Size-Scaling in Optical Trapping of Silicon Nanowires - American

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Size-Scaling in Optical Trapping of Silicon Nanowires Alessia Irrera,† Pietro Artoni,‡,§ Rosalba Saija,|| Pietro G. Gucciardi,† Maria Antonia Iatì,† Ferdinando Borghese,|| Paolo Denti,|| Fabio Iacona,‡ Francesco Priolo,‡,§ and Onofrio M. Marag
o*,† †

CNR-IPCF, Istituto per i Processi Chimico-Fisici, I-98158 Messina, Italy MATIS-IMM CNR, I-95123 Catania, Italy § Dipartimento di Fisica e Astronomia, Universit
a di Catania, I-95123 Catania, Italy Dipartimento di Fisica della Materia e Ingegneria Elettronica, Universit
a di Messina, I-98166 Messina, Italy

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bS Supporting Information ABSTRACT: We investigate size-scaling in optical trapping of ultrathin silicon nanowires showing how length regulates their Brownian dynamics, optical forces, and torques. Force and torque constants are measured on nanowires of different lengths through correlation function analysis of their tracking signals. Results are compared with a full electromagnetic theory of optical trapping developed in the transition matrix framework, finding good agreement. KEYWORDS: Size-scaling, optical trapping, silicon nanowires, electromagnetic scattering theory

ize-scaling is a paradigm of nanoscience.1 It dictates the behavior of solid-state systems for a wealth of applications in the most diverse research fields.2 Crucial properties of materials and interactions change dramatically with size.3 Much progress has already been done in the synthesis, assembly, and fabrication of nanomaterials, and, equally important, toward a wide variety of technological applications.4 Indeed the properties of materials with nanometric dimensions are significantly different from those of atoms or bulk materials, and the suitable control of such properties at this mesoscale have led to new science as well as new products, devices and technologies.1 In this context, the possibility to control, manipulate, characterize, and study nanostructures as a function of size is crucial for the achievement of miniaturized systems with highly advanced functionality. Nanowires (NW) have attracted considerable interest within the scientific community as an innovative material with applications in nanotechnology.5 They are defined as structures with a high aspect ratio, being characterized by two spatial dimensions in the range of tens of nanometers and the third one on a much longer scale, typically micrometers. Moreover, due to their very large surface-to-volume ratio, NW can lead to strongly enhanced surface effects as compared to bulk materials. Silicon is the most important semiconductor since the Si-based devices have dominated integrated circuit devices for many decades.6 With a broad selection of band structures, silicon nanowires (SiNWs) are considered to be the critical components in a wide range of potential nanoscale device applications for microelectronics,7 sensors,8 and photovoltaics.9 The possibility to study and manipulate an individual SiNW in three dimensions is strategic for all these applications.

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r 2011 American Chemical Society

Optical trapping10
12 (OT) of nanostructures13
23 have acquired tremendous momentum in the past few years. Manipulating nanoparticles with OT is generally difficult because radiation forces scale approximately with particle volume12,24 and thermal fluctuations can easily overwhelm trapping forces at the nanoscale.12 However semiconductor13
16 and optical nonlinear17,18 nanowires, carbon nanotubes,19 polymer nanofibers,20 graphene,21 quantum dots,22 and plasmonic nanoparticles23 have now been stably trapped thanks either to their highly anisotropic geometry13
21 or to their intrinsic resonant behavior.22,23 They have been manipulated to build nanoassemblies,13,14 as well as accurately measure forces with resolution at the level of femto-Newtons crucial for photonic force microscopy applications.17,19,21,25,26 Nanotools25 have been specifically designed in order to combine the outstanding force-sensing capabilities of OT with increased nanometric precision and bridge the gap between micro and nanoscale in fluidic environments.26 Furthermore the integration of OT with Raman spectroscopy allowed for ultrasensitive chemicalphysical analysis of trapped particles.21,27,28 In this context, the role of size-scaling is crucial for understanding the interplay between optical forces and hydrodynamic interactions29 that change dramatically with size, hence much affecting both force-sensing and spatial resolution in precision applications.14,17,19,25,26 Here we investigate how optical trapping and Brownian motion of very thin (10 ( 2 nm diameter) SiNWs are dependent on size. We show how their length is the key parameter that Received: August 6, 2011 Revised: September 24, 2011 Published: October 03, 2011 4879

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Nano Letters regulates forces, torques, and hydrodynamics. We fully characterize the three-dimensional translational and angular Brownian motion, measure root-mean-square displacements and show the different size-scaling due to the interplay between radiation forces defining the trapping potential and hydrodynamics. Finally, we compare our findings with a full electromagnetic theory of optical trapping of SiNWs, showing good agreement when taking into account aberration from the coverslip-water interface. SiNWs were prepared by a wet-etching technique assisted by the deposition of an ultrathin gold film on a Si substrate. Silicon nanowires (SiNWs) have been obtained starting from p-type Si single crystal. The resistivity of the p-type silicon wafer was 1.5 Ω cm, corresponding to a boron (B) concentration of about 1016 cm
3. The wafers were cut into 1 cm  1 cm pieces, and a nanometric thin Au layer was deposited on the Si samples at room temperature by electron beam evaporation. Samples were etched at room temperature in an aqueous solution of HF and H2O2. In fact, the Si underneath the ultrathin gold layer acts as the anode and is locally oxidized. The oxidized Si is then immediately and continuously etched down by HF resulting in the formation of SiNWs. By varying the etching time, we prepared five samples in which the SiNWs length, L, is respectively 1.00 ( 0.05, 2.08 ( 0.04, 3.1 ( 0.1, 4.0 ( 0.2, and 5.0 ( 0.5 μm, measured from scanning electron microscopy (field-emission SEM Zeiss Supra 25) analysis of the as-grown samples. In Figure 1a, typical cross-section SEM images of different length of SiNWs obtained by varying the etching time are shown. The images display a dense and uniform distribution of NWs with very small diameters. A mean diameter of 10 ( 2 nm is obtained from Raman spectroscopy analysis. For optical trapping experiments, SiNWs are mechanically scratched from the substrate and then stably dispersed by sonication in watery solution added with sodium dodecyl sulfate (SDS). Although SiNWs can be dispersed in water without the aid of surfactants, we found that using SDS improves the temporal stability of the dispersion toward aggregation and minimize the sticking to glass surfaces. A few tens of microliters are then placed in a sample chamber attached to a piezo-stage with 1 nm resolution. Optical trapping is achieved19,21 by focusing a 830 nm laser beam (from a laser diode Sanyo DL8142-201, 150 mW nominal power) through a 100 oil immersion objective (NA = 1.3) on an inverted microscope (see Figure 1b,c for a sketch of the setup and the geometry). The laser power available at the sample is about 20 mW and was kept constant during the optical force measurements. Optically trapped SiNWs are imaged with a CCD camera (see Figures 1d,e). This enables inspection of the individual trapped SiNW length that is always checked to be consistent with SEM measurements for each sample. Force measurements are performed always at a distance of about 15 μm from the coverslip inside the sample chamber in order to minimize the effect of the glass surface. Positional and angular displacements of the trapped SiNWs are measured by back focal plane (BFP) interferometry,30 where the interference pattern between forward scattered and unscattered light in the microscope condenser BFP is imaged onto a four quadrant photodiode, QPD (Thorlabs PDQ80A). This is oriented with the polarization (x-)axis of the laser beam in order to have sensitivity over polarization effects.16,24 The QPD outputs from each quadrant are then combined as pairwise sums (Sx, Sy) and four-quadrant sum (Sz). These tracking signals are thus acquired at 50 kHz sampling rate for 2 s (105 points). While for a spherical particle the QPD traces are proportional to its center-of-mass displacements, for trapped SiNWs they also

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Figure 1. (a) Cross-section SEM microscopy images of the as-prepared SiNWs with controlled diameter of 10 ( 2 nm and length L. (b) Sketch of the optical tweezers setup: a NIR laser beam (830 nm) is focused by a 1.3 NA oil immersion microscope objective. SiNWs are trapped in a water dispersion placed in a sample chamber. The scattered light from the trapped particle is detected by a QPD through a collection optics yielding electrical signals Sx, Sy, and Sz proportional to the particle translational and angular displacements. (c) Description of the geometry in the trap with relevant coordinates and angles for the characterization of forces and hydrodynamic mobilities of trapped nanowires. (d) Image of a SiNW optically trapped in the focal region of the NIR laser. (e) Image of a SiNW with a length of 3 μm moving in water when the laser is off. Note that these diffraction-limited CCD images result from the interference between scattered and unscattered illumination light from the nanowire yielding a change from brighter to darker region when propagating along the nanowire length.

contain information on angular displacements.19,21,31,32 In particular since radiation pressure aligns the SiNWs with the light propagation15,32
34 (z-)axis (see Figure 1d), positional and angular displacements occur in the small polar angle limit (see geometry in Figure 1c), θ , 1, and the tracking signals are19 Sx
X + aΘx, Sy
Y + bΘy, Sz
Z. X, Y, and Z are the center-ofmass coordinates, a, b, and c are calibration constants, and Θx = θ sin j and Θy = θ cos j are the angular projections on the x- and y-axis, respectively (shown in Figure 1c). Figure 2a,b shows typical (calibrated) tracking signals (Sz) and frequency counts for each recorded trace obtained by trapping individual SiNWs with different length in each sample. The analysis of these signals resulting from Brownian motion in the trap is the key to get optical forces and torques in optical trapping experiments.12,30 In general, a colloidal particle in a fluid is subject to thermal fluctuations.35 In the free-diffusive regime (mean-square displacement is linear with time) the motion is independent of inertia terms (mass, moment of inertia) and it is described by a Langevin equation35 for each relevant dynamic variable (positional or angular). When a colloidal particle is confined by an external potential, for example, an optical trap, the 4880

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Figure 2. (a) Calibrated tracking signals along the propagation (z-)axis and (b) relative frequency counts of the signals for individual trapped SiNWs of lengths L = 1 (black), 2 (red), 3 (blue), 4 (green), and 5 μm (magenta). (c) Exemplar transverse (top) and axial (bottom) autocorrelation functions (ACF) for two individual SiNWs of length L = 1 μm (black) and L = 3 μm (blue). The transverse ACF reveal both the translational and angular fluctuations of the SiNWs in the trap and are well fitted by a double exponential decay yielding the transverse force and torque constants. This is particularly evident at shorter lengths (see also Figure 3). In contrast, axial ACF are well fitted with a single exponential decay with rate ωz that yields the axial (z-)force constant. (d) Relaxation rates ωz measured for different samples of SiNWs. For each length, we trapped about 20 different particles in each sample. Two sets of data points are distinguished corresponding to individual SiNWs and bundles of two SiNWs in the trap. The error bars on the length are extracted from the SEM images of the original samples, while the ones on the rates represents the standard deviation from the mean.

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fluid damps its motion as in the free-diffusive regime, but now the particle explores only a limited region in space.30 Anisotropy largely affects the Brownian dynamics in the trap.16,19,21,32 Thus, in addition to translational coordinates X, Y, and Z, rotational variables, Θx and Θy, have to be considered for a correct calibration of optical forces and torques on SiNWs. Here we consider the SiNWs as a rigid rodlike structure with no bending motion activated. This is reasonable since even for such thin SiNWs the Young’s modulus has been recently measured36 to be about 100 GPa and the corresponding persistence length37 for the thermal activation of bending is about 25 cm, that is much larger than the length of SiNWs explored in this work. Thus for rigid rodlike particles anisotropy yields different viscous damping, γ = (2πηL)/(ln p + δ ), γ^ = (4πηL)/(ln p + δ^), for translations parallel and perpendicular to the SiNW main axis respectively, and γΘ = (πηL3)/[3(ln p + δθ)], for rotations, that are dependent on the SiNW length-to-diameter ratio, p = L/d, water dynamic viscosity, η, and end corrections δi (calculated19,38 as polynomial of (ln 2p)
1). The Brownian dynamics of a trapped SiNW is then well described by a set of uncoupled Langevin equations19,21 d Xi ðtÞ ¼
ωi Xi ðtÞ þ ξi ðtÞ, dt

i ¼ x, y, z

ð1Þ

d Θj ðtÞ ¼
Ωj Θj ðtÞ þ ξj ðtÞ, dt

j ¼ x, y

ð2Þ

where ωi = ki/γi and Ωj = kΘj/γΘ are relaxation frequencies, related to the force and torque constants and viscous damping

components, while ξi(t) are random noise sources. From eqs 1 and 2, the autocorrelation functions (ACF) of the transverse tracking signals Cii(τ) = ÆSi(t)Si(t + τ)æ now contain combined information31,32,39 on center-of-mass and rotational fluctuations, and decay with lag time τ as a double exponential with positional and rotational relaxation frequencies ωi, Ωi (i = x, y). The ACF of the axial tracking signal, that is, the motion parallel to the propagation (z-)axis, is instead unaffected by the rotational motion and decays as a single exponential with relaxation frequency19 ωz. Figure 2c shows ACF analysis for exemplary signals related to trapped SiNWs of 1 μm (black lines) and 3 μm (blue lines) length. The double exponential decay is clearly visible for the transverse signals, while is absent for the axial ones as expected for a strongly aligned linear nanostructure (small angle approximation). For each sample, we trapped about 20 different SiNWs, we then analyzed the corresponding ACF and obtained the relaxation frequencies, force, and torque constants. As an example, in Figure 2d we show the measured axial relaxation frequencies ωz measured for SiNWs of different length. Two sets of data points are distinguished corresponding to individual SiNWs and bundles of two SiNWs in the trap. The error bars on the length are extracted from the SEM images of the original samples, while the uncertainty on the relaxation rates represents the standard deviation from the mean. The functional behavior of relaxation frequencies with SiNWs length is the result of the interplay between optical trapping forces and hydrodynamics. By knowing the hydrodynamic damping for each SiNWs (see also Supporting Information) we extrapolate the functional 4881

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Figure 3. (a) Reconstruction of Brownian fluctuations of the trapped SiNWs for exemplar data sets from each sample. Each plot shows 104 points from the tracking signals in x, y, and z. Transverse (b) root-mean-squared-displacement (RMSD) for x- (red), y-direction (blue) and z-direction (c) as a function of length obtained from each data set shown in (a). The observed monotonic rmsd decrease in (b) is consistent with the increase in trapping force. In contrast, in (c) the axial rmsd along z has a more complex behavior accounting for the contrasting action of trapping and radiation pressure shift yielding a weaker trap for longer SiNW and hence larger positional fluctuations. (d) Angular rmsd as a function of length for fluctuations in the xz- (red) and yz-plane (blue). A larger length for the trapped SiNW results in a stronger optical torque and hence smaller angular fluctuations.

behavior of force and torque constants (as discussed later). Thus from these measurements, we get the full calibration of the optical confining potential. Figure 3a shows the reconstruction of Brownian motion of exemplar individual SiNW in the trap by sampling the calibrated tracking signals in x, y, and z. From each plot the transverse (Figure 3b), axial (Figure 3c), and angular (Figure 3d) rootmean-square displacement (RMSD) are obtained from the ACF zero point value, related to mean-squared-displacements ÆX2i æ = kBT/γiωi and ÆΘ2j æ = kBT/γΘΩj. The observed RMSD decrease in Figure 3b is consistent with an increase in trapping force. In contrast in Figure 3c the axial RMSD has a more complex behavior accounting for the contrasting action of increased trapping and larger radiation pressure shift yielding a weaker trap for longer SiNW and hence larger positional fluctuations. In Figure 3d angular RMSD decreases at larger lengths as a consequence of a stronger optical torque. Optical trapping forces generate from the conservation of momentum in the scattering process of light by a particle.15,24,32
34,40 In a light scattering process by a thin SiNW we have two sizescaling parameters40 defined by its transverse size and length as xd = πnd/λ,1 (xd ≈ 0.05, constant for each SiNW sample) and xL = πnL/λ >1 (varies in the range 5
25 for our SiNW samples), where n is the refractive index of the surrounding medium. The latter size parameter, xL, regulates the interaction between SiNWs and radiation fields in the far-field.40 We then plot in Figure 4 the measured optical force and torque constants and trap parameters as a function of the size parameter xL. In particular, in Figure 4a
d we show the scaling of the measured force and torque constants for individual SiNWs (solid symbols) and bundles of 2 SiNWs (open symbols). Force constants, kx (Figure 4a) and ky (Figure 4b), measured in the transverse plane

Figure 4. Scaling of the measured force and torque constants and trap parameters as a function of the size parameter xL = nπL/λ for individual SiNWs (solid symbols) and bundles (open symbols). Force constants, kx (a) and ky (b), measured in the transverse plane are more than ten times larger than the ones measured along the propagation (z-)axis, kz (c). (d) Torque constants kΘx (red symbols) and kΘy (blue symbols). (e) Polarization anisotropy, kP = 1
(kx/ky), and (f) aspect ratio, (kx + ky)/2kz, of the optical trap. The lines are the calculated values of optical force and torque constants and trap parameters for linear particles modeled as chains of small spheres trapped in an ideal diffraction limited laser spot with no aberration (dotted lines) and in a radiation field aberrated (solid lines) by the glass
water interface of the sample chamber. The theoretical lines have been scaled by only one free parameter in the plots (a
d), while no adjustment was made for the plots (e,f).

are in the range 30
100 pN/μm for the longest SiNWs and are more than 1 order of magnitude larger than the ones measured 4882

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Nano Letters along the propagation (z-)axis, kz (Figure 4c). Transverse optical forces show a flattening for longer SiNWs. This is expected34 when the length is much larger than the Rayleigh range of the focal spot, since the interaction region is approximately the same for all SiNWs and the trapping force should not change dramatically with length. However, the scattering force increases for longer SiNWs pushing and shifting the trapped particles along the propagation axis in regions of the beam where light intensity and the axial trapping are weaker. This justifies the more complex behavior of kz shown in Figure 4c: the axial trapping force increases with size parameter xL up to a maximum and then decreases for longer SiNWs. In contrast, measured torque constants (Figure 4d) kΘx and kΘy show a monotonic increase with size parameter, that is, the longer the SiNWs length the stronger the optical torque on the nanowire. A linear polarized tightly focused laser beam has an anisotropic intensity profile in the focal plane as well as an intrinsic nonunitary aspect ratio of the spot because of light propagation.15,24 This yields asymmetries in the confining potential transverse plane and aspect ratio when trapping submicrometer or nonspherical particles.15
21 This is quantified by two parameters: the trap polarization anisotropy, kP = 1
kx/ky, and trap aspect ratio, (kx + ky)/2kz, that summarize the effects on optical forces from light polarization, propagation, and particle shape. Figure 4e,f shows the scaling behavior of the polarization anisotropy and trap aspect ratio with the SiNW size parameter. In particular the polarization anisotropy, related to the nanometric transverse size15,19 of the SiNWs, appears to be constant, kP ≈ 0.25, over the range of length explored. Instead the trap aspect ratio shows a more complex scaling behavior that is related to the large anisotropy of the SiNWs combined with the shift of the trapping position by the scattering force. We now compare our experimental findings with optical trapping calculations based on electromagnetic scattering theory in the T-matrix formalism.15,24,40 The SiNWs are modeled as linear chains of length L composed of identical spheres of diameter D , λ < L with the refractive index of bulk silicon, nSi = 3.7 at 830 nm (see also Supporting Information). Since the light scattering properties in the far-field are regulated by L, we expect our model to be able to account for a general scaling behavior of optical forces and torques despite the other structural differences at the nanoscale. In Figure 4, the lines are the calculated values of optical force and torque constants and trap parameters for model chains of small (D = 30 nm) spheres trapped in an ideal diffraction limited laser spot with no aberration (dotted lines) and in a radiation field that include the aberration (solid line) by the glass
water interface24 of the sample chamber that blurs the laser spot at the distance where experiments are performed (about 15 μm). The theoretical lines have been scaled by only one free parameter in the plots (a
d) that accounts for the structural differences between the model linear chains15 and the cylindrically shaped nanowires34 used in the experiments, while no adjustment was made for the plots (e,f), whose parameters are obtained through ratios of measured force constants. The theoretical predictions for transverse force (Figure 4a,b) and torque (Figure 4d) constants in the ideal non aberrated case (dotted lines) show a sharp crossover at size parameter xL ≈ 5 from a monotonic increase with length to constant values. However, when aberration is taken into account (solid lines) the theoretical curves agree much better with measured values despite the simplified linear chain model structure used in the calculations. Similarly for the polarization anisotropy

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(Figure 4e) and trap aspect ratio (Figure 4f) including aberration effects in the calculations gives a better quantitative agreement with experimental findings with no free parameter. This shows how the SiNWs anisotropy plays a special role in defining the optical trapping potential yielding highly elongated traps with aspect ratios that range from 10 to 100. In conclusion, SiNWs with transverse size of 10 ( 2 nm and controlled length in the micrometer range have been optically trapped. Such large aspect ratio ensures stable optical trapping of individual SiNWs over thermal fluctuations with transverse optical force constants of the order of 50 and 1
2 pN/μm for the axial ones, as well as granting spatial resolution of the order of tens of nanometers in photonic force microscopy applications. We have shown how length regulates optical forces, optical torques, and their Brownian dynamics in the trap. Correlation function analysis has been used to obtain the force and torque constants, root-mean-square translational and angular displacements, as well as relevant trap parameters. We compared them with calculations on model linear nanoparticles based on electromagnetic scattering theory showing that aberration plays a key role in experiments. Optical trapping of SiNWs with transverse size of few nanometers opens perspectives for their use as nanoprobes with increased spatial and force resolution in photonic force microscopy applications,17,25,26,30 as well as in protocols for laser cooling their center-of-mass and angular motion at the quantum regime in optical traps or cavities in vacuum.41,42

’ ASSOCIATED CONTENT

bS

Supporting Information. Nanowire hydrodynamics, correlation function analysis, optical force and torque on SiNWs in the T-matrix formalism, and additional figures and references. This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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