Journal ArticleMechanical bending of nanoscale thin films can be quite different from that of macroscopic thick films. However, current understanding of mechanical bending of nanoscale thin strained bilayer films is often limited within the Timoshenko model [Timoshenko, J. Opt. Soc. Am. 11, 233 (1925)], which was originally derived for macroscopic thick films. Here, we derive a modified Timoshenko formula by including the prominent effect of surface stress played in the nanofilms, which gives a  much better agreement with the experiments than the classical formula

Liu, Feng

Zang, Ji

English

The University of Utah: J. Willard Marriott Digital Library

APPLIED PHYSICS LETTERS 92, 021905 (2008)Modified Timoshenko formula for bending of ultrathin strained bilayer filmsJi Zang and Feng Liu3’Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA (Received 8 August 2007; accepted 4 December 2007; published online 15 January 2008)Mechanical bending of nanoscale thin films can be quite different from that of macroscopic thick films. However, current understanding of mechanical bending of nanoscale thin strained bilayer films is often limited within the Timoshenko model [Timoshenko, J. Opt. Soc. Am. 11, 233 (1925)], which was originally derived for macroscopic thick films. Here, we derive a modified Timoshenko formula by including the prominent effect of surface stress played in the nanofilms, which gives a much better agreement with the experiments than the classical formula. © 2008 American Institute of Physics. [DOI: 10.1063/1.2828043]Classical bending theory was established a century ago by Stoney1 and Timoshenko' in the context of bimetallic strip as used in a thermostat. The theory has since been ex­tended for stress analysis in many different areas of applica­tions, such as coating, epitaxial growth of thin films, and device interconnection.’” Recently, with the emergence of nanotechnology, the classical bending theory has also been adopted to explain self-assembly of nanostructures via bend­ing of strained nanoscale thin films,8”10 even for films downI I 12to only a few monolayers (MLs, a few angstroms) thick. ' ‘ Despite the fact that mechanical response of nanoscale struc­tures, such as bending,13'14 can be drastically different from that of macroscopic structures, most existin| theoretical analyses of mechanical bending of nanofilms ” are per­formed within the framework of continuum theory, neglect­ing the atomic details of film structure and the intrinsic stress of solid surface. For example, classical Timoshenko formula has been used to calculate the bending curvature (or radius) of rolled-up nanotubes of strained bilayer films. However, there exist apparent discrepancies between the theory and experimental results.11'15'16 Here, we rederive a modified Ti­moshenko formula that allows us to achieve a much better agreement with experiment.To illustrate our point, we first revisit a recent study of Si micro- and nanotubes made by releasing thin partially re­laxed Si layers from their substrate by selective etching,168—10using the “nanomechanical architecture” process. Pure Si tubes of different diameters ranging from 0.25 to 2.5 fini were fabricated from 3 to 20 nm thick Si films, as shown in Fig. 1 (square dots).Usually, the bending curvature of a bilayer film can be analyzed using the classical Timoshenko formula2'17 in the general form as6(Efsmtf) E j 2 7'y .(1 + /S)3,2 n 4  ’1 + 4 afi + 6a{3 + 4a(3 + a f$(la)(lb)where a=Ej!Es, Ej and Es are, respectively. Young's modu­lus of the film and substrate, f3=tf/ts is the ratio of film thickness fy-and substrate thickness r5, and t=tj+ts is the total thickness of the bilayer film. em=(as-cij)/cij is the misfit“ Author to whom correspondence should be addressed. Electronic mail: fliu@eng.utah.edu.strain where as and cij are, respectively, the lattice constant of substrate and film. For the special case of Si tubes (Fig. 1), in which both the film and substrate are made of the same ma­terial of Si (one strained and one relaxed), a=  1 and Eq. (1) reduces to a simpler form as6s(1 + / r (2)Equation (2) was used by Songmuang et al. to predict the Si tube diameters (dashed line in Fig. 1) in comparison with the experiment.16 It was assumed that the strained (partially re­laxed) Si layer (film) having a thickness tj=2 nm and under an average 2.1% tensile strain relative to the unstrained (fully relaxed) Si layer (substrate). However, such a theoret­ical prediction does not agree very well with the experimen­tal results, as shown in Fig. I .16Another example is the fabrication of GaAs/InAs nano­tubes from rolling up released GaAs/InAs bilayer films of only a few MLs thick.11 Nanotubes with diameters ranging from 10 to 1000 nm were fabricated, as shown in Fig. 2 (square dots). The diameters of GaAs/InAs nanotubes have been analyzed by a couple of groups11'15 using the classical Timoshenko formula. However, there is an apparent discrep­ancy between the theory (dashed line) and experiment (square dots), as shown in Fig. 2.11 Further effort using clas­sical Timoshenko formula but adding nonlinear and anhar- monic effects15 failed to resolve the discrepancy.In principle, the Timoshenko formula applies only to relatively thick films for which the effect of surface on me-3.02.5E3  2.0 &- a)1 15 E«  1.0o0.5 0.0 h• Experimental data//" —  - Timoshenko formula /  T/---Modified Timoshenko formula . /  1I2 4 6 8 10 12 14 16 18 20Thickness of Si layer (nm)FIG. 1. (Color online) Diameters of pure Si micro- and nanotubes as a function of thickness of Si layer with fixed thickness of strained Si layer at2 nm. See Ref. 16 for experimental details.20003-6951/2008/92(2)/021905/3/$23.00 92, 021905-1 ©  2008 American Institute of PhysicsDownloaded 16 Feh  9010 to 1 5 5  9 7  1 1 1 8 3 .  Redistribution suh iect to A IP  license or convrinht- see  httn-//anl_ain_orn/anl/convriflht_isn021905-2 J. Zang and F. Liu Appl. Phys. Lett. 92, 021905 (2008)FIG. 2. (Color online) Diameters of InAs/GaAs nanotubes as a function of thickness of GaAs layer with fixed thickness of InAs layer at 2 ML. See Ref. 11 for experimental details.chanical bending can be neglected. An intrinsic property of a solid surface is its non-zero surface stress, which is generally further enhanced by surface reconstruction.18 For ultrathin films that are only a few nanometers thick, surface stress due to surface reconstruction or molecular adsorption has been shown to affect the film bending behavior significantly.19'20 Since misfit strain and instrinsic surface stress drive the beinding together, apparently, the Timoshenko formula that accounts only for misfit strain can not give a good descrip­tion of the bending behavior of nanometer-thick bilayer films (it is not only quantitatively inaccurate but can also be quali­tatively incorrect for some cases19). Therefore, to resolve the discrepancy between the classic theory and experimental re­sults, as shown in Figs. 1 and 2, a modified Timoshenko formula is required for assessing the nanomechanical bend­ing behavior of ultrathin films that are only a few nanometers thick. This has to be achieved by taking into account the effects of surface stress.There are two nanoscale surface-stress effects need to be added: the intrinsic surface stress due to surface reconstruc­tion and the additional surface stress induced by large bend­ing. One usually assumes the intrinsic surface stress remains constant during and after bending. This is approximately true for a thick film of very small bending curvature. But the bending curvature increases with decreasing film thickness. For example, if a film thickness is reduced from 1 ££.m to 1 nm, its bending curvature will increase by six or­ders of magnitude. The very large bending curvature of a nanofilm means a very large bending strain in the film sur­face, which will in turn change the surface stress. Thus, in order to derive a more correct bending curvature formula, the bending induced additional surface stress in the top (bottom) surface must be included.The top and bottom surface stresses of a bilayer film upon bending can be generally calculated as <Xts,bs=o'tso,bsO + Cis,bs£ts,bs- o'tsO ar|d CTbso are, respectively, the intrinsic sur­face stress in top and bottom surface. Cls-bssls-bs are the bend­ing strain (sls and sbs) induced additional surface stress where Cls and Cbs denote the “in-plane” elastic constants of the top and bottom surface layer, marked as the thin blue and pink layers in Fig. 3.The bending induced strain in the top and bottom surface are, respectively, sls=s,„+K(z.Q-tj) = em+ea-KtJ2-Ktj, sbs = -(-f,)] = ea + Ktsl 2, where sa=/c[z0-(-rs/2)] = /cz0FIG. 3, (Color online) Physical and geometric parameters used for the derivation.z=-ts/2. The top and bottom surface strain energies can then be calculated as (assuming Cls=Cbs=C')JEti=A (jlsrfsls =C'tOtsol s,„ + Sa - -y - Ktf)Kt,+ ^-\Sln + Sa- —  -Ktf (3)ZTbs = A I ubide-bs( xts\ C 'Jt ^ b s o l  ea +  9 I +  9 l s a +  9 (4)Here, A is the surface area. Now, for the case of Si tubes (Fig. 1), a=  1 for Ej=Es, then the bending strain energy in the film and substrate can be calculated, respectively, as9Eftj= —£t| — rf + —tetr - KSJr + —t'Jr~ KeJJr+ s'J,'/ 9 ' r V j ' / 'and' xsmtj - Kemtstf +2emeatf + e;ntf j/ j -73 j \:- £ s( ^ * ^  + s;rsJ.*'/(5)(6)Minimization of total energy E=El!t+Eb;t+Efb+E;tb with re­spect to k and sa leads to a modified Timoshenko formula,6(Eseint,) ^  6 (C 'sJ 6(Acr)K —  ^ (1 + /?) +  ^  ^ .E f  + 6C[t E J2 + 6CrJ  E /  + 6CrJ(7)+ Kts/2  is introduced as the average strain of the substrate atDownloaded 16 Feh  2010 to 155.97.11.183. Redistribution suh iectIn Eq. (7), the first two terms account for the misfit strain effect and the third for the intrinsic surface stress effect, with Acr=(jls0-(jbs0 representing the difference of intrinsic sur­face stress between the top and bottom surface. It reduces to Eq. (2) when all the surface stress effects are absent, i.e., Acr=0 and C'=  0.If we examine closely Fig. 1, we notice that the main difference between theory and experiment is that the two curves have a different dependence (slope) of diameters on thickness. This indicates that the classical Timoshenko for­mula fails to correctly predict the scaling of bending curva­ture with the thickness. Comparing Eqs. (7) and (2), we see that in Eq. (2) k scales with f 2, while in Eq. (7) k scalesto AIP license or convrinht; see httn://anl.ain.orn/anr/convrinht.isn021905-3 J. Zang and F. Liu Appl. Phys. Lett. 92, 021905 (2008)with {af + btT1. This different scaling results from the bend­ing induced surface stress contribution (Cts bsets bs) in Eq. (7). Therefore, we will reanalyze the experimental results of pureSi tubes in Fig. 1 by including the bending induced surface stress effect. Assuming e„,=0.021 as in Ref. 16, we can fit the experimental data nicely using Eq. (7) with the fitting parameters C5' = 16.45 eV/A2 and Acr=113 meV/A2 (~1.8J/m2). The fitted surface stress difference (Act) is consistent with the well-known values of Si(001) surface stresss18 as well as the typical values of the surface stress for71 . ' 7most solid materials,‘ which are approximately 1-3 J/nf. However, the fitted surface elastic constant C' seemed to be too large compared to the existing theoretical value in litera­ture with a different sign.“  The reason for such discrepancy needs further study. On the other hand, we note that if C' is set to zero, no satisfactory fitting can be obtained using either Eq. (7) by adjusting sm and/or Act. Thus, the physically im­portant factor is that the diameters of nanotubes scales with film thickness differently due to the atomic-level surface stress effects, which cannot be predicted by the classical bending theory with or without nonlinear and anharmonic effects.Next, we turn our attention to the experimental data of GaAs/TnAs nanotubes in Fig. 2. We notice that the main difference between the experimental data (square dots) and theoretical prediction by classical Timoshenko formula (dashed line) is an almost constant shift of curves. We expect this shift is caused by the difference of intrinsic surface stress between the top GaAs surface and bottom TnAs surface. Whereas the effect of bending induced surface stress, which will affect the slope of the bending curve as discussed above for Si nanotubes, is less important and negligible. Thus, set­ting Cts=Cbs=0 and using a + 1, we derived the following modified Timoshenko formula for this special case:6(EfSjj) + 6(C1crts0 - c 2crbs0) k =  — ------ ;--------- yE r1+2p + a p  1 + /31 + 2 a/3+ap1 1 + /3(8a)(8b)Equation (8) gives an excellent agreement with experiment shown in Fig. 2 as solid red line with <rts0=49.1 meV/A2 and crbs0=54.9 meV/A2. Again, the fitted surface stresses are consistent with typical values of solid surfaces.^1For the most general cases, we will have different elastic constants in the film and substrate (Ey#/y, different intrin­sic surface stress (<xts0 # <xbs0) and different in-plane elastic constants (Cts# C bs) in top and bottom surfaces. Then we must use the most general form of the modified Timoshenko formula derived as the following:______ 6(Efsmtf)F+6(GiO-tso - G2o~bso)£ / 2 + 4f(//iCts + //2Cbs) + 12CtsCbs£7‘(1 +/3)yy, (9a)F=  1 +Cbs(2 + /3) E,t2CtsCbs(1+/3) + Qsg/(1+2/3)/(1+/3)EsEjttf ’(9b)1+2/3+a/^ 2(1+/3 )Cbs ■ + ■ '\ ~¥ (3 EJ1 +2a(3+a(32 2(1 + /3)Cts1 + /3 EJ//, = ( 1 +/3)-2(1 +3/3+3/32+ afi)y ,H2 = ( 1 +/3r2(1 +3a(3+ 3afi2 + afi?,)y.(9c)(9d)This general formula has been shown to agree very well with molecular dynamics simulation results of Si/Ge nanotubes.19In conclusion, we have derived the modified Timosh­enko formula for calculating the bending curvature of nanos­cale strained bilayer films using an energy minimization scheme within the framework of continuum mechanics. The main modification over the classical formula is to include the prominent role of surface stress in two important manifesta­tions. One is the effect of intrinsic surface stress due to dif­ference in surface structure and morphology, which changes the bending magnitude. The other is the effect of bending induced surface stress due to the large bending curvature, which changes the scaling of bending curvature with thick­ness. The modified Timoshenko formula has been used to predict the diameters of Si and GaAs/TnAs nanotubes made from folding of strained bilayer nanofilms, giving a much better agreement with experiments.This work was supported by NSF-CMMT-0652461.'G. G. Stoney, Proc. R. Soc. London. Ser. A 82. 172 (1909).2S. Timoshenko. J. Opt. Soc. Am. 11. 233 (1925).-p. A. Hinn, D. S. Gardner, and W. D. Nix. Ililili Trans. Electron Devices 34, 689 (1987).4L. B. Freund. J. Cryst. Growth 132. 341 (1993).5Y. C. Tsui and T. W. Clyne, Thin Solid Films 306. 23 (1997).(,Z. Suo, Ii. Y. Ma. H. Gleskova. and S. Wagner. Appl. Phys. Lett. 74. 1177(1999).7F. Spaepen, Acta Mater. 48. 31 (2000).8V. Y. Prinz, V. A. Seleznev. A. K. Gutakovsky, A. V. Chehovskiy. V. V. Preobrazhenskii, M. A. Putyato, and T. A. Gavrilova. Physica Ii (Amsterdam) 6. 828 (2000).90. G. Schmidt and K. Hberl, Nature (London) 410. 168 (2001).I0M. Huang. C. Boone. M. Roberts. D. Ii. Savage. M. G. Lagally. N. Shaji.H. Qin. R. Blick. J. A. Naim, and F. Liu. Adv. Mater. (Weinheim, Ger.) 17. 2860 (2005).nC. Deneke, C. Muller. N. Y. Jin-Phillipp. and O. G. Schmidt. Semicond.Sci. Technol. 12. 1278 (2002).I20. Schumacher. S. Mendach, H. Welsch. A. Schramm. Ch. Heyn, and W.Hansen. Appl. Phys. Lett. 86. 143109 (2005).,:<F. Liu. P. Rugheimer, Ii. Mateeva, D. Ii. Savage, and M. G. Lagally.Nature (London) 416, 498 (2002). i4F. Liu. M. Huang. P. Rugheimer. D. Ii. Savage, and M. G. Lagally. Phys.Rev. Lett. 89. 136101 (2002). i5M. Grandmami. Appl. Phys. Lett. 83. 2444 (2003). lflR. Songmuang, Ch. Deneke. and O. G. Schmidt. Appl. Phys. Lett. 89. 223109 (2006).I7M. Huang. P. Rugheimer. M. G. Lagally. and F. Liu. Phys. Rev. B 72. 085450 (2005).i8F. Liu and M. G. Lagally. Phys. Rev. Lett. 76. 3156 (1996).J. Zang. M. Huang, and I-'. Liu. Phys. Rev. Lett. 98. 146102 (2007).20J. Zang and I-'. Liu. Nanotechnology 18. 405501 (2007).2IR. C. Cammarata, Prog. Surf. Sci. 46. 1 (1994).22S. Izumi, S. Hara. T. Kumagai, and S. Sakai. Thin Solid l-'ilms 467. 253(2004).

Modified Timoshenko formula for bending of ultrathin strained bilayer films

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Modified Timoshenko formula for bending of ultrathin strained bilayer films

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