Abstract
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 (WS_{2}) substrate, the strength of the spin–orbit interaction (SOI) in graphene is very strongly enhanced. The induced SOI leads to a pronounced lowtemperature weak antilocalization effect and to a spinrelaxation time two to three orders of magnitude smaller than in graphene on conventional substrates. To interpret our findings we have performed firstprinciple electronic structure calculations, which confirm that carriers in graphene on WS_{2} experience a strong SOI and allow us to extract a spindependent lowenergy effective Hamiltonian. Our analysis shows that the use of WS_{2} substrates opens a possible new route to access topological states of matter in graphenebased systems.
Introduction
Because of the Dirac nature of its charge carriers and the presence of two valleys, graphene is a twodimensional topological insulator^{1,2}. Topological state characteristics have not been observed experimentally, because the strength of the spin–orbit interaction (SOI) intrinsically present in graphene is too weak^{3,4,5}. Various strategies to amplify the SOI strength have been proposed theoretically^{6–8} or explored experimentally^{11}. However, increasing the SOI strength in graphene without drastically affecting other basic aspects of its electronic structure, or the material quality, is proving extremely difficult^{11}. Here we explore whether it is possible to induce strong SOI while preserving the quality of graphene, by exploiting interfacial interactions^{12,13,14} at an atomically sharp interface between graphene and a semiconducting WS_{2} crystalline substrate.
Many semiconducting transition metal dichalcogenides, such as WS_{2}, are ideal substrates for graphene. Like hexagonal boron nitride (hBN)^{15}, they are atomically flat and chemically inert, which is key to preserving highquality transport properties (mobility values as high as μ∼50,000–60,000 cm^{2} V^{–1} s^{–1} have been recently reported for graphene on WS_{2}) (ref. 16). Transition metal dichalcogenide crystals consist of a stack of monolayers having a hexagonal lattice that—like graphene—leads to the presence of two valleys in their electronic structure at the K and K′ point of the Brillouin zone^{17}. The SOI in WS_{2} is extremely strong—several hundreds of millivolts in the valence bands and several tens of millivolts in the conduction band—and in monolayers it pins spin to valley^{18,19,20}. The spins of states near the band edges point in one direction in one of the valleys and in the opposite direction in the other, a behaviour resembling the one expected theoretically in disorderfree graphene^{1,2}. The ability of this substrate material to induce a strong SOI in graphene—as well as the nature of the induced SOI—is, therefore, an important topic that has attracted recent attention^{10}.
Here we address these issues by combining a study of lowtemperature quantum transport in grapheneonWS_{2} devices with ab initio electronic structure calculations. Specifically, we perform systematic magnetotransport measurements to show that when transferred onto WS_{2} substrates graphene exhibits a pronounced and robust weak antilocalization (WAL) effect throughout the explored carrier density and temperature range, down to 250 mK. The detection of WAL provides a direct demonstration of the SOI enhancement in graphene due to the interfacial interactions with WS_{2} substrate. In the attempt to estimate quantitatively the magnitude of SOI enhancement, we show that the magnetotransport data can be fit to the theory of WAL for graphene in the presence of SOI, from which we determine the spinrelaxation time (τ_{so}). We find that the value of τ_{so} (∼2.5–5 ps) in graphene on WS_{2} is 100–1,000 times shorter than τ_{so} in pristine graphene on SiO_{2} or hBN. This very strong enhancement of SOI found experimentally is consistent with the result of our ab initio calculations, which indicate that hybridization with the WS_{2} substrate orbitals is responsible for the SOI induced in graphene, and estimate the SOI strength under the conditions of the experiments to be ∼5 meV. Finally, we show that the results of our calculations close to the K and K′ points can be mapped onto a longwavelength effective Hamiltonian, which, depending on the values of the parameter, describes a topologically insulating state. We therefore conclude that the possibility of using interfacial interactions to induce a strong SOI in graphene while preserving the high quality of the material opens a new possible route to create and investigate a topological insulating state in graphene.
Results
High quality of grapheneonWS_{2} device
We start by characterizing the basic transport properties of the grapheneonWS_{2} devices used in our experiments. The devices are assembled in a multi–terminal Hall bar configuration, placed on a highly doped Si wafer that acts as a gate electrode and is coated with 280 nm SiO_{2} (see Fig. 1a for an optical microscope image, Fig. 1b,c for a schematic of the device structure and the Methods section for the details of the device fabrication). We have realized several such devices. Here we present representative data from one of them (similar data from another device can be found in the Supplementary Note 1 and Supplementary Fig. 1). Figure 1d shows that on ramping up the gate voltage from V_{g}=–40 V, the conductivity σ of graphene on WS_{2} decreases linearly until V_{g}∼8 V, after which it saturates. Saturation occurs because, for V_{g}>8 V, electrons are accumulated at the interface between the SiO_{2} and the WS_{2} crystal, screening the effect of the gate on the graphene layer on top (Fig. 1c, Supplementary Fig. 2 and Supplementary Note 2)^{21}. (Because the mobility of charge carriers in WS_{2} is much smaller than in graphene^{22}, carriers at the WS_{2}/SiO_{2} interface give a negligible contribution to transport.) Sweeping the gate voltage down for V_{g}<8 V, on the contrary, results in an increase (Fig. 1b) of carrier (hole) density in graphene and the conductivity increases. In our devices, therefore, the position of the Fermi level can be gate shifted in the graphene valence band, but accumulation of electrons at the SiO_{2}/WS_{2} interface prevents access to the conduction band. In the following we therefore study only hole transport through graphene. To illustrate this conclusion, and to start assessing the device quality, Fig. 1e–g shows that the halfinteger quantum Hall effect characteristic of monolayer graphene^{23} is clearly observed for different values of V_{g} between –40 and 8 V. The longitudinal resistance measured in this V_{g} range, and plotted versus filling factor ν and magnetic field B (the inset of Fig. 1d), confirms this result. We estimate the carrier mobility from σ using the hole density n extracted from the (classical) Hall effect and by looking at the slope dσ/dV_{g}, and obtain in both cases μ∼13,000 cm^{2} V^{–1} s^{–1} at T=4.2 K. Since no effort has yet been put into optimizing the fabrication process, these values confirm the very good quality of grapheneonWS_{2} devices found in earlier work^{16}.
Robust lowtemperature WAL reveals strong SOI
To demonstrate the presence of SOI in our devices, we probe WAL, which usually manifests itself as a characteristic sharp magnetoconductance (MC) peak at B=0 T (refs 24, 25). In small, fully phasecoherent devices like ours, however, WAL is eclipsed by conductance fluctuations originating from the random interference of electron waves^{25,26}. Indeed, in Fig. 2a, which shows the MC as a function of V_{g} and B, an enhancement in conductance at B=0 T is only faintly visible. No special feature at B=0 T can be detected by looking at a single MC curve measured at a fixed value of V_{g} (see, for example, the top curve shown in Fig. 2b measured at V_{g}=–25 V). The random conductance fluctuations, whose reproducibility is shown in Fig. 2c, can be suppressed through an ensemble averaging procedure in which MC traces measured at different V_{g} values are averaged^{25}. The V_{g} spacing should be chosen to shift the Fermi level by Thouless energy of the system. It is expected that the root mean square amplitude of the fluctuations decreases as N^{−1/2} (N is number of uncorrelated MC traces used to calculate the average), eventually making the sharp conductance peak at B=0 T due to WAL visible, if the strength of SOI is sufficient. This is indeed what the experiments show (Fig. 2b,d,e).
We find that the WAL signal emerging from the ensemble average procedure is robust, and visible in the entire V_{g} range investigated. Its amplitude grows on lowering temperature T (Fig. 3a–c), and reaches ∼0.5 × e^{2}/h at the largest negative V_{g} and T=250 mK (Fig. 3c), where e is electron charge and h is Planck’s constant. The phenomenon is not observed in graphene on conventional substrates such as SiO_{2} (refs 27, 28, 29) (or hBN^{30} or GaAs^{31}), where at subKelvin temperatures only weak localization is measured. This remark is important because in graphene WAL can occur also in the absence of SOI, due only to the Dirac nature of its charge carriers^{32,33,34}. The WAL originating from the Dirac nature of electrons, however, is unambiguously different from what we observe on WS_{2} substrates: it is seen only for T∼10 K or higher, and has small amplitude, because its observation requires the phase coherence time τ_{φ} to be shorter than the intervalley scattering time τ_{iv} (ref. 28). The observation of the lowtemperature MC shown in Fig. 3, therefore, represents a direct, unambiguous demonstration of the presence of SOI in graphene on a WS_{2} substrate.
Very short spin relaxation time in graphene on WS_{2}
To analyse the MC data quantitatively, we use the theory of WAL in graphene that considers the effect of all possible symmetryallowed SOI terms in graphene, and predicts the following dependence of the lowtemperature MC^{35}:
Here is the rate of spin relaxation uniquely due to the SOI terms that break the z→–z symmetry (z being the direction normal to the graphene plane), is the total spin relaxation rate due to all SOI terms present, =4DeB/ħ (D is the carrier diffusion constant) and F(x)=ln(x)+ψ(1/2+1/x) with the digamma function ψ(x). In fitting the data, we constrain all characteristic times to be independent of temperature, except for τ_{φ}, which increases on lowering T, as physically expected in the T range investigated. Equation (1) holds in the limit τ_{φ}>>τ_{iv}, which is the one physically relevant at low T, and reproduces the experimental results quantitatively (solid lines in Fig. 3a–c). The analysis allows us to obtain the relevant characteristic times τ_{so}, τ_{asy} and τ_{φ}, with a precision determined by the residual conductance fluctuations that are not perfectly removed by the ensemble averaging procedure. (These residual effects also cause σ to be nonperfectly symmetric on reversing B, because the conductivity is extracted from the conductance measured in a fourterminal configuration, which in fully phase coherent devices is in general not symmetric^{25}.) We conservatively estimate the error on the characteristic times to be ∼50% in the worst case: although rather large as compared with what can be achieved in more established material systems, such an uncertainty is immaterial for all the considerations that will follow.
We find the spinrelaxation time to be τ_{so}∼2.5–5 ps depending on the value of V_{g} (see the black filled circles in Fig. 3d). Comparable values (within experimental uncertainties) have been obtained on the same devices from the analysis of measurements of nonlocal resistance generated by spinHall and inverse spinHall effect^{36} (see red circles in Fig. 3d, Supplementary Fig. 3 and Supplementary Note 3 for details). The latter technique was used recently in refs 9, 10 to probe SOI in hydrogenated graphene and graphene on WS_{2} in devices analogous to ours (see the Supplementary Note 4 for a comparison). The value of τ_{asy} is approximately three times larger than τ_{so}, consistent with its physical meaning; the phase coherence time τ_{φ}>>τ_{so}, as it must be since a large WAL signal is observed (see Fig. 3e; τ_{φ} decreases on increasing T, as expected). This internal consistency of the hierarchy of characteristic times extracted from fitting the data with equation (1) supports the validity of our analysis. We conclude that in graphene at a WS_{2} interface τ_{so} is 100–1,000 times shorter than τ_{so} for pristine graphene on SiO_{2} (refs 37, 38) or hBN^{39,40} (shown with open circles in Fig. 3d), and that such a large difference in strength must be due to substantially stronger SOI. In contrast to what has been reported in recent studies of graphene on WS_{2} (see ref. 10 and Supplementary Note 4), the larger strength persists throughout the entire gate voltage range investigated.
Determining the precise nature of the WS_{2}induced SOI is not straightforward. A customary way to extract information is to identify the spinrelaxation mechanism by looking at how τ_{so} depends on τ, the transport scattering time. Finding that τ_{so} increases with increasing τ points to the socalled Elliot–Yafet relaxation mechanism (spin relaxation mediated by scattering at impurities)^{41,42}, whereas if τ_{so} decreases with increasing τ, the Dyakonov–Perel mechanism (typical of systems with a strong band SOI) may be invoked^{43}. In graphene on SiO_{2} or hBN substrates, previous work has shown that neither scenario convincingly accounts for the observations^{8}, which has led to both phenomenological approaches to describe the experimental data^{39}, and to the theoretical proposal of new spinrelaxation mechanisms specific to graphene^{44}. For graphene on WS_{2}, despite the much larger strength of SOI, the interpretation of spin relaxation within the canonical schemes poses similar problems: τ_{so} decreases slightly on increasing τ (see Fig. 3d), ruling out Elliot–Yafet as a dominant relaxation mechanism, but the dependence is much weaker than the one predicted by Dyakonov and Perel, τ_{so}∝1/τ, so that the data are not satisfactorily described by this mechanism either. One interesting observation, however, can be made by comparing the spinrelaxation time τ_{so}, with the intervalley scattering time τ_{iv} obtained from the analysis of weak localization in graphene on SiO_{2} (refs 27, 29), hBN^{30} and GaAs^{31}. Literature values of τ_{iv} are shown with empty triangles in Fig. 3d: they are surprisingly narrowly distributed—they all fall within a factor of 2—if we consider that experiments have been performed by different groups and that different substrate materials are used. The values of τ_{iv} match well (again, within a factor of 2 or better) with τ_{so} obtained for graphene on WS_{2}. This close correspondence between two a priori unrelated quantities is remarkable: it strongly suggests that the microscopic processes responsible for scattering between the two valleys are the same processes that cause spin flip in graphene on WS_{2}.
Ab initio calculations of interfaceinduced SOI
To gain a better understanding of our experimental findings we have performed electronic structure calculations, to identify the dominant contributions to the WS_{2}induced SOI in graphene, and to verify that the large enhancement of the SOI strength we have discovered experimentally is indeed expected theoretically. Supercell electronic structure calculations were performed for a large number of crystal approximants to the incommensurate grapheneonWS_{2} system. (See Supplementary Fig. 4 and Supplementary Note 5 for details.) The results demonstrate that hybridization between graphene and substrate orbitals adds SOI terms both to the graphene πband Hamiltonian and to the πband disorder Hamiltonian. For each approximant πband states appear inside the WS_{2} gap E_{g} over a wavevector range of approximately E_{g}/ħv_{F} surrounding both K and K′ Dirac points. Within this range π bands are accurately described by effective Hamiltonians of the form
where σ=(σ_{x}, σ_{y}, σ_{z}) is a Pauli matrix vector that acts on the sub lattice degree of freedom in graphene’s Dirac continuum model Hamiltonian H_{0}, s=(s_{x}, s_{y}, s_{z}) is a Pauli matrix vector that acts on spin and τ_{z}=±1 for K and K′ valleys (the corresponding dispersion relations are illustrated in Fig. 4). All three substrateinduced interaction terms are timereversal invariant and absent by inversion symmetry in isolated graphene sheets. They arise from hybridization between carbon π orbitals and strongly spin–orbit split tungsten d orbitals in both valence and conduction bands of WS_{2}. Unlike the Hamiltonians that describe graphene on hBN^{45}, H in equation (2) is translationally invariant. This is so because the lattice constant difference between graphene and WS_{2} is much larger than for the case of graphene and hBN. As a consequence, the moiré pattern period is short, and superlattice effects couple states near the K and K′ points to states far away in momentum and energy that are outside the range accessible to transport experiments and describable in terms of modified π bands.
In our calculations, the importance of a spatially random intervalley contribution to SOI in real structures, which are not commensurate, is inferred from the observation that the numerical values of the substrate interaction parameters (Δ, λ and λ_{R}) depend on the supercell commensurability between graphene and WS_{2} triangular lattices (that is, the size of the approximant used in the calculation; see Supplementary Fig. 5). Physically, this dependence implies that spindependent terms that vary rapidly in space would be present even if each layer had a perfect, defectfree twodimensional lattice. These terms can scatter graphene electrons between valleys via intermediate WS_{2} states. For this reason the continuum model of equation (2) should be supplemented by including random potential terms. (See the last paragraph of Supplementary Note 5 for a more complete explanation.) An estimate of the strength of these random potentials is given by the difference in the values of the parameters Δ, λ and λ_{R} obtained from calculations performed on approximants of different size (see Supplementary Fig. 5). This random spindependent substrate interaction provides a most plausible explanation for our finding that τ_{so} in graphene on WS_{2} is comparable to τ_{iv}.
As a quantitative estimate of the parameters Δ, λ and λ_{R} in equation (2) we take the values obtained from the calculations performed on the largest supercells that we have considered. The 9:7 ratio between the WS_{2} and graphene lattice constants in these supercells is very nearly in perfect agreement with experiment. We find that the interaction parameters implied by this commensurability are independent of rigid relative translations between graphene and the substrate, providing further support for the translational invariance of equation (2). They are however sensitive to the separation between graphene and substrate layers, which has not yet been accurately determined experimentally, but should be near 3 Å like for graphene on hBN^{46}. For such a separation our calculations give Δ≈0 meV, λ≈5 meV and λ_{R}≈1 meV. The results illustrated in Supplementary Fig. 5 indicate that the value of λ is virtually the same for all approximants while larger relative fluctuations are found for Δ and λ_{R}, implying that it is these terms that dominate the random spatial potential responsible for intervalley scattering. Even though at this stage it is difficult to establish the precise role of the two different SOI contributions (the modification of the bands and the random potentials) to the shortening of the spinrelaxation time extracted from the experiments, our results show that the two larger SOI parameters exceed the scale of the SOI in isolated graphene sheets by two to three orders of magnitude^{3,4,5}, and that sizable spindependent intervalley scattering is present. We therefore can conclude that the results of our calculations are consistent with the experimental observations, pointing unambiguously to a strong enhancement of SOI due to interfacial interactions in graphene on WS_{2}.
Discussion
The general form of the Hamiltonian in equation (2), which yields a gap at charge neutrality because the Rashba term couples conduction and valence band states (compare Fig. 4c,d), deserves comment. It can be shown (see Supplementary Note 6) that when the Fermi energy lies in this gap and when λ is larger than Δ, the system becomes a topological insulator with gap larger by one to two orders of magnitude than that based on the intrinsic SOI in graphene originally proposed by Kane and Mele^{2}. The result of our calculations for the largest supercell that we have considered (yielding Δ≈0 meV, λ≈5 meV and λ_{R}≈1 meV) should be therefore be viewed as a prediction that graphene on WS_{2} is a topological insulator. This is an exciting conclusion, because a SOI strength of a few meV brings the possibility to realize a topologically insulating state in graphene closer to what can be realistically achieved experimentally. Care should be taken, however, because the precise values of the parameters Δ, λ and λ_{R} may depend on the details of the calculations scheme employed. In particular, even if in our calculations the SOI strength was always found to have a few meV scale irrespective of the approximations made, it is difficult to conclude that Δ>λ in all cases (see Supplementary Note 7, where we discuss calculations in which we let the distance between graphene and WS_{2} relax). At this stage it seems that a reliable determination of the parameters in equation (2) with sufficient precision to conclusively determine whether graphene on WS_{2} is a topological insulator or not will require more experiments. Specifically, this may be possible through a careful investigation of the Shubnikov–de Haas oscillations in veryhighmobility devices, capable of detecting a splitting in their frequency and of investigating its precise gate voltage dependence.
The increase in spin–obit interaction strength induced in graphene by proximity with WS_{2} that we observe for all investigated values of gate voltage—and hence position of the Fermi energy—is a large effect, close to two orders of magnitude. It directly shows the relevance of interfacial interactions. Demonstrating the ability to combine the control given by these interactions with the high electronic quality inferred from the experiments is a key result: although clearly more optimization is needed, finding that a drastic increase in SOI strength can be achieved without compromising the electronic quality of graphene offers the possibility to improve the system quality even further. To this end, strategies already exist that largely benefit from the expertise developed for graphene and hBN, such as encapsulating graphene between two WS_{2} crystals^{16}, using different device assembly techniques to avoid contact between graphene and unwanted materials during fabrication^{47}, and optimizing the devicecleaning procedures^{48}. As indicated by both our experimental results and their theoretical modelling, these developments open a possible route to access experimentally topological states^{49} in graphene, and will play an essential role in improving our understanding of the spin dynamics in this system.
Methods
Device fabrication
Thin WS_{2} flakes were exfoliated onto a highly doped silicon substrate acting as a gate covered by a SiO_{2}, from highquality single crystals of WS_{2} grown by chemical vapour transport method. These exfoliated WS_{2} flakes were annealed at 200 degrees for 3 h in an inert atmosphere, after which a specific WS_{2} flake with atomically flat and clean surface was identified through imaging with an atomic force microscope. Transfer of graphene onto the selected WS_{2} flake was achieved by means of (by now) common techniques^{15}. Similarly to the case of other artificial stacks of atomically thin crystals, bubbles were found to form after transfer of graphene (visible as black points in Fig. 1a)^{30}. To minimize the effect of these bubbles on the transport experiments, graphene was etched into a Hallbar geometry, after electrodes consisting of a Ti/Au bilayer (10/70 nm) were defined by means of electronbeam lithography, electronbeam evaporation and liftoff. No postannealing steps or other cleaning processes to further improve the device quality were performed on the devices discussed here. All transport measurements were performed in a He^{3} Heliox system with a base temperature of 250 mK. We investigated in full detail two monolayer devices showing identical results. The data presented in the main text are taken from one of the devices, which has a width of W=2.5 μm, and three pairs of Hall probes (the longest distance between different pairs of Hall probes in this device is 5.5 μm, see Supplementary Fig. 3b). The thickness of WS_{2} is about 26 nm. Data from the second monolayer device, virtually identical to the first one, are shown in the Supplementary Note 1.
Ab initio calculations
Fully relativistic density functional theory calculations were performed using the Vienna abinitio Simulation Package with projectoraugmented wave pseudopotentials under the generalized gradient approximation^{50,51}. The graphene lattice constant was always kept at 2.46 Å. Ionic relaxations were performed for WS_{2} in 1 × 1 unit cells with lattice constants fixed to three different rational multiples (4/3, 5/4 and 9/7) of that of graphene. Supercells with different moiré periodicities were then constructed by repeating these unit cells correspondingly and by aligning the lattice vectors of WS_{2} with that of graphene without further ionic relaxation. The separation between graphene and WS_{2} was fixed to a number of different values ranging from 2.3 to 3.3 Å for each supercell. A MonkhorstPack kpoint mesh^{52} of 6 × 6 × 1 was used for the 4 × 4 and 5 × 5 supercells (in terms of graphene lattice constant), and that of 3 × 3 × 1 was used for the 9 × 9 supercell. The plane wave energy cutoff was set to 400 eV in all calculations.
Additional information
How to cite this article: Wang, Z. et al. Strong interfaceinduced spinorbit interaction in graphene on WS_{2}. Nat. Commun. 6:8339 doi: 10.1038/ncomms9339 (2015).
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Acknowledgements
We gratefully acknowledge technical assistance from A. Ferreira. Z.W., D.K.K. and A.F.M. also gratefully acknowledge financial support from the Swiss National Science Foundation, the NCCR QSIT and the EU Graphene Flagship Project. A.H.M. and H.C. were supported by ONRN000141410330 and Welch Foundation grant TBF1473. H.C. and A.H.M. acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources for the ab initio calculations in this work. H.C. thanks Y. Araki, X. Li and A. Da Silva for valuable discussions.
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Z.W. fabricated the devices and performed the measurements with the collaboration of D.K.K., under the supervision of A.F.M.; H.B. provided the WS_{2} crystals used to realize the devices; Z.W., D.K.K. and A.F.M. analysed the data; H.C. performed the numerical calculations under the guidance of A.H.M.; all authors discussed and interpreted the results, and contributed to writing the paper.
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Supplementary Figures 15, Supplementary Notes 17 and Supplementary References (PDF 669 kb)
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Wang, Z., Ki, D., Chen, H. et al. Strong interfaceinduced spin–orbit interaction in graphene on WS_{2}. Nat Commun 6, 8339 (2015). https://doi.org/10.1038/ncomms9339
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