4a and b. By fitting the measured spectra using Eq. (2), we obtain 1/d_F dependence of $1/{\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$ for the Ni₈₁Fe₁₉/TiN/PtO_x films with different oxidation levels, as shown in Fig. 4c–f. Figure 4c–f show that $1/{\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$ changes linearly with 1/d_F, enabling us to determine the DL torque efficiency ${\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ using Eq. (4). Here, the DL-torque efficiency ${\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ can be determined from the ST-FMR signals regardless of the transport mechanism in the PtO_x layer. The reason for this is that ${\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ is obtained from the measurement of the effective fields acting on the magnetization of the Ni₈₁Fe₁₉ layer, and the detection of ST-FMR signals relies on the magnetoresistance of the Ni₈₁Fe₁₉ layer; the ST-FMR model does not assume a specific transport mechanism in the non-magnetic layer.

Fig. 4: Spin-torque ferromagnetic resonance for devices with different thickness.

Spin-torque ferromagnetic resonance (ST-FMR) spectra for the Ni₈₁Fe₁₉/TiN/PtO_x films with a Q = 0% and b Q = 6% at 6.5 GHz, where d_F is the thickness of the Ni₈₁Fe₁₉ layer. ${H}_{{{{{{{{\rm{res}}}}}}}}}$ is the FMR field. The solid circles are the experimental data and the solid curves are the fitting results using Eq. (2). 1/d_F dependence of $1/{\xi }_{{{{{{{{\rm{FMR}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}$ for the devices with c Q = 0%, d Q = 6%, e Q = 7%, and f Q = 10%. The open circles are the experimental data (exp.). Error bars, which represent the standard deviation of the fitting procedure, are smaller than the symbols. The solid lines are the linear fitting results.

Discussion

Using the ST-FMR result, we investigate the variation of the spin Hall conductivity σ_SH in the PtO_x films, where the longitudinal electric conductivity σ_N is varied by three orders of magnitude. Since the SHE in the PtO_x layer dominates the observed DL torque, the effective spin Hall conductivity ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ of the PtO_x layer can be determined directly from the extracted values of ${\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ using

$${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }=\left(\frac{\hslash }{2e}\right){\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}.$$

(5)

The effective spin Hall conductivity ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ is related to the spin Hall conductivity σ_SH as ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}=(1/{T}_{{{{{{{{\rm{int}}}}}}}}}){\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$, where T_int is the spin transparency. We show the variation of ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ in Fig. 5 (see the open circles in red). This result shows that ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ is clearly suppressed by increasing the oxidation level when Q ≥ 5%, while ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ is almost unchanged by changing Q from 0–1%. The significant change in ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ when Q ≥ 5% indicates that the spin Hall conductivity σ_SH is suppressed with increasing the oxidation level of the PtO_x layer.

Fig. 5: Scaling of spin Hall conductivity.

The spin Hall conductivity σ_SH of Pt and Pt-based alloys plotted as a function of the longitudinal electric conductivity σ_N. The solid circles in red are the experimental result of this work. The effective spin Hall conductivity ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }=\left(\frac{\hslash }{2e}\right){\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}$ is also plotted (open circles in red). The error bars are the standard deviation. Other data are taken from published papers: Pt^9,10,25,49, PtN_x⁵⁰, Pd_xPt_1−x³⁹, Pt_x(MgO)_1−x¹¹, and Au_xPt_1−x^40,51. The open circles and open squares represent the data obtained from spin Hall effect (SHE) and inverse SHE measurements, respectively. For results where σ_SH is not explicitly shown, we estimated σ_SH from the spin Hall angle. The gray line is a guide for the eyes. The different background colors represent different regimes of the SHE: the superclean regime (blue), moderately dirty metallic regime (light blue), dirty metallic regime (orange), and disordered insulating regime (pink).

For the metallic PtO_x (Q = 0 and 1%), the spin transparency T_int in the Ni₈₁Fe₁₉/TiN/PtO_x film is described by the framework based on the spin diffusion model (see also Supplementary Note 3):

$${T}_{{{{{{{{\rm{int}}}}}}}}}=\frac{1-{{{{{{{\rm{sech}}}}}}}} \left({\delta }_{{{{{{{{\rm{N}}}}}}}}}\right)}{{G}_{{{{{{{{\rm{I}}}}}}}}}\sinh \left({\delta }_{{{{{{{{\rm{I}}}}}}}}}\right)+{G}_{{{{{{{{\rm{N}}}}}}}}}\cosh \left({\delta }_{{{{{{{{\rm{I}}}}}}}}}\right)\tanh \left({\delta }_{{{{{{{{\rm{N}}}}}}}}}\right)}\frac{{G}_{{{{{{{{\rm{F}}}}}}}}}\tanh {\delta }_{{{{{{{{\rm{F}}}}}}}}}}{1+\left({G}_{{{{{{{{\rm{F}}}}}}}}}/{G}_{{{{{{{{\rm{ext}}}}}}}}}^{\prime}\right)\tanh {\delta }_{{{{{{{{\rm{F}}}}}}}}}},$$

(6)

where δ_i = d_i/λ_i, G_i = σ_i/λ_i, and ${G}_{{{{{{{{\rm{ext}}}}}}}}}^{\prime}={G}_{{{{{{{{\rm{I}}}}}}}}}\left[\frac{{G}_{{{{{{{{\rm{I}}}}}}}}}\coth \left({\delta }_{{{{{{{{\rm{N}}}}}}}}}\right)+{G}_{{{{{{{{\rm{N}}}}}}}}}\coth ({\delta }_{{{{{{{{\rm{I}}}}}}}}})}{{G}_{{{{{{{{\rm{I}}}}}}}}}\coth \left({\delta }_{{{{{{{{\rm{N}}}}}}}}}\right)\coth ({\delta }_{{{{{{{{\rm{I}}}}}}}}})+{G}_{{{{{{{{\rm{N}}}}}}}}}}\right]$. Here, λ_i is the spin diffusion length and σ_i is the longitudinal electric conductivity of the i(=F, I, and N) layer, where F, I, and N correspond to Ni₈₁Fe₁₉, TiN, and PtO_x, respectively. Using^27,28 G_F = 1.1 × 10¹⁵ Ω⁻¹ m⁻² and G_N = 1.6 × 10¹⁵ Ω⁻¹ m⁻² with measured values of σ_i, we obtain T_int = 0.35 for Q = 0%. By increasing Q, the transparency T_int increases due to suppression of the spin conductance G_N because G_N is proportional to the carrier density and independent of the carrier scattering time¹¹. By taking into account the change of the carrier density, determined from Hall measurements, and assuming a dominant Elliott-Yafet spin relaxation mechanism, we obtain T_int = 0.44 for Q = 1%. In Fig. 5, we plot the spin Hall conductivity, determined from ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}=(1/{T}_{{{{{{{{\rm{int}}}}}}}}}){\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{* }$ (see the solid circles in red). Figure 5 shows that σ_SH is almost independent of σ_N when σ_N is in the range of around 2 × 10⁴ Ω⁻¹ cm⁻¹ to 8 × 10⁴ Ω⁻¹ cm⁻¹. This result is consistent with the prediction of the SHE in the moderately dirty regime, where the dominant mechanism of the SHE is the intrinsic mechanism and the spin Hall conductivity is insensitive to the electric conductivity².

The situation is clearly changed by further decreasing the electric conductivity of the PtO_x layer. For the heavily oxidized PtO_x devices, a full description of the spin transparency requires a theoretical framework describing hopping spin transport. However, developing such a framework has been a major challenge for theoretical study and is beyond the scope of the present work. In this work, for simplicity, we adopt the spin-diffusion model for the insulating regime, as well as for the metallic regime. This assumption is supported for organic semiconductors where the transport is dominated by carrier hopping; recent experimental and theoretical studies show that the spin-diffusion model describes well the hopping-dominated spin transport^29,30,31. However, for the insulating PtO_x, the validity of describing the hopping spin transport by the spin-diffusion model is unclear. Thus, we estimate T_int of the insulating PtO_x systems by extrapolating T_int of the metallic PtO_x systems, instead of calculating T_int based on the spin-diffusion model using parameters for the PtO_x devices in the hopping regime. Under this assumption, T_int = 0.44, for Q = 1%, is the lower bound of the transparency for Q ≥ 5% because the decrease of the carrier density decreases G_N, which increases T_int (see Eq. (6)). In Fig. 5, we plot the spin Hall conductivity σ_SH obtained under the assumption of T_int = 0.44 for Q≥5%. This result demonstrates that σ_SH decreases rapidly with decreasing σ_N for Q ≥ 5%, showing ${\sigma }_{{{{{{{{\rm{SH}}}}}}}}}\propto {\sigma }_{N}^{\gamma }$ with γ = 0.8. Since T_int is expected to increase with decreasing Q under the above assumption, γ = 0.8 is the lower bound; γ can be larger than unity if T_int(Q = 10%)/T_int(Q = 5%) > 1.9.

Notable is that the scaling behavior of the spin Hall conductivity prevails not only in the dirty metallic regime but also in the disordered insulating regime. Figure 5 shows that the observed variation of the spin Hall conductivity of the PtO_x film in the insulating hopping regime (σ_N < 10³ Ω⁻¹ cm⁻¹) is similar to that for Pt and Pt-based alloys in the dirty metallic regime (see the data obtained by the present and previous studies in the range of around 2 × 10³ Ω⁻¹ cm⁻¹ to 2 × 10⁴ Ω⁻¹ cm⁻¹ in Fig. 5). This result is reminiscent of the variation of the anomalous Hall conductivity in the dirty metallic and disordered insulating regimes. In the dirty metallic regime, the anomalous Hall conductivity is suppressed by the disorder due to the influence of finite-lifetime disorder broadening on the intrinsic contribution³², which explains the scaling of the anomalous Hall conductivity in this regime. This mechanism is also responsible for the scaling of the spin Hall conductivity in the dirty metallic regime¹¹. For the AHE, experimental studies have uncovered that the same scaling holds even in the disordered insulating regime, despite the distinct difference in the transport mechanism². In the insulating hopping regime, the AHE arises from interference between direct (i → j) and indirect (i → k → j) hoppings in a triad, where i and j are pairs of hopping sites, and k is the intermediate hopping site¹³. A similar mechanism can also give rise to the SHE in the insulating regime; the SHE in this regime arises when the hopping via an intermediate site is considered in addition to the hopping between pairs of sites³³. The observed scaling of the spin Hall conductivity over the wide range of electric conductivity demonstrates an important correspondence between the SHE and the AHE in both the metallic and insulating regimes.

Here, we note that, even for the AHE, the microscopic mechanism in the insulating regime is still not fully understood despite the long history of the experimental and theoretical studies². In contrast to the insulating regime, there is a consensus that the AHE and SHE in the metallic regime can be understood by the intrinsic and extrinsic mechanisms. In establishing the understanding of the AHE and SHE in the metallic regime, extensive experimental studies on the scaling behavior of the anomalous Hall and spin Hall conductivities have played a crucial role. Since the exploration of spin transport in insulating systems has been challenging both experimentally and theoretically, we believe that our experimental demonstration of the scaling of the SHE will stimulate theoretical and computational studies on the spin transport in the insulating hopping regime.

In summary, we have demonstrated the crossover between the metallic and insulating regimes of the SHE by tuning the oxidation level of PtO_x. We found that the spin Hall conductivity in the lightly oxidized PtO_x is almost independent of the electric conductivity, which is consistent with the prediction of the intrinsic SHE in the moderately dirty metallic regime. By further increasing the oxidation level, the PtO_x film enters the dirty regime, where the spin Hall conductivity decreases with decreasing the electric conductivity. We found that the spin Hall conductivity varies systematically despite the drastic change of the transport mechanism; the scaling of the spin Hall conductivity with the electric conductivity prevails not only in the dirty metallic regime but also in the insulating hopping regime. This result is reminiscent of the scaling of the anomalous Hall conductivity. Here, we note that although the SHE and AHE share the similar mechanisms, the relation between the two phenomena is non-trivial. The relation between the SHE and AHE has recently been investigated for ferromagnetic metals³⁴. In ferromagnets, both spin and charge accumulations can exist and are detected as the SHE and AHE, respectively^35,36. Since a charge flow in ferromagnets is spin polarized, an anomalous Hall current is accompanied by a spin current. Thus, it would be natural to expect that the spin polarization relates the AHE and SHE in ferromagnets. In fact, the anomalous Hall conductivity and the spin Hall conductivity, defined by spin-dependent Hall conductivities, are related by the spin polarization in the two-current model, where the Hall current consists of two independent channels formed by the majority and minority spins^36,37. This simple relation, derived from the model often used to describe the spin transport in ferromagnets, might hold in the limit of diffusive transport with an isotropic spin polarization³⁶. However, this simple relation is not valid in general^36,37. A recent experiment has shown that the relation between the SHE and AHE is complex in ferromagnetic metals³⁶. This complication can be attributed to the difference in the spin characters of the bands responsible for the SHE and AHE³⁸. Since the SHE and AHE behave differently even in the same system³⁶, it is not obvious whether there should be a direct correspondence between the scaling relation of the AHE in ferromagnets and that of the SHE in non-magnets in general. Thus, we believe that the observed crossover of the SHE provides essential information for a deeper fundamental understanding of spin-orbit physics and stimulates in-depth theoretical studies of the physics of spin transport in disordered systems. We also note that one of the main challenges of spintronics is exploring approaches to improve the spin-orbit torque efficiency because current-induced spin-orbit torques play a crucial role in a variety of spintronics applications, such as nonvolatile magnetic memories, reconfigurable logics, and neuromorphic computing devices^{8,17,24,39,40,41,42,43,44,45,46,47}. The observed scaling of the spin Hall conductivity provides important information for the development of efficient spin-orbit torque generators.

Methods

Device fabrication

The Ni₈₁Fe₁₉/TiN/PtO_x films were deposited on SiO₂ substrates by radio frequency (RF) magnetron sputtering. We first deposited the PtO_x layer on the SiO₂ substrate in a mixed argon and oxygen atmosphere. The amount of the oxygen gas in the reactive mixture Q was varied between 0 and 10%. After the PtO_x deposition, the chamber was evacuated to 2 × 10⁻⁵ Pa. Then, the TiN layer was fabricated by introducing 10% nitrogen into the argon gas flow. On the top of the TiN layer, the Ni₈₁Fe₁₉ film and a SiO₂ capping layer were sputtered in a pure argon atmosphere. The resistivity of the TiN layer was determined from measured resistance of Ni₈₁Fe₁₉(10 nm)/TiN(2 nm) and Ni₈₁Fe₁₉(10 nm) films by assuming a two layer parallel circuit model. For the ST-FMR measurement, the Ni₈₁Fe₁₉/TiN/PtO_x films were patterned into rectangular strips with a width of 10 μm and length of 100 μm using the photolithography and lift-off techniques.