Nanoplate elasticity under surface reconstructionHyun Woo Shim, L. G. Zhou, Hanchen Huang, and Timothy S. Cale Citation: Applied Physics Letters 86, 151912 (2005); doi: 10.1063/1.1897825 View online: http://dx.doi.org/10.1063/1.1897825 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theoretical calculations for structural, elastic, and thermodynamic properties of c-W3N4 under high pressure J. Appl. Phys. 114, 063512 (2013); 10.1063/1.4817904 Mechanical and piezoresistive properties of thin silicon films deposited by plasma-enhanced chemical vapordeposition and hot-wire chemical vapor deposition at low substrate temperatures J. Appl. Phys. 112, 024906 (2012); 10.1063/1.4736548 Thermal properties of nanotubes and nanowires with acoustically stiffened surfaces J. Appl. Phys. 111, 034319 (2012); 10.1063/1.3682114 Surface effects on the elastic modulus of nanoporous materials Appl. Phys. Lett. 94, 011916 (2009); 10.1063/1.3067999 Off-diagonal elastic constant and s p 2 -bonded graphitic grain boundary in nanocrystalline-diamond thin films Appl. Phys. Lett. 86, 231904 (2005); 10.1063/1.1946920

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Nanoplate elasticity under surface reconstructionHyun Woo Shim, L. G. Zhou, and Hanchen Huanga!

Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy,New York 12180

Timothy S. CaleDepartment of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy,New York 12180

sReceived 4 October 2004; accepted 17 February 2005; published online 7 April 2005d

Using classical molecular statics simulations, we show that nanoplate elasticity strongly depends onsurface reconstruction and alignment of bond chains. Because of its well-established surfacereconstructions and the readily available interatomic potential, diamond-cubic silicon is theprototype of this study. We focus on silicon nanoplates of high-symmetry surfaces,h111j andh100j;with 737 and 231 reconstructions. Nanoplates with unreconstructedh111j surfaces are elasticallystiffer than bulk. In contrast, the same nanoplates with 737 reconstructedh111j surfaces areelastically softer than bulk. Onh100j surfaces, the 231 surface reconstruction has little impact. Thebond chains are along one of the twok110l directions, making the twok110l directionsnonequivalent. The alignment of the bond chains on the opposite surfaces of a nanoplate dictates itselastic anisotropy. The sensitivity of nanoplate elasticity on details of surface atomic arrangementsmay impact the application of nanoplatessor nanocantileversd as sensors.© 2005 American Institute of Physics. fDOI: 10.1063/1.1897825g

Nanostructures—such as nanoplates, nanobeams, andnanowires—are enabling elements of emerging nanotechnol-ogy. For example, silicon nanocantilevers are capable ofsensing minute objects.1,2 Because of the low dimensionality,nanostructures respond to external stimuli differently fromtheir microscopic and macroscopic counterparts. Under me-chanical loading, the elastic response is known to be differ-ent. Experimental studies have shown that nanoplates or freestanding films can be stiffer3 or softer4 than their bulk coun-terparts. Simple discrete models predict softer nanoplates,5

while more reliable atomistic simulations of Si and Al haveindicated that nanobeams can be softer or stiffer.6 The under-standing of the softening and stiffening phenomena hasevolved over the years. Wolf has shown, using linear elastic-ity analyses and molecular statics simulations, that surfacestress leads to decrease of atomic volume;7 such a decrease iscorrelated with stiffening phenomena.8 In a similar sprit, Sunet al.9 have attributed the stiffening effects to bond contrac-tion; at smaller atomic volume, atomic bonds contract andthereby become stiffer. Our recent studies focus on the redis-tribution of electrons or bonding electrons, instead of atomicvolume or atomic bond length. Using a combination ofab initio and molecular statics simulations, we have shownthat the stiffening or softening of metal nanoplates is theresult of competition between bond loss, and bond saturationor electron redistribution.10 Segallet al.11 have suggested thecorrelation of elastic stiffening and electron distribution,based on less well-defined atomic elastic constants. In addi-tion to these two competing factors, surface reconstructionmay impact nanoplate elasticity, and is prominent indiamond-cubic crystals, such as silicon.

In this letter, we investigate the effects of surface recon-struction and bond chain alignments on the nanoplate elas-

ticity, by using molecular statics simulations. The prototypesof study are silicon nanoplates of high-symmetry surfaces,h111j and h100j. Atoms in silicon interact with each otheraccording to the bond-order potential,12 which correctly pre-dicts sad the bulk elastic constants andsbd the surface recon-structions, 737 on h111j, 231 on h111j, and 231 on h100jsurfaces.13 In our simulations, each nanoplate consists of aslab of silicon atoms in diamond-cubic structure, as shown inFig. 1sad. Periodic boundary conditions are applied along thex and they directions, leaving free surfaces along thez di-rection. Normal strains«x and «y are fixed and imposedalong thex and they directions, while the free surfaces fa-cilitate relaxations along thez direction. For each set ofstrains«x and«y, the atomic positions are relaxed using thequenching method. The total energy of each nanoplate is afunction of the strains«x and«y, as shown in the inset of Fig.1sbd. For each strain«x, there is an«y that minimizes the totalenergy. The minimum energyEs«xd as a function of«x isshown in Fig. 1sbd. It is worthy noting that the minimumenergy occurs at a nonzero«x, due to relaxation along thezaxis. Based on this energy curve, the Young’s modulus alongthe x direction is determined according toY=s1/Vd3fd2Es«xd /d«x

2g, whereV is the volume of the fully relaxedsimulation cell.

Several aspects of the numerical calculations deserveelaboration. First, the simulation cell dimensions along thexand they directions are at least three cut-off distances of theinteratomic potential, to avoid artificial effects from thesimulation cell size and to enable the necessary surface re-construction. Second, strains from −0.5% to 0.5% aremeshed into 21321 points, so the mesh size of strain is0.05%. Third, the average potential energy of each atom in asimulation converges to within 10−8 eV after the relaxationfor each strain condition, so that the error of total energy issmaller than 1% of the energy variation between two neigh-boring meshing points. Fourth, in determining the volumeV,

adAuthor to whom correspondence should be addressed; electronic mail:hanchen@rpi.edu

APPLIED PHYSICS LETTERS86, 151912s2005d

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we assume that each atomic layer occupies an equal volumeon its two sides. In the case of nanoplates withh111j sur-faces, each atomic layer consists of atoms from two face-centered-cubic sublattices; in other words, the atomic layerarrangement according to this convention is ABCABC…packing like in face-centered-cubic crystals. Finally, thesecond-order derivative of energy with respect to strain iscalculated near the bottom of the energy curvefFig. 1sbdg,which represents the fully relaxed equilibrium state. This de-rivative converges to within 1% when the local meshing sizeis halved.

Figure 2sad shows the Young’s moduli of the siliconnanoplate withh111j surfaces as a function of thickness,along two high-symmetry directionsk110l and k112l. With-out surface reconstruction, the nanoplate is elastically stifferthan bulk; the isotropy along the two directions remains. TheYoung’s moduli increase as the nanoplate thickness de-creases. The linear dependence on the inverse of thickness isconsistent with earlier observations.8 We attribute the moduliincrease to bond saturation near surfaces.10 In contrast, thenanoplate is softer than bulk after the 737 surface recon-struction, and the Young’s moduli decrease as the nanoplatethickness decreases; the isotropy along the two directionsremains. Without the reconstruction, surface atoms have nomissing bonds parallel to the surface; the only missing bondis perpendicular to the surface. On the 737 reconstructedsurfaces, surface bonds are distorted and periodic holes exist,as shown in Fig. 2sbd. The distortions and holes reduce the

elastic stiffness of the surface. Meanwhile, the 231 surfacereconstruction destroys the elastic isotropy, making themoduli different along thek110l and k112l directions. Al-though bond distortion is still a factor under the surface re-construction, thep-bond chains along thek110l directionfFig. 2scdg are strong. Because of these chains, the nanoplateis elastically stiffer than bulk along thek110l direction, andthe corresponding Young’s modulus increases as the nano-plate thickness decreases. At the same time, the formation ofp-bond chains also leads to gaps between surface atomsalong the k112l direction. Consequently, the nanoplate iselastically softer along thek112l direction, and the corre-sponding Young’s modulus decreases as the nanoplate thick-ness decreases, due to the bond distortion and gap formation.

Figure 3 shows Young’s moduli of the silicon nanoplatewith h100j surfaces as a function of thickness, along thehigh-symmetryk110l directions. The twok110l directions arenonequivalent, and bond chains exist along one of the twodirections; we refer to this direction as the chainedk110ldirection. When the chainedk110l direction on the top sur-face and that on the bottom surface of the nanoplate areorthogonal, the Young’s moduli along the twok110l direc-tions are identical and slightly softer than bulkfFig. 3sadg.The 231 surface reconstruction has little effect. When the

FIG. 1. sColor onlined. sad Simulation cell of a silicon nanoplate withh100jsurfaces,sbd strain energy as a function of«x. The inset shows the strainenergy as a function of«x and«y.

FIG. 2. sColor onlined. sad Normalized Young’s moduli, with respective tothe bulk values, alongk110l and k112l of silicon nanoplates withh111jsurfaces, as a function of thicknessH; “UR,” “7 37,” and “231” indicateunreconstructed, 737 reconstructed, and 231 reconstructed surfaces, re-spectively. Atomic configurations of 737 sbd and 231 scd reconstructions;darker spheres represent atoms closer to the top.

151912-2 Shim et al. Appl. Phys. Lett. 86, 151912 ~2005!

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chainedk110l direction on the top surface and that on thebottom surface of the nanoplate are parallel, the nanoplate isstiff along the chainedk110l direction and soft along theorthogonal direction. The Young’s modulus along thechainedk110l direction increases as the nanoplate thicknessdecreases. In contrast, the modulus along the orthogonal di-rection decreases as the nanoplate thickness decreases. The231 surface reconstruction leads to formation of dimers, asshown in Fig. 3sbd. The dimer bonds make the nanoplatestiffer perpendicular to the chainedk110l direction. At thesame time, the formation of dimers also distorts the bondchains, making the nanoplate softer along the chainedk110ldirectionfFig. 3sadg. In general, the 231 surface reconstruc-tion on h100j has little effect.

It is interesting to examine various stiffening and soften-ing mechanisms. The softening effects are primarily from thebond loss.10 In contrast, the stiffening effects have been at-tributed to bond saturation,10 and bond contraction or volumecontraction.7,9 Due to the prominent stiffening phenomenonof the nanoplate with unreconstructedh111j surfaces, wechoose it as the prototype for discussions of stiffeningmechanisms. In contrast to the tensile stress in metalsurfaces,7 the surface stress in the silicon nanoplate is com-pressivesabout 1.5 J/m2d. As a result, the nanoplate expandsin plane, and contracts out of plane. It might appear that such

expansion and contraction would lead to volume expansion.In fact, the volume still contracts, because of the largeramount of out-of-plane contraction before the relaxation ofin-plane surface stresses. As a reference, the surface bondsare 0.2312 nm, shorter than the bulk value of 0.2352 nm, fora fully relaxed 2.5 nm thick nanoplate with unreconstructedh111j surfaces. This shorter surface bond is consistent withthe bond contraction scenario. Or, more precisely, the shortersurface bond indicates a higher electron densitysassumingbonding electrons are localized in covalent systemsd, consis-tent with the bond saturation concept.10

In summary, our classical molecular statics simulationsreveal two phenomena. First, surface reconstruction affectsthe elastic stiffness of silicon nanoplates. This stiffening ef-fect is particularly prominent for silicon nanoplates withh111j surfaces. Second, the alignment of atomic bond chainson surfaces destroys the bulk elastic isotropy and dictates thestiffness of silicon nanoplates. This effect is particularlyprominent for silicon nanoplates withh100j surfaces.

Before closing, we comment on technological implica-tions of the simulation results. Our results show that elasticmoduli of a nanoplate, or a nanocantilever, sensitively de-pend on details of surface structures; such as reconstruction,or alignment of bond chains, or oxidation. In a nanocantile-ver, the natural frequencyf depends on both the Young’smodulusY and the massm according tof ~ÎY/m. For dif-ferential variations, dm/m=dY/Y−2sdf / fd. Usually, thevariation of the natural frequency is related to the variationof mass; the varied mass is that of a detected object. Such acorrelation is rigorous only when possible variations of elas-tic moduli, as a result of surface structure changes, are takeninto account.

The authors gratefully acknowledge the financial supportfrom the Focus Center-New York, Rensselaer: Interconnec-tions for Gigascale Integration.

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FIG. 3. sColor onlined. sad Normalized Young’s moduli, with respective tothe bulk values, along two possiblek110l directions with bond chains on topand bottom being parallel or orthogonal, as a function of thicknessH; “UR”and “231” indicate unreconstructed and 231 reconstructed surfaces, re-spectively.sbd Atomic configurations of 231 reconstruction; darker spheresrepresent atoms closer to the top.

151912-3 Shim et al. Appl. Phys. Lett. 86, 151912 ~2005!

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