A Two-Dimensional Liquid Structure Explains the Elevated Melting

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A Two-Dimensional Liquid Structure Explains the Elevated Melting Temperatures of Gallium Nanoclusters Krista G. Steenbergen*,†,‡ and Nicola Gaston*,¶ †

Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University, Auckland Campus, Private Bag 102904, North Shore City, 0745 Auckland New Zealand ‡ Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, United States ¶ MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, P.O. Box 600, 6140 Wellington, New Zealand S Supporting Information *

ABSTRACT: Melting in finite-sized materials differs in two ways from the solid− liquid phase transition in bulk systems. First, there is an inherent scaling of the melting temperature below that of the bulk, known as melting point depression. Second, at small sizes changes in melting temperature become nonmonotonic and show a sizedependence that is sensitive to the structure of the particle. Melting temperatures that exceed those of the bulk material have been shown to occur for a very limited range of nanoclusters, including gallium, but have still never been ascribed a convincing physical explanation. Here, we analyze the structure of the liquid phase in gallium clusters based on molecular dynamics simulations that reproduce the greater-than-bulk melting behavior observed in experiments. We observe persistent nonspherical shape distortion indicating a stabilization of the surface, which invalidates the paradigm of melting point depression. This shape distortion suggests that the surface acts as a constraint on the liquid state that lowers its entropy relative to that of the bulk liquid and thus raises the melting temperature. KEYWORDS: Two-dimensional materials, finite-size effects, Ga nanoclusters, surface stability, solid−liquid phase transition that approximate well those of a first order transition) and nonmelters.12 The thermodynamic behavior of nonmagic melters arises from a series of solid−solid phase transitions in the clusters prior to melting.11 Nonmagic melters undergo two premelting structural transitions between approximates of the bulk β, γ, or δ-phases, each having adatoms that disrupt the surface structure such that the transition to the liquid state occurs without significant latent heat. This previous work includes detailed comparison of melting characteristics between cluster sizes, as well as a characterization of the changes between the solid to liquid states through atomic mobility measures.11−14 However, none of the extensive studies have yet explained the greater-than-bulk melting temperatures in clusters.4−8 In this Letter, we describe how quasi-two-dimensional liquid structures lower the entropy of the liquid cluster phase, resulting in systematically greater-than-bulk melting temperatures for gallium nanoclusters. Using density functional theory (DFT) molecular dynamics (MD) simulations (as implemented in VASP 5.2),15−22 we model the effect of temperature (200−1100 K) on gallium cluster cations sized 32−36 atoms.

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he existence of the liquid phase is due to a version of the Goldilocks principle: when not too hot, and not too cold, the liquid state achieves balance between the extremes of energetic stabilization and entropic disorganization. It is a remarkably complex phenomenon despite the apparent simplicity of solid−liquid and liquid−vapor phase transitions.1 Surprises remain to be discovered, such as the challenge to theory represented by the greater-than-bulk melting temperatures of small nanoclusters of tin and gallium.2−7 More recent work on gallium clusters has found that this phenomenon gives way at sizes greater than 94 atoms8 to the usual paradigm of melting point depression,9 which arises from the relative lability of the generalized surface and its increasing dominance at small cluster sizes. The discovery of greater-than-bulk melting in clusters raised two very interesting questions: What causes the large variation in melting temperature and specific heat curve features between cluster sizes? What is the physical origin of the anomalously high melting temperatures at such small dimensions? Recent theoretical work has addressed the first question, demonstrating that different structural phases (β, δ, and γ-Ga) are present in the clusters at temperatures below the melting temperature (Tm), where solid−solid transitions are clearly observed.10,11 Geometric features, such as adatoms, account for the difference between “magic” melters (clusters that melt with a clearly defined melting temperature, and thermodynamic signatures © XXXX American Chemical Society

Received: June 2, 2015 Revised: December 1, 2015

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DOI: 10.1021/acs.nanolett.5b02158 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters Given the complex structural landscape known for gallium, we utilize a parallel tempering algorithm to enhance ergodicity23 and calculate specific heat curves by the multiple histogram method.24,25 Our computational simulations of the melting behavior of small gallium clusters reproduce the greater-thanbulk melting temperatures.11−14 This computational model has been validated for the clusters by direct comparison to the experimental results4−6 with the similarity between the detailed features of the simulated and experimental specific heat curves demonstrating our agreement.12 For additional validation of the expected level of agreement with experiment, we have simulated the melting of the bulk gallium structure at the same first-principles level of theory using the “interface pinning” method developed by Pedersen.26,27 This results in a melting temperature of 212 ± 10 K for bulk gallium, verifying that the melting temperatures of the clusters are increased by a factor of 2 or more with respect to the bulk Tm in the simulations. Compared to experiment, clusters and bulk generally melt at temperatures ∼90 K lower, which is attributed to the limitations of the density functional used in the simulations.12 Additional details of the simulation methods can be found in the Supporting Information (Sections S1 and S2). It is something of a truism that these clusters have lower cohesive energy than the bulk in both the solid and liquid states due to their higher ratio of surface-to-internal atoms. As illustrated in Figure 1, the latent heat of melting for clusters in

In order for the cluster melting temperatures to nearly double that of the bulk, it is clear from eq 1 that ΔSfus must scale even smaller than ΔHfus in comparison to the bulk values. Our simulation and analysis methods allow us to calculate both ΔHfus and ΔSfus (Supporting Information, Section S3), confirming that the cluster latent heats are 1/2−1/3 of the bulk latent heat, while the cluster entropies of fusion are 1/3− 1/6 of the bulk value. The consistently smaller-scaling of the solid−liquid entropy change (in the denominator of eq 1) results in the systematically greater-than-bulk melting temperatures for the clusters, consistent with experimental observations.8 The entropy of fusion may be reduced relative to the bulk due to two causes. First, the entropy of the solid clusters may be increased relative to the bulk due to the dominance of the surface with its higher inherent lability compared to a bulk environment. Evidence for this exists in the form of the solid− solid transitions that result in a higher entropy of the solid structure at melting.11 Second, the entropy of the ensemble of liquid clusters could be reduced. The liquid phase is inherently difficult to analyze due to its highly fluxional nature. However, statistical analysis of the cluster coordinate data (Supporting Information, Section S4) has allowed us to identify the maximum orthogonal dimensions of the liquid cluster structure at each MD time step: 1 (longest dimension), 2 (next-longest), and 3 (shortest).10,11 Breaking the structural analysis down to each time step allows us to visualize the structural changes in the liquid state, where numerous fleeting configurations coexist and the typical analysis methods (structure averaging or quenching structural subsets) would be inadequate. Calculating the aspect ratio, as α = 1/ 3, reveals extreme distortions of the cluster shape that persist for significant periods of time in the liquid phase. For all temperatures at least 300 K above the cluster Tm, the fraction of simulation time with α ≥ 1.6 is between 4 and 31% (Supporting Information, Section S5), depending on cluster size. This surprising result strongly contradicts the expectation that spherical geometries (α = 1) would be most stable for liquid cluster structures due to the minimization of surface area. As this assumption underlies the paradigm of melting point depression in finite-size materials, we have a proximate explanation of the greater-than-bulk melting for these clusters. Figure 2a,b presents the cluster dimensions and aspect ratios for a portion of the 33-atom simulation trajectory at 330 K above the cluster melting temperature. A large variation in cluster dimensions and aspect ratio is clear. Taking the most extreme prolate geometry at a value of α = 2.4, labeled (i), we see that the dimensions are in fact all distinct. Although we refer to a structure as prolate due to the extreme α, it is clear that the ratio of the second-longest cluster dimension to the shortest, as β = 2/ 3, is also greater than one in these structures. With α and β both >1, these structural shapes might be better described as “flattened prolate”, or “triaxial”. The structure labeled (iii) is more spherical in shape with 1 ≈ 2 ≈ 3, while (ii) is nonspherical but with lower α. Further analysis confirms that structures with high α and β correlate consistently with smaller cohesive energies, as illustrated by the potential energy surface in Figure 3 (PES for additional cluster sizes and temperatures in Supporting Information, Section S6). The 33 and 36-atom liquid structures exhibit these flattened-prolate geometries more persistently than the 32, 34, and 35-atom clusters (Supporting Information, Section S5). We note that both the 33 and 36-atom clusters have a closed geometric shell at the lowest simulated

Figure 1. Caloric curves for each of the Ga+n clusters (n = 32−36) compared to that of bulk α-gallium. For illustrative purposes, the highand low-temperature lines have been straightened with matching slopes; however, the melting temperatures,5 latent heats of melting (Supporting Information, Section S2),12,28 and cohesive energies (yintercept without zero point energy correction)12,29 are quantitatively represented.

this size-regime are also consistently smaller than bulk, by a factor of ∼1/3−1/2. However, these small clusters have counterintuitively higher melting temperatures. This mystery, as long since posed by the authors of the original experimental findings,5 must lie in the nature of the change in entropy between the solid and liquid phases (ΔSfus). We know that Tm is related to both ΔSfus and the latent heat of melting (ΔHfus) as

Tm =

ΔHfus ΔSfus

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Figure 2. A portion of the MD trajectory for the 33-atom cluster at Tm+330 K. Changes in the structural properties as a function of simulation time are captured by (a) the cluster aspect ratio, α, and (b) the corresponding cluster dimensions. (Center) Visualizations of the (i), (ii), and (iii) cluster structures from two perspectives: the left column illustrate the longest cluster dimension, while the right structure is rotated 90° in order to illustrates the shortest (each scale is given in Å). (Right) The PDOS for the points labeled (i), (ii), and (iii). The bottom, right panel (avg) illustrates the timeaverage PDOS for 11 liquid structures spanning a range of α, spaced at 0.6 ps intervals. This averaged PDOS include structures (i)−(iii), as well as the additional eight points marked by an “×” in panels (a) and (b).

the 35 and 36 atom clusters to the proximity to an electronic shell closing,11 because it is known that temperature effects will smear electronic shell closures over several sizes due to the effect of electronic entropy.33 Inspection of the projected density of states (PDOS) gives additional insight into the relationship between electronic shell structure and the relative sphericity of the cluster structures. The PDOS is calculated by projecting the electron density onto cluster-centered angular harmonic functions,30,31 permitting the assessment of the extent to which the electronic bands remain delocalized and consistent with the electronic shell structure observed in the solid clusters.11 Electronic shell models assume that the ionic positions are of secondary importance to the overall shape of the cluster and as such are applicable to structures found in the liquid state. The PDOS for structures (i), (ii), and (iii) are illustrated in the right column of Figure 2. As expected, a high value of α correlates with strong breaking of the degeneracies in the superatomic electronic structure and an indistinct separation of the electronic shells in the density of states with a consequently indistinct character of the states near the Fermi energy (EF). On the other hand, as the cluster becomes more compact or spherical, there is higher degeneracy in the electronic states and the distinct angular momentum character of each band persists until closer to EF. For all cluster shapes, the angular momentum character contains some mixing of the states between −2 eV and the Fermi level. At −2 eV, the density of states changes from states with an atomic 4s character to those arising from the atomic 4p orbitals. The time-averaged PDOS is also presented in Figure 2, calculated for 11 cluster structures evenly spaced through the MD trajectory, demonstrating that the distinct nature of the delocalized 1S, 1P, 1D, and 1F orbitals is preserved in the liquid state. It has been previously demonstrated that the electronic entropy, which describes the variation of electronic states that are accessible at finite temperature, has a significant effect on the persistence of electronic shell effects at finite temperature.33 However, we note in the current case that these two effects are strongly coupled: the distortion of cluster shape away from

Figure 3. Potential energy surface for Ga+36 liquid clusters of the three highest simulated temperatures, ranging from 290−420 K above Tm. The horizontal and vertical axes represent the cluster dimensional ratios, α and β. The lower-left corner of the plot, where α ≈ β ≈ 1, represent the spherical clusters, while the upper right where α and β are maximum are the flattened-prolate clusters. Despite the complex energy landscape, the overall trend is clear: the potential energy increases with increasing α and β.

temperatures. On the other hand, the electronic shell structure is also important for the behavior of these clusters.12−14,30,31 Using the spherical jellium model,32 we have therefore considered whether the proximity to a closed electronic shell at 34 and 35 atoms may encourage greater sphericity, given that electronic shell numbers presuppose spherical potentials. The total number of valence electrons is 95, 98, 101, 104, and 107 for the 32−36 atom cations, respectively. According to the jellium model, the nearest electronic shell closings occur at 92 and 106 electrons32 and therefore none of the clusters investigated here has an electronic ‘magic number.’ This does not, however, mean that electronic shell structure does not influence the stability of these clusters. We have previously ascribed the change in ground-state structural motif between C

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Nano Letters sphericity leads to a lifting of degeneracy of the electronic states, which can be clearly seen in Figure 2, for example, in the splitting of the 1P shell as the cluster moves from a quasispherical shape (Figure 2iii) to a triaxially distorted shape (Figure 2i). The average PDOS presented in Figure 2 for the aggregate liquid state demonstrates this increase in electronic entropy, and the sense in which the lifting of electronic degeneracies enhances metallicity in these clusters. Nonspherical jellium models have also been studied33,34 and demonstrate the broad applicability of electronic shell models to clusters of all shapes and sizes. However, due to the relatively large number of electrons (∼100) and the high temperatures of interest in the liquid state, these model refinements do not add much predictive ability to the spherical jellium states for the systems of interest here. The 36-atom cluster, a so-called “magic-melter”, is that way because of a closed-shell geometric structure in the solid state even at temperatures just prior to melting, as has been previously described.11 The fact that the ground-state structure has a highly symmetric, geometric closed shell, may itself be due to the proximity of this cluster to the electronic shell-closing. Other common electronic structure quantities, such as EF and the band gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), ΔEgap, exhibit some correlation with α. There is an observable trend for less-spherical clusters to have a smaller EF and ΔEgap (Supporting Information, Section S7), indicating a tendency toward increased metallic character in the liquid clusters. This metallic nature of the flattened prolate structures lends itself to further reflection. Experimental work has demonstrated that the (010)-surface of α-gallium is obtained by slicing through the center of the covalent bonds, thereby exposing a layer of the metallic buckled planes of gallium.35−37 This surface has been experimentally shown to be more metallic than the bulk structure, while the particular stability of this bilayer under dynamic conditions is demonstrated by its reappearance in only slightly modified form in the process of GaN growth,38,39 at temperatures of around 1000 K. In that case, the bilayer forms on the hexagonal GaN(0001) surface, controlling adatom diffusion and thus the kinetics of film growth,40 resulting in improved morphology. Thus, this twodimensional bilayer of metallic planes seems a candidate on which to model the strange distortion of the liquid phase. For all liquid structures other than the most spherical, the length of the shortest axis is close to 6 Å, suggestive of a bilayer structure. Structural analysis revealed a strong tendency for the liquid clusters to promote all internal atoms to surface sites with the majority of highly distorted structures (α ≥ 2) having only 0 or 1 internal atoms (Supporting Information, Section S8). With increasing temperature, the number of structures with >2 internal atoms drops off quickly. Moreover, an increasing number of clusters have both a high aspect ratio and zero highly coordinated internal atoms at the higher temperatures. In Figure 4a, we compare the pair distribution functions of an optimized bilayer of the α-Ga(010) surface with a liquid cluster of each size. Figure 4b,c illustrates the visualization of the relaxed surface bilayer and a cluster structure with the dominant structural trend highlighted for comparison. Despite considerable disorder of the highly fluxional clusters, the correspondence between the flattened prolate structures and the surface bilayer is clear: in effect, we are now dealing with cluster shapes in which all atoms are at the surface.

Figure 4. (a) The pair distribution functions, g(r), for the relaxed surface of bulk α-Ga compared with a representative g(r) for a liquid cluster at each size. Visualizations of the (b) the relaxed surface of bulk α-Ga and (c) a 33-atom liquid cluster structure with an aspect ratio of 2.2 with the dominant structural feature highlighted in blue.

Analysis of the atomic mobility, through the mean square displacement (MSD) of individual atoms, reveals further detail about these liquid cluster states. Generally, the MSD averages the atomic displacement for a given time interval (Δt) over a series of time-origins. Employing the same statistical technique that allows us to identify the maximum orthogonal cluster dimensions (Supporting Information, Section S4), we have been able to complete the MSD analysis using the clusterdimension axes as our coordinate system, yielding the meansquared atomic displacements along each of the cluster dimensions 1, 2, and 3 (Supporting Information, Section S9). For the triaxial clusters, we observe high atomic mobility in the plane of the cluster surface defined by the longer axes 1 and 2 (the flattened-prolate orientation of the cluster), while the mobility in the short-axis ( 3) direction is very low. The atoms easily diffuse, as would be expected of a liquid; however, they are constrained to move primarily in the direction of the flattened-prolate surface. Compared with the MSD analysis of the spherical clusters, where the atoms diffuse isotropically, we have some evidence for how the surface acts as a constraint on the behavior of liquid atoms in finite-size clusters. We can now describe melting in these clusters in the following terms. Once distorted into flattened-prolate shapes, the liquid clusters enforce a substantial energetic barrier to reentry of an atom into an internal site due to the need to reorganize the entire bonding network when such changes occur, an effect that is not present in the bulk liquid environment. These distorted shapes thereby destabilize the liquid phase energetically relative to the solid phase to the extent that they are able to dominate the liquid aggregate state. Alternatively, the liquid state may be viewed as destabilized in the clusters relative to the bulk, due to the constraints on mobility provided by the surface. D

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greater-than-bulk melting ends at around 94 atoms suggests that a 6 × 7 atom bilayer (6 × 7 × 2 atoms) with an aspect ratio of 3.5 may represent a limit on the extent to which these structures are accessible in the liquid state.

These constraints reduce the total number of accessible states in the liquid phase, compared to a more typical liquid where there would be no preference for a particular shape. This is best demonstrated in considering the microcanonical entropy, as S(E) = kB ln(Ω(E)), where kB is the Boltzmann factor and Ω(E) represents the total density of states. In the classical regime (ignoring contributions from electronic entropy), the total density of states is calculated as the convolution of the kinetic and configurational density of states: Ω(E) = ∫ E0 ΩK ΩCdK.25 It is clear that the demonstrated preference for α-gallium bilayer-like structures will diminish the configurational density of states, compared to a liquid where there is no structural preference. From our cluster-dimensional MSD analysis, it would appear that the accessible kinetic modes (kinetic density of states) would also be reduced in the flattened-prolate structures owing to the diminished kinetic motion in the short-axis direction. Both effects collude to greatly reduce the entropy of the aggregate liquid state for the clusters, resulting in a diminished entropy of fusion. Intriguingly, a previous experimental study has also demonstrated surface ordering for bulk liquid gallium.41,42 The characteristics of this layering are nearly identical to what we observe for the clusters: the liquid-surface layer was noted to be ∼6 Å thick, consisting of three atomic layers. Remarkably, the thickness of this surface layering remained consistent through a wide temperature range from a supercooled liquid sample41 up to ∼440 K (140 K above the bulk melting temperature).42 Comparing the liquid surface and liquid bulk regions, it was observed that the increased atomic ordering within the surface layer diminished the entropy of that region of the liquid,41 consistent with our observations. A final question is the extent to which this behavior is specific to gallium. The key ingredient in our recipe for greater-thanbulk melting is the resemblance of the liquid structures to the α-gallium bilayer. Stabilization of the configurations that approximate the gallium surface bilayer results in the dominance of these quasi-two-dimensional configurations in the liquid state, making re-entry of an atom into an internal site (which would lower the energy of the liquid) more difficult. These intrinsic nanoscale effects energetically destabilize the liquid state, lowering the change in entropy between the solid and liquid in the clusters, relative to that in bulk gallium. Therefore, we might expect greater-than-bulk melting behavior in the elements most closely related to gallium, where similar patterns of bonding are possible. It has been previously observed that the low melting temperature of gallium is more anomalous than the melting temperature of the clusters, in comparison to similar elements, and that this can be explained by the “molecular” structure of the metal and the nature of the bulk liquid. In this Letter, we have solved the secondary paradox, which is how the labile nature of the surface does not automatically lead to a lower Tm in the clusters; the preference for the flattened α-gallium surface-like structure lowers the total density of states (entropy) of the liquid phase. Our findings therefore reveal the importance of both shape and dimensionality in the melting of nanoscale systems. This observation that finite size itself acts as a constraint is an important general principle in analogy to the substrate stabilization of clusters which has previously been demonstrated to increase the melting temperature.43 It is also clear that the energetic cost of the quasi-2D structures will increase unfavorably with size. The experimental observation that

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b02158. Bulk melting simulations, cluster entropy/enthalpy quantification, simulation and analysis detail, high-α persistence, energy and electronic trends, additional PES, internal atom analysis. (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the Marsden Fund of the Royal Society of New Zealand under contract IRL0801. We thank the New Zealand eScience Infrastructure (NeSI), particularly the BlueFern (University of Canterbury: nesi67) and the Pan (University of Auckland: nesi7, nesi144 and nesi224) supercomputer teams, for computational time and support, with special thanks to Gene Soudlenkov, Francois Bissey, Jordi Blasco, and Ben Roberts.

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REFERENCES

(1) Samanta, A.; Tuckerman, M.; Yu, T.-Q.; Weinan, E. Science 2014, 346, 729−732. (2) Shvartsburg, A. A.; Jarrold, M. F. Phys. Rev. Lett. 2000, 85, 2530− 2532. (3) Shvartsburg, A.; Hudgins, R.; Dugourd, P.; Jarrold, M. Chem. Soc. Rev. 2001, 30, 26−35. (4) Breaux, G. A.; Benirschke, R. C.; Sugai, T.; Kinnear, B. S.; Jarrold, M. F. Phys. Rev. Lett. 2003, 91, 215508. (5) Breaux, G. A.; Hillman, D. A.; Neal, C. M.; Benirschke, R. C.; Jarrold, M. F. J. Am. Chem. Soc. 2004, 126, 8628−8629. (6) Breaux, G. A.; Cao, B.; Jarrold, M. F. J. Phys. Chem. B 2005, 109, 16575−16578. (7) Krishnamurty, S.; Chacko, S.; Kanhere, D. G.; Breaux, G. A.; Neal, C. M.; Jarrold, M. F. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 045406. (8) Pyfer, K. L.; Kafader, J. O.; Yalamanchali, A.; Jarrold, M. F. J. Phys. Chem. A 2014, 118, 4900−4906. (9) Pawlow, P. Z. Phys. Chem. 1909, 65, 1−35. (10) Steenbergen, K. G.; Gaston, N. J. Chem. Phys. 2014, 140, 064102. (11) Steenbergen, K. G.; Gaston, N. Chem. - Eur. J. 2015, 21, 2862− 2869. (12) Steenbergen, K. G.; Gaston, N. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 161402(R). (13) Steenbergen, K. G.; Schebarchov, D.; Gaston, N. J. Chem. Phys. 2012, 137, 144307. (14) Steenbergen, K. G.; Gaston, N. Phys. Chem. Chem. Phys. 2013, 15, 15325−15332. (15) Kresse, G.; Hafner, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47, 558−561. (16) Kresse, G.; Hafner, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 49, 14251−14269. E

DOI: 10.1021/acs.nanolett.5b02158 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters (17) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15−50. (18) Kresse, G.; Furthmüller, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169−11186. (19) Blöchl, P. E. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 17953−17979. (20) Kresse, G.; Joubert, D. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758−1775. (21) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. Rev. B: Condens. Matter Mater. Phys. 1992, 46, 6671−6687. (22) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 48, 4978−4978. (23) Calvo, F.; Neirotti, J. P.; Freeman, D. L.; Doll, J. D. J. Chem. Phys. 2000, 112, 10350−10357. (24) Labastie, P.; Whetten, R. L. Phys. Rev. Lett. 1990, 65, 1567− 1570. (25) Calvo, F.; Labastie, P. Chem. Phys. Lett. 1995, 247, 395−400. (26) Pedersen, U. R. J. Chem. Phys. 2013, 139, 104102. (27) Pedersen, U. R.; Hummel, F.; Kresse, G.; Kahl, G.; Dellago, C. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 094101. (28) CRC Handbook of Chemistry and Physics, 81st ed.; Lide, D. P., Ed.; CRC Press: Boca Raton, FL, 2000. (29) Kittel, C. Introduction to Solid State Physics, 7th ed.; Wiley: New York, 1996. (30) Schebarchov, D.; Gaston, N. Phys. Chem. Chem. Phys. 2011, 13, 21109−21115. (31) Schebarchov, D.; Gaston, N. Phys. Chem. Chem. Phys. 2012, 14, 9912−9922. (32) Cohen, M. L.; Chou, M. Y.; Knight, W. D.; de Heer, W. A. J. Phys. Chem. 1987, 91, 3141−3149. (33) Yannouleas, C.; Landman, U. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51, 1902−1917. (34) Yannouleas, C.; Landman, U. Phys. Rev. Lett. 1997, 78, 1424− 1427. (35) Walko, D. A.; Robinson, I. K.; Grütter, C.; Bilgram, J. H. Phys. Rev. Lett. 1998, 81, 626−629. (36) Moré, S.; Soares, E. A.; VanHove, M. A.; Lizzit, S.; Baraldi, A.; Grütter, C.; Bilgram, J. H.; Hofmann, P. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 075414. (37) Altfeder, I. B.; Chen, D. M. Phys. Rev. Lett. 2008, 101, 136405. (38) Northrup, J.; Neugebauer, J.; Feenstra, R.; Smith, A. Phys. Rev. B: Condens. Matter Mater. Phys. 2000, 61, 9932−9935. (39) Adelmann, C.; Brault, J.; Mula, G.; Daudin, B.; Lymperakis, L.; Neugebauer, J. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 165419. (40) Bruno, G.; Losurdo, M.; Kim, T.-H.; Brown, A. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 075326. (41) Regan, M. J.; Kawamoto, E. H.; Lee, S.; Pershan, P. S.; Maskil, N.; Deutsch, M.; Magnussen, O. M.; Ocko, B. M.; Berman, L. E. Phys. Rev. Lett. 1995, 75, 2498−2501. (42) Regan, M.; Pershan, P.; Magnussen, O.; Ocko, B.; Deutsch, M.; Berman, L. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 9730− 9733. (43) Schebarchov, D.; Hendy, S. C. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 085407.

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