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Young’s Modulus and Size-Dependent Mechanical Quality Factor of Nanoelectromechanical Germanium Nanowire Resonators Damon A. Smith, Vincent C. Holmberg, Doh C. Lee,† and Brian A. Korgel* Department of Chemical Engineering, Texas Materials Institute, Center for Nano- and Molecular Science and Technology, UniVersity of Texas at Austin, Austin, Texas 78712 ReceiVed: February 4, 2008; ReVised Manuscript ReceiVed: May 2, 2008
Germanium cantilever nanoelectromechanical resonators were fabricated using chemically grown nanowires with diameters ranging from 50 to 140 nm. Single nanowires were mechanically positioned at the edge of a copper transmission electron microscope (TEM) grid and then pinned to the grid with local platinum deposition. Oscillating cantilevers were induced into electromechanical resonance with an applied AC voltage, and the frequency response of the vibrational amplitude was measured. From this data, the Young’s modulus of the nanowires was determined to be insensitive to diameter in this size range with an average value of 106 GPa (with 95% confidence limits of (19 GPa), which is on par with the literature values for bulk Ge (100-150 GPa). The mechanical quality factors (Q) of the nanowire cantilevers were also measured and found to decrease with decreasing diameter. The data indicate that energy dissipation from the oscillating cantilevers occurs predominantly via surface losses, which increase in magnitude with increasing surface area-to-volume ratio of the nanowires. Introduction Atomic bonding, point defects, and extended defects like dislocations and grain boundaries, determine the mechanical properties of crystalline materials. External surfaces do not play a role. In contrast, nanomaterials have a significant fraction of atoms at the surface, which have a different bonding environment than atoms within the bulk solid and therefore the mechanical properties become highly sensitive to surface reconstructions and surface species like adsorbed molecules and coatings.1–5 The extended defect densities that underlie many mechanical properties of materials (like yield stress and plasticity in particular) are also different since the limited volume of the nanostructure cannot sustain typical defect densities found in bulk materials. This leads to qualitatively different properties, such as tolerance of much greater elastic strain.6 Semiconductor nanowires are expected to have particularly interesting mechanical properties:7 they are radially confined, long cylindrical crystals with nearly infinite periodicity in one dimension and thus have low extended defect densities and very high surface area-to-volume ratios that depend on diameter.8–14 Their unique mechanical properties also make them interesting candidates for constructing ultrahigh frequency and low power nanoelectromechanical resonators for a variety of different applications.15 There have been many reported mechanical property measurements of nanowires, but with widely varying results. For example, the size-dependence of the elastic modulus of nanowires in similar size ranges have been shown to increase (single and multiwall carbon nanotubes1,16,17 and Ag, Pb, and ZnO nanowires18,19), decrease (Cr, Si, and GaN nanowires20–22) and remain constant (for multiwall carbon nanotubes and Au and Ge nanowires23–25) with decreasing diameter. Nanowires must be fabricated with varying diameter without changes in crystal * To whom correspondence should be addressed. Phone: +1-512-4715633. Fax: +1-512-471-7060. E-mail:
[email protected]. † Present address: Chemistry Division, C–PCS, Los Alamos National Laboratory, Los Alamos, NM 87545.
quality or surface chemistry and then must be effectively integrated into test platforms. The wide variability in mechanical property measurements shows that the structural qualitysand thus the mechanical propertiessof the nanowires and nanotubes is highly sensitive to the synthesis process. Several methods have been utilized to measure mechanical properties of nanostructures, including atomic force microscopy (AFM) to obtain force-displacement measurements from three-point bend tests1,17,18,22–28 and mechanical- or electric field-induced oscillations of suspended beams.16,19–21,29–33 Other methods include tensile testing between two AFM cantilevers and MEMS platforms, and nanoindentation.34 These methods each have advantages and disadvantages with respect to one another and some of the variation in mechanical property measurements is the result of the wide range of experimental methods used and not necessarily from the material quality. Anyhow, the contrasting and sometimes conflicting reports of nanoscale mechanical properties points to the need for increased study in this area. In this study, Ge nanowire cantilevers were induced to vibrate by applying a sinusoidal voltage. The natural resonance frequency and the vibrational amplitude were measured and then used with elastic beam theory to estimate Young’s modulus, E. The mechanical quality factor Q, was also calculated. E was found to be independent of diameter and Q decreased with decreasing diameter. Experimental Methods Ge Nanowire Cantilever Fabrication. Ge nanowires were synthesized by gold nanocrystal-seeded supercritical fluidliquid-solid (SFLS) growth in benzene followed by isoprene surface passivation, using established procedures (See Supporting Information).11,12,36 Nanowires were dispersed in chloroform and then deposited onto a lacey carbon TEM grid by drying a drop of the nanowire suspension. The TEM grid was mounted on an SEM stage and placed in an FEI Strata DB235 dual beam scanning electron microscope/focused ion beam (SEM/FIB) system equipped with a Zyvex S100 nanomanipulator. In SEM
10.1021/jp8010487 CCC: $40.75 2008 American Chemical Society Published on Web 07/01/2008
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Figure 1. (a,b) SEM and (c) TEM images of isoprene-passivated Ge nanowires produced by Au nanocrystal-seeded SFLS growth.
mode, Ge nanowires were identified that extended partially over the copper grid and partially over the lacey carbon. The lacey carbon underneath the nanowire was then physically removed using the electrochemically etched tungsten nanoprobes of the Zyvex S100 nanomanipulator. Platinum was deposited by electron beam-induced deposition (EBID) at the junction between the copper substrate and the Ge nanowire to eliminate slippage of the nanowire during oscillation. Mechanical Measurements. The nanowire cantilevers were excited into oscillation by applying an AC voltage between two tungsten nanoprobes of the Zyvex S100 nanomanipulator attached to a Solartron 1260A frequency response analyzer (FRA) connected to the probes through a breakout panel outside of the SEM/FIB vacuum chamber. One probe was contacted to the copper grid and the second was held within a micrometer of the nanowire tip. Measurements were carried out by applying a sinusoidal voltage sweep with amplitude of 100 mV. The AC frequency was scanned from 1 kHz to 32 MHz, initially with a 500 Hz resolution until the nanowire was observed to oscillate. The frequency range and resolution were then repeatedly adjusted until the full frequency-dependent amplitude could be observed and approximately 15-25 images of the oscillating nanowire were obtained by SEM. The vibrational amplitude was determined visually using Scion image processing software The cantilever length was determined from SEM images acquired at several different tilt angles (-15° to 52°). The nanowire diameters were determined by transmission electron microscopy (TEM) on a JEOL 2010F TEM (at a 200kV accelerating voltage). Results Figure 1 shows SEM and TEM images of the Ge nanowires used to construct the cantilevers. The nanowires are crystalline diamond cubic Ge coated with a covalently bound monolayer of hydrocarbon that prevents surface oxidation.11,12,36 The predominant growth direction of the nanowires is with a small (∼10%) proportion of nanowires with and growth directions.12 The Ge nanowires have very few extended defects,8 and correspondingly the nanowires can sustain large amounts of flexural strain without fracture. Figure 2 shows an example of enhanced elasticity of a Ge nanowire being flexed by two scanning tunneling microscopy (STM) tips (see Supporting Information for a movie showing a nanowire being flexed between two STM tips). The Ge nanowire cantilevers were constructed using nanowires ranging from 50 to 140 nm in diameter.37 Figure 3 shows a typical Ge nanowire cantilever. An AC field is applied between an STM tip positioned close to the free end
Figure 2. SEM image of a Ge nanowire flexed between two STM tips.
Figure 3. (a) SEM image of a Ge nanowire cantilever: the nanowire is glued to the Cu substrate with Pt deposited by electron beam-induced deposition in the SEM/FIB tool. (b) SEM image of an Ge nanowire vibrating in response to a sinusoidal potential applied by the nearby tungsten probe. (Inset) Device schematic.
of the nanowire cantilever as shown in Figure 3b and an STM tip positioned at the clamped end of the nanowire. As the frequency of the AC field applied between the two STM tips is scanned, the cantilever begins to oscillate, as illustrated in Figure 3b, as it nears the fundamental resonance frequency of the cantilever (see Supporting Information for a movie of an oscillating nanowire). The vibrational amplitude of the cantilever as a function of applied frequency was determined from SEM images of the oscillating nanowire. Figure 4 shows a sample data set for an 88 nm diameter Ge nanowire. The vibrational amplitude varies with frequency and peaks at the natural resonance frequency f1, which was
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β21 f1 ) 2π
EI mL4
(1)
The subscript “1” refers to the first resonance mode (i.e., the first eigenmode) of the cantilever, for which β1 ) 1.875. For a cylindrical beam, m ) F(π/4)d2, where F is the density of Ge, and I ) πd4/64, which yields the expression38
f1 ) Figure 4. Oscillation amplitude determined from SEM images (() versus frequency measured for an 88 nm diameter Ge nanowire. The curve is a four-parameter Lorentzian fit to the data: y ) y0 + a/(1 + ((x - x0)/b)2) used to determine f1 and Q. For this nanowire, Q ) 542 ( 16 and f ) 181.6 kHz.
(
β21 8π
EF ) × Ld
2
(2)
A plot of f1 versus d/L2 in Figure 5a gives a straight line, indicating that E and F do not vary significantly with nanowire diameter in this size range. Fitting eq 2 to the data in Figure 5a gave an average value of E)106 GPa with 95% confidence limits of (19 GPa. These values correspond to the Young’s modulus of Ge down the length of the nanowire, which is predominantly in the direction for these nanowires.11 This is within the range of the reported values of bulk Ge (E ) 103-150 GPa) in the literature.39–42 Figure 5b shows E calculated individually for each nanowire using the relation
( )( )
E ) 64F
πf1 d
2
L β1
4
(3)
E calculated in this way does not vary with nanowire diameter and has an average value of 97 ( 37 GPa, which is slightly less than the value of E determined from the plot of f1 versus d/L2. Ngo, et al.25 recently reported three-point loading measurements using an atomic force microscope (AFM) tip to flex individual suspended Ge nanowires (in the same size range) and also observed diameter-independent values of E close to the bulk value of Ge. The mechanical quality factors of the nanowire cantilevers, Q ) f1/∆f1,43 are plotted in Figure 6a. Q decreases systematically with decreasing nanowire diameter. The plot of Q versus the
Figure 5. (a) Measured values of f1 plotted versus the nanowire dimensions, d/L2. A best fit of eq 2 (s) to the data yields an average value of E ) 106 GPa with 95% confidence limits of (19 GPa. (Inset) Diameter-dependence of the measured values of f1. (b) The Young’s modulus, E, plotted as a function of nanowire diameter. The shaded region represents the range of values of E for bulk Ge reported in the literature. The value of E was 97 ( 37 GPa. (Error bars represent experimental error as described in the Supporting Information).
determined along with the full-width-at-half-maximum (fwhm) ∆f1, by fitting the data points with a four-parameter Lorentzian curve. In the range of oscillation amplitudes measured in this study, f1 is independent of the AC voltage since the amplitudes are more than an order of magnitude less than the cantilever length and therefore satisfies the requirements for application of the Euler-Bernouli equation.33,38,56 Figure 5a shows the natural resonance frequency f1 measured for 14 Ge nanowire cantilevers plotted as a function of the nanowire diameter divided by its length squared. For a clampedfree beam (i.e., the Euler-Bernoulli model), f1 depends on the Young’s modulus E, of the nanowire, the beam’s diameter d, length L, cross-sectional area moment of inertia I, and mass per unit length m:38
Figure 6. (a) Q measured for 14 Ge nanowires of different diameter. The dashed line is a a linear extrapolation of the data, provided as a guide to the eye. (b) Q plotted versus the surface-to-volume ratio of the nanowires assuming a cylindrical shape. (Error bars represent experimental error as described in the Supporting Information).
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cantilever surface area-to-volume ratio (Figure 6b) shows the trend of decreasing Q with increasing surface area-to-volume ratio, which is well-known for cantilevers fabricated with submicrometer diameters, as Carr et al.31 and others15,44–46 have reported in the literature. The precise reason that Q decreases with decreasing diameter of cantilevers in this size range remains a topic of study but it is clear that the energy dissipation related to the surfaces of the crystal is important.
coated by dodecene. And Feng, et al.47 reported recently extremely high Q values of 13 100 for “pristine” 80 nm diameter Si nanowire cantilevers. The Ge nanowires studied here are coated by isoprene during their synthesis to provide a surface passivation layer that prevents oxidation.36 The covalently bound surface passivation layer has also been shown to greatly reduce surface states/defects that lead to very low field effect mobilities and hysteresis in nanowire transistor structures.36,53 Nonetheless, the Q values measured here are 2 orders of magnitude lower than those of the “pristine” Si nanowires of similar size studied by Feng et al.47 Since the crystal quality and surface faceting54,55 of those Si nanowires and the Ge nanowires studied here are not expected to be significantly different, it is uncertain as to what gives rise to the distinctly different low Q values for the Ge nanowires in this study and the very high Q values of the Si nanowires studied by Feng, et al.47 Certainly, the Si and Ge surfaces are quite different, as Si forms a good oxide with few surface defects, whereas Ge does not. Additional systematic studies of the influence of surface chemistry on nanomechanical resonator properties are needed to fully understand the energy dissipation pathways and how to eliminate these from nanomechanical resonators to obtain ultrahigh Q values.
Discussion
Conclusions
The Ge nanowire resonators studied here confirm that Q decreases with decreasing diameter for cantilevers of equal length and that this size dependence is a fundamental trend of nanometer-scale resonators. This fact does not bode well for the construction of ultra high Q nanomechanical resonators. However, the absolute limit on Q is a much more complicated issue that deserves more discussion. Q relates inversely to the energy dissipated from the oscillating nanowire.44 Energy can be lost via multiple pathways i, including the support, atmospheric damping, thermoelastic losses, Ohmic losses, and surface losses, each dissipating a certain amount of energy ∆Wi, per cycle of vibration,44,45,47,48
The fundamental resonance frequency and mechanical quality factor were measured for Ge nanowire cantilevers with diameters varying from 50 to 140 nm. The Young’s modulus was found to be independent of diameter and Q decreased with decreasing diameter. The nanowires of differing diameter studied here exhibit no observable difference in crystal quality or surface chemistry. Thus, the mechanical properties related to the Ge-Ge bonding in the core of the wire, such as E, do not differ from that of the bulk material when the diameters are in this size range of larger than ∼10 nm in diameter, but properties that are very sensitive to the surface, such as the energy dissipation of an oscillating nanowire cantilever (i.e., 1/Q), change significantly with the increasing fraction of exposed crystal surface with decreasing nanowire diameter. High Q values are desired for force-sensing applications and therefore surface losses that decrease Q with decreasing nanoscale cantilever diameters may be a significant limiting factor in the miniaturization of suspended beam resonators.15,32,44–46 The Ge nanowires studied here clearly indicate that smaller diameter decreases Q, and that this is undoubtedly a fundamental trend of nanometer-scale resonators. The absolute value of Q, however, depends strongly on the surface chemistry and surface structure of the cantilever material and wide variations in Q have been observed, such as the recent ultrahigh Q values of 13 000 measured by Feng, et al.47 for “pristine” 80 nm diameter Si nanowire cantilevers that are 2 orders of magnitude higher than the Q values of the Ge nanowires studied here. Clearly, the underlying surface chemistry and surface structure are crucial to how energy dissipates from oscillating nanoelectromechanical resonators and this is currently a research topic that requires much greater study and understanding.
Figure 7. 1/Q versus length of the nanowire cantilevers.
Q-1 ∝
∑ ∆Wi
(5)
i
In the case of the nanowire cantilevers studied here, the support losses from these high aspect ratio cantilevers are not expected to dominate dissipation.44,45 Q-1 plotted versus cantilever aspect ratio in Figure 7 confirms this expectation, as there is no correlation between Q and the aspect ratio; however, the shorter cantilevers that are ∼10 µm long or less are perhaps exhibiting slightly higher losses from the support than the longer cantilevers.47 “Atmospheric” damping due to the adsorption of molecules from the surroundings should be negligible because the measurements were conducted in vacuum. Thermoelastic losses (sometimes called “internal friction”) should be negligible since the characteristic time for thermal diffusion across the diameter of these thin nanowires greatly exceeds the resonance frequency (R/d2 . 2πf1, where R is the Ge thermal diffusion coefficient), and the oscillating nanowire retains thermal equilibrium.48,49 Therefore, the dominant pathway for energy dissipation must relate to the surfaces. Q depends strongly on the surface chemistry and surface structure of the cantilever material and can be significantly altered by the presence of surface defects and adsorbed species and surface layers.44,45,50 For example, Yang, Ono and Esashi51 showed that UHV annealing of Si cantilevers for 30 s at 1000 °C increased Q in their devices by an order of magnitude. Wang, et al.52 showed that methyl-terminated micrometer-diameter Si cantilevers had Q values that were 50% greater than cantilevers
Acknowledgment. This work was supported by the Robert A. Welch Foundation, the Advanced Materials Research Center in collaboration with International SEMATECH, the Advanced Processing and Prototype Center (AP2C: DARPA NR0011-061-0005) and the Office of Naval Research (N00014-05-1-0857). V.C.H. gratefully acknowledges financial support from a Hertz Foundation Graduate Fellowship. We also thank Mike Tiner and William Lackowski of the Center for Nano- and Molecular
Nanoelectromechanical Germanium Nanowire Resonators Science and Technology for their instrument training and insightful discussions. Supporting Information Available: Movies of a Ge wire flexed by two STM tips (“Flexed Ge nanowire movie”) and a vibrating Ge nanowire cantilever (“Oscillating Ge nanowire cantilever”); Ge nanowire synthesis details; error analysis calculations. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Lee, K.; Lukic, B.; Magrez, A.; Seo, J. W.; Briggs, G. A. D.; Kulik, A. J.; Forro, L. Nano Lett. 2007, 7, 1598–1602. (2) Broughton, J. Q.; Meli, C. A. Phys. ReV. B 1997, 56, 611–618. (3) Miller, R. E.; Shenoy, V. B. Nanotech. 2000, 11, 139–147. (4) Sharma, P.; Ganti, S.; Bhate, N. Appl. Phys. Lett. 2003, 82, 535– 537. (5) Lee, B.; Rudd, R. E. Phys. ReV. B 2007, 75, 195328. (6) Herring, C. The Physics of Powder Metallurgy; McGraw-Hill Book Company, Inc.: New York, 1951. (7) Herring, C.; Galt, J. K. Phys. ReV. 1952, 85, 1060–1061. (8) Davidson III, F. M.; Lee, D. C.; Fanfair, D. D.; Korgel, B. A. J. Phys. Chem. C 2007, 111, 2929–2935. (9) Holmes, J. D.; Johnston, K. P.; Doty, R. C.; Korgel, B. A. Science 2000, 287, 1471–1473. (10) Korgel, B. A. Nat. Mater. 2006, 5, 521–522. (11) Hanrath, T.; Korgel, B. A. J. Am. Chem. Soc. 2002, 124, 1424– 1429. (12) Hanrath, T.; Korgel, B. A. AdV. Mater. 2003, 15, 437–440. (13) Shah, P. S.; Hanrath, T.; Johnston, K. P.; Korgel, B. A. J. Phys. Chem. B 2004, 108, 9574–9587. (14) Law, M.; Joshua, G.; Yang, P. Annu. ReV. Mater. Res. 2004, 34, 83–122. (15) Ekinci, K. L.; Roukes, M. L. ReV. Sci. Instrum. 2005, 76, 061101. (16) Poncharal, P.; Wang, Z. L.; Ugarte, D.; de Heer, W. A. Science 1999, 283, 1513–1516. (17) Lukic, B.; Seo, J. W.; Bacsa, R. R.; Delpeux, S.; Beguin, F.; Bister, G.; Fonseca, A.; Nagy, J. B.; Kis, A.; Jeney, S.; Kulik, A. J.; Forro, L. Nano Lett. 2005, 5, 2074–2077. (18) Cuenot, S.; Fretigny, C.; Demoustier-Champagne, S.; Nysten, B. Phys. ReV. B 2004, 69, 165410–165414. (19) Chen, C. Q.; Shi, Y.; Zhang, Y. S.; Zhu, J.; Yan, Y. J. Phys. ReV. Lett. 2006, 96, 075505. (20) Li, X.; Ono, T.; Wang, Y.; Esashi, M. Appl. Phys. Lett. 2003, 83, 3081–3083. (21) Nam, C.-Y.; Jaroenapibal, P.; Tham, D.; Luzzi, D. E.; Evoy, S.; Fischer, J. E. Nano Lett. 2006, 6, 153–158. (22) Nilsson, S. G.; Borrise, X.; Montelius, L. Appl. Phys. Lett. 2004, 85, 3555–3557. (23) Wong, E. W.; Sheehan, P. E.; Lieber, C. M. Science 1997, 277, 1971–1975. (24) Wu, B.; Heidelberg, A.; Boland, J. J. Nature 2005, 4, 525–529. (25) Ngo, L. T.; Almecija, D.; Sader, J. E.; Daly, B.; Petkov, N.; Holmes, J. D.; Erts, D.; Boland, J. J. Nano Lett. 2006, 6, 2964–2968. (26) Heidelberg, A.; Ngo, L. T.; Wu, B.; Phillips, M. A.; Sharma, S.; Kamins, T. I.; Sader, J. E.; Boland, J. J. Nano Lett. 2006, 6, 1101–1106. (27) Hoffmann, S.; Utke, I.; Moser, B.; Michler, J.; Christiansen, S. H.; Schmidt, V.; Senz, S.; Werner, P.; Gosele, U.; Ballif, C. Nano Lett. 2006, 6, 622–625. (28) Salvetat, J.-P.; Briggs, G. A. D.; Bonard, J.-M.; Bacsa, R. R.; Kulik, A. J.; Stockli, T.; Burnham, N. A.; Forro, L. Phys. ReV. Lett. 1999, 82, 944–947. (29) Dikin, D. A.; Chen, X.; Ding, W.; Wagner, G.; Ruoff, R. S. J. Appl. Phys. 2003, 93, 226–230. (30) Chen, X.; Zhang, S.; Wagner, G. J.; Ding, W.; Ruoff, R. S. J. Appl. Phys. 2004, 95, 4823–4828.
J. Phys. Chem. C, Vol. 112, No. 29, 2008 10729 (31) Sazonanova, V.; Yalsh, Y.; Ustunel, H.; Roundy, D.; Arias, T. A.; McEuen, P. L. Nature 2004, 431, 284–287. (32) Carr, D. W.; Evoy, S.; Sekaric, L.; Craighead, H. G.; Parpia, J. M. Appl. Phys. Lett. 1999, 75, 920–922. (33) Liu, K. H.; Wang, W. L.; Xu, Z.; Liao, L.; Bai, X. D.; Wang, E. G. Appl. Phys. Lett. 2006, 89, 221908. (34) Tan, E. P. S.; Lim, C. T. Compos. Sci. Technol. 2005, 66, 1102– 1111. (35) Nayfeh, A. H.; Mook, D. T. Nonlinear Oscillations; Wiley: New York, 1979. (36) Hanrath, T.; Korgel, B. A. J. Am. Chem. Soc. 2004, 126, 15466– 15472. (37) Nanowires smaller than 50 nm in diameter are obtained in the synthesis; however, these nanowires were unable to withstand the force of pulling away the underlying carbon film and would break during this process and therefore were not studied. (38) Meirovitch, L. Analytical Methods in Vibrations; The Macmillan Company: New York, 1967. (39) Borchi, E.; Gennaro, S. D.; Macii, R.; Zoli, M. J. Phys. D: Appl. Phys. 1988, 21, 1304–1305. (40) McSkimin, H. J. J. Appl. Phys. 1953, 24, 988–997. (41) Wortman, J. J.; Evans, R. A. J. Appl. Phys. 1965, 36, 153–156. (42) Some of the variation observed in E between samples might relate to a difference in nanowire growth direction. High resolution TEM could not be used to directly determine the growth direction of the nanowires after the cantilever was fabricated because the nanowire is not sufficiently stable under the electron beam for high resolution imaging. Statistical analysis of the nanowire growth directions in the sample showed that the majority of the wires used in the study had growth directions, however, some of the nanowires in the sample exhibit and growth directions and it is possible that some of the cantilevers were constructed with nanowires with these other growth directions. Calculations of the Young’s modulus have shown a difference of 17 GPa for the and crystal directions in bulk Ge.40 (43) Fowles, G. R.; Cassiday, G. R. Analytical Mechanics; Hartcourt, Inc.: Orlando, FL, 1999. (44) Yasumura, K. Y.; Stowe, T. D.; Chow, E. M.; Pfafman, T.; Kenny, T. W.; Stipe, B. C.; Rugar, D. J. Microelectromech. S. 2000, 9, 117–125. (45) Yang, J.; Ono, T.; Esashi, M. J. Microelectromech. S. 2002, 11, 775–783. (46) Gaspar, J.; Chu, V.; Conde, J. P. Appl. Phys. Lett. 2004, 84, 622– 624. (47) Feng, X. L.; Rongrui, H.; Yang, P.; Roukes, M. L. Nano Lett. 2007, 7, 1953–1959. (48) Cimalla, V.; Niebelschutz, F.; Tonisch, K.; Foerster, C.; Brueckner, K.; Cimalla, I.; Friedrich, T.; Pezoldt, J.; Stephan, R.; Hein, M.; Ambacher, O. Sens. Actuators, B 2007, 126, 24–34. (49) Zener, C. Phys. ReV. 1938, 53, 90–99. (50) Ibach, H. J. Vac. Sci. Technol. A 1994, 12, 2240–2245. (51) Yang, J.; Ono, T.; Esashi, M. Appl. Phys. Lett. 2000, 77, 3860– 3862. (52) Wang, Y.; Henry, J. A.; Debodhonyaa, S.; Hines, M. A. Appl. Phys. Lett. 2004, 85, 5736–5738. (53) Hanrath, T.; Korgel, B. A. J. Phys. Chem. B 2005, 109, 5518– 5524. (54) Hanrath, T.; Korgel, B. A. Small 2005, 1, 717–721. (55) Feng et al.47 state that their Si nanowires had growth directions and faceted {112} surfaces. The Ge nanowires studied here have predominantly growth directions, which have predominantly {111} and {100} faceted surfaces. (56) An additional requirement for using the Euler-Bernoulli model to calculate the Young’s modulus, is to distinguish the fundamental resonance frequency from other higher-order vibrational modes. Forced resonance occurs at ω ) ω0 and ω ) ω0/2, while parametric resonance occurs at ω ) ω0/n, where ω is the angular frequency of the applied voltage (ω ) 2πf), ω0 is the natural angular resonance frequency of the nanowire, and n is an integer greater than one.34 By applying the potential with the counterelectrode transverse to the nanowire, where forced excitation is dominant,18,31 the natural resonance frequency was determined.
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