Strong interface-induced spin–orbit interaction in graphene on WS2

Interfacial interactions allow the electronic properties of graphene to be modified, as recently demonstrated by the appearance of satellite Dirac cones in graphene on hexagonal boron nitride substrates. Ongoing research strives to explore interfacial interactions with other materials to engineer targeted electronic properties. Here we show that with a tungsten disulfide (WS2) substrate, the strength of the spin–orbit interaction (SOI) in graphene is very strongly enhanced. The induced SOI leads to a pronounced low-temperature weak anti-localization effect and to a spin-relaxation time two to three orders of magnitude smaller than in graphene on conventional substrates. To interpret our findings we have performed first-principle electronic structure calculations, which confirm that carriers in graphene on WS2 experience a strong SOI and allow us to extract a spin-dependent low-energy effective Hamiltonian. Our analysis shows that the use of WS2 substrates opens a possible new route to access topological states of matter in graphene-based systems.

. R NL is larger than R ohm in the entire V g range explored. (b) At a fixed V g = -27 V, the measured R NL (black dots) exhibits a very rapid decay upon increasing L, consistent with an exponential behavior (quantitatively not consistent with the behavior expected for the Ohmic signal, represented by the red dashed line). As illustrated in the inset, R NL = V NL /I c is obtained by injecting a charge current I c (red arrow) from source to drain contacts (denoted by I S and I D , respectively), and measuring the non-local voltage V NL = V + -Vbetween two probes (denoted correspondingly) separated from the injecting contacts by a distance L. Supplementary Note 1: Weak anti-localization effect measured in different devices.
As discussed in detail in the main text, the observation of weak anti-localization (WAL) at low temperature unambiguously demonstrates the presence of strong SOI in graphene-on-WS 2 . To illustrate the reproducibility of our observations, here we show data from a different device measured at the lowest temperature of our cryostat (T = 250 mK), which are virtually identical to those obtained from the device discussed in the main text. Supplementary Figure 1a shows that the conductivity varies linearly with V g only for negative gate voltages, and the inset illustrates the V g -dependence of n. The quantum Hall effect data in Supplementary Figure 1b clearly confirms that holes in the device behave as Dirac fermions 1 . Since the device dimension is similar to that of the device discussed in the main text, we reveal WAL by ensemble averaging the measured magnetoconductance around three different values of gate voltage; the results of the averaging are shown in Supplementary Figure 1c. It is apparent that a sharp conductance peak at zero magnetic field appears in all gate-voltage ranges explored, with an amplitude as large as ~0.5e 2 /h at the largest negative gate-voltages. In the main text, we show that the conductivity () of graphene-on-WS 2 devices saturates for V g > ~8 V, because upon changing the gate voltage in this range, charges are accumulated at the WS 2 /SiO 2 interface (where the carrier mobility is low) and not in graphene. Here, we confirm this conclusion by measuring the Hall effect in different gatevoltages ranges, to show that the Hall density (n) remains constant in the interval of V g where  saturates. Supplementary   Figure 1a-b show the Hall resistance (R Hall ) as a function of magnetic field (B), measured for values of V g below and above 0 V, respectively. It is apparent that the slope of this curve, which measures the charge density (n) accumulated in graphene, changes upon varying V g below 0 V, and remains unchanged for positive V g . A comparison between Supplementary Figure 1c and 1d, in which the carrier density and the conductivity are shown as a function of V g between -40 V and 40 V, demonstrates that the saturation of n parallels the saturation of , as expected.

Supplementary Note 3: Non-local resistance as a signature of SOI.
We note that demonstrating the presence of strong SOI by measuring WAL effect at low temperatures, an entirely established method, has not been achieved previously in graphene. Related earlier work, including a recent one (further discussed in the section Supplementary Note 4) also focusing on graphene-on-WS 2 3 , has relied on another transport phenomenon, namely the measurement of non-local resistance (R NL ) generated through the combination of spin-Hall and inverse spin-Hall effect (often referred to as the non-local spin-Hall effect) 4 . This method works when the contribution to the non-local resistance due to this phenomenon is larger than the Ohmic contribution (which can be estimated to be R ohm = e -L/W ,  is the resistivity and L/W is the device aspect ratio). The non-local signal due to the spin-Hall and inverse spin Hall effect decays exponentially away from the contacts used to inject current on a characteristic scale determined by the spin-relaxation length  so = (D so ) 1/2 . Although our devices are not intentionally designed to optimally perform non-local measurements, the multi-terminal Hall bar geometry nevertheless allows us to probe non-local signals. The results of these non-local measurements for V g < 0 V (i.e., away from the V g -region where the conductivity of graphene saturates) are shown in Supplementary Figure 3. It is apparent that R NL in this regime is larger than R Ohm (by approximately a factor of 2 to 3, depending on V g , Supplementary Figure 3a) and that it decays very rapidly with increasing L, the distance between the contacts used to inject current and those used to detect voltage (Supplementary Figure 3b). The data are compatible with an exponential decay (black line in Supplementary Figure 3b) and using the formula R NL =  2 W/2 so e -L/so for the expected behavior of the non-local signal due to the spin-Hall effect (γ is the spin-Hall coefficient), we estimate values of  so (see the filled red dots in Fig. 3d of the main text) that are very close to those inferred from the analysis of WAL (the factor 2 difference is certainly compatible with the errors associated to the non-ideal device geometry for the analysis of non-local effect and with the precision of the WAL analysis; as mentioned in the main text, such level of uncertainty in  so does not affect the conclusions of our study).

Supplementary Note 4: Discussions of a recent study of SOI in graphene-on-WS 2 .
In previous work by A. Avsar et al. 3

Supplementary Note 5: Electronic Structure of Graphene on WS 2 .
The role of spin-orbit coupling in the electronic structure of graphene on WS 2 can be addressed theoretically using the tools of electronic structure theory. This approach is complicated by the lattice constant mismatch between the two materials, and by the fact that the separation between layers may not be reliably predicted on the basis of purely theoretical considerations. To make progress we have performed electronic structure calculations (details are given in the Methods section) for super cells with three different ratios between the lattice constants of WS 2 and graphene, and for a wide range of layer separations. We find that in all cases a set of states whose wavefunctions that are strongly peaked in the graphene layer appear inside the WS 2 gap. The electronic structure of these states is always well described by a Hamiltonian in which the Dirac continuum model of isolated graphene is supplemented by three momentum and position independent substrate interaction terms, Here H 0 is the spin-independent Dirac Hamiltonian of isolated graphene and ,  z , and s act respectively on its sublattice, Similarly all three terms are absent in an isolated graphene sheet because of inversion symmetry, which maps  z to - z and transforms the  matrices like  x . Although we expect a term in the low energy model Hamiltonian that is proportional to  z  z s z , which is symmetry allowed even in the isolated graphene case 5 , this term is evidently too weak for it to be manifested in the microscopic supercell calculations.
In Supplementary Figure 5  insensitive for a given commensurate structure to rigid displacements of the substrate relative to the graphene sheet, we can conclude that they reflect the band Hamiltonian. Our calculations demonstrate however that these band parameters, including those for the spin-dependent terms, are dependent on the commensurate structure examined. Since the actual structure of graphene on WS 2 is likely incommensurate, this sensitivity will be manifested as a disorder contributions to the low-energy effective Hamiltonian. where B + and ∂B + are respectively a half Brillouin zone and its boundary, and A(k) and Ω z (k) are respectively the Berry connection and the Berry curvature summed over filled bands. Z 2 =1 for a topological insulator and Z 2 =0 for a trivial insulator. In the present case the half Brillouin zone can be identified with an area around a single valley and a boundary that is sufficiently far from the centre of the valley. We calculated the Z 2 number for  much larger than  R and Δ and found that Z 2 is always 1. Therefore in the ideal situation in which the low energy physics of the graphene/WS 2 system is completely determined by Supplementary Eq. 1 the system should be a topological insulator. It is worth noting is that in the original Kane-Mele model the topologically nontrivial gap is opened by the intrinsic spin-orbit coupling of graphene which preserves inversion symmetry, whereas in the present model the gap is opened by two spin-orbit coupling terms, both of which break inversion symmetry and are absent in freestanding graphene.

Supplementary Note 7: Distance between graphene and WS 2 calculated with vdW functional
As the distance between graphene and WS 2 substrate is not experimentally determined yet, in this article we take it to be around 3Å (distance measured between graphene and h-BN 6 ) and conclude strong SOI in the system. To avoid serious bias on it, we also relax the graphene-WS 2 distance with van der Waals functionals though this method has limited predictive power. We performed relaxation of the 5:4 graphene/WS 2 supercell using the optB86b-vdW functional 9 implemented in VASP 9-11 , with a k-point mesh of 7×7×1 and the stopping criterion being the force on each atom smaller than 10 -3 eV/Å. We found the graphene-WS 2 separation increases to 3.36 Å, with a slight corrugation of the graphene sheet of 0.05 Å at its maximum. We have also tried other van der Waals functionals and found that they give graphene-WS 2 separations different from one another with a variation of 0.3 Å. Despite the larger graphene-WS 2 separation and the graphene corrugation, we found the DFT band structure for the graphene can still be well fitted by the effective Hamiltonian Supplementary Eq. 1. The spin orbit coupling parameter λ is 1.7 meV, still two orders of magnitude larger than that in freestanding graphene. However, the sublattice splitting Δ also increases to be comparable with λ, mainly due to the slight corrugation of the graphene which enhances local matching between the graphene and the WS 2 lattices and hence the sublattice splitting. Since the effective Hamiltonian Supplementary Eq. 1 is a topological insulator only when Δ<λ, the closeness between the fitted values of Δ and λ makes the topological character of the graphene more delicate.
However, as these values are sensitively dependent on the graphene-WS 2 separation and the supercell periodicity, which cannot be reliably determined from first principles, whether the graphene/WS 2 system is a topological insulator or not should best be left for future experimental verification. Nevertheless we stress that the much larger spin-orbit splitting than that in freestanding graphene is still robust. (2)