Letter pubs.acs.org/NanoLett
Environment-Controlled Dislocation Migration and Superplasticity in Monolayer MoS2 Xiaolong Zou, Mingjie Liu, Zhiming Shi, and Boris I. Yakobson* Department of Materials Science and NanoEngineering, Department of Chemistry, and the Smalley Institute for Nanoscale Science and Technology, Rice University, Houston, Texas 77005, United States S Supporting Information *
ABSTRACT: The two-dimensional (2D) transition metal dichalcogenides (TMDC, of generic formula MX2) monolayer displays the “triple-decker” structure with the chemical bond organization much more complex than in wellstudied monatomic layers of graphene or boron nitride. Accordingly, the makeup of the dislocations in TMDC permits chemical variability, depending sensitively on the equilibrium with the environment. In particular, first-principles calculations show that dislocations state can be switched to highly mobile, profoundly changing the lattice relaxation and leading to superplastic behavior. With 2D MoS2 as an example, we construct full map for dislocation dynamics, at different chemical potentials, for both the M- and X-oriented dislocations. Depending on the structure of the migrating dislocation, two different dynamic mechanisms are revealed: either the direct rebonding (RB) mechanism where only a single metal atom shifts slightly, or generalized Stone−Wales (SWg) rotation in which several atoms undergo significant displacements. The migration barriers for RB mechanism can be 2−4 times lower than for the SWg. Our analyses show that within a range of chemical potentials, highly mobile dislocations could at the same time be thermodynamically favored, that is statistically dominating the overall material property. This demonstrates remarkable possibility of changing material basic property such as plasticity by changing elemental chemical potentials of the environment. KEYWORDS: Two-dimensional, transition-metal dichalcogenides, dislocation dynamics, chemical potential, first-principles theory
T
For relatively simple single-atom-layer carbon and h-BN materials, their dislocation structures,22−27 dynamics,28−31 and yield behaviors32−35 have been studied extensively in both theory and experiments. Especially, at high temperature and under electron irradiation, different levels of plasticity have been observed in CNT32,36 or h-BN,37 which can be attributed to the migration of dislocations in CNT or graphene31 or a particular armchair row glide in case of 60° GB in h-BN.37 Different from the case of monoelemental graphene, the heteroelemental composition of complex three-atom-layer TMDC brings about extra variability in the dislocation cores, due to their effective interaction with surrounding point defects38 or direct exchange with gaseous environment.39 Depending on the thermodynamic conditions, this can alter the stable structure of the dislocation, among the several possible derivative cores (within the same topological invariant Burgers vector b⃗). This also alters the organization and strengths of local bonds. One thus might expect that the dislocation dynamics and the related physical behavior of polycrystalline samples can be manipulated, in well-designed experiments with controlled chemical potential μ. While such environmentcontrolled dislocation mobility is of fundamental and practical (as perhaps tunable plasticity) interest, it is challenging to
he intriguing electronic structures of two-dimensional (2D) transition-metal dichalcogenides (TMDCs)1 have motivated researchers to explore effective ways to control their electronic and optical properties, in particular using strain2−4 among others,5−9 or further to attempt new concept of device architectures. For instance, nanoindents in MoS2 as artificial atoms have been proposed as broad-spectrum solar funnels,10 where elastic strain acts as a key agent to continuously tune the optoelectronic properties.11 Further development of various nanodevices based on 2D TMDCs1,12 depends critically on the large-scale chemical vapor deposition (CVD) samples,13,14 where the nanopatterned strain engineering could be applied, as demonstrated for graphene earlier.15,16 Topological defects, such as dislocations and grain boundaries (GB), play critical roles in determining the characteristics of these polycrystalline samples, not only changing the structural integrity, but also influencing a variety of physical properties.13,14,17 Essentially, the mechanical deformation of materials, which determines the feasibility and performance of various device architectures based on these 2D layered structures, is significantly influenced by dislocation dynamics and their response to external stimuli. However, there are only few reports on the details of chemical makeup, bond organization and consequently mechanics of TMDC nanostructures so far, mainly focused on simple perfect matrix18,19 or point defects only,20,21 leaving the crucial behaviors of dislocation dynamics unexplored. © XXXX American Chemical Society
Received: March 4, 2015 Revised: April 9, 2015
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DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 1. (a) The B5|7 + N• and N5|7 + B• cores energies are shown relative to the basic B5|7 and N5|7 dislocations; the numbers above the arcs are corresponding glide barriers, all in eV. (b) Energies of radical-dislocations relative to basic B5|7 (red line) and N5|7 (blue line) versus the chemical potential of boron. Insets show the structures of the derivative cores. The chemical potential for bulk α-B is set to zero. The left three panels in c−d show the initial, saddle-point, and final configurations for basic B5|7 and N5|7 dislocations, respectively, while the corresponding states for their radical cores are shown in the right three columns. Dislocation cores and surrounding hexagons are gray-shaded to highlight the migration event. Red and blue spheres represent B and N atoms, respectively. Pink or cyan circles highlight the main rebonding atom, and green dashed lines indicate the rhomb structures in the rebonding (RB) process.
thermodynamic stability of various derivative dislocation cores is accessed by their formation energies, calculated as
depict and understand the detailed dynamics involved. Here we present such analysis based on our recent successful determination of the dislocation structures.40 Here we employ first-principles calculations to obtain the underlying quantities (formation energies, reaction paths, activation barriers), and to lay out a strategy to control the dislocation dynamics by environmental chemical potentials. We use MoS2 as a representative 2D TMDC material to reveal two migration mechanisms possible for different dislocations: either through the direct rebonding (RB) when only one Mo atom experiences a minor shift, or a concerted shift of atoms in opposite directions resembling a bond-rotation41 as in generalized Stone−Wales mechanisms (SWg). The activation energy barriers for these two mechanisms show significant difference, with those for the RB mechanism remarkably lower. Then, the detailed analysis shows that some dislocations possess high mobility while are also thermodynamically favored (and in this sense “stable”) under certain range of chemical potentials. This clearly demonstrates that the dynamic behavior of the dislocations at the microscopic level, and therefore, the macroscopic plasticity of 2D-sheet can be controlled by the environment conditions. Density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP),42,43 with the Perdew−Burke−Ernzerhof parametrization (PBE)44 of the generalized gradient approximation (GGA) and projector-augmented wave (PAW) potentials.45,46 Our model structures, of either nanoribbon (NR) or periodic boundary condition (PBC) configurations, embed dislocations in small-angle (9°) GBs (see schematic illustration for the models in Supporting Information of ref 40) Hereafter we report results obtained from NR models unless noted. Adopting the supercell approach, we choose the vacuum layer larger than 10 Å to keep the spurious interaction negligible. Using total energy minimization,47 all structures are relaxed until the force on each atom is below 0.01 eV/Å. The
Ef = Etot − Eref − NMoμMo − NSμS
where Etot and Eref are the total energies of the derivative and reference basic dislocation cores respectively; NMo/S is the number of added Mo/S atoms. Under thermodynamic equilibrium, the chemical potentials are restricted by μMo + 2μS = μMoS2, where μMoS2 is the chemical potential of MoS2 unit in their pristine single-layer 2H phase. The range of the chemical potential of sulfur is chosen within −1.3 eV < μS < 0 eV, where MoS2 can remain stable with respect to the segregation of the bulk Mo (μS = −1.3 eV) or bulk α-S (used as the reference, μS = 0 eV).48,49 The same procedure is applied to h-BN for comparison, with the equilibrium balance then being μB + μN = μBN, and the range of the chemical potential of B chosen by the requiring h-BN stability against the formation of N2 molecules or bulk α-B phase. The climbing image nudged elastic band (CI-NEB)50 was used to identify the migration paths for structural transition. To illustrate the concept of environment-controlled dislocation dynamics, we start from structurally simpler heteroelemental 2D material, single-layer h-BN. The glide of basic dislocations follows the Stone−Wales (SW) bond rotation, more known for carbon.31,33,51 In the honeycomb h-BN lattice computations for B5|7 and N5|7 dislocations yield the barriers as high as 5.4 and 4.4 eV, respectively (Figure 1a). We find that barriers could be greatly reduced if the basic dislocation structure is changed to its “radical” derivative, through the adsorption of one additional atom. Hereafter, the derivative cores are named as B5|7 + N• or N5|7 + B•, with a dot mark for such extra atom-“radical”. Due to the local disruption of the sp2 network near such dislocation core, the extra atom carries one unsaturated dangling bond (inset in Figure 1b), to warrant a name of radical-dislocation. The unsaturated bond suggests that B
DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 2. In a−c, left, middle, and right panels present the corresponding starting, saddle-point (intermediate state for S5|7), and final configurations for (a) S5|7, (b) S5|7 + VS, and (c) 4|6, respectively. Dislocation cores and surrounding hexagons are gray-shaded to different degree to highlight the migration step. Black, small white, and large white spheres represent Mo, front-layer, and back-layer S atoms, respectively. The red spheres highlight the main migrating atoms, while the blue arrows in (b) indicate the two back-layer S atoms having counter movement (to left) with respect to that of the front-layer red S atom (to right) in SWg process (see text for discussion). (d) Minimum energy paths during migrations for different S-oriented dislocations.
thermodynamically depends on the environment chemical potential. In a glide step of 5|7 the atomic movements depend on specific dislocation core compositions. For S-oriented dislocations family, these include “normal” S5|7 (Figure 2a) or its element-deficient derivatives (Figure 2b−c), examined below in order of increasing complexity. Under S-poor conditions, the favored S-oriented dislocation is 4|6, which does not include any homoelemental bond. Its glide follows the RB mechanism discussed above, as shown in Figure 2c. The red Mo atom in the rhomb structure (left panel) breaks its bonds with S2 dimer in the middle of 4|6 and then rebonds with S2 to its right side (right panel), while the 4|6 glides one step to the right. The middle panel shows the saddlepoint structure. Although it contains a homoelemental Mo−Mo bond, and the two Mo atoms are under-coordinated, the entire process does not involve significant strain of the surrounding lattice. The calculated activation barrier, E* = 0.5 eV for such a glide path is remarkably low, in sharp contrast to the highbarriers (∼5−10 eV) for dislocation glide in carbon materials and h-BN, where SW rotation is only permissible at high temperature and/or under electron beam, on a longer time scale.28−31,51 The high mobility of 4|6 is progressively reduced if the S atoms are “returned”, forming S5|7 + VS or further S5|7. In the S5|7 + VS glide, the main moving atom is one S of the two in the front layer, forming across the dislocation a S−S bond, shared by the pentagon and heptagon (small red S atom in Figure 2b). This S atom breaks its S−S bond across the core and shifts to the right to form a new S−S bond (right panel in Figure 2b). At the same time, two S atoms in the back layer indicated by arrows, as well as Mo atom (red) in-between, all shift to the lef t. This concerted counter-movement of atoms resembles the SW rotation, albeit is more intricate, and the concomitant structural distortion raises the barrier to E* ≈ 2.2 eV. This higher barrier can be additionally understood by examining the saddle-point structure (middle panel in Figure 2b), which can be seen as adding one bridged S of double coordination to the saddle-point structure of 4|6 (midpanel in Figure 2c). Not only is one more under-coordinated S atom introduced (compared to those in pure matrix), but the saddlepoint structure experiences high stress, manifested by the lateral
radical-dislocation formation energy is positive relative to their basic 5|7 forms, but it varies with the chemical potential, Figure 1b. Importantly, the migration mechanism of these radical cores qualitatively changes to direct RB, with just a single atom moving slightly. For instance, during the migration of B5|7 + N• (right three panels in Figure 1c), the B atom (pink) breaks away (highlighted bond) from N on the right side of the octagon, and then rebonds to the interstitial N•. In the process, the B5|7 + N• glides one lattice unit to the right. Compared with SW process (left three panels in Figure 1c), much smaller distortion in saddle-point structure significantly reduces migration barriers, to 2.6 eV for B5|7 + N• and to 1.7 eV for N5|7 + B•, greatly increasing the dislocation mobility. Overall, the departure from the canonic high-barrier SW process is 2fold here: (i) on the one hand, the introduced radical-atom shifts the dislocation core energy, up by ∼1 eV, and (ii) on the other hand, the migration barrier of thus formed radical-cores is much reduced, down by ∼2.7 eV (see Figure 1a schematics). Nevertheless, owing to the rigid sp2 hybridization in h-BN, the formation of rhombs (marked by green dashes in Figure 1c−d) with four-coordinated atoms in the saddle-point structures is forbidden, and thus effectively three dangling bonds keep the barriers still significant. When turning now from h-BN to TMDCs, some qualitative differences are important. Not only does the flexible coordination of S atom (such as, two in H2S molecule, three in pristine MX2, and four in the rhombs of 60° GB in MX240) facilitate the stabilization of various derivative cores, but also the relatively weak metal−ligand bond could potentially reduce the energy of the migration barriers. Previously, we have successfully predicted the structures of various dislocation cores,40 later confirmed by several independent experiments,13,14,52 which thus can serve as reliable starting foothold. In brief, there are two basic dislocations with the smallest b⃗ = (1, 0) or (0, 1), that is Mo5|7 or S5|7. The effective interaction between the dislocation cores and intrinsic point defects generates derivative families, each within the same topological invariant b⃗. An S5|7 may react with a single or a double Svacancy (VS) yielding S5|7 + VS or S5|7 + 2 VS = 4|6. Similarly, a Mo5|7 can interact with single or double interstitial S, forming 6|8 with single S (6|8+VS) or 6|8. Which structure is favored C
DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 3. In a−c, left, middle, and right panels present the corresponding starting, saddle-point (intermediate state for Mo5|7), and final configurations for (a) Mo5|7, (b) 6|8 + VS, and (c) 6|8, respectively. Dislocation cores and surrounding hexagons are gray-shaded in different scales to illustrate the migration path. Black, small white, and large white spheres represent Mo, front-layer, and back-layer S atoms, respectively. The red spheres highlight the main migrating atoms. Red dashed lines in (c) indicate the rhomb structures in the rebonding (RB) process. (d) Minimum energy path during migrations for different Mo-oriented dislocations.
displacement of the same-site S atom pairs (one in front, another at back layers), most visible for the S2 sites on the sides of the shaded area (marked by the arrows). Both effects contribute to higher energy of the migration barrier. As for S5|7, its glide step is similar to S5|7 + VS but involves sequentially moving the atoms of the same S2 site, front (small red) and then back-plane S (large red), to the right in Figure 2a. Accordingly, the energy profile in Figure 2d is twin-peaked, with the intermediate-state structure depicted in the mid panel of Figure 2a. At the same time, Mo atom in between shifts to the lef t. Again, this counter-movement of atoms resembles the SW atomic-dumbbell rotation and both cases we tag as generalized SWg mechanism, in Figure 2a−b. Similar results are obtained for Mo-oriented dislocations family of Mo5|7 and shown in Figure 3. NEB computations show that 6|8, favored at S-rich conditions, moves via the RB mechanism (highlighted Mo atom disconnects from the S2 on its right and rebonds to another S2 on its left, with a barrier of only 0.7 eV). With less sulfur, the 6|8+VS glide follows the SWg mechanism involving substantial atomic displacement (S atom in the frontal plane) and accordingly an almost twice higher barrier of 1.3 eV. When one more VS is introduced (i.e., S is removed) recovering basic Mo5|7, sequential moves of two S atoms from the side of the heptagon and a counter-shift of Mo are needed for one glide step, that is again SWg mechanism. The intermediate state of the Mo5|7 glide (middle panel of Figure 3a) is composed of four hexagons and a rhomb with the two S atoms in different sulfur layers. The migration barrier is 1.1 eV (Figure 3d). Our analysis thus far shows how the glide barrier greatly depends on the dislocation structure, affected of course by its chemical content. The latter, in turnowing to the binary composition of MoS2varies with the elemental chemical potentials in the environment. The computed formation energies in Figure 4 show that the most mobile types of Sor Mo-orientated dislocations (4|6 and 6|8) can be thermodynamically favored, at the left and right sides of the μS range (color-shaded areas in the plot). This is rather counterintuitive or even surprising since one normally associates thermodynamic preference with lower energy caused by stronger bonding, and consequently lower mobility, while
Figure 4. Formation energies for the derivative cores relative to Mo5|7 or S5|7, as functions of the chemical potential of sulfur. The orange and blue shaded areas indicate the chemical potential range where the two most mobile dislocation cores, 4|6 and 6|8, can be thermodynamically favorable, respectively. The gray shaded area shows the estimated range of chemical potential in experiments.
the highly mobile, “excited-state” configurations should be typically less probable. In order to relate the formal μS range with practical conditions of temperature T and gas pressure P,13 one can assume that the latter is mainly due to partial pressure of sulfur most stable molecule, S8. Accordingly, its μS(P, T) is calculated by adding up the contributions of translational, rotational, and vibrational degrees of freedom,53 the detailed formula given in the SI. In the experimental (P, T) range from (500 Pa, 1000 °C) to (120 KPa, 550 °C), the calculated μS interval is [−1.0, −0.5] eV as shown in the gray-shaded area in Figure 4. The resultant range of μS can even be somewhat lower (to the left) since the partial pressure of S8 is less than the full pressure in experiments. Overall this suggests that 4|6 (i.e., S5|7 + 2 VS) should be most common, which agrees with the fact that neither S5|7+VS nor S5|7 have been seen frequently (although in experiments, the removal of S could be nonthermodynamic but also caused by electron beam52,54). We see that, in TMDC, the dislocation can be thermodynamically favorable and highly mobile at once, remarkably different from graphene or h-BN where the formation energies of such dislocations appear D
DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
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obtained through the Einstein equation, v = f·(D/kT) = σb· (D⊥/kT). Note the value of the two-dimensional elastic modulus obtained here from DFT computations, Y ≈ 100 N/m. Next, to very roughly estimate the dislocation density in a polycrystalline sample, one can count all the dislocations comprising the grain boundaries. For a typical grain size L ∼ 50 μm with the dislocations spaced by l ∼ 1 nm, one gets ρ = (Ll)−1 = 2 × 1013 m−2. Putting now ρ and v (at T = 300 K and elastic strain ε = 0.01) together into Orowan’s equation above, one can estimate the plastic deformation rates due to presence or either 4|6 (E* ∼ 0.5 eV) or 6|8s (E* ∼ 0.7 eV) to be near 1 s−1 or 4 × 10−5 s−1, respectively, while for samples with other dislocations dominantly present, the plastic response is practically nonobservable. To additionally demonstrate the chemical-potential controlled dislocation migration, we perform finite-temperature molecular dynamics for two representative dislocations, 4|6 and S5|7 + VS, using nonself-consistent-charge DFT-based tightbinding (DFTB)56−58 approximation with PBC models, which contain about 350 atoms. The temperature 1000 K of Anderson thermostat59 and shear strain 0.005 were used. With a time step of 1 fs, we do observe one glide event for 4|6 in 35 ps following the exact RB mechanism, while no glide at all is detected for S5| 7 + VS. The trajectory movies are provided in the SI. In conclusion, the systematic analyses show that the dislocation structure and consequently dynamics in heteroelemental TMDCs is highly sensitive to the chemical potentials imposed by their gaseous environment. Different dislocations move (glide) by distinctly different migration mechanisms through either direct rebonding (RB) or generalized Stone− Wales rotation (SWg), with very different barriers. In particular, according to the first-principles calculations, the RB barrier can be as low as 0.5 eV. The possibility of thermodynamically favorable dislocations with high mobility implies that by simply changing the environmental conditions one can induce dramatic change in the macroscopic mechanical properties such as plasticity, which suggests intriguing possibility of experimental verification. This also suggests a possibility to anneal the defects in polycrystalline 2D TMDC samples not (or not only) by usual high temperature, but by carefully changing the chemical potentials to bring the dislocations into their mobile states. Although our results are presented for freestanding MoS2, the introduction of typical substrate SiO2 (with high sublimation energy) does not change the whole picture here. It may be worth mentioning that although our specific quantitative analysis and computations here are focused on MoS2 as example, the general picture and possibility to switch dislocations into highly mobile state and enhance the plasticity is valid for all 2D TMDC materials.
prohibitive, as shown above. This renders the control of dislocation dynamics through chemical potential quite feasible. The low-barrier migration of 6|8 and 4|6 can be easily activated thermally, even at room temperature. It is important to note now, that the physics of dislocations in 2D materials is qualitatively different from 3D: while in normal 3D crystal the dislocation lines move as a result of cooperative formation and propagation of the kinks, changing their loops shapes and glide directions, in 2D case the “dislocation line” spans just angstroms, from one side of the layer to the opposite. Consequently, its simpler dynamics is akin to diffusion of an atom, and the coefficient can be similarly evaluated as D⊥ ∼ a2(kT/h)e−E*/kT, where a is the lattice parameter (in this case a = |b|), kT is thermal energy, and h is the Planck’s constant). Estimates show that the Brownian motion (confined of course to a glide planein 2D case, to a line along b) is rapid enough to make an observation of dislocations with E* ∼ 0.5 eV difficult for high resolution microscopy. However, their presence at sufficient concentration ρ can result in macroscopically observable greatly increased plasticity of the material. Its deformation rate dε/dt, proportional to the applied tensile stress σ ∼ Yε (Y is the elastic modulus), is defined by the drift velocity v of individual dislocations, given by the Orowan equation, dε/dt = ρbv. The nonzero drift velocity v is acquired due to the Peach−Koehler (PK) force exerted on dislocation by the stress field, generally f ⃗ = (b⃗·σ) × n⃗ (n⃗ is normal to the layer). In geometry of Figures 2 and 3, if a shear stress σxy = σ is applied, one simply has f = σb, and the energy reduction upon one glide step must be ΔE = σba. This analytical PK value can be compared with the direct DFT computations (under applied shear), to validate the latter. The energy difference ΔE and the migration barriers are shown in Figure 5, as functions of shear strain. The overall linear
Figure 5. Energy difference ΔE upon one glide step obtained with DFT (empty circles) and by PK equation (solid line), versus applied shear strain. The higher set of solid circles, guided by solid lines, represent the glide barriers as computed with DFT NEB, for the 6|8 and 4|6 cores, also as functions of shear strain. A small positive energy difference ΔE at zero strain results from the interaction between dislocations.
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ASSOCIATED CONTENT
S Supporting Information *
Derivation of the relation between deformation rate (or drift velocity) and strain, determination of the chemical potential corresponding to practical P and T, and MD movies for 4|6 and S5|7 + VS. This material is available free of charge via the Internet at http://pubs.acs.org.
dependence holds well both for the barriers E* and the energy difference ΔE. This proportionality of the “reaction” barrier to its exothermicity (here, δE* ≈ 0.54·δΔE) is a great example of classic Evans−Polanyi rule.55 From the computed barriers, which the applied shear makes unequal for the forward and backward glide steps, one can easily calculate the drift velocity as a function of strain ε (see SI). For relatively small stress, it can be linearized to yield v ∼ (Yεba2/h)·e−E*/kT. This linear-response limit can also be
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. E
DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters Notes
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This work was supported by the U.S. Army Research Office MURI Grant W911NF-11-1-0362 and by the Robert Welch Foundation (C-1590). The computations were performed at the Data Analysis and Visualization Cyberinfrastructure funded by NSF under Grant OCI-0959097.
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DOI: 10.1021/acs.nanolett.5b00864 Nano Lett. XXXX, XXX, XXX−XXX
