2D Nanomechanical Force Field Sensing: How to Use Nanowires

Built with a very large aspect ratio (up to 5000) with a diameter and length of around 150 nm/100 μm, respectively, and effective masses that fall in the picogram range, suspended nanowires are the much-preferred force sensors.

Suspended nanowires oscillate in unison along the two transverse directions, which permit users to operate them as vectorial 2D force sensors with sensitivities in the range of attonewton/√Hz at room temperature [1].

This article gives an account of how these characteristics, along with the HF2LI phase-locked loop (PLL), can be used to attain 2D force measurements based on the optical readout of SiC-nanowires’ vibration.

2D—Force Field Sensing

Nanowire Mechanics

Applying the force field at the very edge of a singly clamped suspended nanowire means its dynamics can be attributed to one of its two central, transverse eigenmodes. The deflection δr of the nanowire edge is expressed as follows:

(1)

with the eigenmode frequencies Ω_1,2/2π, the innate damping rate Γ, and the effective mass M_eff of the oscillator. Lacking any external force, the driving force F is a stochastic Langevin force vector δF_th(t) at the origin of the Brownian motion of the oscillator, as shown in Figure 1.

Figure 1. The scheme illustrates the vibration pattern of the two first fundamental mechanical-modes of a suspended nanowire. Due to a slight geometrical asymmetry, the nanowire presents two perpendicular modes (labeled 1 and 2) with around 1% relative frequency splitting. The right part shows typical Brownian motion trajectories in 2D of the nanowire tip (2 s acquisition).

This exposes the nanowire to an extra external force F_ext (r₀ + δr), which is analyzed at the position of nanowire extremity r₀ + δr, when, during the assessment, it is saturated in the local force field. In a first-order approximation, it can be calculated as

(2)

Thus, the first term denotes a static local force while the second is the gradient that the nanowire experiences along with its oscillations. The results show that the former experiences a static deflection of the nanowire’s rest position (equal to a few nm), while the latter adds to the restoring forces and thus, alters the mechanical properties of the nanowire. With this linearization, Equation 1 can be expressed in Fourier space by

(3)

where the integration of an external driving force, with the dressed mechanical susceptibility matrix _χ reads in the base of the bare eigenmodes:

(4)

Introducing the four components of the nearby 2D reduced force field gradients matrix: g_ij given by 1/M_eff · ∂_iF_j were and are relative to the derivative along the direction i of the force projected along the direction j.

This render does not alter the Langevin force vector δF_th. Therefore, it is possible to assess the force field by the effect it has on the nanowire thermal noise in 2D, as was done from the outset in the group when establishing the force field gradients.[2] Acquisition of this measurement can also be achieved with an external driving force that will act to accomplish driven response measurements to establish the dressed mechanical susceptibility of the nanowire. To obtain that, the dual PLL scheme, as illustrated in this article, will be used.

To present two dressed eigenmodes of the perturbed system in Equation 4, diagonalizing the matrix is required. This leads to the creation of eigenmode frequency shifts and rotations of the eigenvectors using the 2D force field gradients. It has been found [2] that the measurement of those four quantities enables a full determination of the local force field gradient matrix (g_ij).

Hence, to determine the variance between the two eigenmode orientations, it is necessary to use a 2D readout of the nanowire motion.

Optical Readout Along Two Dimensions

Due to their high refractive index, which causes the phenomenon of internal Mie resonances, the SiC nanowires harbor a large optical scattering cross-section. This allows for an efficient optical position readout even though their diameter is comparably very small to the wavelength of visible light. Located in the waist of a focused laser beam, capturing the light reflected off the nanowire is possible through the focusing objective. It is also documented on a split photodiode delivering the sum and difference signals defined from the amplitudes measured on the two segments of the photodiode.

This readout technique is illustrated in Figure 2 and the two maps procured on each measurement channels are displayed when scanning the nanowire in the waist area: δV_⊖,⊕(r₀). When pulsating, the nanowire controls the photocurrents as δV_⊖,⊕(t) = δr(t) · ∇V_⊖,⊕, so that projective measurement carried by the photodiode output of the nanowire motion along a measurement vector is distinguished by the local gradients of the reflected maps. Positioning the nanowire at a location marginally offset from the waist on the optical axis allows for the operation of two quasi-perpendicular measurement channels.

Figure 2. Optical readout scheme of the nanowire position in the xz plane. The intensity maps (right) display the reflection signals measured during a scan of the nanowire in the xz-plane. Their local gradients determine the readout vectors. The highlighted position provides two quasi-perpendicular measurement vectors.

After fixing the working point in place, the LabOne Spectrum Analyzer module is utilized to compute the spectra of the sum and difference signals to record and determine the nanowire noise spectra. Figure 3 shows a nanowire’s thermal noise spectrum with the eigenmodes at a 45° slant towards the readout vector so that each mechanical polarization is effectively taken on each measurement channel. The two diverse eigenmode frequencies are 27.80 kHz and 27.94 kHz.

Figure 3. Nanowire thermal noise spectra for eigenmode calibration as measured along the ⊕ and ⊖ readout channels. The spectrum has been acquired using the LabOne API.

Dual Resonance Tracking

Experimental Realization

For the experiment, a dual resonance tracking method was used, which requires two autonomous PLLs. Compared to a thermal noise-based force gradient measurement [2], this methodology based on regulated driven trajectories (and not on arbitrary thermal noise trajectories) does not require the fitting of spectra.

The rapid measurements are enabled and the speed is only restricted by the finite linewidth of the mechanical resonances, which relates to how long it takes for the nanowire to adapt to the variation in the external force landscape.

Figure 4 further illustrates the experimental setup. Decreasing the frictional damping of the nanowire is achieved with a vacuum chamber and isolates the setup from environmental unrests. An XYZ piezo stage controls the position of the experiment’s sample and nanowire relative to the fixed laser beam. A second XYZ Piezo stage regulates the position of the sample in relation to the nanowire. Taking on the role of the test driving force δF_drive in Equation 3, with an efficiency of approximately 1 fN/μW the optical force of an amplitude modulated green laser, co-injected with the red probe beam, works to drive the nanowire.

Figure 4. Experimental setup illustrating the 2D optical readout channels (red probe laser beam), the intensity modulated optical drive (green pump laser), the voltage bias and their connections to the HF2LI instrument.

This green laser beam has the capability of managing both nanowire transverse modes as they are not essentially aligned with the optical axis. The driven trajectories, which are used to sense eigenmode rotations and mechanical frequency changes, are recorded using the split photodiode. This is achieved by directing the sum and difference signals to the HF2PLL signal inputs. One signal output controls the concentration of the pump laser via an acousto-optical modulator (AOM). Thus, the AOM control signal is amplitude-modulated simultaneously with both mechanical frequencies.

To exploit the driven signals along both measurement vectors, the nanowire is rotated until its bare eigenmodes form an angle of around 45° in relation to the optical axis. This offers the assurance that the pump laser can capably drive the two modes.

Illustrated in Figure 5, the measured response is in the described configuration. This response curve gives the set-point parameters for the PLL. They can easily be read in the LabOne Sweeper module. When measured, four of the six available demodulators assess the two input signals at both resonance frequencies. Thus, all the quantities necessary to reconstruct eigenmode orientations and eigenfrequency changes are measured at the same time.

Figure 5. Amplitude and phase of sum and difference signals measured with the LabOne Sweeper in response to the optical drive. These response measurements serve to set the dual PLL parameters that will subsequently track both mechanical resonances.

Force Gradient Imaging

The imaging proficiency of the setup are determined using a test sample that produces an electrostatic force field with features that are known. The sample is made up of a structured conducting surface on which a bias voltage can be applied.

When performing the imaging process, the sample is scanned beneath the extremity of the nanowire while recording the frequency variations and amplitude changes, which are the result of eigenmode rotations.

A TTL signal on the HF2LI’s DIO connector correlates the acquisition with the movement of the Piezo stage. By transmitting the results to a measurement interface written in Python for this experiment, the Data Acquisition Module of the LabOne Application Programming Interface (API) appropriately coordinates spatial scans and acquisition of the task. The direct inclusion of the measurement into the current sample positioning setup allows for direct imaging of the local force field gradient without elaborate post-processing being called for. Extensive explorations of the sample under study are enabled because the force field gradients are computed in quasi-real-time (approximately 10 measurements per second).

One of these factors is the 2D force divergence ∇ · F = M_eff(g₁₁ + g₂₂), which is illustrated in Figure 6 for the electrostatic force above an arrangement of holes. The fact that the nanowire is drawn towards the border of the holes, the observations determine that the repulsive force field of circular shape can be comprehended this way. By recording and mapping all four force field gradients beyond the sample under study, the 2D force field can be reconstructed with a 2D spatial integration (refer [2]).

Figure 6. Measured force field planar divergence about 500 nm above an array of approx. 400 nm large holes drilled in a conducting surface. The bright central feature is located above a hole in the center of the scanned area. A bias of -0.2 V has been applied between surface and nanowire. Due to the attraction towards the periphery of the holes, the measured projected force field is repulsive from the center of the hole (positive planar divergence).

Outlook

Integrating the dual PLL of the HF2LI has enhanced the measurement speed up to 50 times compared to earlier measurements [2] based on thermal noise assessment in 2D. Moreover, due to directly observing the resonance frequencies and projected driven oscillation amplitudes on each channel enables force fields to be imaged in quasi-real-time without the need for laborious post-processing.

Subsequently, implementation of a bidirectional drive will be included using both HF2LI signal outputs to vastly improve experimental control and permit signal post-processing. Making the most of these benefits along with the intrinsic sensitivity and adaptability of nanowires, measurements based on the defined method paves the way towards original explorations in physics based on ultrasensitive nano-mechanical vectorial force field sensors. For example - imaging the Casimir force in non-standard geometries, sensing weak magnetic structures such as Skyrmions, or examining the coupled dynamics of a nanowire and a single spin.

References

[1] A. Gloppe, P. Verlot, E. Dupont-Ferrier, A. Siria, P. Poncharal, G. Bachelier, P. Vincent, and O. Arcizet. Bidimensional nano-optomechanics and topological backaction in a non-conservative radiation force field. Nature Nanotechnology, 9(11):920–926, Nov 2014.

[2] Laure Mercier de Lépinay, Benjamin Pigeau, Benjamin Besga, Pascal Vincent, Philippe Poncharal, and Olivier Arcizet. A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields. Nature Nanotechnology, 12(2):156–162, 2016.

This information has been sourced, reviewed and adapted from materials provided by Zurich Instruments AG.

For more information on this source, please visit Zurich Instruments AG.