This document is the Accepted Manuscript version of a Published Work that appeared in final form in Nano Letters, copyright © American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see: https://dx.doi.org/10.1021/acs.nanolett.8b04368. Room temperature spin Hall effect in graphene/MoS2 van der Waals heterostructures C. K. Safeer1,‡, Josep Ingla-Aynés1,‡, Franz Herling1,‡, José H. Garcia2, Marc Vila2,3, Nerea Ontoso1, M. Reyes Calvo1,4, Stephan Roche2,5, Luis E. Hueso1,4, Fèlix Casanova1,4,*. 1 CIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain. 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra, Catalonia, Spain. 3 Department of Physics, Universitat Autònoma de Barcelona, Campus UAB, 08193 Bellaterra, Catalonia, Spain. 4 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain. 5 ICREA – Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Catalonia, Spain. ‡ These authors contributed equally to this work *E-mail: f.casanova@nanogune.eu Keywords: Graphene, Transition metal dichalcogenides, Spintronics, Spin Hall effect, Rashba- Edelstein effect. Graphene is an excellent material for long distance spin transport but allows little spin manipulation. Transition metal dichalcogenides imprint their strong spin-orbit coupling into graphene via proximity effect, and it has been predicted that efficient spin-to-charge conversion due to spin Hall and Rashba-Edelstein effects could be achieved. Here, by combining Hall probes with ferromagnetic electrodes, we unambiguously demonstrate experimentally spin Hall effect in graphene induced by MoS2 proximity and for varying temperature up to room temperature. The fact that spin transport and spin Hall effect occur in different parts of the same material gives rise to a hitherto unreported efficiency for the spin- to-charge voltage output. Remarkably for a single graphene/MoS2 heterostructure-based device, we evidence a superimposed spin-to-charge current conversion that can be indistinguishably associated with either the proximity-induced Rashba-Edelstein effect in graphene or the spin Hall effect in MoS2. By comparing our results to theoretical calculations, the latter scenario is found the most plausible one. Our findings pave the way towards the combination of spin information transport and spin-to-charge conversion in two-dimensional materials, opening exciting opportunities in a variety of future spintronic applications. The efficient transport and the manipulation of spins in the same material are mutually exclusive as they would require simultaneously weak and strong spin-orbit coupling (SOC) respectively. Graphene, due to its low intrinsic SOC, is proven to be an outstanding material that can transport spins over long distance of tens of micrometres1-10. For the same reason, the generation and tuning of spin currents in graphene are out of reach, limiting its capability to active spintronic device functionalities and related applications. To solve this issue, methods to artificially induce SOC in graphene have been explored. For instance, the SOC in graphene has been enhanced by chemical doping11-17 or by proximity-induced coupling with materials possessing large SOC18-35. The latter method is more convenient since the chemical properties of graphene are not altered, whereas its high-quality electronic transport properties are preserved. Transition metal dichalcogenides (TMDs) with chemical formula MX2 (M=Mo, W and X=S, Se) are layered materials of semiconducting nature displaying unique combined electronic, optical, spintronic and valleytronic properties36-50. They possess a strong intrinsic SOC of tens of meV, few orders larger than that of pristine graphene36,37. Accordingly, a large intrinsic spin Hall effect (SHE) has been theoretically predicted in TMDs38. However, its experimental observation remains elusive because of the technical difficulties to inject and detect spin information into these materials49,50. Only recently, a spin-orbit torque experiment has shown a large spin-to-charge conversion in monolayer MoS2 and WSe2 interfaced with a ferromagnet, although it has been attributed to Rashba- Edelstein effect (REE)51. Interestingly, TMDs can be hybridized to graphene forming van der Waals heterostructres18-35 Theoretical calculations show that TMDs imprint their strong SOC into graphene via proximity effect, with an induced SOC of the order of few meV17,18. The strong induced SOC in TMD-proximitized graphene has been confirmed by many weak anti-localization measurements21-24. Another predicted feature of this proximity effect is the induced spin-valley coupling in graphene, leading to a large spin lifetime anisotropy25 which has also been recently confirmed experimentally26,27. Further interesting theoretical predictions for TMD-proximitized graphene include the emergence of the SHE29,30 and the Rashba-Edelstein effect (REE)30,31. The SHE and the REE are exotic phenomena that convert charge currents into spin currents (in case of SHE)52-56 or spin densities (in case of REE)57-62. Their reciprocal effects [inverse SHE (ISHE) and inverse REE (IREE)] convert spin currents or spin densities into charge currents. Since these mechanisms lead to highly efficient electrical generation and detection of spins, they can play an important role towards the realization of all-electrically controlled spintronic devices, such as spin-orbit torque memories63- 65 or spin-orbit logic circuits66,67. Whereas the SHE has been mostly studied in heavy metals with strong SOC52-56, the REE has been observed in two-dimensional electron gas systems58-60 as well as interfaces lacking inversion symmetry61,62. In graphene, previous measurements of the SHE using nonlocal Hall bar geometries have been reported15,16,20. However, a variety of effects unrelated to spin can contribute to such nonlocal measurements, questioning the interpretation of the results and therefore impeding a further optimization and implementation of those spin effects into practical applications68-72. A convincing experimental proof of spin-to-charge conversion due to the SHE or the REE in graphene is thus crucially lacking. In this letter, by carefully designing graphene/TMD- based lateral spin valves (LSVs), we solve this long-standing experimental challenge, and report an unprecedented manifestation of spin-to-charge conversion in graphene/MoS2 van der Waals heterostructures that can be unambiguously attributed to the SHE in graphene. Additionally, a second contribution, which can be induced by either the REE in graphene or the SHE in MoS2 is also clearly observed. Our theoretical calculations indicate that the latter is the most plausible source, thus providing the first all-electrical measurement of spin Hall effect in MoS2. Realization of spin transport and spin-to-charge conversion in the same material (in this case graphene) a long awaiting milestone for the field of spintronics. Our device design consists of MoS2 placed in the middle of cross-shaped graphene as shown in Figure 1. This design enables studying spin-charge interconversion stemming from three distinct sources: 1) Proximity-induced SHE in graphene30,53 (Figure 1a); 2) Proximity-induced REE in graphene30,31,62 (Figure 1b); 3) SHE in MoS2 55,56,76,77 (Figure 1c). For the first case, a charge current applied along the graphene/MoS2 stripe results in the generation of spin current with out-of-plane spin polarization along the graphene channel. In the second and third cases, a charge current applied along the graphene/MoS2 stripe gives rise to a spin current with the same in-plane polarization along the graphene channel. The signal induced by the SHE in graphene can thus be separated from the other two effects by verifying the spin polarization directions. The Onsager reciprocal equivalents, ISHE and IREE, can also be studied and separated in the same manner. Figure 1. Description of our device configuration and possible spin-charge interconversion scenarios. a) Sketch of proximity-induced spin Hall effect in graphene. A charge current applied along the graphene/MoS2 stripe (𝑦-axis) results in a spin current with out-of-plane (along 𝑧) spin polarization in the graphene channel along 𝑥. b) Sketch of proximity-induced Rashba-Edelstein effect in graphene. A charge current applied along the graphene/MoS2 stripe (𝑦-axis) creates a spin accumulation with in-plane (along 𝑥) spin polarization which then diffuses to the graphene channel along 𝑥. c) Sketch of spin Hall effect in MoS2. A charge current applied along the 𝑦-axis of MoS2 creates an out-of-plane (along 𝑧) spin current with in-plane (along 𝑥) spin polarization. When this MoS2 is placed on top of graphene, this spin current diffuses to the graphene channel along 𝑥. d) Optical microscope image of one of our devices (sample A). It contains graphene shaped into two Hall bars connected each other (blue). The ends of the graphene stripes are connected to Ti/Au contacts (yellow). The MoS2 flake (light blue) lies on one of the graphene Hall bars. Four Co/TiOx electrodes (grey) are placed on top of the graphene channel. To study and quantify the different spin-to-charge conversion contributions, we fabricated devices as the one shown in Figure 1d (sample A). Each device contains exfoliated few-layer graphene shaped into a narrow channel (𝑊gr = 350 nm) with a double Hall bar (𝑊H = 1.2 μm). A multilayer MoS2 flake lies on top of one of the graphene Hall bars. Four ferromagnetic (FM) Co electrodes (with interface resistances ranging from 10 to 70 k due to a TiOx underlayer) are placed on the graphene channel forming three different LSVs. A detailed description of the device fabrication process is given in methods. As shown in Figure 1d, the LSV with FM electrodes 1 and 2 and the LSV with electrodes 3 and 4 can both be used as reference LSVs. The LSV with FM electrodes 2 and 3 contains MoS2 placed on top of the graphene Hall bar between the FM electrodes. The spin-to-charge conversion is measured between the graphene/MoS2 vertical stripe and FM electrode 3. Figure 2. Spin transport in LSVs. a) Sketch of the nonlocal resistance measurement at the reference graphene LSV. b) Nonlocal resistance measurements performed by applying field along the in-plane easy axis (𝐵𝑦 ). The black and green curves correspond to the reference graphene LSV (with FM electrodes 1 and 2) and the LSV with MoS2 in between (with FM electrodes 2 and 3), respectively. Baselines are subtracted in both curves. c) Nonlocal symmetric Hanle curves measured at the reference graphene LSV by applying a magnetic field along the in-plane hard axis (𝐵𝑥 ) for the initial parallel (𝑅𝑃 , black) and antiparallel (𝑅𝐴𝑃 , red) configuration of the FM electrodes. Baselines are subtracted in both curves. d) Spin signal calculated using 𝑅𝑃 − 𝑅𝐴𝑃 from the Hanle data in c) (black open circles) with the corresponding fit (blue solid line) using the Bloch equations and the extracted parameters. Data in all panels correspond to sample A at 10 K. An electrical current 𝐼C =10 µA is used. First, we study the reference LSV with FM electrodes 1 and 2 to extract the spin transport properties of pristine graphene. The measurements are done in the standard nonlocal configuration73,74. A charge current (𝐼C ) is injected from a Co electrode into the graphene channel, creating a spin accumulation at the Co/graphene interface. This spin accumulation diffuses toward both sides of the graphene channel, creating a pure spin current out of the current path, which is detected by another Co electrode as a nonlocal voltage (𝑉NL ), see Figure 2a. The nonlocal resistance P AP 𝑅NL = 𝑉NL /𝐼C is high (𝑅NL , parallel) or low (𝑅NL , antiparallel) depending on the relative orientation of the magnetization of the two electrodes, which can be set by applying an in-plane magnetic field in the 𝑦−direction (𝐵𝑦 ) due to the different shape anisotropies of the electrodes (Figure 2b). The spin P AP signal, which is defined as ∆𝑅NL = 𝑅NL − 𝑅NL , is ~60 mΩ at 10 K. Moreover, a Hanle precession measurement has been performed to characterize the spin transport properties of the graphene channel (Figure 2c). Since the injected spins are oriented along the 𝑦−direction, a perpendicular in- plane magnetic field 𝐵𝑥 is applied. The precession and decoherence of the spins cause the oscillation and decay of the signal. In addition, the rotation of the Co magnetizations with 𝐵𝑥 tends to align the polarization of the injected spin current with the applied field, restoring the 𝑅NL signal to its zero- field value when the Co electrodes reach parallel magnetizations along the 𝑥−direction at high enough 𝐵𝑥 . By the proper combination of Hanle measurements for initial parallel and antiparallel states of the Co electrodes, the contribution from spin precession and decoherence (Figure 2d) and the rotation angle β of the Co magnetization (Note S1 in the Supporting Information) can be obtained. The Hanle curve for the spin precession and decoherence is then fitted to the solution of the Bloch equations75 as shown in Figure 2d (for details, see Note S1 in the Supporting Information). From this fitting, the spin lifetime of the pristine graphene 𝜏𝑠𝑔𝑟 =300±20 ps, the spin polarization of the Co/graphene interface 𝑃𝑖 =2.9±0.2% and the spin diffusion constant of the pristine graphene 𝑔𝑟 𝐷𝑠 =(5.7±0.5)× 10−3 m2/s are extracted. Next, we show the nonlocal resistance in the LSV with the middle MoS2 flake (between FM electrodes 2 and 3). ∆𝑅NL in this case is expected to be lower than ref that of the reference LSV (∆𝑅NL ) due to the well-known spin absorption effect33,34 and the proximity-induced spin relaxation in the graphene channel26,27,32. Depending on the quality of the graphene/MoS2 interface, the reduction of the spin signal can vary. In our case, the measured spin signal for graphene/MoS2 LSV at 10 K is smaller than the noise level (~5 m), as shown in Figure 2b. Once we confirmed that the spin signal diminishes at the graphene/MoS 2 region, we inspect the possibility of spin-to-charge conversion in the same region. For this, we initially align the magnetization of the Co electrodes along the easy axis by applying a magnetic field (±𝐵𝑦 ). Subsequently, while sweeping the field along the in-plane hard axis (±𝐵𝑥 ), 𝑉NL is measured across the graphene/MoS2 bar (Figure 3d and 3f) while injecting a constant current 𝐼C =10 µA from the Co electrode 3 to the graphene channel. As mentioned above, a voltage across the graphene/MoS2 branch placed along the 𝑦−direction can be obtained either due to the spin-to-charge conversion of i) out-of-plane spins with spin polarization along 𝑧 or ii) in-plane spins with spin polarization along 𝑥. For the first case, as the initial spin polarization (𝑦) and the applied field (𝑥) are orthogonal, the spins precess in the 𝑦 − 𝑧 plane around the field. This process results in an antisymmetric Hanle signal exhibiting either a maximum or minimum at certain values of ±𝐵𝑥 , when the spins reaching the graphene/MoS2 region point out of plane (𝑧). The antisymmetric Hanle curve is also reversed when the initial magnetization direction is switched, since the injected spins are opposite (Figure 3d) 30,46,77. For the second case, as the required spin polarization and the applied field are along the same direction (𝑥), 𝑅NL increases or decreases while the electrode magnetization rotates towards ±𝑥 and saturates at values of ±𝐵𝑥 that correspond to the saturation magnetization of the Co electrode (Figure 3f). This results in an S-shaped signal that is odd with 𝐵𝑥 and independent of the initial magnetization direction of the Co electrode76,77. As shown in Figure 3a, we clearly observe antisymmetric Hanle curves which reverse by switching the initial magnetization direction. This confirms that, at the graphene/MoS2 region, the out-of-plane spins are converted into a charge current, which in open circuit condition gives rise to 𝑉NL. As explained in Figure 1, the only possible source of out-of-plane spin-to-charge conversion in our experimental design is the ISHE in graphene due to the SOC induced by MoS2 (Figure 1a). Therefore, we unambiguously conclude the observation of proximity-induced ISHE in graphene in sample A. For further confirmation, we repeat the same experiment in a different sample (sample B) obtaining similar results. In sample B, we also performed measurements where the spin current direction is reversed by injecting current with the Co electrode on the right and the left side of the graphene/MoS2 region, respectively (Note S3 in the Supporting Information). We find that the spin-to-charge conversion signal changes sign with reversing the spin current direction, which further confirms the proximity-induced ISHE in graphene as the source of the signal, since the ones generated by the other possible sources (REE in graphene and SHE in MoS2) do not depend on the spin current direction in the graphene channel. Figure 3. Spin-to-charge conversion measurement and analysis. a) Nonlocal spin-to-charge conversion curves obtained by applying a charge current between Co electrode 3 and the right Ti/Au electrode and measuring the voltage across the graphene/MoS2 stripe. The magnetic field is applied along the in-plane hard axis direction (𝐵𝑥 ) for initial positive (black) and negative (red) magnetization directions of the Co electrodes. b) Same measurements obtained by applying a charge current between Co electrode 2 and the right Ti/Au electrode and measuring the voltage across the pristine graphene stripe in the reference LSV. c) Net antisymmetric ↑ ↓ Hanle signal (open circles) obtained by subtracting the two Hanle curves (∆𝑅SH = 𝑅𝑁𝐿 − 𝑅𝑁𝐿 ) in a), which is fitted with the solution of the Bloch equation (blue curve) and the extracted parameters. d) Sketch of the measurement configuration that results in the antisymmetric Hanle curve shown in c). When 𝐵𝑥 is low enough, the Co magnetization is in its easy axis and the injected (red) spins are polarized along 𝑦, leading to a spin precession along the 𝑦−𝑧 plane. e) Background signal (open circles) extracted by averaging the two ↑ ↓ Hanle curves [𝑅𝑆𝐶𝐶 = (𝑅𝑁𝐿 + 𝑅𝑁𝐿 )/2] in a). The magnitude of this spin-to-charge conversion signal (∆𝑅𝑆𝐶𝐶 ) is quantified by calculating the zero-field extrapolation using linear fittings to the signal at high positive and negative fields. f) Sketch of the measurement configuration that results in the S-shaped curve shown in e). When 𝐵𝑥 saturates the Co magnetization in its hard axis, the injected (red) spins are polarized along 𝑥 and do not precess. Data in all panels correspond to sample A at 10 K. To quantify the overall spin-to-charge conversion due to the ISHE in proximitized graphene, the two antisymmetric Hanle curves for the initial positive and the negative magnetizations direction ↑ ↓ of the Co electrodes were subtracted (∆𝑅SH = 𝑅𝑁𝐿 − 𝑅𝑁𝐿 ) and the resulting curve is shown in Figure 3c. Fitting this resulting curve to the solution of the Bloch equation (Note S2 in the Supporting 𝑇𝑀𝐷/𝑔𝑟 Information), we extract a spin Hall angle 𝜃𝑆𝐻 = −4.5±0.6%, an effective spin lifetime 𝑒𝑓𝑓 𝑒𝑓𝑓 𝜏𝑠 =72±7 ps, and an effective spin diffusion constant 𝐷𝑠 =(1.2±0.1)× 10−3 m2/s at 10 K. Comparing the variation of 𝑉NL with respect to the Co magnetization direction and the direction of 𝐵𝑥 , the spin Hall angle is found to be negative. The robustness of this value is confirmed in sample 𝑇𝑀𝐷/𝑔𝑟 B, where 𝜃𝑆𝐻 = −4.8±0.9%, 𝜏𝑠𝑒𝑓𝑓 =70±9 ps, and 𝐷𝑠𝑒𝑓𝑓 =(1.5±0.1)× 10−3 m2/s are obtained at 10 K. 𝑇𝑀𝐷/𝑔𝑟 These values are an order of magnitude larger than 𝜃𝑆𝐻 ~0.1% predicted theoretically for MoS2 (Ref. 29). We understand such difference by noting that those calculations have been limited to a phenomenological description of disorder using intrinsic broadening of the electronic states. Here by performing spin Hall conductivity calculations for monolayer graphene/MoS2 heterostructures, we traced back the origin of the SHE to the combination of the valley-Zeeman interaction, the intrinsic SOC, and the staggered potential. Using this model, and taking into account 𝑇𝑀𝐷/𝑔𝑟 the broadening effects and temperature, we predict a 𝜃𝑆𝐻 ranging from 1 to 10%, depending on 𝑇𝑀𝐷/𝑔𝑟 the position of the Fermi Level. 𝜃𝑆𝐻 is larger close to the charge neutrality point and changes from negative to positive when going from the valence to the conduction band. This last result would then imply that, in the current experiment, graphene is p-doped. Details are explained in Note S8.1 in the Supporting Information. To confirm that the spin-to-charge conversion occurs strictly in the graphene regions in proximity with MoS2, and not in the pristine graphene itself, we perform a control experiment in the reference LSV by injecting current with Co electrode 2 and measuring 𝑉NL across the vertical pristine graphene stripe placed next to it (Figure 1d). Figure 3b shows no evidence of Hanle spin precession effect. Instead, the fact that 𝑅NL varies linearly with the magnetic field confirms that the spin-to- charge conversion measurement in graphene/MoS2 is not due to any experimental spurious effect. Ideally, we expect the linear background in the reference LSV to be also present in the background of the antisymmetric Hanle curve (Figure 3a) measured in graphene/MoS2. To check this, we extract the background signal of the spin-to-charge conversion measurements in graphene/MoS2 by ↑ ↓ excluding the antisymmetric component [𝑅SCC = (𝑅𝑁𝐿 + 𝑅𝑁𝐿 )/2] as shown in Figure 3e (see Note S4.1 in the Supporting Information). Surprisingly, we found an S-shaped background signal of amplitude 11.5 mΩ with changing slopes at magnetic field 𝐵𝑥 ~ ±240 mT, the same field that saturates the magnetization of the Co electrodes. This indicates that an in-plane spin-to-charge conversion signal in graphene/MoS2 is also measured. As explained above, there are two possible sources for in-plane spin-to-charge conversion in our device configuration: the proximity-induced IREE in graphene (Figure 1b) and the ISHE in MoS2 (Figure 1c). In our experiment, both are indistinguishable, jeopardizing a separate quantification of the efficiencies of the two effects. However, we can separately calculate the limits of the efficiencies of each phenomenon by assuming that the signal is entirely obtained from a single effect. For the proximity-induced IREE in graphene, we quantify the spin-to-charge conversion efficiency to be 𝛼𝑅𝐸 ~ +0.85% (a detailed derivation is given in Note S4.2 in the Supporting Information). One of the major differences between the REE and the SHE is that the former produces a non-equilibrium spin density instead of a spin current (as is the case of the latter), which complicates the definition of a figure of merit. An equivalent of a spin Hall angle 𝛼𝑅𝐸 can be nevertheless proposed by assuming that the spin accumulation transforms entirely into a spin current by diffusion, and divide it by the injected charge current density. To obtain such information, numerical simulations based on Kubo methods are performed29,30 together with the analytical theory 𝑇𝑀𝐷/𝑔𝑟 of Offidani et al.31. Using the same spin-orbit parameters used to compute 𝜃𝑆𝐻 , we estimate that the REE dimensionless efficiency 𝛼𝑅𝐸 would be at maximum 0.1% (see Note S8.2 in the Supporting Information for details), which is one order of magnitude smaller than the value extracted from the experiment. More remarkably, the theoretical analysis predicts that the sign of the proximity-induced 𝑇𝑀𝐷/𝑔𝑟 REE is the same as for the proximity-induced SHE. Since 𝜃𝑆𝐻 is negative, 𝛼𝑅𝐸 should also be negative. Therefore, the positive sign of the in-plane spin-to-charge conversion rules out the proximity-induced REE as its origin. On the other hand, if the in-plane spin-to-charge conversion occurs due to ISHE in MoS2, it requires spin absorption from graphene into MoS2, which depends on the resistivities and spin diffusion lengths of both materials and their interface resistance 33,34. As graphene and MoS2 are stamped together in our device, it is not possible to separately quantify the resistivity of MoS2 (𝜌𝑀𝑜𝑆2 ). Therefore, we examine the possible strength of ISHE in MoS2 by varying 𝜌𝑀𝑜𝑆2 (see Note S4.3 in the Supporting Information). Here, the spin-to-charge conversion voltage measured across the graphene/MoS2 channel depends on two competing contributions that vary in opposite way with 𝜌𝑀𝑜𝑆2 : 1) an increase in 𝜌𝑀𝑜𝑆2 decreases the spin absorption and, thus, the spin-to-charge conversion voltage; 2) the effective resistance of the graphene/MoS2 region increases with 𝜌𝑀𝑜𝑆2 and, thus, increases the spin-to-charge conversion voltage until it saturates due to the shunting effect of the graphene channel. In the optimal resistivity value (𝜌𝑀𝑜𝑆2 ~7×10-4 m), a minimum spin Hall angle of MoS2 𝜃𝑆𝐻𝑀𝑜𝑆2 ~3.3% is required to obtain the spin-to-charge conversion signal of 11.5 mΩ. In this 𝑀𝑜𝑆 scenario, we can estimate a spin Hall conductivity 𝜎𝑆𝐻 2 ~0.47 Ω-1cm-1, which is reasonable for a bulk n-doped MoS2 flake, as theoretically calculated in Ref. 38. It thus seems more plausible that the observed in-plane spin-to-charge conversion arises from the ISHE in MoS2. We finally performed additional spin-to-charge conversion measurements at 100 K, 200 K and 300 K (see Notes S5 and S6 in the Supporting Information). As shown in Figure 4a for sample A, the out-of-plane component of the spin-to-charge conversion signal clearly decreases with increasing temperature, while the in-plane component persists up to room temperature, indicating that the former is strongly influenced by the temperature in contrast to the latter. Unfortunately, we could not measure for both magnetization directions of the Co injector, which prevents us from a full quantification of the ISHE in the proximitized graphene as done at 10 K. For sample B, we could properly extract the spin Hall angle, which strongly decays with temperature from −4.8±0.9% at 10 K down to −0.33±0.04% at 300 K (Figure 4b). For this estimation, we also need to take into account the temperature dependence of the graphene/MoS2 sheet resistance (inset in Figure 4b), which we observe to decrease with increasing temperature, a feature reported in graphene when the Fermi level is near the Dirac point77. Theoretically, the spin Hall conductivity of proximitized graphene is also expected to decrease with temperature close to the neutrality point due to the change of sign when crossing from the valence to the conduction band. In Note S8.1 in the supporting information, we show that the spin Hall conductivity is inversely proportional to the temperature, which combined with the decrease of the sheet resistance with temperature leads to a fast decay of the spin Hall angle. Concerning the in-plane spin-to-charge conversion, it can be quantified up to room temperature in sample A because in this case one curve at a single magnetization direction of the Co injector already gives reliable information. This component slightly decreases from 11.5 mΩ at 10 K to 7.4 mΩ at 300 K. On the one hand, if we assume the origin of the signal is IREE in proximitized graphene, the conversion efficiency 𝛼𝑅𝐸 increases from 0.85±0.03% at 10 K to 3.0±0.2% at 300 K. This is in sharp contrast with our theoretical calculations, which predict that 𝛼𝑅𝐸 strongly decays with temperature due to the antisymmetric behavior between the valence and conduction bands which will be mixed in the presence of temperature (see Note S8.2 in the Supporting Information for details). On the other hand, if we assume the origin of the signal is ISHE in MoS2, we need a 𝑀𝑜𝑆 minimum 𝜃𝑆𝐻 2 ~2.2% to explain the signal at 300 K, a value close to the ~3.3% at 10 K. Such a weak temperature variation in the spin Hall angle is expected for a bulk material, which is the case of our MoS2 flake. Although we cannot fully rule out any scenario, the temperature dependence of the in-plane spin-to-charge conversion suggests again that spin absorption and subsequent ISHE in the MoS2 flake is a more plausible option. It is worth mentioning that no clear in-plane spin-to-charge conversion is observed in sample B (see Notes S4.1 and S5 in the Supporting Information). The lack of signal in sample B can be again better understood if one assumes that its origin is the ISHE in MoS2, rather than the IREE in proximitized graphene: whereas the spin absorption into MoS2 depends on details of the graphene and the MoS2 flake that are independent of the presence of the ISHE in the proximitized graphene, IREE in proximitized graphene should be present whenever the proximity-induced ISHE occurs. Figure 4. Temperature dependence of the spin-to-charge conversion. a) Nonlocal spin-to-charge conversion curves for initial positive magnetization direction of Co electrode 3 at different temperatures in sample A. b) Spin Hall angle of the proximitized graphene at different temperatures for sample A (blue solid circle) and sample B (red solid squares). Inset: Square resistance as a function of temperature from two-point measurements along the graphene/MoS2 stripe in samples A and B. Finally, it is also worth noting that, although the spin Hall angle in proximitized graphene is smaller than that of the best SHE materials such as heavy metals or topological insulators, the overall spin-to-charge conversion efficiency can be very high. A proper way to quantify this efficiency is the ratio 𝑅𝑒𝑓𝑓 between the output voltage (∆𝑉SH) and the injected spin current reaching the spin Hall region (𝐼S ), which has units of resistance (see Note S7 in the Supporting Information). In the device 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 ̅ geometry used in this work, 𝑅𝑒𝑓𝑓 = 2𝜃𝑆𝐻 𝑅𝑠𝑞 (𝐼𝑠 /𝐼𝑠 ), where 𝐼𝑠̅ is the average spin 𝑇𝑀𝐷/𝑔𝑟 current at the spin Hall region. Since all parameters are known, we can estimate 𝑅𝑒𝑓𝑓 = 13.4 . Previous reports of the largest spin-to-charge conversion output were performed in graphene/Pt LSVs76,77 in which graphene was used to transport spins into Pt where the spin-to-charge conversion 𝑃𝑡 occurs. In this case, the efficiency is defined as 𝑅𝑒𝑓𝑓 = (2𝜃𝑃𝑡 ̅ 𝑃𝑡 𝑆𝐻 𝜌𝑃𝑡 𝑥𝑃𝑡/𝑔𝑟 /𝑊𝑃𝑡 )(𝐼𝑠 /𝐼𝑠 ), where 𝜃𝑆𝐻 , 𝜌𝑃𝑡 , 𝑊𝑃𝑡 are the spin Hall angle, resistivity and width of Pt, and 𝑥𝑃𝑡/𝑔𝑟 a correction factor that considers the current in Pt shunted through the graphene channel. Taking the parameters from Ref. 76, we 𝑃𝑡/𝑔𝑟 obtain 𝑅𝑒𝑓𝑓 =0.27 . In other words, the unique feature of spin transport and spin-to-charge 𝑇𝑀𝐷/𝑔𝑟 conversion in graphene itself (with 𝜃𝑆𝐻 = − 4.5%) results in an efficiency 50 times larger than when spin-to-charge conversion occurs in a different material after spin absorption, even if Pt (with 𝑃𝑡 𝜃𝑆𝐻 = 23%) is used. To conclude, an unprecedented and unambiguous experimental demonstration of the proximity-induced ISHE has been found in graphene together with a manifestation of another spin- to-charge conversion phenomenon either due a proximity-induced IREE in graphene or an ISHE in MoS2, being the latter more likely. We obtained the largest spin-to-charge conversion efficiency as compared to previous reports in LSVs. The device concept proposed here, a LSV with a cross- shaped graphene channel, can easily integrate a gate voltage which is expected to tune the spin-to- charge conversion in graphene29,30, enabling an extra functionality with a lot of potential which is not possible in prototypical spin Hall metals. The device concept can also be extended to study spin-to- charge conversion in a variety of material combinations of graphene with heavy metals, oxides, different TMDs, and topological insulators, which have been also suggested to be suitable companions for generating giant REE79. Most importantly, our proof of concept in which spins can be generated, transported and detected in the same material (graphene) is a propitious feature for developing future spintronics devices. Methods. Device fabrication. The graphene/MoS2 van der Waals heterostructures are fabricated by mechanical exfoliation followed by dry viscoelastic stamping. We first exfoliate graphene from bulk graphitic crystals (supplied by NGS Naturgraphit GmbH) using a Nitto tape (Nitto SPV 224P) onto Si substrates with 300 nm SiO2. Few-layer graphene flakes are identified by optical contrast under an optical microscope. Then a MoS2 crystal (supplied by SPI supplies) is exfoliated using the Nitto tape and transferred on to a piece of poly-dimethyl siloxane (Gelpak PF GEL film WF 4, 17 mil.). After identifying a short and narrow MoS2 flake using an optical microscope, it is stamped on top of graphene using visco-elastic stamping tool where a three-axes micrometer stage is used to accurately position the two flakes. The flake is then structured into graphene double cross-shape using electron-beam lithography followed by reactive ion etching. After etching, the sample is annealed for 1 h at 400 °C in ultrahigh vacuum (10−9 torr) to possibly remove the contamination. The graphene is then connected with Ti(3nm)/Au(40nm) contacts fabricated using electron-beam lithography followed by electron beam deposition in ultrahigh vacuum and lift-off. Using the same procedure, the 35 nm thick Co electrodes are fabricated on top of the graphene channel. Before this deposition, a TiOx tunnel barrier is fabricated by depositing 3Å of Ti and subsequent natural oxidation in air. The widths of the Co electrodes vary between 250 nm to 400 nm leading to different coercive fields for each electrode. The exact dimensions of the devices are extracted from scanning electron and atomic force microscopy after they have been measured (Note S9 in the Supporting Information). Electrical measurements. Charge and spin transport measurements are performed in a Physical Property Measurement System by Quantum Design, using a ‘DC reversal’ technique with a Keithley 2182 nanovoltmeter and a 6221 current source at temperatures ranging from 10 to 300 K. 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Standard model To extract the spin transport parameters of the pristine graphene channel and determine the spin injection efficiency of the ferromagnetic (FM) electrodes, we have performed conventional symmetric Hanle precession experiments in the pristine graphene region. These are performed by measuring the nonlocal resistance of the reference graphene lateral spin valve (LSV) while having a parallel (𝑅𝑃 ) or antiparallel (𝑅𝐴𝑃 ) orientation of the FM electrode magnetizations and applying a magnetic field in the 𝑥 direction (𝐵𝑥 ) (i.e., perpendicular to the easy axis of the FM), as shown in Figure S1a for the case of Sample A at 10 K. Finally, the spin transport parameters have been obtained from fitting to the following equation [S1]: 𝐿 𝑔𝑟 − 𝑔𝑟 √1−𝑖(𝜔−𝜔0 )𝜏𝑠 2 𝑔𝑟 𝑔𝑟 𝑃𝑠𝑦𝑚 cos2 (𝛽)𝑅𝑠𝑞 𝜆𝑠 𝑒 𝜆𝑠 ∆𝑅𝑁𝐿 = 𝑅𝑃 − 𝑅𝐴𝑃 = 𝑅𝑒 { } + 𝑅0 (S1) 𝑊𝑔𝑟 √1−𝑖(𝜔−𝜔0 )𝜏𝑠𝑔𝑟 2 Here, 𝑃𝑠𝑦𝑚 = 𝑃𝑖 𝑃𝑑 is the product of the injector and detector spin polarization respectively, 𝛽 is the 𝑔𝑟 angle between the contact magnetization and the easy axis [see Figure S1b], 𝑅𝑠𝑞 is the square 𝑔𝑟 𝑔𝑟 𝑔𝑟 𝑔𝑟 resistance of the graphene channel, 𝜆𝑠 = √𝐷𝑠 𝜏𝑠 is the spin relaxation length, 𝐷𝑠 is the spin 𝑔𝑟 diffusion coefficient, 𝜏𝑠 is the spin lifetime, 𝑊𝑔𝑟 is the channel width, 𝜔 = 𝑔𝜇𝐵 𝐵𝑥 /ħ is the Larmor frequency, 𝑔 = 2 is the Landé factor, 𝜇𝐵 is the Bohr magneton, and ħ is the reduced Plank constant. Finally, 𝜔0 = 𝑔𝜇𝐵 𝐵0 /ħ accounts for a small remanescense from the magnet and 𝑅0 is the background signal. Equation S1 assumes that spin transport is 1D and the contact resistances are 𝑔𝑟 𝑔𝑟 𝑔𝑟 much higher than the channel spin resistance (𝑅𝑁 = 𝑅𝑠𝑞 𝜆𝑠 /𝑊𝑔𝑟 ) [S2]. 20 a) b) 1.0 c) 90 fit=300 20 ps -3 2 RP-RAP 0 Dfit=(5.7 0.5)10 m /s Fit 0.5 Pcorr=2.9 0.2 % 60 RNL () -20 RNL (m)  (/2) 0.0 30 -40 RP -0.5 0 -60 RAP -1.0 -80 -30 -0.6 -0.3 0.0 0.3 0.6 -0.6 -0.3 0.0 0.3 0.6 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Bx (T) Bx (T) Bx (T) Figure S1. (a) Conventional Hanle precession data in the parallel and antiparallel contact magnetization configurations for sample A at 10 K. (b) Angle between the contact magnetization and the easy axis extracted from the Hanle data in a) as described in the text below. (c) Spin signal calculated using 𝑅𝑃 − 𝑅𝐴𝑃 from the Hanle data in (a) with the corresponding fit using Eq. S1 and extracted parameters. 𝑃𝑐𝑜𝑟𝑟 is the spin polarization obtained using the model with an extra arm described in Section S2 and the fit parameters as inputs. Since 𝐵𝑥 is applied perpendicular to the easy axis of a FM, it pulls its magnetization by an angle 𝛽 in the field direction [S3]. To determine 𝛽, we use the fact that the measured nonlocal resistance [Figure S1a] does not include only the precessing component described above, but also includes a signal which is generated by the spins which are injected parallel to the 𝐵𝑥 direction as the contact magnetizations are pulled. This term is proportional to sin2 (𝛽) and appears in both 𝑅𝑃 and 𝑅𝐴𝑃 with the same sign. Because the precessing component has opposite sign for 𝑅𝑃 and 𝑅𝐴𝑃 , 𝑅𝑃 + 𝑅𝐴𝑃 is proportional to sin2 (𝛽). To obtain the data plotted in Figure S1b, we have normalized 0 < 𝑅𝑃 + 𝑅𝐴𝑃 < 1, and taken the arcsine of its square root to obtain 𝛽 as a function of the magnetic field. Finally, to guarantee that the extracted 𝛽 at 𝐵𝑥 = 0 is zero and that it reaches 𝜋/2 at high 𝐵, we have renormalized the result of this operation. At 𝐵 = 0.4 T, we see a jump in 𝛽, which we attribute to a switch of one of the magnetizations. To prevent this step from affecting our analysis, we have used only the negative 𝐵 result for the fits, such as the one shown in Figure S1c where the spin signal 𝑅𝑃 − 𝑅𝐴𝑃 is fit to Eq. S1. The obtained fit parameters are shown in the blue rows of Table S1 (sample A) and Table S2 (sample B). In the case of sample A, the reference graphene LSV has two extra arms (the Hall bar) connected to the main channel, which implies that spins are also diffusing towards these arms. To account for the effect of the arms on the spin polarization we have assumed that spin propagation is 1D and the arms are much longer than the spin relaxation length. As a consequence, the shape of the Hanle curve (determined by 𝐷𝑠𝐺𝑟 and 𝜏𝑠𝐺𝑟 ) is not affected within our model by the presence of the extra arms and only the spin injection and detection efficiencies have to be adjusted. The spin transport parameters extracted from the fit are used as inputs to the three-arm model described in Section S1.2 to determine a better estimate of the contact spin polarization, which we call 𝑃𝑐𝑜𝑟𝑟 . For Sample B, the symmetric Hanle curves were taken in a reference LSV without any cross between spin injector and detector, hence, 𝑃𝑖 is extracted directly from the fit of ∆𝑅𝑁𝐿 = 𝑅𝑃 − 𝑅𝐴𝑃 to Eq. S1. S1.2. Three-arm model The original geometry we want to model for the reference graphene LSV used in Sample A is shown in Figure S2a. Figure S2. (a) 1D approximation of the reference device geometry for Sample A. (b) Simulated device geometry. The width of the top arm is two times the width of the arms in the device and in (a) and the white numbers correspond to the different modelling regions (see discussion below). However, because we are interested in the spins that diffuse towards the middle arm and the spin transport is fully diffusive, the two arms can be simplified down to one with twice the width [see Figure S2b]. The most important assumptions here are 1D spin transport and non-invasive contacts. The left contact is the injector and the right one the detector. To calculate the spin signal, one needs to define the spin accumulation, which is a solution of the Bloch equations considering a magnetic field applied in the 𝑧 direction [S4]: 𝑔𝑟 𝑑 2 𝜇𝑥 𝜇𝑥 𝐷𝑠 − − 𝜔𝜇𝑦 = 0 𝑑𝑥 2 𝜏𝑠𝑔𝑟 𝑔𝑟 𝑑 2 𝜇𝑦 𝜇𝑦 𝐷𝑠 − + 𝜔𝜇𝑥 = 0 𝑑𝑥 2 𝜏𝑠𝑔𝑟 Note that here we have considered that the magnetic field is applied in the 𝑧 direction for analogy with the conventional Hanle precession, but we are applying a magnetic field along 𝑥. We have done this because the solution for 𝜇𝑦 is the same in both cases for an isotropic system such as pristine graphene. The general solution for 𝜇𝑦 for each region can be written as: 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑦 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝜇𝐼 = 𝜆 𝐴𝑒 𝑠 + 𝜆 𝐵𝑒 𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑦 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝜇𝐼𝐼 = 𝜆 𝐶𝑒 𝑠 + 𝜆 𝐷𝑒 𝑠 + 𝜆 𝐸𝑒 𝑠 + 𝜆 𝐹𝑒 𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝑦 𝜆 𝜆 𝜇𝐼𝐼𝐼 = 𝐺𝑒 𝑠 + 𝐻𝑒 𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝑦 𝜆 𝜆 𝜇𝐼𝑉 = 𝐼𝑒 𝑠 + 𝐽𝑒 𝑠 From the Bloch equations we obtain an expression for 𝜇𝑥 : 𝑔𝑟 𝐷𝑠 𝑑 2 𝜇𝑦 𝜇𝑦 𝜇𝑥 = − + 𝜔 𝑑𝑥 2 𝜔𝜏𝑠𝑔𝑟 which results, at the different regions, 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝜇𝐼𝑥 = −𝑖𝐴𝑒 𝜆𝑠 + 𝑖𝐵𝑒 𝜆𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑥 𝑔𝑟 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝑥 𝜆 𝜆 𝜆 𝜆 𝜇𝐼𝐼 = −𝑖𝐶𝑒 𝑠 + 𝑖𝐷𝑒 𝑠 − 𝑖𝐸𝑒 𝑠 + 𝑖𝐹𝑒 𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝑥 𝜆 𝜆 𝜇𝐼𝐼𝐼 = −𝑖𝐺𝑒 𝑠 + 𝑖𝐻𝑒 𝑠 𝑥 𝑔𝑟 𝑥 𝑔𝑟 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 𝑥 𝜆 𝜆 𝜇𝐼𝑉 = −𝑖𝐼𝑒 𝑠 + 𝑖𝐽𝑒 𝑠 To determine the spin accumulation in the system we need to determine the coefficients 𝐴 − 𝐻. This is done by applying the following boundary conditions. First, we impose the continuity of 𝜇 𝑥 and 𝜇 𝑦 at 𝑥 = 0 (intersection point between regions I and II) and 𝑥 = 𝑥1 (intersection point between regions II, III, and IV). From these conditions, we obtain 6 equations. Second, we define the spin current as 𝑊𝑔𝑟 𝑑𝜇𝑠𝑖 𝐼𝑠𝑖 = − 𝑔𝑟 where 𝑖 refers to the spin polarization direction of the spin accumulation (𝑥 or 𝑦). 𝑒𝑅𝑠𝑞 𝑑𝑥 𝑦 Next, we impose the continuity of 𝐼𝑠𝑥 and 𝐼𝑠 at 𝑥 = 𝑥1 and the continuity of 𝐼𝑠𝑥 and discontinuity of 𝑦 𝐼𝑠 of 𝑃𝑖 𝐼𝑐 , where 𝑃𝑖 is the injector polarization and 𝐼𝑐 the applied current, at 𝑥 = 0. This gives the 4 equations that we are missing. Finally, the solution for the spin signal is: 𝑦 𝐿 𝑔𝑟 𝐿 𝑔𝑟 2𝑃𝑑 𝜇𝐼𝑉 (𝐿) 2𝑃𝑑 − 𝑔𝑟 √1+𝑖𝜔𝜏𝑠 − 𝑔𝑟 √1−𝑖𝜔𝜏𝑠 ∆𝑅𝑁𝐿 = 𝑅𝑃 − 𝑅𝐴𝑃 = = (𝐼𝑒 𝜆𝑠 + 𝐽𝑒 𝜆𝑠 ) 𝑒𝐼𝑐 𝑒𝐼𝑐 In the final solution, both 𝐼 and 𝐽 are proportional to 𝑃𝑖 and, hence, the spin signal is proportional to 2 𝑃𝑖 𝑃𝑑 = 𝑃𝑐𝑜𝑟𝑟 . Here, we use the above equation together with the spin transport parameters in the pristine graphene region obtained from the fit to Eq. S1 and the magnitude of the measured spin signal to obtain 𝑃𝑐𝑜𝑟𝑟 , that is the average spin polarization of the ferromagnetic contacts. S2. Analysis of the (antisymmetric) Hanle precession measurements with spin Hall detection Figure S3. Sketch of the measured geometry with the positive magnetic field and the “up” direction of the magnetization indicated with arrows. The green area is covered by a TMD and is where the spin-to-charge conversion takes place. When a spin current density 𝐼𝑠 /𝑊𝑔𝑟 enters a region with high spin-orbit coupling, it generates a 𝑇𝑀𝐷/𝑔𝑟 transverse charge current density 𝐼𝑐𝑇 /𝑊𝑐𝑟 = 𝜃𝑆𝐻 𝐼𝑠 /𝑊𝑔𝑟 via the inverse spin Hall effect, here 𝜃𝑆𝐻 is the spin Hall angle and 𝑊𝑐𝑟 the length of the high spin-orbit coupling region (see Figure S3). This generates a transverse voltage 𝑉𝑆𝐻 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑉𝑆𝐻 𝐼𝑐𝑇 𝜃 𝐼𝑠 𝜃 𝑅𝑔𝑟 𝑊𝑐𝑟 𝐼𝑠 = = 𝑆𝐻 → 𝑉𝑆𝐻 = 𝑆𝐻 𝑊𝑐𝑟 𝑅𝑔𝑟 𝑊𝑐𝑟 𝑊𝑔𝑟 𝑊𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 where 𝑅𝑔𝑟 = 𝑅𝑠𝑞 𝑊𝑔𝑟 /𝑊𝑐𝑟 is the resistance for the transverse current that determines the 𝑇𝑀𝐷/𝑔𝑟 conversion from an induced current to a voltage. Note that 𝑅𝑠𝑞 is the square resistance of the TMD-covered graphene region. Therefore, 𝐺𝑟 𝑇𝑀𝐷/𝑔𝑟 𝜃𝑆𝐻 𝑅𝑠𝑞 𝑊𝑔𝑟 𝑊𝑐𝑟 𝐼𝑠 𝐺𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑉𝑆𝐻 = = 𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑠 𝑊𝑐𝑟 𝑊𝑔𝑟 To obtain the spin currents propagating in our device, we assume that it is homogeneous and isotropic, and the propagation is 1D. Because the transport properties of the TMD-covered region are 𝑒𝑓𝑓 not the same as those of the pristine graphene region, we will extract effective parameters (𝜏𝑠 , 𝑒𝑓𝑓 𝑒𝑓𝑓 𝐷𝑠 , and 𝜆𝑠 ) from the homogeneous fit. These parameters are an average for both the pristine and the TMD-covered graphene regions. The out-of-plane (𝑧 direction) spin accumulation (𝜇𝑧 ) at a distance 𝐿 from the spin injector, which induced by spin precession around a magnetic field applied in the 𝑥 direction is written as [S4, S5]: 𝐿 𝑒𝑓𝑓 − 𝑒𝑓𝑓 √1−𝑖𝜔𝜏𝑠 𝑔𝑟 𝑒𝑓𝑓 𝜆 𝑃𝑖 𝑒𝐼𝑐 𝑅𝑠𝑞 𝜆𝑠 𝑒 𝑠 𝜇𝑧 = ± 𝐼𝑚 2𝑊𝑔𝑟 √1 − 𝑖𝜔𝜏𝑠𝑒𝑓𝑓 { } Here, 𝑃𝑖 the spin injection efficiency of the injector, 𝐼𝑐 the bias current. The ± corresponds to the up and down direction of the injector magnetization defined as in Figure S3. The spin current is defined as: 𝐿 𝑒𝑓𝑓 𝑊𝑔𝑟 𝑑𝜇𝑧 𝑃𝑖 𝐼𝑐 − 𝑒𝑓𝑓 √1−𝑖𝜔𝜏𝑠 𝐼𝑠 = 𝐼↑ − 𝐼↓ = 𝐺𝑟 =± 𝐼𝑚 {𝑒 𝜆𝑠 } 𝑒𝑅𝑠𝑞 𝑑𝑥 2 Then the signal induced by this spin current is: 𝐺𝑟 𝑇𝑀𝐷/𝑔𝑟 𝐺𝑟 𝑇𝑀𝐷/𝑔𝑟 𝐿 𝑒𝑓𝑓 𝑉𝑆𝐻 𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑠 𝜃𝑆𝐻 𝑃𝑖 𝑅𝑠𝑞 − 𝑒𝑓𝑓 √1−𝑖𝜔𝜏𝑠 𝑅𝑆𝐻 = = =± 𝜆 𝐼𝑚 {𝑒 𝑠 } 𝐼𝑐 𝐼𝑐 2 However, in our geometry, we have a cross with a width 𝑊𝑐𝑟 (Figure S3) that is comparable to the spin relaxation length. Therefore, 𝐼𝑠 needs to be replaced by the average spin current 𝐼̅𝑠 = 1 𝐿+𝑊𝑐𝑟 ∫𝐿 𝐼𝑠 𝑑𝑥. The result from this operation is: 𝑊𝑐𝑟 𝐿 𝑒𝑓𝑓 𝐿+𝑊𝑐𝑟 √ 𝑒𝑓𝑓 − 𝑒𝑓𝑓 √1−𝑖𝜔𝜏𝑠 − 𝑒𝑓𝑓 1−𝑖𝜔𝜏𝑠 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 ̅ 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 𝜆 𝜆𝑠 𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑠 𝜃𝑆𝐻 𝑃𝑖 𝑅𝑠𝑞 𝜆𝑠 𝑒 𝑠 𝑒 𝑅𝑆𝐻 = =± 𝐼𝑚 − 𝐼𝑐 2𝑊𝑐𝑟 √1 − 𝑖𝜔𝜏𝑠𝑒𝑓𝑓 √1 − 𝑖𝜔𝜏𝑠𝑒𝑓𝑓 { } To remove any signal which does not come from the 𝑦-component of the injector magnetization in our measurement we subtract both injector configurations to 𝐿 √ 𝑒𝑓𝑓 − 1−𝑖(𝜔−𝜔0 )𝜏𝑠 𝐿+𝑊𝑐𝑟 √ 𝑒𝑓𝑓 2 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 𝑒𝑓𝑓 − 1−𝑖(𝜔−𝜔0 )𝜏𝑠 𝑃𝑎𝑛𝑡𝑖 𝑅𝑠𝑞 𝜆𝑠 𝜆𝑠 𝜆𝑠 ↑ ↓ 𝑒 𝑒 ∆𝑅𝑆𝐻 = 𝑅𝑆𝐻 − 𝑅𝑆𝐻 = 𝐼𝑚 { − } + 𝑅0 (S2) 𝑊𝑐𝑟 √1−𝑖(𝜔−𝜔0 )𝜏𝑠𝑒𝑓𝑓 √1−𝑖(𝜔−𝜔0 )𝜏𝑠𝑒𝑓𝑓 ↑ ↓ where 𝑅𝑆𝐻 and 𝑅𝑆𝐻 correspond to the spin Hall component of the nonlocal resistance of the graphene/TMD lateral device (Figure S3) while orienting the FM injector along the +𝑦 and −𝑦 direction, respectively, and applying B in the 𝑥 direction, which leads to antisymmetric Hanle ↑ ↓ precession curves. The measured signal with these injector configurations is called 𝑅𝑛𝑙 and 𝑅𝑛𝑙 . Note that we have also added the term cos(𝛽) to account for the effect of the magnetic field on the magnetization direction of the FM injector. Equation S2 has been used to fit to the antisymmetric Hanle precession curves such as the ones shown in Figure S4. From the fit we obtain the effective 𝑒𝑓𝑓 𝑒𝑓𝑓 𝜏𝑠 , 𝐷𝑠 and 𝑃𝑎𝑛𝑡𝑖 of the channel, which are averaged over the pristine graphene and TMD- 𝑇𝑀𝐷/𝑔𝑟 covered regions. Note that 𝑃𝑎𝑛𝑡𝑖 = √𝑃𝑖 𝜃𝑆𝐻 depends on both the injector spin polarization 𝑃𝑖 and 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 the spin Hall angle 𝜃𝑆𝐻 . To determine 𝜃𝑆𝐻 we assume that 𝑃𝑖 = 𝑃𝑐𝑜𝑟𝑟 for Sample A and 𝑃𝑖 = 𝑃𝑠𝑦𝑚 for sample B obtained from the symmetric Hanle curve (Section S1). The fitted parameters are shown in the orange rows of Table S1 (Sample A) and Table S2 (Sample B). 90 a)  RNL b) RSH=RNL-RNL   20  RNL Fit 80 RSH (m) RNL (m) 0 70 fit=72 7 ps -3 2 Dfit=(1.19 0.09)10 m /s -20 Pfit=3.6 0.2 % 60 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Bx (T) Bx (T) Figure S4. (a) Antisymmetric Hanle precession measurements for the two different orientations of the FM injector for ↑ ↓ Sample A at 10 K. Note that we have called 𝑅𝑁𝐿 and 𝑅𝑁𝐿 the signals measured with the magnetization of the spin injector aligned in the +𝑦 and −𝑦 direction respectively. (b) ∆𝑅𝑆𝐻 and its fit to Eq. S2 with the extracted parameters. S3. Measurement of spin Hall signals obtained with the spin injector placed at the right and left side of the Hall cross Since the spin signal is induced by the spin current, which points in the graphene plane, it should change sign when the spin injector is changed from the right to the left side of the Hall cross. Note that, if the spin signal would be induced by spin absorption in the TMD, it would not change sign. This control experiment has been performed in sample B, where the two measurements shown in Figure S5 confirm that the signal is induced by the spin current propagating in the graphene plane. 5 fit=72 6 ps 20 fit=70 9 ps a) b) RSH -3 Dfit=(2.1 0.1)10 m /s 2 RSH Dfit=(1.5 0.1)10 m /s -3 2 4 15 Fit Pfit=1.9 0.1% Fit Pfit=3.4 0.3% 3 10 2 RNL(m) RNL(m) 5 1 0 0 -5 -1 Left Right -2 -10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 Bx (T) Bx (T) Figure S5. Antisymmetric Hanle curves obtained by subtracting the measurements obtained for both magnetic configurations of the FM injector at 10 K in sample B, for the (a) left and (b) right spin injectors. We see that, as expected, the spin transport parameters (𝜏𝑓𝑖𝑡 and 𝐷𝑓𝑖𝑡 ) extracted from both Hanle curves are very similar. However, because the magnitude of the spin signals is different, the spin polarizations extracted from both curves show a discrepancy of almost a factor of two. This is caused by the fact that, in both cases we are using different spin injectors which have different efficiencies. 𝑇𝑀𝐷/𝑔𝑟 In Table S2, this has consequences on the determination of 𝜃𝑆𝐻 . To avoid this discrepancy, we rely on the measurements performed using the right injector, in which the spin injector is the same as for the reference Hanle curves. S4. Analysis of the 𝝁𝒙 -induced spin-to-charge conversion signal S4.1. Extraction of the signal As explained above, when a magnetic field 𝐵𝑥 is applied perpendicular to the easy axis of a FM electrode, its magnetization rotates an angle 𝛽 in the direction of the field. In our geometry, this results in the injection of spins in the direction of the magnetic field independently of the initial ↑ magnetization direction and, therefore, this component can be extracted calculating 𝑅𝑆𝐶𝐶 = (𝑅𝑁𝐿 + ↓ 𝑅𝑁𝐿 )/2. The result from this operation is shown in Figure S6 for the case of Sample A at 10 K. 85   RSCC=(RNL+RNL)/2 RSCC=11.5±0.4 m 80 75 RSCC (m) 70 65 60 -1.0 -0.5 0.0 0.5 1.0 Bx (T) Figure S6. 𝜇𝑥 -induced spin-to-charge conversion signal 𝑅𝑆𝐶𝐶 obtained by averaging the antisymmetric Hanle precession curves measured for both magnetic configurations of the FM injector at 10 K for Sample A [Figure S4a]. The magnitude of the spin-to-charge conversion signal (∆𝑅𝑆𝐶𝐶 ) is quantified by calculating the zero-field extrapolation using linear fittings to the spin signal at high positive and negative fields. The S-shaped signal saturates at around ±240 mT, which is the saturation field for the magnetization of the FM injector [see Figure S1b]. This result indicates that the source of this signal depends on the 𝑥 component of the spin accumulation (𝜇𝑥 ). Because there is a background which has different slope for positive and negative magnetic fields, we extrapolate the high 𝐵 data to 𝐵𝑥 = 0 to extract the magnitude of this component. As discussed in the main manuscript, this means that it can be caused either by the Rashba-Edelstein effect in the graphene layer or the spin Hall effect in the MoS 2 semiconducting channel after spin absorption. Left Right 2 a)   b) 3 RSCC=(RNL+RNL)/2 0 0 RSCC (m) RSCC (m) -3 -2 RSCC= 2.9 m -6 -9 -4 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Bx (T) Bx (T) Figure S7. 𝑅𝑆𝐶𝐶 signal at 10 K for Sample B measured with spins injected by a FM electrode placed on (a) the right and (b) the left side of the TMD-covered region. We have performed the same operation on the measurements obtained from Sample B and the results at 10 K are plotted in Figure S7. In contrast with the left injector, the right one does not show any 𝜇𝑥 - induced spin-to-charge conversion feature. S4.2. Quantification of the Rashba-Edelstein effect For the Rashba-Edelstein case, a spin accumulation in the x direction (𝜇𝑥 ) induces a charge current density in the y direction 𝐽𝑐𝑇 = 𝐼𝑐𝑇 /𝑊𝑐𝑟 with an efficiency 𝛾 that is the conversion efficiency: 𝜇𝑥 𝛾 = 𝐼𝑐𝑇 𝑉𝑅𝐸 1 𝐿+𝑊𝑐𝑟 =𝑊 . As a consequence, 𝑉𝑅𝐸 = ̅̅̅𝛾𝑅 𝜇𝑥 𝑠𝑞 𝑊𝑔𝑟 where 𝜇̅𝑥 = 𝑊 ∫𝐿 𝜇𝑥 (𝑥)𝑑𝑥 is the average 𝑊𝑐𝑟 𝑐𝑟 𝑅𝑔𝑟 𝑐𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 𝑥 𝑒𝑃𝑖 𝐼𝑅𝑠𝑞 𝜆𝑠 − 𝑒𝑓𝑓 spin accumulation at the cross, which is defined as 𝜇𝑥 = ± 𝑒 𝜆 𝑠 where + and – 2𝑊𝑔𝑟 correspond to the positive and negative magnetization directions of the spin injector corresponding to positive and negative magnetic field respectively. 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 1 𝐿+𝑊𝑐𝑟 𝑒𝑃𝑖 𝐼𝑐 𝑅𝑠𝑞 𝜆𝑠 1 𝐿+𝑊𝑐𝑟 −𝑥/𝜆𝑒𝑓𝑓 𝜇̅𝑥 = ∫ 𝜇𝑥 (𝑥)𝑑𝑥 = ± ∫ 𝑒 𝑠 𝑑𝑥 𝑊𝑐𝑟 𝐿 2𝑒𝑊𝑔𝑟 𝑊𝑐𝑟 𝐿 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 𝑥 𝐿+𝑊𝑐𝑟 𝑒𝑃𝑖 𝐼𝑐 𝑅𝑠𝑞 𝜆𝑠 1 𝑒𝑓𝑓 − 𝑒𝑓𝑓 =± [−𝜆𝑠 𝑒 𝜆𝑠 ] 2𝑊𝑔𝑟 𝑊𝑐𝑟 𝐿 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 2 𝐿 𝐿+𝑊𝑐𝑟 𝑒𝑃𝑖 𝐼𝑐 𝑅𝑠𝑞 𝜆𝑠 − 𝑒𝑓𝑓 𝜆𝑠 − 𝑒𝑓𝑓 𝜆𝑠 =± (𝑒 −𝑒 ) 2𝑊𝑔𝑟 𝑊𝑐𝑟 𝑒𝑓𝑓 2 𝐿 − 𝑒𝑓𝑓 − 𝐿+𝑊𝑐𝑟 𝑉𝑅𝐸 𝜇𝑥 𝑠𝑞 𝑊𝑔𝑟 ̅̅̅𝛾𝑅 𝛾𝑅𝑠𝑞 𝑊𝑔𝑟 𝑒𝑃𝑖 𝐼𝑅𝑠𝑞 𝜆𝑠 𝑒𝑓𝑓 𝑅𝑅𝐸 = = =± (𝑒 𝜆𝑠 −𝑒 𝜆𝑠 ) 𝐼𝑐 𝐼𝑐 𝐼𝑐 2𝑊𝑔𝑟 𝑊𝑐𝑟 2 𝑒𝑓𝑓 2 𝐿 − 𝑒𝑓𝑓 − 𝐿+𝑊𝑐𝑟 𝑒𝑃𝑖 𝛾𝑅𝑠𝑞 𝜆𝑠 𝑒𝑓𝑓 =± (𝑒 𝜆𝑠 −𝑒 𝜆𝑠 ) 2𝑊𝑐𝑟 𝐿 𝐿+𝑊𝑐𝑟 2 𝜆 𝑒𝑓𝑓 2 − 𝑒𝑓𝑓 − → ← 𝑒𝑃𝑖 𝛾𝑅𝑠𝑞 𝑠 𝜆 𝑒𝑓𝑓 𝜆𝑠 ∆𝑅𝑅𝐸 = 𝑅𝑅𝐸 − 𝑅𝑅𝐸 = (𝑒 𝑠 −𝑒 ) (S3) 𝑊𝑐𝑟 We also want to quantify the Rashba-Edelstein effect using a dimensionless parameter. For this purpose, we assume that the spin currents are the source of the transverse charge currents and not the spin accumulations. For this, we use the formula derived in [S6]. 𝐿 𝐿+𝑊𝑐𝑟 𝛼𝑅𝐸 𝑃𝑖 𝑅𝑠𝑞 𝜆𝑠 𝑒𝑓𝑓 − 𝑒𝑓𝑓 − 𝑒𝑓𝑓 → ← 𝜆 𝜆𝑠 ∆𝑅𝑅𝐸 = 𝑅𝑅𝐸 − 𝑅𝑅𝐸 = (𝑒 𝑠 −𝑒 ) (S4) 𝑊𝑐𝑟 Finally, to obtain an estimation for 𝛾 and 𝛼𝑅𝐸 , we assume that the entire observed signal arises from Rashba-Edelstein effect, i.e., ∆𝑅𝑆𝐶𝐶 = ∆𝑅𝑅𝐸 . The extracted parameters from these operations are shown in the green rows of Table S1 for Sample A. For Sample B we did only see a clear signal for one of the configurations at 10 K. S4.3. Quantification of the spin Hall effect in MoS2 Because of the reduced spin lifetime of the MoS2-covered graphene region it is not straightforward to determine the spin current which is absorbed by the semiconductor. However, we can estimate an upper limit of the signal expected if the spin current diffusing from the graphene channel into the MoS2 is converted to a charge current due to the spin Hall effect in the MoS2 channel. Figure S8. Sketch of the modeled geometry. The spin injector is grey, the Ti/Au contact is yellow, the graphene channel is blue, and the MoS2 is green. The thickness of the graphene is assumed to be zero. To evaluate the effect of spin absorption in a 1D approach which we can solve analytically, we assume that the width of the MoS2 flake is significantly smaller than the spin relaxation length in the 𝑔𝑟 graphene channel (𝑊𝑐𝑟 ≪ 𝜆𝑠 ). We also ignore the role of the cross and assume that the graphene channel is 1D through all its length. With these assumptions, the modeled geometry is shown in Figure S8. To model for this geometry, we break the device into different regions: I. The right side of the spin injector. II. From 𝑥 = 0 to 𝑥 = −𝐿, III. MoS2, from 𝑧 = 0 to 𝑧 = 𝑡𝑀𝑜𝑆2 , and IV. Graphene for 𝑥 < −𝐿. We write the spin accumulation in the different parts of the sample as follows: 𝑥 − 𝑔𝑟 𝜇𝐼 = 𝜆 𝐴𝑒 𝑠 𝑥 𝑥 𝑔𝑟 − 𝑔𝑟 𝜇𝐼𝐼 = 𝜆 𝐵𝑒 𝑠+ 𝐶𝑒 𝑠𝜆 𝑧 𝑧 𝑀𝑜𝑆2 − 𝑀𝑜𝑆 2 𝜇𝐼𝐼𝐼 = 𝐷𝑒 𝜆𝑠 + 𝐸𝑒 𝜆𝑠 𝑥 𝑔𝑟 𝜆 𝜇𝐼𝑉 = 𝐹𝑒 𝑠 𝑀𝑜𝑆 where 𝜆𝑠 2 is the spin relaxation length at the MoS2 and we have imposed that 𝜇𝑠 (𝑥 → ±∞) → 0. To obtain the spin accumulation in the system we need to determine the coefficients 𝐴 − 𝐹. For this purpose, we need 6 equations. We obtain three of them from imposing the continuity of 𝜇𝐼𝑉 at 𝑥 = 0 and 𝑥 = −𝐿. To obtain the other 3 equations, we need to define the spin currents in the graphene and MoS 2 channels respectively: 𝑔𝑟 𝑊𝑔𝑟 𝑑𝜇𝑠𝑔𝑟 𝐼𝑠 = − 𝑔𝑟 𝑒𝑅𝑠𝑞 𝑑𝑥 𝑀𝑜𝑆 𝑊𝑔𝑟 𝑊𝑐𝑟 𝑑𝜇𝑠𝑀𝑜𝑆2 𝐼𝑠 2 =− 𝑒𝜌𝑀𝑜𝑆2 𝑑𝑧 Here, 𝜌𝑀𝑜𝑆2 is the resistivity of the MoS2 channel, and 𝑊𝑐𝑟 is the width of the MoS2 flake in Figure 𝑔𝑟 S8. By imposing the discontinuity of 𝐼𝑠 at 𝑥 = 0, which is of 𝑃𝑖 𝐼𝑐 as discussed in section S2, the 𝑔𝑟 𝑀𝑜𝑆 𝑀𝑜𝑆 continuity of 𝐼𝑠 and 𝐼𝑠 2 at 𝑥 = 𝐿, and 𝐼𝑠 2 (𝑧 = 𝑡𝑀𝑜𝑆2 ) = 0, that guarantees that there is no spin current leaving the MoS2 flake, we obtain the remaining 3 equations. From the derivation above, we obtain the spin currents and accumulations in the MoS2 channel. However, we still need to convert them into voltages to determine the measured signal. A spin 𝑀𝑜𝑆2 𝑀𝑜𝑆2 𝐼 current density in the MoS2 channel (𝐽𝑠 = 𝑊𝑠 ) induces a transverse charge current density 𝑔𝑟 𝑊𝑐𝑟 2 𝑀𝑜𝑆 𝐼𝑐𝑇 𝑀𝑜𝑆2 𝐼𝑠 𝑀𝑜𝑆2 𝐽𝑐𝑇 = 𝑊 = 𝜃𝑆𝐻 where 𝜃𝑆𝐻 is the spin Hall angle of the MoS2. 𝑐𝑟 𝑡𝑀𝑜𝑆2 𝑊𝑔𝑟 𝑊𝑐𝑟 𝑀𝑜𝑆 Because we are measuring the voltage in an open circuit configuration, we use: 𝑉𝑆𝐻 2 = 𝐼𝑐𝑇 𝑅𝑒𝑓𝑓 , where 𝑅𝑒𝑓𝑓 is the effective resistance of the MoS2 and graphene. Because the resistance of the MoS2 is considerably higher than that of the graphene channel, we need to take the effect of shunting by the graphene into account to determine 𝑅𝑒𝑓𝑓 . We assume that the system behaves like if the MoS2 channel would be connected in parallel with the graphene: −1 −1 −1 𝑊𝑐𝑟 𝑡𝑀𝑜𝑆2 𝑊𝑐𝑟 𝑅𝑒𝑓𝑓 = 𝑅𝑀𝑜𝑆 + 𝑅𝑔𝑟 = + 2 𝑊𝑔𝑟 𝜌𝑀𝑜𝑆2 𝑅𝑠𝑞 𝑊𝑔𝑟 Finally, 𝑀𝑜𝑆2 𝑀𝑜𝑆 𝑀𝑜𝑆2 𝐼𝑠 𝑡𝑀𝑜𝑆2 𝑅𝑒𝑓𝑓 𝑉𝑆𝐻 2 = 𝐼𝑐𝑇 𝑅𝑒𝑓𝑓 = 𝜃𝑆𝐻 𝑊𝑔𝑟 𝑀𝑜𝑆2 Because the spin current is not constant along the MoS2 thickness we replace 𝐼𝑠 in the equation above by: 𝑡𝑀𝑜𝑆2 𝑊𝑐𝑟 𝑊𝑔𝑟 𝑀𝑜𝑆2 1 𝑀𝑜𝑆2 𝑀𝑜𝑆2 𝑀𝑜𝑆2 𝐼𝑠̅ = ∫ 𝐼𝑠 (𝑧)𝑑𝑧 = [𝐷 (1 − 𝑒 𝑡𝑀𝑜𝑆2 /𝜆𝑠 ) + 𝐸 (1 − 𝑒 −𝑡𝑀𝑜𝑆2 /𝜆𝑠 )] 𝑡𝑀𝑜𝑆2 0 𝑒𝑡𝑀𝑜𝑆2 𝜌𝑀𝑜𝑆2 Using the two equations above, we determine the voltage induced by the spin Hall effect in MoS 2, which we divide by the current applied to the spin injector to obtain the nonlocal resistance: 𝑀𝑜𝑆2 𝑀𝑜𝑆 ̅ 2 𝑡𝑀𝑜𝑆 𝑅𝑒𝑓𝑓 𝑀𝑜𝑆 𝑉 𝑀𝑜𝑆2 𝐼𝑠 𝑅𝑆𝐻 2 = 𝑆𝐻 = 𝜃𝑆𝐻 2 𝐼𝐶 𝐼𝑐 𝑊𝑔𝑟 𝑀𝑜𝑆 As explained above, the sign of 𝑅𝑆𝐻 2 changes with the magnetization direction. Because our → ← 𝑀𝑜𝑆 definition of the spin-to-charge conversion signal is ∆𝑅𝑆𝐶𝐶 = 𝑅𝑆𝐶𝐶 − 𝑅𝑆𝐶𝐶 , we define ∆𝑅𝑆𝐻 2 = 𝑀𝑜𝑆 𝑀𝑜𝑆 𝑀𝑜𝑆 2𝑅𝑆𝐻 2 . The inputs of our model are 𝜆𝑠 2 , 𝜌𝑀𝑜𝑆2 , and 𝜃𝑆𝐻 2 and are unknown to us, while the 𝑀𝑜𝑆 𝑀𝑜𝑆 𝑀𝑜𝑆 output is ∆𝑅𝑆𝐻 2 and we know its value. To give an estimate for 𝜃𝑆𝐻 2 , we assume that 𝜆𝑠 2 = 20 𝑀𝑜𝑆 nm, as estimated in Ref. [S7], we leave 𝜌𝑀𝑜𝑆2 as a free parameter and extract the 𝜃𝑆𝐻 2 that gives a 𝑀𝑜𝑆 signal ∆𝑅𝑆𝐻 2 = ∆𝑅𝑆𝐶𝐶 . The results from this model are shown in Figure S9 for sample A at 10 K. Because of the effect of shunting and the reduction of the spin currents which get absorbed by the 𝑀𝑜𝑆 MoS2 as 𝜌𝑀𝑜𝑆2 increases, 𝑅𝑆𝐻 2 presents a maximum value of 11.5 mΩ at 𝜌𝑀𝑜𝑆2 = 7.1 × 10−4 Ωm. This helps us to give an estimate of the best-case scenario for spin absorption that requires the lowest 𝑀𝑜𝑆 𝑀𝑜𝑆 𝑀𝑜𝑆 𝜃𝑆𝐻 2 to achieve the measured signal. We have adjusted 𝜃𝑆𝐻 2 so that the maximum 𝑅𝑆𝐻 2 corresponds to the measured signal. From this process we conclude that the minimal spin Hall angle required to achieve ∆𝑅𝑆𝐶𝐶 at 10 K in sample A is 3.3%. To estimate the temperature dependence of 𝑀𝑜𝑆 𝜃𝑆𝐻 2 for sample A, we have performed the same analysis described above assuming that the spin relaxation length in MoS2 does not depend on the temperature, which is the case if the Dyakonov- Perel mechanism is the dominant source of spin relaxation [S8]. The results are shown in Table S1. 0.984 a) b) c) 160 10 0.983 120 RSH (m) 0.982 MoS2 total Reff(m) /Is MoS2 80 5 0.981 Is 0.980 40 0 0.979 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 -3 -3 -3 MoS2(10 m) MoS2(10 m) MoS2(10 m) Figure S9. (a) Estimation of the spin signal induced by the spin Hall effect in the MoS 2 channel as a function of the 𝑀𝑜𝑆 MoS2 resistivity using the parameters of Sample A at 10 K and by assuming 𝜃𝑆𝐻 2 = 3.3 %. (b) Ratio between the spin 𝑀𝑜𝑆 current diffusing into the MoS2 (𝐼𝑠 2 ) and spin current entering the graphene-MoS2 junction in the model (𝐼𝑠𝑡𝑜𝑡𝑎𝑙 ). The ratio decreases as 𝜌𝑀𝑜𝑆2 increases. (c) Effective resistance of the MoS2 and graphene channel. As 𝜌𝑀𝑜𝑆2 increases, 𝑅𝑒𝑓𝑓 increases before saturating when the MoS2 resistance becomes significantly higher than that of the graphene channel. S5. Analysis of the measurements at different temperatures S5.1. Sample A Unlike the case of 10 K (Fig. 3a of the main text), we only have one magnetic injector configuration ↑ for the antisymmetric Hanle precession curves (𝑅𝑁𝐿 ) measured at 100 K, 200 K, and 300 K (Figure 4a of the main text). Hence, we cannot quantify the spin Hall angle in graphene for Sample A at these temperatures. In contrast, the 𝜇𝑥 -induced spin-to-charge conversion signals can be obtained ↑ from direct extrapolation of 𝑅𝑁𝐿 from high fields, as explained in section S3.1. The extracted signal ∆𝑅𝑆𝐶𝐶 is shown in Table S1. S5.2. Sample B ↑ For sample B, we have both magnetic configurations of the antisymmetric Hanle curves (𝑅𝑁𝐿 and ↓ ↑ ↓ 𝑅𝑁𝐿 ) at 100, 200 and 300 K and, consequently, ∆𝑅𝑆𝐻 = 𝑅𝑁𝐿 − 𝑅𝑁𝐿 can be fit to Eq. S2. The extracted parameters are shown in Table S2. However, no clear 𝜇𝑥 -induced spin-to-charge conversion signal is observed at these temperatures. As an example, the measurements at 300 K are shown in Figure S10.  fit=72 16 ps RNL 6 a) 3 b) 9 -3 Dfit=(2.8 0.4)10 m /s 2 c)  RNL 4 Pfit=1.4 0.1% 2 RSH (m) 2 RNL (m) 1 6 RSCC(m) 0 0 -1 3 -2 RSH   -2 -4 RSCC=(RNL+RNL)/2 Fit -0.6 -0.3 0.0 0.3 0.6 0.9 -0.6 -0.3 0.0 0.3 0.6 -0.6 -0.3 0.0 0.3 0.6 Bx (T) Bx (T) Bx (T) Figure S10. (a) Antisymmetric Hanle precession measurements for the two different orientations of the FM injector for sample B at 300 K. (b) ∆𝑅𝑆𝐻 obtained as defined above by subtracting the components shown in (a) and its fit to Eq. S2 with the extracted parameters. (c) 𝑅𝑆𝐶𝐶 . In this case ∆𝑅𝑆𝐶𝐶 is hard to extract because it is highly sensitive to the 𝐵 field range selected for the extraction. However, it can be estimated to be between 3 and 5 mΩ. S6. Extracted parameters for sample A and B 𝑔𝑟 𝑔𝑟 Table S1. Extracted parameters for sample A. 𝐷𝑠 , 𝜏𝑠 , and 𝑃𝑠𝑦𝑚 are extracted from fits to the symmetric Hanle 𝑒𝑓𝑓 𝑒𝑓𝑓 precession data, 𝐷𝑠 , 𝜏𝑠 , and 𝑃𝑎𝑛𝑡𝑖 are extracted from fits to the antisymmetric Hanle precession data, and ∆𝑅𝑆𝐶𝐶 is extracted from fits to the S-shaped background of the antisymmetric Hanle precession data. 𝛾 and 𝛼𝑅𝐸 are obtained using ∆𝑅𝑆𝐶𝐶 and Eqs. S3-S4 with the spin transport parameters extracted from the antisymmetric Hanle precession data. The uncertainty ranges here are obtained considering only the uncertainty in ∆𝑅𝑆𝐶𝐶 . Sample A 10 K 100 K 200 K 300 K 𝑔𝑟 𝑅𝑠𝑞 (Ω) 543 454 347 295 𝑇𝑀𝐷/𝑔𝑟 497 430 335 288 𝑅𝑠𝑞 (Ω) 𝑔𝑟 𝐷𝑠 (10−3 m2/s) 5.7 ± 0.5 8.8 ± 0.5 8.8 ± 0.5 20 ± 4 𝑔𝑟 𝜏𝑠 (ps) 300 ± 20 221 ± 8 221 ± 8 152 ± 6 𝑃𝑠𝑦𝑚 (%) 1.4 ± 0.1 1.35 ± 0.06 1.35 ± 0.06 2.3 ± 0.1 𝑃𝑐𝑜𝑟𝑟 (%) 2.9 ± 0.2 2.9 ± 0.1 2.9 ± 0.1 4.8 ± 0.2 𝑒𝑓𝑓 1.2 ± 0.1 𝐷𝑠 (10−3 m2/s) 𝑒𝑓𝑓 72 ± 7 𝜏𝑠 (ps) 𝑃𝑎𝑛𝑡𝑖 (%) 3.6 ± 0.2 𝑇𝑀𝐷/𝑔𝑟 4.5 ± 0.6 𝜃𝑆𝐻 (%) 𝐺𝑟 𝜎𝑆𝐻 (Ω ) -1 (9 ± 1) × 10−5 ∆𝑅𝑆𝐶𝐶 (mΩ) 11.5 ± 0.4 12 ± 0.5 6.7 ± 0.2 7.4 ± 0.5 𝛾(A/(Jm)) (2.9 ± 0.1) × 1020 (1.63 ± 0.07) × 1021 (3.02 ± 0.09) × 1021 (3.3 ± 0.2) × 1021 𝛼𝑅𝐸 (%) 0.85 ± 0.03 2.7 ± 0.1 3.32 ± 0.1 3.0 ± 0.2 𝑀𝑜𝑆 𝜃𝑆𝐻 2 (%) 3.3 ± 0.1 4.1 ± 0.2 2.98 ± 0.09 2.2 ± 0.1 Table S2. Extracted parameters for sample B. The “Left” and “Right” labels refer to the data obtained with the spin injector placed at the left and right sides of the TMD-covered region, respectively. Sample B 10 K 100 K 200 K 300 K 𝑔𝑟 𝑅𝑠𝑞 (Ω) 𝑇𝑀𝐷/𝑔𝑟 282 266 231 215 𝑅𝑠𝑞 (Ω) 𝑔𝑟 𝐷𝑠 (10−3 m2/s) 7.0 ± 0.3 11 ± 1 15 ± 2 16 ± 3 𝑔𝑟 𝜏𝑠 (ps) 252 ± 8 146 ± 14 139 ± 2 114 ± 1 𝑃𝑠𝑦𝑚 (%) 2.4 ± 0.1 7.2 ± 0.1 7.7 ± 0.1 6.9 ± 0.1 𝑒𝑓𝑓 1.5 ± 0.1 1.6 ± 0.1 1.4 ± 0.1 2.8 ± 0.4 𝑅𝑖𝑔ℎ𝑡 𝐷𝑠 (10−3 m2/s) 𝑒𝑓𝑓 70 ± 9 49 ± 3 44 ± 3 72 ± 16 𝑅𝑖𝑔ℎ𝑡 𝜏𝑠 (ps) 𝑅𝑖𝑔ℎ𝑡 𝑃𝑎𝑛𝑡𝑖 (%) 3.4 ± 0.3 3.8 ± 0.3 3.1 ± 0.3 1.5 ± 0.1 𝑇𝑀𝐷/𝑔𝑟 4.8 ± 0.9 2.0 ± 0.5 1.2 ± 0.2 0.33 ± 0.04 𝑅𝑖𝑔ℎ𝑡 𝜃𝑆𝐻 (%) 𝑅𝑖𝑔ℎ𝑡𝜎𝑆𝐻 (Ω-1) (17 ± 3) × 10−5 (7.5 ± 2) × 10−5 (5.2 ± 0.9) × 10−5 (1.5 ± 0.2) × 10−5 𝑒𝑓𝑓 2.1 ± 0.1 1.5 ± 0.1 𝐿𝑒𝑓𝑡 𝐷𝑠 (10−3 m2/s) 𝑒𝑓𝑓 73 ± 6 53 ± 2 𝐿𝑒𝑓𝑡 𝜏𝑠 (ps) 𝐿𝑒𝑓𝑡 𝑃𝑎𝑛𝑡𝑖 (%) 1.9 ± 0.1 3.0 ± 0.2 𝑇𝑀𝐷/𝑔𝑟 1.5 ± 0.2 1.3 ± 0.2 𝐿𝑒𝑓𝑡 𝜃𝑆𝐻 (%) 𝐺𝑟 𝐿𝑒𝑓𝑡 𝜎𝑆𝐻 (Ω ) -1 (5.3 ± 7) × 10−5 (4.9 ± 0.8) × 10−5 S7. Comparison between spin Hall effect in Pt and in TMD/graphene devices S7.1. TMD/graphene normalized conversion efficiency As shown in Section S2, the signal induced by the spin Hall effect in the case of TMD/graphene heterostructures is determined as: ∆𝑉𝑆𝐻 ̅ 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝐼𝑠 ∆𝑅𝑆𝐻 = = 2𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑐 𝐼𝑐 𝑇𝑀𝐷/𝑔𝑟 We want to determine an efficiency factor which is a trans-resistance defined as follows: 𝑅𝑒𝑓𝑓 = ∆𝑉𝑆𝐻 /𝐼𝑠 where 𝐼𝑠 is the spin current entering the TMD/graphene heterostructure and ∆𝑉𝑆𝐻 = ↑ ↓ (𝑅𝑁𝐿 − 𝑅𝑁𝐿 )𝐼𝑐 is the transverse voltage output measured as shown in Figure S3. Since the measured signal depends on 𝐼𝑠̅ , the average spin current at the region where the conversion takes place, we rewrite the equation above in the following way: 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 2𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑠̅ 𝑇𝑀𝐷/𝑔𝑟 ∆𝑉𝑆𝐻 = ( ) 𝐼𝑠 = 𝑅𝑒𝑓𝑓 𝐼𝑠 𝐼𝑠 The efficiency factor is defined as follows: 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 2𝜃𝑆𝐻 𝑅𝑠𝑞 𝐼𝑠̅ 𝑅𝑒𝑓𝑓 = 𝐼𝑠 𝑇𝑀𝐷/𝑔𝑟 To obtain 𝑅𝑒𝑓𝑓 we need to determine the correction factor 𝐼𝑠̅ /𝐼𝑠 , which can be done by considering that the spin current entering the TMD-covered graphene region is: 𝐼𝑠 (𝑥) = 𝑒𝑓𝑓 (𝑃𝑖 𝐼𝑐 𝑒 −𝑥/𝜆𝑠 )/2 and the average spin current at the region of interest reads: 𝑒𝑓𝑓 1 𝐿+𝑊𝑐𝑟 𝑃𝑖 𝐼𝑐 𝐿+𝑊𝑐𝑟 −𝑥/𝜆𝑒𝑓𝑓 𝑃𝑖 𝐼𝑐 𝜆𝑠 𝑒𝑓𝑓 𝑒𝑓𝑓 𝐼𝑠̅ = ∫ 𝐼𝑠 (𝑥)𝑑𝑥 = ∫ 𝑒 𝑠 𝑑𝑥 = (𝑒 −𝐿/𝜆𝑠 − 𝑒 −(𝐿+𝑊𝑐𝑟 )/𝜆𝑠 ) 𝑊𝑐𝑟 𝐿 2 𝑊𝑐𝑟 𝐿 2 𝑊𝑐𝑟 Now we can obtain the 𝐼𝑠̅ /𝐼𝑠 factor: 𝑒𝑓𝑓 𝑒𝑓𝑓 𝑒𝑓𝑓 𝑒𝑓𝑓 𝐼𝑠̅ 𝜆𝑠 𝑒 −𝐿/𝜆𝑠 − 𝑒 −(𝐿+𝑊𝑐𝑟 )/𝜆𝑠 𝜆𝑠 𝑒𝑓𝑓 = 𝑒𝑓𝑓 = (1 − 𝑒 −𝑊𝑐𝑟 /𝜆𝑠 ) 𝐼𝑠 𝑊𝑐𝑟 𝑒 −𝐿/𝜆𝑠 𝑊𝑐𝑟 which for sample A at 10 K is 0.3. Now we finally write the efficiency factor for the TMD/graphene case: 𝑇𝑀𝐷/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 𝑒𝑓𝑓 𝑇𝑀𝐷/𝑔𝑟 2𝜃𝑆𝐻 𝑅𝑠𝑞 𝜆𝑠 𝑒𝑓𝑓 𝑅𝑒𝑓𝑓 = (1 − 𝑒 −𝑊𝑐𝑟 /𝜆𝑠 ) 𝑊𝑐𝑟 𝑇𝑀𝐷/𝑔𝑟 This expression gives a numerical value of 𝑅𝑒𝑓𝑓 ≈ 13.4 Ω for sample A at 10 K. Note that, because we have measured with magnetic fields applied in the 𝑥 direction, there is still a correction factor required to account for the diffusive broadening of the spin precession, which we have estimated to be of about 0.62 using Eq. S2. S7.2. Pt/graphene normalized conversion efficiency 𝑇𝑀𝐷/𝑔𝑟 We would like to compare the expression obtained above for 𝑅𝑒𝑓𝑓 with a properly normalized efficiency for Pt/graphene devices in which the spin Hall signal is determined by [S9]: 𝑃𝑡 𝑃𝑡 2𝜃𝑆𝐻 𝜌𝑃𝑡 𝑥𝑃𝑡/𝑔𝑟 𝐼𝑠̅ ∆𝑅𝑆𝐻 = 𝑊𝑃𝑡 𝐼𝑐 Therefore, the efficiency factor can be obtained from this relation: 𝑃𝑡 𝑃𝑡 2𝜃𝑆𝐻 𝜌𝑃𝑡 𝑥𝑃𝑡/𝑔𝑟 𝐼𝑠̅ 𝑃𝑡/𝑔𝑟 𝑉𝑆𝐻 =( ) 𝐼𝑠 = 𝑅𝑒𝑓𝑓 𝐼𝑠 𝑊𝑃𝑡 𝐼𝑠 𝑃𝑡 where, 𝐼𝑠 is the spin current entering the Pt layer, 𝜃𝑆𝐻 the spin Hall angle, 𝜌𝑃𝑡 the resistivity, and 𝑊𝑃𝑡 the width of the Pt wire. 𝑥𝑃𝑡/𝑔𝑟 is a parameter that accounts for the parallel conduction channel provided by the graphene that can reduce the measured voltage. In the case of graphene, however, because its sheet resistance is higher than Pt, 𝑥𝑃𝑡/𝑔𝑟 ≈ 1. The 𝐼𝑠̅ /𝐼𝑠 factor is [S9, S10]: 𝐼𝑠̅ 𝜆𝑃𝑡 1 − 𝑒 −𝑡𝑃𝑡/𝜆𝑃𝑡 = 𝐼𝑠 𝑡𝑃𝑡 1 + 𝑒 −𝑡𝑃𝑡/𝜆𝑃𝑡 where 𝑡𝑃𝑡 is the thickness of the Pt wire and 𝜆𝑃𝑡 its spin diffusion length. Now we can write the efficiency for Pt as: 𝑃𝑡 𝑃𝑡/𝑔𝑟 2𝜃𝑆𝐻 𝜌𝑃𝑡 𝑥𝑃𝑡/𝑔𝑟 𝜆𝑃𝑡 1 − 𝑒 −𝑡𝑃𝑡/𝜆𝑃𝑡 𝑅𝑒𝑓𝑓 = 𝑊𝑃𝑡 𝑡𝑃𝑡 1 + 𝑒 −𝑡𝑃𝑡/𝜆𝑃𝑡 𝑃𝑡/𝑔𝑟 𝑃𝑡 We estimate 𝑅𝑒𝑓𝑓 ≈ 0.27 Ω using the following parameters from Ref. [S9]: 𝜃𝑆𝐻 = 0.23, 𝜌𝑃𝑡 = 134 𝜇Ωcm, 𝜆𝑃𝑡 = 2 nm, 𝑊𝑃𝑡 = 200 nm, 𝑡𝑃𝑡 = 21 nm. 𝑃𝑡/𝑔𝑟 𝑇𝑀𝐷/𝑔𝑟 The conversion efficiencies can now be directly compared by looking at the 𝑅𝑒𝑓𝑓 and 𝑅𝑒𝑓𝑓 𝑇𝑀𝐷/𝑔𝑟 𝑃𝑡/𝑔𝑟 values. Since 𝑅𝑒𝑓𝑓 is ~50 × 𝑅𝑒𝑓𝑓 , we conclude that the TMD-covered graphene is 50 times more efficient than highly resistive Pt for spin detection despite the fact that the spin Hall angle in Pt is an order of magnitude higher than in TMD-covered graphene. Note that, accounting for the spin 𝑇𝑀𝐷/𝑔𝑟 𝑃𝑡/𝑔𝑟 precession used in our measurement leads to, 𝑅𝑒𝑓𝑓 /𝑅𝑒𝑓𝑓 = 31, still a large difference. S8. Theoretical calculation of spin-to-charge conversion in TMD/graphene heterostructures To obtain a theoretical estimate to the magnitude of SHE and REE signals in our devices, we use a simplified model that captures the physics of a monolayer graphene/TMD heterostructure by using the following Hamiltonian [S11, S12]: 𝐻 = 𝐻0 + 𝐻𝐼 + 𝐻𝑅 where 𝐻0 = ħ𝑣𝑓 (𝜎𝑥 𝑘𝑥 + 𝜏𝑧 𝜎𝑦 𝑘𝑦 ) + ∆𝜎𝑧 𝐻𝐼 = (𝜆𝐼 𝜎𝑧 + 𝜆𝑉𝑍 )𝜏𝑧 𝑠𝑧 𝐻𝑅 = 𝜆𝑅 (𝑠𝑦 𝜎𝑥 𝑘𝑥 − 𝜏𝑧 𝑠𝑥 𝜎𝑦 ) with 𝜎𝛼 , 𝜏𝛼 and 𝑠𝛼 the Pauli matrices on the 𝛼 direction acting on the pseudospin, valley and spin degrees of freedom, respectively. 𝐻0 represents the orbital part, which is described by the Dirac’s Hamiltonian with Fermi velocity 𝑣𝑓 and a staggered potential with strength ∆, the latter appearing because of the broken sub-lattice symmetry of the TMD. 𝐻𝐼 represents the intrinsic spin-orbit coupling of the heterostructure, modelled with a Kane-Mele term with strength 𝜆𝐼 and a valley- Zeeman coupling characterized by the parameter 𝜆𝑉𝑍 , both appearing because of the honeycomb structure and the broken sub-lattice symmetry. This term is responsible for the SHE, as we will show ahead. Finally, 𝐻𝑅 represents the Rashba spin-orbit coupling arising from the interface between graphene and the TMD, with strength 𝜆𝑅 . This last term is the source of the REE. In this model we are excluding the so-called pseudospin inversion asymmetry terms [S12], because they do not contribute to any of the effects of interest at low energy, and we use the tight-binding parameters from Ref. [S12]. S8.1. The spin Hall effect Despite the presence of the Rashba spin-orbit coupling, the origin of the spin Hall effect was traced back to the intrinsic spin-orbit coupling [S11], described in these heterostructures by a valley- Zeeman and Kane-Mele spin-orbit coupling, which interacts with the staggered potential for producing a net berry phase and, therefore, a finite SHE. For simplicity, we can neglect the Rashba spin-orbit coupling, and consider the spin-up and spin-down bands as independent. Moreover, if we also neglect intervalley scattering, the Dirac's cone can also be considered as independent, leading then to 8 almost identical bands described by the dispersion relation 𝜀(𝒌, 𝜏, 𝑠) = 𝑠𝜆𝑉𝑍 ± √ħ(𝑣𝐹 𝑘)2 + Δ𝑠,𝜏 2, where Δ𝑠,𝜏 = ∆ − 𝜏𝑠𝜆𝐼 . Each of these bands will contribute to the total spin Hall conductivity, and the calculation can be done following Refs. [S13, S14], leading to the following result: 𝑒2 Θ(|Δs,τ |−|𝜀𝐹 −𝜏𝑠𝜆|) 𝜎𝑆𝐻 (𝜀𝐹 ) = 2ℎ ∑𝑠,𝜏=±1 𝜏 [Δs,τ + Θ(|𝜀𝐹 − 𝜏𝑠𝜆| − Δs,τ )] (S5) √Δs,τ 2 +(𝜀𝐹 −𝜏𝑠𝜆)2 where Θ is the Heaviside function. The comparison between this model and the curve simulated by solving numerically the Kubo formula for graphene/MoS2 heterostructure is presented in Figure S11, where the small difference arises from the difference between the broadening functions used in the model (Lorentzian) and in the simulation (Lorentzian-like) [S11]. Figure S12a shows the spin Hall conductivity over a range of Fermi energies for three different temperatures. At 𝑇 = 0 one can see that the maximum of the spin Hall conductivity is achieved at two points at around ±1 eV, changing sign when crossing the charge neutrality point (CNP). Therefore, a negative spin Hall angle (as found in the main text) would be an indication of the Fermi level being at the valence band of graphene but close to the CNP. In addition, the spin Hall conductivity maximum is highly susceptible to temperature effects, where we see a linear variation with 𝛽 = 1/𝑇 at low and high temperature (Figure S12b). In this sense, the maximum spin Hall conductivity achieved in this system at low temperature is 0.2𝑒 2 /ℎ. This value is also suppressed by intervalley scattering [S15], so it is difficult to establish an exact value, but assuming weak intervalley scattering and using the 𝑇𝑀𝐷/𝑔𝑟 square resistance (𝑅𝑠𝑞 ) extracted from sample A, one obtains a conversion efficiency in the 𝑇𝑀𝐷/𝑔𝑟 range of 𝜃𝑆𝐻 = 1 − 10%, which is consistent with the experimental observation. Figure S11. Comparison between the simulation of the spin Hall effect for graphene/MoS2 heterostructure and the model based on individual spin bands. We have introduced a gaussian broadening of 20 meV to match the intrinsic broadening of the Kernel Polynomial Method (KPM) simulations performed at Ref. [S11]. The small difference is due to the broadening which is not energy independent in the KPM method. Figure S12. (a) Spin Hall conductivity as a function of Fermi energy for three different temperatures and (b) the evolution of the maximum of the spin Hall conductivity computed as a function of temperature, both computed using Eq. S5. Both results are obtained at zero gaussian broadening. S8.2. The Rashba-Edelstein effect and its reciprocal The REE is a phenomenon where the combined effect of a momentum dependent spin texture, and an external electric field 𝑬0 produces a nonequilibrium spin density 𝑺(𝑛𝑒𝑞) . It is usually described by the electrical spin susceptibility 𝜒𝑅𝐸 , which under the linear response limit allow for writing the following constitute relation 𝑺(𝑛𝑒𝑞) = 𝜒𝑅𝐸 (𝑬0 × 𝑘̂)𝑺 ̂(𝑛𝑒𝑞) . The polarization of the spin density is perpendicular to both the external electric field and the out-of- plane direction and is defined in units of a charge density. The REE is a dissipative effect and, as shown by Offidani et al. [S16], for pure Rashba-like systems this implies that the spin susceptibility is proportional to the DC conductivity (𝜎𝐷𝐶 ) 𝜒𝑅𝐸 1 =𝑣 , 𝜎𝐷𝐶 𝑅𝐸 where 𝑣𝑅𝐸 is a parameter with units of velocity characterizing the spin-to-charge conversion efficiency. There are different estimations for this parameter. We have, for instance the result of Dyrdal [S13]: 𝑣𝐹 1 2+𝑥 𝑥 = 2 1+𝑥 Θ(2 − 𝑥) + 𝑥 2 −1 Θ(𝑥 − 2), with 𝑥 ≡ 𝜀𝐹 /𝜆𝑅 , 𝑣𝑅𝐸 the result of Offidani et al. [S16]: 2𝑣𝐹 = Θ(2 − 𝑥) + 2𝑥𝑔(𝐷𝑐 )Θ(𝑥 − 2), with 𝑥 ≡ 𝜀𝐹 /𝜆𝑅 , 𝑣𝑅𝐸 where 𝑔(𝐷𝑐 ) is a dimensionless parameter that depends of the disorder, and the result of Garcia et al. [S11], which takes into account all the spin-orbit parameters leading to a higher spin density for high energies but to a small 1/𝑣𝑅𝐸 due to the broadening effects present in the system, which will increase 𝜎𝐷𝐶 close to the CNP. Figure S14. Figure of merit 2𝑣𝐹 /𝑣𝑅𝐸 of the Rashba Edelstein effect for graphene/MoS 2 heterostructure considering a broadening of 5 meV. Using 2𝑣𝐹 /𝑣𝑅𝐸 as a figure of merit, it is possible to extract a conversion efficiency 𝛼𝑅𝐸 by computing the transformation from spin density to spin current using Fick's law [S1]: 𝑦 𝐽𝑠 𝐷𝑠 𝛼𝑅𝐸 = = 𝐽𝑥 𝜆𝑠 𝑣𝑅𝐸 where 𝐷𝑠 and 𝜆𝑠 are the spin diffusion constant and the spin relaxation length of graphene. One should keep in mind that in this process there is no electric field affecting the spin current; therefore, the charge current density is obtained solely from the spin density. Using 𝐷𝑠 and 𝜆𝑠 obtained from sample A, one obtains a conversion efficiency 𝛼𝑅𝐸 ~0.1% in the best-case scenario, decreasing down to ~0.01% at higher temperatures. One should also keep in mind that there is an additional suppression term which appears due to the averaging over the channel width, which will further suppress this value. The REE conversion efficiency 𝛼𝑅𝐸 changes sign following the same trend as the spin Hall angle 𝑇𝑀𝐷/𝑔𝑟 𝜃𝑆𝐻 , because both effects originate from the proximity-induced spin-orbit coupling of graphene. 𝑇𝑀𝐷/𝑔𝑟 Therefore, both 𝛼𝑅𝐸 and 𝜃𝑆𝐻 have the same sign. This is not observed in our experiment, ruling out the REE as the origin of the in-plane spin-to-charge conversion signal. S9. Scanning electron and atomic force microscopy imaging of sample A After the electrical characterization we imaged our device with scanning electron microscopy (SEM), see Figure S15. We extracted the length and width of the graphene channels as well as the width of the ferromagnetic electrodes and their distances from these measurements. As the samples were in contact with air after the electrical characterization in vacuum, the Co electrodes are oxidized in the images. Figure S15. False-colored scanning electron microscopy image of Sample A. The width of the horizontal graphene (in blue) channel is 350 nm. The width of the MoS2 flake (in green) and the two vertical graphene channels are 0.9 µm and 1.2 µm respectively. After the electrical characterization and SEM imaging we measured the topography of device with an atomic force microscope (Agilent 5500) in tapping mode (Figure S16). The main result of this measurement is the height profile of the graphene/MoS2 stack, as we extracted all lateral distances from the SEM image. The thickness of the graphene flake was determined to be roughly 5 nm. The MoS2 flake shows a ramp-like shape on top and is between 35 and 75 nm thick. Figure S16. Atomic force microscopy characterization of Sample A. (a) Area scan showing the topography of the device. The scale bar is 2 µm long. (b) Line profile taken along the marked line in (a) across the graphene flake, where the thickness of the graphene flake is extracted to be roughly 5 nm, equivalent to around ten layers [S18]. 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