## 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.

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## 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 ago^{7}. 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 phenomena^{3}.

The SHE in metallic systems arises from intrinsic and extrinsic contributions, the mechanisms that are also responsible for the anomalous Hall effect (AHE) in ferromagnets^{2}. 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 regimes^{2}, 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 regime^{2}. 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 SiO_{2}(4 nm)/Ni_{81}Fe_{19}(*d*_{F})/TiN(2 nm)/PtO_{x}(10 nm)/SiO_{2}-substrate, where the numbers in parentheses represent the thickness (for details, see “Methods”). The separation between the Ni_{81}Fe_{19} 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 oxidation^{19}, 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 SOC^{20}. In the Ni_{81}Fe_{19}/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.

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*: *d**R*_{s}/*d**T* > 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%: *d**R*_{s}/*d**T* < 0.

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) mechanism^{21}:

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 Ni_{81}Fe_{19}(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 Ni_{81}Fe_{19} 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}:

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.

The ST-FMR signals observed for the Ni_{81}Fe_{19}/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 Ni

_{81}Fe

_{19}layer and the spin polarization direction of the spin current generated by the SHE in the PtO

_{x}layer, respectively

^{22}. We also note that the

*V*

_{DC}signal disappears in a Ni

_{81}Fe

_{19}(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 Ni

_{81}Fe

_{19}layer, as well as the spin-orbit torques generated by the TiN layer is negligible in the Ni

_{81}Fe

_{19}/TiN/PtO

_{x}films. The interfacial spin-orbit torques originating at the TiN/PtO

_{x}interface also play a minor role in the Ni

_{81}Fe

_{19}/TiN/PtO

_{x}film because interfacial SOC effects are notable only when PtO

_{x}is directly contacted with Ni

_{81}Fe

_{19}

^{18}. These results indicate that the sizable symmetric voltage

*V*

_{DC}observed for the Ni

_{81}Fe

_{19}/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 Ni_{81}Fe_{19}/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}

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 Ni_{81}Fe_{19} layer is not affected by the change of the oxidation level of the PtO_{x} layer, supporting that the Ni_{81}Fe_{19} 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}

To determine \({\xi }_{{{{{{{{\rm{DL}}}}}}}}}^{E}\) using Eq. (4), we measured the ST-FMR for the Ni_{81}Fe_{19}/TiN/PtO_{x} films with different Ni_{81}Fe_{19} 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 Ni_{81}Fe_{19}/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 Ni_{81}Fe_{19} layer, and the detection of ST-FMR signals relies on the magnetoresistance of the Ni_{81}Fe_{19} layer; the ST-FMR model does not assume a specific transport mechanism in the non-magnetic layer.

## 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

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.

For the metallic PtO_{x} (*Q* = 0 and 1%), the spin transparency *T*_{int} in the Ni_{81}Fe_{19}/TiN/PtO_{x} film is described by the framework based on the spin diffusion model (see also Supplementary Note 3):

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 Ni_{81}Fe_{19}, TiN, and PtO_{x}, respectively. Using^{27,28} *G*_{F} = 1.1 × 10^{15} Ω^{−1} m^{−2} and *G*_{N} = 1.6 × 10^{15} 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 time^{11}. 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 × 10^{4} Ω^{−1} cm^{−1} to 8 × 10^{4} 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 conductivity^{2}.

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} < 10^{3} 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 × 10^{3} Ω^{−1} cm^{−1} to 2 × 10^{4} 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 contribution^{32}, 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 regime^{11}. 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 mechanism^{2}. 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 site^{13}. 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 sites^{33}. 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 studies^{2}. 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 metals^{34}. 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 polarization^{36}. 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 metals^{36}. This complication can be attributed to the difference in the spin characters of the bands responsible for the SHE and AHE^{38}. Since the SHE and AHE behave differently even in the same system^{36}, 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 Ni_{81}Fe_{19}/TiN/PtO_{x} films were deposited on SiO_{2} substrates by radio frequency (RF) magnetron sputtering. We first deposited the PtO_{x} layer on the SiO_{2} 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 Ni_{81}Fe_{19} film and a SiO_{2} capping layer were sputtered in a pure argon atmosphere. The resistivity of the TiN layer was determined from measured resistance of Ni_{81}Fe_{19}(10 nm)/TiN(2 nm) and Ni_{81}Fe_{19}(10 nm) films by assuming a two layer parallel circuit model. For the ST-FMR measurement, the Ni_{81}Fe_{19}/TiN/PtO_{x} films were patterned into rectangular strips with a width of 10 μm and length of 100 *μ*m using the photolithography and lift-off techniques.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported by JST FOREST Program (Grant Number JPMJFR2032), JSPS KAKENHI (Grant Number 19H00864), Canon Foundation, Asahi Glass Foundation, and Spintronics Research Network of Japan (Spin-RNJ).

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### Contributions

H.M. fabricated devices. H.M., A.M., and S.H. collected and analyzed the data. K.A. designed the experiments. K.A. and H.M. developed the explanation and wrote the manuscript. All authors discussed results and reviewed the manuscript.

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*Communications Physics* thanks Pranaba muduli and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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### Cite this article

Moriya, H., Musha, A., Haku, S. *et al.* Observation of the crossover between metallic and insulating regimes of the spin Hall effect.
*Commun Phys* **5**, 12 (2022). https://doi.org/10.1038/s42005-021-00791-1

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DOI: https://doi.org/10.1038/s42005-021-00791-1

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