Experimental observation of spin-split energy dispersion in high-mobility single-layer graphene/WSe2 heterostructures

Proximity-induced spin-orbit coupling in graphene has led to the observation of intriguing phenomena like time-reversal invariant $\mathbb{Z}_2$ topological phase and spin-orbital filtering effects. An understanding of the effect of spin-orbit coupling on the band structure of graphene is essential if these exciting observations are to be transformed into real-world applications. In this research article, we report the experimental determination of the band structure of single-layer graphene (SLG) in the presence of strong proximity-induced spin-orbit coupling. We achieve this in high-mobility hBN-encapsulated SLG/WSe2 heterostructures through measurements of quantum oscillations. We observe clear spin-splitting of the graphene bands along with a substantial increase in the Fermi velocity. Using a theoretical model with realistic parameters to fit our experimental data, we uncover evidence of a band gap opening and band inversion in the SLG. Further, we establish that the deviation of the low-energy band structure from pristine SLG is determined primarily by the valley-Zeeman SOC and Rashba SOC, with the Kane-Mele SOC being inconsequential. Despite robust theoretical predictions and observations of band-splitting, a quantitative measure of the spin-splitting of the valence and the conduction bands and the consequent low-energy dispersion relation in SLG was missing -- our combined experimental and theoretical study fills this lacuna.

Before one conceives increasingly complex graphene/TMDC heterostructures and considers their potential applications, it is imperative to understand the impact of the proximity of TMDC on the electronic properties of graphene. Prominent amongst these are the breaking of inversion symmetry, breaking of sub-lattice symmetry, and hybridization of the d -orbitals of the heavy element in TMDC with the p-orbitals of SLG, leading to strong SOC in SLG.
This proximity-induced SOC in SLG has three primary components, all of which contribute to spin splitting of the bands -(i) valley-Zeeman (also called Ising) term, which couples the spin and valley degrees of freedom, (ii) Kane-Mele term [36,37] which couples the spin, valley and sublattice components and opens a topological gap at the Dirac point [18,38], and (iii) Rashba term [39] which couples the spin and sublattice components.
In the presence of a strong Ising SOC, the electronic band dispersion of graphene is predicted to be spin-split [17,24,[39][40][41][42], as was observed recently in bilayer graphene/WSe 2 heterostructures [10,43]. Consequences of this induced SOC include the appearance of helical edge modes and quantized conductance in the absence of a magnetic field in bilayer graphene [5] and of weak antilocalization [19], and Hanle precession in SLG [27,28,44,[44][45][46][47][48]. Despite these advances, a quantitative study of the effect of a strong SOC on the electronic energy band dispersion of SLG is lacking.
In this research article, we report the results of our studies of quantum oscillations in high-mobility heterostructures of SLG and trilayer WSe 2 . Careful analysis of the oscillation frequencies shows spin-splitting of the order of ∼ 5 meV for both the valence band (VB) and the conduction band (CB). We find that the bands remain linear down to at least 70 meV (corresponding to n ∼ 2 × 10 11 cm −2 ). Close to zero energy, the lower energy branches of the CB and the VB overlap, leading to band inversion and opening of a band gap in the energy dispersion of SLG. We fit our data using a theoretical model that establishes that, to the zeroth-order, the magnitude of the spin-splitting of the bands and that of the band gap are determined by only the valley-Zeeman and Rashba spin-orbit interactions.

A. Experimental Observations
Heterostructures of single-layer graphene and trilayer WSe 2 , encapsulated by hexagonal boron nitride (hBN) (see device schematic Fig. 1(a)) of thickness ∼ 20-30 nm, were fabricated using dry transfer technique [49,50]. One-dimensional Cr/Au electrical contacts  were created by standard nanofabrication techniques -note that this method completely evades contacting the WSe 2 thus avoiding parallel channel transport (see Supplementary Information for details). Electrical transport measurements were performed using a lowfrequency ac lock-in technique in a dilution refrigerator at the base temperature of 20 mK unless specified otherwise. Multiple devices of SLG/WSe 2 were studied, and the data from all of them were qualitatively very similar. All the data presented here are from a device labeled B9S6. The data for two other similar devices are presented in the Supplementary Information. The extracted impurity density from the four-probe resistance of the device as a function of gate voltage (see Fig. 1(b)) was ∼ 2.2 × 10 10 cm −2 , and the mobility was ∼140,000 cm 2 V −1 s −1 . The four-probe resistance response as a function of the gate voltage were identical for different measurement configurations (see Supplementary Information), indicating that the fabricated device is spatially homogeneous. Fig. 1(c) shows the quantum Hall data at 3 T -the presence of plateaus at ν = ± 2, ± 6, ± 10 confirms it as SLG.
Along with the signature plateaus of SLG, one can see a few of the broken symmetry states appearing already at 3 T, confirming it to be a high-quality device.
Representative data of the Shubnikov-de Haas (SdH) oscillations measured at 20 mK are  is the case. We find similar splitting in the SdH oscillation frequency in all the SLG/WSe 2 devices studied by us -the data for two additional similar devices are presented in the Supplementary Information. There may be a legitimate concern that the observed beating can be caused by device inhomogeneities which lead to different charge carrier density in different regions of the graphene channel. We rule out this artifact from measurements of the four-probe resistance and SdH oscillations in multiple contact configurations -we find that the data are identical in each case (see Supplementary Information).
Recall that the SdH oscillation frequency, B F is directly related to the cross-sectional area A (k) at the Fermi energy by the relation B F = A (k) /2πe [51]. For an isotropic dispersion in which the Fermi energy E F is a function of k F = k 2 x + k 2 y (where (k x , k y ) are defined with respect to one of the Dirac points, K or K , of the SLG), the cross-sectional area of the Fermi surface is given by A(k) = πk 2 F . Fig. 2(c) shows the charge carrier density (n) dependence of B F . The appearance of two closely spaced frequencies at all n (or E F ) implies that for each value of the Fermi energy, there are two distinct values of k F . This is a direct proof of the energy splitting of both the CB and the VB of the SLG.
From the temperature dependence of the amplitude of the SdH oscillations ( Fig. 3), we extracted the effective charge carrier mass m * , using the the Lifshitz-Kosevich relation [52,53]: Here, R 0 the longitudinal resistivity at B = 0. On fitting the effective mass m * versus n using the relation m * = ( √ π/v F )n α [54] (see Supplementary Information), we obtain α = 0.5 ± 0.02 and Fermi velocity v F = 1.29 ± 0.04 × 10 6 ms −1 . The value of α being 0.5 establishes the dispersion relation between energy and momentum in SLG on WSe 2 to be linear [51]. We observe that, on extrapolating the E − k plots to E = 0, the low-energy branches of the spin-split bands of both the CB and the VB bands enclose a finite area in the k-space at E = 0. This leads us to expect that there will be an overlap between the lower branches of the CB and VB, ultimately leading to band inversion near the K (and K ) points. A verification of this assertion requires further measurements in extremely high quality devices that will allow measurements of SdH oscillations near E = 0.
To summarize our experimental observations, we have quantified the spin-splitting of the energy bands in SLG in proximity to WSe 2 and mapped out the dispersion relation of the spin-split bands of SLG. We find that till a certain energy, the dispersion remains linear; below this energy scale, we observe a deviation from linearity.

B. Theoretical calculations
Using the experimental data, we fit a theoretical model to obtain the dispersion relation close to the Dirac points. The continuum Hamiltonian near the Dirac points for SLG with WSe 2 has the following terms (see, for instance, Ref. [39]): In Eq. 2, the Pauli matrices σ i and the S i represent the sublattice and spin degrees of freedom, respectively. The first term denotes the linear dispersion near the Dirac points, where v F is the Fermi velocity, k x and k y are the momenta with respect to the Dirac point, and η = ±1 denotes the valleys K (K ) respectively (We note that v F = 3ta/2, where t is the nearestneighbor hopping amplitude and the nearest neighbour carbon carbon distance a is 1.42 Å). The second term represents a sublattice potential of strength ∆. We have considered the four possible spin-orbit couplings: (i) Kane-Mele SOC with strength λ KM , (ii) valley-Zeeman SOC with strength λ VZ , (iii) Rashba SOC with strength λ R , and (iv) pseudo-spin asymmetric SOC with strengths λ A PIA and λ B PIA for sublattices A and B respectively. Since this Hamiltonian results in the same dispersion at both the valleys, we only consider the case η = +1 (K point). The Hamiltonian in Eq. 2 is invariant under a simultaneous rotation of (k x , k y ), (σ x , σ y ) and (S x , S y ) by the same angle; this implies that the dispersion is isotropic in momentum space, and it is sufficient to take k x = k and k y = 0. The data for the four bands shown in Fig. 4(a) are fitted to the Hamiltonian with t, ∆, λ KM , λ VZ and λ R as the fit parameters. The best fit gives t = 3979.10 ± 3.99 meV implying a large Fermi velocity in this device of 1.286 × 10 6 ms −1 (compared to about 0.86 × 10 6 ms −1 in pristine SLG [55]). The parameters in the Hamiltonian which give the spin-split band gap in both conduction and valence bands are λ V Z and λ R . We find that the best fit gives the values of λ VZ and λ R to lie on a circle of radius 2.51 meV, such that λ VZ = 2.51 cos θ meV, and λ R = 2.51 sin θ meV. ( where θ can take any value from 0 to 2π, and ∆ = λ KM = 0. Eq. 3 can be understood by looking at the first-order perturbative effect of the valley-Zeeman and Rashba terms in the we find that the zeroth order spin-degenerate dispersion E 0 = ± v F k in the positive and negative energy bands receives first-order corrections given by for both the bands, thus giving the general relation in Eq. 3. This gives a gap equal to twice  Finally, a comment on the relative magnitudes of λ VZ and λ R : The spin relaxation mechanism in graphene/TMDC heterostructures is extraordinary. It relies on intervalley scattering and can only occur in materials with spin-valley coupling. In such systems, the lifetime τ and relaxation length λ of spins pointing parallel to the graphene plane (τ s , λ s ) can be markedly different from those of spins pointing out of the graphene plane (τ ⊥ s , λ ⊥ s ). Realistic modeling of experimental studies indicate that the spin lifetime anisotropy ratio can be as large as a few hundred in the presence of intervalley scattering [41,44,47]. Recall that the λ VZ provides an out-of-plane spin-orbit field and affects the in-plane spin relaxation time, τ s . On the other hand, λ R generates an in-plane spin-orbit field and is relevant for determining τ ⊥ s [47]. The large spin lifetime anisotropy ratio (τ s /τ ⊥ s 1) seen both from experiments and theory [41,44,47] show that the value of λ VZ can indeed be significantly larger as compared to λ R .
In conclusion, we have experimentally determined the band structure of single-layer graphene in the presence of proximity-induced SOC. We find both the VB and the CB spin-

Device Fabrication
The SLG, WSe 2 , and hBN flakes were obtained by mechanical exfoliation on SiO 2 /Si wafer using scotch tape from the corresponding bulk crystals. The thickness of the flakes was verified from Raman spectroscopy. Heterostructures of SLG and WSe 2 , encapsulated by single-crystalline hBN flakes of thickness ∼20-30 nm was fabricated by dry transfer technique using a home-built transfer set-up consisting of high-precision XYZ-manipulators.
The heterostructure was then annealed at 250 • C for 3 hours. Electron beam lithography followed by reactive ion etching (where the mixture of CHF 3 and O 2 gas were used with flow rates of 40 sccm and 4 sscm, respectively, at a temperature of 25 • C at the RF power of 60 W) was used to define the edge contacts. The electrical contacts were fabricated by depositing Cr/Au (5/60 nm) followed by lift-off in hot acetone and IPA.
All electrical transport measurements were performed using a low-frequency AC lock-in technique in a dilution refrigerator (capable of attaining a lowest temperature of 20 mK and maximum magnetic field of 16 T).

Data availability
The authors declare that the data supports the findings of this study are available within the main text and its supplementary Information. Other relevant data are available from the corresponding author upon reasonable request.

SUPPLEMENTARY MATERIALS Appendix A: Device Fabrication and characterization
We fabricated heterostructures of single-layer graphene (SLG) and trilayer WSe 2 , encapsulated by single-crystalline hBN flakes of thickness ∼20-30 nm. The SLG, WSe 2 , and hBN flakes were obtained by mechanical exfoliation on SiO 2 /Si wafer using scotch tape from the corresponding bulk crystals. The thickness of the flakes was verified both from optical contrast under an optical microscope and Raman spectroscopy.
The Raman data for SLG and trilayer WSe 2 flake are shown in Figure. S6(a) and (b) respectively. For the graphene, the high intensity of the Lorentzian G' peak confirms it to be a single-layer. The flakes were formed into a heterostructure using dry transfer technique [49] using a home-built transfer set-up consisting of high-precision XYZ-manipulators, the entire process being performed under an optical microscope.Briefly, the hBN was first picked up using a Polycarbonate (PC) film at 90 o C. This combination was then used to pick up the SLG followed by WSe 2 and hBN. The prepared stack was transferred on a clean Si/SiO 2 wafer at 180 o C and cleaned using chloroform to remove the PC, and this was followed by cleaning with acetone and isopropyl alcohol.The heterostructure was then annealed at 250 • C for 3 hours.
Electron beam lithography was used to define the edge contacts. The edge contacts were made by reactive ion etching (where the mixture of CHF 3 and O 2 gas was used with flow rates of 40 sccm and 4 sscm, respectively, at a temperature of 25 • C at the RF power of 60 W) [56]. The electrical contacts were finally created by depositing Cr/Au (5/60 nm) followed by lift-off in hot acetone and IPA. Cr/Au was chosen as it forms a very high quality ohmic contact with graphene [56], but at the same time it does not form any contact with WSe 2 due to high Schottky barrier and large difference in work functions [57,58]. Finally, the device was etched into a Hall bar geometry. An optical image of the final device is shown in the main text (inset of Fig. 1(b)).
To estimate the impurity density (n 0 ) and field-effect mobility (µ) of the device, the gate-voltage dependent resistance data were fitted by the equation where R c is the contact resistance, L and W are the channel length and width, respectively, originate from spatial inhomogeneity of charge carrier density.
We measured SdH in several devices, the results from all of them were similar. Here, we present the data from two such devices: (1) Device B6S1, which is heterostructure of SLG and single layer WSe 2 encapsulated in hBN with a graphite back gated and (2) device B6S3, which is an SLG/few-layer-WSe 2 heterostructure encapsulated by hBN, this device is on SiO 2 . Supplementary Figure S9 of the SdH oscillations and two frequencies in the FFT.

Appendix C: Calculation of the dispersion relation
We extract the effective mass m * by fitting the normalized amplitude of longitudinal resistivity to the relation [52,53]: where ∆R xx is the amplitude of longitudinal resistivity and R 0 longitudinal resistivity at zero magnetic fields.
The effective mass can be written as For A k = πk 2 and for a linear dispersion of single layer graphene (E = v F k), we have One can fit the experimentally obtained dependence of m * on n using the relation m * = √ π n α /v F [54] keeping α and v F as fitting parameters. The fits to the experimental data shown in main text (Fig. 3(c)) yield α = 0.5 ± 0.02 and v F = 1.29 ± 0.04 × 10 6 ms −1 . The value of α is nearly 0.5, establishing the dispersion relation between energy and momentum for SLG/WSe 2 to be linear [51].

Appendix D: Theoretical Modelling
The continuum Hamiltonian near the Dirac points used for fitting the experimental data has the following terms, The best fit gives hopping parameter t = 3979.10 ± 3.99 meV implying a large Fermi velocity v F = 3ta/2 in this sample. We further note that the parameters in the Hamiltonian which give the spin-split band gap in both conduction and valence bands are λ VZ and λ R . The other parameters do not significantly alter the dispersion in the region of experimental data.
We also find that the best fit gives the values of λ VZ and λ R to lie on a circle of radius  Rashba SOC. This is demonstrated in Supplementary Figure S11 Thus, although it is tempting to explain the trend of increase in the energy gap between the spin-split bands as one approaches the Dirac point to originate from a large λ KM , we refrain from doing so.