Strain Engineering in Monolayer Materials Using Patterned Adatom

Jul 22, 2014 - pattern produces strains of as large as 5%, close to the strain in the adsorbed region. Also, nonzero maximum shear strain (∼4%)...
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Strain Engineering in Monolayer Materials Using Patterned Adatom Adsorption Yao Li,† Karel-Alexander N. Duerloo,‡ and Evan J. Reed*,‡ †

Department of Applied Physics, Stanford University, Stanford, California 94305, United States Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States

‡

S Supporting Information *

ABSTRACT: We utilize reactive empirical bond order (REBO)-based interatomic potentials to explore the potential for the engineering of strain in monolayer materials using lithographically or otherwise patterned adatom adsorption. In the context of graphene, we discover that the monolayer strain results from a competition between the in-plane elasticity and out-of-plane relaxation deformations. For hydrogen adatoms on graphene, the strain outside the adsorption region vanishes due to outof-plane relaxation deformations. Under some circumstances, an annular adsorption pattern generates homogeneous tensile strains of approximately 2% in graphene inside the adsorption region, approximately 30% of the strain in the adsorbed region. We find that an elliptical adsorption pattern produces strains of as large as 5%, close to the strain in the adsorbed region. Also, nonzero maximum shear strain (∼4%) can be introduced by the elliptical adsorption pattern. We find that an elastic plane stress model provides qualitative guidance for strain magnitudes and conditions under which strain-diminishing buckling can be avoided. We identify geometric conditions under which this effect has potential to be scaled to larger areas. Our results elucidate a method for strain engineering at the nanoscale in monolayer devices. KEYWORDS: Monolayer materials, strain engineering, buckling analysis, molecular dynamics (MD), AIREBO potential, graphene

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monolayers with their substrates may limit such approaches from being employed for these materials. For example, the strain in monolayer graphene on a poly(ethylene terephthalate) (PET) substrate is reported to achieve up to 1.2%−1.6% due to substrate friction.15 Ideally, one would like methods of strain engineering for monolayer materials that may be employed using standard lithographic techniques. Furthermore, one would like to maximize the area over which strain can be achieved for devices. Here, we explore the potential for the control of strain over large areas in monolayer materials using patterned adatom adsorption. This work explores a fundamentally new approach to strain engineering that is applicable to monolayers with weak substrate interaction. We find that the strain magnitudes and states demonstrated using this approach are potentially larger than achievable using flexible substrates and have the potential to be scaled up to areas larger than demonstrated in our calculations. Deposition of adatoms on graphene can introduce strain in the deposited area on the order of several percent. In the Supporting Information of ref 13, in-plane tensile strain due to atom adsorption on graphene (single adatom on a 1 × 1 graphene, 2 C atom unit cell) has been reported to range up to

train engineering has been used to improve properties of materials for use in electronic devices.1 One example is strained silicon technology, which has been widely employed to enhance electron or hole mobility in metal-oxide-semiconductor field-effect transistors (MOSFETs).2−4 One might expect strain engineering to be at least as relevant in monolayer materials because they can sustain large strains unachievable in bulk materials. For example, the elastic tensile strain limit of a graphene monolayer has been experimentally measured to be as high as 10%.5 The engineering of strain in monolayer materials has been proposed as the basis for a variety of electronic and optical devices.6−11 Strain in graphene has been shown to generate quantized Landau-like electronic levels, as if a magnetic field were applied.7,8 Strain-induced phonon softening and bandgap modulation have been observed in MoS29,10 and strained monolayer WS2 regions have been identified as effective catalysts for hydrogen evolution.11 Also, strain engineering in piezoelectric monolayer materials may have potential applications in electronic and nanoelectromechanical systems (NEMS).12−14 Recent computational results predict that some monolayer transition metal dichalcogenides (TMDs) exhibit stronger piezoelectricity than bulk wurtzite structures like GaN.12 Monolayer graphene sheets with selective surface adsorption of atoms on one side can exhibit piezoelectric magnitudes comparable to some bulk piezoelectric materials.13,14 Strain engineering can be achieved in bulk materials using heterostructures,2,3 but the relatively weak interaction of © 2014 American Chemical Society

Received: March 14, 2014 Revised: June 17, 2014 Published: July 22, 2014 4299

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In Table 1, Young’s modulus and Poisson’s ratio are calculated from the elastic modulus C11 and C12. The elastic moduli are calculated at 0 K by doing quadratic curve fitting for strain energy U = (1/2)Cijεiεj (Voigt notation). Due to the symmetry of the two-dimensional graphene layer at 0 K, the Young’s modulus E and Poisson’s ratio ν can be simplified to28 E = (C112 − C122)/C11 and ν = C12/C11. Symmetry also gives C11 = C22 and C12 = C21. Therefore, strain energy in graphene can be further simplified to U = (1/2)C11(ε12 + ε22) + C12ε1ε2, when only normal strain ε1 = εxx and ε2 = εyy components are present. The curve fitting is performed on an 11 × 11 grid in (εxx, εyy), where εxx and εyy range from −1% to 1% with an increment of 0.2%. The quadratic curve fitting results are C11 = 385.73 ± 2.80 N/m and C12 = 53.86 ± 1.75 N/m. The fit errors suggest the quadratic function fits the MD results reasonably for small strain. The bending rigidity is derived from calculations of the bending energy of single-walled carbon nanotubes (SWCNTs) as a function of the radius using the approach of ref 27. In this work, the graphene sheet is placed on a relaxed graphite substrate. van der Waals binding with the substrate is expected to play a role in limiting out of plane wrinkles in the monolayer. Two parameters relate to the interaction between graphene and graphite−the elastic modulus C33 of graphite and the interlayer cohesive energy of graphite. The elastic modulus C33 can be calculated by fitting to the elastic strain energy of graphite U = (1/2)C33εzz2.The present potential predicts a C33 value of 39.2 GPa, compared to the experimental29 value of 38.7 GPa. The interlayer cohesive energy of graphite calculated using the present potential is 51 meV/atom, compared to the experimental30 value of 47−57 meV/atom. These potential parameters and the elastic parameters in Table 1 are in reasonable agreement with reference values, indicating that this version of the AIREBO potential provides a reasonable description of the relevant mechanical properties. Friction on the substrate could also potentially play a role in the resulting strain. The AIREBO potential employed here has been reported to underestimate the interplanar sliding energy barrier height for graphite by approximately a factor of 10 relative to van der Waals density functional (vdW-DF) calculations.31 Rotationally misaligned graphene layers are reported to have significantly lower friction than the aligned layers found in crystals.32 Graphene synthesis and experiments have been performed on a number of different substrates, ranging from SiO2,33 Si,34 SiC,35 and graphite36 to various metals.37−39 The results here are expected to be applicable to cases of weak graphene substrate friction. Annealing molecular dynamics is utilized to evolve the systems from initial configuration to final strained states. Periodic boundary conditions are employed in all MD simulations and a vacuum space greater than 5 nm exists in the direction perpendicular to the graphene sheet and graphite substrate. Annealing simulations are performed using a constant temperature canonical ensemble (NVT) with a time step of 1 fs at room temperature. A Nosé−Hoover thermostat is used. Total simulation time ranges from 50 to 100 ps and simulation stops when the average potential energy shift is smaller than 5 μeV/ps per atom. The calculation of strain in monolayer materials has been discussed in previous works.28,40 In this work, the Green− Lagrangian strain tensor28 is used to describe local deformation in the graphene layer. The Green−Lagrangian strain tensor is calculated in Cartesian coordinates and converted to polar

6.3%, for various adatom types. There is potential for this strain to extend well beyond the doped region into the pristine monolayer region. Some methods of spatially patterning the adsorption of hydrogen and fluorine atoms have been demonstrated using electron-beam lithography16 and other techniques.17,18 Metal electrical contacts on monolayers can also potentially introduce strained regions adjacent to the contact and play a role in contact resistances that are known to be important in nanoscale devices.19,20 We choose to focus on the specific case of a graphene sheet modified by adsorbed hydrogen adatoms on the top layer of a graphite (ABAB stacking sequence) substrate. Adsorption of hydrogen results in a local expansion of the graphene lattice.13 We expect these results to be qualitatively analogous to other monolayer materials, substrates, and adatom types. We explore the strain magnitude that can be generated using a variety of annular and elliptical adatom adsorption patterns that can be generated using lithography and other techniques.16−18 We find that the resulting monolayer mechanics is a competition between out-of-plane strain relaxation deformations, in-plane elasticity, and substrate interaction. We find that filled circular adsorption patterns have little or no effect on the strain in the surrounding graphene sheet, whereas other doping patterns, such as an annulus, have the potential to introduce tensile strains of a few percent over areas of multiple nanometer scale dimensions. We find that the elastic plane stress model, which assumes that the monolayer is planar and there is no substrate friction, predicts qualitatively correct behavior in some conditions and provides guidance regarding the geometries under which strain-reducing out-of-plane deformations occur. Methods. Molecular dynamics (MD) simulations are performed using the adaptive intermolecular reactive empirical bond order (AIREBO) potential21 excluding the torsional term as implemented in the classical molecular dynamics software package, Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).22 On the basis of the second generation reactive empirical bond order (REBO) potential,23−26 the AIREBO potential has been widely used to simulate graphene. To study the performance of this interatomic potential for the present application to graphene, we compute the elastic and other energetic properties that are expected to be most relevant. In Table 1, we calculate and compare four parameters Table 1. Computed AIREBO Potential (without Torsion Term) Parameters of Graphene this work reference

a (Å)

E (N/m)

2.42 2.46 (exp.a)

377 290−390 (exp.b)

ν 0.14 0.16 (exp.c)

D (eV) 1.49 1.46 (ab initiod)

a Reference 46. bReference 5. cThe experimental value of ν is the Poisson’s ratio for graphite in the basal plane.47 dReference 27.

of graphene: lattice constant a, Young’s modulus E, Poisson’s ratio ν, and bending rigidity D. We have calculated these four parameters of graphene with and without the torsional term and find that presence of the torsional term only impacts the bending rigidity. Omitting the torsional term leads to a bending rigidity value of 1.49 eV, close to the reported ab initio calculation value of 1.46 eV,27 whereas introducing the torsional term leads to a much smaller value of 0.97. Therefore, this torsion term is disabled in our simulations. 4300

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coordinates through a rotation matrix in some cases. Strain is calculated at the end of each simulation when the system is relaxed and at equilibrium. Details are described in the Supporting Information. A relaxed, unstrained graphene sheet is used as the reference to calculate the local strain. There is no requirement for a Cauchy−Born type of assumption41,24 because the calculation is fully atomistic. Curvature of the sheet is a higher order effect that has potential to introduce error in this approach when displacements due to curvature are large on the scale of a bond length. We expect these curvature effects to be negligible for the graphene regions of our calculations. To help interpret and guide our atomistic calculations, we have also applied the elastic plane stress model42 for an elastic material constrained to a plane. This model (described in Supporting Information) assumes that the monolayer materials are strictly planar with no substrate friction and can shed light on the role of out-of-plane distortion observed in the atomistic simulations. The magnitude of the lattice constant change in the modified graphene region plays a critical role in the resulting strain of surrounding pristine graphene. Here, we focus on the case of the adsorption of one hydrogen adatom on every 1 × 1 graphene, 2 C atom unit cell (C2H). The lattice constant change, or strain upon adsorption, calculated using the present potential is 5.8%, compared to the DFT13 value of 2.6%. According to plane stress model, all strain components observed in the graphene are proportional to the strain induced upon atom adsorption (ΔRI/RI in eq 3 and 4 in Supporting Information). One might expect to observe smaller strain magnitudes in C2H experiments than those reported here because the DFT values of strain are smaller than those of AIREBO. However, the plane stress model indicates that the ratio of the strain in the nonmodified graphene region to the strain upon adsorption is expected to be independent of ΔRI/RI and, therefore, independent of the magnitude of strain induced by adatom adsorption. DFT-based results for in-plane strain due to atom adsorption on graphene have been reported13 to vary from 2.5% to 6.3%, for a variety of adatom types including lithium (Li), potassium (K), hydrogen (H), and fluorine (F) atoms. The results obtained with the potential here are expected to be quantitatively applicable to adatom cases that fall on the higher end of this range of strains. Results and Discussion. The first case is investigated in Figure 1. A circular graphene sheet is placed on a relaxed graphite substrate with the bottom layer frozen and its central region is modified on one side by hydrogen atoms (C2H). The equilibrium lattice constant change is 5.8% in the chemically modified region. The radius of the graphene sheet is RII = 20 nm and the radius of the functionalized region is RI = 3 nm. Periodic boundary conditions are employed for the substrate with the computation cell length L = 44 nm, large enough to ensure that the graphene sheet on it does not interact with periodic images. Figure 1B shows the strain distribution at the end of an annealing MD simulation “3D MD” (blue lines) and Figure 1C shows the graphene at the end of the simulation. Expansion of the modified region results in a large circular out-of-plane ripple around the boundary, which can potentially impact the strain distribution in the graphene around the chemically modified region. In fact, Figure 1B shows that the strain is nearly negligible outside the modified region.

Figure 1. Effect of adatom adsorption on strain in the graphene sheet outside the adsorption region. (A) Shown are a side view and top view of the initial configuration of the system before annealing by molecular dynamics (MD) simulation. Hydrogen atoms are adsorbed on one side in the circular central region of a circular graphene sheet (C2H), which is placed on a relaxed graphite substrate of three layers with the bottom layer frozen. (B) Comparison of the strain in the graphene sheet outside the adsorption region in three cases. Blue lines are the results of an unconstrained MD simulation of the system shown in (C); red lines are MD results of the same system but in which the graphene sheet is constrained in the plane, that is, not allowed to relax in the direction perpendicular to the sheet; green lines are the prediction of the elastic plane stress model.42 In “3D MD”, the out-ofplane ripple causes strain to vanish quickly outside the adsorption region, whereas in the other two cases, the strain decreases much more slowly with radius. (C) Shown is a zoom-in snapshot of the hydrogen adsorption region in the “3D MD” simulation. A circular out-of-plane ripple is formed around the boundaries.

To shed some light on the role of the out of plane wrinkle in this result, Figure 1B shows a constrained MD simulation “2D Constrained MD” in which the graphene sheet is constrained to be planar and not allowed to relax in the direction perpendicular to the graphene sheet (red lines). Also shown in Figure 1B are predictions of the plane stress model42 for the strain in the graphene sheet outside the adatom adsorption region (green lines). From Figure 1B, we can see that the simulation results of “2D Constrained MD” are consistent with the prediction of the elastic plane stress model, and are quite different from the “3D MD” results. In “3D MD”, both radial and tangential strains vanish rapidly outside the adsorption region, whereas in the other two cases, the strain decreases much more slowly. This difference suggests that the relaxation in the direction perpendicular to the graphene sheet plays a key role in determining the magnitude of the strain in the graphene sheet outside the adsorption region. In Figure 2, we investigate the effect of adatom adsorption on the strain inside the adsorption region by considering an annular pattern. One might expect expansion of the adsorbed region to generate tensile strain in the central graphene region. Two unconstrained annealing MD simulations have been performed for the system with initial configurations shown in Figure 2A. In the center of an infinite graphene sheet on a relaxed graphite substrate, an annular region is modified on a single side by hydrogen atoms (C2H). The inner and outer radii of the adsorption region are RI and RII respectively. Periodic boundary conditions are employed for the substrate and the 4301

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to provide some qualitative guidance in this case. The radial ripples, however, will result in some out-of-plane displacements inside the modified region and greatly relax the strain in this region. Comparison of the two simulations here suggests that radial buckling is more likely to occur in a system with a larger value of RI/RII and there might be a critical ratio of RI/RII, above which radial buckling will start to appear. To study the formation of radial buckling, we performed five simulations with the same outer radius RII = 10 nm, but different inner radii RI = 2, 3, 4, 5, 6 nm. The zoom-in snapshots of the adatom adsorption region at the end of MD simulations are shown in Figure S1 of Supporting Information. At RII = 10 nm, when RI/RII ≤ 0.3, there is only a circular buckling mode around the outer edges of the adsorption region. When RI/RII > 0.3, strain-diminishing radial modes appear in the adsorption region. Some insight into the onset of radial buckling modes for RI/ RII > 0.3 can be obtained by comparing the elastic energy density to the binding energy of the monolayer with the substrate. The substrate is expected to play a role here, unlike the buckling of a free-standing plate. When RI/RII increases, the elastic plane stress model predicts the tangential stress component in the adsorption region σθII(r) increases and is compressive (see Figure S2 and eq 1 and 4 in Supporting Information). Beyond some critical value, the elastic energy can be partly relaxed through the formation of radial buckling and possible sliding. At critical ratio of RI/RII = 0.3 (RII = 10 nm), the strain energy density at r = RI is approximately 0.3 J/m2 given by the plane stress model. The binding energy density between the graphene and graphite substrate is approximately 0.5 J/m2 (calculated from the interlayer cohesive energy of graphite using AIREBO potential ∼51 meV/atom). If there is no sliding, one might expect this to be an upper bound on the elastic energy that could exist without spontaneous wrinkle formation. If the monolayer can slide on the substrate, one could expect that wrinkles will form at lower elastic energy densities as observed in the MD. These results suggest that one might be able to engineer the size and ratio of the adatom adsorption region to limit the existence of these radial ripples and, therefore, the strain. Inspired by the level of experimental control over the shape of the adsorbed region achievable by lithographic techniques,16 Figure 3 shows a comparison of the strain inside a circular adsorption region and an elliptical adsorption region. The structures of these two systems are similar to the annular case in Figure 2A, but sizes are scaled up. Both systems employ periodic boundary conditions with the same computational cell length of L = 120 nm and the same outer radius of the adsorption region RII = 30 nm. For the top system, the inner radius RI = 6 nm; for the bottom system, the major radius of the inner elliptical region is RIa = 6 nm and the minor radius is RIb = 2 nm. There is no radial bucking mode in the adsorption region in either cases. In Figure 3, we calculate and compare the strain inside the adsorption region. The normal strain along the major axis of the elliptical adsorption region has a larger value (5%) than the short axis (1%), close to the strain upon adsorption (5.8%). This calculation suggests that the elliptical adsorption pattern may be employed to engineer large anisotropic strain. The stress state in this interior region is not equi-biaxial. Shear strain in monolayer materials has been predicted to introduce changes in electronic properties.43,44 One would like to know the magnitude of shear strain generated in the

Figure 2. Effect of adatom adsorption on strain in the graphene sheet inside the annular adsorption region. (A) Shown are a side view and top view of the initial configuration of the system used for the MD simulation. Hydrogen atoms are adsorbed on one side in an annular region in the center of a graphene sheet (C2H), which is placed on a relaxed graphite substrate of three layers with the bottom layer frozen. (B) (C) Shown are the out-of-plane displacement Δz and the strain in the graphene sheet inside the adsorption region for two simulations of different RI. Smaller RI leads to larger tensile strain in the center region (average of 2.1%). Larger RI leads to radial wrinkles shown in (D) that reduce the strain in the interior. In the top snapshot in (D), only a circular ripple around the outer edges is clearly seen.

modified graphene. The two systems simulated have the same RII but differ in RI, as shown in Figure 2. Figure 2B shows the out-of-plane displacement Δz in this region. In the case of smaller RI, there is almost no out-of-plane displacement anywhere in this region, from the center to the edges. However, in the bottom system with larger RI, Δz around the edges can be as large as 0.2−0.3 nm. This suggests that the outside adsorption region has caused some ripples inside, which could affect the strain in this region. Figure 2C shows the strain in the radial coordinate calculated for this region. In the top system, the average of both radial and tangential strain is approximately 2%, approximately 34% of the strain in the adsorbed region. These strain magnitudes compare favorably with the predictions of the elastic plane stress model of eq 4 in Supporting Information(1.9%). [We employ the plane stress model here, which assumes the stress is zero at RII, because we find that the circular out-of-plane ripple at RII relaxes the stress and strain in the region r > RII. The magnitude of relaxation can be seen in Figure 1. This is an approximation.] In all cases where radial wrinkles are not observed, the strain in the interior graphene region is macroscopically homogeneous as predicted by the plane strain model. Relatively small microscopic inhomogeneities are caused by thermal fluctuations at 300 K. In the bottom system with larger RI, the average of radial and tangential strain is only approximately 0.5%, approximately 58% smaller than the prediction of the elastic plane stress model (1.2%). These differences with the plane stress model suggest that there are two different mechanisms determining the strain in these two systems. Figure 2D shows the zoom-in snapshots of the adatom adsorption region at the end of MD simulations. In the top snapshot, there is a large circular ripple around the outer edges, whereas in the bottom snapshot, several radial ripples have replaced the large circular ripple. The large circular ripple around the outer edges does not cause large out-of-plane displacements inside and the elastic plane stress model appears 4302

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Figure 3. Comparison of strain in annular and elliptical adatom adsorption patterns. The top case has the same geometry as the top system in Figure 2C, but RI, RII, and L are scaled up by a factor of 3 (RII = 30 nm, RI/RII = 0.2, and L = 120 nm.). The system of the bottom case has the same size of RII, and L as the top case, but the shape of the pristine graphene sheet inside the adsorption region is elliptical. The major radius of the inner elliptical region is RIa = 6 nm and the minor radius is RIb = 2 nm. Calculated and shown are the strains in the graphene sheet inside the adsorption region. In both cases, there is no radial buckling across the adsorption region. In the top annular pattern, εxx and εyy has a value of ∼2%. In the bottom elliptical case, the strain along the major axis of the elliptical adsorption pattern has a larger value of (5%) than the short axis (1%), close to the strain in the adsorption region (5.8%). Maximum shear strain is ∼4%. The elliptical pattern provides non-equi-biaxial strain tunable with ellipticity.

Figure 4. Study of the potential for scaling up the annular deposition pattern without radial strain-reducing wrinkle formation. Depicted are the strains in the interior graphene regions for four different size patterns with the same ratio RI/RII = 0.2. The top system has exactly the same size as the top system shown in Figure 2 (RI = 2 nm, RII = 10 nm, L = 40 nm). From the second row to bottom, the system scales up by factors of 2, 3, and 4. In all these four scales, strain εr and εθ is of similar magnitude ∼2%.

adsorption patterns that can be generated by spatial patterning of adsorbates. Here, we consider the maximum shear strain (γmax)/2 = (((εxx − εyy)/2)2) + ((γxy)/2)2)1/2, where (γxy)/2 is the off-diagonal term in Green−Lagrangian strain tensor. In the case of the circular and annular adsorption patterns, εxx ≈ εyy and γxy is negligible. Therefore, the maximum shear strain is approximately zero. In the case of the elliptical adsorption pattern shown in Figure 3, we find that γxy is negligible, but εxx is much larger than εyy. The maximum shear strain γmax has the magnitude of 4%. This nonzero maximum shear strain can be tuned by controlling the aspect ratio of the ellipse. Many lithographic approaches are capable of creating adsorption patterns at the microscale, rather than the nanoscale considered here, and one would like to know how these results scale up. The plane stress model (eq 4 in Supporting Information) suggests that the strains and stresses should be scale independent for fixed RI/RII, that is, σr = σr(r/RII). Therefore, one might expect that an annular pattern with RI/RII chosen such that no radial wrinkles are present will have the best chance of being scaled up to larger sizes without wrinkles forming. Inspired by this observation, we choose the top system in Figure 2 where no wrinkles are observed, an annular adsorption pattern in which RI = 2 nm, RII = 10 nm (RI/RII =0.2), and L = 40 nm, and scale it up by factors of 2, 3, and 4. The strain in the pristine graphene sheet inside the adsorption region has been calculated and compared in Figure 4. The strain εr and εθ in all four scales has similar magnitude ∼2%, consistent with the plane stress model discussed in Supporting Information. The absence of radial wrinkles in these calculations suggests that increasing the size while maintaining fixed RI/RII is a strategy for producing larger areas of strained graphene. Other wrinkling effects may play a role at larger length scales beyond our simulations. When the total elastic energy becomes large, sliding of the monolayer on the substrate is likely to enable wrinkle formation. For a case where radial

wrinkles are observed at small scales, Figure S3 of the Supporting Information shows how the wrinkles persist as the system size is scaled up. Conclusion. Advances in the isolation and fabrication of monolayer materials combined with their excellent mechanical strength have generated a lot of interest in their potential for use in novel electronic and optical devices, ranging from field effect transistors19,45 to solar cells10 and effective catalysts.11 Strain engineering has been proposed as the basis of a variety of such devices. One would like to produce strain over large areas in these monolayer materials using methods comparable to standard lithographic techniques. In this work, we explore the potential for the control of strain at the nanoscale in monolayer materials through patterned adatom adsorption, which has been employed in the laboratory. Our results show that the resulting monolayer strain is a competition between the in-plane elasticity and out-of-plane relaxation deformations. Outside the adatom adsorption region, the strain quickly vanishes due to expansion of the adsorbed region and the out-of-plane relaxation deformation. Inside the annular adsorption region, however, large strains of a few percent can be generated. We find that an elliptical adsorption pattern proves to have the potential to produce larger strain along a specific direction and nonzero maximum shear strain (∼4%) can be generated. We find that an elastic plane stress model provides qualitative guidance for strain magnitudes and conditions under which buckling can be avoided. We identify geometric conditions under which this effect has potential to be scaled to larger areas. These conclusions are expected to be qualitatively applicable to other monolayer materials, including monolayer TMDs, which 4303

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are of interest in part due to their nonzero bandgap. We expect our investigation to provide guidance for the engineering of strain in electronic and optical devices made from monolayer materials.

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ASSOCIATED CONTENT

S Supporting Information *

Elastic plane stress model, calculation of strain, and scale effect (in the case of radial buckling). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Tel: [+1] (650) 723-2971 Fax: [+1] (650) 725-4034. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Our work was supported in part by the U. S. Army Research Laboratory, through the Army High Performance Computing Research Center, Cooperative Agreement W911NF-07-0027. This work was also partially supported by DARPA YFA grant N66001-12-1-4236, and used resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Some calculations were performed in part using the Stanford NNIN Computing Facility (SNCF), a member of the National Nanotechnology Infrastructure Network (NNIN), supported by the National Science Foundation (NSF). We thank Prof. D. Barnett of Stanford for his generous assistance and insights with the elastic model employed here.

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