Optical trapping of porous silicon nanoparticles

IOP PUBLISHING

NANOTECHNOLOGY

Nanotechnology 22 (2011) 505704 (8pp)

doi:10.1088/0957-4484/22/50/505704

Optical trapping of porous silicon nanoparticles Maria G Donato1 , Marco A Monaca1,2 , Giuliana Faggio2 , Luca De Stefano3, Philip H Jones4 , Pietro G Gucciardi1 and Onofrio M Marag`o1 1 IPCF-CNR, Istituto per i Processi Chimico-Fisici, Viale F Stagno d’Alcontres 37, I-98158 Messina, Italy 2 Dipartimento Meccanica e Materiali (MECMAT), Facolt`a di Ingegneria, Universit`a ‘Mediterranea’ di Reggio Calabria, L.t`a Feo di Vito, I-89060 Reggio Calabria, Italy 3 IMM-CNR, Istituto per la Microelettronia e Microsistemi-Unit`a di Napoli, via P Castellino 111, I-80131 Napoli, Italy 4 Department of Physics and Astronomy, University College London, London WC1E 6BT, UK

E-mail: [email protected]

Received 6 August 2011, in final form 12 October 2011 Published 23 November 2011 Online at stacks.iop.org/Nano/22/505704 Abstract Silicon nanoparticles obtained by ball-milling of a 50% porosity silicon layer have been optically trapped when dispersed in a water–surfactant environment. We measured the optical force constants using linearly and radially polarized trapping beams finding a reshaping of the optical potential and an enhanced axial spring constant for the latter. These measurements open perspectives for the control and handling of silicon nanoparticles as labeling agents in biological analysis and fluorescence imaging techniques. (Some figures may appear in colour only in the online journal)

applications of force sensing and tracking at the nanoscale in biosystems. The ability to exploit light forces for the trapping and handling of microparticles was pioneered by Ashkin [8] in the 1970s. Some years later, the first optical tweezers (OT) were realized [9] using a laser beam strongly focused by a high numerical aperture objective lens. In these systems a particle is trapped in the focal region of the lens by the forces arising from the scattering of light by the particle [10]. Since then, OT have been extensively used for applications in cellular and molecular biology, soft matter and nanotechnology. In biology, OT are used to make micromechanical experiments on cells and micro-organisms both in vitro and in vivo [11–13], where the use of a near-infrared wavelength (800–1100 nm) laser prevents photodamage and thus the death of micro-organisms and cells [14]. In physics, the ability to apply forces in the range of piconewtons to micro- and nanoparticles, and to measure their displacements with nanometer precision is crucial for the investigation of colloidal [15] and condensed matter systems [16]. Optical trapping of nanoparticles is not an easy task. Optical forces scale with the particle volume, thus stable

1. Introduction Bridging the gap between silicon electronics and photonics has been the goal of intense research for the last 20 years [1]. In this context, the light emission from silicon nanosized domains represented a major breakthrough [2]. The origin of photoluminescence emission can be associated to both the quantum confinement effect [2] and/or to surface effects that increase with the surface-to-volume ratio of nanocrystals [3]. Besides being used as the building blocks for several photonic devices [4], silicon nanoparticles (SiNPs) have been recently proposed as ideal candidates for substituting for fluorescent dyes in biological analysis and fluorescence imaging techniques, thanks to their tunable fluorescence emission, high photoluminescence quantum efficiency and stability against photobleaching [5]. Furthermore SiNPs have been used as fluorescent biological staining labels [5], as a luminescent label to DNA [6] and even as biodegradable nanoparticles for in vivo imaging and drug delivery applications [7]. In this scenario, the careful control of the position and the contactless manipulation of SiNPs during biological analysis is extremely interesting for potential 0957-4484/11/505704+08$33.00

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we show how optical trapping of nanoparticles is improved by using radially polarized laser beams.

trapping against Brownian motion is often achieved by exploiting a resonant enhancement of the radiation forces that arises from the nanoparticle properties, e.g. in metal nanoparticles through the occurrence of plasmon resonances [17–19] or in quantum dots through their inherent quantum resonant trapping [20]. OT have also been used to manipulate, rotate and assemble a variety of nanostructures, such as carbon nanotubes [21–24], nanowires [25, 26], polymer nanofibers [27], graphene flakes dispersed in water [28] and for the calibration of nanotools [29, 30]. Polarization plays an important role in shaping the trapping potential. First, the tight focusing of a linearly polarized trapping laser yields an intrinsic anisotropy of the focal spot in the plane orthogonal to light propagation [31]. This can be exploited to orient and rotate micro- and nanoparticles such as microdiscs [32], gold nanorods [24], graphene [28] and particles with shape [33] or optical [34] anisotropy that generate polarization torques. Similarly the optical trapping of particles in an optically anisotropic surrounding medium, such as a liquid-crystal fluid [35], is now gaining increased attention because of the resulting polarization-dependent interaction of the focused laser beam with colloidal inclusions, defects and structures of long-range molecular order, that opens perspectives for novel non-contact optical control. Recently, great interest has also been devoted to the peculiar characteristics of radially and azimuthally polarized cylindrical vector beams [36–39]. These beams satisfy the Maxwell equations with cylindrical boundary conditions both in amplitude and polarization, i.e. their intensity profile and polarization patterns are cylindrically symmetric with respect to the optical axis. The electric field of radially and azimuthal polarized beams is locally linear but describe a complete rotation on a trajectory about the optical axis. As the polarization direction is thus not defined on the optical axis (there is a polarization singularity), such beams have a dark spot in the center that is enclosed by a bright ring [36]. Moreover, at variance with ordinary Gaussian beams, when a radially polarized beam is tightly focused by a high numerical aperture lens, a strong longitudinal component of the electric field, which may be even larger than the maximum intensity of the transverse field, is created on axis [37]. Thus, a strong gradient force is present not only in a transverse direction with respect to the optical axis, but also in the axial direction. This fact, together with the fact that the flux of the Poynting vector along the optical axis is identically zero and so the scattering force in this direction vanishes [36], increases the axial trapping efficiency of optical tweezers based on radially polarized beams. This is likely to be of benefit particularly for trapping nanoparticles, since rather than exploiting a nanoparticle property to enhance the forces, we now can exploit the focusing property of the radially polarized beam. In this paper we show that optical forces can be used to trap silicon nanoparticles obtained by a ball-milling process from porous silicon samples. Particles below 200 nm diameter are stably manipulated with optical tweezers. The samples show a photoluminescence in the red that can be exploited for labeling applications in biological systems. Furthermore

2. Experimental details SiNP samples have been obtained after a multi-step procedure. First, crystalline p-doped Si has been electrochemically etched for 25 min in 30% hydrofluoric acid (HF) using a current density of approximately 40 mA cm−2 . The estimated porosity is approximately 50%. Afterward, the porous layer has been made free-standing [40]. Third, the free-standing porous sample has been pulverized by 2 h mechanical ball-milling. Powder samples have been studied by means of scanning electron microscopy (SEM), Raman spectroscopy and photoluminescence (PL). The Raman scattering measurements have been carried out at room temperature with a Jobin Yvon Ramanor U1000 double monochromator, equipped with an Olympus BX40 microscope for micro-Raman analysis, and with an electrically cooled Hamamatsu R943-02 photomultiplier for photon-counting detection. The 514.5 nm (2.41 eV) and 457.9 nm (2.71 eV) lines of an Ar+ ion laser (Coherent Innova 70) have been used to excite Raman scattering. The laser beam has been focused to a diameter of approximately 100 μm. Care has been taken to minimize heating of the sample by choosing low laser power (below 2.5 mW at its surface). Room-temperature photoluminescence measurements (λexc = 457.9 nm) have been carried out by means of the same apparatus used for Raman spectroscopy. Before being tested in the optical tweezers set-up, SiNPs have been dispersed in water with a surfactant (sodium dodecyl sulfate) and sonicated to separate any agglomerate. Then the whole dispersion was filtered to ensure that the particle diameter was below 200 nm. The optical tweezers apparatus (see the sketch in figure 1) has as its central element an inverted microscope [28]. Optical trapping is achieved by focusing a laser beam on the sample, through an oil immersion high numerical aperture (NA = 1.3) objective (Olympus UPLFLN 100X). The beam from a laser diode with wavelength in the near-infrared (830 nm) is collimated by an aspherical lens and circularized by a pair of anamorphic prisms. A telescope is used to enlarge the beam and overfill the aperture of the objective in order to achieve a diffraction-limited spot; thus the high intensity gradient within the focal spot traps the particles. The samples are loaded on a slide with a small round sample chamber with a volume of approximately 80 μl mounted on a piezoelectric translator which has a displacement resolution equal to 1 nm (Physik Instrumente). A CCD camera is used to view and record images of the trapped particles via the same objective lens that is used for optical trapping. Optical force measurements and calibration of optical tweezers are obtained through interferometry of forward- and unscattered laser light in the back focal plane of the condenser of the microscope (figure 1). The focal plane of the condenser is projected by a collection lens onto a four-quadrant photodiode (QPD) [41] and the analog outputs from each quadrants are combined to generate signals proportional to the spatial displacements x, y, z of 2

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Figure 1. (a) Optical tweezers set-up. An infrared (830 nm) beam from a laser diode is circularized through a pair of anamorphic prisms and magnified through a telescope to overfill the back aperture of an oil immersion objective (NA = 1.3) mounted on an inverted microscope. This results in a diffraction-limited spot that creates an optical trap in a sample chamber. The radial polarizer is inserted into the beam path to convert linearly polarized light to radial. The light scattered by the trapped particles is collected by the condenser lens and projected onto a quadrant photodiode in order to get the particle tracking signals. Images of the trapped particles are obtained by a CCD camera. (b) SEM micrograph of Si powder obtained by ball-milling. The nanoparticles appear to be spherical in shape. (c) Distribution of particle sizes obtained from SEM measurements showing an average size of 155 ± 36 nm.

have been carried out to assess the light-emission property of the material. The analysis of SEM micrographs revealed that the particles are roughly spherical with an average diameter of approximately 155 nm (figure 1(b)) and a standard deviation of 36 nm estimated from the size distribution shown in figure 1(c). Extensive Raman measurements have been carried out on the SiNPs. Raman spectroscopy is a powerful nondestructive technique for the structural analysis of semiconductor materials as silicon. In crystalline silicon (c-Si) samples, only optical phonons at the center of the Brillouin zone (namely, having q ≈ 0 wavevector) may contribute to the Raman scattering process [44]. The Raman scattering from the threefold-degenerate optical phonon at q ≈ 0 gives rise to a single Lorentzian line at approximately 519 cm−1 from laser line, having a full width at half-maximum (FWHM) of ≈4 cm−1 at room temperature [45]. In silicon nanocrystals, the q ≈ 0 selection rule does not apply due to the loss of long-range order. Thus, not only phonons with zero wavevector, but also those with q = 0 take part in the Raman scattering process, resulting in a frequency downshift, broadening and asymmetry of bulk Si Raman line [46]. To interpret the effect of the reduced size of Si crystals on their Raman spectra, Campbell and Fauchet introduced the phonon confinement model [38]. For spherical

the trapped particle (particle tracking). Several procedures can be used to generate radially polarized light [36]. In our experimental set-up, a linearly polarized laser beam is switched to a radially polarized beam by means of a commercial liquidcrystal θ cell (ARCoptix) [42]. The trapping laser power during force measurements was about 20 mW at the sample when using linear polarized beams, while it was about 15 mW during measurements performed with the radial polarized beam because of the conversion efficiency of the liquid-crystal θ cell.

3. Results and discussion 3.1. Sample characterization In order to assess the structural and optical properties of the samples, several characterization techniques have been used. Scanning electron microscopy has been applied to the measurement of the average size of the particles as this is of crucial importance for the calculation of the trap force constants and for the reconstruction of the Brownian motion of the trapped particles. Raman spectroscopy has been used to evaluate the extent of the nanocrystalline domains by means of the well-known phonon confinement model of Campbell and Fauchet [43] and to highlight the possible presence of amorphous phases. Finally, photoluminescence measurements 3

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nanocrystals of diameter L , the intensity of the first-order Raman spectrum I (ω, L) is given by [43]

 I (ω, L) ∝

 2 2 q L d 3q exp − 16π 2 [ω(q) − ω]2 + ( 20 )2

(1)

where 0 is the natural linewidth for crystalline Si, ω(q) is the phonon dispersion relation [47], q is expressed in units of 2π/a and L in units of lattice constant a (a = 0.543 nm for c-Si). As demonstrated by SEM analysis, the ball-milled Si powder consists of nanograins having different sizes. Therefore, in order to take into account the effect of such a distribution in crystallite size on the Raman spectrum of the powder, we have adopted the crystallite size distribution (CSD) model that Islam and Kumar [48] proposed to calculate the Raman spectrum of porous silicon. According to this model, the intensity of first-order Raman spectrum for a single crystallite (equation (1)) is integrated over the appropriate CSD. Thus, if ϕ(L) represents a Gaussian distribution of crystallite size with mean size L 0 and the standard deviation σ , the total Raman intensity becomes [43]  I (ω) = ϕ(L)I  (ω, L) d L (2) where ϕ(L) is

  1 (L − L 0 )2 ϕ(L) = √ exp − . 2 σ2 2πσ 2 1

(3) Figure 2. (a) Raman spectrum (circles) of Si powder and fit of the data (solid line) according to the CSD model (λexc = 457.9 nm; laser power density = 40 mW mm−2 ). In the inset, the band due to amorphous Si phase in the sample is also indicated. (b) Photoluminescence spectra (laser power density = 300 mW mm−2 ) registered on the native 50% porous Si layer (orange line) and on the corresponding powder sample obtained by ball-milling. The SiNP powder shows a broader and more intense (about seven times) photoluminescence than the native p-Si layer due to both quantum confinement and surface effects.

The application of this model to a typical Raman spectrum of Si powder is shown in figure 2. It is worth noting that in very small systems heat dissipation is inefficient and, thus, sample heating is an important issue to be assessed during Raman measurements of powders [49]. For this reason, an accurate investigation of the value of laser power avoiding heating effects in our samples has been done. The very low laser power density of 40 mW mm−2 has been found not to perturbate the Raman response of our samples. The fit of the data has been carried out after the subtraction of a broad band at approximately 470 cm−1 and with an FWHM of 85 cm−1 (inset of figure 2) which is compatible with an amorphous Si phase [50] probably induced by the mechanical milling. After the fitting procedure, a size distribution of Si nanocrystallites with a mean value of about 13 nm and a standard deviation of 3 nm is estimated. The comparison with SEM data suggests that each particle is composed of several smaller nanocrystalline domains and amorphous phases. In order to further characterize the optical properties of the milled powders, extensive photoluminescence measurements have been carried out. Figure 2 (bottom) shows roomtemperature PL spectra (λexc = 457.9 nm, laser power density 300 mW mm−2 ) of the native 50% porous silicon sample (orange line) compared to the corresponding powder obtained by ball-milling (black line). The porous silicon sample exhibits a weak PL band centered at approximately 2.47 eV, whereas the powder shows a broad and intense (∼7 times larger) band extending from 1.6 to 2.6 eV. Since the

observation of photoluminescence in the visible from highly porous silicon by Canham [2], a variety of possible models have been proposed to individuate the origin of the visible PL. Quantum confinement increases the optical gap in Si nanocrystals and shifts PL from the near-IR into the visible range [2]. On the other hand, surface effects in porous Si layer [3] have also been proposed as a possible origin for visible luminescence emission. On the basis of these two phenomena, the results shown in figure 2 can be explained as follows. In a native 50% porous layer the weak PL emission observed could be associated with only the surface effects [3] and/or the presence of amorphous Si phases [51]. Quantum confinement effects are not considered here due to the low porosity of the sample [2]. In contrast, when the sample is ballmilled, both the confinement effect and the enhanced surfaceto-volume ratio with respect to the native layer could be the origin of the higher photoluminescence observed. Moreover the ball-milling process might induce a partial amorphization of the nanocrystalline structure [52], further contributing to the PL emission observed. 4

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3.2. Optical trapping of silicon nanoparticles A crucial element in force measurements with optical tweezers is Brownian motion. For optically trapped particles, position fluctuations due to Brownian motion are tracked by a QPD, appearing as randomly fluctuating signal voltages. The statistical analysis of these fluctuations enables force measurements [41, 53]. The starting point for optical tweezers calibration is the Langevin in a confining harmonic  1 equation 2 potential, V (x i ) = k x . For a spherical particle of 2 i i diameter d in the heavily overdamped regime applicable to trapped micro- and nanoparticles, inertial effects may be neglected and the equation for the particle dynamics can be written as d x i (t) = −ωi x i (t) + ξi (t), i = x, y, z dx

(4)

where the relaxation frequencies ωi = kγi are related to the force constants and hydrodynamic viscous damping γ = 3πηd, η being the dynamic viscosity of the surrounding liquid (water). The terms ξi (t) describe random uncorrelated fluctuations with zero mean, ξ(t) = 0, and temporal correlation related to particle diffusion, i.e. ξ(t + τ )ξ(t) = 2 kB T δ(τ ) [36, 48]. In order to extract the optical trap γ force constants, we rely on a time domain analysis of the experimental particle tracking signals where we consider the autocorrelation functions of the position fluctuations, Cii (τ ) = x i (t)x i (t + τ ) . As can be deduced from equation (4) they obey first-order uncoupled differential equations with the lag time τ that can be easily integrated giving exponential decays [22, 24, 28] with relaxation frequencies ωi = kγi and

zero-point value Cii (0) = kBkiT . In experiments, the tracking signals are obtained from the outputs of QPD and, thus, the autocorrelation functions of the voltage signals must be connected to the autocorrelation functions of the particle position through a calibration procedure. More specifically, the signals from the QPD yield the signal autocorrelation functions CiiV (τ ) = βi2 Cii (τ ) = βi2 Cii (0)e−ωi τ , where βi are calibration factors relating the QPD voltage to particle displacement. Thus, from the fit of CiiV (τ ) the relaxation  frequencies ωi and the calibration

Figure 3. (a)–(c) Normalized signal autocorrelation functions (logarithmic scale) in the x (a), y (b) and z (c) directions, respectively. These are obtained when trapping with a linearly (red) and radially (blue) polarized beam. The single exponential fits of the data (solid lines) are also reported. Note how the change in decay constants in the transverse plane ((a) and (b)) is mainly due to the drop of power at the sample when converting linear to radial polarization. However, despite the drop of power, the radial polarized beam generates a much stronger trapping force with respect to the linear polarization along the axial propagation direction (c) yielding an enhanced stability to the trapping process.

C V (0)k

i ii are obtained, allowing the parameters βi (V /m) = kB T measure of the force constants ki and the reconstruction of the Brownian motion of the particles in the trap. In figure 3, representative normalized autocorrelation functions of the particle displacement are shown. SiNPs have been trapped by linearly (red line) and radially (blue line) polarized laser beams. In both cases, the data are well fitted with a single exponential decay. Due to the presence of a broad distribution of shapes and sizes in our p-Si powder, several (tens of) trapping measurements have been made in order to obtain a good statistical analysis. In figure 4 the histograms for the spring constants in the x , y , z directions and under linearly polarized trapping beams are shown at two different depths in the sample. It is interesting to note that, at a depth of ∼15 μm, k x and k y have a distribution centered at ∼2.5 and ∼3.2 pN μm−1 , respectively, whereas k z ∼ 0.8 pN μm−1 is much smaller. On the other hand, at a depth of 6 μm we

measure k z ∼ 2.8 pN μm−1 that is closer to k x ∼ 3 pN μm−1 and k y ∼ 3.8 pN μm−1 . This weakening of the axial trap force is caused by the aberration of the focused laser beam when passing through the coverslip of the sample chamber [31]. For comparison, 2 μm diameter latex beads have been also trapped in the same set-up. In table 1, the force constants normalized to trapping power are summarized for both kinds of samples. It is interesting to note that both in linear and radial polarization the force constants measured when trapping SiNPs are smaller than those measured with latex beads, mainly because of their much smaller volume. However, the aspect k +k ratio of the trap (i.e. the ratio x2kz y between the mean lateral to the axial stiffness) is quite comparable. 5

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Figure 5. Temporal evolution of the Brownian motion ( yz projection) of an SiNP trapped with linearly (red) and radially (blue) polarized laser beam. Sy and Sz are the displacement from equilibrium in the y and z directions, respectively. The nanoparticle can explore more freely the z direction when the trapping beam has linear polarization, while the introduction of the radial polarizer produces a more isotropic trap.

the statistical analysis, average spring constants have been obtained both for latex beads and SiNPs. The axial trapping efficiency is enhanced with respect to linearly polarized light and the measured force constants are summarized in table 1. k +k The ratio between mean lateral and axial stiffness is x2kz y ≈ 0.8 for both kinds of particles, showing that the axial spring constant is slightly larger than the transverse one, in contrast to what is measured for linear polarization. Thus, the temporal evolution of the Brownian motion of exemplar SiNPs (figure 5) under both linear and radial polarized trapping beam clearly points out that the particle can more freely explore the z direction when the trapping beam has linear polarization, while the introduction of the radial polarizer produces a more isotropic trap. This experimental observation confirms the beneficial effect on the trapping efficiency of the strong z component of the focused light field of a radial polarized beam. Finally, we compare the values of the measured spring constants for SiNPs to a theoretical estimate calculated in the Rayleigh regime [10], i.e. when the size of the trapped particle is much smaller than the trapping wavelength. In this

Figure 4. Histograms of the values of the spring constants k x , k y and k z (linear polarization) when trapping SiNPs at different focus depths, depth 15 μm (data peaked at lower values of k ); depth 6 μm (data peaked at higher values of k ). The solid line is the Gaussian distribution having the same mean and standard deviation of the data. While in the transverse plane the change of depth has not a dramatic effect, we see a substantial decrease (about four times) in axial trapping when we increase the trapping depth.

We have also used a radial polarized beam for the optical trapping of SiNPs. In figure 3 the corresponding signal autocorrelation functions (blue triangles) are shown. From 6

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Table 1. Measured force constants k x , k y , k z divided by laser power at the sample P and trap aspect ratio under both linearly and radially polarized trapping beams.

Latex beads Linear polarization Radial polarization SiNPs Linear polarization Radial polarization

k x /P k y /P k z /P (pN μm−1 mW−1 ) (pN μm−1 mW−1 ) (pN μm−1 mW−1 )

kx +k y 2k z

1.35 ± 0.26 0.47 ± 0.1

1.06 ± 0.12 0.51 ± 0.1

0.42 ± 0.03 0.66 ± 0.13

3. 0 ± 0. 7 0. 7 ± 0. 3

0.13 ± 0.07 0.06 ± 0.03

0.16 ± 0.08 0.05 ± 0.03

0.04 ± 0.02 0.07 ± 0.04

3. 6 ± 2. 5 0. 8 ± 0. 6

approximation, the gradient force acting on the particle is [10]

Fgrad (r ) = where Re{α} = 4πε0

Re{α} ∇ E 02 4

 3  2  d m −1 2 m2 + 2

Acknowledgments

(5)

The authors acknowledge C Gentile of the Department of Physics, University of Messina for providing the SEM micrographs. MGD, PHJ, PGG and OMM acknowledge support from the Royal Society.

(6) n

is the real part of the particle polarizability α [10] and m = n mp is the ratio between the refractive indices of the particle (n p ≈ 3.7 for Si at NIR wavelengths) and the medium (n m ≈ 1.33 for water), respectively. For the calculation of the gradient force at the focus (z = 0), we consider a Gaussian beam with a λ (diffraction-limited spot) and beam waist radius w0 ≈ 0.6 NA

References [1] Jalali B and Fathpour S 2006 Silicon photonics J. Lightwave Technol. 24 4600 [2] Canham L T 1990 Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers Appl. Phys. Lett. 57 1046 [3] Kanemitsu Y 1994 Luminescence properties of nanometer-sized Si crystallites: core and surface states Phys. Rev. B 49 16845 [4] Daldosso N and Pavesi L 2009 Nanosilicon photonics Laser Photon. Rev. 3 508 [5] Li Z F and Ruckenstein E 2004 Water-soluble poly(acrylic acid) grafted luminescent silicon nanoparticles and their use as fluorescent biological staining labels Nano Lett. 4 1463 [6] Wang L, Reipa V and Blasic J 2004 Silicon nanoparticles as a luminescent label to DNA Bioconjug. Chem. 15 409 [7] Park J-H, Gu L, von Maltzahn G, Ruoslahti E, Bhatia S N and Sailor M J 2009 Biodegradable luminescent porous silicon nanoparticles for in vivo applications Nature Mater. 8 331 [8] Ashkin A 1970 Acceleration and trapping of particles by radiation pressure Phys. Rev. Lett. 24 156 [9] Ashkin A, Dziedzic J M, Bjorkholm J E and Chu S 1986 Observation of a single-beam gradient force optical trap for dielectric particles Opt. Lett. 11 288 [10] Jonas A and Zemanek P 2008 Light at work: the use of optical forces for particle manipulation, sorting, and analysis Electrophoresis 29 4813 [11] Ashkin A, Dziedzic J M and Yamane T 1987 Optical trapping and manipulation of single cells using infrared laser beams Nature 330 769 [12] Wang M D, Schnitzer M J, Yin H, Landick R, Gelles J and Block S M 1998 Force and velocity measured for single molecules of RNA polymerase Science 282 902 [13] De Luca A C, Rusciano G, Ciancia R, Martinelli V, Pesce G, Rotoli B, Selvaggi L and Sasso A 2008 Spectroscopical and mechanical characterization of normal and thalassemic red blood cells by Raman tweezers Opt. Express 16 7943–57 [14] Liu Y, Cheng D K, Sonek G J, Berns M W, Chapman C F and Tromberg B J 1995 Evidence for localized cell heating induced by infrared optical tweezers Biophys. J. 68 2137 [15] Dholakia K, Reece P and Gu M 2007 Optical micromanipulation Chem. Soc. Rev. 37 42–55 [16] Preece D, Warren R, Evans R M L, Gibson G M, Padgett M J, Cooper J M and Tassieri M 2011 Optical tweezers: wideband microrheology J. Opt. 13 044022

π w2

Rayleigh length z R = λ 0 [10]. Thus we obtain the transverse spring constant kr by linearizing the average force in the radial direction:  3  2  d m −1 Re{α}E 02 16 P kr = = (7) m2 + 2 w02 cn m w04 2 where we have used the relation E 02 = π w24nP ε c between the 0 m 0 electric field amplitude and the laser power P [10]. Thus, by using the parameters of our experiment, we estimate a transverse spring constant of about 10 pN μm−1 , which is in fair agreement with the measured value of about 3 pN μm−1 on SiNPs. The discrepancy can be accounted by the fact that we have not considered any effect from the scattering force that pushes the particles away from the focal spot, hence weakening the trap.

4. Conclusions In conclusion, optical trapping of Si nanoparticles, obtained by ball-milling of porous Si layers, has been demonstrated. They have been trapped with both linearly and radially polarized laser beams, aiming at comparing the trapping efficiencies in both cases. We have found that trapping is possible in both configurations and that, while radial polarization enables higher trapping efficiencies in the propagation direction, the conversion efficiency of the liquid-crystal device used to create these beams reduces the available power, yielding smaller trapping efficiencies in the transverse plane of the trap. The Raman and photoluminescence analysis of the p-Si powder shows that these samples are fluorescent. The combination of fluorescence excitation and optical trapping open perspectives for both manipulation and labeling applications in biological analysis. 7

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