The electronic thickness of graphene

Zero-density lines

The numerical conductance dG/ dV_tg as a function of V_tg and V_bg is shown in Fig. 2A for sample A and Fig. 2B for sample B. In both cases, two pronounced curved lines are observed, corresponding to a dip in the conductance G. The lines cross at zero gate voltages, and the splitting between these lines increases with increasing difference in V_tg and V_bg. One line (following the yellow dashed line) is affected more strongly by the top-gate voltage and therefore corresponds to the condition for charge neutrality in the upper graphene layer, whereas the other line (red dashed) indicates charge neutrality in the lower layer.

Analytical formulas for the zero-density lines [i.e., V_bg( V_tg)∣ and V_bg( V_tg)∣] can be calculated using the ideal density of states of defect-free graphene and are depicted in Fig. 2 (A and B) for the different electrostatic configurations (i.e., with or without hBN between the graphene sheets). The formulas and details of the calculation are given in the Supplementary Materials. Fitting these curves to the data allows us to extract C_m, which is the only free fitting parameter. The other capacitances in the problem are given by the thickness of the top and bottom hBN, i.e., C_tg = ϵ _hBN/ d_t with ϵ _hBN = 3.3ϵ ₀. A discussion for the precision of this method is given in the Supplementary Materials.

For sample A, we obtain an interlayer capacitance of C_m = 0.81 μF cm ⁻², which corresponds to the expected value for a plate separation of d = 3.5 nm and the hBN dielectric constant of ϵ _hBN = 3.3ϵ ₀. For sample B, we determine a large interlayer capacitance C_m = 7.5 ± 0.7 μF cm ⁻². This value is three times larger than the capacitance between two thin plates, separated by vacuum and an interlayer distance of , which is the expected distance between two graphene layers (, ). Consistent with our findings, large interlayer capacitance values have been reported in () in large perpendicular magnetic fields (quantum Hall regime) with a capacitance model that is only valid for n_t = − n_b. A detailed explanation for the large value of C_m has not been given so far.

The finite thickness of graphene

To understand the origin of such a large effective interlayer capacitance, we need to take into account the finite thickness of graphene, as this reduces the effective distance between the capacitor plates, leading to an enhanced interlayer capacitance. Therefore, we have estimated the extent of the p orbitals of carbon atoms in graphene from first-principles calculations (details are given in the Supplementary Materials). We calculated the integrated local density of states profile ρ( z) of single-layer graphene in the energy range E ∈ [−3, 3] eV from the charge neutrality point at = 0 eV. In this energy range, the bands are of pure p orbital character without contributions from the s-, p- and p-like orbitals). The calculated integrated local density of states ILDOS( z) as a function of distance from the center of the carbon atom is shown in Fig. 2D. From the charge distribution, we then calculated the expectation value of the position operator 〈 z〉 for one lobe of p orbital (positive z). The values are shown as black dashed lines in the figure. Since there is a substantial amount of charge at ∣ z∣ > 〈 z〉, we have to take into account the induced charge density Δρ = ρ( E = 0) − ρ( E) in an external electric field E, which determines the dielectric thickness of graphene (), defined as the distance from the center of carbon atoms to the point at which the dielectric constant of graphene ϵ = 6.9ϵ ₀ decays to the vacuum permittivity. The dielectric thickness is the relevant quantity if considering a single layer of graphene to be a nanocapacitor on its own. The dielectric thickness of graphene d_g is indicated by the blue shaded region in Fig. 2D, with values according to ().

To check whether twisted bilayer graphene (tBLG) displays a qualitatively different electrostatic behavior than AA- and AB-stacked BLGs, we performed first-principles calculations of tBLG with a twist angle of 22° (details of computations are given in the Supplementary Materials). In Fig. 2E, we show the comparison of the induced charge density Δρ( z) ≔ ρ(0) − ρ( E_z) for tBLG, AA BLG, and AB BLG under an external electric field E_z = 1 V/nm perpendicular to the BLG lattice. The interlayer distance of AA and AB BLGs was set to to fit the average distance between tBLG layers. Nevertheless, the results are representative and insensitive to small deviations of interlayer distance from the optimized value or to the choice of the dispersive correction due to van der Waals forces (see the Supplementary Materials). One can see that the responses of the different BLGs to the external electric field E_z are almost the same on the outer side of the BLG, while they are very different in the interlayer region. For , we observe a flattening of Δρ( z) in the case of tBLG compared to AA and AB BLGs. Within this region, the amplitude of Δρ( z) for tBLG is 15 times smaller than for AB BLG and 50 times smaller for AA BLG, demonstrating a qualitatively different electrostatic picture.

Our analysis is generally valid in the large angle regime (>5°). If the twist gets reduced, then the bands of the upper and lower layers start to hybridize at smaller energies, leading to a reduction of Fermi velocity and an increase of quantum capacitance. We expect to observe a stronger splitting in this case. For small twists, once the layers are coupled at low enough energies, there will be only one line in the gate-gate map at zero total density. Regarding the interlayer capacitance, in Fig. 2E, one can see that the interlayer charge distribution is different for AA-stacked graphene and tBLG. This indicates a modification of the interlayer capacitance toward smaller angles.

Decoupled Fabry-Pérot interferences

In the next step, we use a Fabry-Pérot interferometer to measure the layer density of sample B for arbitrary gate voltages and compare the results to tight-binding simulations based on an elaborate electrostatic model. The analysis of the Fabry-Pérot resonance pattern will allow us to determine the Fermi wavelength in the individual layers and will reveal that the graphene layers are indeed electronically decoupled. In Fig. 3 (A and B), we show dG/ dV_tg for top gates, sized L = 190 and 320 nm, respectively. For both cases, the cavity width W ≫ L. The zero-density lines are depicted in yellow for the top layer and dark red for the bottom layer.

The Fabry-Pérot resonator exhibits a pattern that can be qualitatively understood by considering the layer densities in the regions underneath and outside the top gate, as depicted in Fig. 3C. The density in the single-gated outer regions is affected only by V_bg. Since V_bg < 0, the outer regions are p-doped (blue colored). For small voltages, labeled (1) and (2) in Fig. 3 (A and C), the density of each of the two layers is comparable, i.e., there is only a small energy difference U between the two layers (see Fig. 3D). A p-n-p cavity below the top gate is formed for a sufficiently positive top-gate voltage (2) in both layers. Given a large energy difference between the layers, it becomes possible to create a p-n-p cavity in only one layer (3) or also in both (4).

As soon as a p-n-p cavity is formed, the conductance is modulated by standing waves, leading to the observed resonance pattern in Fig. 3 (A and B). In the inner region (3), only one set of Fabry-Pérot resonances, related to zero density in the upper layer, is observed. For densities beyond the zero-density line of the lower layer (dark red line in Fig. 3A), a more complex resonance pattern appears.

The resonance pattern is determined by the Fabry-Pérot condition, where the jth resonance is j = 2 L/λ _F = k_FL/π, where L is the cavity size and λ _F is the Fermi wavelength. Note that is given by the density in the top and bottom layer. As expected, we observe a finer spacing of the resonance pattern for the larger cavity ( Fig. 3B with L = 320 nm) as compared to the smaller cavity ( Fig. 3A with L = 190 nm). In the region between the zero-density lines, 6 resonances are observed at large U for L = 190 nm and even 10 resonances for L = 320 nm, i.e., it is possible to fill 10 modes in the upper resonator while there is still no cavity formed in the lower layer. By assuming that L is given by the lithographic size, it follows that λ _{F, bottom} = 640 nm and λ _{F, top} = 64 nm once the first mode fits into the cavity in the bottom layer at large U. Therefore, the wavelength can differ by an order of magnitude between two graphene layers despite the fact that those layers are atomically close.

In the measurement, especially for the larger cavity ( Fig. 3B), it can also be seen that the oscillation amplitude is largest for either small values of V_bg or close to the zero-density lines. Under these conditions, either the graphene part tuned only by V_bg or the cavity below the top gate is close to zero density; therefore, the density profile along the junction is especially flat, leading to a smooth transition between the cavity and the outer region. The enhanced oscillation amplitude can be understood by considering that smooth p-n interfaces act as strong angular filters (, ).

Simulation of density and transport

We now compare the resonance pattern to tight-binding simulations. The underlying density profiles n_t( x) and n_b( x) are obtained from a self-consistent electrostatic model where we assume that the dispersion relation remains linear, such that the carrier density formulas () derived for single-layer graphene with quantum capacitance (, ) taken into account can be readily applied. The extremely thin spacing between the two graphene layers leads to a notable electrostatic coupling. Effectively, the channel potential of the top layer plays the role as a gate for the bottom layer and vice versa. For the twisted bilayer sample B (see Fig. 4A), the electrostatic coupling between the layers is significant, as can be seen by comparing to the classical density profiles (dashed lines). In Fig. 4B, we calculate the interlayer energy difference U( V_tg, V_bg) for sample B. The maximum value that we can reach is U = 80 meV in our device. We note here that the formula given in (, ) for the displacement field [i.e., D = 1/2( C_tgV_tg − C_bgV_bg)] only holds under the condition n_t = − n_b. Apparently, lines of constant U (white lines in Fig. 4B) do not have a constant slope in the ( V_tg, V_bg) map. A more detailed comparison is given in the Supplementary Materials.

To see whether the electrostatic model is in agreement with the experiment, we perform transport simulations based on a real-space Green's function approach, considering two dual-gated, electronically decoupled graphene layers. To optimize the visibility of the Fabry-Pérot interference fringes, we implement periodic boundary hoppings along the transverse dimension (), equivalent to the assumption of infinitely wide graphene samples. This is justified since W ≫ L in our device. The normalized conductances g_t( V_tg, V_bg) and g_b( V_tg, V_bg) for the top and bottom graphene layers, respectively, are calculated using carrier density profiles n_t( x) and n_b( x). The numerical derivative of the results is shown in Fig. 4C. To compare with the measurement, we consider the numerical derivative ∂g_tot/ ∂V_tg of the sum g_t + g_b = g_tot ( Fig. 4D). The excellent agreement to the measurement ( Fig. 3B) is a strong indication that the wave functions of the top and bottom layers are essentially decoupled and individually tunable.

The tight-binding theory allows us to compare the electrostatic model to the experiment and to estimate the precision of the obtained value for the graphene interlayer capacitance C_m. For the cavity L = 320 nm and for V_bg = −10 V, we observe N = 11 ± 1 modes between the two zero-density lines in the experimental data ( Fig. 3B) and N = 11 ± 0.5 modes in the tight-binding data ( Fig. 4D). Since the splitting of zero-density lines is proportional to C_m, we estimate the error to be ≈10% for C_m, and therefore, we estimate the dielectric thickness of graphene .