Research ArticleCONDENSED MATTER PHYSICS

The electronic thickness of graphene

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Science Advances  13 Mar 2020:
Vol. 6, no. 11, eaay8409
DOI: 10.1126/sciadv.aay8409

Abstract

When two dimensional crystals are atomically close, their finite thickness becomes relevant. Using transport measurements, we investigate the electrostatics of two graphene layers, twisted by θ = 22° such that the layers are decoupled by the huge momentum mismatch between the K and K′ points of the two layers. We observe a splitting of the zero-density lines of the two layers with increasing interlayer energy difference. This splitting is given by the ratio of single-layer quantum capacitance over interlayer capacitance Cm and is therefore suited to extract Cm. We explain the large observed value of Cm by considering the finite dielectric thickness dg of each graphene layer and determine dg ≈ 2.6 Å. In a second experiment, we map out the entire density range with a Fabry-Pérot resonator. We can precisely measure the Fermi wavelength λ in each layer, showing that the layers are decoupled. Our findings are reproduced using tight-binding calculations.

INTRODUCTION

The van der Waals stacking technique allows scientists to bring two conductive crystalline layers into atomically close proximity (1). This has been exploited in a variety of experiments, including the formation of layer-polarized, counterpropagating Landau levels (2) and experiments that build on strong capacitive coupling such as Coulomb drag measurements (3) or interlayer exciton condensation (4, 5).

There are two main approaches to bring two conductive layers in close proximity while suppressing an overlap of the layer wave functions: One approach introduces a thin layer of hexagonal boron nitride (hBN) [see, e.g., (3, 4, 6)] as depicted in Fig. 1 (A and B), and the other twists the layers by a large angle (θ > 5) [see Fig. 1 (C and D)] (2, 79). In the former case, decoupling is achieved by spatial separation. In the latter case, the layers are ultimately close, but they remain decoupled because of a large momentum mismatch (KtKb) between the upper and lower layers (Fig. 1D). Experimental signatures of decoupling are an increased interlayer resistance (10, 11) and layer-polarized Landau levels at large magnetic fields (2, 9).

Fig. 1 Experimental design.

(A) Top and side views of two aligned layers of graphene that are decoupled in the middle (blue part) by a thin intermediate layer of hBN. A graphite back gate and a local top gate allow controlling the density and, thereby, the carrier wavelength in the upper and lower layers individually. (B) Using atomic force microscopy (AFM), we measured the encapsulated hBN layer to be 3.5 nm thick (sample A). (C) Alternatively, the decoupling wave functions can be achieved by twisting two graphene layers (sample B). (D) For large twist angles, the valleys in the upper/lower layer (Kt, Kb) are separated by a large momentum, leading to an effective electronic decoupling of the layers.

In this work, we perform quantum transport experiments to monitor precisely the coupling, coherence, and tunability of two graphene layers that are in close proximity to each other. In one device, we separate the two layers by a thin layer of hBN with thickness d = 3.5 nm (sample A), and in the other device, we twist the layers by 22° to decouple them (sample B).

In the first experiment, we observe a splitting of the charge neutrality points of the two layers in the parameter plane of top- and back-gate voltage (Vtg and Vbg). By analyzing the splitting, we extract a geometric capacitance Cm between the graphene layers. For sample A, we obtain the expected value given the thickness and dielectric constant of the intermediate hBN layer. However, for sample B, Cm is three times larger than the geometric capacitance between two ideal capacitor plates, separated by the interlayer distance between carbon atoms dCC=3.4 Å, assuming vacuum in between. We argue that, because of the finite electronic thickness of graphene, the plates of the capacitor are effectively closer than dCC, leading to the enhanced Cm. We find good agreement with a capacitive model where we take the electronic thickness of graphene into account.

In the second experiment, on sample B, we use a gate-defined Fabry-Pérot cavity to monitor the layer densities, coherence, and interlayer coupling of wave functions. The cavities are formed by gate-defined p-n junctions, which act as semitransparent lateral “mirrors” of the interferometer (1216). Either only one or both layers can be tuned to the bipolar p-n-p regime. In both layers, we observe the lowest energy Fabry-Pérot mode, corresponding to λ = 600 nm, while the wavelength in the other cavity can be shorter by a factor of 10. We model the observed interference pattern using tight-binding calculations, assuming completely decoupled layers. This second experiment confirms the assumed electronic decoupling and, for arbitrary gate voltages, the electrostatic model that considers thick graphene.

RESULTS

Zero-density lines

The numerical conductance dG/dVtg as a function of Vtg and Vbg 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 Vtg and Vbg. 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.

Fig. 2 Zero-density lines.

Numerical derivative of the two-terminal conductance dG/dVtg(Vtg, Vbg) for a device where the graphene layers are (A) separated by a thin hBN layer (sample A) or (B) in atomic vicinity but twisted by a large angle (sample B). Zero-density lines in the upper (yellow) and lower (red) graphene layers are obtained from numerical calculations. (C) Schematic electrostatic configuration of sample B. (D) Calculated integrated local density of states ILDOS(z) of pz-like orbital of carbon atoms in graphene (red) and the induced charge density Δρ(z) ≔ ρ(0) − ρ(Ez) per carbon atom under an external electric field Ez. The geometry of graphene is shown on the background picture. The graphene sheet is placed at z = 0 and extends in the xy plane. Positions of black dotted lines mark the effective thickness of graphene calculated from the expectation value of the position operator z=0.66 Å. The blue shaded region shows the dielectric thickness of graphene extracted from the dielectric permittivity (19). (E) Comparison of Δρ(z) for bilayer graphene (BLG) in AA stacking configuration (gray line), AB Bernal (blue line), and twisted BLG (tBLG) (red line). The position of graphene layers is marked by vertical dashed lines, and the blue shaded regions depict the dielectric thickness of single-layer graphene. a.u., arbitrary units.

From electrostatic considerations, we find that the zero-density condition can be expressed as (details are given in the Supplementary Materials) Vtg Vbg|nb=0CbgCtg(1+CqtCm) Vbg Vtg|nt=0CtgCbg(1+CqbCm)where Cbg (Ctg) is the geometric capacitance of the bottom (top) graphene to the bottom (top) gate (see Fig. 2C) and the density in the bottom (top) graphene layer is nb (nt). The capacitance measured between the two graphene plates is Cm. The quantum capacitance Cqt = e2𝒟t(ℰF) of the top layer is proportional to the density of states at the Fermi energy in the top layer (the analog relation holds for the bottom layer). For a single sheet of graphene, the slope of the zero-density line in a (Vtg, Vbg) map is given by the ratio −Cbg/Ctg (prefactor in the above equations). For the two-layer system, the deviations from linearity of the constant-density line are governed by the ratio between quantum capacitance and Cm, respectively. Therefore, the splitting is smaller in sample B, where Cm is larger, as compared to sample A, where Cm is smaller.

Analytical formulas for the zero-density lines [i.e., Vbg(Vtg)∣nt = 0 and Vbg(Vtg)∣nb = 0] 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 Cm, 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., Ctg = ϵhBN/dt with ϵhBN = 3.3ϵ0. A discussion for the precision of this method is given in the Supplementary Materials.

For sample A, we obtain an interlayer capacitance of Cm = 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 ϵhBN = 3.3ϵ0. For sample B, we determine a large interlayer capacitance Cm = 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 dCC=3.4 Å, which is the expected distance between two graphene layers (17, 18). Consistent with our findings, large interlayer capacitance values have been reported in (9) in large perpendicular magnetic fields (quantum Hall regime) with a capacitance model that is only valid for nt = −nb. A detailed explanation for the large value of Cm 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 pz 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 Ec = 0 eV. In this energy range, the bands are of pure pz orbital character without contributions from the s-, px- and py-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 pz 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 (19), defined as the distance from the center of carbon atoms to the point at which the dielectric constant of graphene ϵ = 6.9ϵ0 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 dg is indicated by the blue shaded region in Fig. 2D, with values according to (19).

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(Ez) for tBLG, AA BLG, and AB BLG under an external electric field Ez = 1 V/nm perpendicular to the BLG lattice. The interlayer distance of AA and AB BLGs was set to 3.51 Å 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 Ez are almost the same on the outer side of the BLG, while they are very different in the interlayer region. For z=13±0.7 Å, 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.

These calculations motivate a simplified capacitance model where the measured capacitance Cm (between the center of charge of each layer) contains two dielectric materials coupled in series: graphene with ϵg = 6.9ϵ0 (19) and thickness dg and an interlayer region of vacuum with thickness dinter = dCCdg and a dielectric constant of vacuum. Therefore, 1/Cm = dgg + (dCCdg)/ϵ0. With dCC=3.4 Å and the measured capacitance, we determine a dielectric thickness of dg=2.6±0.2 Å from our measurements, which is in agreement with theoretical predictions in single-layer graphene exhibiting 2.4 Å (19). Using a similar model for the hBN device with 1/Cm = dgg + dhBNhBN, we find dhBN=35 Å, which is in excellent agreement with the thickness measured with the atomic force microscopy (AFM). However, the correction by the thickness of graphene (≈1 Å) in this case is of the order of the measurement accuracy of our AFM.

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/dVtg for top gates, sized L = 190 and 320 nm, respectively. For both cases, the cavity width WL. The zero-density lines are depicted in yellow for the top layer and dark red for the bottom layer.

Fig. 3 Fabry-Pérot interference pattern.

Differential conductance dG/dVtg(Vtg, Vbg) for gates of length (A) L = 190 nm and (B) L = 320 nm for sample B. (C) Sketches of the local density in the two Fabry-Pérot layers. Blue regions are p-doped, and red regions are n-doped. The sketches (1) to (4) show different gating configurations, marked correspondingly in (A). (D) As the difference in gate voltages increases, the energies in the top and bottom layers will shift by the interlayer energy difference U.

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 Vbg. Since Vbg < 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 = 2LF = kFL/π, where L is the cavity size and λF is the Fermi wavelength. Note that kF=πn 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 λF, bottom = 640 nm and λF, 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 Vbg or close to the zero-density lines. Under these conditions, either the graphene part tuned only by Vbg 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 (13, 15).

Simulation of density and transport

We now compare the resonance pattern to tight-binding simulations. The underlying density profiles nt(x) and nb(x) are obtained from a self-consistent electrostatic model where we assume that the dispersion relation remains linear, such that the carrier density formulas (20) derived for single-layer graphene with quantum capacitance (21, 22) 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(Vtg, Vbg) 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 (2, 9) for the displacement field [i.e., D = 1/2(CtgVtgCbgVbg)] only holds under the condition nt = −nb. Apparently, lines of constant U (white lines in Fig. 4B) do not have a constant slope in the (Vtg, Vbg) map. A more detailed comparison is given in the Supplementary Materials.

Fig. 4 Simulations.

(A) Numerically calculated layer density profiles nt(x) (blue) and nb(x) (red), shaped by a top gate with a size of 320 nm and a global back gate for sample A with an hBN spacer (left) and for sample B with a large-angle tBLG (right). The depicted gating condition is (Vtg, Vbg) = (5,5) V. The dashed lines show the classical result (neglecting the quantum capacitance). (B) Interlayer energy difference U(Vtg, Vbg). Zero-density lines are marked with yellow and red lines. (C) Numerical derivative of the calculated normalized conductance, ∂g/Vtg. Using the obtained nt and nb, the conductance g(Vtg, Vbg) of the top layer (top) and the bottom layer (bottom) are calculated individually using a real-space Green’s function approach. (D) The sum of the two differential conductances ∂gtot/∂Vtg reproduces the experimental data in Fig. 3B.

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 (23), equivalent to the assumption of infinitely wide graphene samples. This is justified since WL in our device. The normalized conductances gt(Vtg, Vbg) and gb(Vtg, Vbg) for the top and bottom graphene layers, respectively, are calculated using carrier density profiles nt(x) and nb(x). The numerical derivative of the results is shown in Fig. 4C. To compare with the measurement, we consider the numerical derivative ∂gtot/∂Vtg of the sum gt + gb = gtot (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 Cm. For the cavity L = 320 nm and for Vbg = −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 Cm, we estimate the error to be ≈10% for Cm, and therefore, we estimate the dielectric thickness of graphene dg=2.6±0.2 Å.

DISCUSSION

We have performed transport experiments for two representative cases of decoupled layers of graphene. We investigated two devices: one where decoupling is achieved by a thin hBN layer (sample A) and the other where the decoupling is given by the large momentum mismatch between graphene layers due to a large twist angle (sample B). In both cases, we observed a clear splitting of the charge neutrality points in a two-terminal measurement with the strength of the splitting given by Cq/Cm. By comparing to a self-consistent electrostatic model, we extracted a very large geometric interlayer capacitance Cm = 7.5 ± 0.7 μF cm−2 for the tBLG sample, which we explained by taking into account an effective dielectric thickness of graphene of dg=2.6±0.2 Å. In a further step, we investigated Fabry-Pérot fringes that originate from p-n-p cavities created with a local top gate and a global back gate. We were able to form a p-n-p cavity in only one of the layers and could tune the wavelength in each layer individually. In an L = 320 nm cavity, we observed the first mode in the bottom layer, while we had already filled 10 modes in the top layer. The measurements are in very good agreement with the results from tight-binding simulations based on two graphene layers electronically decoupled but electrostatically coupled through their quantum capacitances. Our work emphasizes that the finite thickness of two-dimensional materials is relevant for the electronic properties of van der Waals heterostructures where conducting layers are in close proximity.

MATERIALS AND METHODS

To achieve ballistic transport, we encapsulate (24) either tBLG (sample B) or graphene–3.5-nm hBN–graphene between hBN layers (sample A) and use a graphite bottom gate (25, 26). The alignment of the graphene layers is controlled by the method described in (27, 28), and we used twist angles (between the graphene layers) θ ≈ 0° for sample A and θ ≈ 22° for sample B. The thickness of the top, bottom, and intermediate hBN layers is determined by AFM. Electrical one-dimensional contacts are achieved by reactive ion etching and evaporation of Cr/Au. Top gates of sizes 320 and 190 nm are defined by electron beam lithography. By adjusting the top-gate voltage Vtg and the back-gate voltage Vbg, a Fabry-Pérot cavity can be formed below the top gate. Two-terminal linear conductance measurements are performed using a low-frequency lock-in technique (177 Hz) at the temperature T = 1.5 K.

Supplementary Materials

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/11/eaay8409/DC1

Fig. S1. Measurement on two further devices.

Fig. S2. Accuracy of the fitting procedure.

Fig. S3. Interlayer energy difference U.

Fig. S4. Induced density profile for AA and AB bilayer graphene.

Fig. S5. Schematics of the crystalline structure of the unit cell of 22° tBLG with 196 carbon atoms.

Fig. S6. Comparison of the induced charge density Δρ(z) for AA and AB bilayer graphene.

Fig. S7. Schematic model for gated, decoupled BLG.

Table S1. Calculated interlayer distances for AA and AB BLGs for different types of dispersion corrections.

References (2938)

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REFERENCES AND NOTES

Acknowledgments: Funding: We acknowledge financial support from the European Graphene Flagship, the Swiss National Science Foundation via NCCR Quantum Science and Technology, the Deutsche Forschungsgemeinschaft through SFB 1277 project A07, and the Taiwan Ministry of Science (107-2112-M-006-004-MY3 and 107-2627-E-006-001) and Ministry of Education (Higher Education Sprout Project). This work is also supported by the National Science Center under the contract DEC-2018/29/B/ST3/01892 and, in part, by PAAD Infrastructure cofinanced by Operational Programme Innovative Economy, Objective 2.3. Growth of hBN crystals was supported by the Elemental Strategy Initiative conducted by MEXT, Japan and the CREST (JPMJCR15F3), JST. Author contributions: P.R. fabricated the device and performed measurements and data analysis. Tight-binding calculations and density simulations were provided by M.H.-L. Density functional theory calculations were performed by M.K. A.K., Y.L., H.O., M.E., and R.P. were involved in the fabrication and measurements. T.T. and K.W. provided hBN crystals. K.R., K.E., and T.I. supervised the work. All authors contributed to the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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