Chemical physics – Carbon Chemist

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Strong-field phenomena play an important part in our understanding of the quantum world. Light–matter interactions beyond the perturbative limit can substantially distort the energy landscape of a quantum system, which forms the basis of many strong-field effects⁸ and provides opportunities for efficient quantum control schemes¹¹. Moreover, resonant strong coupling induces rapid Rabi cycling of the level populations¹², enabling complete population transfer to a target state². The development of intense extreme ultraviolet (XUV) and X-ray light sources has recently led to the investigation of related phenomena beyond valence electron dynamics, in highly excited, multi-electron and inner-shell electron states^{9,10,13,14,15,16,17}. Yet in most of these studies, the dressing of the quantum systems was induced by intense infrared fields overlapping with the XUV and X-ray pulses. In contrast, the alteration of energy levels directly by short-wavelength radiation is more difficult. So far, only a few studies have reported XUV-induced AC-Stark shifts of moderate magnitude (≲100 meV), difficult to resolve experimentally^9,18,19,20.

Another important step in exploring and mastering the quantum world is the active control of quantum dynamics with tailored light fields^21,22,23. At long wavelengths, sophisticated pulse-shaping techniques facilitate the precise quantum control and even the adaptive-feedback control of many light-induced processes, in both weak- and strong-field regimes^{24,25,26,27,28}. Several theoretical studies have pointed out the potential of pulse shaping in XUV and X-ray experiments^29,30,31. As an experimental step in this direction, phase-locked monochromatic and polychromatic pulse sequences have been generated^32,33,34,35. Using this tool, coherent control demonstrations in the perturbative limit^32,35,36 and the generation of intense attosecond pulses were achieved³⁷. Moreover, ultrafast polarization shaping at XUV wavelengths³⁸ and chirp control for the temporal compression of XUV pulses³⁹ were recently demonstrated. However, spectral phase shaping, which forms the core of pulse-shaping techniques, has not been demonstrated for the control of quantum phenomena at short wavelengths. Here we establish spectral phase shaping of intense XUV laser pulses and demonstrate high-fidelity quantum control of the Rabi and photoionization dynamics in helium.

In the experiment, He atoms are dressed and ionized by intense coherent XUV pulses (I > 10¹⁴ W cm⁻²) delivered by the seeded FEL FERMI (Fig. 1a). The high radiation intensity causes a strong dressing of both the bound states in He and the photoelectron continuum, whereas the dynamics of the quantum system are still in the multiphoton regime (Keldysh parameter γ = 11). By contrast, the dynamics of a system dressed with near-infrared (NIR) radiation of comparable intensity would be dominated by tunnel and above barrier ionization (γ = 0.35) (ref. ⁸). Hence, the use of short-wavelength radiation provides access to a unique regime, in which the interplay between strongly dressed bound states and a strongly dressed continuum can be studied.

**Fig. 1: XUV strong-field coherent control scheme.**

To dress the He atoms, we induce rapid Rabi cycling of the 1s² → 1s2p atomic resonance with a near-resonant field E(t). The generalized Rabi frequency of this process is $\varOmega ={\hbar }^{-1}\sqrt{{(\mu E)}^{2}+{\delta }^{2}}$, where μ denotes the transition dipole moment of the atomic resonance, δ the energy detuning and $\hbar $ the reduced Planck constant. In the dressed-state formalism, the eigenenergies of the bound states depend on the field intensity and show the characteristic Autler–Townes (AT) energy splitting ΔE = ħΩ (ref. ⁴⁰). The observation of this phenomenon requires the mapping of the transiently dressed level structure of He while perturbed by the external field⁴¹. This is achieved by immediate photoionization over the course of the femtosecond pulses, thus projecting the time-integrated energy level shifts onto the electron kinetic energy (eKE) distribution (Fig. 1b).

Analogous to the bound-state description, the dressed continuum states are obtained by diagonalization of the corresponding Hamiltonian. The hybrid electron–photon eigenstates consist of a mixing of partial waves with different angular momenta, which alters the coupling strength to the dressed bound states of the He atoms (Fig. 1a).

Figure 2 demonstrates experimentally the dressing of the He atoms. The build-up of the AT doublet is visible in the raw photoelectron spectra as the XUV intensity increases (Fig. 2a). The evolution of the AT doublet splitting is in good agreement with the expected square-root dependence on the XUV intensity $\Delta E=\mu \sqrt{2{I}_{{\rm{eff}}}/({{\epsilon }}_{0}c)}$. Here, I_eff denotes an effective peak intensity, accounting for the spatially averaged intensity distribution in the interaction volume, ϵ₀ denotes the vacuum permittivity and c denotes the speed of light. The data can be thus used for gauging the XUV intensity in the interaction volume, a parameter otherwise difficult to determine. At the maximum XUV intensity, the photoelectron spectrum shows an energy splitting exceeding 1 eV, indicative of substantial AC-Stark shifts in the atomic level structure. The large AT splitting further implies that a Rabi flopping within 2 fs is achieved, offering a perspective for rapid population transfer outpacing possible competing intra- and inter-atomic decay mechanisms, which are ubiquitous in XUV and X-ray applications.

**Fig. 2: Build-up of the AT splitting in He atoms.**

Figure 2b,c shows the photoelectron yield as a function of excitation photon energy. For high XUV intensity (Fig. 2b), the photoelectron spectra show an avoided level crossing of the dressed He states as they are mapped to the electron continuum (see also Fig. 4). Accordingly, at lower XUV intensity (Fig. 2c), the avoided crossing is not visible anymore. In the latter, the eKE distribution centres at 17.9 eV. In Fig. 2b, a similar contribution appears at the same kinetic energy that overlays the photoelectrons emitted from the strongly dressed atoms. Likewise, a notable portion of photoelectrons at eKE ≈ 17.9 eV in Fig. 2a does not show a discernible AT splitting. We conclude that a fraction of He atoms in the ionization volume are excited by much lower FEL intensity, which is consistent with the aberrated intensity profile of the FEL measured in the ionization volume (Extended Data Fig. 1). This overlapping lower intensity contribution does not influence the interpretation of the results in this work. For better visibility of the main features, we thus subtract this contribution from the data shown in Figs. 3 and 4.

**Fig. 3: Strong-field quantum control of dressed He populations.**

**Fig. 4: Energy-domain representation of the quantum control scheme.**

The demonstrated dressing of He atoms provides the prerequisite for implementing the strong-field quantum control scheme (Fig. 1b,c). The main mechanism underlying the control scheme is described in the framework of the selective population of dressed states (SPODS), which is well established in the NIR spectral domain²⁸. Here, we extend SPODS to the XUV domain and include a new physical aspect—that is, the transition of the bound atomic system into a strongly dressed continuum. In SPODS, a flat phase leads to an equal population of both dressed states in the excited state manifold of helium; a positive phase curvature results in a predominant population of the lower dressed state and a negative phase curvature results in a predominant population of the upper dressed state (Fig. 1c). The scheme has been experimentally demonstrated with long-wavelength radiation⁴², in which pulse-shaping techniques are readily available. However, the opportunities for pulse-shaping technologies are largely unexplored for XUV and X-ray radiation.

We solve this problem by exploiting the potential of seeded FELs to allow for the accurate control of XUV pulse properties^39,43. These demonstrations have been so far limited to applications of temporal compression and amplification of the FEL pulses. By contrast, the deterministic control of quantum dynamics in a material system involves many more degrees of freedom, which makes the situation considerably more complex. The seeded FEL FERMI operation is based on the high-gain harmonic generation (HGHG) principle⁴⁴, in which the phase of an intense seed laser pulse is imprinted into a relativistic electron pulse to precondition the coherent XUV emission at harmonics of the seed laser (Fig. 1d). For FEL operation in the linear amplification regime, the phase ϕ_nH(t) of the FEL pulses emitted at the n’th harmonic of the seed laser follows the relationship³⁹

$${\phi }_{n{\rm{H}}}(t)\approx n[{\phi }_{{\rm{s}}}(t)+{\phi }_{{\rm{e}}}(t)]+{\phi }_{{\rm{a}}}.$$

(1)

Here, ϕ_s denotes the phase of the seed laser pulses, which can be tuned with standard pulse-shaping technology at long wavelengths (Methods); ϕ_e accounts for the possible phase shifts caused by the energy dispersion of the electron beam through the dispersive magnet and is negligible for the parameters used in the experiment; and ϕ_a accounts for the FEL phase distortion due to the amplification and saturation in the radiator and has been kept negligibly small by properly tuning the FEL (Methods). Although complex phase shapes may be implemented with this scheme, for the current objective of controlling the strong-field induced dynamics in He atoms, shaping the quadratic phase term (group delay dispersion (GDD)) is sufficient⁴². Therefore, we focus on the GDD control in the following discussion.

Figure 3 demonstrates the quantum control of the dressed He populations. The eKE distribution shows a pronounced dependence on the GDD of the XUV pulses (Fig. 3a). At minimum chirp (GDD = 135 fs²), we observe an almost even amplitude in the AT doublet, whereas for GDD < 0, the higher energy photoelectron band dominates; for GDD > 0, the situation is reversed. These changes directly reflect the control of the relative populations in the upper and lower dressed states of the He atoms. We obtain an excellent control contrast and the results are highly robust (Extended Data Fig. 2), which is remarkable given the complex experimental setup.

The experiment is in good agreement with the theoretical model (Fig. 3b) numerically solving the time-dependent Schrödinger equation for a single active electron (TDSE-SAE; Methods). To account for experimental broadening effects, we calculated the photoelectron spectra for a single intensity (corresponding to the experimental I_eff) and including the focal intensity average present in the experiment (Methods). All salient features of the experiment are well reproduced. The control of the dressed-state populations is in very good qualitative agreement. The different widths and shapes of the photoelectron peaks are qualitatively well-matched between the experiment and the calculations. The difference in the AT energy splitting between the experiment (ΔE_exp ≈ 1.02 eV) and theory (ΔE_theo = 0.74 eV) is in good agreement with the fact that the model underestimates the transition dipole moment of the 1s² → 1s2p transition by a factor of 1.4 (Methods).

The high reproducibility, the excellent control contrast and the good agreement with theory confirm the feasibility of precise pulse shaping in the XUV domain and of quantum control applications, even of transient strong-field phenomena. This is an important achievement in view of quantum optimal control applications at short wavelengths.

The implemented control scheme is not restricted to adiabatic processes²⁸. In our experiment, the dynamics are adiabatic only for the largest frequency chirp (GDD = −1,127 fs²) (Extended Data Fig. 3). However, this also shows that the condition for rapid adiabatic passage² can be generally reached with our approach, offering a perspective on efficient population transfer in the XUV and potentially in the soft X-ray regime.

The active control of quantum dynamics with tailored light fields is an asset of pulse shaping. As another asset, systematic studies with shaped laser pulses can be used to uncover underlying physical mechanisms that are otherwise hidden. Here, we demonstrate this concept for pulse shaping in the XUV domain. The high XUV intensities used in our study lead to a peculiar scenario in which both bound and continuum states are dressed and a complex interplay between their dynamics arises. Hence, for a comprehensive understanding of the strong-field physics taking place, the bound-state dynamics and the non-perturbative photoionization have to be considered. This is in contrast to the strong-field control at long wavelengths, for which the continuum could be described perturbatively⁴².

Figure 4a,b shows the avoided crossing of the photoelectron bands for different spectral phase curvatures applied to the XUV pulses. The experimental data show a clear dependence of the AT doublet amplitudes on the detuning and the GDD of the driving field, in good agreement with the theory. In the strong dressing regime, the bound–continuum coupling marks a third factor that influences the photoelectron spectrum. As predicted by theory, the strong-field-induced mixing of continuum states (Fig. 1a) leads to different photoionization probabilities for the upper and lower dressed states of the bound system⁴⁵. This is in agreement with the prevalent asymmetry of the AT doublet amplitudes observed in our data and calculations (Fig. 4a,b). An analogous effect is observed for the strong-field bound–continuum coupling in solid state systems⁴⁶.

To disentangle this strong-field effect from the influence of the detuning and spectral phase of the driving field, we evaluate the amplitude ratio between the upper and lower photoelectron bands at detuning δ = 0 eV (Fig. 4c). Interpolation to GDD = 0 fs² isolates the asymmetry solely caused by the strong-field bound–continuum coupling. We find reasonable agreement with our model when including the dressing of the ionization continuum (blue curve), in stark contrast to the same model but treating the continuum perturbatively (yellow curve). Hence, the dressing of the He atoms provides a probe of the strong-field dynamics in the continuum. This property is otherwise difficult to access and becomes available through our systematic study of the spectral phase dependence on the photoelectron spectrum.

Another possible mechanism for a general asymmetry in the AT doublet amplitudes could be the interference between ionization pathways through resonant and near-resonant bound states as recently suggested for the dressing of He atoms with XUV^20,47 and for alkali atoms with bichromatic NIR fields⁴⁸. In our experiment, we study the energetically well-isolated transition 1s² → 1s2p, in which the contributions from neighbouring optically active states should be negligible. This provides us with a clean two-level system and greatly simplifies the data interpretation. For confirmation, we performed a calculation with a modified model in which any two-photon ionization through near-resonant states (except for the 1s2p state) was suppressed and, thus, possible photoionization interference effects were eliminated. Still, we observe a pronounced asymmetry in the AT doublet amplitudes (Extended Data Fig. 4). Moreover, owing to the large Keldysh parameter (γ = 11) and the low ponderomotive potential (U_p < 100 meV) in our study, other strong-field effects are expected to play a negligible part in the observed dynamics. We thus assign the experimental observation to the coupling of the dressed atom dynamics with a dressed ionization continuum induced by intense XUV driving fields.

A comprehensive understanding of the strong-field-induced dynamics in the system lays the basis for another quantum control effect, that is, the suppression of the ionization rate of the system, as proposed theoretically⁴⁵. The excitation probability for one-photon transitions is generally independent of the chirp direction of the driving field. However, if driving a quantum system in the strong-field limit, its quasi-resonant two-photon ionization rate may become sensitive to the chirp direction. We demonstrate the effect experimentally in Fig. 4d. A substantial reduction of the He ionization rate by 64% is achieved, solely by tuning the chirp of the FEL pulses while keeping the pulse area constant. The good agreement with the TDSE-SAE calculations confirms the mechanism. This control scheme exploits the interplay between the bound-state dynamics and the above-discussed selective coupling of the upper and lower dressed states to the ionization continuum. We note a stabilization mechanism of the dressed states in He was recently proposed, effectively causing also a suppression of the ionization rate⁴⁷. This mechanism requires, however, extreme pulse parameters, difficult to achieve experimentally. By contrast, our approach based on shaped pulses is more feasible and applies to a broader parameter range.

With this work, we have established a new tool for the manipulation and control of matter using XUV light sources. The demonstrated concept offers a wide pulse shaping window regarding pulse duration, photon energy and more complex phase shapes. In particular, the recent progress in echo-enabled harmonic generation^49,50 promises to extend the pulse-shaping concept to the soft X-ray domain (up to the 600 eV range) in which localized core electron states can be addressed. As such, we expect our work will stimulate other experimental and theoretical activities exploring the exciting possibilities offered by XUV and soft X-ray pulse shaping: first theory proposals in this direction have already been made^29,30,31. The demonstrated scheme already sets the basis for highly efficient adiabatic population transfer^1,2 and an extension to cubic or sinusoidal phase shaping would open up many more interesting control schemes^26,27. This may find applications, for example, in valence-core-stimulated Raman scattering or efficient and fast qubit manipulation with XUV and soft X-ray light. Furthermore, selective control schemes may reduce the influence of competing ionization processes ubiquitous in XUV and X-ray spectroscopy and imaging experiments, for which our work provides experimental demonstration. The generation of coherent attosecond pulse trains, with independent control of amplitude and phases, has been demonstrated at seeded FELs³⁷, bringing pulse shaping applications on the attosecond time scale within reach. This paves the way for the quantum control of molecular and solid state systems with chemical selectivity and on attosecond time scales.

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Control of proton transport and hydrogenation in double-gated graphene

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Device fabrication

Apertures 10 µm in diameter were etched in silicon nitride substrates (500 nm SiN_x) using photolithography, wet etching and reactive ion etching, as previously reported¹. Source and drain electrodes (Au/Cr) were patterned using photolithography and electron-beam evaporation. Mechanically exfoliated monolayer and bilayer graphene crystals were transferred over the apertures and on the electrodes (Extended Data Fig. 1a). We selected crystal flakes with a rectangular shape, with their long side being several tens of micrometres, to form a conducting channel between the source and drain electrodes. The width of the flake was chosen to be only a couple of micrometres wider than the aperture (Extended Data Fig. 1b), which ensured that a whole cross-section of the flake could be gated with the two gates effectively. The cross-section of the flake became the active area of the device, with the non-gated areas acting as electrical contacts to the gated section. An SU-8 photo-curable epoxy washer with a hole 15 μm in diameter was transferred over the flake and on the source and drain electrodes³ with the hole in the washer aligned with the aperture in the silicon nitride substrate (Extended Data Fig. 1b). The polymer seal ensured that the electrodes were electrically insulated from the electrolyte. The electrolyte used was 0.18 M HTFSI dissolved in polyethylene glycol (number-averaged molecular mass, M_n, of 600)¹⁴, never exposed to ambient conditions. This electrolyte was drop-cast on both sides of the device in a glovebox containing an inert gas atmosphere. For reference, we measured devices prepared in the same way, except HTFSI was substituted for LiTFSI in the electrolyte. Palladium hydride foils (around 0.5 cm²) were used as gate electrodes. The device was placed inside a gas-tight Linkam chamber (HFS600E-PB4) filled with argon for electrical measurements.

Transport measurements

For electrical measurements, a dual-channel Keithley 2636B sourcemeter was used to bias both gates. The applied voltages yielded two proton current signals, top (I_t) and bottom (I_b) channel current, also recorded with a Keithley sourcemeter. These two signals quantified the two halves of the proton-transport circuit: proton transport from one gate electrode towards graphene, and then from graphene towards the other gate electrode. Because of the proton permeability of graphene, these two currents were effectively identical (Extended Data Fig. 3), differing by only about 1%. For this reason, it is sufficient to use only one of them to unambiguously characterize proton transport in the device, I. A second Keithley 2614 A sourcemeter was used both to apply drain-source bias (V_ds) and to measure the electronic conductance (σ_e).

To confirm that the two gates are independent, we connected the electrolyte in the top channel with a reference electrode (PdHx foil, the same size as the gate electrodes), and monitored its potential, V_t^ref, with a Keithley 2182 A nanovoltmeter (Extended Data Fig. 2a). In the first experiment, we swept V_b for various fixed V_t. Extended Data Fig. 2b shows that V_t^ref = V_t for all values of V_b within the experimental scatter of less than 4 mV. This demonstrates that sweeping V_b does not affect V_t. In the second experiment, we swept V_t for various fixed V_b. This measurement also showed that V_t^ref = V_t (Extended Data Fig. 2c), which demonstrates that a fixed V_b does not affect V_t either. These experiments therefore demonstrate that the gates are independent from each other.

To obtain maps of the proton and electronic systems, we measured I and σ_e simultaneously as a function of V_t and V_b. The maps were obtained using software that allowed us to control V_t − V_b and V_t + V_b as independent variables. We swept V_t − V_b (for a fixed V_t + V_b) at a rate of 10 mV s⁻¹ for each gate and stepped V_t + V_b with intervals of 10 mV. The maximum V_t or V_b applied was ±1.4 V, which resulted in a maximum V_t + V_b and V_t − V_b of ±2.8 V. We normally did not apply V bias beyond these voltage ranges to avoid damaging the devices.

Raman spectroscopy

For Raman measurements, the graphene devices were left in the same gas-tight Linkam chamber (HFS600E-PB4) used for electrical measurements. It has an optical window. The Raman spectra of devices were measured as a function of applied V bias using a 514 nm laser. The background signal from the electrolyte was removed for clarity, resulting in relatively weak Raman spectra for pristine graphene (Extended Data Fig. 5). After hydrogenation, a strong D peak appeared and the intensity of the G peak increased while the 2D peak became broader, in agreement with previous work¹⁴. The density of adsorbed hydrogen atoms in hydrogenated graphene was estimated from the ratio of peak-height intensities of the D and G bands^40,41, I_D/I_G. In our devices, I_D/I_G ≈ 1, which corresponds to a distance between hydrogen atoms in graphene of L_D ≈ 1 nm. An equivalent analysis using the integrated-area ratio in these peaks⁴², A_D/A_G ≈ 2, yields L_D < 1.2 nm. Both estimates yield a density of hydrogen atoms of around 1 × 10¹⁴ cm⁻², in agreement with previous reports on hydrogenated graphene^13,14.

Electrolyte characterization

To characterize the limiting conductivity of our devices, we measured devices similar to those described above but in which the aperture in the silicon nitride substrate was not covered with graphene (‘open hole device’). Extended Data Fig. 9 shows that the I–V characteristics of these open-hole devices were linear in all the V-bias range used in this work. This demonstrates that the field effect we observed in graphene devices did not arise from changes in the electrolyte conductivity at high V, consistent with the known large electrochemical window of this electrolyte (4–5 V; ref. ⁴³).

To characterize the capacitance of the electrolyte, we patterned two gold electrodes on a silicon nitride substrate using photolithography and electron-beam evaporation. The electrodes were connected in an electrical circuit and a polymer mask was used to cover all the electrodes, except for an active area that was exposed to the electrolyte. The area of the electrodes differed by a factor of around 50, which ensured that the total capacitance was dominated by the smaller one and allowed us to observe differences, where present, in the response of the devices under positive and negative potentials⁴⁴. Cyclic voltammetry (CV) measurements with scan speeds in the range 1–40 mV s⁻¹ were performed over the voltage range −0.1 V to 0.1 V. Extended Data Fig. 6a shows that the CV curves displayed no redox peaks or asymmetry between the positive and negative voltage branches. The area-normalized capacitance of the electrolyte, C, could then be obtained from the CV curves from the expression^44,45 C = (A × ΔV × ν )⁻¹ ∫ I dV, where A is the active area of the electrode, ΔV is the voltage range in the CV, I is the current and v is the scan speed. Extended Data Fig. 6b shows the extracted C as a function of v. For the smallest scan rate (1 mV s⁻¹), we found C ≈ 30 µF cm⁻². This value decreases with v increases, as expected. Because our measurements use v = 10 mV s⁻¹, we used the value obtained at such v, C ≈ 20 µF cm⁻², in our estimates involving C.

Estimation of E and n

The Debye length in our electrolyte (0.18 M salt and solvent dielectric constant, ε_r ≈ 10)^46,47 can be estimated as λ_D ≈ 0.3 nm. Given this and the relatively large gate bias used in this work, the electrical potential across the graphene–electrolyte interface in our devices dropped almost entirely across the Stern layer. Hence, each of the gate potentials can be described using a parallel plate capacitor model, which we used to derive the relations between the gate potentials and E and n, as shown in refs. ^{25,26,27,48,49}.

To derive the relation between V_t + V_b and n, we note that if only one gate (top) operates, the charge induced is neC⁻¹ = (V_t − V_t^NP), where the superscript NP marks the neutrality point and e is the elementary charge constant. An equivalent relation holds for the top gate. Hence, the total charge from both gates is given by the addition of their contributions: neC⁻¹ = (V_t + V_b) − Δ^NP, where Δ^NP ≡ V_t^NP + V_b^NP. To consider the quantum capacitance of graphene, we note that ${\mu }_{e}={\hbar v}_{{\rm{F}}}\,\sqrt{{\rm{\pi }}n}$, where v_F ≈ 1 × 10⁶ m s⁻¹ is the Fermi velocity in graphene and ħ is the reduced Planck constant. This changes the relation to⁵⁰:

$$\left({V}_{{\rm{t}}}+{V}_{{\rm{b}}}\right)-{\Delta }^{{\rm{NP}}}={neC}^{-1}+{\hbar v}_{{\rm{F}}}{e}^{-1}{\left(\pi n\right)}^{1/2}$$

(1)

From the estimate of C above, we get (V_t + V_b) − Δ^NP ≈ (0.8 × 10⁻¹⁴ V cm²) n + (1.16 × 10⁻⁷ V cm) n^1/2. Note that this description is accurate only if the Fermi energy of the system is outside a bandgap³¹. Hence, we use it only when graphene is conductive. After the hydrogenation transition, we cannot assess n, as indicated by the breaks in the top axes in Fig. 2.

To derive the relation between V_t − V_b and E, we note that if only one gate (top) operates, the electric field induced by the gate is ${E}_{{\rm{t}}}={en}_{{\rm{t}}}{(2\varepsilon )}^{-1}={C(2\varepsilon )}^{-1}({V}_{{\rm{t}}}-{V}_{{\rm{t}}}^{{\rm{NP}}})$, where ε is the dielectric constant of the solvent. Note that the electric field points in the direction between graphene and its corresponding electrical double layer, which we define as +x for the top gate. An equivalent relation holds for the bottom gate, except that E_b points in the −x direction (towards its corresponding electrical double layer). The total electric field in graphene is then:

$$E={E}_{{\rm{t}}}-{E}_{{\rm{b}}}={C(2\varepsilon )}^{-1}\left({V}_{{\rm{t}}}-{V}_{{\rm{b}}}\right).$$

(2)

This yields E ≈ 1.13 × 10⁹ m⁻¹ (V_t − V_b).

Analytical model of proton transport and hydrogenation in double-gated graphene

We used an analytical model to illustrate how the gate voltages affect proton transport and hydrogenation in double-gated graphene. The energy barrier for proton transport through the centre of the hexagonal ring in graphene is modelled using a Gaussian function: V_p = G₀ × exp(−x/W)², where G₀ = 0.8 eV is the barrier height determined experimentally in the low-electric-field limit¹ and W = 0.5 Å is the barrier width (Extended Data Fig. 7b). For the hydrogenation process, the proton is directed towards the top of a carbon atom in graphene.

The potential energy profile for the hydrogenation process consists of two parts: the energy barrier (V_Hb) and the adsorption well (V_Ha). The energy barrier is modelled with a Lorentzian-type function: V_Hb = V₀ [((x − |x₀|)/d₀)³ + 1]⁻¹, where V₀ = 0.2 eV is the barrier height, x₀ = 1.7 Å is the distance between the barrier and graphene, and d₀ = 0.4 Å is the barrier width. The third power in the denominator models long-range van der Waals interactions. The adsorption well is modelled with a Lorentzian: V_Ha = V₁ [((x − |x₁|)/d₁)² + 1]⁻¹, where V₁ = −0.8 eV is the well depth, x₁ = 1.1 Å is the distance between the well and graphene, and d₁ = 0.25 Å is the width of the well. Note that the well is modelled to be strongly repulsive at x = 0 to capture the repulsion between the carbon and hydrogen atoms at very short distances. The parameters for these functions are taken from DFT calculations^14,51 and the total potential for the hydrogenation process is then V_Hb + V_Ha (Extended Data Fig. 7a).

The gate potential profiles (V_t and V_b) are modelled with a Guoy–Chapman–Stern model²⁴, using dielectric constant ε_r = 10 for the solvent, an electrolyte concentration of 0.18 M and a Stern-layer thickness of 0.4 nm. The resulting gate potentials drop almost exclusively over the Stern layer (Extended Data Fig. 7) and, as a result, the graphene–electrolyte interface behaves as a capacitor, as discussed above. The qualitative findings of our model are relatively insensitive to the specific parameters of the Guoy–Chapman–Stern model if the Stern layer exceeds 0.3 nm. The superposition of the potentials for each of the processes with the gate potentials model the behaviour of the devices.

To illustrate the role of the gates in the hydrogenation process, we set them to yield n = 1.2 × 10¹⁴ cm⁻² but E = 0. Extended Data Fig. 7a shows that this distorts the potential energy profile for hydrogenation, such that the hydrogenation barrier is now easily surmounted by incoming protons, which become trapped in the adsorption well and hydrogenate graphene. To illustrate the role of E in the proton-transport process, we set the gates to produce large E = 1.7 V nm⁻¹ but n = 0 cm⁻². Extended Data Fig. 7b shows that this distorts the potential energy profile for proton transport, such that the barrier is now easily surmounted by a proton moving in the direction of the electric field (from the −x to the +x). To illustrate the role of electron doping in proton transport, we set the gates to give large n = 1 × 10¹⁴ cm⁻² but E = 0.67 V nm⁻¹. Extended Data Fig. 7c shows that this also distorts the potential energy profile for the incoming protons, resulting in facilitated transmission over the barrier. The model illustrates that the distortion of the energy profile for incoming protons due to E and n in these devices is comparable to the barrier height. For this reason, these variables dominate the response of the devices, and previously identified effects, such as strain and curvature, should have a secondary role.

Electrochemical description of the hydrogenation process

The transport data are described using the variables E and n. However, it is equivalent to describe the system using the electrochemical potential of electrons in graphene with respect to the NP, μ_e, instead of n. Indeed, one important property of graphene is that n and µ_e are related by the formula ${\mu }_{e}={\hbar v}_{{\rm{F}}}\,\sqrt{{\rm{\pi }}n}$, where v_F ≈ 1 × 10⁶ m s⁻¹ is the Fermi velocity in graphene. This relation is fundamental, arising from the density of states in the material, and holds exactly in experimental systems^52,53. Moreover, this relation is valid independently of whether the material is gated or not. Hence, the top x axis in the hydrogenation map in Fig. 2a can be re-expressed in terms of µ_e, which illustrates that the hydrogenation process is driven by μ_e. This is consistent with the well-established notion that electrochemical charge-transfer processes are driven by this variable. Note that although the relation between n and μ_e is fixed, applying a gate voltage to graphene shifts both variables^31,54. However, these variables are not independent, as discussed above. To determine their dependence on the gate voltage, we need to establish the electrostatic gate capacitance, C (Extended Data Fig. 6). When C is determined, the dependence of n (or μ_e) on the gate voltage is described by equation (1) above.

Hydrogenation transition

It is instructive to compare our results with previous work on plasma-hydrogenated graphene^13,55. In those earlier studies, plasma-hydrogenated graphene typically displayed around 100-times higher electronic resistivity than in the non-hydrogenated state. By contrast, in our work and in ref. ¹⁴, this factor is about 10⁴, yielding an insulating state that was mostly insensitive to the gate voltage. There are at least two possibilities for this difference. The first is that the hydrogen-atom densities obtained by the different methods are different. Indeed, although the Raman spectra of the current work and ref. ¹³ yield I_D/I_G ≈ 1, this could arise from a hydrogen-atom density of less than 10¹² cm⁻² or around 10¹⁴ cm⁻², because of the bell-shape^40,41 of the graph of I_D/I_G against defect density. To decide which one applies, it is therefore necessary to look for further evidence of disorder in the spectra. The Raman spectra in ref. ¹³ displayed a sharp 2D band, which is typical of ordered samples and indicates that the hydrogen density was likely to be less than 10¹² cm⁻². This contrasts with the 2D band in our spectra, which is smeared, consistent with a figure of around 10¹⁴ cm⁻². The second possibility is that both systems had the same hydrogen density. In this case, the higher resistivity could arise from a more disordered hydrogen-atom distribution. Indeed, hydrogen atoms in plasma-hydrogenated graphene are known to cluster^33,56, which reduces the number of effective scattering centres proportionally to the number of atoms in the cluster. The reduction could be considerable because the scattering radius around each hydrogen atom extends to second neighbours (nine carbon atoms)^33,56,57. The electrochemical system could be less prone to clusters, perhaps because the electrolyte stabilizes the proton as it adsorbs on graphene, making the reaction more likely to happen than in a vacuum, thus yielding a more random distribution.

Another difference between the two hydrogenation methods is their reversibility. According to ref. ¹³, the plasma-hydrogenation process could be almost completely reversed by annealing the material in an argon atmosphere. However, a D band was still notable after annealing and some of the electronic properties of graphene were not fully recovered¹³. This imperfect reversibility was attributed¹³ to the presence of vacancy defects introduced during the plasma exposure. In both our work and in ref. ¹⁴, the hydrogenated transition is fully reversible, with no D peak apparent in the Raman spectra of dehydrogenated samples.

Another difference with plasma-hydrogenated samples is that electrochemical hydrogenation allows the dependence of the transition on n to be studied. This has revealed that the transition is sharp. We attribute this sharpness to a percolation-type transition⁵⁸ triggered both by the high density of adsorbed hydrogen atoms in the samples and to the carrier scattering associated with them^59,60. We propose that the insulating state in the samples is therefore a consequence of their high disorder, as suggested previously^59,60, rather than a bandgap. This is consistent with experimental studies reporting that a bandgap in plasma-hydrogenated samples typically requires either the patterned distribution of hydrogen atoms⁶¹ or a much higher hydrogen-atom density⁶² than in the samples in this work.

DFT calculations of graphene hydrogenation

Graphene hydrogenation was simulated using the Vienna Ab initio Simulation Package (VASP)^63,64,65,66. Electron–ion interactions were modelled using the projector augmented wave method, and the exchange correlations of electrons were modelled with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation functional⁶⁷. Spin polarization was considered and the van der Waals interactions were incorporated by using the Grimme’s DFT-D3 method⁶⁸. Initial crystal-structure relaxation was performed with a force criterion of 0.005 eV Å⁻¹ and an electronic convergence of 10⁻⁶ eV, accelerated with a Gaussian smearing of 0.05 eV. The energy cut-off was set at 500 eV, and Monkhorst–Pack k-point mesh with a reciprocal spacing of 2π × 0.025 Å⁻¹ was implemented, which ensured energy convergence to 1 meV. We constructed a cubic simulation, consisting of a 4 × 7 orthogonal supercell with 112 carbon atoms placed at the centre in the z direction (perpendicular to the 2D plane) and with a vacuum slab to prevent interactions between adjacent periodic images. After relaxation, the energy barriers for a proton to be adsorbed on top of a carbon atom under vacuum conditions were calculated by ab initio molecular-dynamics simulations using the microcanonical ensemble and the same convergence criteria as mentioned above. We used a time step of 0.1 fs and a minimal initial kinetic energy for the proton in the direction perpendicular to the 2D layer, as previously reported^1,69. A dipole correction was implemented to study the influence of an external electric field perpendicular to the 2D layer (in the z direction)⁷⁰. Owing to the periodic boundary conditions, this dipole is repeatedly inserted in all the simulation boxes in the z direction, yielding a constant electric field in the direction perpendicular to graphene⁷⁰.

Extended Data Fig. 8 shows the calculated potential energy curves for the proton–graphene system. The curves were calculated as a function of distance between the proton and the top of a carbon-atom site with a fully relaxed lattice. The potential energy curves display a minimum (adsorption well) at around 1.14 Å (C–H bond) and a small adsorption barrier around 2 Å, in agreement with previous studies¹⁴. We find the electric field distorts the potential energy profile for hydrogenation, favouring the process in agreement with the analytical model. For reference, we also performed calculations using the non-local optB88-vdW and the hybrid functional HSE06. These resulted only in minor differences (<0.1 eV) in the hydrogenation-barrier height compared with PBE.

DFT calculations of proton transport through graphene

The DFT calculations of proton transport through graphene were performed using VASP^63,64,65,66 and the plane-wave self-consistent field (PWscf) package with Quantum Espresso (QE). We used the optB88-vdW⁷¹ functional, with a 3 × 3 × 3 Γ-centred k-point grid, a 1,000 eV energy cut-off with hard pseudopotentials^72,73, and a force-convergence criterion of 0.03 eV Å⁻¹. We used a 4 × 4 unit cell with a vacuum separating periodically repeating graphene sheets of 12 Å for pristine graphene and around 23 Å for hydrogenated graphene. The zero electric field energy profiles were computed using the climbing-image nudged elastic band method⁷⁴ with VASP. Charged cells were used to describe the protons in the simulations with a uniform compensating background. In the model, proton transfer was simulated from a water molecule on one side of graphene to another one on the opposite side. Using these two water molecules minimizes spurious charge transfer from the graphene sheet to the proton, as confirmed with a Bader⁷⁵ charge analysis. To incorporate the electric field, we modelled the system using QE. Here, we used the optB88-vdW functional^{71,76,77,78,79}, a 3 × 3 × 3 Γ-centred k-point grid and a 600 Ry energy cut-off. We confirmed that the VASP zero electric field energy barriers were reproduced within around 15 meV in QE. The electric field in QE was simulated as a saw-like potential added to the ionic potential, together with a dipole correction implemented according to ref. ⁸⁰. The saw-like potential increased in the region from 0.1 a₃ to 0.9 a₃, where a₃ is the lattice vector perpendicular to the graphene sheet, which was placed at the centre of the cell (0.5 a₃), then decreased to 0 at a₃ and 0. The discontinuity of the sawtooth potential was placed in the vacuum region. The electric field was applied in the perpendicular direction to the graphene basal plane (the z direction). For reference, we also performed calculations using the PBE-D3 functional, which gave comparable results.

We first calculated the energy profile for proton transport through graphene in the absence of an electric field and for two different levels of hydrogen-atom coverage of the lattice (0% and 20%). The choice of 20% hydrogenation was to take into account the fact that adsorbed hydrogen atoms typically form dimer structures consisting of two hydrogen atoms per eight-carbon-atom sublattice^33,56,81, which correspond to a local lattice coverage of about 25%. In agreement with ref. ¹⁸, we observed that the energy barrier for pristine graphene reduced by around 30% for 20% hydrogen-atom coverage. The barrier at zero field we found, Γ₀ ≈ 3.1–3.4 eV for the different functionals, is larger than the typically found values⁷ of Γ₀ ≈ 1–2 eV because in our approach the computed proton trajectory involved a chemisorption state, as described previously¹⁸. However, we note that the absolute values of the barriers in these simplified models are not especially informative, as discussed in ref. ⁶⁹. These models aim to provide only qualitative insights into the influence of E and hydrogenation in proton transport through graphene. Next, we computed the energy profiles along the same pathway used in the zero-E calculations, but now including a perpendicular electric field, E, along the direction of motion of the proton. Extended Data Fig. 10 shows the energy profiles along the reaction path for the two different levels of hydrogenation of the lattice for various electric fields. Regardless of the extent of hydrogenation, we observed a roughly linear barrier reduction when the electric field was switched on, achieving an approximately 20% reduction with E at around 1 V nm⁻¹.

Logic and memory measurements

For logic and memory measurements, we defined V_t and V_b as the IN1 and IN2 signals, respectively, and, guided by the maps of the devices, we systematically explored their proton and electronic responses to different input signals. To test the stability of the memory states as a function of time, the electronic system was pre-programmed into a conducting (dehydrogenated) or insulating (hydrogenated) state applying V_t + V_b = −2.8 V and +2.8 V, respectively. The retention of the insulating state was measured for more than a day with a constant IN1 = IN2 = 0 V, and a reading in-plane V_ds of 0.5 mV was applied for 20 s every 1,000 s. During logic-and-memory measurements, the electronic system was pre-programmed into a conducting or insulating state as described above. We then applied the input signals. The optimal parameters were found to be 0 V and +1.0 V for both IN1 and IN2 signals, because this yields high E but low n and thus enables strong modulation of the proton channel with minimum disruption of the electronic memory state. We found that in these measurements, the potentials at which graphene became hydrogenated were larger than in our transport maps. We attribute this to the fact that the fast sweeping of the gates may be altering the composition of the electrochemical double layer, probably resulting in lower concentrations of protons in the graphene–electrolyte interface and thus requiring higher potentials to hydrogenate graphene given the short timescales of this measurement. To implement the logic-and-memory application, the input signals were applied as a function of time in squared waveform patterns. Low and high gate voltages were defined as the logic inputs 0 and 1, respectively, yielding continuous cycles of different input combinations (00, 01, 11, 10).

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