## Introduction

Graphene-based van der Waals (vdW) heterostructures became one of the most studied physical systems in material science in recent years, which led to the emergence of designer electronics1,2. Since the electrons are localized at the surface for a single layer of graphene by definition, their properties can be easily modified by combining it with other few layer crystals leading to remarkable changes in its band structure. A prominent example is the moiré effect caused by the rotation (and possible small lattice mismatch) of the graphene and the underlying other lattice. This led to the Hofstadter physics and formation of secondary charge neutrality points (CNPs) when graphene is placed on hexagonal boron nitride (hBN)3,4,5,6,7,8,9,10, whereas correlated phases including superconductivity correlated insulators or ferromagnetic states have been found if it is placed on another graphene sheet11,12,13,14. Graphene-based heterostructures are also promising building blocks for spintronic devices15,16,17.

Although graphene is known to provide very long spin lifetimes18,19, the absence of spin–orbit coupling (SOC) also hinders electrical control and charge to spin conversion in it. However, a large SOC can be induced in graphene by proximity effect if placed on a transition metal dichalcogenide (TMDC) flake20,21,22 which can lead to topologically nontrivial states and the quantum spin Hall effect23. Recently, a wide range of experiments demonstrated the presence of proximity-induced SOC in various heterostructures by weak localization (WL), capacitance, or spin transport measurements, and it has been found that Rashba and valley–Zemann-like SOC is induced in graphene leading to a large spin relaxation anisotropy24,25,26,27,28,29,30,31,32,33,34,35. Since this enhancement of SOC originates from the hybridization of graphene’s π orbitals with the TMDC layer’s outer orbitals, the strength of the SOC depends strongly on the overlap of the orbital wavefunctions and, therefore, on the interlayer distance, which is determined by the vdW force. Compressing such a heterostructure by applying an external pressure (see Fig. 1a) is expected to increase the SOC, which can be captured by WL measurements, as shown by simulated magneto-conductivity curves in Fig. 1b21,36.

In this work, we present experimental evidence of manipulation of the interlayer coupling and the proximity SOC in a graphene/WSe2 heterostructure using hydrostatic pressure. The pressure control adds another knob with which the electronic properties of 2D materials can be engineered, allowing to more robust proximity states or even engineer novel states of matter.

## Results and discussion

### General characterization of the device

The illustration of the studied device is shown in Fig. 1a, whereas an optical image is given in the inset of Fig. 2. It consists of a monolayer graphene, which is on top of a very thin (3 nm) WSe2 flake providing the SOC, and covered with a hBN flake to protect it from the kerosene pressure medium37 (see Fig. 1a). The device is also equipped with a top gate extending over the major part of the measured segment but it was grounded during the measurements. Another sample showing similar behavior is shown in Supplementary Fig. 1. Low-temperature (1.5 K) measurements have been carried out at four different hydrostatic pressure settings in increasing order (no pressure, 0.6 GPa, 1.2 GPa, 1.8 GPa). Each pressure change involved warming up the sample to room temperature, applying the pressure using a hydraulic press, clamping the pressure cell, and cooling the sample down again.

We characterized our devices by measuring the two-terminal conductance as a function of back gate voltage at each pressure, as shown in Fig. 2. The measured segment is highlighted in the inset. The CNP position was found at slightly different Vg back gate voltages in each case but remained between −5.2 and −0.6 V, which was corrected by shifting the curve minima to Vg = 0 for further use. The curves are quite similar in shape, with a minimum around the CNP. Using a simple parallel plate capacitor model for the estimation of the charge carrier density n(Vg), field effect mobility was calculated based on a linear fit on the two-terminal conductance. Electron mobility values were found between 11,000 and 24,000 cm2 V−1 s−1 without any systematic dependence on the applied pressure. A change in the scattering processes and the observed field effect mobility is not unusual in case of vdW heterostructures during subsequent cooldowns even without applied pressure and can be attributed to the rearrangement of scattering centers. Thus, we conclude that the sample conductance and quality are not significantly affected by the applied pressure.

### WL measurements

Now we turn to low-field magneto-conductance measurements. WL is a low-temperature quantum correction to the magnetoconductance of diffusive samples and expected to show a conductance minimum (WL) or maximum (weak antilocalization, WAL) at zero field depending on the strength of SOC38.

We have recorded the two-terminal conductance curves as the B field was swept in a range of ±30 mT for several gate voltages, both in the up and down magnet ramping direction. At each gate voltage, we symmetrized the curves, then subtracted the zero-field conductance from the measured curve leading to 2D conductance maps ΔG(Vg, B). The zero-field conductance against the gate voltage for the no pressure case is plotted in Fig. 3a. The corresponding conductance map is shown in Fig. 3b, where a vertical dark gray line along B = 0 appears due to the correction method, and lighter gray tones on each side are a sign of positive magnetoconductance corresponding to WL. At higher magnetic fields, universal conductance fluctuations38 (UCF) of amplitude up to 0.4e2/h across the gate voltage axis are also visible on the map as saturated horizontal lines. To increase visibility of WAL signal, an averaging on a gate range of ±1.5 V was applied on the conductance, noted as ΔGavg(Vg, B). Cuts of ΔGavg at fixed Vg values are shown in Fig. 3c. All these cuts show a WL dip with slight changes in the curve shape as a function of the gate voltage. Here, the absence of a WAL peak suggests a weak SOC, which will be discussed later. This is in agreement with previous measurements on this device given in the Supplementary Material of ref. 31, which made this sample an ideal candidate for the pressure experiment.

After having performed the above procedure for each pressure, we selected the curves at 4 ± 1.5 V, i.e., averaging the ΔG(Vg, B) curves between 2.5 and 5.5 V for all the pressures. We assume the characteristic times of the scattering processes do not change much across the gate range where the averaging is performed, while the contribution of UCF is reduced. The conductance was converted to conductivity for the extraction of scattering time scales by curve fitting. Background signal was also recorded simultaneously with the localization measurement and found to be constant, see Supplementary Fig. 2 for details.

Our main findings are presented in Fig. 4a, where the averaged magneto-conductivity Δσavg(Vg = 4 V, B) for all four pressures is plotted for low magnetic fields. The initial curve, at ambient pressure, shows a wide conductivity dip, which gets wider as the pressure increases to 0.6 GPa, and a sharp central peak appears at higher pressures. This transition of WL to WAL is a clear signature of the increasing proximity-induced SOC in the graphene layer. This tendency is robust for the entire gate voltage range as shown by the cuts at different gate voltages in Fig. 4b. See Supplementary Figs. 3 and 4 for the full comparison of all pressures at various gate voltages. Releasing the pressure again restores the WL dip, as presented in Supplementary Fig. 5.

### Curve fitting

To quantitatively demonstrate this transition, the data were fitted using the McCann–Falko WAL formula39

$${{\Delta }}\sigma (B)=-\frac{1}{2}\frac{{e}^{2}}{\pi h}\left(F\left(\frac{{\tau }_{B}^{-1}}{{\tau }_{\varphi }^{-1}}\right)-F\left(\frac{{\tau }_{B}^{-1}}{{\tau }_{\varphi }^{-1}+2{\tau }_{\,{{\mbox{asy}}}\,}^{-1}}\right)\right.\left.-2F\left(\frac{{\tau }_{B}^{-1}}{{\tau }_{\varphi }^{-1}+{\tau }_{\,{{\mbox{asy}}}}^{-1}+{\tau }_{{{\mbox{sym}}}\,}^{-1}}\right)\right),$$
(1)

where Δσ(B) = σ(B) − σ(B = 0) is the correction to the magneto conductivity. We introduced the function $$F(x)={{\mathrm{ln}}}\,(x)+\psi (0.5+{x}^{-1})$$ with ψ being the digamma function and h being the Planck’s constant. The rate $${\tau }_{B}^{-1}=4eDB/\hslash$$ is associated to the magnetic field with D, the diffusion constant. τφ is the phase-breaking time, τasy is the scattering time due to SOC terms that are asymmetric upon z/-z inversion, and τsym is the corresponding time due to terms invariant under z/-z inversion. Using the previously calculated charge carrier density n(Vg), the Fermi velocity of graphene, and the Einstein relation, the diffusion constant and the momentum relaxation time can be extracted from the gate voltage curves and were found between D = 0.07–0.12 m2 s−1 and τm = 1.3–2.4  10−13 s for all pressure measurements without any systematic pressure dependence, but in correlation with the field effect mobility. Since τm is expected to be the shortest amongst the time scales, we set it as the lower bound for all fitting time parameters. Equation (1) is valid if the intervalley scattering time τiv is much shorter than the times corresponding to SOC, therefore the contribution of the effect of the charge carriers’ chiral nature can be neglected. We also performed fits with a more complex, five-parameter fitting formula including the τiv intervalley and τia intravalley scattering times as fitting parameters, which suggest that τiv is indeed at least an order of magnitude smaller than the other fitted time scales, supporting the validity of Eq. (1) (see Supplementary Note 1).

In Fig. 4a, fit curves are plotted using solid blue lines. The extracted fit parameters are summarized in Fig. 4c, d. A clear tendency of the reduction of τasy can be observed as its value decreases from 7.4  10−12 s to 2.7  10−12 s as the pressure increases to 1.8 GPa. This is a reduction by a factor of 2.7 and corresponds to an increasing SOC. This is the main proof to our initial expectation of increasing interlayer coupling. The phase-breaking time, τφ, also shows a moderately decreasing but fluctuating trend with pressure, obtaining values from 9.2  10−12 s to 5.5  10−12 s, staying well above τasy at all pressures. The reason for this decreasing tendency is unclear at the moment. The third fit parameter, τsym, obtains fit values that are much longer than τφ and the fit errors are more than an order of magnitude large (see Fig. 4d), which means that it has negligible effect on the magneto-conductance curve. Therefore, although there seems to be a decreasing trend in the lifetime, we cannot extract any reliable value for it at any pressure. Discussion on this result will be given later.

The spin relaxation times can be connected to band structure parameters via spin relaxation mechanisms. It has been established that two relevant spin orbit terms are formed in graphene/TMDC heterostructures. One of them is the Rashba term, described by the Hamiltonian HR = λR(κσxsy − σysx)40, which corresponds to the breaking of the lateral mirror symmetry due to the difference of the hBN and TMDC neighboring layers to the graphene sheet. Its strength is set by the λR parameter, κ = ±1 for the K and K′ valleys, the Pauli matrices σ are acting on the lattice pseudospin, and kx, ky are the electron wave vector components measured from the K (K′) points. This term leads to relaxation processes that are contained in τasy via the Dyakonov–Perel mechanism, from which the Rashba parameter can be expressed as $${\lambda }_{{{\mbox{R}}}}=\frac{\hslash }{2}\sqrt{{\tau }_{{{\mbox{m}}}}^{-1}{\tau }_{{{\mbox{asy}}}\,}^{-1}}$$41.

The increasing scattering rates lead to an increase in the Rashba parameter, which clearly demonstrates that we were able to tune the strength of the induced SOC. The values are summarized in Fig. 4e, shown by the orange curve. We have also plotted the value of the Rashba parameter from the five-parameter fitting formula in green (see Supplementary Note 1), which is in agreement with the values from the simplified formula within the error limits.

The second dominant spin orbit term in these systems is the HVZ = λVZκσ0sz valley–Zeeman term, where λVZ characterizes the coupling strength. This term corresponds to an effective Zeeman magnetic field which is opposite in the two valleys due to inversion symmetry breaking, but still preserving time reversal symmetry. It leads to relaxation contained in the τsym via a modified Dyakonov–Perel mechanism, where the intervalley scattering time, τiv, enters instead of the momentum scattering time $${\lambda }_{{{\mbox{vz}}}}=\frac{\hslash }{2}\sqrt{{\tau }_{{{\mbox{iv}}}}^{-1}{\tau }_{{{\mbox{sym}}}\,}^{-1}}.$$

The strength of the valley–Zeeman coupling cannot be extracted reliably in our experiments due to the uncertainty in τsym. The mean values from the fit show a slightly increasing trend and would give 4 and 23 eV for the lowest and highest pressure, respectively, but even taking the smallest obtained values from τsym, we arrive at values between 140 and 440 eV.

In order to quantify changes in the Rashba parameter and to compare it to expectations, we have performed ab initio calculations following our previous results in ref. 20,21. Using a structural supercell model of 4 × 4 graphene and 3 × 3 WSe2 with relaxed lateral atomic positions, 5%/GPa compressibility for the layer distance between the graphene and WSe2 layers was found. The applied 1.8 GPa pressure induces 9% compression in the z direction that leads to an increment of the calculated Rashba energy from 600 eV to 1.8 meV and of the calculated valley–Zeeman energy from 1.2 to 3.0 meV.

As plotted in Fig. 4e, the value of the Rashba energy λR increases from 0.3 to 0.5 meV, which is in the order of magnitude of the expected values, although slightly lower than them. The values of the valley–Zeeman energy are much smaller than expected theoretically.

Several reasons might be behind these low SOC values relative to the theoretical expectations originating from the imperfection of the stacking process. One possibility is that the interfaces are not clean enough and some contamination is trapped in-between, see Supplementary Fig. 8, which would lower the strength of the SOC. Another difference from other samples31 might come from the relative orientation of the graphene and the WSe2. It has been theoretically found42,43 that the rotation angle strongly modulates the strength of the SOC and for certain angles the SOC almost disappears. Since we did not control the twist angles between the layers during fabrication, they may have aligned such a way that it would lead to reduced SOC. The rotation angle affects the Rashba and the valley–Zeeman coupling in a different way, which can explain why we see stronger suppression for the λVZ than for λR, and why the former deviated between measured and simulated values. Finally, we note that the strength of SOC obtained from these formulas might also depend on how well the UCF is removed during averaging. This can be seen in Fig. 4b, where the different gate values lead to slightly different magneto-conductance trend. We stress here that although the absolute value of the extracted parameters has to be taken with care, the tendency visible in Fig. 4a clearly shows that the pressure indeed leads to an increase of the SOC. We neither expect that the twist angle is modulated by the applied pressure44 nor the functional form of the angle dependence is modified by it, only the overall strength of the SOC is affected.

In conclusion, we studied the effect of hydrostatic pressure on WL signal in graphene/TMDC heterostructure. We demonstrated the enhancement of the proximity-induced SOC using hydrostatic pressure. Our analysis of the measured signals supports an increasing SOC and is in qualitative agreement with theoretical expectations. The strength of SOC is an important parameter in graphene spintronics, since it determines charge to spin conversion efficiencies and also plays a central role in graphene–TMDC optoelectronic devices20. Our work also points out that hydrostatic pressure can be generally used to change the interlayer distance up to a remarkable 10%, which provides an experimental knob to boost proximity effects in vdW heterostructures, e.g., exchange interaction induced into graphene45,46,47,48 or into TDMCs49,50, furthermore, a strong proximity effect on correlated magic angle twistronics devices51,52, or to stabilize fragile states such as the topological insulator phase in graphene–TDMC heterostructures33.

## Methods

### Device fabrication

The heterostructure is built on a Si/SiO2 substrate using the dry stacking assembly method53, where the heavily doped silicon layer was used as a global back gate. The WSe2 crystal is from HQ Graphene, whereas the hBN crystal was grown at the National Institute for Materials Science, Japan. It is then shaped into a Hall bar and contacted using 1D Cr/Au electrodes54 using electron beam lithography and reactive ion etching (see Fig. 2 inset). The total length of the graphene segment was L = 8.3 m and the width was W = 1.1 m.

### Sample preparation

The wafer carrying the device was cut tightly and bonded on a special high-pressure sample holder, then placed in kerosene environment in a piston-cylinder hydrostatic pressure cell. The setup is designed to overcome the technical difficulties of electronic measurements of nanocircuits in a hostile environment37. Measurements were carried out by standard lock-in technique at f = 177 Hz with an AC bias voltage VAC = 100 V and an external low-noise current amplifier.