NANO LETTERS
Ultimate-Strength Germanium Nanowires Lien T. Ngo,†,‡ Dorothe´e Alme´cija,†,‡ John E. Sader,§ Brian Daly,‡,| Nikolay Petkov,| Justin D. Holmes,‡,| Donats Erts,⊥ and John J. Boland*,†,‡
2006 Vol. 6, No. 12 2964-2968
School of Chemistry, Trinity College Dublin, Dublin 2, Ireland, Centre for Research on AdaptiVe Nanostructures and NanodeVices (CRANN), Dublin, Ireland, Department of Mathematics and Statistics, UniVersity of Melbourne, ParkVille, Victoria 3010, Australia, Department of Chemistry, UniVersity College Cork, Cork, Ireland, and Institute of Chemical Physics, UniVersity of LatVia, Raina BouleVard 19, LV-1586 Riga, LatVia Received August 18, 2006; Revised Manuscript Received November 8, 2006
ABSTRACT Semiconducting nanowires (NWs) are important “building blocks” for potential electrical and electromechanical devices. Here, we report on the mechanical properties of supercritical fluid-grown Ge NWs with radii between 20 and 80 nm. An analysis of the bending and tensile stresses during deformation and failure reveals that while the NWs have a Young’s modulus comparable to the bulk value, they have an ultimate strength of 15 GPa, which is the maximum theoretical strength of these materials. This exceptional strength is the highest reported for any conventional semiconductor material and demonstrates that these NWs are without defect or flaws that compromise the mechanical properties.
In the drive to achieve higher integration densities and to fabricate ever-smaller nanoscale device structures, there is enormous interest in finding ways to build devices from the bottom up rather than fabricate from the top down. Potential “building blocks” need to be characterized and their properties well understood before they can be deployed in circuits and other structures. Since these “building blocks” will be on the nanoscale, their behavior and properties may differ significantly from their bulk counterparts,1-4 making the complete elucidation of these properties especially important. Nanowire (NW) systems have attracted particular attention because of potential applications in the area of interconnects and the active components of device structures and for this reason the electrical and mechanical performance of these systems are critical to device operation. The former determines the mobility of charge carriers while the latter is important for materials processing and potential electromechanical device applications.5 Germanium NW systems are particularly attractive for device applications. Germanium has a significantly higher mobility compared with silicon6 but is not widely used in conventional microelectronics due to the inferior properties of its oxide.7 Ge wafers are also exceptionally brittle and notoriously difficult to handle. However, on the nanoscale, * Author to whom correspondence should be addressed. † School of Chemistry, Trinity College Dublin. ‡ Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN). § Department of Mathematics and Statistics, University of Melbourne. | Department of Chemistry, University College Cork. ⊥ Institute of Chemical Physics, University of Latvia. 10.1021/nl0619397 CCC: $33.50 Published on Web 11/17/2006
© 2006 American Chemical Society
carrier depletion by interface states is unavoidable even for silicon,8,9 so it will be necessary to develop strategies to control the electrical properties of nanoscale interfaces regardless of the NW system. In this work we report on the mechanical properties of Ge NWs that range in size from 20 to 80 nm. We demonstrate that although these wires exhibit elastic moduli comparable to the bulk, their mechanical strength approaches the theoretical value for the material and is significantly larger than that reported for any bulk or whisker semiconductor material. Although near-theoretical strengths have been reported for composite whiskers systems,10 in the case of brittle semiconductors such as Si, most fracture strengths observed so far are only 15% of the theoretical limit, even for the best nanoscale materials.10-12 Higher fracture strengths of 12 GPa in VLS-grown Si NWs have been reported recently,13 but this is still less than 50% of the theoretical limit. There are no comparative data available for Ge, but the present results indicate that it is possible to synthesize Ge NWs that are without defects or flaws that compromise the mechanical properties. When combined with their known electrical properties,5,6 this exceptional strength suggests that Ge NWs may be ideal building blocks for nanoscale device applications. Ge nanowires from 20 to 80 nm in radius were synthesized by a supercritical fluid (SCF)-liquid-solid mechanism, using sputtered gold nanoparticles on a porous aluminum oxide (AAO) support (nominal pore size 200 nm), similar to that reported earlier.14 The reaction was performed in a batch reactor loaded with the porous support coated with
Figure 1. TEM images and diffraction pattern of Ge NW. (a) TEM image and electron diffraction pattern of a supercritical-fluid-grown Ge nanowire of 200 nm radius. Scale bar ) 500 nm. (b) HRTEM image of a 20 nm radius Ge NW showing an amorphous Ge oxide layer on the surface and lattice fringes with a d-spacing of about 0.325 nm, corresponding to the (111) set of planes. Scale bar ) 10 nm.
the Au seeds, 10 mL of hexane, and 0.1 mL of diphenyl germane (GePh2) as a germanium precursor. The reaction was carried out at 365 °C (ramping 10 °C/min) and a pressure of 5.5 × 104 mbar for 30 min. Except for the possibility of doping by the Au catalyst, the nanowires are undoped. Figure 1a shows that the NWs are well-formed single crystals. The growth direction (shown with arrow on the SEAD) is along the [112] direction and was found to be the preferred growth direction for almost 90% of the nanowires. Figure 1b shows a high-resolution transmission electron microscopy (HRTEM) image of a single Ge NW of 20 nm radius. The wire is clearly single crystalline, showing lattice fringes with a d-spacing of about 0.325 nm, which corresponds to the (111) set of planes. To disperse the NWs for mechanical measurement, pieces of the AAO template were first placed into isopropyl alcohol (IPA) and gently sonicated. Drops from the resulting suspension were pipetted onto a prepatterned substrate containing trenches 300-500 nm deep and of varying widths, and the IPA was allowed to evaporate. To minimize sagging, smaller diameter NWs were analyzed only in instances where they were found across narrow trenches. In all cases, the ratio of trench width to NW diameter was greater than 10. Once suitable NW-trench samples were identified by scanning electron microscopy, the NWs were pinned to the surface by e-beam-assisted deposition of Pt, with the pinning points placed as close to the edge as possible (see Figure 2). The double-clamped NW samples prepared in this matter were subject to lateral manipulation using atomic force microscopy (AFM) (see Figure 2a). All measurements were performed with a Digital Instruments Nanoman System with a closed-loop x-y-z scanner.1,15 NWs were imaged in tapping mode prior to and directly after each manipulation, the latter being performed with the cantilever oscillation turned off. During NW manipulation both the normal and lateral force (F) signals are recorded as a function of displacement (d). Here we focus on the latter since the normal component contributes less that 5% of the total force signal.15 However, the normal force signal is carefully monitored to identify possible slippage between tip and NW. The resulting Nano Lett., Vol. 6, No. 12, 2006
Figure 2. Experimental setup and AFM images. (a) Schematic showing the lateral loading of the NW by the AFM tip and the laser/photodiode setup that allows us to monitor both lateral and normal force on the cantilever. (b) Ge NW pinned over a trench, before manipulation. (c) The NW after manipulation, with the failure point at the point of loading. The scale bars for (b) and (c) indicate 1 µm.
Figure 3. F-d curve from a typical NW manipulation experiment. The black points are manipulation data and the red line is the fit to the nonlinear equation (eq 3). Since Ge is a brittle material, the curve shows elastic behavior up until NW failure, with no plastic deformation regime. The fit gives a Young’s modulus of 137.6 GPa and a fracture strength of 17.1 GPa.
F-d curve (Figure 3) reveals that during manipulation the force F initially increases linearly with the NW displacement, consistent with an initial bending deformation of the wire. However, at larger displacements F increases much more rapidly, ultimately exhibiting an approximately cubic dependence on d. This apparent increase in the stiffness of the wire is due to tensile stretching.15 Figure 3 shows that the mechanical behavior over this entire range is well described by a generalized elastic continuum model (red curve) that explicitly includes bending and stretching deformations and which provides an accurate measure of the modulus E.15 Figure 4 (upper panel) shows the results of applying this model to Ge NWs that range in size from 20 to 80 nm in radius. The measured modulus is 112 GPa ( 43 GPa, which compares very favorably with the known bulk values which range from 103 to 150 GPa, depending on crystal orientation.16-18 The observed modulus is essentially independent of wire diameter over the range of values shown. However, it is important to emphasize that the diameter values used in this analysis were obtained from averages of AFM measure2965
determined using elastic analysis and is given by the maximum stress in the beam at the point of fracture. Accounting for both axial and bending stresses yields the following relation for the maximum stress σmax )
Figure 4. Young’s modulus (upper panel) and ultimate strength (lower panel) vs Ge NW radius. The dashed lines show the average values: 15.0 GPa for the ultimate strength and 112 GPa for the Young’s modulus. The hashed box shows the range of values for the Young’s modulus of Ge from refs 8-10. There are more data points for the E-modulus because the E-modulus can be calculated using F-d curves that strain but do not break the NW. Multiple such manipulations can be done on a single NW, yielding multiple evaluations of E. The ultimate strength calculation requires the maximum force delivered at the point of NW failure, so there is only one such calculation for each NW.
| |
d2w T + ER πR2 dx2
(2)
x)L/2
where the first and second terms are contributions due to stretching and bending, respectively, E is the Young’s modulus of the material along the axis of the beam, w is the deflection normal to the axis of the beam, x is the coordinate along the axis of the beam, x ) L/2 is the load point, and T is the induced tension in the beam. Solving the governing equation for the deformation of the beam16,21 gives explicit expressions for the induced tension and bending moment. From eq 2 and the above discussion, we then obtain the required generalized result for the ultimate strength of the material σy )
FcenterL 2πR3
g(R)
(3)
where ments along the wire lengths and in all cases includes a contribution from the surface oxide. Since the modulus of the oxide is substantially less than that of Ge,19 and since the oxide contribution becomes increasingly important for smaller wires, the value of 112 GPa represents a lower bound at these smaller diameters. Figure 3 shows that the wire fractures at large displacements without plastic deformationsa phenomenon that was observed for all NWs in this study. This is interesting on two levels. First, the absence of ductile behavior indicates that the compressive stresses in the AFM-probe-induced notched zone are insufficient to generate dislocations and failure is due to crack propagation.20 Second, because of this brittle failure these data provide a direct measure of the ultimate strength of the Ge NWs. In the limiting case that fracture is exclusively due to bending, there is a well-known relationship between the ultimate strength, the applied force and the dimensions of the wire σy0 )
FcenterL 2πR3
(1)
where R and L are the radius and length of the double clamped beam that is loaded at its central point by force Fcenter. This relationship does not take account of the detailed tip-NW contact or the tensile stresses that occur at large deformations. The latter can be accounted for by noting that the maximum tensile stress in the beam occurs at the surface at the load point and is identical to the maximum stress at the clamped ends. Since the beam fails by brittle fracture in a uniaxial state of stress with no plastic deformation, the true ultimate strength of the material σy can be directly 2966
g(R) )
( )
R1/2 4 tanh + 4 R1/2
(
2 + cosh(R1/2/2) - 6
R)
)
)
sinh(R1/2/2)
R cosh2(R1/2/4)
R1/2
1/2
6(140 + ) 350 + 3
(
)
2∆zcenter R
2
Equation 3 can be used directly to determine the ultimate strength σy by measuring the applied load Fcenter and resulting displacement ∆zcenter at the central point required for brittle fracture, i.e., when ∆zcenter ) ∆zmax. Note that knowledge of Young’s modulus is not needed. In the limit where the displacement ∆zcenter , R, g(R) f 1 and we recover the result of classical beam theory that includes bending stresses only, i.e., eq 1. A plot of g(R) as a function of ∆zmax/R is shown in Figure 5 and is equivalent to the ratio of the actual maximum stress σmax to that predicted by the classical leading order equation, σmax0. It is clear that significant deviation from the leading order formula exists only if fracture occurs at displacements that are much larger than the radius of the beam. This is always the case with NW systems (see Figure 3), and therefore the classical bending-only formula (eq 1) dramatically overestimates the true maximum tensile stress in the beam. With the generalized approach (eq 3), the failure data in Figure 3 were analyzed to determine the ultimate strength of the Ge NWs. Figure 4 (lower panel) shows that the Nano Lett., Vol. 6, No. 12, 2006
Figure 5. A plot of the function g(R) as a function of∆zmax/R, which is equivalent to the ratio of the actual maximum stress σmax to that predicted by the classical bending-only formula σomax (i.e., eq 1). At displacements larger than one nanowire radius, neglecting tensile stress in the analysis of the F-d curve gives rise to significant deviation from the true ultimate strength, leading to large overestimates of the ultimate strength.
average maximum strength of these materials is 15 ( 4 GPa, which is the highest recorded strength for any semiconductor material system. This value is also in excellent agreement with the predicted theoretical strength of ∼E/2π for a perfect defect-free material,22 which in the present case corresponds to a theoretical ultimate strength of 17 GPa. Moreover, the measured value of 15 GPa agrees well with recent DFT simulations that reported an ideal tensile strength of 14 GPa.23 It is important to point out that the classical beam bending analysis based on eq 1 gives an average ultimate strength of 70 GPa for these NWs, so that failing to adequately account for the apparent stiffening of these wires due to tensile stretching leads to vastly overestimated ultimate strengths. The fact that measured ultimate strength is so close to the theoretical value suggests the NWs are near perfect materials, with few physical flaws that compromise their intrinsic mechanical strength. This is consistent with TEM analysis and is likely a consequence of the SCF growth condition.14 The latter ensures that growth is not transport limited and that the growth rate is significantly larger than the nucleation rate. These conditions promote the growth of uniform singlecrystal NWs during which the vast majority of defects are expelled from the material. An important figure of merit for mechanical applications is the ability of a material to deform elastically prior to failure. Figure 6 shows the displacement at failure as a function of the wire radius for the Ge NWs in this study (normalized to their length), together with the behavior predicted by elasticity theory for a perfect material with an ultimate strength of 15 GPa (solid line). The agreement between theory and experiment is excellent and demonstrates that an accurate description of NW mechanical strength necessitates the generalized form (eq 3). For comparison, an analysis based on classical bending eq 1 (dashed line) dramatically overestimates the mechanical performance of these wires. Despite this, smaller diameter wires are remarkably flexible and in the present double clamped configuration 20 nm wires spanning a 300 nm trench can be elastically deformed up to 15 times their radii prior to failure. The extraordinary resilience of these Ge NWs and their ability Nano Lett., Vol. 6, No. 12, 2006
Figure 6. Displacement at failure in radii vs NW radius (normalized to length). The solid line shows the predicted behavior, from eq 3, of a theoretical, perfect NW with Young’s modulus and ultimate strength equal to the average values reported in Figure 3 and is in excellent agreement with the data from this work (circles). The dashed line shows the behavior of the same perfect NW as calculated using the first-order, beam-bending-only method (eq 1), which predicts that the NW would need to be bent to much larger displacements, for the same Young’s modulus and ultimate strength values, than is actually observed experimentally.
to store elastic energy suggest potential applications as sensors and electromechanical oscillators. In conclusion, we have developed a generalized model for NW strength and used AFM lateral manipulation to demonstrate that Ge NWs grown using SCF methods exhibit the maximum theoretical strength. This exceptional strength is the highest reported for any conventional semiconductor material and demonstrates that these NWs are without defects or flaws that compromise the mechanical properties. These NW materials are expected to be valuable building blocks in the development of future nanoscale devices. Acknowledgment. The work at the School of Chemistry and the Centre for Research on Adaptive Nanostructures and Nanodevices at Trinity College Dublin is supported by Science Foundation Ireland under Grants 00/PI.1/C077A.2 and 03/CE3/M406. This research was supported by the Human Frontier Science Program RGY17/2002, Science Foundation Ireland Research Grant (01/PI.2/C033), and the Particulate Fluids Processing Center of the Australian Research Council and by the Australian Research Council Grants Scheme. B.D. acknowledges the Irish Research Council for Science, Engineering and Technology (IRCSET). D.E. acknowledges the Council of Science of Latvia. References (1) Wu, B.; Heidelberg, A.; Boland, J. J. Nat. Mater. 2005, 4, 525529. (2) Gao, H.; Ji, B.; Jager, I. L.; Arzt, E.; Fratzi, P. Pro. Natl. Acad. Sci. U.S.A. 2003, 100, 5597-5600. (3) Prorok, B. C.; Zhu, Y.; Espinosa, H. D.; Guo, Z.; Bazant, Z. P.; Zhao, Y.; Yakobson, B. I. Encyclopedia of Nanoscience and Nanotechnology; Nalwa, H. S., Ed.; American Scientific Publishers: Stevenson Ranch, CA, 2004; Vol. 5, pp 555-600. (4) Wu, B.; Heidelberg, A.; Boland, J. J.; Sader, J. E.; Sun, X.-M.; Li, Y.-D. Nano Lett. 2006, 6, 468-472. (5) Ziegler, K. J.; Lyons, D. M.; Holmes, J. D.; Erts, D.; Polyakov, B.; Olin, H.; Svensson, K.; Olsson, E. Appl. Phys. Lett. 2004, 84, 40744076. 2967
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(15) 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, 11011106. (16) McSkimin, H. J. J. Appl. Phys. 1958, 24, 988-997. (17) Wortman, J. J.; Evans, R. A. J. Appl. Phys. 1965, 36, 153-156. (18) Borchi, E.; De Gennaro, S.; Macii, R.; Zoli, M. J. Phys. D: Appl. Phys. 1988, 21, 1304-1305. (19) Spinner, S.; Cleek, G. W. J. Appl. Phys. 1960, 31, 1407-1410. (20) Kearney, C.; Zhao, Z.; Bruet, B. J. F.; Radovitzky, R.; Boyce, M. C.; Ortiz, C. Phys. ReV. Lett. 2006, 96, 255505. (21) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon: Oxford, 1970. (22) Barrett, C. R.; Nix, W. D.; Tetelman, A. S. The principles of engineering materials; Prentice Hall: Englewood Cliffs, NJ, 1973. (23) Roundy, D.; Cohen, M. L. Phys. ReV. B 2001, 64, 212103.
NL0619397
Nano Lett., Vol. 6, No. 12, 2006
