NANO LETTERS
A Generalized Description of the Elastic Properties of Nanowires
2006 Vol. 6, No. 6 1101-1106
Andreas Heidelberg,† Lien T. Ngo,† Bin Wu,† Mick A. Phillips,† Shashank Sharma,‡ Theodore I. Kamins,‡ John E. Sader,§ and John J. Boland*,† Department of Chemistry and the Center for Research on AdaptiVe Nanostructures and NanodeVices (CRANN), Trinity College Dublin, Dublin 2, Ireland, Quantum Science Research, Hewlett-Packard Laboratories, 1501 Page Mill Road, MS 1123, Palo Alto, California 94304, and Department of Mathematics and Statistics, UniVersity of Melbourne, Victoria 3010, Australia Received January 6, 2006; Revised Manuscript Received April 3, 2006
ABSTRACT We report a model of nanowire (NW) mechanics that describes force vs displacement curves over the entire elastic range for diverse wire systems. Due to the clamped-wire measurement configuration, the force response in the linear elastic regime can be linear or nonlinear, depending on the system and the wire displacement. For Au NWs the response is essentially linear since yielding occurs prior to the onset of the inherent nonlinearity, while for Si NWs the force response is highly nonlinear, followed by brittle fracture. Since the method describes the entire range of elastic deformation, it unequivocally identifies the yield points in both of these materials.
Knowledge and understanding of the mechanical properties of nanowires (NWs) are extremely important for emerging applications in the areas of nanocomposite strengtheners, nanoscale interconnects, and the active components in nanoelectromechanical (NEMS) devices. In recent years, researchers have investigated the mechanical properties of various nanowire systems using different techniques.1-18 These include the use of a nanostressing stage within a scanning electron microscope,1,2 vibrational studies within a transmission electron microscope (TEM),3-5 and atomic force microscope (AFM) manipulations.6-18 Each involves pinning the nanowire at a particular location and measuring its response to thermal excitation or mechanical deformation. The clamped-clamped beam configuration has been widely used to study the mechanical properties of NWs.6-14 Using this configuration, we recently reported a new AFM-based method that enables the full spectrum of NW mechanical properties to be measured, including the elastic properties, plastic deformation, and failure.11 This technique involved clamping nanowires over trenches on a SiO2 substrate and then measuring their response under the lateral load from an AFM tip (Figure 1). This method has several advantages over those reported earlier in that it employs well-defined clamping points and eliminates wire-substrate friction effects. * To whom correspondence may be addressed. E-mail:
[email protected]. † Department of Chemistry and the Center for Research on Adaptive Nanostructures and Nanodevices, Trinity College Dublin. ‡ Quantum Science Research, Hewlett-Packard Laboratories. § Department of Mathematics and Statistics, University of Melbourne. 10.1021/nl060028u CCC: $33.50 Published on Web 05/13/2006
© 2006 American Chemical Society
Figure 1. Schematic of the experimental setup for lateral nanowire manipulation: (a) before the manipulation and (b) during the manipulation. The nanowire under investigation is suspended over a trench in the substrate and fixed by Pt deposits at the trench edges. During the manipulation the lateral and normal cantilever deflection signals are simultaneously recorded.
Despite the simplicity and widespread use of the clampedbeam configuration, vastly different force (F) vs displacement (d) curves have been reported for different systems. For example, in the case of metallic NWs, we reported F-d curves that were linear for small displacements after which there was an abrupt change in slope associated with the yield point in these materials.11 Linear F-d curves were also reported for single-wall carbon nanotube bundles,7,10,14 multiwalled carbon nanotubes,6,9 MoS2 ropes,8 and even lithographically defined Si beams.12,13 Nonlinear F-d curves have also been reported in measurements involving carbon
nanotubes15-17 and for poly(ethylene oxide) and glass nanofibers.18 These observations have led to the use of models that treat F-d curves as being exclusively linear or nonlinear. The limitations of using these exclusively linear or nonlinear models are yet to be investigated in the literature. Furthermore, a comprehensive model that accounts for the detailed shape of these F-d curves over the entire elastic region is yet to be described and validated for nanowire systems. Here we provide a complete description of the elastic properties in a double-clamped beam configuration over the entire elastic regime for diverse wire systems. Within this model we show that it is possible to perform a comprehensive analysis of force-displacement (F-d) curves using a single closed-form analytical description and not only to extract linear material constants such as the Young’s modulus (E) but also to describe the entire elastic range and hence to identify the yield points for dramatically different systems such as Si and Au nanowires. The E-modulus obtained for the Si nanowires is diameter independent and comparable to that of bulk Si. We demonstrate that the linear F-d curves previously reported for Au NWs are a consequence of mechanical yielding at small displacements prior to the onset of a nonlinear geometric effect associated with tensile stretching. In the case of Si NWs the F-d curves are highly nonlinear at large deformations and exhibit brittle failure without measurable plastic deformation. Si nanowires were grown epitaxially in 〈111〉 directions on (100) oriented Si substrates using chemical vapor deposition catalyzed by Au nanoparticles.19 The nanowires were then sonicated off the supporting substrate and dispersed in IPA. Au NWs were prepared by electrochemical deposition into pores of highly ordered anodized aluminum oxide (AAO) templates11,20 and were dispersed in toluene after being liberated from the template. Both types of wires were then deposited over trenches on a SiO2 substrate and clamped at the trench edges by electron beam induced deposition of Pt. The clamping points are robust and, although they carry the largest tensile or compressive stresses, do not show failure or movement after manipulations. After sample preparation, AFM lateral manipulations were carried out using a Digital Instruments Nanoman System with closed-loop x-y-zscanner. During the manipulation the lateral and normal force signals were recorded. A more detailed description of the substrate preparation and manipulation procedures can be found elsewhere.11 The upper panel in Figure 2 shows a typical F-d curve during the bending test for a Si nanowire with a radius r of 75 nm. On approach of the cantilever to the nanowire, the lateral force acting on the AFM cantilever is essentially zero as seen in Figure 2. At the point of contact between AFM tip and Si nanowire, the force increases linearly at first, but after a displacement d, of about 75 nm a nonlinear increase can be observed. (The normal load applied by the AFM tip to the nanowire was also measured, but it had no significant influence on the nanowire mechanics and is omitted for clarity.) Increased loading of the nanowire ultimately results in failure without plastic deformation, consistent with a brittle material like Si. The lower panel in Figure 2 shows the F-d curve obtained for a Au NW (r ) 1102
Figure 2. Comparison of typical F-d curves obtained by bending experiments for a Si and a Au nanowire. (upper panel) F-d curve recorded during manipulation of a 75 nm radius Si nanowire. After an initial linear rise upon contact of the AFM tip, the F-d curve clearly shows the nonlinear elastic behavior of the Si wire, followed by brittle failure of the structure. (Lower panel) F-d curve taken during a manipulation of a 113 nm radius Au wire. A pseudolinear elastic region is visible followed by a yield point after which plastic deformation occurs. The wire was unloaded prior to failure.
113 nm). The latter is characterized by a linear region followed by an abrupt change in slope previously attributed to the yield point of the material, followed by plastic deformation. In contrast with the Si case, there is little evidence for nonlinear behavior in the elastic region of the F-d curve. To account for these different behaviors, we have to consider that in linear elastic beam bending theory, the Young’s modulus is normally obtained from a measurement of the beam displacement as a function of the applied load. For a clamped-clamped beam used here, the resulting curve is described by the following well-known equation21 Fcenter )
192EI ∆zcenter L3
(1)
where Fcenter is the load applied to the center of the beam, ∆zcenter is the resulting displacement of the beam at the load point, E is the Young’s modulus of the beam, I ) πR4/4 is the moment of inertia (assuming a cylindrical beam), and L is the beam length. Importantly, this equation predicts a linear relationship between the applied load and the resulting displacement and thus requires that measurements also exhibit this property. However, as the beam is displaced, an axial tensile force is inherently induced due to the stretching of the beam. This force affects the total stress experienced by the beam, leading to an enhancement of its rigidity.21 Here we present a simple closed form solution to this problem that is valid for any beam displacement, enabling the determination of the Young’s modulus irrespective of the applied load. Nano Lett., Vol. 6, No. 6, 2006
To begin we note that the governing beam equation,21 which includes the effects of axial forces, is 4
EI
asymptotic solutions
R)
2
dw dw - T 2 ) Fcenterδ(x - (L/2)) 4 dx dx
(2)
{
12 3 2 , f0 5 875 2 , f∞
(8)
where where w is the deflection of the beam, x is the spatial coordinate along the beam length, T is the tension along the beam, and δ is the Dirac delta function. Note that ∆zcenter ) w(L/2). The tension T is determined from the strain along the axis of the beam, and is given by T)
EA 2L
∫0L (dw dx )
2
dx
(3)
) ∆z2center
(4)
Solving eqs 2-4 gives the required solution Fcenter )
192EI f(R)∆zcenter L3
(5)
where R
f(R) ) 48 -
R)
where R is related to the maximum deflection ∆zcenter by the following transcendental equation
2 + cosh(xR /2) - 6
sinh(xR /2)
(
)
tanh(xR /4)
xR
xR ∆z2center
2
)
()
A (7) I
We note that f(R) g1, so that inclusion of the tensile force increases the effective rigidity of the beam. Also, R is directly connected to the tension in the rod via the relation R)
TL2 EI
and represents the ratio of axial stresses due to extension and bending. Whereas eqs 5-7 present an exact analytical solution,21 it is complex in nature, requiring the numerical solution of a transcendental equation. To overcome this limitation, we note that the solution to eq 7 possesses the following Nano Lett., Vol. 6, No. 6, 2006
(10)
which exhibits a maximum error of only 2.1% over the entire range of . Equations 5, 6, and 10 give the generalized form we seek and enable the applied force to be determined for any beam displacement. Equation 9 indicates that the solution, eq 1, is valid provided the maximum displacement ∆zcenter is less than the radius of the beam. As ∆zcenter is increased to a value larger than the beam radius, the solution becomes progressively nonlinear with respect to the displacement, leading finally to a cubic dependence on ∆zcenter. Thus, an approximate solution can be obtained by superimposing the small and large deflection limits, namely Fcenter )
xR
1-4
6(140 + ) 350 + 3
(6)
192 tanh(xR /4)
R cosh2(xR /4)
(9)
An approximate and explicit expression for R is then obtained by constructing a Pade´ approximant using eq 8, i.e.
Equations 2 and 3 are solved subject to the usual clamped boundary conditions at both ends of the beam, i.e. dw ) 0 at x ) 0 and L w) dx
(AI)
192EI A ∆z2center ∆zcenter 1 + 3 24I L
(
)
(11)
a result which exhibits a maximum error of 18%. This approximate solution (which is not used in any analysis detailed here) contains both linear and nonlinear terms widely reported in the literature. The linear dependence accounts for elastic bending of the beam, whereas the cubic dependence describes the tensile stresses that are induced along the length of the beam at large displacements. Figure 3a shows a plot of the rigidity enhancement function f(R) as a function of x (which is directly proportional to the maximum displacement, see eq 9). In parts b-d of Figure 3 the effect of this nonlinearity on the measured force curves for a cylindrical beam is illustrated. First, we consider the case where the maximum deflection is one radius (Figure 3b). It can clearly be seen that for deflections smaller than half of one radius, the standard theory (eq 1) accurately predicts the behavior of the nanowire, with observable deviations occurring as the deflection approaches one radius. Note that the results have been normalized so that a slope of unity corresponds to the standard linear theory. Next, we show the force curve for a slightly larger extension up to twice the radius of the nanowire (Figure 3c). As can be seen, the curve becomes increasingly nonlinear as the deflection is increased past one radius. For larger deflections, the nonlinearity grows, leading 1103
Figure 3. (A) Plot of the rigidity enhancement function f(R) as a function of x. (B) Calculated F-d curve for a circular cylinder showing the nonlinearity for a maximum deflection of one radius. The force values have been normalized so that a slope of unity corresponds to the standard linear theory. (C) Simulated and normalized F-d curve for a maximum displacement of two radii. (D) F-d curve for large deflections showing a growing nonlinearity leading to the ultimate cubic dependence on the deflection.
to an ultimate cubic-like dependence on the deflection (see Figure 3d). However, it is important to emphasize that the gradual change in the slope of the F-d curve makes it difficult to exclusively identify linear and nonlinear regions and necessitates the use of the generalized eqs 5, 6, and 10, which are valid for all displacements in the elastic region. Whenever the linear or nonlinear term is used by itself, as is presently the case in the literature, it provides a very poor description of the mechanical properties. For example, an exclusively linear treatment fails to recognize that the F-d data beyond the linear region is still the result of elastic deformation, except that the dominant stresses are now tensile and associated with bending the wire beyond one radius. This model can be directly compared with the force curves recorded using Si and Au nanowires.22 Figure 4a shows a fit (using the generalized eqs 5, 6, and 10) for the F-d curve for the Si wire shown in Figure 2a, and which gives an E-modulus of 190 GPa. The only adjustable parameter in the fit was the E-modulus. It is important to note that the excellent fit in Figure 4a covers the entire displacement range, from the first contact point between tip and nanowire to the sharp force-drop point, which indicates failure without plastic deformation, hence demonstrating that Si nanowires are brittle materials. In Figure 5 the E-moduli determined for nanowires with radii between 50 and 100 nm are shown. The average modulus E ) 158 GPa is essentially independent of the radius over the range investigated and close to the value reported for bulk Si(111) (169 GPa).23 The scatter in the data presented here can be explained by the fact that the measured E-modulus is dominated by estimates of the physical dimensions of the wire and AFM cantilever/tip11 1104
and is not a reflection of the model itself. Indeed the accuracy of the model is significantly better than that of present experimental methods, which are limited by physical dimension measurements, but is expected to become more important as better nanomechancial techniques are developed. The real importance of this method is the ability to unambiguously identify the elastic deformation regime for a particular material. For example, in the case of the Au nanowire system, we previously demonstrated that the force response is essentially linear and suggested that the change in slope after this initial linear region was due to yielding.11 This analysis is borne out by the application of the present model to this system. The lower curve in Figure 2 shows the F-d curve for an Au nanowire (r ) 113 nm) that is essentially linear up to an obvious break point, which occurs at a wire displacement of about 110 nm. Figure 4b shows that these data can be fit using eqs 5, 6, and 10 up to the break point, thus demonstrating the break point is indeed the yield point of the material. Given the data for a particular Au NW, and assuming the modulus is a materials constant for the system under study, this model can be used to predict the generalized elastic behavior for any Au nanowire. The fact that the Au NW in Figure 4b yielded at a wire displacement of 110 nm reflects the detailed microstructure of the wire. Hardening of the wire, whether by grain boundary or impurity hardening, will increase the yield point but the F-d data must still follow the model (red curve) in Figure 4b, except that the point where the force data deviates from the model occurs at larger displacements (shown as open circles in Figure 4b). Using this approach, it is possible to accurately study the evolution Nano Lett., Vol. 6, No. 6, 2006
Figure 5. Plot of the E-modulus versus the nanowire radius in the range of 50-100 nm for Si nanowires, showing that the observed E-modulus is radius independent (within the limits of the scatter in the data), with an average value of 158 GPa is close to the modulus for bulk Si(111) of 169 GPa (solid line).24
and hence facilitates studies of the evolution of NW mechanical properties during work hardening and annealing.
Figure 4. Curve fits for Si and Au F-d curves to eqs 5, 6, and 10. The only adjustable parameter in the fits is the E-modulus. (A) For a Si nanowire with a radius of 75 nm, the curve fit gives an E-modulus of 190 GPa. (B) For a Au wire with a radius of 113 nm, an E-modulus of 75 GPa is obtained by fitting to the initial part of the F-d curve. The extrapolated fit to the curve directly confirms that the Au wire yields before the onset of appreciable nonlinear behavior. Predicted F-d behavior of a hypothetical hardened Au NW, with the same E-modulus, is shown with open circles.
of the mechanical properties of a family of NWs through a series of work-hardening or annealing treatments.25 In conclusion we have introduced a model which accurately accounts for the mechanical properties of nanowires in a clamped-clamped beam configuration over the entire elastic range and provides a comprehensive methodology for the analysis of a broad range of nanowire systems. A nonlinear response is expected for small wires such as single carbon nanotubes for all significant displacements whereas an initial linear response followed by an approximate cubic dependence is expected for larger diameter wires. In general, however, a detailed description of such F-d curves requires a solution of the generalized equations developed here. Whether this nonlinear behavior is observed depends on the intrinsic mechanical properties of the wire, and for systems such as metals that are susceptible to mechanical yielding the nonlinear response is suppressed. Even in this case, however, for an accurate measurement of elastic properties it is necessary to account for nonlinear effects that result from build-up of tensile stresses in the wire. Finally, this model allows the yield point to be identified in NW systems Nano Lett., Vol. 6, No. 6, 2006
Acknowledgment. The work at the Department of Chemistry and the Center for Research on Adaptive Nanostructures and Nanodevices at Trinity College Dublin is supported by Science Foundation Ireland under Grant 00/ PI.1/C077A.2. The research by J.E.S. is supported by a Science Foundation Ireland Walton Fellowship and was performed while on leave at Trinity College Dublin. The work at Hewlett-Packard is partially supported by the U.S. Defense Advanced Research Projects Agency (DARPA). References (1) Yu, M.-F.; Lourie, O.; Dyer, M. J.; Moloni, K.; Kelly, T. F.; Ruoff, R. S. Science 2000, 287, 637-640. (2) Yu, M.-F.; Files, B. S.; Arepalli, S.; Ruoff, R. S. Phys. ReV. Lett. 2000, 84, 5552-5555. (3) Krishnan, A.; Dujardin, E.; Ebbesen, T. W.; Yianilos, P. N.; Treacy, M. M. J. Phys. ReV. B 1998, 58, 14013-14019. (4) Poncharal, P.; Wang, Z.-L.; Ugarte, D.; de Heer, W. A. Science 1999, 283, 1513-1516. (5) Chopra, N. G.; Zettl, A. Solid State Commun. 1998, 105, 297-300. (6) Wong, E.-W.; Sheehan, P. E.; Lieber C. M. Science 1997, 277, 19711975. (7) Kis, A.; Kasas, S.; Babic, B.; Kulik, A. J.; Benoit, W.; Briggs, G. A. D.; Schonenberger, C.; Catsicas, S.; Forro, L. Phys. ReV. Lett. 2002, 89, 248101. (8) Kis, A.; Mihailovic, D.; Remskar, M.; Mrzel, A.; Jesih, A.; Piwonski, I.; Kulik, A. J.; Benoit, W.; Forro, L. AdV. Mater. 2003, 15, 733736. (9) Salvetat, J.-P.; Kulik, A. J.; Bonard, J. M.; Briggs, G. A. D.; Stockli, T.; Metenier, K.; Bonnamy, S.; Beguin, F.; Burnham, N. A.; Forro, L. AdV. Mater. 1999, 11, 161-165. (10) Salvetat, J.-P.; Bonard, J. M.; Thomson, N. H.; Kulik, A. J.; Forro, L.; Benoit, W.; Zuppiroli, L. Appl. Phys. A 1999, 69, 255-260. (11) Wu, B.; Heidelberg, A.; Boland, J. J. Nat. Mater. 2005, 4, 525529. (12) Namazu, T.; Isono, Y.; Tanaka, T. J. Microelectromech. Syst. 2000, 9, 450-459. (13) Sundararajan, S.; Bushan, B.; Namazu, T.; Isono, Y. Ultramicroscopy 2002, 91, 111-118. (14) Salvetat, J.-P.; Briggs, G. A. D.; Bonard, J. M.; Bacas, R. R.; Kulik, A. J.; Stockli, T.; Burnham, N. A.; Forro, L. Phys. ReV. Lett. 1999, 82, 944-47. (15) Walters, D. A.; Ericson, L. M.; Casavant, M. J.; Liu, J.; Colbert, D. T.; Smith, K. A.; Smalley, R. E. Appl. Phys. Lett. 1999, 74, 380305. 1105
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1106
found these equations to be valid. Au nanowires with very large length-to-diameter ratios (>20) can be pushed more than one diameter before they yield, and F-d curves of these nanowires show nonlinear behavior before onset of inelastic deformation. For the Au nanowires studied here, this onset is obvious, but for materials that yield less dramatically or for wire configurations that have wide range of aspect ratios, this analysis may be exceptionally useful. (23) Wortman, J. J.; Evans, R. A. J. Appl. Phys. 1965, 36, 153-156. (24) Callister, W. D., Jr. Materials Science and Engineering; Wiley: New York, 1994. (25) Wu, B.; Heidelberg, A.; Boland, J. J.; Sader, J. E.; Sun, X.-M.; Li, Y.-D. Nano Lett. 2006, 6, 468-472.
NL060028U
Nano Lett., Vol. 6, No. 6, 2006
