Size-independent Young’s modulus of inverted conical GaAs nanowire resonators
We explore mechanical properties of top down fabricated, singly clamped inverted conical GaAs nanowires. Combining nanowire lengths of m with foot diameters of nm yields fundamental flexural eigenmodes spanning two orders of magnitude from kHz to MHz. We extract a size-independent value of Young’s modulus of GPa. With foot diameters down to a few tens of nanometers, the investigated nanowires are promising candidates for ultra-flexible and ultra-sensitive nanomechanical devices.
The rise of nanotechnologies in basic research as well as the applied sciences goes along with the development of more and more compact and sensitive devices. For example, nanomechanical systems (NEMS) are promising candidates for ultra-responsive mass Chaste et al. (2012); Hanay et al. (2012), force Mamin and Rugar (2001); Arlett et al. (2006); Hallstrom et al. (2010) or biosensors Burg et al. (2007); Waggoner and Craighead (2007); Arlett et al. (2011), as well as accelerometers Krause et al. (2012) or oscillators Feng et al. (2008). The successful realization of such devices is enabled by a combination of three key factors: integrated architectures, reliable fabrication and high sensitivity. Particularly for integrated sensing devices the vertical arrangement of dense arrays of micro- or nanowires Hallstrom et al. (2010) is considered beneficial. A compact sensor design with higher functionality and reproducible control over device parameters is facilitated by top down fabrication. Finally, the device’s sensitivity is closely related to its spring constant, which is a function of both geometry and material properties such as Young’s modulus , as, quite generally, a softer spring allows resolving smaller signals. Thus, detailed knowledge of the mechanical properties is crucial to predict the performance of a device. Typically, Young’s moduli are determined by relating the measured resonance frequency of the resonator’s fundamental flexural mode to its geometrical dimensions Treacy et al. (1996); Li et al. (2003); Chen et al. (2006); Gavan et al. (2009); Lexholm et al. (2009); Qin et al. (2012). A prominent example is the simple singly-clamped cylindrical beam for which Euler Bernoulli beam theory Weaver et al. (1990) yields the well-known relation
for aspect ratios with mass density , beam radius and length .
Here we present a nanomechanical resonator which is an excellent candidate for a nanomechanical sensing device. Figure 1 shows nanowires etched into an ()-oriented GaAs substrate, combining the benefits of top down fabrication with an ultrasoft mechanical response, allowing for immediate integration into a sensing array. Notably, the nanowires are not cylindrical, but of inverted conical shape. This is apparent from the magnified nanowire in the right part of Fig. 1 featuring a length of m, a head radius of nm and a foot radius of only nm, which corresponds to a taper angle of . While the narrow nanowire foots enable high force sensitivities Rast et al. (2000); Lee et al. (2005), the relatively large nanowire heads can easily be resolved in an optical microscope. As a consequence, inverted conical nanowire arrays are highly promising devices to study e.g. cellular force exertion, as benchmark sensitivities can be obtained without the need to functionalize and thus modify the nanowires using fluorescent markers, which also avoids the frequently encountered bleaching of fluorescent dyes. In addition, the comparatively large heads allow for precise and reproducible nanowire definition using electron-beam lithography. After the evaporation of a nickel etch mask and a lift-off process, the nanowire is etched using inductive coupled plasma reactive ion etching (ICP RIE) with SiCl. Following a thorough conditioning of the etching chamber, nanowires can be fabricated reproducibly with chosen sidewall roughness and taper angle. For a given taper angle the nanowire length is a function of the etching time and thus relies on the resilience of the etching mask. The presented nanowires have been processed using an rf power of W, an ICP power of W, sccm of SiCl flow at a pressure of mTorr with an etch rate of nmmin in an Oxford PlasmaLab 100 etcher and feature head radii between nm and m. The nanowire length is varied between m and m, giving rise to foot radii from nm to nm.
As a last step, all nanowires are imaged using a high resolution scanning electron microscope (SEM) to determine , and with an accuracy of about nm resulting from finite pixel size and edge effects in SEM images (see right part of Fig. 1). Subsequently, the chips are mounted in a vacuum chamber to characterize the nanowires’ eigenfrequencies at pressures below mbar and at room temperature. To this end, a shear piezo is glued to the nanowire chip for piezo-actuation Favero et al. (2009) as indicated in the inset of Fig. 2. Nanowire vibration with amplitudes in the range of a few nanometers is probed via direct optical detection Sanii and Ashby (2010) by detecting the reflected light of a laser ( nm) vertically focused on the device Paulitschke et al. , resulting in a displacement sensitivity of pm. The obtained resonance spectrum is fitted with a Lorentzian to extract both eigenfrequency and quality factor. The eigenfrequencies of the investigated nanowires cover the frequency range between kHz and MHz. The observed quality factors lie between and , while generally higher quality factors are observed for increasing foot radii. A typical dataset featuring an eigenfrequency of MHz and a quality factor of is depicted in Fig. 2.
An important consequence of the gradually increasing cross section of the inverted conical nanowire is that the simple Euler-Bernoulli formula from eq. (1) relating eigenfrequency, geometry and material properties for the cylindrical beam does not hold anymore. To correctly interpret the measured resonance frequencies, the appropriate functional relationship for an inverted conical nanowire has to be obtained. This is accomplished by solving the general Euler-Bernoulli differential equation Weaver et al. (1990)
with being the axis along the nanowire length, the eigenfunction of the bent nanowire, the mass density, the area moment of inertial and the cross sectional area of the non-prismatic nanowire at height . For the inverted conical nanowire, and with = . Under these assumptions, a general solution of eq. (S1) can be obtained using Kirchhoff’s approach Kirchhoff (1879) which is constructed from Bessel functions Conway (1946); Conway and Dubil (1965). However, this method does not enable an analytic solution of the corresponding boundary value problem. For the boundary conditions of the inverted conical nanowire,
only tabulated numerical values for selected nanowire dimensions are found in literature Conway and Dubil (1965).
However, for the case of the piezo-actuated, inverted conical nanowire an analytic solution can be derived which is detailed in the supplemental material bib . The result can be summarized as
where the function is a real number describing the full geometry dependence of the nanowire (see eq. (S4) of the supplemental material. bib ). We would like to note that this solution is valid for all aspect ratios and for positive as well as negative angles . In the limit the result approximates the well-known of the singly-clamped cylinder (c.f. eq. (1)). The numerical solutions for four pre-selected aspect ratios given in the literature Conway (1946); Conway and Dubil (1965) are in almost perfect agreement with the respective values of eq. (4).
Equation (4) allows to determine the nanowires’ Young’s modulus from the experimentally determined eigenfrequencies and geometry factors . Figure 3 displays the optically measured eigenfrequencies of the nanowires’ fundamental flexural mode as a function of the geometry factor determined from the nanowire dimensions , and obtained via SEM analysis. The data displays a linear slope across the entire range of , covering a frequency range from kHz to MHz. This implies that the complete set of nanowires down to the most flexible ones (see inset for a magnified view of datapoints including error bars) is characterized by the same, global value of Young’s modulus . Using a constant mass density kgm for GaAs Bateman et al. (1959), the fitted line through origin included in Fig. 3 as solid line yields GPa.
In the following, the observed global value of is compared to the existing literature. For the presented nanowires etched into ()-GaAs, the flexural vibrations are governed by Young’s modulus along the ()-direction Hopcroft et al. (2010) which has a bulk value of GPa Burenkov et al. (1973). Lacking a reference value for nanostructures along this crystal direction we compare our results to GaAs nanowires epitaxially grown along the ()-direction, for which a broad spectrum of literature can be found. Experimentally obtained values range from GPa for nanowires with diameters between nm and nm Wang et al. (2011); Kallesoe et al. (2012); Alekseev et al. (2012), yielding a somewhat unspecific picture compared to a bulk value of Young’s modulus for (111)-GaAs of GPa Burenkov et al. (1973).
A possible explanation for this vast spread could be a dependence of Young’s modulus on the geometric dimensions of the underlying nanowires, which is actively discussed in part of the research literature Li et al. (2003); Chen et al. (2006); Gavan et al. (2009); Lexholm et al. (2009); Nam et al. (2006); Ngo et al. (2006); Zhang et al. (2008); Löffler et al. (2011). In order to elucidate possible size effects in our data, Young’s modulus is additionally determined for each individual nanowire, again using eq. (4), now applied to the respective datapoints in Fig. 3 rather than fitting their overall slope. The resulting values denoted are plotted as a function of in the left part of Fig. 4 as black circles. A histogram of the observed values of shown in the right part of Fig. 4 yields a normal distribution with expected value GPa (dotted line) which agrees well with the global value GPa. However, an increasingly broad distribution of values is observed in the left part of Fig. 4 for decreasing geometry factor . In order to rule out a possible influence of size effects, the nanowires are segmented into five subsets containing datapoints each, which are indicated by alternating background color outlining the increasing relative error in determining the nanowires’ dimensions for decreasing . For each of these subsets the above statistical analysis is repeated, yielding both average () and standard deviation, indicated by filled squares including error bars. Clearly, even though the errors strongly increase for the subsets featuring smaller , all agree with within their error limits. Furthermore the observed values of are statistically distributed around and there is no systematic increase or decrease for smaller . Thus we conclude that the determined Young’s modulus is not dependent on the geometric dimensions of the nanowires and remains constant even for the smallest nanowire foots of nm. The increasingly broad scatter of datapoints for decreasing nanowire size (and thus ) is thus fully attributed to the increasing relative error in determining , and by SEM inspection.
In conclusion, we investigate singly clamped inverted conical nanowires with lengths between to m, foot radii between nm and nm and a taper angle of about that have been etched into a ()-oriented GaAs wafer by inductive coupled plasma reactive ion etching. The nanomechanical eigenfrequencies of the fundamental flexural nanowire vibration is experimentally characterized, spanning two orders of magnitude in frequency from kHz to MHz. This comprehensive set of inverted conical nanowire resonators allows for a detailed analysis of this previously ill-explored type of nanomechanical system. Along with an analytic eigenfrequency solution derived from the Euler-Bernoulli boundary value problem for the inverted conical nanowire described by equations (S1) and (S2), a single, global value for Young’s modulus of GPa is determined. This value applies to all investigated nanowire geometries, and does not exhibit size effects.
The presented work sets the foundations for further investigations of inverted conical nanowires as nanomechanical systems. The combination of an optically detectable nanowire head with an extremely narrow nanowire foot makes inverted conical nanowires a promising candidate for ultra-sensitive devices, capable to probe minute forces exerted on a nanowire via its head deflection Melli et al. (2010). Precise process control and top down nanofabrication techniques allow to tailor the nanowires’ spring constant for the desired application Waggoner and Craighead (2007); Arlett et al. (2011) and simultaneously facilitates the realization of large, custom-coupled nanowire arrays Spletzer et al. (2008). Besides future applications as force sensors, such ultraflexible nanowire arrays are ideal model systems to explore the nonlinear dynamics and coupling phenomena in nanomechanical arrays Buks and Roukes (2002); Lifshitz et al. (2012), for which synchronization Perlikowski (2008) and Q-boosting Lin et al. (2007) have been predicted. Metallized nanowire heads will allow to integrate plasmonic functionality Bora et al. (2010) or a transport degree of freedom Wiersig et al. (2008). Finally, the integration of photonic elements such as Bragg reflectors or quantum dots can be envisioned Lauhon et al. (2004); Schneider et al. (2012), which may lead to additional functionality incorporating optomechanical arrays Heinrich et al. (2011); Xuereb et al. (2012).
Acknowledgements.Financial support from the Volkswagen Foundation, the German Excellence Initiative via the Nanosystems Initiative Munich (NIM), LMUinnovativ as well as LMUexcellent, and the Center for NanoScience (CeNS) is gratefully acknowledged. The authors appreciate ongoing support and stimulating discussion with J. P. Kotthaus.
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Euler-Bernoulli theory for the inverted conical nanowire
The following discussion is dedicated to the solution of the Euler-Bernoulli problem of an inverted conical mechanical resonator, which is singly clamped at its narrow end. Its equation of motion, the general Euler-Bernoulli differential equation 
is included in the main text as eq. (2). Here, denotes the axis along the nanowire length, the eigenfunction of the bent nanowire, the mass density, the area moment of inertial and the cross sectional area of the non-prismatic nanowire at height . For the inverted conical nanowire, both the area moment of inertia and the cross sectional area depend on a linear radius increment = using the designation assigned in the inset of Fig. 1 of the manuscript.
Solving the boundary value problem for the case of distributed uniform loading
The boundary conditions of the inverted conical nanowire
are summarized in eq. (3) of the manuscript. In the following, an analytic solution of equations (S1) and (S2) will be derived for the piezo-actuated inverted conical nanowire. The right hand side of eq. (S1) represents a force per unit length acting on the cross section of the wire at height . Assuming this force is mediated by a shear piezo (see inset of Fig. 2 of manuscript), its position-dependent part can be considered as , eliminating the -dependence of the actuating force. This assumption of a distributed uniform load is justified as long as the condition is fulfilled for the acoustic travel time along the nanowire length , with m/s being the velocity of sound of the transverse wave in ()-direction in GaAs . In other words, is constant as long as the retardation time until the free end of the nanowire experiences the acceleration of the driven substrate is negligibly small compared to the vibration period of the resonator. We would like to note that this condition is generally fulfilled for nanoresonators with regardless of the underlying material’s .
For the special case of a distributed uniform load, i.e. a constant , the boundary value problem consisting of equations (S1) and (S2) is solved analytically for a singly clamped inverted conical nanowire with using Wolfram Research Mathematica, yielding the eigenfunction for the fundamental flexural mode
with . Note that in the limit the solution converges to the well-known result for the cylindrical beam  .
Determining the eigenfrequency
The eigenfrequency of the inverted conical nanowire is determined by equating the maximum potential energy and the maximum kinetic energy of the conservative system. For a harmonic oscillator with this results in the following analytic expression for the angular eigenfrequency of the nanowire
Figure S1 displays the resulting eigenfrequencies of the fundamental mode as a function of for a set of nanowire lengths (solid lines). The analytically derived solutions are complemented by numerical eigenfrequency analysis for the experimentally explored nanowire geometries using COMSOL Multiphysics (hollow symbols). Excellent agreement between the analytical solution from eq. (S4) and the numerical simulations is found for the full range of geometric parameters explored, corresponding to nanowire eigenfrequencies spanning more than two orders of magnitude. The maximum relative deviation remains below for nanowire aspect ratios and moderate taper angles between to (dependence on taper angle not shown). Only for aspect ratios , the numerical eigenfrequency results start differing from eq. (S4), as apparent for the rightmost upright hollow triangle corresponding to nm and m which exhibits a deviation from the solid line. This is a consequence of the increasing influence of shearing deformations , which go beyond the standard Euler-Bernoulli beam theory approach and are not included in the above derivation.
|||W. Weaver, S. P. Timoshenko, and D. H. Young, Vibration problems in engineering, 5th ed. (John Wiley &|
|||T. B. Bateman, H. J. McSkimin, and J. M. Whelan, Journal of Applied Physics 30, 544 (1959).|
|||A. N. Cleland, Foundations of Nanomechanics (Springer Verlag Berlin Heidelberg New York, 2003).|