AA stacking, tribological and electronic properties of double-layer graphene with krypton spacer

Structural, energetic and tribological characteristics of double-layer graphene with commensurate and incommensurate krypton spacers of nearly monolayer coverage are studied within the van der Waals-corrected density functional theory. It is shown that when the spacer is in the commensurate phase, the graphene layers have the AA stacking. For this phase, the barriers to relative in-plane translational and rotational motion and the shear mode frequency of the graphene layers are calculated. For the incommensurate phase, both of the barriers are found to be negligibly small. A considerable change of tunneling conductance between the graphene layers separated by the commensurate krypton spacer at their relative subangstrom displacement is revealed by the use of the Bardeen method. The possibility of nanoelectromechanical systems based on the studied tribological and electronic properties of the considered heterostructures is discussed.


Belarus
Structural, energetic and tribological characteristics of double-layer graphene with commensurate and incommensurate krypton spacers of nearly monolayer coverage are studied within the van der Waals-corrected density functional theory. It is shown that when the spacer is in the commensurate phase, the graphene layers have the AA stacking. For this phase, the barriers to relative in-plane translational and rotational motion and the shear mode frequency of the graphene layers are calculated. For the incommensurate phase, both of the barriers are found to be negligibly small. A a) Electronic mail: popov-isan@mail.ru b) Electronic mail: liv_ira@hotmail.com c) Electronic mail: lozovik@isan.troitsk.ru d) Electronic mail: poklonski@bsu.by considerable change of tunneling conductance between the graphene layers separated by the commensurate krypton spacer at their relative subangstrom displacement is revealed by the use of the Bardeen method. The possibility of nanoelectromechanical systems based on the studied tribological and electronic properties of the considered heterostructures is discussed.

I. INTRODUCTION
The discovery of graphene 1 sparked tremendous efforts in development of graphene-based nanometer-scale systems. The most important systems realized recently include AA-stacked bilayer 2 and multilayer 3 graphene and double-layer graphene, [4][5][6][7][8][9][10] i.e. the system consisting of two graphene layers separated by a dielectric spacer. While in bilayer graphene, the interlayer distance is about 3.4 Å and is close to that in graphite, the distance between the layers in doublelayer graphene is determined by the thickness of the dielectric spacer.
Both AA-stacked bilayer graphene and double-layer graphene represent significant interest for studies of fundamental phenomena and practical applications. Electronic and optical properties of AA-stacked bilayer graphene were predicted to be very different from those of bilayer graphene with the ordinary AB Bernal stacking. [11][12][13][14][15] The quantum spin Hall effect, 11 spontaneous symmetry violations, 12 low-energy electronic spectra 13 and magneto-optical absorption spectra 14 for AA-stacked bilayer graphene and metal-insulator transition 15 for doped AA-stacked bilayer graphene were considered. A field-effect transistor consisting of two graphene layers with the nanometer-scale dielectric spacer between the layers was implemented. 4,5 A tunable metal-insulator transition was observed in double-layer graphene heterostructures. 6 Electron tunneling between graphene layers separated by an ultrathin boron nitride barrier was investigated. 7 Double-layer graphene heterostructures were also used to determine Fermi energy, Fermi velocity and Landau level broadering. 8 Measurements of Coulomb drag of massless fermions in double-layer graphene heterostructures were reported 9,10 and the theory of this phenomenon was considered. [16][17][18][19][20][21][22] Theoretical studies of electron-hole pairs condensation in a double-layer graphene were presented. [23][24][25][26][27][28][29][30] Since the electronic properties of double-layer graphene with a thin dielectric spacer depend strongly on the stacking of the graphene layers, production of double-layer graphene with the controllable stacking is essential for its possible applications. Several types of double-layer graphene heterostructures with different dielectric spacers between the layers have been realized up to now. Namely, graphene layers can be separated by a few-nanometer Al2O3 8 or SiO2 9 spacers, by one 5,7 or several 5-7,10 atomic boron nitride layers, and by a layer of adsorbed molecules. 4 However, most of these implementations do not allow to control the stacking of graphene layers. Al2O3 8 or SiO2 9 spacers are not layered materials and, therefore, they do not make it possible to produce double-layer graphene not only with a given stacking of graphene layers but also with a given interlayer distance. As for the boron nitride spacer, the lattice constant of this material is 2% greater than the lattice constant of graphene. 31 The recent study of the commensurate-incommensurate phase transition in bilayer graphene with one stretched layer 32 revealed that even such a small mismatch of 2% between the lattice constants is sufficient for the transition to the incommensurate phase with a pattern of alternating commensurate and incommensurate regions so that the same relative position of the layers in the whole system is feasible only for a small size of the layers. In fact, the spatially inhomogeneous biaxial compressive strain has been observed lately in graphene/hexagonal boron nitride heterostructure. 33 As for the AA-stacked bilayer graphene without any spacer, observations of such a structure are restricted to the cases of bilayer graphene with a common folded edge 2 and AAAA-stacked regions of multilayer graphene on the C-terminated SiC substrate. 3 In the case of nearly coincident edges of a graphene flake and a graphene layer, stable and metastable positions of the flake differing both from the AA and AB stacking were found to be possible as a result of the trade-off between the edge-edge interaction and the van der Waals interaction. 34 In the present paper, we suggest that a new type of graphene-based heterostructures, doublelayer graphene with controllable stacking of the graphene layers, can be produced by the use of the layer of adsorbed atoms or molecules commensurate with the graphene layers as a spacer. It is well known that krypton can form commensurate layers on graphite (Ref. 35, 36 and references therein). Therefore, we consider this inert gas as a candidate for the commensurate spacer. The van der Waals-corrected density functional theory is applied to reveal the stacking of graphene layers separated by the commensurate krypton spacer and to calculate the barriers to relative motion of these layers and the shear mode frequencies. We show that the AA stacking of graphene layers can be realized in this heterostructure for an arbitrary substrate or in the suspended system and for an arbitrary size of neighbouring graphene layers, structure and relative position of their edges. Recently we have revealed frictionless tribilogical behavior for double-layer graphene with the incommensurate argon spacer. 37 However, our calculations have shown that the commensurate phase of the argon spacer between graphene layers is much less stable than the incommensurate phase and, therefore, should be difficult to obtain. The present paper is devoted to double-layer graphene with the commensurate spacer, i.e. to the heterostructure with radically different tribological and electronic properties.
The previous calculations demonstrated that the electronic structure of twisted bilayer graphene changes considerably with changing the twist angle. 38 The tunneling conductance between the layers of bilayer graphene changes by several times upon relative displacement of the layers 39,40 and by an order of magnitude upon relative rotation of the layers. 39 Here we use the Bardeen method to calculate 2D maps of tunneling conductance between graphene layers of bilayer graphene and double-layer graphene with the commensurate krypton spacer as a function of coordinates describing relative in-plane displacements of the layers. Possible applications of the revealed tribological and electronic properties of double-layer graphene with the commensurate and incommensurate krypton spacers in nanoelectromechanical systems (NEMS) are discussed.
The paper consists of the following parts. In Sec. II, we give the details of van der Waalscorrected density functional theory calculations. Sec. III is devoted to the analysis of structural and tribological properties of double-layer graphene with the krypton spacer. Sec. IV presents the results of calculations of tunneling conductance between krypton-separated graphene layers.
In Sec. V, we consider the possibility of experimental realization and application of the studied heterostructure in NEMS and summarize our conclusions.

II. COMPUTATIONAL DETAILS
Analysis of structural and tribological properties of krypton-separated double-layer graphene has been performed using the VASP code. 41 The performance of three approaches with the correction for the van der Waals interaction has been compared: (1) the DFT-D2 method 42 with the generalized gradient approximation (GGA) density functional of Perdew, Burke, and Ernzerhof 43 corrected with the dispersion term (PBE-D), (2) the vdW-DF method 44 with the optPBE-vdW exchange functional, 45,46 and (3) the vdW-DF2 method. 46,47 The basis set consists of plane waves with the maximum kinetic energy of 500-800 eV. The interaction of valence electrons with atomic cores is described using the projector augmented-wave method (PAW). 48 A second-order Methfessel-Paxton smearing 49 of the Fermi surface with a width of 0.1 eV is applied. The energy convergence tolerance for electronic self-consistent loops is 10 -7 eV.
Two krypton spacers of different structure are considered. The spacer A is a krypton layer commensurate with the graphene layers (FIG. 1) and corresponds to the double-layer graphene with the krypton to carbon ratio Kr:C = 1:12. In this spacer, the distance between adjacent krypton atoms is 7% greater than the equilibrium distance in the isolated krypton layer of 4.00 Å for the DFT-D2 method, 5% greater than the equilibrium distance in the isolated krypton layer of 4.07 Å for the vdW-DF2 method and 4% greater than the equilibrium distance in the isolated krypton layer of 4.12 Å for the optPBE-vdW method. The spacer B has krypton atoms with inequivalent positions of krypton atoms on the graphene lattice within the model cell (FIG. 1) and corresponds to the krypton to carbon ratio Kr:C = 9:100. In this spacer, the distance between adjacent krypton atoms is only 3% greater than the equilibrium krypton-krypton distance according to the DFT-D2 method, only 1% greater than the equilibrium krypton-krypton distance for the vdW-DF2 method and very close to the equilibrium krypton-krypton distance for the optPBE-vdW method. The results below show that 9 krypton atoms with inequivalent positions for the spacer B are sufficient for the dramatic change of tribological characteristics of the system. Thus, we consider the spacer B as a prototype of the incommensurate spacer since we are not able to simulate incommensurate krypton-graphene heterostructures under the periodic boundary conditions (PBCs) directly. the graphene layers, between each of the graphene layers and the krypton spacer, and inside the krypton spacer to the total interaction energy of the krypton-separated double-layer graphene (the interactions inside the graphene layers are excluded from this quantity) for found minimum energy positions are also considered. To evaluate these contributions the energies of the systems consisting of two graphene layers at the same interlayer distance and the in-plane relative position as in krypton-separated double-layer graphene but without the krypton spacer, the isolated graphene layer and the isolated krypton spacer are found.
The convergence on the number of k-points in the Brillouin zone and the maximum kinetic energy of plane waves was tested previously for bilayer graphene. 57 Additional convergence tests have been also performed for krypton-separated double-layer graphene. Increasing the number of k-points from 12 x 12 x 1 to 24 x 24 x 1 for the spacer A and from 7 x 7 x 1 to 14 x 14 x 1 for the spacer B and simultaneously increasing the maximum kinetic energy of plane waves from 500 to 800 eV leads to changes in the total interaction energy of double-layer graphene by less than 0.004 meV per carbon atom of one of the graphene layer. At the same time, the barrier to relative motion of the commensurate krypton spacer A and each of the graphene layers changes by less than 0.01 meV per carbon atom. Stretching or compressing the graphene layers by 0.5% from the ground state results in changes of the total interaction energy of krypton-separated graphene by less than 0.1 meV per carbon atom and changes of the barrier to relative motion of the commensurate krypton spacer A and each of the graphene layers by less than 0.01 meV per carbon atom. In Sec. III, we give the data obtained using the maximum kinetic energy of plane waves of 500 eV and the k-point grids of 12 x 12 x 1 and 7 x 7 x 1 for the spacers A and B, respectively.

GRAPHENE LAYERS
The structural characteristics and the total interaction energy of krypton-separated double-layer graphene as well as different contributions to this energy calculated using DFT-D2, optPBE-vdW and vdW-DF2 methods are listed in Table I. The data obtained for bilayer graphene using the same methods are also given for comparison.
Let us first discuss the results on structural properties of double-layer graphene. The equilibrium distances between the graphene layers separated by the spacers A and B calculated using all the methods considered lie in the range from 6.8 to 7.2 Å. These values are in qualitative agreement with the experimental value of ~6 Å for undetermined adsorbed molecules 4 and previous calculations for argon. 37 Nevertheless, the optPBE-vdW and vdW-DF2 methods predict slightly larger interlayer distances than the DFT-D2 method. To address the accuracy of the methods let us compare the results on structural properties of bilayer graphene (Table I)  Thus, the DFT-D2, optPBE-vdW and vdW-DF2 methods have comparable accuracy for description of structural properties of the van der Waals-bound systems considered here. The optPBE-vdW and vdW-DF2 methods tend to overestimate equilibrium distances, while the DFT-D2 method tends to underestimate equilibrium distances. Therefore, the interval of the equilibrium interlayer distances from 6.8 to 7.2 Å calculated for krypton-separated double-layer graphene using different methods should enclose the experimental data. are given. The data for bilayer graphene are also given for reference. Let us now proceed with the discussion of relative stability of double-layer graphene with the spacers A and B and bilayer graphene. All the methods considered here agree that both for the spacers A and B, the total interaction energy of double-layer graphene is higher in magnitude than for bilayer graphene without any spacer (Table I) The krypton-krypton interaction energy in the krypton commensurate layer with the same krypton coverage as in the spacer A can be deduced from the the exponential factor in the temperature dependence of the critical pressure for the commensurate-incommensurate phase transition to be around -50 meV per krypton atom. 63 The optPBE-vdW method gives the krypton-krypton interaction energy of -107 meV per krypton atom for the spacer A and of -110 meV per krypton atom for the spacer B, i.e. much greater in magnitude than the values estimated from the experimental data. According to the DFT-D2 and vdW-DF2 methods, the kryptonkrypton interaction energy is -78 meV and -81 meV per krypton atom for the spacer A and -84 meV and -86 meV per krypton atom for the spacer B, respectively. So these methods also overestimate the krypton-krypton interaction energy but the discrepancy with the experimental data is smaller.
Let us finally compare the accuracy of the methods considered with respect to the interlayer interaction in bilayer graphene. The latest experimental measurements for graphite 64 gave the interlayer binding energy of -52±5 meV per carbon atom. It is seen from Table I that the optPBE-vdW strongly overestimates the interlayer binding energy in bilayer graphene, while the DFT-D2 and vdW-DF2 methods provide reasonable values of this energy. From all these considerations, the DFT-D2 approach seems to be the most reliable for the analysis of relative energies of graphene-based heterostructures among the methods used here.
Let us now address the relative stability of the commensurate and incommensurate phases of the krypton spacer. All the methods considered predict that the total interaction energy is higher in magnitude for the spacer B with the higher krypton coverage than for the spacer A (Table I).
According to the DFT-D2 calculations, the incommensurate spacer B is preferred in energy compared to the spacer A only by 3 meV per carbon atom of one of the graphene layers.
Extrapolating the krypton-graphene and graphene-graphene binding energies to the limit of krypton coverage corresponding to the equilibrium krypton-krypton distance in the isolated krypton layer of 4.0 Å, we can estimate that the incommensurate phase should be preferred over the commensurate one by no more than 4 meV per carbon atom of one of the graphene layers.
This energy difference between the commensurate and incommensurate spacers is more than 3 times less than in the case of argon between graphene layers, for which the incommensurate phase is much more favorable than the commensurate one. 37 The similarly small energy difference between the heterostructures with the krypton spacers A and B of 4.4 meV per carbon atom of one of the graphene layers follows from the relatively accurate vdW-DF2 method (Table   I) The magnitudes of corrugation of the potential energy relief and the barriers to translational motion of each of the graphene layers relative to the krypton layer are given in Table I. These data show considerable scatter for different methods. As it was shown in our previous publication, though the correction for the van der Waals interaction does not contribute much to the roughness of the potential energy relief, this relief is very sensitive to the interlayer distance. 57 Due to the differences in the interlayer distances, the calculated barriers vary by 40% from the DFT-D2 method to the optPBE-vdW method. The same as for interlayer distances, the barriers calculated using different methods mark the interval that should enclose the experimental data. To address the accuracy of different methods with respect to the description of tribological properties of layered graphene-based structures let us compare the results for bilayer graphene. On the basis of experimentally measured shear mode frequencies for few-layer graphene and graphite, the barrier to relative motion of graphene layers and the magnitude of corrugation of the potential energy relief for bilayer graphene were estimated to be 1.7 meV and vdW-DF2 methods underestimate these quantities by 40% and 50%, respectively, whereas the DFT-D2 method overestimates these quantities by only 25% (Table I) x, y, z U z cos k x cos k y cos k y U z  As shown previously for argon spacers, 37 the contribution of graphene-graphene interaction to corrugations of the potential energy relief in double-layer graphene does not exceed 0.003 meV per carbon atom of one of the layers. This is more than two orders of magnitude smaller than the magnitude of corrugation obtained for the interaction between the graphene layers and the krypton spacer A (Table 1). Therefore, the variation of the total interaction energy upon relative motion of the graphene layers can be obtained as the sum of variations of interaction energies between the krypton layer and each of the graphene layers, both of which can be approximated by Equation 1. The minimal energy of the system with the relative position of graphene layers 0 0 0 x , y , z can, therefore, be calculated as The potential energy relief calculated using Equation 2 is given in FIG. 3. We should emphasize that as opposed to bilayer graphene with the AB stacking of layers, the AA stacking of graphene layers is found here for the ground state of the double-layer graphene with the commensurate krypton spacer A.
In this case of relative displacement along the energy favourable zigzag direction (see FIG. 2   and FIG. 3), the expression (2) where N is an integer. From this formula, it is seen that the barrier to relative motion of the graphene layers in double-layer graphene with the commensurate krypton layer is exactly equal to the barrier to relative motion of the krypton layer on one of the graphene layers and in the case of parametrization on the basis of the DFT-D2 calculations is gr-gr gr-Kr 1 per carbon atom of one of the graphene layers (Table I).
For double-layer graphene with the incommensurate krypton spacer, the potential energy relief for relative motion of the graphene layers is extremely smooth due to the absence of barriers to relative motion of the incommensurate krypton spacer and each of the graphene layers. Therefore, the static friction force for relative motion of the graphene layers is negligibly small.
The expression (3) can be also used to estimate the shear mode frequency of graphene with n layers separated by the commensurate krypton spacers corresponding to in-plane vibrations of adjacent graphene layers in opposite directions. Assuming that the krypton atoms do not move in this mode, its frequency can be found as   2 gr-gr  , m is the mass of a carbon atom and p is an integer. This formula looks just the same as for graphene bilayer. 57,68 However, here 1 U characterizes the krypton-graphene interaction instead of the graphenegraphene interaction. For double-layer graphene with the commensurate krypton spacer A, the formula parameterized on the basis of the DFT-D2 calculations gives the frequency -1 10 5 cm f .  , which is three times smaller than for the graphene bilayer. 57,68,69,72 This can be explained by the one-order difference in the magnitudes of corrugation of the potential energy reliefs for krypton-separated double-layer graphene and graphene bilayer. 57,68,69 For the material consisting of alternating graphene layers and commensurate krypron spacers A, the formula (4) parameterized on the basis of the DFT-D2 calculations gives the shear mode frequency -1 , three times smaller than for graphite. 68 The shear modulus of this material can be estimated as where S is the overlap area between the graphene layers, 0 m is the electron mass in vacuum,  In the tight-binding approximation, the wave function of the bottom/top graphene layer takes the form 75 Here G N is the number of elementary unit cells of graphene, near K-points of the Brillouin zone,    r is the Slater 2px-orbital of a carbon atom Here  dS dxdy , Y are the coordinates of the g-th elementary unit cell of the top graphene layer, i.e.    respectively. It is seen that the tunneling conductance between the graphene layers strongly depends on their relative position at the subnanometer scale, similar to the results obtained previously for bilayer graphene 40 and for double-walled carbon nanotubes. [77][78][79] In the both considered systems, the tunneling conductance reaches its maximum for the AA stacking, in which atoms of the graphene layers are located at the smallest distances to each other, while the minima of the tunneling conductance correspond to the SP stacking. However, the difference in the ground state stacking leads also to qualitatively different changes of the tunneling conductance at subangstrom in-plane relative displacements of the graphene layers from the ground state. Namely, for bilayer graphene with the AB stacking at the ground state, a decrease or an increase of the tunneling conductance is possible depending on the direction of the displacement, whereas for double-layer graphene with the commensurate krypton spacer, the AA stacking at the ground state corresponds to the maximal tunneling conductance and any in-plane relative displacement of the graphene layers causes a decrease of the conductance. Possible NEMS that can be based on this qualitatively different behavior of the tunneling conductance at the ground state are discussed below. The contributions of interaction between the krypton atoms, between the krypton spacer and the graphene layers, and between the graphene layers into the total interaction energy of the system have been obtained for both of the spacers. All the methods used agree that both of the considered structures of double-layer graphene are more stable than bilayer graphene, i.e. the escape of krypton atoms from these structures is not energetically favorable.  (2) the energy required to compress the krypton layer in order to form the area of double-layer graphene free of krypton atoms. We denote the diameter of the critical island of bilayer graphene as L and assume that to stick together both of the graphene layers get curved and the distance between them decreases by 2 80 The second contribution to the activation energy for nucleation of the critical island of bilayer graphene is related to the fact that to form this island it is necessary to free this area from krypton atoms and, correspondingly, to compress the krypton layer. The krypton coverage in the spacer A (the ratio of krypton atoms to carbon atoms in one of the graphene layers) is com For this size of the critical island, the activation energy can be estimated as For double-layer graphene with the commensurate krypton spacer, considerable corrugations of the potential energy relief describing the relative in-plane displacements of the graphene layers are revealed. On the basis of the DFT-D2 calculations, the barrier for relative motion of the graphene layers in this heterostructure is found to be 1.44 meV per carbon atom, which is about 70% of the corresponding barrier in bilayer graphene. 57 Simultaneously, the changes in the tunneling conductance between the graphene layers at their relative displacement through this barrier are calculated here to be up to 90%. A set of NEMS based on the interaction and subangstrom relative motion of layers of bilayer graphene was proposed, including the nanoresonator 69 , two different schemes of the force sensor 40,81 and the floating gate memory cell. 81 NEMS with analogous or other operational principles based on krypton-separated graphene layers can be elaborated. The Q-factor for subangstrom relative vibrations of the layers of bilayer graphene is rather small Q = 30 -150. 69 The presence of the krypton spacer should lead to additional channels of energy dissipation of the mechanical oscillations arising after switching or measurement events in NEMS based on krypton-separated graphene layers. Thus, the considered heterostructure is perspective for elaboration of fast-acting sensors and other NEMS.
For double-layer graphene with the incommensurate krypton spacer, the potential energy relief describing the relative in-plane displacements of the graphene layers is shown to be extremely smooth and, therefore, the static friction force for relative motion of the graphene layers is negligibly small. Thus, double-layer graphene with the incommensurate krypton spacer is also suitable for NEMS based on free relative translational or rotational motion of the graphene layers proposed recently for argon-separated double-layer graphene. 37 In particular, the revealed static-friction-free relative in-plane motion of the graphene layers separated by the incommensurate krypton spacer allows us to propose that such a heterostructure can be perspective for elaboration of variable capacitors with the capacitance proportional to the overlap area of the layers. The tunneling conductance between the krypton-separated graphene layers is found to be seven orders of magnitude smaller than the tunneling conductance between the layers of bilayer graphene. The distance between the graphene layers separated by the incommensurate argon spacer consisting of two atomic layers was calculated to be 9.97 Å. 37 For this distance between the graphene layers, the tunneling conductance between the layers can be estimated to be 18 orders of magnitude less than for bilayer graphene. Thus, the leakage current of capacitors based on double-layer graphene with the krypton and argon spacers is sufficiently small to apply them in fast-acting nanodevices.
Up to now the AA stacking of graphene layers is found only for bilayer graphene with common folded edge 2 or local regions of multi-layer graphene on the C-terminated SiC substrate. 3 We have found that at the ground state of double-layer graphene with the commensurate krypton spacer, the AA stacking of graphene layers takes place. The AA-stacking can be realized for this heterostructure for an arbitrary size of neighbor graphene layers, structure and relative positions of their edges. A set of various phenomena was predicted for the AAstacked bilayer graphene. [11][12][13][14][15] We believe that the considered AA-stacked double-layer graphene with the dielectric spacer holds great promise both for studies of fundamental phenomena and for the use in nanoelectronics. The ground state of this heterostructure with the AA stacking of graphene layers is found to correspond to the maximum in the dependence of the tunneling conductance between the graphene layers on their in-plane relative displacement. Therefore, any relative displacement of the layers causes a decrease of the tunneling conductance and, thus, this heterostructure can be perspective for elaboration of nanoresonator-like sensors based on measurements of the amplitude of relative in-plane vibrations of the graphene layers through measurements of the tunneling conductance between them (such sensors were proposed on the basis of double-walled carbon nanotubes 71,82 ). This is different from the bilayer graphene with the AB stacking in the ground state where the direction of relative in-plane displacement of the layers determines whether a decrease or an increase of the tunneling conductance takes place.
Based on the potential energy relief describing relative in-plane displacements of the graphene layers with the commensurate krypton spacer calculated within the DFT-D2 method, the shear mode frequency for this heterostructure is estimated to be -1 10 5 cm f .  . Recently Raman measurements of the shear mode of few-layer graphene were reported. 72 The analogous measurements for the considered heterostructure could be used to test the calculations performed and, therefore, to test the adequacy of the van der Waals-corrected density functional theory for consideration of the interaction between graphene and inert gases.