DOI: 10.1039/C8CP05408H
(Paper)
Phys. Chem. Chem. Phys., 2019, Advance Article

Zhuoqun
Zheng
^{ab},
Haifei
Zhan
*^{bc},
Yihan
Nie
^{b},
Arixin
Bo
^{b},
Xu
Xu
*^{a} and
Yuantong
Gu
^{b}
^{a}College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, China. E-mail: xuxu@jlu.edu.cn
^{b}School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology (QUT), Brisbane QLD 4001, Australia. E-mail: zhan.haifei@qut.edu.au
^{c}School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia

Received
25th August 2018
, Accepted 22nd October 2018

First published on 22nd October 2018

Nanowires (NWs) are one of the fundamental building blocks for nanoscale devices, and have been frequently utilized as mechanical resonators. Earlier studies show that ultra-sensitive vectorial sensing toolkits can be fabricated by changing the flexural mode of NWs to oscillation doublets along two orthogonal directions. Based on in silico studies and the Timoshenko beam theory, this work finds that the dual orthogonal flexural mode of NWs can be effectively controlled through the proper selection of their growth direction. It is found that metallic NWs with a directional-independent shear modulus possess a single flexural mode. However, NWs with a directional-dependent shear modulus naturally exhibit flexural mode doublets, which do not disappear even with increasing slenderness ratio. Further studies show that such a feature generally exists in other NWs, such as Si NWs. Mimicking a pendulum configuration as used in NW-based scanning force microscopy, the cantilevered 〈110〉 Si NW demonstrates zeptogram mass resolution and a force sensitivity down to the order of 10^{−24} N Hz^{−1/2} (yN Hz^{−1/2}) in both transverse directions. The findings in this work open up a new and facile avenue to fabricate 2D vectorial force sensors, which could enable ultra-sensitive and novel detection devices/systems for 2D effects, such as the anisotropy strength of atomic bonds.

Along with the broad applications of mechanical nano-resonators, there is a continuous interest to fabricate novel multifrequency sensing toolkits.^{28} By virtue of slight asymmetries in the geometry, a very recent work developed vectorial scanning force microscopy, which could enable the imaging of a sample surface by a single NW.^{6} Different from the one-dimensional (1D) dynamics lateral force microscopy that relies on the torsional mode of cantilevers, vectorial scanning force microscopy uses the orthogonal flexural mode doublets of NWs as triggered by their geometrical asymmetries. A similar scheme was realized earlier by Gil-Santos to establish mass sensing and stiffness spectroscopy.^{29} Actually, researchers have established an analogous concept before to fabricate novel nanosensors based on the so-called coupled resonance, which is originated from the coupled geometry (i.e., a coupled oscillator made from two doubly clamped beams).^{30–32} It is noticed that previous studies have focused on the flexural mode doublets as induced by the geometrical asymmetries of NWs, which require a decent control for practical applications.^{33} It is thus of great interest to explore other ways to trigger the mode doublets of NWs.

Interestingly, our previous work found that the 〈110〉 orientated face-centred cubic (FCC) metallic NW exhibits two first-order flexural modes (or beat phenomenon).^{34} Unlike the experimentally reported dual flexural modes as triggered by cross-sectional asymmetry,^{6} the 〈110〉 FCC NW has a uniform circular cross-section and the dual flexural modes are originated from the asymmetric lattice arrangements in lateral directions. It is found that the 〈110〉 FCC NW dominantly vibrates in two orthogonal axes. For external excitation along any other directions (different from orthogonal axes), the vibration will split into doublets oscillating along these two orthogonal directions. Apparently, such an intriguing flexural mode doublets feature of 〈110〉 FCC NWs makes them attractive for the usage in the vectorial force microscopy, which does not require the introduction of cross-sectional asymmetry. Given the diversity of metallic NWs, an instant question to answer is whether or not such a feature is only limited to the 〈110〉 FCC NWs? Further, will such flexural mode doublets diminish when the NW size increases?

To answer these questions, we conduct a series of in silico studies by taking Cu NWs as the representative sample. It is found that the flexural mode is governed by the shear modulus that corresponds to different lateral crystalline directions of NWs. For NWs with a uniform shear modulus along lateral crystalline directions, the vibration is always a single mode. However, for NWs with a directional-dependent or anisotropic shear modulus, they always exhibit flexural mode doublets.

(1) |

Here, x is the axial position (along the length direction) and t is the time. u(x,t) and φ(x,t) are the NW's transverse and angular displacement, respectively. p is the external load induced by the surface stress. Here, the NW is approximated as a core–shell beam to include the surface effect.^{22} The surface elasticity is not included in the calculation of the flexural rigidity (EI) as it was found to induce a negligible effect on the mechanical behaviour.^{36}G and k are the shear modulus and shear coefficient, respectively. For the NW with a circular cross-section, the external load p equals to 2τD∂^{2}u/∂x^{2} (with D and τ as the diameter and surface stress, respectively). Assuming that the surface stress is a constant along the periphery, it is obvious that altering the lateral crystalline direction may change the shear modulus and Poisson's ratio of the NW, and thus results in different natural frequencies. However, it is noticed that although some studies discussed the shear deformation on the vibrational behaviours of NWs, almost all of these studies assumed a constant shear modulus, i.e., a directional-independent shear modulus and Poisson's ratio.

For a crystalline material, the directional dependency of the shear modulus and Poisson's ratio can be easily calculated from its second-order elastic stiffness tensors.^{37,38} For illustration, Fig. 1 shows the shear modulus of the 〈110〉 Cu NW based on the elastic constants predicted by the employed empirical potential in this work.^{39} As is seen, the shear modulus is highly orientation dependent. It should be noted that polar coordinates are utilized in Fig. 1, where the radius represents the value of the shear modulus and the angle signifies the crystalline direction in the considered planes. The maximum shear modulus is about 76 GPa, which nearly triples the minimum value (∼25 GPa). Specifically, the shear modulus of the 〈110〉 NW achieves its minimum and maximum values at the 〈110〉 and 〈100〉 crystalline directions, respectively. Such observation is in line with the two orthogonal directions that the vibration decomposes into. For comparison, we calculate the shear modulus of three other types of NWs, including 〈111〉, 〈100〉, and 〈112〉 orientated Cu NWs. As shown in Fig. 1, both 〈111〉 and 〈100〉 orientated NWs have a uniform shear modulus, i.e., independent of the lateral crystalline direction. In comparison, the shear modulus for the 〈112〉 orientated NWs exhibits an elliptical profile with a smaller difference between the maximum and minimum shear moduli. Based on these results, it is speculated that either 〈111〉 or 〈100〉 orientated NWs will exhibit a single flexural mode, whereas 〈110〉 and 〈112〉 orientated NWs will exhibit a dual flexural mode.

Since the shear modulus of the 〈111〉 NW is independent of the lateral crystalline directions, its vibration is expected to always follow the excitation direction. To verify, we examine its vibrational behavior by altering the excitation angle from 0 to 90° but keeping the same excitation amplitude. It is found that all testings almost yield an overlapped external energy profile, and the trajectory of the mass center is also highly consistent with its excitation direction (see the ESI,† S1 for detailed results). As shown in Fig. 3a, the acquired natural frequency is nearly a constant around 4.27 GHz while altering the excitation angle, with a marginal standard deviation of ∼0.25 GHz. These observations verify the assumption that the 〈111〉 NW will only exhibit a single mode vibration and is independent of the excitation angle.

From continuum mechanics, the influence of shear deformation on the vibrational behavior of the beam diminishes when its slenderness increases, thus, we further test the NW with a slenderness ratio λ ranging from 5 to 16.7 (by keeping a constant length of 100 nm and varying the diameter). Here, the slenderness ratio is defined as the ratio of length to the diameter of the NW. For such purpose, excitations along both [11] and [10] crystalline directions are tested for each NW. It is uniformly observed that all NWs exhibit a single natural frequency and the mass center trajectory follows the excitation direction. As shown in Fig. 3b, the natural frequencies of the NW as obtained from the two different excitation directions are nearly overlapped with each other with varying the slenderness ratio. Overall, these results confirm that the 〈111〉 NW only exhibits a single mode vibration due to its directional-independent shear modulus.

To further verify the influence of shear deformation on the vibrational properties of 〈110〉 NWs, a slenderness ratio ranging from 5 to 16.7 is examined. All examined NWs have the same length of 100 nm but different diameters. The actuations along both [10] and [001] directions are considered. In order to minimize the nonlinear effect from the axial deformation (especially for a nanowire with a large slenderness ratio),^{41,42} a smaller excitation amplitude of 0.4 Å ps^{−1} is chosen. As shown in Fig. 4b, the difference between the two normalized natural frequencies of the 〈110〉 NW increases when the slenderness decreases, which is reasonable as the shear deformation plays a stronger role on the thicker NWs. Here, the natural frequency is normalized by dividing the frequency of the NW at a slenderness ratio of 16.7. For comparison, we also incorporated the directional-dependent shear modulus in the Timoshenko beam model in eqn (1). Specifically, E along the 〈110〉 direction is estimated to be ∼250 GPa based on the elastic stiffness tensor. τ equals to 1.353 J m^{−2}, ρ is 8933 kg m^{−3}, ν is 0.35 and k = (6 + 12ν + 6ν^{2})/(7 + 12ν + 4ν^{2}).^{39} The absolute value calculated from eqn (1) is higher than the simulation results. For instance, at a slenderness ratio of ∼16.7, MD simulation shows a natural frequency around 2.14 GHz in the 〈110〉 direction, while the theoretical predicted value is around 2.97 GHz. Several factors are responsible for the gaps between the simulation results and theoretical predictions, such as the constant Poisson's ratio being assumed and the axial extension effect being ignored in the theoretical model. Despite the difference between the absolute values, the changing tendency of the normalized natural frequency as obtained from the theoretical prediction is highly consistent with that of the simulation results. Such a result justifies the role of shear deformation in shaping the vibrational behavior of the 〈110〉 NWs.

One interesting indication from Fig. 4b is that the 〈110〉 NW will exhibit different vibration behaviors when its slenderness ratio changes. The role of shear deformation is expected to diminish along with the increase of the slenderness ratio, i.e., a transition from a thick beam to a thin beam vibration. As such, one may expect the degeneration of the dual flexural mode to a single flexural mode. Surprisingly, our simulation results show that such degeneration does not happen and the NW always undergoes dual lateral/transverse vibrations. Fig. 5 compares the vibrational properties of the NW with different slenderness ratios, but the same excitation angle of 45°. Clearly, for the NW with a slenderness ratio of ∼8.3, the external energy shows an obvious beat phenomenon (top panel of Fig. 5a). The two natural frequencies have a relatively large difference (around 0.5 GHz), and thus can be easily resolved by using FFT. When the slenderness ratio increases to 12.5 or higher, the two natural frequencies become indistinguishable (for the considered time domain), and the amplitude oscillation possesses a much smaller frequency. As such, only a single natural frequency is detectable from the frequency spectrum (see the ESI,† S3). Despite that, the trajectory of the mass center illustrates a clear two-dimensional motion (see bottom panels of Fig. 5). Such results suggest that the dual flexural mode will always exist in the 〈110〉 NWs, which do not diminish with the increasing slenderness ratio.

Fig. 5 Vibrational behaviors of 〈110〉 NWs. The time history of external energy (upper panel) and the corresponding trajectory of mass center (bottom panel) for the NW with a slenderness ratio of: (a) 8.3; (b) 12.5, and (c) 16.7. The actuation is imposed along the [11] direction, i.e., an actuation angle of 45°. The trajectory is colored based on the time, the same as in Fig. 2c. |

From the experimental perspective, the natural frequency is normally extracted from the displacement trajectory of the NW along the excitation direction (as recorded by scanning electron microscopy),^{43} rather than the external energy as used above. As discussed below, both signals will yield to the same results. By assuming a harmonic vibration in two orthogonal directions, the displacement of the mass center can be expressed as u = a_{1}cos(2πf_{1}t) and v = a_{2}cos(2πf_{2}t) along the two orthogonal axes, respectively. f_{i} and a_{i} are the corresponding angular frequency and amplitude (with no phase lag), respectively. Thus, assuming that the excitation has an angle of φ with the [10] orthogonal direction, the trajectory of the mass center along the excitation direction can be described by

U_{cm} = a_{1}cosφcos(2πf_{1}t) + a_{2}sinφcos(2πf_{2}t) | (2) |

It is apparent from eqn (2) that one would obtain two natural frequencies if their difference is larger than the detection resolution by tracking the NW images along the excitation direction (different from the orthogonal directions). However, if the two natural frequencies are too close to each other, experimental measurements will yield to a single frequency, which might mistakenly be regarded as a 1D vibration. Specifically, for a small Δf = f_{1} − f_{2}, eqn (2) can be approximated as

U_{cm} = (a_{1}cosφ + a_{2}sinφ)cos(πf_{1}t − πf_{2}t)cos(πf_{1}t + πf_{2}t) | (3) |

The above equation describes a nearly harmonic vibration, with the amplitude oscillating under a low frequency of Δf/2, which corresponds to the amplitude oscillation. In other words, for smaller Δf, a longer time is required for the mass center to return to its initial position, as shown in the bottom panel of Fig. 5.

To further probe the flexural mode of NWs, we also test the 〈100〉 and 〈112〉 orientated Cu NWs. NW samples with a circular cross-section and a slenderness ratio of 16.7 are tested (length of ∼100 nm). The initially applied excitation has an angle of 45° with the [110]-axis and [010]-axis of the two examined samples, respectively. As shown in the ESI,† S4, the amplitude of the external energy of the 〈112〉 NWs exhibits a clear oscillation, which is well reflected by the two-dimensional trajectory of the mass center. Such results agree with the fact that the 〈112〉 NW has a directional-dependent shear modulus. In comparison, the trajectory of the mass center for the 〈100〉 NW follows well with the excitation direction, signifying a single mode vibration. Fig. 6 compares the estimated natural frequency of the 〈112〉 and 〈100〉 NWs with varying excitation angles. Similar to the 〈110〉 NWs (Fig. 4a), the 〈112〉 NW shows two natural frequencies. The difference between the two natural frequencies is much smaller compared with that of the 〈110〉 NWs, which is acceptable as the ratio of the maximum to minimum shear modulus of the 〈110〉 NWs is much larger. Meanwhile, only a single natural frequency is detected for the 〈100〉 NWs, which remains nearly a constant while altering the actuation angle (bottom panel of Fig. 6). Overall, the above results suggest that shear deformation plays a pivotal role in determining the vibrational behaviors of FCC NWs with a relatively small slenderness ratio. Unlike the flexural mode doublets as originated from the geometry asymmetries,^{6,29} the ratio between the two resonance frequencies is well depicted by the modified beam model (in eqn (1)). More importantly, the dual flexural mode feature does not disappear when the slenderness ratio increases. Comparing with the control of the geometry asymmetries of the NWs, the vibration mode of the NW can be effectively tuned through the control of its growth directions.

As shown in Fig. 7a, the mass center for the 〈110〉 Si NW exhibits a clear two-dimensional trajectory. Unlike the 〈110〉 NW, the trajectory of the 〈111〉 Si NW aligns well with the excitation angle (Fig. 7b). By further altering the excitation angle between the two orthogonal directions, it is found that the 〈111〉 NW uniformly show a single mode vibration, and the estimated natural frequency is a constant around 9.00 GHz. In comparison, the 〈110〉 NW possesses two natural frequencies, corresponding to its two orthogonal directions (see the ESI,† S5 for detailed results). These results signify that the shear deformation plays the same role in semiconductor NWs, and the dual flexural mode is generally existed for NWs with a directional-dependent shear modulus.

To explore the implications for experiments, we further consider the vibration of a cantilevered 〈110〉 Si NW, mimicking a pendulum configuration as used in NW-based scanning force microscopy.^{6} The NW has a slenderness ratio of 8.3 with a diameter of 6 nm and a length of 50 nm. An initial excitation in the form of v(x) = ax/L (approximating a bent cantilever profile) is applied along the direction with a 45° angle to the orthogonal directions. The vibration is continued sufficiently long for 80 ns. As expected, the NW shows two resonance peaks at f_{1} = 2.79 GHz and f_{2} = 2.83 GHz, corresponding to the two flexural eigenmodes polarized along two orthogonal directions (Fig. 8 and ESI,† S6). The modes are split by Δf = 40 kHz. To estimate the quality factor, we further simulate the vibration of the Si NW under its two orthogonal directions (i.e., under one mode vibration). According to the kinetic energy trajectory, the quality factor Q of the examined Si NW is on the order of 10^{8} (see the ESI,† S6). According to the results in Fig. 8, the displacement amplitude along each orthogonal direction is about 4 nm for both flexural modes. With these values, the spring constants of each flexural mode can be calculated from k_{i} = k_{B}T/〈a_{i}^{2}〉, where i = 1, 2, T is the temperature, and k_{B} is the Boltzmann constant. It is found that the examined Si NW has a spring constant in the order of 10 μN m^{−1}. These parameters yield mechanical dissipations Γ_{i} = k_{i}/(2πf_{i}Q_{i}) and thermally limited force sensitives around 5 × 10^{−24} kg s^{−1} (5 zg s^{−1}) and 5.2 × 10^{−23} N Hz^{−1/2} (52 yN Hz^{−1/2}), respectively. These estimations indicate that ultra-sensitive vectorial force sensors can be constructed by utilizing the dual flexural mode feature of the NWs as originated from their directional-dependent shear modulus.

Fig. 8 The time history of the displacement for the cantilevered Si NW under the excitation with an angle of 45° with its orthogonal [10] direction. |

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## Footnote |

† Electronic supplementary information (ESI) available: The vibrational behaviors of 〈111〉 and 〈110〉 Cu NWs; the frequency spectrum for the 〈110〉 Cu NWs with different slenderness ratios; the vibrational behaviors of 〈112〉 and 〈100〉 Cu NWs; the vibrational behaviors of 〈111〉 and 〈110〉 Si NWs; and the vibrational behaviors of cantilevered 〈110〉 Si NWs. See DOI: 10.1039/c8cp05408h |

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