Maier M, M.Mattheakis, Kaxiras E, Luskin M, Margetis D. Homogenization of plasmonic crystals: Seeking the epsilon-near-zero behavior. Proceedings of the Royal Society A. 2019;475 (2230). Publisher's VersionAbstract
By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell{\textquoteright}s equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host.\  This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. By using a Drude model for the surface conductivity of the sheet, we construct a Lorentzian function that describes the effective dielectric permittivity tensor of the plasmonic crystal as a function of frequency.
Choukroun J, Pala M, Fang S, Kaxiras E, Dollfus P. High performance tunnel field effect transistors based on in-plane transition metal dichalcogenide heterojunctions. NANOTECHNOLOGY. 2019;30 (2).Abstract
In-plane heterojunction tunnel field effect transistors based on monolayer transition metal dichalcogenides are studied by means of self-consistent non-equilibrium Green's functions simulations and an atomistic tight-binding Hamiltonian. We start by comparing several heterojunctions before focusing on the most promising ones, i.e. WTe2-MoS2 and MoTe2-MoS2. The scalability of those devices as a function of channel length is studied, and the influence of backgate voltages on device performance is analyzed. Our results indicate that, by fine-tuning the design parameters, those devices can yield extremely low subthreshold swings (<5 mV/decade) and I-ON/I-OFF ratios higher than 10(8) at a supply voltage of 0.3 V, making them ideal for ultra-low power consumption.
Ma Q, Xu S-Y, Shen H, MacNeill D, Fatemi V, Chang T-rong, Valdivia AMM, Wu S, Du Z, Hsu C-H, et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. NATURE. 2019;565 (7739) :337+.Abstract
The electrical Hall effect is the production, upon the application of an electric field, of a transverse voltage under an out-of-plane magnetic field. Studies of the Hall effect have led to important breakthroughs, including the discoveries of Berry curvature and topological Chern invariants(1,2). The internal magnetization of magnets means that the electrical Hall effect can occur in the absence of an external magnetic field(2); this `anomalous' Hall effect is important for the study of quantum magnets(2-7). The electrical Hall effect has rarely been studied in non-magnetic materials without external magnetic fields, owing to the constraint of timer-eversal symmetry. However, only in the linear response regime-when the Hall voltage is linearly proportional to the external electric field-does the Hall effect identically vanish as a result of time-reversal symmetry; the Hall effect in the nonlinear response regime is not subject to such symmetry constraints(8-10). Here we report observations of the nonlinear Hall effect(10) in electrical transport in bilayers of the non-magnetic quantum material WTe2 under time-reversal-symmetric conditions. We show that an electric current in bilayer WTe2 leads to a nonlinear Hall voltage in the absence of a magnetic field. The properties of this nonlinear Hall effect are distinct from those of the anomalous Hall effect in metals: the nonlinear Hall effect results in a quadratic, rather than linear, current-voltage characteristic and, in contrast to the anomalous Hall effect, the nonlinear Hall effect results in a much larger transverse than longitudinal voltage response, leading to a nonlinear Hall angle (the angle between the total voltage response and the applied electric field) of nearly 90 degrees. We further show that the nonlinear Hall effect provides a direct measure of the dipole moment(10) of the Berry curvature, which arises from layer-polarized Dirac fermions in bilayer WTe2. Our results demonstrate a new type of Hall effect and provide a way of detecting Berry curvature in nonmagnetic quantum materials.
Hoyt RA, Montemore MM, Sykes ECH, Kaxiras E. Anhydrous Methanol and Ethanol Dehydrogenation at Cu(111) Step Edges. JOURNAL OF PHYSICAL CHEMISTRY C. 2018;122 (38) :21952-21962.Abstract
Oxidative methanol dehydrogenation is a major industrial reaction with global formaldehyde production exceeding 30 million tonnes per year. Unfortunately, oxidative dehydrogenation produces water aldehyde mixtures that require subsequent distillation. Anhydrous alcohol dehydrogenation is a promising alternative that produces H-2 instead of water. Pursuant to recent experimental work showing that highly stepped Cu(111) surfaces exhibit anhydrous dehydrogenation activity, we present first-principles density functional theory calculations for methanol and ethanol dehydrogenation at Cu(111) step edges to provide an atomistic understanding of the catalytic mechanism; these sites stabilize all intermediates while reducing activation energies. We find that van der Waals contributions to the energy account for more than 50% of adsorption energies, and essential in achieving good agreement with experimental desorption temperatures. Furthermore, vibrational zero-point energy corrections significantly reduce the activation energy for all reaction steps considered here. Hydrogen bonding among ethanol intermediates at step edges is weakened by geometric frustration. These insights lead us to propose several suggestions for further research on undercoordinated Cu sites as anhydrous alcohol dehydrogenation catalysts.
You J-S, Fang S, Xu S-Y, Kaxiras E, Low T. Berry curvature dipole current in the transition metal dichalcogenides family. PHYSICAL REVIEW B. 2018;98 (12).Abstract
We study the quantum nonlinear Hall effect in two-dimensional (2D) materials with time-reversal symmetry. When only one mirror line exists, a transverse charge current occurs in the second-order response to an external electric field, as a result of the Berry curvature dipole in momentum space. Candidate 2D materials to observe this effect are two-dimensional transition metal dichalcogenides (TMDCs). First, we use an ab initio based tight-binding approach to demonstrate that monolayer T-d-structure TMDCs exhibit a finite Berry curvature dipole. In the 1H and 1T' phase of TMDCs, we show the emergence of a finite Berry curvature dipole with the application of strain and an electrical displacement field, respectively.
Chen W, Cubuk ED, Montemore MM, Reece C, Madix RJ, Friend CM, Kaxiras E. A Comparative Ab Initio Study of Anhydrous Dehydrogenation of Linear-Chain Alcohols on Cu(110). JOURNAL OF PHYSICAL CHEMISTRY C. 2018;122 (14) :7806-7815.Abstract
The catalytic behavior of Cu surfaces in the anhydrous production of aldehydes from alcohols, a process of industrial significance, is puzzling: the two simplest alcohols (methanol and ethanol) show dramatically different decomposition behavior on Cu. Here, we study the thermodynamic and kinetic processes involved in the anhydrous dehydrogenation of linear-chain alcohols including methanol, ethanol, 1-propanol, and 1-butanol on the Cu(110) surface using multiscale approaches. First, we obtain the adsorption structures and energies of the reaction intermediates, in which van der Waals (vdW) interactions play a crucial role. Then, we determine the kinetic barriers for the two dehydrogenation steps, namely, the O-H and the subsequent C-H bond-breaking on Cu. The reaction of methoxy-to-formaldehyde has a rather high-energy transition state, in contrast to that of alkoxide-to-aldehyde in the longer-chain systems. This difference qualitatively explains the lower production efficiency of formaldehyde on Cu. Finally, we simulate the production rates of aldehydes based on which we optimize reaction conditions and propose possible avenues for enhancing the production of anhydrous formaldehyde using Cu-based catalysts.
Bakhta A, Cances E, Cazeaux P, Fang S, Kaxiras E. Compression of Wannier functions into Gaussian-type orbitals. COMPUTER PHYSICS COMMUNICATIONS. 2018;230 :27-37.Abstract
We propose a greedy algorithm for the compression of Wannier functions into Gaussian-polynomials orbitals. The so-obtained compressed Wannier functions can be stored in a very compact form, and can be used to efficiently parameterize effective tight-binding Hamiltonians for multilayer 2D materials for instance. The compression method preserves the symmetries (if any) of the original Wannier function. We provide algorithmic details, and illustrate the performance of our implementation on several examples, including graphene, hexagonal boron-nitride, single-layer FeSe, and bulk silicon in the diamond cubic structure. (C) 2018 Elsevier B.V. All rights reserved.
Cao Y, Fatemi V, Demir A, Fang S, Tomarken SL, Luo JY, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Kaxiras E, et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. NATURE. 2018;556 (7699) :80+.Abstract
A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked two-dimensional building blocks held together by the van der Waals forces between the layers. This design means that the properties of van der Waals heterostructures can be engineered precisely, even more so than those of two-dimensional materials(1). One such property is the `twist' angle between different layers in the heterostructure. This angle has a crucial role in the electronic properties of van der Waals heterostructures, but does not have a direct analogue in other types of heterostructure, such as semiconductors grown using molecular beam epitaxy. For small twist angles, the moire pattern that is produced by the lattice misorientation between the two-dimensional layers creates long-range modulation of the stacking order. So far, studies of the effects of the twist angle in van der Waals heterostructures have concentrated mostly on heterostructures consisting of monolayer graphene on top of hexagonal boron nitride, which exhibit relatively weak interlayer interaction owing to the large bandgap in hexagonal boron nitride(2-5). Here we study a heterostructure consisting of bilayer graphene, in which the two graphene layers are twisted relative to each other by a certain angle. We show experimentally that, as predicted theoretically(6), when this angle is close to the `magic' angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling. These flat bands exhibit insulating states at half-filling, which are not expected in the absence of correlations between electrons. We show that these correlated states at half-filling are consistent with Mott-like insulator states, which can arise from electrons being localized in the superlattice that is induced by the moire pattern. These properties of magic-angle-twisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic many-body quantum phases in two dimensions in the absence of a magnetic field. The accessibility of the flat bands through electrical tunability and the bandwidth tunability through the twist angle could pave the way towards more exotic correlated systems, such as unconventional superconductors and quantum spin liquids.
Kolesov G, Kaxiras E, Manousakis E. Density functional theory beyond the Born-Oppenheimer approximation: Accurate treatment of the ionic zero-point motion. PHYSICAL REVIEW B. 2018;98 (19).Abstract
We introduce a method to carry out zero-temperature calculations within density functional theory (DFT) but without relying on the Born-Oppenheimer (BO) approximation for the ionic motion. Our approach is based on the finite-temperature many-body path-integral formulation of quantum mechanics by taking the zero-temperature limit and treating the imaginary-time propagation of the electronic variables in the context of DFT. This goes beyond the familiar BO approximation and is limited from being an exact treatment of both electrons and ions only by the approximations involved in the DFT component. We test our method in two simple molecules, H-2 and benzene. We demonstrate that the method produces a difference from the results of the BO approximation which is significant for many physical systems, especially those containing light atoms such as hydrogen; in these cases, we find that the fluctuations of the distance from its equilibrium position, due to the zero-point motion, is comparable to the interatomic distances. The method is suitable for use with conventional condensed-matter approaches and currently is implemented on top of the periodic pseudopotential code SIESTA.
Wen M, Carr S, Fang S, Kaxiras E, Tadmor EB. Dihedral-angle-corrected registry-dependent interlayer potential for multilayer graphene structures. PHYSICAL REVIEW B. 2018;98 (23).Abstract
The structural relaxation of multilayer graphene is essential in describing the interesting electronic properties induced by intentional misalignment of successive layers, including the recently reported superconductivity in twisted bilayer graphene. This is difficult to accomplish without an accurate interatomic potential. Here, we present a new, registry-dependent Kolmogorov-Crespi-type interatomic potential to model interlayer interactions in multilayer graphene structures. It consists of two parts, representing attractive interaction due to dispersion and repulsive interaction due to anisotropic overlap of electronic orbitals. An important new feature is a dihedral-angle-dependent term that is added to the repulsive part to describe correctly several distinct stacking states that the original Kolmogorov-Crespi potential cannot distinguish. We refer to the new model as the dihedral-angle-corrected registry-dependent interlayer potential (DRIP). Computations for several test problems show that DRIP correctly reproduces the binding, sliding, and twisting energies and forces obtained from ab initio total-energy calculations based on density-functional theory. We use the new potential to study the structural properties of a twisted graphene bilayer and the exfoliation of graphene from graphite. Our potential is available through the OpenKIM interatomic potential repository at https://openkim org.
Wang Z-T, Hoyt RA, El-Soda M, Madix RJ, Kaxiras E, Sykes ECH. Dry Dehydrogenation of Ethanol on Pt-Cu Single Atom Alloys. TOPICS IN CATALYSIS. 2018;61 (5-6, SI) :328-335.Abstract
The non-oxidative dehydrogenation of ethanol to acetaldehyde and hydrogen is an industrially relevant chemical conversion. Although Cu-based catalysts show high reactivity toward oxidative ethanol dehydrogenation, the flat Cu(111) surface is rather inactive for ethanol dehydrogenation in the absence of water, surface oxygen or defects. Herein we show, using experimental and theoretical studies of model systems, that adding 1% Pt into the surface of Cu(111) to form dilute Pt-Cu single atom alloys (SAAs) increases the activity of Cu(111) for ethanol dehydrogenation sixfold. The mechanism of ethanol dehydrogenation was investigated at the molecular level using scanning tunneling microscopy, temperature programmed experiments and density functional theory calculations. Our results demonstrate that Pt-Cu SAAs are much more active than Cu(111) for converting ethanol to acetaldehyde and hydrogen in the absence of surface oxygen and water. Specifically, the O-H bond of ethanol is activated at Pt sites below 160 K, followed by ethoxy spillover to Cu sites which results in a significant increase of the ethoxy intermediate yield. The C-H bond of ethoxy is then activated at 310 K, and the final product, acetaldehyde, desorbs from Cu(111) in a reaction rate limited process. Finally, we show that the Cu model surfaces exhibit stability with respect to poisoning as well as 100% selectivity in the alcohol dehydrogenation to acetaldehyde and hydrogen.
Fang S, Carr S, Cazalilla MA, Kaxiras E. Electronic structure theory of strained two-dimensional materials with hexagonal symmetry. PHYSICAL REVIEW B. 2018;98 (7).Abstract
We derive electronic tight-binding Hamiltonians for strained graphene, hexagonal boron nitride, and transition-metal dichalcogenides based on Wannier transformation of ab initio density functional theory calculations. Our microscopic models include strain effects to leading order that respect the hexagonal crystal symmetry and local crystal configuration and are beyond the central force approximation which assumes only pairwise distance dependence. Based on these models, we also derive and analyze the effective low-energy Hamiltonians. Our ab initio approaches complement the symmetry group representation construction for such effective low-energy Hamiltonians and provide the values of the coefficients for each symmetry-allowed term. These models are relevant for the design of electronic device applications since they provide the framework for describing the coupling of electrons to other degrees of freedom including phonons, spin, and the electromagnetic field. The models can also serve as the basis for exploring the physics of many-body systems of interesting quantum phases.
Mattheakis M, Tsironis GP, Kaxiras E. Emergence and dynamical properties of stochastic branching in the electronic flows of disordered Dirac solids. EPL. 2018;122 (2).Abstract
Graphene as well as more generally Dirac solids constitute two-dimensional materials where the electronic flow is ultra-relativistic. When a Dirac solid is deposited on a different substrate surface with roughness, a local random potential develops through an inhomogeneous charge impurity distribution. This external potential affects profoundly the charge flow and induces a chaotic pattern of current branches that develops through focusing and defocusing effects produced by the randomness of the surface. An additional bias voltage may be used to tune the branching pattern of the charge carrier currents. We employ analytical and numerical techniques in order to investigate the onset and the statistical properties of carrier branches in Dirac solids. We find a specific scaling-type relationship that connects the physical scale for the occurrence of branches with the characteristic medium properties, such as disorder and bias field. We use numerics to test and verify the theoretical prediction as well as a perturbative approach that gives a clear indication of the regime of validity of the approach. This work is relevant to device applications and may be tested experimentally.
Defo RK, Zhang X, Bracher D, Kim G, Hu E, Kaxiras E. Energetics and kinetics of vacancy defects in 4H-SiC. PHYSICAL REVIEW B. 2018;98 (10).Abstract
Defect engineering in wide-gap semiconductors is important in controlling the performance of single-photon emitter devices. The effective incorporation of defects depends strongly on the ability to control their formation and location, as well as to mitigate attendant damage to the material. In this study, we combine density functional theory, molecular dyamics (MD), and kinetic Monte Carlo (KMC) simulations to study the energetics and kinetics of the silicon monovacancy V-Si and related defects in 4H-SiC. We obtain the defect formation energy for V-Si in various charge states and use MD simulations to model the ion implantation process for creating defects. We also study the effects of high-temperature annealing on defect position and stability using KMC and analytical models. Using a larger (480-atom) supercell than previous studies, we obtain the temperature-dependent diffusivity of V-Si in various charge states and find significantly lower barriers to diffusion than previous estimates. In addition, we examine the recombination with interstitial Si and conversion of V-Si into CSiVC during annealing and propose methods for using strain to reduce changes in defect concentrations. Our results provide guidance for experimental efforts to control the position and density of V-Si defects within devices, helping to realize their potential as solid-state qubits.
Yang Y, Fang S, Fatemi V, Ruhman J, Navarro-Moratalla E, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P. Enhanced superconductivity upon weakening of charge density wave transport in 2H-TaS2 in the two-dimensional limit. PHYSICAL REVIEW B. 2018;98 (3).Abstract
Layered transition-metal dichalcogenides that host coexisting charge-density wave (CDW) and superconducting orders provide ideal systems for exploring the effects of dimensionality on correlated electronic phases. Dimensionality has a profound effect on both superconductivity and CDW instabilities. Here we report a substantial enhancement of the superconducting T-c to 3.4 K for 2H-TaS2 in the monolayer limit, compared to 0.8 K in the bulk. In addition, the transport signature of a CDW phase transition vanishes in the two-dimensional limit. In our analysis of electronic and vibrational properties of this material, we show that a reduction of the CDW amplitude results in a substantial increase of the density of states at the Fermi energy, which can boost T, by an amount similar to that seen in experiment. Our results indicate competition between CDW order and superconductivity in ultrathin 2H-TaS2 down to the monolayer limit, providing insight toward understanding correlated electronic phases in reduced dimensions.
Xu Y, Chen W, Kaxiras E, Friend CM, Madix RJ. General Effect of van der Waals Interactions on the Stability of Alkoxy Intermediates on Metal Surfaces. JOURNAL OF PHYSICAL CHEMISTRY B. 2018;122 (2, SI) :555-560.Abstract
The critical role of noncovalent van der Waals (vdW) interactions in determining the relative thermodynamic stability of alkoxy intermediates has been demonstrated for the Cu(110) surface using a combination of experiment and theory. The results may be significant for the selectivity control of copper-based reactions of alcohols. Previous examination of this effect on Au(110) was also extended to include higher molecular weight alcohols; on Cu(110) and Au(110) the hierarchy for the strength of binding of the alkoxys was found to be the same within experimental accuracy, with alkoxy species of greater chain length being more stable. The equilibrium constants governing the competition of alcohol pairs for binding sites of the alkoxys are also similar on the two surfaces. These results reveal the generality of such vdW effects. This work expands the understanding of the role of vdW interactions on the binding efficacy of key reactive intermediates on metal surfaces, a key factor in the rational design of complex and selective catalytic processes.
Bediako DK, Rezaee M, Yoo H, Larson DT, Zhao SYF, Taniguchi T, Watanabe K, Brower-Thomas TL, Kaxiras E, Kim P. Heterointerface effects in the electrointercalation of van der Waals heterostructures. NATURE. 2018;558 (7710) :425+.Abstract
Molecular-scale manipulation of electronic and ionic charge accumulation in materials is the backbone of electrochemical energy storagel(1-4). Layered van der Waals (vdW) crystals are a diverse family of materials into which mobile ions can electrochemically intercalate into the interlamellar gaps of the host atomic lattice(5,6). The structural diversity of such materials enables the interfacial properties of composites to be optimized to improve ion intercalation for energy storage and electronic devices(7-12). However, the ability of heterolayers to modify intercalation reactions, and their role at the atomic level, are yet to be elucidated. Here we demonstrate the electrointercalation of lithium at the level of individual atomic interfaces of dissimilar vdW layers. Electrochemical devices based on vdW heterostructures(13) of stacked hexagonal boron nitride, graphene and molybdenum dichalcogenide (MoX2;X=S, Se) layers are constructed. We use transmission electron microscopy, in situ magnetoresistance and optical spectroscopy techniques, as well as low-temperature quantum magneto-oscillation measurements and ab initio calculations, to resolve the intermediate stages of lithium intercalation at heterointerfaces. The formation of vdW heterointerfaces between graphene and MoX2 results in a more than tenfold greater accumulation of charge in MoX2 when compared to MoX2/MoX2 homointerfaces, while enforcing a more negative intercalation potential than that of bulk MoX2 by at least 0.5 V. Beyond energy storage, our combined experimental and computational methodology for manipulating and characterizing the electrochemical behaviour of layered systems opens new pathways to control the charge density in two-dimensional electronic and optoelectronic devices.
O'Connor CR, Hiebel F, Chen W, Kaxiras E, Madix RJ, Friend CM. Identifying key descriptors in surface binding: interplay of surface anchoring and intermolecular interactions for carboxylates on Au(110). CHEMICAL SCIENCE. 2018;9 (15) :3759-3766.Abstract
The relative stability of carboxylates on Au(110) was investigated as part of a comprehensive study of adsorbate binding on Group IB metals that can be used to predict and understand how to control reactivity in heterogeneous catalysis. The binding efficacy of carboxylates is only weakly dependent on alkyl chain length for relatively short-chain molecules, as demonstrated using quantitative temperature-programmed reaction spectroscopy. Corresponding density functional theory (DFT) calculations demonstrated that the bidentate anchoring geometry is rigid and restricts the amount of additional stabilization through adsorbate-surface van der Waals (vdW) interactions which control stability for alkoxides. A combination of scanning tunneling microscopy (STM) and low-energy electron diffraction (LEED) shows that carboxylates form dense local islands on Au(110). Complementary DFT calculations demonstrate that adsorbate-adsorbate interactions provide additional stabilization that increases as a function of alkyl chain length for C-2 and C-3 carboxylates. Hence, overall stability is generally a function of the anchoring group to the surface and the inter-adsorbate interaction. This study demonstrates the importance of these two important factors in describing binding of key catalytic intermediates.
Onat B, Cubuk ED, Malone BD, Kaxiras E. Implanted neural network potentials: Application to Li-Si alloys. PHYSICAL REVIEW B. 2018;97 (9).Abstract
Modeling the behavior of materials composed of elements with different bonding and electronic structure character for large spatial and temporal scales and over a large compositional range is a challenging problem. Cases in point are amorphous alloys of Si, a prototypical covalent material, and Li, a prototypical metal, which are being considered as anodes for high-energy-density batteries. To address this challenge, we develop a methodology based on neural networks that extends the conventional training approach to incorporate pre-trained parts that capture the character of different components, into the overall network; we refer to this model as the ``implanted neural network'' method. We show that this approach works well for the Si-Li amorphous alloys for a wide range of compositions, giving good results for key quantities like the diffusion coefficients. The method is readily generalizable to more complicated situations that involve two or more different elements.
Larson DT, Fampiou I, Kim G, Kaxiras E. Lithium Intercalation in Graphene-MoS2 Heterostructures. JOURNAL OF PHYSICAL CHEMISTRY C. 2018;122 (43) :24535-24541.Abstract
Two-dimensional (2D) heterostructures are interesting candidates for efficient energy storage devices due to their high carrier capacity by reversible intercalation. We employ here density functional theory calculations to investigate the structural and electronic properties of lithium intercalated graphene/molybdenum disulfide (Gr/MoS2) heterostructures. We explore the extent to which Li intercalates at the interface formed between graphene (Gr) and molybdenum disulfide (MoS2) layers by considering the adsorption and diffusion of Li atoms, the energetic stability, and the changes in the structural morphology of MoS2. We investigate the corresponding electronic structure and charge distribution within the heterostructure at varying concentrations of Li. Our results indicate that the maximum energetically allowed ratio of Li to Mo (Li to C) is 1:1 (1:3) for both the 2H and 1T' phases of MoS2. This is double the Li concentration allowed in graphene bilayers. We find that there is 60% more charge transfer to MoS2 than to Gr in the bilayer heterostructure, which results in a maximum doping of Gr and MoS2 of n(C) = 3.6 x 10(14) cm(-2) and n(MoS2) = 6.0 x 10(14) cm(-2), respectively.