Surface stress effects on the resonant properties of silicon nanowires

Transcription

1 JOURNAL OF APPLIED PHYSICS 103, Surface tre effect on the reonant propertie of ilicon nanowire Harold S. Park a Department of Mechanical Engineering, Univerity of Colorado, Boulder, Colorado 80309, USA Received 15 February 2008; accepted 11 April 2008; publihed online 16 June 2008 The purpoe of the preent work i to quantify the coupled effect of urface tree and boundary condition on the reonant propertie of ilicon nanowire. We accomplih thi by uing the urface Cauchy Born model, which i a nonlinear, finite deformation continuum mechanic model that enable the determination of the nanowire reonant frequencie including urface tre effect through olution of a tandard finite element eigenvalue problem. By calculating the reonant frequencie of both fixed/fixed and fixed/free 100 ilicon nanowire with unrecontructed 100 urface uing two formulation, one that account for urface tree and one that doe not, it i quantified how urface tree caue variation in nanowire reonant frequencie from thoe expected from continuum beam theory. We find that urface tree ignificantly reduce the reonant frequencie of fixed/fixed nanowire a compared to continuum beam theory prediction, while mall increae in reonant frequency with repect to continuum beam theory are found for fixed/free nanowire. It i alo found that the nanowire apect ratio, and not the urface area to volume ratio, i the key parameter that correlate deviation in nanowire reonant frequencie due to urface tree from continuum beam theory American Intitute of Phyic. DOI: / I. INTRODUCTION Nanowire have been among the mot tudied nanomaterial in recent year. The intene interet in nanowire ha emerged for a variety of reaon, foremot becaue their mall ize often lead to unique phyical propertie that are not oberved in the correponding bulk material. Nonbulk phenomena have been oberved in the mechanical, electrical, thermal, and optical propertie of both metallic and emiconducting nanowire. 1 Thee unique propertie have therefore generated ignificant interet in uing nanowire a the baic building block of future multifunctional nanoelectromechanical ytem NEMS. 2 With the recent exploion in NEMS reearch, ilicon nanowire, or ilicon-baed compound SiC, SiN, have been utilized mot frequently a the baic building block of NEMS. There are multiple reaon for thi, including the ability to micromachine nanometer-cale ilicon nanowire, 3 and the fact that ilicon i the fundamental material in the microelectronic indutry coupled with it potential a an optoelectronic material. 4 Silicon nanowire-baed NEMS have been tudied in a wide variety of application, including high frequency reonator for next-generation wirele device, 2 force enor, 5 electrometer for charge meaurement, 6 and chemical and biological ma enor. 7 Many of thee application 2,5,7 require precie knowledge of the nanowire reonant frequency, or variation in the reonant frequency due to environmental change that are to be meaured, i.e., adorbed ma for ma ening. However, the undertanding and deign of uch NEMS i ignificantly complicated by the fact that the elatic propertie, and thu the reonant frequencie of nanowire are known to differ from thoe of the bulk material. a Electronic mail: harold.park@colorado.edu. Nanowire elatic propertie are different from thoe of bulk material due to the preence of free urface. In particular, urface atom are ubject to urface tree Surface tree on nanomaterial arie due to an imbalance in the force acting on urface atom due to their lack of bonding neighbor. Becaue urface atom have a different bonding environment than atom that lie within the material bulk, the elatic propertie of urface differ from thoe of the bulk material, and the effect of the difference between urface and bulk elatic propertie on the effective elatic propertie of the nanowire become magnified with decreaing tructural ize or increaing urface area to volume ratio. The elatic propertie of ilicon nanowire have been theoretically predicted to how a trong ize dependence. 11,12 Many experimental tudie, either through reonant-baed teting 7,13 or atomic force microcrope AFM -baed bending 14,15 of the elatic propertie of ilicon ue nanowire with cro-ectional ize larger than 50 nm, where urface effect are not expected to be ignificant. However, experiment uing nanowire with cro-ectional ize le than 20 nm Ref agree with theoretical prediction 11 that the Young modulu of ilicon decreae with decreaing ize. We note that uch ize dependence ha alo been experimentally oberved in metallic nanowire. 19,20 Therefore, the purpoe of thi work i to quantify how urface tree impact the reonant propertie of ilicon nanowire. In particular, we tudy 100 ilicon nanowire with unrecontructed 100 urface that have cro-ectional ize ranging from 10 to 30 nm uing both the fixed/fixed and fixed/free beam-type boundary condition that are prevalent in NEMS deign. The ize range for the nanowire cro ection i deliberately choen to bridge the gap between ize cale that can be tudied uing atomitic calculation, and thoe which are commonly tudied experimentally /2008/ /123504/10/$ , American Intitute of Phyic

2 Harold S. Park J. Appl. Phy. 103, While it i known that the 100 urface of ilicon tend to undergo dimerized recontruction, 21 we focu on the unrecontructed 100 urface in the preent work. In doing o, we note that the ideal and 2 1 dimerized 100 urface of ilicon both experience a compreive urface tre 21 leading to tenile train. However, the compreive urface tre experienced by the 2 1 dimerized ilicon nanowire i not a large a that of the unrecontructed 100 urface due to the fact that the dimerized urface atom have three nearet neighbor, a compared to only two for the unrecontructed urface atom. Therefore, it i expected that the trend dicued in the preent work would hold qualitatively for dimerized ilicon nanowire, with the effect of the urface tre being lightly mitigated a compared to the unrecontructed urface. We addre thi goal by utilizing the recently developed urface Cauchy Born SCB model. 22,23 The uniquene of the SCB model a compared to other urface elatic model i that it enable, by correctly accounting for urface tre effect on nanomaterial, the olution of threedimenional nanomechanical boundary value problem for diplacement, tree and train in nanomaterial uing tandard nonlinear finite element FE technique, where the nonlinear material contitutive repone obtained directly from realitic interatomic potential uch a the Teroff potential. 28 The reonant propertie of the ilicon nanowire are determined by olving a tandard FE eigenvalue problem for the reonant frequencie and aociated mode hape, with full accounting for urface tre effect through the FE tiffne matrix. We quantify the effect of urface tre on the fundamental reonant frequency for both fixed/free and fixed/fixed boundary condition a function of geometry, ize, and urface area to volume ratio. We further compare the reult to thoe obtained on the ame geometrie uing the tandard bulk Cauchy Born material, which doe not account for urface tre effect, to quantify how urface tree caue the reonant frequencie of ilicon nanowire to deviate from thoe predicted uing continuum beam theory. II. THEORY A. Bulk Cauchy Born model for ilicon The bulk Cauchy Born BCB model i a hierarchical multicale aumption that enable the calculation of continuum tre and moduli directly from atomitic principle. 29 The BCB formulation in thi work for ilicon cloely mirror that of Tang et al. 30 and Park and Klein. 31 Becaue the SCB model for ilicon i much eaier to undertand once the bulk formulation i preented, we preent an abbreviated verion of the BCB formulation later. In the preent work, we utilize the T3 form of the Teroff potential 28 and the reulting parameter. The T3 i named a uch due to the fact that two earlier verion of the Teroff potential uffered from variou hortcoming, including not predicting diamond a the ground tate of ilicon, inaccuracie in the bulk elatic contant, and inaccurate modeling of the 100 urface of ilicon. 21 The T3 potential energy U can be written a U = 1 2 i j V ij, V ij = f C r ij f R r ij + b ij f A r ij, where r ij i the ditance between atom i and j, f C i a cutoff function, which i ued to enure that the Teroff potential i effectively a nearet neighbor potential, f R i a repulive function, f A i an attractive function, and b ij i the bond order function, which i ued to modify the bond trength depending on the urrounding environment. The variou function all have analytic form, which are given a where f R r ij = Ae r ij, f A r ij = Be r ij, b ij = 1+ n n ij 1/2n, ij = f C r ik g ijk, k i,j and g ijk =1+ c2 d 2 c 2 d 2 + h co ijk 2, where ijk repreent the angle between a triplet of atom i j k. In order to turn the atomitic potential energy into a form uitable for the BCB approximation, two tep are taken. Firt, the potential energy i converted into a train energy denity through normalization by a repreentative atomic volume 0 ; 0 can be calculated for diamond cubic DC lattice uch a ilicon by noting that there are eight atom in a DC unit cell of volume a 0 3, where a 0 =5.432 Å i the ilicon lattice parameter. Thu, 0 =8/a 0 3 for a 100 oriented ilicon crytal. Second, the neighborhood urrounding each atom i contrained to deform homogeneouly via continuum mechanic quantitie uch a the deformation gradient F, or the tretch tenor C=F T F. It i critical to note that due to the uage of nonlinear kinematic through F and C, the BCB model i a finite deformation, nonlinearly elatic contitutive model that explicitly repreent the tretching and rotation of bond undergoing large deformation. Silicon i well known to occur naturally in the DC lattice tructure, which i formed through two interpenetrating facecentered-cubic fcc lattice, where the two fcc lattice are offet by a factor of a 0 /4,a 0 /4,a 0 /4. The DC lattice i hown in Fig. 1, which illutrate the interpenetrating fcc lattice. The complication in modeling DC lattice, which will be reolved later, i that the interpenetrating fcc lattice mut be allowed to tranlate with repect to each other. Thi key retriction can be accommodated through a five-atom

3 Harold S. Park J. Appl. Phy. 103, =0, 12 and change the final expreion for the PK2 tre to S =2 C. 13 The patial tangent modulu can be imilarly calculated uing tandard continuum mechanic relation, and can be written a FIG. 1. Color online Illutration of the diamond cubic lattice tructure of ilicon. Black atom repreent tandard fcc unit cell atom, while green atom repreent the interpenetrating fcc lattice. The drawn bond connect atom in fcc lattice B to atom in fcc lattice A. unit cell, i.e., atom A and it four neighbor in Fig. 1 b, for which the correponding Teroff train energy denity can be written a r 1j C = V 1j r 1j C, 5 j=2 where i=1 in Eq. 7 becaue atom i i conidered the center of the unit cell ee Fig. 1, and the ummation goe over the four nearet neighbor bond j =2,3,4,5. The full expreion for the train energy denity r 1j can be written a r 1j C = V 1j 2 0 = Ae r 1j C Be r 1j C 1+ n g 1jk n 1/2n, 8 k i,j where again the multibody effect of the bonding environment are captured through the g 1jk term. We enable the interpenetrating fcc lattice to tranlate with repect to each other by introducing an internal degree of freedom aociated with all neighboring atom of atom A in Fig. 1 b through the modified bond length r 1j a 7 r 1j = r 1j = F R 1j +, j = 2,3,4,5, 9 where r 1j i the deformed bond vector, R 1j i the undeformed bond vector between atom 1 and j and i the hift introduced between the two interpenetrating fcc lattice i.e., lattice A and B in Fig. 1 in the undeformed configuration. The incorporation of the internal degree of freedom and writing the bond length in term of F reult in a modified train energy denity function a C = C, C. 10 Uing tandard continuum mechanic relationhip, we can calculate the econd Piola Kirchoff tre PK2 a 1 2 S = C = C + C. 11 To keep the crytal at an energy minimum, the internal degree of freedom are contrained to deform according to, which lead to the following relationhip: where C IJKL = M IJKL A IJp A KLq D 1 pq, M IJKL =4, C IJ C KL D pq = p q, A IJp =2 C IJ. p B. Surface Cauchy Born model for ilicon In thi ection, we preent the formulation by which urface tree are accounted for through an extenion of the BCB model we call the SCB model. The SCB model wa developed previouly for both fcc crytal 22,23 and for DC lattice. 31 We therefore briefly ummarize the relevant apect of the SCB model for ilicon 31 in thi ection. We firt note that the total energy of a nanotructure can be written a the um of bulk and urface term n atom U C d C d, 16 bulk =1 r where U r repreent the potential energy for each atom, C i the bulk energy denity previouly defined in Eq. 8, and C i the urface energy denity. The iue then i to determine a repreentation for the urface unit cell that will be ued to calculate the urface energy denity C. We accomplih thi through the nine atom urface unit cell for unrecontructed 100 ilicon urface hown in Fig. 2. The rationale for thi particular unit cell arie becaue atom 2 and 6 both have a full complement of neighbor, and thu repreent a ditinct fcc lattice B. The atom neighboring atom 2 and 6 therefore mut be part of the interpenetrating fcc lattice A, and thu hould be able to tranlate with repect to atom 2 and 6. Therefore, we aign an internal degree of freedom, where the upercript deignate an internal urface degree of freedom, to all the black atom 1,3,4,7,8,9 of fcc lattice A in Fig. 2. The reulting train energy denity for the urface unit cell een in Fig. 2 can thu be written a

4 Harold S. Park J. Appl. Phy. 103, FIG. 3. Nanowire geometry conidered for numerical example. where C IJKL M IJKL = M IJKL =4 A IJp C IJ C KL D pq = p, q, A KLq D 1 pq, 21 FIG. 2. Color online Illutration of the nine atom urface unit cell for the urface with a 010 normal of a diamond cubic crytal. Black atom repreent fcc lattice A, while green atom repreent the interpenetrating fcc lattice B. The drawn bond connect atom in fcc lattice B to atom in fcc lattice A. A IJp =2 C IJ. p 22 = 1 0 V 1j r 1j + j=2,6 + V 2m r 2m, m=1,3,4,5 k=1,7,8,9 V 6k r 6k 17 where 0 i the area per atom on the urface. Following Eq. 9, we expre the bond length for the urface unit cell a r 1j = r 1j = F R 1j +, j = 2,6, r 6k = r 6k = F R 6k +, k = 1,7,8,9, r 2m = r 2m = F R 2m +, m = 1,3,4,5. 18 Incorporating the bond length that have been modified by the deformation gradient F and the internal degree of freedom in Eq. 18 create a modified urface energy denity C from Eq. 17, where the urface energy denity can be modified analogouly to the procedure outlined previouly for the bulk energy denity in Eq. 11 and 12 to enforce the energy minimizing condition =0, 19 where, imilar to the meaning in the bulk cae in Eq. 12, repreent the deformation of the urface internal degree of freedom neceary to minimize the urface energy. Uing the modified urface energy denity C, we arrive at the expreion for the urface PK2 tre S C, where the upercript here and later indicate urface value S C =2 C C. 20 Similarly, the urface tangent modulu can be written a C. Finite element eigenvalue problem for nanowire reonant frequencie The equation decribing the eigenvalue problem for continuum elatodynamic i written a K 2 M u =0, 23 where M i the ma matrix and K i the tiffne matrix of the dicretized FE equation; the olution of the eigenvalue problem decribed in Eq. 23 give the reonant frequencie f, where f = /2 and the correponding mode hape u. We note that the tiffne matrix K contain the effect of both material and geometric nonlinearitie through a conitent linearization about the finitely deformed configuration. We emphaize that the addition of the urface energy term in Eq. 16 lead naturally to the incorporation of the urface tree in the FE tiffne matrix K, which then lead to the dependence of the reonant frequencie f on the urface tree. The eigenvalue problem wa olved uing the Sandia-developed package TRILINOS, 32 which wa incorporated into the imulation code TAHOE. 33 III. NUMERICAL EXAMPLES All numerical example were performed on threedimenional, ingle crytal ilicon nanowire of length l that have a quare cro ection of width a a illutrated in Fig. 3. Three different parametric tudie are conducted in thi work, which conider nanowire with contant croectional area CSA, contant length, and contant SAV SAV ; the geometrie are ummarized in Table I. All wire had a 100 longitudinal orientation with unrecontructed 100 tranvere urface and had either fixed/ free cantilevered boundary condition, where the left x urface of the wire wa fixed while the right +x urface of the wire wa free, or fixed/fixed boundary condition, where both the left x and right +x urface of the wire were fixed. All FE imulation were performed uing the tated

5 Harold S. Park J. Appl. Phy. 103, TABLE I. Summary of geometrie conidered: contant SAV ratio, contant length, and contant CSA. All dimenion are in nanometer nm. Contant SAV Contant length Contant CSA boundary condition without external loading and utilized regular mehe of eight-node hexahedral element. The bulk and urface energy denitie in Eq. 8 and 17 were calculated uing Teroff T3 parameter, 28 while the bulk and urface FE tree were found uing Eq. 13 and 20. Regardle of boundary condition, the nanowire are initially out of equilibrium due to the preence of the urface tree. For fixed/free nanowire, the free end expand in tenion to find an energy minimizing configuration under the influence of urface tree. To illutrate thi, we compare the energy minimized poition of the nm fixed/free nanowire uing the SCB model to a benchmark molecular tatic MS calculation performed uing the Teroff T3 potential with LAMMPS Ref. 34 MS code. A een in Fig. 4, the SCB model, which required only FE node, give a very accurate decription of the minimum energy configuration due to urface tree a compared to the MS calculation, which required more than atom. We note that the tenile train induced in the nanowire due to the urface tree i about 0.1%. A noted previouly, no external force were applied to obtain the reult een in Fig. 4; all deformation i olely due TABLE II. Summary of contant CSA nanowire fundamental reonant frequencie for fixed/free boundary condition a computed from: 1 The analytic olution given by Eq. 24, 2 BCB, and 3 SCB calculation. All frequencie are in megahertz MHz, the nanowire dimenion are in nm. Geometry Eq. 24 BCB SCB to urface tree. In analyzing the reult in Fig. 4, we emphaize that the SCB model accurately predict the tenile expanion of the free end due to urface tree, in addition to capturing the inhomogeneou nature of the tenile expanion, which occur due to the noncentroymmetric nature of the DC ilicon lattice. The reult are in agreement with firt principle calculation, 11,21 which alo indicate that 100 ilicon nanowire with 100 urface have compreive urface tree that caue the nanowire to expand. Fixed/fixed nanowire, on the other hand, are contrained uch that the nanowire i unable to expand due to the boundary condition. The boundary condition contraint therefore caue the minimum energy configuration of fixed/ fixed nanowire to be a tate of compreion, which we will demontrate i critical to undertanding how urface tree and boundary condition couple to alter the reonant propertie of fixed/fixed nanowire a compared to continuum beam theory prediction. Once the minimum energy configuration for either boundary condition i known, the eigenvalue problem decribed in Eq. 23 i olved uing the FE tiffne matrix from the equilibrated deformed nanowire configuration to find the reonant frequencie. Reonant frequencie were alo found uing the tandard BCB model without urface tree on the ame geometrie for comparion to quantify how urface tree change the reonant frequencie a compared to the bulk material for a given geometry and boundary condition. For all reonant frequencie reported in thi work, the fundamental or lowet mode frequencie correponded to a tandard bending mode of deformation. A. Contant cro-ectional area To validate the accuracy of the calculation for the BCB material, we compare in Table II and III the BCB and SCB TABLE III. Summary of contant CSA nanowire fundamental reonant frequencie for fixed/fixed boundary condition a computed from: 1 The analytic olution given by Eq. 25, 2 BCB, and 3 SCB calculation. All frequencie are in MHz, the nanowire geometry i in nm. Geometry Eq. 25 BCB SCB FIG. 4. Color online Minimum energy configuration of a fixed/free nm ilicon nanowire due to urface tree a predicted by top MS calculation, bottom SCB calculation

6 Harold S. Park J. Appl. Phy. 103, reonant frequencie to thoe obtained uing the well-known analytic olution for the fundamental reonant frequency for both fixed/free cantilevered and fixed/fixed beam. 35 For the fixed/free beam f 0 = B l 2 EI A, 24 where B 0 =1.875 for the fundamental reonant mode, E i the modulu for ilicon in the 100 direction, which can be found to be 90 GPa, 28 I i the moment of inertia, l i the nanowire length, A i the cro-ectional area, and i the denity of ilicon. The FE calculation ued to calculate the BCB and SCB reonant frequencie involved regular mehe of eight-node hexahedral element; the meh ize ranged from about 8000 to node for the contant CSA nanowire conidered. The BCB reonant frequencie compare quite well to thoe predicted by the analytic formula, with increaing accuracy for increaing apect ratio l/a, a would be expected from beam theory. We note that the SCB reonant frequencie are conitently larger than the BCB reonant frequencie and thu the analytic olution; reaon for thi trend will be dicued later. For the fixed/fixed beam, the analytic olution i given a 35 f 0 = i2 2l 2 EI A, 25 where i 1.5 i a mode hape factor for fixed/fixed beam. Table III how that the BCB and analytic olution again agree nicely. However, in contrat to the fixed/free cae, the SCB reonant frequencie are found to decreae with increaing apect ratio relative to the bulk material; again, reaon for thi will be dicued later. A key point to emphaize here i that due to the accuracy of the BCB reult for both boundary condition a compared to the analytic olution, normalizing the SCB reonant frequencie by the BCB reonant frequencie can be conidered to be equivalent to normalizing by the olution expected from continuum beam theory. Figure 5 a how the normalized reonant frequencie f cb / f bulk plotted againt both the nanowire apect ratio l/a and the SAV ratio, for both fixed/fixed and fixed/free boundary condition. A can be oberved, the urface tre effect on the reonant frequencie depend trongly upon the correponding boundary condition. For the fixed/free nanowire, the reonant frequencie predicted uing the SCB model are about 2% higher than thoe of the BCB model for all apect ratio. In contrat, the fixed/fixed nanowire how completely different behavior. In that cae, the reonant frequencie predicted by the SCB model dramatically decreae with increaing apect ratio l/ a, with the reonant frequencie due to urface tre decreaing to nearly 20% lower than the correponding bulk material when the apect ratio l/a 30. The reonant frequency calculation are alo plotted with repect to the SAV ratio in Fig. 5 b. A can be een, the FIG. 5. Color online Normalized reonant frequencie for contant CSA ilicon nanowire. fixed/free nanowire how little variation with the SAV ratio, while the fixed/fixed nanowire how a decreae in reonant frequency with decreaing SAV ratio. B. Contant length We next invetigate the reonant frequencie of nanowire in which the length of the nanowire wa fixed at 240 nm, while the quare cro ection wa varied in ize. The FE calculation to determine the reonant frequencie required meh ize ranging from about node for the mallet 8 nm cro ection to about node for the larget 30 nm cro ection conidered. A with the contant CSA nanowire, we plot the f cb / f bcb ratio againt both the apect ratio l/a and the SAV ratio in Fig. 6 a. When plotted againt the apect ratio l/a, the trend for the contant length nanowire are imilar to thoe of the contant CSA nanowire, particularly for the fixed/fixed boundary condition, for which urface tree caue the reonant frequencie to decreae rapidly with increaing l/a. In fact, for l/a=30 for the 8 nm cro-ection nanowire, urface tree caue the reonant frequency to be le than 65% of the bulk value. The urface tree caue a

7 Harold S. Park J. Appl. Phy. 103, FIG. 7. Color online Normalized reonant frequencie for contant SAV ilicon nanowire. FIG. 6. Color online Normalized reonant frequencie for contant length ilicon nanowire. lightly different trend for the fixed/free cae. There, the reonant frequencie are oberved to increae lightly with repect to the bulk value with increaing l/a, while the trend wa a very minute decreae in the contant CSA cae. However, when plotted againt the SAV ratio, a in Fig. 6 b, the reult for contant length nanowire differ trongly from the contant CSA nanowire. In particular, the reult i mot noticeable for the fixed/fixed nanowire; in the contant CSA cae, urface tree caued an increae in reonant frequency with increaing SAV ratio. However, for the contant length nanowire, the oppoite trend i oberved; the urface tree caue the reonant frequencie to decreae with increaing SAV ratio. The trend are alo revered, though not a dramatically, for the fixed/free boundary condition. C. Contant urface area to volume ratio Due to the variation in urface tre and boundary condition effect on the nanowire reonant frequencie, we conider thoe coupled effect for nanowire that have the ame SAV ratio, 0.28 nm 1. The FE meh ize ranged in thi cae from about node for the 15.2 nm cro-ection nanowire to about node for the 14.5 nm croection nanowire. Becaue the SAV ratio i kept contant, we plot the reonant frequencie for both boundary condition only againt the nanowire apect ratio l/a in Fig. 7. Figure 7 thu how one of the fundamental finding of thi work, in that the reonant frequencie of fixed/fixed ilicon nanowire do not, due to urface tree, depend on the SAV ratio. The reult for the fixed/free nanowire are more ambiguou judging olely from Fig. 7. However, Fig. 5 a and 6 a indicate that the reonant frequencie of fixed/free ilicon nanowire, imilar to fixed/fixed ilicon nanowire, do not cale according to SAV ratio. In particular, in all cae, it appear that the nanowire apect ratio l/a i a much tronger predictor of how the boundary condition and urface tree couple to vary the reonant frequencie a compared to the correponding bulk material than the SAV ratio. Thi finding correpond to reult recently publihed by Verbridge et al. 36 and Petrova et al. 37 The Verbridge work analyzed the reonant propertie of SiN nanotring, with cro-ectional dimenion around 100 nm. While the urface tre effect oberved in the preent work are unlikely to have a ignificant impact on 100 nm cro-ection nanowire, it i intereting that even when urface effect become ignificant, a they do for the nanowire conidered in the preent work, that the reonant frequencie, and thu the elatic propertie, are largely independent of SAV ratio. The Petrova work offer a comparion at a different length cale cro ection on the order of nm and for a different material, gold. However, that work alo found weak dependence of the reonant frequencie and thu elatic propertie on the SAV ratio; thee reult, on different material at different ize, lend credibility to the reult obtained in the preent work. IV. DISCUSSION AND ANALYSIS We now preent an analyi of the boundary condition and urface tre effect on the nanowire elatic propertie,

8 Harold S. Park J. Appl. Phy. 103, FIG. 8. Color online Normalized Young modulu for both fixed/fixed and fixed/free boundary condition for contant CSA nanowire. and in particular the Young modulu. To calculate the Young modulu, we utilize the beam theory expreion that relate the reonant frequencie to the modulu in Eq. 24 and 25. The beam theory expreion for the modulu are utilized a they are alo ubiquitou in the experimental literature to calculate the Young modulu for nanotructure. 3,7,16,36 38 Figure 8 depict the variation in the Young modulu, a normalized by the bulk value, for both the fixed/fixed and fixed/free contant CSA nanowire. A can be oberved, the Young modulu for the fixed/free cae how about a 5% variation from the bulk value, which can be expected from the fact that the fixed/free reonant frequencie, when bulk normalized, alo howed a mall increae with repect to the bulk reonant frequencie. However, there i a dramatic variation in the fixed/fixed Young modulu with increaing apect ratio l/a, a hown in Fig. 8. In particular, due to the nature of the reonance formula in Eq. 25, the modulu that i calculated i actually ignificantly reduced a compared to the bulk Young modulu than the bulk-normalized reonant frequencie. For example, f cb / f bulk =0.81 for l/a=30, a een in Fig. 5 a. However, when urface tree are accounted for, the Young modulu drop to only 65% of the bulk modulu when l/a =32. Furthermore, thi oberved reduction of the Young modulu ha been oberved in other theoretical tudie for fixed/fixed ilicon nanowire. In particular, we note the recent denity functional theory tudie by Lee and Rudd for ultramall 4 nm fixed/fixed ilicon nanowire, 11 which alo predicted a decreae in Young modulu due to the fact that the urface tree in conjunction with the fixed/fixed boundary condition caue the nanowire to exit in a tate of compreion; we note that the variation of the Young modulu accounting for length wa not performed in that work. Molecular dynamic imulation of the reonant frequencie of fixed edge ilicon oxide nanoplate by Broughton et al. 39 alo revealed a ditinct reduction in the reonant frequencie with decreaing ize. FIG. 9. Color online Normalized reonant frequencie for fixed/fixed contant CSA, SAV, and length nanowire plotted againt the nanowire apect ratio. We alo eek to quantify the variation due to urface tree in the reonant frequencie for the fixed/fixed cae. To do o, Fig. 9, which plot the normalized reonant frequencie f cb / f bulk for all fixed/fixed nanowire contant CSA, length, SAV againt the nanowire apect ratio l/a, demontrate one of the major finding of thi work. A can be een, for the nanowire ize conidered in thi work, the reonant frequencie for all nanowire a compared to the reonant frequencie of the bulk material overlap on a imilar curve a a function of the apect ratio, with the trend being a decreaing reonant frequency with increaing apect ratio. The enhanced effect of urface tree for the contant length nanowire with apect ratio of l/a=30 i likely due to the fact that it wa the mallet cro ection conidered, i.e., 8 nm, where the urface tre effect are particularly trong. Figure 9 can therefore erve a a deign guide for predicting how urface tree will change the reonant frequencie of nanowire a compared to the continuum beam theory in Eq. 25 which doe not account for urface effect. We attempted to determine imilar relationhip for the fixed/free nanowire in linking the oberved variation of the nanowire reonant frequencie due to urface tree to geometric parameter. Unfortunately, a illutrated in Fig. 10, uch a relationhip wa not found in thi work. We alo tudied the variation in reonant frequencie due to urface tree a compared to the tenile train in the nanowire, but a imilarly inconcluive reult wa obtained. However, Fig. 10 doe indicate that urface tree are likely not to trongly impact more than 2% the reonant frequencie of fixed/free nanowire unle very mall cro ectional area 10 nm and large apect ratio are utilized. A. Comparion to experiment An extenive literature earch ha revealed that mot tudie utilizing reonating ilicon nanowire involve nanowire with cro-ectional ize generally exceeding 50 nm. 3,7,38 At thoe ize, both the experimental reult and extrapolation of the current SCB reult indicate that urface

9 Harold S. Park J. Appl. Phy. 103, theoretical reult 11,39 in predicting a relative decreae in reonant frequencie, and thu Young modulu, for fixed/ fixed nanowire. V. CONCLUSIONS FIG. 10. Color online Normalized reonant frequencie for fixed/free contant CSA, SAV, and length nanowire plotted againt the nanowire apect ratio. effect will not have a dominant role on the reonant frequencie, and that continuum beam theory hould be valid for interpreting the reonant propertie. We did find one tudy involving the reonant propertie of ub-30 nm cro-ection ilicon nanowire, that of Li et al. 16 The ilicon nanowire in that work were of the 110 orientation and were fabricated in the fixed/fixed configuration. The nanowire were found to have a harp decreae in Young modulu, with a 53 GPa Young modulu reported for 12 nm diameter nanowire. In comparion, uing the reult in Fig. 8, we find that the SCB model predict a 58.5 GPa Young modulu for a 16 nm cro-ection 100 nanowire. We note that a direct comparion cannot be made due to the fact that the nanowire in the preent work were axially aligned in the 100 direction. Two other tudie involving the mechanical propertie of ub-30 nm cro-ection ilicon nanowire were found, with both involving tenile deformation. Kizuka et al. 17 ued an AFM to perform tenile elongation of ingle crytal 100 ilicon nanowire with cro ection le than 10 nm; the meaured Young modulu wa on the order of 18 GPa, which i coniderably maller than the 90 GPa Young modulu for bulk 100 ilicon. More recently, Han et al. 18 alo performed in itu tranmiion electron microcopy obervation of the tenile failure of 110 ilicon nanowire. Nanowire ize down to 15 nm cro ection were conidered; uing the activation energy for dilocation nucleation, they alo obtained a trong ize dependence in the Young modulu, with a modulu value of 55 GPa reported for 15 nm cro-ection nanowire. Again, the 58.5 GPa modulu obtained uing the SCB model for 16 nm cro-ection 100 nanowire agree well, though a before, a direct comparion cannot be made due to the different crytallographic orientation. Depite the mall amount of experimental data to which to compare the preent reult, the preent reult are qualitatively conitent with available experimental data and In concluion, we have utilized the recently developed urface Cauchy Born model to quantify how boundary condition and urface tree couple to caue variation in the reonant propertie of ilicon nanowire a compared to thoe expected from continuum beam theory. The reonant propertie were found through olution of a tandard finite element eigenvalue problem, where the effect of urface tree are naturally captured within the finite element tiffne matrix. The uage of a three-dimenional nonlinear finite element formulation thu efficiently enabled the analyi of a variety of nanowire geometrie to quantify urface tre effect on nanowire reonant propertie. With regard to the effect of urface tree on the ilicon nanowire reonant frequencie, we have found that: 1 Surface tree caue ignificant deviation in the reonant frequencie of nanowire a compared to thoe that are found uing tandard continuum beam theory with bulk material propertie, with the deviation having a different trend depending on whether fixed/fixed or fixed/free boundary condition are ued. We find that the reonant frequencie of nanowire with cro-ectional length greater than about 30 nm how little deviation from thoe predicted from continuum beam theory. We alo find that urface tree mot trongly impact the reonant propertie of fixed/fixed ilicon nanowire, which are found to decreae ubtantially a compared to prediction from continuum beam theory. In contrat, urface tree do not caue ubtantial deviation from beam theory for fixed/free ilicon nanowire unle nanowire with very mall cro-ectional length 10 nm and large apect ratio are conidered. 2 For fixed/fixed ilicon nanowire, accounting for the compreive tate of tre reulting from the coupled effect of urface tree and boundary condition i critical to capturing the oberved reduction in the reonant frequencie a compared to continuum beam theory. 3 The deviation that urface tree caue in the reonant propertie of fixed/fixed nanowire a compared to beam theory cale proportional to the nanowire apect ratio l/a. 4 No uch caling relationhip wa found for urface tre effect on the reonant propertie of fixed/free nanowire. 5 The preent finding that the reonant propertie of fixed/fixed ilicon nanowire, and therefore the elatic propertie uch a the Young modulu decreae with repect to the bulk value qualitatively agree with recent experimental and theoretical 11,39 reult. ACKNOWLEDGMENTS H.S.P. gratefully acknowledge upport of the NSF through Grant No. CMMI Y. Xia, P. Yang, Y. Sun, Y. Wu, B. Mayer, B. Gate, Y. Yin, F. Kim, and H. Yan, Adv. Mater. 15, H. G. Craighead, Science 290, A. N. Cleland and M. L. Rouke, Appl. Phy. Lett. 69, L. T. Canham, Appl. Phy. Lett. 57,

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