NANO LETTERS
Force-Deflection Spectroscopy: A New Method to Determine the Young’s Modulus of Nanofilaments
2006 Vol. 6, No. 9 1904-1909
Qihua Xiong,†,‡,| N. Duarte,§ S. Tadigadapa,§ and P. C. Eklund*,†,‡ Department of Physics, Department of Materials Science and Engineering, and Department of Electrical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 Received May 1, 2006; Revised Manuscript Received June 28, 2006
ABSTRACT We demonstrate the determination of Young’s modulus of nanowires or nanotubes via a new approach, that is, force-deflection spectroscopy (FDS). An atomic force microscope is used to measure force versus deflection (F−D) curves of nanofilaments that bridge a trench patterned in a Si substrate. The FD data are then fit to the Euler−Bernoulli equation to determine Young’s modulus. Our approach provides a generic platform from which to study the mechanical and piezoelectric properties of a variety of materials at the nanoscale level. Young’s modulus measurements on ZnS (wurtzite) nanowires are presented to demonstrate this technique. We find that the Young’s modulus for rectangular cross section ZnS nanobelts is 52 ± 7.0 GPa, about 30% smaller than that reported for the bulk.
Accurate determination of mechanical constants and properties of nanoscale materials, for example, nanotubes or nanowires (NTs/NWs), are important for their application in nanomechanics,1 micro/nanoelectromechanical systems,2,3 and sensors.4 However, mechanical or electromechanical measurements are quite challenging at the nanoscale level. Several recent approaches to study nanomechanical properties have relied on atomic force microscopy (AFM). For example, Wong et al.1 first reported the utilization of lateral force microscopy (LFM) to determine the Young’s modulus (E) of silicon carbide nanorods and multiwalled carbon nanotubes (MWNTs). Song et al.5 also used LFM to measure the elastic modulus of vertically aligned ZnO nanowires. In part due to the uncertainty in the measurements of the nanofilament dimensions (i.e., cross section and length), the reported values of Young’s modulus relative to values for the same bulk material have exhibited a large variation.5 A particular problem for the LFM approach is the torsional mode of the AFM cantilever, which is not easy to calibrate. Despite this drawback, LFM nonetheless remains an important method for vertically aligned nanostructures with short length and low density on the supporting substrate. The bending of MWNTs6 or single-walled carbon nanotube (SWNT) ropes7 supported on nanopores of an anodic alumina membrane * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Physics. ‡ Department of Materials Science and Engineering. § Department of Electrical Engineering. | Present address: Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA 02138. 10.1021/nl060978f CCC: $33.50 Published on Web 08/03/2006
© 2006 American Chemical Society
have been measured under contact-mode AFM. In these particular cases, a combination of elastic and shear moduli of the nanotubes was measured. Recently, nanoindentation has emerged as an alternative technique to measure Young’s modulus, E, and hardness at the nanoscale level. This method has been applied to ZnS nanobelts8 and silver nanowires.9 Nanoindentation is also an AFM-based method, and it employs a stiff cantilever with a diamond tip to make an indentation under a known applied force. Force-displacement curves are obtained, and the morphology of the indent is subsequently imaged by the same cantilever. The “hardness” is formally defined as the maximum force per unit area to make the indent without inducing plastic deformation. In bulk materials, there is an empirical relationship between Young’s modulus, E, and hardness. However, it is not obvious that this bulk relationship will hold at the nanoscale level. We therefore suggest that a combination of bending and hardness measurements be made to determine if it holds at the nanoscale level. Electromechanical resonance excited in situ with a transmission electron microscope (TEM) is another technique that has been used to measure the mechanical properties of nanoscale filaments such as MWNTs,10 ZnO nanobelts,11 and GaN nanowires.12 The dimensions of the filament and the resonant frequency are observed in TEM images; the latter is recognized by the maximum amplitude of vibration of the nanofilament. In this letter, we report on a new approach to measure the Young’s modulus, E, of nanofilaments. We have developed
Figure 1. (a) Schematic diagram showing the geometry behind FDS measurement of Young’s Modulus (E) of a nanofilament. Nanofilaments are deposited on a Si substrate, and processing including XeF2 etching is used to fabricate a trench under the nanofilament. The ends of the nanofilament have been fixed by a thin overlaying film (Cr-Au). PSD represents position-sensitive detector. (b) FESEM image of a fabricated ZnS nanobridge indicated in part a.
a fabrication protocol involving photolithography and subsequent XeF2 etching to produce trenches in a Si substrate located underneath the nanofilaments to be mechanically evaluated. The “nanobridge” (clamped at both ends) is then subjected to downward deflection by a calibrated AFM cantilever. Depending on the position of the point of contact relative to the midpoint of the nanobridge, a different forcedeflection curve can be obtained. The slope of these curves is related to E for the nanofilament. A spectrum of such curves can be obtained for a single bridge, and this spectrum defines a self-consistent set of data leading to Young’s modulus. Thus, we refer to our method as force-deflection spectroscopy (FDS). Nanowire bridges, as shown schematically in Figure 1a, were fabricated by dispensing nanowires by spin coating onto a clean silicon substrate with the native oxide removed. The nanowires can be grown elsewhere and put into a metastable solution in ethanol via ultrasonic agitation. The Si substrate with ZnS nanowires was patterned with photoresist, and a lift-off process was used to define two mechanical clamping Nano Lett., Vol. 6, No. 9, 2006
pads at the ends of the nanowires. The use of the lift-off process ensures that the evaporated film consisting of 10nm-thick Cr followed by 40-nm-thick Au metal film was not deposited anywhere else on the nanowire except for the clamped ends. It is expected that the Cr adhesion layer forms a chemical bond at the interface and clamps the two ends of the wire firmly. Finally, XeF2 etching was carried out to form a trench with a depth of ∼500 nm in the Si substrate underneath the nanowire. The XeF2 is a selective etch for Si. Figure 1b shows a field emission scanning electron microscope (FESEM) image (Leo 1530) that shows that the etched surface in the trench is uniform and the nanowire “bridge” is fully released and free of residue. If reliable clamping of the nanowire bridge by an overlaying metal film were the only consideration, then the metal film should be very thick (e.g., micrometers). However, we have used a 50-nm-thick metal film to maintain a reasonably small height difference between the clamp and nanowire surface. This facilitates the safe navigation of the AFM tip near the clamp. It should be noted that if either clamp were to fail, that is, if the nanowire were to slip in a clamp, then the slope of the F-D curve during that particular deflection would exhibit a discontinuity. Second, if slipping were to occur, then we would expect that subsequent F-D curves would not have repeatable slopes. Therefore, the experimental criterion for a functioning clamp is the repeatability of the F-D data when the contact point for the applied force is not changed. In all of our experiments on several nanowires, we never observed problems duplicating the F-D data. It should also be noted that the actual typical deflection of our nanowire bridge is less than 50 nm. This is only ∼1% of the trench width (i.e., bridge or beam length); a small deflection minimizes the possibility of the nanowire slipping in the clamp. The principles behind FDS are illustrated schematically in Figure 1a. The nanowire bridge was first located using contact-mode AFM (XE-100, PSIA, Inc.). Then, the AFM cantilever was brought into contact with the nanobridge, which was achieved by the x-y closed-loop piezo scanner. Force-deflection (F-D) curves can then be obtained. To avoid breaking the nanobridge, we limited the amount of total force and the z-scanner range. The F-D curves show the deflection of the free end of the AFM cantilever as the cantilever is brought toward (increasing force) and away (decreasing force) from the sample surface. The downward movement of the z scanner (deflection Z of the nanobridge) was deduced by the internal strain gauge of the AFM, whereas the force was measured from the output of the AFM’s position-sensitive detector (PSD). Silicon AFM cantilevers with calibrated force constants ranging from 0.13 to 0.17 N/m (MikroMash, Inc.) were used for our FDS experiments. The x-y and z scanners for the AFM used in FDS measurements were calibrated according to the procedures provided by PSIA, Inc. Figure 2 shows a schematic diagram of the linear part of F-D curves obtained when (1) the AFM tip pushes on the rigid Si substrate (solid line) and (2) the tip pushes on the deformable nanobridge (dashed line). To obtain the F-D 1905
inertia, and f is the applied force at position x0. The use of the δ function in eq 3 requires that the contact area of the tip be negligible. Formulae for I for filaments with different cross sections can be found in ref 14. For example, for a cylindrical beam of diameter d, I ) (πd4)/64 and for a rectangular beam with width w and thickness h, I ) (wh3)/ 12.14 The nanofilament to be mechanically evaluated is clamped at both ends by deposited metal films. The bridges are pinned down by these metal films just outside the trench; thus, the appropriate boundary condition is ∆Z(0) ) ∆Z(L), where L is the length of the bridge without deflection. The solution to eq 3 is given by13
∆Z(x0) ) Figure 2. (a) Schematic diagram of a force-deflection (F-D) curve (linear part). The solid line represents the curve obtained from tip contact on the rigid substrate, and the dashed curve represents that obtained from tip contact on the nanobridge. The deflection of the nanobridge at a certain force, for example, f, is given by (Z ) Z′( f) - Z( f), where a point force of load f is applied at point x0. (b) Schematic nanobeam deflection (normalized to the maximum deflection at the middle point) at the load position as a function of the position t (t ) x0/L) according to eq 5. The inset shows the coordinates used in eqs 3-5.
for (2), the z scanner must move further upon the application of the same force. As a result, the F-D curves obtained from a deformable nanobridge exhibit a smaller F-D slope, as indicated by the dashed line in the figure. The deflection, ∆Z, of the nanobridge for a given applied force, f, is given by ∆Z ) Z'( f) - Z( f) )
f f tan R tan β
(1)
where tan R and tan β are, respectively, the slopes of the F-D curves in the linear regime obtained from experiments (1) and (2) described above. Rearranging eq 1, we find that the force per unit deflection is given by f tan R tan β ) ∆Z tan β - tan R
The expected deflection of the clamped nanobridge upon the application of a concentrated point load, f, at a certain position, x0, on the nanobridge by the AFM cantilever can be modeled using the Euler-Bernoulli beam equation.13 The coordinates are defined in the inset of Figure 2b. The outof-plane deflection, ∆Z, for a homogeneous beam is given by the equation13 EI
d4∆Z ) fδ(x - x0) dx4
(3)
where E is the Young’s modulus, I is the area-moment of 1906
(4)
The negative sign indicates downward deflection. Using a dimensionless quantity t ) x0/L, eq 4 becomes ∆Z(t) ) -
fL3 [t(1 - t)]3 3EI
(5)
This result indicates that the deflection has a simple dependence on the normalized position of the applied force (t) (Figure 2b). At the midpoint of the nanobeam, that is, t ) 0.5, the first derivative of deflection versus position is zero. This means that near the midpoint the resulting deflection is relatively insensitive to a small change in the position of the applied force. Thus, if measurements of E are to be performed only at one contact point, then the midpoint should be chosen for this purpose. This is the experimental approach we have taken in this initial publication. In a subsequent paper, we will attempt to demonstrate that a self-consistent set of data may be obtained from one value of E and many different contact points x0. Evaluating eq 5 at x0 ) L/2, Young’s modulus, E, can be expressed in terms of experimentally determined quantities as E)
(2)
3 3 f x0 (L - x0) 3EI L3
f L3 ∆Z 192I
(6)
The first term in eq 6 is the force per unit deflection, measured by the slope of the F-D curve, and the second term is a geometric factor associated with the area-moment and length of the nanobridge. To put these ideas into practice, we have used the F-D curves at the bridge midpoint (x0 ) L/2) to measure E for rectangular cross section ZnS (wurtzite) nanobridges. Detailed characterization about the morphology (including cross section) and crystalline structure of the ZnS nanowires can be found in refs 15 and 16. Details of the nanowire growth, characterization, and mechanical device fabrication appear in the Supporting Information. Equation 6 for a rectangular cross-section beam is given by (cf., Figure 2a) Nano Lett., Vol. 6, No. 9, 2006
E)
f L3 L3 tan R tan β ) 3 ∆Z 16wh tan β - tan R 16wh3
(7)
Figure 3a shows an AFM image of a ZnS nanobridge taken in the contact mode. The ZnS nanofilament and the processed trench in the Si wafer below the nanofilament can be seen easily. Filament height (h) or thickness measurements were made by AFM z scans. Because the width of the rectangular bridge is broadened in the AFM image due to convolution with the tip, we measured the width of the nanobridge by FESEM (see the Supporting Information). Figure 3b shows typical F-D curves obtained for tip contact on the rigid Si substrate (spot A in Figure 3a, blue curve) and for tip contact at the midpoint of the nanobridge (spot B, red curve). The decrease in slope in the linear regime of the F-D curve from tip contact on the substrate to tip contact on the nanobridge is evident. Least-squares analyses were carried out to extract the slope for both of these F-D curves. E was obtained using these slopes (tan R and tan β) and eq 7. To assess the reproducibility of measured values for E, we studied a series of ZnS filaments with different cross sections and spanning trenches of width L ≈ 2-10 µm. The results are compiled in Table 1 including the geometric factor for each nanobridge, as determined by AFM and FESEM, the least squares slope for the F-D curve with the tip contact near the midpoint of the nanobridge and also on the substrate, the calculated f/∆Z, and the resulting value for E (eq 7). Averaged over all samples, we find E ) 52 ( 7.0 GPa. This value is ∼30% less than that reported for the bulk ZnS (76 GPa).17 It is worthwhile to address two additional characteristics observed for our F-D curves. First, our F-D curves show negligible effects of capillary forces. The primary adhesion force is therefore identified with a van der Waals interaction, which is described by a nonretarded Hamaker constant19 between the cantilever and the Au film/nanobridge. Second, the adhesion force between Au film and Si cantilever is about 60 nN, that is, much stronger than that found between Si cantilever and ZnS nanobridge. The adhesion force observed between the cantilever and the Au film was found to be very repeatable and consistent with the well-characterized SiAu Hamaker constant.19 However, the adhesion forces observed between Si cantilever and the ZnS nanobridges were found to vary considerably from bridge to bridge; most were in the range of 0-15 nN. Some of the nanobridges even showed negligible adhesion forces. This variation is most likely due to the variable surface conditions from nanowire to nanowire, perhaps due to oxides, variety of amorphous structure at the surface, adsorbed gases, and so forth.20 The small adhesion forces observed between the Si cantilever and ZnS nanobridge provides us with two advantages: (1) the nanobridge is not broken easily as the tip approaches or leaves the nanobridge (“snap-on” or “snapoff”). Thus, FDS measurements can be repeated; and (2) the deflection due to the adhesion force is so small that FDS can be performed in the linear regime. Also, to minimize breakage, we used low x-y scan speeds (∼0.5 µm/s). From Nano Lett., Vol. 6, No. 9, 2006
Figure 3. (a) AFM image of a ZnS nanobridge taken in contact mode. The trench is vertical; the nanowire is indicated by B and is covered by Au/Cr film on either side of the trench. (b) Forcedeflection curves obtained on the substrate (blue) and at the middle point of the nanobridge (red). Table 1. Tabulated Results for Several Nanobridgesa tanβ tan R f/∆Z E sample geometric factor (nN/µm) (nN/µm) (nN/nm) (GPa) ID ×1016 (1/µm) A03 A11 C1_02 C2_01 E1_01 E2_01 a
5.76 5.40 17 230 5.81 3.84
144 137 148 143 155 162
125 119 106 20 129 142
0.94 0.93 0.38 0.03 0.77 1.13
54 50 64 52 45 43
The geometric factor was determined using the second term in eq 7
eq 7, we can see that the Young’s modulus is determined by two terms. The first term relates to the applied force to deflect the nanobridge to achieve a unit deflection, as calculated from the slopes of the F-D curves (Figure 2), cf. eq 2. To improve the accuracy of our E measurement, each F-D curve was actually averaged from 5 to 10 measurements. Least-squares fitting was then carried out to extract the average slope of the various F-D curves. The standard deviation from the least-squares fitting of repeated measurements on the same ZnS bridge was found to be ∼1-2%, which is surprisingly small. The second term in eq 7 is a geometric factor determined by the length, thickness, and 1907
Figure 4. (a) FESEM Image of a nanobridge, showing the width variation along the nanowire indicated by white lines. (b) AFM thickness profile of a ZnS nanobridge before dry etching. The arrows indicate the average thickness, h, and thickness variation, ∆h (∼2-5%). Both the width and thickness variation will introduce uncertainty in the geometric factor in eq 7.
width of the nanobridge. Although we have used an AFM to measure the thickness and length of the nanobridge and an FESEM to measure the width, the geometric factor is responsible for most of the total uncertainty in the measurement of E. This is true for any method that requires the beam or filament to bend.1,5,21 As discussed earlier, eq 4 is the solution to eq 3 only under the assumption of a homogeneous beam, that is, constant E and I along the beam length. In nanowires, it has been observed that the diameter or cross section can modulate along the length.16,22 Microscopically, in our FESEM and AFM images, we have observed that the cross section (width in FESEM and thickness in AFM) of the nanobridge can exhibit some significant variation along the bridge length. For example, Figure 4a shows a magnified FESEM of a small section of a nanobridge. Two sets of white lines have been used to highlight the edges of the bridge. In this case, a variation of 5% in the width of the bridge can be seen. One would assume a similar variation in the thickness. Thus the area varies by 10%. For the PSIA XE-100 system,
the z scanner noise is less than 0.2 nm and x-y scanner noise is about 6 nm. For a nanobridge 100-200 nm thick and 2-10 µm long, this noise will account for only ∼0.1% variation both in thickness and length. Figure 4b shows a typical AFM thickness profile of a nanobridge. The vertical arrows indicate the variation in thickness; we suppose that the lateral width exhibits a similar fluctuation. This discussion indicates that the primary source of error in the measurement of E should stem from the modulation of the cross-sectional dimensions. This prediction could be explored if more uniform nanowires (better cross-section stability) can be found and studied. A controversial issue in nanomechanics is the length scale at which Young’s modulus might be enhanced or suppressed relative to the value in the bulk. For example, it has been reported recently that the Young’s modulus of Au nanowires (40-250 nm) is independent of diameter, whereas the yield strength was found to be diameter-dependent and up to 100 times larger than that of the bulk material values.21 In GaN semiconducting nanowires, a recent report showed that the Young’s modulus is diameter-dependent; that is, the Young’s modulus decreases significantly as the diameter decreases.12 Recently, the Young’s modulus for semiconducting nanowires measured by LFM5 and nanoindentation8 were reported to exhibit a reduction in E compared with that of the bulk material. The measured values of Young’s modulus for ZnS nanowires reported here (FDS) are consistent with these observations,5,8,12 showing a 10-40% reduction in comparison with the bulk values. In Table 2, we present a comparison of values for Young’s modulus of bulk semiconductors and their nanofilament counterparts determined by different methods. We collected the results from six studies on semiconducting nanofilaments with various minimum lateral dimensions, d, in the range of 20-200 nm. Except for the results on SiC, all of the values for the measured modulus, E, of nanofilaments exhibit a significant reduction relative to the corresponding values in the bulk. Values for the bulk modulus, B, are included in the table for completeness, and formulas connecting the elastic constant Cij, E, and B appear as footnotes to the table. In conclusion, we have developed a new approach to measure Young’s modulus for nanofilaments using a calibrated AFM to measure a downward force-deflection (F-
Table 2. Comparison of Young’s Modulus (E) and Bulk Modulus (B) of Bulk Semiconductors and Their Nanofilament Counterparts j,k SiC a(GPa)
Ebulk Bbulk b (GPa) Enano (GPa) d (nm)
ZnO
GaN
492c
138c
267c
224c
135c
202c
610-660d 23
38-65e 20-55
29 ( 8f ∼45
227-305g 36-84
ZnS 81c 76c 36 ( 3h ∼40
43-64i 50-200
a The “average” Young’s modulus of an anisotropic bulk material has been proposed to be evaluated using a Voigt average E ) [(C - C + 3C )(C 11 12 44 11 + 2C12)]/(2C11 + 3C12 + C44) (ref 23). b The bulk modulus is related to the elastic constants Cij via B ) [C33(C11 + C12) - 2C132]/(C11 + C12 - 4C13 + 2C33) (ref 24). Reference 25 gives another relationship: B ) (2/9)[(C11 + C12 + 2C13 + (C33/2)]. Both of these equations yield similar values of B for the same set of Cij. c Elastic constants Cij used to evaluate the “average” Young’s modulus (Voigt average) and the bulk modulus are from refs 17 and 23-25. d Reference 1 by lateral force microscopy (LFM). e Reference 11 by electromechanical resonance in situ TEM. f Reference 5 by LFM on a vertical array of nanowires on a Si substrate. g Reference 12 by electromechanical resonance in situ TEM; the GaN nanowire has a triangle cross section. h Reference 8 by nanoindentation. i Present work by force-deflection spectroscopy at the midpoint of the nanobridge j Note: For an isotropic solid, E and B are related by by E ) 3B(1-2σ) (ref 26), where σ is the Poisson ratio. σ typically ranges between 0.1 and 0.5 for most materials. k d is the nanofilament diameter or minimum lateral width.
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D) curve. In our approach, the filament is suspended over a trench and clamped at both ends. The AFM tip pushes the filament down a small distance into the trench. We have created this physical situation by first depositing the nanofilaments on a Si substrate via spin coating and then using deposited Au-Cr films to clamp their ends. XeF2 dry etching was used to undercut a trench in the Si substrate beneath the nanowire. In the measurements reported here, we have tested our approach by contacting the midpoint of the nanofilaments and measured the resulting F-D curves. We have applied our method to several rectangular cross-section ZnS (wurtzite) nanowires grown by pulsed laser vaporization. Using the classical Euler-Bernoulli equation for the deflection of a beam to interpret our F-D data, we obtain the value E ) 52 ( 7.0 GPa for “nano” ZnS which is ∼30% lower than reported for bulk wurtzite ZnS. We believe that our approach and analysis shows considerable promise as a general technique for the study of mechanical properties of nanofilaments. In the present study, we have applied the force only at the midpoint of the nanofilament. However, by varying the position of the contact point, a complete set of different F-D “spectra” can be obtained resulting in one self-consistent value for the Young’s modulus of the nanofilament. Thus, we refer to our approach as FDS, or force-deflection spectroscopy. Besides the inherent self-consistency in the FDS approach, our method has three important advantages when compared to the lateral force AFM approach. First, the bending force constant of the AFM cantilever can be calibrated more accurately and routinely than the lateral force constant.18 Second, the total deflection of the nanobeam in FDS is small (
