Unambiguous Electrical Detection of Spin-Charge Conversion in Lateral Spin-Valves

TL;DR

This paper introduces a robust electrical detection scheme for spin-charge conversion in lateral spin-valves, enabling clear differentiation of inverse spin Hall and spin galvanic effects through symmetry analysis.

Contribution

It presents a novel detection geometry that disentangles spin-orbit phenomena in diffusive channels, improving accuracy and applicability in spintronics research.

Findings

The detection scheme accurately separates ISHE and SGE signals.

The method is robust against magnetization tilting and disorder.

Applicable to various spintronic device architectures.

Abstract

Efficient detection of spin-charge conversion is crucial for advancing our understanding of emergent phenomena in spin-orbit-coupled nanostructures. Here, we provide proof of principle of an electrical detection scheme of spin-charge conversion that enables full disentanglement of competing spin-orbit coupling transport phenomena in diffusive lateral channels i.e. the inverse spin Hall effect (ISHE) and the spin galvanic effect (SGE). A suitable detection geometry in an applied oblique magnetic field is shown to provide direct access to spin-charge transport coefficients through a simple symmetry analysis of the output non-local resistance. The scheme is robust against tilting of the spin-injector magnetization, disorder and spurious non-spin related contributions to the non-local signal, and can be used to probe spin-charge conversion effects in both spin-valve and hybrid optospintronic devices.

Equations

$$R_{\text{ISHE(SGE)}}=\frac{1}{2}[\Delta R_{\text{nl}}(\mathbf{B})\pm \Delta R_{\text{nl}}({\mathbf{B}}^{\ast })],$$

$$[\nabla_{t}s]^{a}+\big{[}\nabla_{i}\mathcal{J}_{i}\big{]}^{a}=-\frac{s^{a}}{\tau_{s}^{a}}-\kappa_{i}^{a}J_{i},$$

$$J_i=-D_c\left({\partial }_i\rho +{\kappa }_i^a{s}^a\right)+{\gamma }_{ij}^a{\mathcal{J}}_j^a,$$

$${\mathcal{J}}_i^a=-D_s{\left[{\nabla }_{i}s\right]}^a+{\gamma }_{ij}^a{J}_j.$$

$$\bar{\mathcal{D}}\cdot \mathbf{s}(x)+\gamma \left(\mathbf{s}\left(x\right)\times \mathbf{B}\right)=0,$$

$$\boldsymbol{\mathcal{\bar{D}}}=D_{s}\left(\begin{array}[]{ccc}\partial_{x}^{2}-l_{s,x}^{-2}&0&\kappa_{R}\partial_{x}\\ 0&\partial_{x}^{2}-l_{s,y}^{-2}&0\\ -\kappa_{R}\partial_{x}&0&\partial_{x}^{2}-l_{s,z}^{-2}\end{array}\right)\,$$

$$V_{\text{nl}}(x)=-\frac{\bar{W}{D}_s}{{\sigma }_{\text{2D}}}\left[l_{\text{SGE}}^{-1}{s}^x\left(x\right)+{\theta }_{\text{SHE}}{\partial }_x{s}^z\left(x\right)\right]+\mathcal{O},$$

$$\Delta R_{\text{nl}}=R_0\text{Im}\left[e^{-qL}\left({\theta }_{\text{SHE}}q{l}_s{s}_B\cos \varphi -\frac{l_s}{l_{\text{SGE}}}\sin \varphi \right)\right],$$

$${\Delta R_{\text{nl}}|}_{\text{ISHE(SGE)}}^{\mathbf{B}\perp \hat{\mathbf{y}}}=\frac{1}{2}\left({\Delta R_{\text{nl}}|}_B\pm {\Delta R_{\text{nl}}|}_{-B}\right),$$

$$R_{\text{nl}}^{\text{YZ}}$$

$$\Delta R_{\text{nl}}^z$$

$$\hat{\mathbf{n}}(B,\varphi )=\hat{x}\cos \left[\gamma (B,\varphi )\right]\sin \left[\beta (B,\varphi )\right]+p\hat{y}\cos \left[\gamma (B,\varphi )\right]\sin \left[\beta (B,\varphi )\right]+\hat{z}\sin [\gamma (B,\varphi )],$$

$$\beta (B,\varphi )=\arctan (B_x/B_{\text{sat;x}}),\gamma (B,\varphi )=\arctan (B_z/B_{\text{sat;z}}),$$

$$\Delta R_{\text{SHE(SGE)}}=\frac{1}{2}\left[\Delta R_{\text{nl}}(B)\pm \Delta R_{\text{nl}}(-B)\right]|\equiv \frac{1}{2}\left[\Delta R_{\text{nl}}(\varphi )\pm \Delta R_{\text{nl}}(-\varphi )\right]|.$$

Full text

The generation and manipulation of nonequilibrium spins at interfaces are key goals in the operation of spintronic devices, with recent research heavily focused on using spin-orbit coupling (SOC) for achieving both. As the spin dynamics are sensitive to symmetry-breaking effects, it is conspicuous that transport measurements provide a powerful probe of emergent spin-orbit phenomena. As such, a fundamental understanding of spin transport at interfaces is currently a major enterprise towards the effective control over the spin degree of freedom (Review_Interfaces_15, ; Review_Interfaces_16, ; Review_Interfaces_17, ).

The conversion of spin current into a charge current is also essential for an all-electrical readout. SOC has been shown or predicted to provide a suitable means to achieve this in a number of systems with broken inversion symmetry ranging from two-dimensional (2D) electron gases in oxide-oxide (1stP_SCC_2DEGOxide, ), metal-semiconductor (1stP_SCC_Ge-metal, ) and metal-metal (1stP_SCC_metal_metaloxide, ) interfaces to surface states of topological insulators (1stP_SCC_metal_TI, ) and spin-split bulk states in polar semiconductors (1stP_BulkBReffect_BiTeI, ). More recently, bilayers of 2D crystals have emerged as highly-controllable testbeds for exploring interfacial SOC phenomena due to their gate-tunable charge carriers and interplay between spin and lattice-pseudospin degrees of freedom (GSOC_Rashba_09, ; GSOC_TMDs_Zhu_11, ; GSOC_TMDs_Xiao_12, ; GSOC_TMDs_Muniz_15, ; GSOC_Review_Garcia_18, ). This novel class of Dirac materials have further extended the breadth of interfacial phenomena to encompass proximity-induced SOC within atomically thin crystals (GTMD_Avsar_14, ; GTMD_Wang_15, ; GTMD_Wang_16a, ; GTMD_Volkl_16b, ; GTMD_Yang_17, ; GTMD_Wakamura_18a, ; GTMD_Omar_18b, ), all-optical spin-current injection (GTMD_Optical_Luo_NanoLett2017, ; GTMD_Optical_Avsar_ACSNano2017, ), large spin lifetime anisotropy (GTMD_SRTA_Cummings, ; GTMD_SRTA_Ghiasi_17, ; GTMD_SRTA_Benitez18, ; GTMD_SRTA_Offidani_18, ) and the co-existence of SHE and inverse SGE (Milletari_ConservationLaws_17, ; GSOC_Offidani_17, ). Theoretical studies of 2D systems have also envisioned unconventional spin-orbit scattering mechanisms, including robust skew scattering from spin-transparent impurities (Milletari_ConservationLaws_17, ) and a direct magneto-electric coupling (DMC) effect—arising from quantum interference between distinct components of the single-impurity SOC potential—which generates a nonequilibrium spin polarization (GSOC_Huang_16, ).

The abundance of microscopic spin-orbit mechanisms in materials with broken inversion symmetry motivates the search for device geometries that can enable electrical detection of spin-charge interconversion effects. The H-bar scheme in Refs. (GTMD_Avsar_14, ; GTMD_Wang_15, ) employs the SHE for spin-current generation together with its Onsager reciprocal, the inverse spin Hall effect (ISHE), for electrical read-out. This method can be further extended to the detection of the SGE, the Onsager-reciprocal of current-induced spin polarization, as shown in Ref. (Huang_NonLocalResistance_17, ). However, the two-step process, at the heart of the quadratic dependence of the nonlocal resistance with the spin-charge conversion rate, has serious drawbacks. Primarily it makes the H-bar approach prone to noise and variability, as demonstrated by the experiments on graphene with adatoms (AdGraph_Balakrishnan_13, ; AdGraph_Kaverzin_15, ; AdGraph_Volkl_19, ; AdGraph_Wang_15, ). Secondly, it precludes the unambiguous determination of the spin Hall angle ($\theta_{\textrm{SHE}}$) and SGE efficiency ($\theta_{\textrm{SGE}}$) in samples where these effects compete (SHE_SGE_2D_1, ; SHE_SGE_2D_2, ; SHE_SGE_2D_3, ). Further progress in this flourishing field will require alternative lateral-device detection schemes where competing SOC effects can be simultaneously detected and quantified.

In this Letter, we propose a measurement protocol for the unambiguous linear detection of ISHE and SGE in lateral spin devices. An opto-spintronic analogue, where the initial spin accumulation is achieved by purely optical means, is also presented. To illustrate the general principle of the detection scheme, we focus on a nonlocal setup comprising a spin-injector, a high-fidelity graphene spin-channel (SpinChannel_Tombros_07, ; SpinChannel_Kamalakar_15, ; SpinChannel_Gurram_18, ) and a cross-shaped junction, where the graphene is covered by a high-SOC material (see Fig. 1). This layout is particularly well suited to the measurement protocol because one can separate the spin-channel from the SOC-active (spin-charge conversion) region. We stress, however, that the detection scheme is general to any lateral spin-injection device providing that the breaking of inversion symmetry away from the detection region is sufficiently weak, such that the channel length ($L)$ $\gtrsim$ spin diffusion length ($l_{s}$) $\gg$ mean free path ($l$), which corresponds to the experimentally relevant diffusive regime. As such, the proposed scheme can potentially shed light onto spin-charge conversion effects in different material systems, such as oxide heterostructures (SID_Reyren, ), metallic bilayers (SID_Isasa, ) and doped semiconductors (SID_Kamerbeek, ; SID_Lou, ), for which spin injection has been recently established by nonlocal transport methods, but the precise interplay of ISHE and SGE is yet to be uncovered.

The general principle is akin to electrical detection of ISHE (ISHE_Valenzuela, ). Spin-polarized carriers, injected by applying a current $I$ at a ferromagnet (FM) contact, generate lateral charge accumulation via SOC, which is then detected as a nonlocal voltage at the Hall cross. The possibility unveiled here to isolate SGE and ISHE contributions on the output signal hinges on a fundamental distinction between the non-equilibrium spin-polarization density induced Hall effect (SGE) and the most familiar spin-current-induced Hall effect (ISHE). In the former, the component of the diffusive spin accumulation, with spin moment collinear to the propagation direction, generates a transverse charge current $J_{i}=\epsilon_{ij}\,\theta_{\textrm{SGE}}\,\mathcal{J}_{j}^{j}\>$($i,j=x,y$), where $\mathcal{J}_{i}^{a}$ is the spin current density in direction $i$ with the spin polarized in $a$ ($a=x,y,z$). Conversely, in the ISHE, the spin-moment, the spin current density and the charge current density are mutually orthogonal such that $J_{i}=\epsilon_{ijk}\,\theta_{\textrm{SHE}}\,\mathcal{J}_{j}^{k}$, where $\epsilon_{ij}$ ($\epsilon_{ijk}$) stands for the Levi-Civita symbol and summation over repeated indices is implied. Note that the steady-state spin current is linked to the spin polarization in the channel as $\mathcal{J}_{i}^{a}=-D_{s}\partial_{i}s^{a}$, where $D_{s}$ is the diffusion constant (a more accurate definition is given below). The crucial role played by the active polarization channel—out-of-plane ($s^{z}$) for ISHE and in-plane ($s^{x}$) for SGE—is borne out when coherent spin precession is induced by an oblique magnetic field normal to the spin-injector easy-axis, $\mathbf{B}=(B_{x},0,B_{z})$, forcing diffusive spins to undergo ISHE and SGE simultaneously (Fig. 1). Oblique spin precession is a powerful probe of spin relaxation anisotropy (Raes_SRTA_Graphene_16, ; GTMD_SRTA_Ringer, ) and here we show that a suitable detection scheme in applied oblique field provides access to the charge-spin transport coefficients $\theta_{\textrm{SHE}}$ and $\theta_{\textrm{SGE}}$. Our main result is a linear filtering protocol for the ISHE ($+$)/SGE($-$) nonlocal resistance

[TABLE]

where $\Delta R_{\textrm{nl}}=\frac{1}{2I}(V_{\textrm{nl};n_{y}>0}-V_{\textrm{nl};n_{y}<0})$, with $\hat{\boldsymbol{n}}=\mathbf{M}/|\mathbf{M}|$, is the output transresistance difference between opposite configurations of the spin-injector and $\mathbf{B}^{*}=(B_{x},0,-B_{z})$ is the mirror-image of $\mathbf{B}$. The spin-charge conversion efficiency parameters ($\theta_{\textrm{SHE}}$, $\theta_{\textrm{SGE}}$) can be accurately determined using a simple model of the spin-injector magnetization tilting $\mathbf{M}(\mathbf{B})$. (For optical spin-injection, the detection is carried out with standard Hanle technique.)

Theory.—We give an intuitive proof of Eq. (1). In the narrow channel limit ($W\ll l_{s}$), the spin dynamics of a typical disordered sample with weak SOC ($l_{s}\gg l$) are well captured by the generalized 1D Bloch model $\partial_{t}\mathbf{\bar{s}}=D_{s}\partial_{x}^{2}\mathbf{\bar{s}}+\gamma\,(\mathbf{\bar{s}}\times\mathbf{B})-\hat{\Gamma}\,\mathbf{\bar{s}}$ (Raes_anisotropic, ), where $\mathbf{\bar{s}}\equiv(\mathbf{s}_{n_{y}>0}-\mathbf{s}_{n_{y}<0})/2$ is the spin-density difference between opposites configurations of the spin-injector, $\hat{\Gamma}=\textrm{diag}\{1/\tau_{s}^{x},1/\tau_{s}^{y},1/\tau_{s}^{z}\}$ and $\gamma$ is the gyromagnetic ratio. The spin-relaxation matrix $\hat{\Gamma}$ encapsulates the effects of spin-dephasing (Cummings_16, ) and irreversible spin relaxation mechanisms (which need not to be isotropic due to SOC (GTMD_SRTA_Offidani_18, )). The output signal $\Delta R_{\textrm{nl}}$ is proportional to the transverse current generated at the cross. A simple dimensional analysis yields: $\Delta R_{\textrm{nl}}\propto D_{s}\,[\theta_{\textrm{SHE}}\partial_{x}\bar{s}^{z}(x)+(\theta_{\textrm{SGE}}/l_{s})\bar{s}^{x}]_{x=L}$. Crucially, since the injected spins follow the contact magnetization, the boundary term satisfies $\mathbf{\bar{s}}(x=0)||\hat{y}$, for any $\mathbf{B}\perp\hat{y}$. This implies that the solution of the 1D Bloch equation for the spin polarization density difference $\bar{\mathbf{s}}$ transforms as $(\bar{s}^{x}(x),\bar{s}^{z}(x))\rightarrow(-\bar{s}^{x}(x),\bar{s}^{z}(x))$, under the operation $B_{z}\rightarrow-B_{z}$. This shows that $\Delta R_{\textrm{nl}}(\mathbf{B})=\Delta R_{\textrm{nl}}(\mathbf{B}^{*})$ for SHE and $\Delta R_{\textrm{nl}}(\mathbf{B})=-\Delta R_{\textrm{nl}}(\mathbf{B}^{*})$ for SGE, thus proving the generality of the filtering scheme Eq. (1).

We formalize our results with a theory of coupled spin-charge diffusive transport for disordered 2D conductors subject to random SOC sources (Lin_LongPaper, ). The charge and spin observables, coarse-grained over typical distances longer than the mean free path, satisfy the continuity-like equation:

[TABLE]

the continuity relation $\partial_{t}\rho=-\partial_{i}J_{i},$ and the generalized constitutive relations

[TABLE]

[TABLE]

The coefficients appearing in Eqs. (2)-(4) can be obtained from microscopic calculations (Lin_LongPaper, ; Shen_PRB2014, ; Burkov_04, ; Tokatly_Sherman_10, ). The DMC efficiency $\kappa_{i}^{a}$ (units $m^{-1}$) and spin Hall angle $\gamma_{ij}^{a}$ quantify the spin-charge conversion efficiency of random SOC sources. In this compact formalism, the effects of all spatially uniform spin-dependent interactions are captured by the SU(2) gauge field $\mathcal{A}_{\mu}=A_{\mu}^{a}\,\sigma^{a}$ ($\mu=t,i$), with $\sigma^{a}$ the spin-1/2 Pauli matrices. The gauge field is implicit in the covariant derivative $[\nabla_{\mu}O]^{a}=\partial_{\mu}O^{a}\pm\epsilon^{abc}A_{\mu}^{b}O^{c}$, with $O^{a}=\{s^{a},\mathcal{J}^{a}\}$ a spin observable and the sign $\pm$ holding for a space ($+$)/time($-$) derivative. The time component $A_{0}^{a}=g\mu_{B}B^{a}$ reproduces the spin precession in the external Zeeman field and $A_{i}^{a}$ encodes the symmetry-breaking SOC (Tokatly_08, ). We specialize to uniform SOC of the Bychkov-Rashba (BR) form, which is ubiquitous in interfaces with broken inversion symmetry. For 2D Dirac fermions, its non-zero components read as $A_{y}^{x}=-A_{x}^{y}=\lambda/v_{F}$, with $v_{F}$ the Fermi velocity and $\lambda$ the BR-spin gap, whereas for parabolic 2D electron gases, $A_{y}^{x}=-A_{x}^{y}=2m^{*}\alpha$, with $m^{*}$ the effective mass and $\alpha$ the BR coupling. The BR effect contributes to the total spin-charge conversion rates with terms proportional to the momentum scattering time $\tau$, namely (i) an ISHE-like term $\gamma_{ij}^{a;\textrm{BR}}=\epsilon_{ij}\,\omega_{\textrm{BR}}\tau$, where $\omega_{\textrm{BR}}=\lambda^{2}/\epsilon$ with $\epsilon$ the Fermi energy (2D Dirac gas) and $\omega_{\textrm{BR}}=2m\alpha^{2}$ (2D electron gas); and (ii) an indirect coupling between spin polarization density and charge current that modifies the DMC coefficient according to $\kappa_{i}^{a}\rightarrow\kappa_{i}^{a}-\epsilon_{ia}\omega_{\textrm{BR}}\tau\,l_{R}^{-1}$, where $l_{R}\equiv\hbar/|A_{x}^{y}|$ is the length scale of spin precession about the BR field.

Armed with this formalism, we can evaluate the spin density profile in applied oblique field and establish the range of validity of the linear filtering protocol [Eq. (1)]. Beyond the SU(2) gauge field and impurity scattering, we note that the spin-charge conversion rates can also receive contributions from Berry curvature and phonons (RMP_SHE, ; Gorini_15, ). For ease of notation, we lump all these contributions under the effective ISHE and SGE rates $\gamma_{ij}^{a}=\theta_{\textrm{SHE}}\epsilon_{ij}\delta_{az}$ and $\kappa_{i}^{a}=l_{\textrm{SGE}}^{-1}\epsilon_{ia}$, where $\delta_{ij}$ is the Kronecker delta symbol*. *For disordered samples in the typical weak SOC regime $(l_{s}/l_{\textrm{SGE}},\ \theta_{\textrm{SHE}})\ll 1$, the build up of a nonlocal voltage in the cross-shaped device involves two independent steps as described below.

Spin precession and spin-charge conversion.—The effective Bloch equation governing the steady-state spin accumulation is obtained from Eqs. (2)-(4) after eliminating the spin current in favor of the spin polarization density. To leading (linear) order in the spin-charge conversion rates, we find

[TABLE]

with $\gamma=g\mu_{B}/\hbar$ and

[TABLE]

where $l_{s,a}=\sqrt{D_{s}\tau_{s}^{a}}$ ($a=x,y,z$). The diffusion matrix $\boldsymbol{\mathcal{\bar{D}}}$ displays the standard Fick terms (diagonal), accounting for spin relaxation processes (Raes_anisotropic, ), and a conspicuous nondiffusive term proportional to the wavelength of inversion symmetry breaking within the 1D channel $\kappa_{R}=2l_{R}^{-1}$. This term is usually neglected in the analysis of spin precession (Raes_anisotropic, ), since it is assumed that the main effect of SOC is a renormalization of spin lifetimes. However, the more general Bloch equation (5) shows that the BR gauge field effectively induces a coherent coupled precession of $s_{x}$ (SGE) and $s^{z}$ (ISHE) components in materials with $\kappa_{R}\gtrsim l_{s}^{-1}$. The protocol, which relies on the decoupling of SGE and ISHE precessions using suitable oblique fields $\mathbf{B},\mathbf{B}^{*}\perp\hat{y}$, is essentially exact for $l_{R}\gg l_{s}$, since in this limit Eq. (5) reduces to its standard diffusive form. Remarkably, the scheme is still accurate in channels with moderate to strong BR effect, where $l_{R}$ can be only few times larger than $l_{s}$; a detailed discussion is given in the supplemental material (SM) (SM, ).

We now turn our attention to the detection region, where SOC generates a transverse current $J_{y}$. The nonlocal voltage build-up is determined by the open circuit condition $J_{y,\textrm{total}}=J_{y}+(\sigma_{\textrm{2D}}/\bar{W})V_{\textrm{nl}}=0$, where $\sigma_{\textrm{2D}}$ is the DC conductivity. From Eqs. (3)-(4), we find

[TABLE]

where $\mathcal{O}$ are next order corrections in $\theta_{\textrm{SHE}}$ and $l_{\textrm{SGE}}^{-1}$, thus validating the expression of $\Delta R_{nl}(x)$ obtained earlier via a dimensional analysis argument.

Detection.—We will now show how the detection is carried out in a real setup. We first consider the spin-valve layout in Fig. 1. The injected spin polarization is parallel to the spin-injector quantization axis (defined by $\mathbf{M}$) and thus is very sensitive to the applied field (SpinChannel_Gurram_18, ). First, a field $\mathbf{B}_{\textrm{in}}\parallel\pm\hat{y}$ is applied to set the injector configuration either “up” ($n_{y}>0$) or “down” ($n_{y}<0$). Subsequently, the field is removed and an oblique field in the $Oxz$ plane is swept, $\boldsymbol{B}=(|B|\cos\phi\thinspace\hat{\boldsymbol{x}}+B\sin\phi\thinspace\hat{\boldsymbol{z}})$, across first and fourth quadrants ($\phi\in[0,90^{\circ}]$). The nonlocal resistance difference between opposite configurations of the spin injector, $\Delta R_{\textrm{nl}}\equiv[\left.V_{\textrm{nl}}(L)\right|_{n_{y}>0}-\left.V_{\textrm{nl}}(L)\right|_{n_{y}<0}]_{x=L}/2I$ is obtained from Eq. (7) upon solving the effective spin Bloch equation [Eq. (5)] for the spin density profile $s^{a}(x)$. After a straightforward calculation, we find

[TABLE]

where $q=l_{s}^{-1}\sqrt{1+\imath B\gamma\tau_{s}}$ is the complex spin-precession wavevector, $s_{B}=\textrm{sign}(B)$, and $R_{0}=s^{y}(0)(\bar{W}D_{s}/l_{s}I\sigma_{\textrm{2D}})$. We have also assumed an isotropic spin lifetime $\tau_{s}^{a}=\tau$, which is characteristic of single-layer graphene (Raes_SRTA_Graphene_16, ). The two contributions in Eq. (8) have opposite parity under mirror reflection $B\rightarrow-B$ (or equivalently, $\phi\rightarrow-\phi$). This in turn implies that the bona fide ISHE ($+$)/SGE($-$) spin-transresistance satisfies

[TABLE]

which is simply a special case of Eq. (1). The signal subtraction $\Delta R_{\textrm{nl}}\equiv\frac{1}{2}(\left.R_{\textrm{nl}}\right|_{n_{y}>0}-\left.R_{\textrm{nl}}\right|_{n_{y}<0})$ ensures that non-spin-related contributions, such as the local Hall effect due to stray fields generated by the FM contact, are filtered out. Moreover, the spin-transresistance scale $R_{0}\propto s^{y}(0)\equiv s^{y}(0,\mathbf{B})$ defines the maximum achievable nonlocal-signal. Contacts with low magnetic anisotropy saturate easily ($R_{0}\propto s^{y}(0,\mathbf{B})\rightarrow 0$ for $B_{x,z}\gg B_{x,z}^{\textrm{sat}}$), which shrinks the $\Delta R_{\textrm{nl}}$-lineshape with respect to an ideal spin-injector with magnetization pinned along the easy axis $\mathbf{M}\parallel\pm\hat{y}$. The typical lineshapes for an isotropic channel with an ideal contact ($\hat{\mathbf{n}}||\pm\hat{\mathbf{y}}$) are shown in Fig.  2(a). For field applied perpendicularly to the plane ($\phi=90^{\circ}$), spins precess in the plane, there is no available out-of-plane spin density and thus ISHE is absent, while SGE achieves is highest magnitude. When the field is tilted towards the 2D plane ($\phi<90^{\circ}$), a symmetric ISHE component appears. The realistic lineshapes are obtained by a trivial rescaling $\Delta R_{\textrm{nl}}\rightarrow\mathcal{V}_{B}\,\Delta R_{\textrm{nl}}$, where $\mathcal{V}_{B}=s^{y}(0,B)/s^{y}(0,0)$ is the “visibility” factor that takes into account response of the spin-injector (SM, ). As aforementioned, since $\mathcal{V}_{B}=\mathcal{V}_{-B}$ (for a homogeneous contact) the only effect of the FM response within the protocol is a shrinkage of the filtered signals. We briefly discuss an alternative detection scheme with an oblique field applied in the $Oyz$-plane (Raes_SRTA_Graphene_16, ). The nonlocal resistance for a fixed spin-injector configuration, $R_{\textrm{nl}}^{\textrm{YZ}}=V_{\textrm{nl}}^{\textrm{YZ}}/I$, is found as

[TABLE]

where $r_{0}=[s^{y}(0)]_{\mathbf{B}}\times(\bar{W}D/l_{s}I\sigma_{\textrm{2D}})$ is the typical transresistance factor and $f(q)=[e^{-L/l_{s}}-Re\left(ql_{s}e^{-qL}\right)]/2$. The ISHE/SGE components can be extracted via Fourier analysis if the $\phi$-dependence of $r_{0}$ is known with good resolution. Alternatively, for typical channels ($L\gtrsim l_{s}$), $\theta_{\textrm{SHE}}$ can be determined directly from nonlocal signal saturation at large field $R_{\textrm{nl}}^{YZ}\rightarrow(r_{0}/2)\theta_{\textrm{SHE}}\sin(2\phi)e^{-L/l_{s}}$. Once $\theta_{\textrm{SHE}}$ is determined, the SGE coefficient is easily retrievable by fitting the full lineshape to Eq. (10).

We now discuss an optospintronic analogue, in which a transition metal dichalcogenide monolayer replaces the FM contact as a spin injector (GSOC_TMDs_Muniz_15, ). Optical spin-injection has been recently demonstrated in graphene heterostructures (GTMD_Optical_Avsar_ACSNano2017, ; GTMD_Optical_Luo_NanoLett2017, ). Since optically-pumped spin currents from atomically thin semiconductors with spin-valley coupling are polarized normal to the 2D plane, an in-plane field $\boldsymbol{B}=B\hat{y}$ can be used to detect ISHE and SGE simultaneously. The spin-transresistance $\Delta R_{\textrm{nl}}^{z}\equiv[\left.V_{\textrm{nl}}(L)\right|_{n_{z}>0}-\left.V_{\textrm{nl}}(L)\right|_{n_{z}<0}]_{x=L}/2I$ is easily computed as

[TABLE]

where $q$ is defined as before and $r_{0}^{z}=s^{z}(0)\bar{W}D/(l_{s}I\sigma_{\textrm{2D}})$. Similar to in-plane spin injection configuration, the ISHE and SGE signals have different parity under mirror reflection $B\rightarrow-B$. Typical lineshapes are shown in Fig. 2(b).

Experimental feasibility. The measurement protocol is directly applicable to devices with anisotropic spin dynamics, e.g. due to BR effect or spin-valley locking (Yang15, ). Different material systems and device layouts can be used providing (i) the injection/detection regions are well separated to avoid the use of large precession fields ($L\gtrsim l_{s}$) and (ii) the inversion symmetry breaking within the spin channel is weak enough ($l_{R}\apprge l_{s}$). Such a moderate SOC regime also guarantees that the spin-charge conversion is within the linear regime (Eq. (7)), which is indeed observed for most devices (SID_Reyren, ; SID_Isasa, ; SID_Lou, ; SID_Kamerbeek, ; SpinChannel_Gurram_18, ; SpinChannel_Kamalakar_15, ; SpinChannel_Tombros_07, ; SHE_SGE_2D_1, ; SHE_SGE_2D_2, ; SHE_SGE_2D_3, ). For example, in monolayer graphene, one finds $l_{R}\approx\hbar v_{F}/\lambda\approx 14$ $\mu\textrm{m}\gg l_{s}$ for a typical $\lambda=50$ $\mu$eV (Min_06, ; Konschuh_10, ). For interfaces hosting 2D electron gases with weak or moderate BR effect, the situation is similar. For example, using representative values for the Au(111) surface, \alpha=$$0.03 eV$\cdot\mathring{\textrm{A}}$ and $m^{*}=0.26m_{e}$ (Popovic05, )), one obtains $l_{R}\approx 100$ nm $\gg l_{s}$. Beyond this regime, the interpretation of experimental data requires explicit modeling (using a self-consistent solution of Eqs. (2)-(4)), hindering the unambiguous detection of ISHE/SGE.

We conclude with a remark on the spin-injector magnetization response to the applied field. As shown in this work, the tilting of the magnetization $\mathbf{M}$ away from its preferred easy axis reduces the ISHE/SGE nonlocal resistance by a nonlinear factor (“visibility”) given by $\mathcal{V}_{\mathbf{B}}=[s^{y}(0)]_{\mathbf{B}}/[s^{y}(0)]_{\textrm{ideal}}$. The field dependence of the initial spin accumulation $[s^{y}(0)]_{\mathbf{B}}$ can be accurately described by a Stoner-Wohlfarth model with parameters determined from separate Hanle measurements. According to the realistic simulations provided in the SM (SM, ), the visibility can be as large as 20% at low fields $\sim 0.1$ T for typical FM saturation parameters. We expect that the measurement scheme introduced in this work will enable accurate experimental determination of spin-charge conversion parameters ($\theta_{\textrm{SHE}}$, $\theta_{\textrm{SGE}}=l_{s}/l_{\textrm{SGE}}$). We note that a similar scheme can in principle be employed for electrical detection of spin Hall effect and Edelstein effect, the Onsager reciprocal phenomena of ISHE and SGE. For example, this could be achieved by injecting a current perpendicular to the channel and measuring the resulting spin-electrochemical accumulation at the FM contact.

In compliance with EPSRC policy framework on research data, this publication is theoretical work that does not require supporting research data.

*Acknowledgements.–*A.F. gratefully acknowledges the financial support from the Royal Society, London through a Royal Society University Research Fellowship. M.O. and A.F. acknowledge funding from EPSRC (Grant Ref: EP/N004817/1). Y.-H. L., C. H. , and M.A.C. acknowledge support from the Ministry of Science and Technology of Taiwan through grants 102-2112-M-007- 024-MY5 and 107-2112-M-007-021-MY5, as well as from the National Center for Theoretical Sciences (NCTS, Taiwan).

S1. Nonlocal Resistance: Injector Magnetization Tilting

In this section, we illustrate the application of the filtering protocol with realistic simulations of the nolocal resistance that take into account the FM magnetization tilting. The $\mathbf{B}$-dependent polarization $\hat{\boldsymbol{n}}$ of injected spins is parameterized as follows

[TABLE]

where $p=\pm 1$ for initial \hat{\boldsymbol{n}}=$$\pm\hat{y} magnetic configurations of the ferromagnetic injector. The field-dependence of the angle variables ($\beta,\gamma$) is usually described by a Stoner-Wohlfarth model (Raes_SRTA_Graphene_16, ; SpinChannel_Tombros_07, ). For our purposes, it suffices to consider a heuristic model for the tilting angles

[TABLE]

where $B_{\textrm{sat;}i}$ is the saturation field normal to the easy axis $\hat{i}\perp\hat{y}$ ($B_{\textrm{sat;z}}\gg B_{\textrm{sat;x}}$ due to shape anisotropy (SpinChannel_Tombros_07, ; Raes_SRTA_Graphene_16, )).

Figure 3 shows the nonlocal resistance of the simulated spin-valve device displaying ISHE and SGE. The bare signal for “parallel” ($\hat{\boldsymbol{n}}=+\hat{y}$) and “antiparallel” ($\hat{\boldsymbol{n}}=-\hat{y}$) initial configurations is shown in the left panel. The plateaus at large field $|B|$ signal the saturation of the ferromagnetic contact. The middle panel shows the spin-transresistance between parallel and antiparallel configurations. The right panel shows the ISHE/SGE lineshapes isolated using the protocol presented in the main text, i.e.,

[TABLE]

The ISHE/SGE signal has even/odd parity with respect to mirror reflection ($\phi\rightarrow-\phi$). This shows that the filtering scheme accurately separates the two independent components. Furthermore, the lineshapes have the same exact form than those presented in Fig. 2, main text (in which the tilting correction had been neglected by setting $R_{0}=$constant).

Importantly, the unambiguous ISHE/SGE detection is possible even for channels with anisotropic spin relaxation ($\tau_{\parallel}\neq\tau_{\perp}$), i.e. devices where the spin channel itself is characterized by strong SOC. This is shown in Fig.4 (left and middle panels). The filtering is still accurate (albeit not exact) in channels with strong BR effect ($l_{R}$ a few times $l_{s}$) and breaks down for $l_{R}\apprle l_{s}$. An example with $l_{R}=5l_{s}$ is shown in Fig. Fig.4 (right panel). In graphene, this would correspond to a proximity-induced BR field on the order of 1 meV.

S2. Hanle Precession: Role of BR field

Here, we show that the coherent precession around a (strong) BR field is also detectable in standard nonlocal Hanle measurements with spin-injector (FM1) and spin-detector (FM2). Figure 5 contrasts the Hanle curve of an isotropic channel ($l_{R}=\infty$ and $l_{s,\parallel}=l_{s,\perp}=l_{s}$ (Raes_SRTA_Graphene_16, )) with that of a pure BR channel ($l_{R}<\infty$ and $l_{s,\parallel}=\sqrt{2}l_{s,\perp}$ (GTMD_SRTA_Offidani_18, )). The impact of a short $l_{R}\propto l_{s}$ is perceptible for applied field with $\phi<90^{\circ}$, enabling diffusive spins to precess in the $Oxz$ plane (comment_SP; ZhangWu_11, ). (This effect was previously noted in Ref. (ZhangWu_11, ), where it was shown that the BR field induces damped oscillations in $s^{a}(x)$ when injected spins are oriented in the $Oxz$-plane.) Interestingly, the BR precession boosts the effective spin diffusion length (Lin_LongPaper, ), which explains the enhanced $\hat{y}$-spin density accumulation away from the detector as compared to a hypothetical BR channel with the same $l_{s,\parallel}=\sqrt{2}l_{s,\perp}$ but with $l_{R}=\infty$.