Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2

Van der Waals interactions with transition metal dichalcogenides were shown to induce strong spin-orbit coupling (SOC) in graphene, offering great promises to combine large experimental flexibility of graphene with unique tuning capabilities of the SOC. Here, we probe SOC-driven band splitting and electron dynamics in graphene on WSe2 by measuring ballistic transverse magnetic focusing. We found a clear splitting in the first focusing peak whose evolution in charge density and magnetic field is well reproduced by calculations using the SOC strength of ~ 13 meV, and no splitting in the second peak that indicates stronger Rashba SOC. Possible suppression of electron-electron scatterings was found in temperature dependence measurement. Further, we found that Shubnikov-de Haas oscillations exhibit a weaker band splitting, suggesting that it probes different electron dynamics, calling for a new theory. Our study demonstrates an interesting possibility to exploit ballistic electron motion pronounced in graphene for emerging spin-orbitronics.

Supplementary Figure 4. TMF spectra near zero density.1D cuts of the data shown in Figs.2a,b for sample 1 (a) and 2 (b).No TMF signals are observed below around 2~3 × 10 11  −2 which matches roughly with the density below which negative non-local resistance disappears (the inset of Fig. 1e) and large non-local signal appears (Fig. 4b).It indicates the crossover between the diffusive spin-Hall and ballistic TMF effects.

Supplementary Note 1. Quantum transport simulation
Current density maps (the inset of Fig. 1c and Supplementary Fig. 3e) and conductance spectra (Fig. 1d, right) were simulated using Kwant 6 , an open-source python package for quantum transport simulations based on tight-binding models.To model transport in graphene on TMDCs, we adopt the effective tight-binding Hamiltonian 7  =  0 +  1 +   +   +   , (S1) where the first two terms are spin-independent and the rest describe spin-orbit couplings of different origins.The first term, , is composed of nearest-neighbor kinetic hopping of strength , and is typically used to describe a spinless graphene lattice.Here, ,  are lattice site indices,  is the spin index,   † (  ) is the creation (annihilation) operator that creates (annihilates) an electron of spin  at site , and Σ 〈,〉 sums over site indices that are nearest to each other.Except  0 , all the rest of the terms in Eq. (S1) arise from the effect of the neighboring TMDC lattice.First, the symmetry between the sublattices A and B of the graphene lattice is broken, giving rise to an effective energy difference experienced by electrons on atoms of the two sublattices, which can be described by , where   is the sublattice index of site  ,   = +1 (−1) when   =  (   =  ), and Δ characterizes the strength of such a staggered potential energy.The rest of the three terms in Eq.
(2) of the main text include the Rashba spin-orbit coupling, where   the Rashba coupling strength,  = (  ,   ,   ) is a vector of Pauli matrices acting on spin, and   is a unit vector pointing from site j to , the valley-Zeeman term, , where Σ 〈〈,〉〉 sums over site indices ,  that are second nearest to each other, the sign factor   = +1 (−1) when the resulting hopping path is counterclockwise (clockwise), and     is the sublattice-resolved valley-Zeeman coupling strength, and finally the pseudospin-inversionasymmetry term   that does not influence the band structure at  and ′.Neglecting   and setting    = −   =  for both sublattices   = ,  , the eigenenergy of tight-binding model Hamiltonian Eq. ( 3) is given by 8 with ,  = ±1, whose low- expansion simplifies to the eigenenergy of Eq. (1) of the main text, which is adopted in several previous studies 1, [8][9][10] .Note that to account for micron-sized graphene systems, we have adopted the scaled tight-binding model 11 which is compatible with the spin-orbit coupling terms as remarked in the recent study 12 .All quantum transport simulations presented here are based on the scaling factor   = 8.
To simulate our TMF experiment done on the multi-terminal graphene Hall bar (Fig. 1b of the main text), considering the geometry as close to the real device as possible while keeping the computation affordable, we calculate the conductance G between the injector and collector (those reported in Figs.1d and 3f), i.e.,  = ( 2 /ℎ) where T is the corresponding transmission function (from the injector lead to the collector lead), based on the Landauer formula 13 .On the other hand, the experimentally measured TMF signal is based on the so-called four-point resistance (such as Fig. 2 of the main text), which can also be simulated using the Büttiker formula 13 .Computationally, however, the Büttiker formula requires computing ( − 1) transmission functions between any pair of two different leads for an -terminal system.
In our case (see Fig. 1b of the main text), there are 10 × 9 = 90 transmission functions to compute, which is beyond our computation limit, even if we could afford to model the full size of our experimental device.In fact, instead of modeling the full device of our experiment, we used an effective three-terminal device of smaller area and shorter probe spacing ( = μm) for our transport simulations, in order to lower the computation burden.Nevertheless, our conductance calculations (for the effective three-terminal Hall bar) reveal consistent behaviors of the TMF peaks, compared to those from our four-point resistance measurement, including the splitting of the first peaks.

Supplementary Note 2. Semi-classical ray tracing
In order to check if the absence of the splitting in the second focusing peak (Figs.1d and 2) is due to the inter-band scattering at the edge, we have employed the semi-classical ray tracing by solving the following equation of motion 14 : where one can manually add or remove scattering conditions like the inter-band transition at the edge.Supplementary Fig. 3 shows the results that clearly prove that the absence of the splitting in the second focusing peak is from the scattering between the two bands  + and  − at the sample edge.

Supplementary Note 3. Discussions on TMF and SdH oscillations
As discussed briefly in the main text, in TMF, the electrons make only half of the cyclotron motion while in SdH oscillations, it needs to make a full circle without losing its phase coherence.This can give rise to several differences in the two phenomena.First, the TMF is more sensitive to the electron scattering than the SdH oscillations as TMF is from the ballistic motion of electrons that can be destroyed by elastic scattering.Thus, in our sample, we could see the splitting in TMF focusing peak only on the hole side where we find higher sample quality while in SdH oscillations we found no obvious differences in both hole and electron side.Second, as electrons need to make a full cyclotron motion, SdH oscillations probe the total Fermi surface area only, whereas the TMF probes the partial trajectory along the Fermi surface.This can make a difference when the band structure is shifted in one momentum direction while its area.This might be the reason why the studies on 2DEG with SOC including ours showed different splitting in TMF and SdH oscillations.Lastly, the SdH oscillations generally occur at larger magnetic fields than the TMF.This may result in a stronger effect of the Zeeman energy that couples to out-of-plane spin components.It could thus affect the spin-valley Zeeman and Rashba terms in Eq. (1) differently, that may lead to different band splitting in SdH oscillations compared with TMF.Nonetheless, the microscopic process that governs TMF and SdH oscillations is different and our study, together with other studies on 2DEG with SOC 15,16 , shows that it might be important to consider their differences when analysing the relevant experimental results.