Optical Epitaxial Growth of Gold Nanoparticle Arrays - Nano Letters

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Optical Epitaxial Growth of Gold Nanoparticle Arrays Ningfeng Huang1, Luis Javier Martínez1†, Eric Jaquay1‡, Aiichiro Nakano2, and Michelle L. Povinelli1 1

Ming Hsieh Department of Electrical Engineering, University of Southern California,

Los Angeles, California 90089, USA 2

Department of Physics & Astronomy, University of Southern California, Los Angeles,

California 90089, USA

Abstract: We use an optical analogue of epitaxial growth to assemble gold nanoparticles into 2D arrays. Particles are attracted to a growth template via optical forces and interact through optical binding. Competition between effects determines the final particle arrangements. We use a Monte Carlo model to design a template that favors growth of hexagonal particle arrays. We experimentally demonstrate growth of a highly stable array of fifty, 200-nm-diameter gold particles with spacing of 1.1 m.

Keywords: Optical trapping; gold nanoparticles; self-assembly; photonic crystal; optical binding; epitaxial growth

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Optical fields induce mechanical forces on matter, capable of pulling objects into precise positions1-3. Single-particle optical traps based on this principle have been demonstrated for biological cells4-7, dielectric particles8,

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and metal particles10-12 and

have widespread application in physics and biology. For small particles, the optical force on a particle is proportional to the gradient of the light intensity. Arrays of optical traps can be created by varying the optical field intensity in space, creating a periodic optical potential (Fig. 1a). Holographic optical traps13, interference optical lattices14, 15, microoptics arrays16 and microphotonic near-field approaches17-19 have all been used to trap arrays of objects. These techniques suggest the exciting possibility of on-demand assembly of photonic matter “from the bottom up.” Nano- and micro- scale constituents such as dielectric particles20, particle-molecule complexes18, biological cells16,

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and

microlenses22 have been assembled into arrays to achieve various functionalities. Although initial attempts have been made to assemble metal particles23,

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, optical

trapping of a 2D periodic array has not been achieved. Periodic metallic nanoparticle arrays have unique and tunable optical properties with applications in Raman amplification25, sensing26 and lasing27, and the ability to assemble such structures optically would enable far-reaching dynamic control. Metallic particles interact strongly through coherent light scattering, complicating optical trapping techniques. Scattering produces interparticle forces, a phenomenon known as optical binding14. Optical binding (Fig. 1b) drives the formation of certain welldefined particle arrangements within an optical trap14, 28-31; for extended, multi-particle patterns, the arrangements are in general aperiodic. To assemble metallic particles in a periodic optical trap array, the optical binding effect between particles in different


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trapping sites must be considered. In recent experiments, we used a photonic crystal slab to create a 2D periodic trap array17, 32 (Fig. 1c). While polystrene nanoparticles trap at every site in the array17,

gold nanoparticles trap in one-dimensional chains23. This

difference can be attributed to stronger optical binding in the metallic nanoparticle system, which competes with trapping forces to prevent the filling of every trap site. However, because previous work on optical binding has not studied binding effects within multiple traps, no general theory is available to explain or predict the observed behavior. Here we suggest that the behavior of strongly interacting particles within a periodic array of optical traps can be viewed as an optical analogue to epitaxial growth. The array serves as the growth substrate, with individual particles able to trap at any site. Particles also interact with one another via optical binding. Both effects determine the final arrangement of particles. To grow particular structures of interest, one must determine how to design the substrate so that optical trapping and optical binding forces cooperate, rather than compete. We introduce a Monte Carlo model to study the energetics of growth. For square lattices, we find that 1D particle chains form perpendicular to the incident light polarization, in agreement with previous experiments23. For hexagonal lattices, we predict that a 2D particle array will be formed. Drawing upon this prediction, we design and fabricate a photonic crystal template with hexagonal symmetry. Using the template, we demonstrate low power optical trapping of a 2D periodic array of over fifty gold nanoparticles with spacing comparable to the wavelength. Due to optical binding interactions, the stability of the 2D array is much greater than for a single, trapped particle. To our knowledge, this is the first experimental demonstration of a periodic array of gold nanoparticles via optical trapping methods.


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Figure 1 | Pattern formation due to optical trapping and optical binding. (a) A periodic optical potential traps weakly interacting particles at intensity maxima, due to the optical gradient force. (b) Optical binding causes strongly interacting particles to arrange themselves into well-defined patterns, even in a nearly uniform light field. (c) Competition between a periodic optical potential and optical binding can give rise to pattern formation, preventing particles from trapping at every site. Here, particles form 1-D chains.

In our model, we assume that each hole in the photonic-crystal lattice is a possible single-particle trapping site. Square and hexagonal lattices are shown in Fig. 2a. Due to interparticle interactions, different configurations of trapped particles have different energies. Low-energy particle configurations will have the greatest stability. The energy shift of particle i induced by all other particles can be written (see Supplementary Information) as





1 U i   Re  i   Re E*i G  R, k   j E j 2 j i ,

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to particle j, k is the wave vector, and G(R, k ) is the dyadic Green’s function. We approximate the Green’s function by that of a uniform dielectric slab. The numerical approach is described in the Supplementary Information. The interaction between particles described by Equation 1 is long range and oscillatory. We use a Monte Carlo approach to identify low-energy configurations. We begin by randomly placing the particles at trapping sites. The energy shift for the particle ensemble ( U total   U i ) is calculated from Equation 1. One particle is then selected at random. i

The particle is either moved to one of the nearest neighbor sites or released from the surface, with each possible option having equal probability. If a particle is released from the surface or hops outside of the device region, it is relocated at a random empty site. If new old the total energy of the new configuration U total is lower than the old one U total , the

new configuration will be unconditionally accepted. Otherwise, the new configuration





new old will be accepted with the probability exp   U total  U total  / kBT , where kB is the

Boltzmann constant and T is the temperature. The average energy shift per particle is plotted in Fig. 2b as a function of lattice constant, for both the square and hexagonal arrays. The energy strongly depends on the lattice constant. Several local minima (marked with arrows) can be seen. For the hexagonal lattice, there is a pronounced dip when the lattice constant is similar to the wavelength (a/ ~ 1). Particle configurations corresponding to each of the local minima in Fig. 2b are plotted in Figs. 2c-h. The configurations change dramatically with lattice type and lattice -5ACS Paragon Plus Environment

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constant. For the square template (Figs. 2c-e), we observe vertical chains with different spacings. The simulation result agrees well with our previously reported experiments23, in which we observed chain formation on a lattice with a/ ~ 0.94. In both simulation and experiments, the chains are oriented perpendicular to the polarization direction. For the hexagonal template, oblique chains are observed at a/ = 0.85 and a/ = 0.90. However, at the lowest-energy local minimum (a/ = 1.00), we observe a close-packed array, or cluster. We have verified that an array is also formed for a wide range of lattice constants nearby, from ~0.94 to 1.1. The results suggest that for appropriate choice of lattice type and lattice constant, interparticle interactions act to stabilize the array, which is the lowest energy particle configuration of any shown in Fig. 2b. For the Monte Carlo simulation shown in Fig. 2, we note that particle size plays an important part in determining the assembled particle pattern. This is due to fact that the optical trapping and optical binding potentials scale differently with particle polarizability. We expect that pattern formation will be more difficult to observe for smaller particle sizes. A detailed discussion of this point may be found in the Supplementary Information.


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Figure 2 | Strong interaction-induced pattern formation during optical epitaxial growth. (a) Square (left) and hexagonal (right) lattices, each with lattice constant a. (b) Monte-Carlo simulation results of energy shift per particle as a function of lattice constant. The solid lines are guides to the eye. Energy is plotted in arbitrary units. (c-e) Particle configurations obtained for square lattices with varying lattice constants. (f-h) Particle configurations for hexagonal lattices with varying lattice constants.

Following the prediction of the simulation, we designed a photonic crystal slab with hexagonal symmetry and a/ ~ 1 and performed optical epitaxial growth experiments. The lattice constant was 1166 nm, and the laser wavelength in vacuum was 1550 nm; adjusting for the refractive index of water, a/ = 0.99. Figure 3 shows a series of optical microscope images from a typical experiment. The diameter of the gold nanoparticles is 200 nm, and a red arrow indicates incident light polarization. When the laser power was turned on, particles immediately began to trap near the slab (Figs. 3a,b). The trapping sites are not close-packed, and release and re-trapping of particles were frequently observed. As time passed, the trapping area became smaller and the cluster became more filled-in (Figs. 3c-f). Particles also appeared to be trapped more stably. After 40 minutes, a highly stable, close-packed array of nanoparticles was formed. When the laser is turned -7ACS Paragon Plus Environment

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off, the particles released immediately and diffused back into solution (Figs. 3g,h). This process can be observed in Supplementary Video 1 and 2. The final array shows evidence of extremely strong optical trapping. Movement of particles within each trapping site was negligible when seen by the naked eye. Analysis of video data for the experiment found that the stiffness of an individual trap was greater than 82.5 pN/nm/W, two orders of magnitude higher than values reported in previous experiments on polystyrene particles17. The total power used in the experiment was only 30 mW, for a maximum power of 85 W per optical trap. The power per trap is orders of magnitude lower than for previously reported metallic nanoparticle traps10-12, 29, 30, 33. In particular, it is around 2,000 times lower than for standard optical tweezers, given similar particle size11. a

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Figure 3 | Optical epitaxial growth of a gold particle array. Optical microscope images show 200-nm diameter gold nanoparticles trapped on the photonic crystal slab, visible in the background. (a-f), Snapshots taken with the laser power on; elapsed time is shown below image. (g,h), Snapshots taken with laser power off. The scale bar indicates 5 m.


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To further understand the energetics of the growth process, we calculated the energy shift as a function of time. Particle positions were extracted from the video, and the energy shift was calculated from Equation 1. Figures 4a-c show the energy shift for each particle at three different snapshots in the growth process, corresponding to the microscope images in Figs. 3d-f. At early times (Fig. 4a), the particles sparsely attached to the substrate over a large area, and the energy was high (light red). Some oblique chains were formed, and the particles at the center of the chains had lower energy. As time passed (Fig. 4b), the trapping area became smaller and the cluster became more filled-in. Meanwhile, the energy was greatly reduced (dark red), especially for particles at the center of the cluster (Fig. 4c). This result correlates with the observation of greater trapping stability as the experiment proceeded. The average energy shift per particle is plotted in Fig. 4d, which confirms that the formation of the nanoparticle array was accompanied by minimization of the energy. (An animated version of Fig. 4 is available as Supplementary Video 3.)


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Figure 4 | Energy lowering during the growth process. (a-c) The energy shift of each particle. (d) Average energy shift per particle as a function of time.

We note that the time required for growth of an array is limited by diffusion of particles to the trapping region. Increasing the particle concentration in solution and/or using flow both reduce the assembly time. A detailed discussion can be found in the Supplementary Information. We have performed an experiment in a microfluidic flow channel using higher particle concentration and flow that yielded a 120-particle array in six minutes. In summary, we have studied a system in which strong optical binding interactions between particles drive pattern formation within a periodic optical trapping potential. Monte Carlo simulations predict that patterns such as 1-D chains and 2-D arrays may be formed, depending on the lattice constant and lattice symmetry. Using this prediction, we successfully demonstrate growth of a periodic, hexagonal array of closely-spaced gold nanoparticles on top of a photonic crystal template. In this system, the optical force of the


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template on the particle cooperates with the optical binding interaction to produce a pattern that is energetically favorable and highly stable. The analogy between our system and traditional epitaxial growth suggests various strategies for improving the quality of the photonic material. Optimization of the “growth temperature” (laser intensity) and growth rate (particle delivery rate), along with the use of annealing, may increase the size and quality of the particle array. An expanded range of complex materials could be grown using different types of nanoparticles (dielectric, metal, semiconductor, magnetic) as building blocks, offering a wide range of valuable physical properties. Moreover, the ability to observe the optical epitaxial growth process directly in both space and time suggests that our system could form a versatile model for studying interaction-driven dynamics in microscopic systems, such as cold atoms in optical lattices34.


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Methods Monte Carlo simulation parameters. The simulations used a 50 m diameter area lattice with fixed 5% site coverage. The unperturbed field values were set assuming horizontal polarization, constant phase at all trapping sites, and overall amplitude given by a two-dimensional Gaussian envelope. The total incident power is assumed to be 30 mW, similar to experiment. The electric field intensity on each trapped site is assumed to be 50× higher than the incident plane wave, based on electromagnetic simulations (see Supplementary Information). Each simulation is run for 5 million steps. An initial transient is observed in the energy shift, corresponding to loss of memory of the initial configuration. The energy then fluctuates around an equilibrium value. The average energy shift is calculated over the last 4.5 million steps. Photonic crystal template design. The holes in the photonic crystal slab are arranged in hybrid triangular-graphite lattice35, which supports low-dispersion photonic bands with high quality factors. The spacing between the larger holes is chosen to be 1.166 m, and the diameter of the larger and smaller holes are 156 nm and 110 nm, respectively. The thickness of the slab is 250 nm. (See also Supplementary Information.) Device fabrication and sample preparation. The device was fabricated using standard electron beam lithography and inductively-coupled plasma reactive ion etching (ICPRIE). We used a silicon-on-insulator wafer (SOITEC) with a 250 nm thick silicon layer on top of a 3 m silica layer. Full details are provided elsewhere17. A 500 nm thick PDMS microfluidic chamber was fabricated on a glass slide one inch in diameter. The square chamber was 5 mm on a side. The chamber was used to cover on top the photonic -12ACS Paragon Plus Environment

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crystal device and seal 200 nm diameter gold particles colloidal solution (Sigma-Aldrich) inside. Experimental setup. A laser with a tuning range from 1500 nm to 1620 nm was connected to an erbium-doped fiber amplifier (EDFA), collimated by a lens (f = 11 mm), and focused by a second lens (f = 30 mm) onto the photonic crystal slab device from the back side. Polarization of incident laser light was controlled by polarizers and half wave plates. A 20× objective was used in conjunction with a beam splitter to collect the light from the top surface and to image the particle motion on a CMOS camera. Prior to each experiment, the transmission was measured in cross-polarization mode, and the wavelength of the transmission peak was identified. The laser was then tuned to the peak wavelength to carry out the trapping experiment. Stiffness analysis. After the assembly of a particle array, we recorded a video with exposure time of 1/30 s at 15 frames per second. The video is 1000 frames in length. Each pixel on the video represents a 70 nm × 70 nm area of the sample. The coarse particle positions were obtained using the scale-space blob detection algorithm36 and refined by calculating the brightness weighted centroid proposed by Crocker and Grier37, enabling subpixel positional accuracy. We took statistics of the variance in particle position across the trap and used the equipartition theorem to estimate the stiffness of our traps as described in reference17. The peak power per trap is estimated to be 85 W.


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Associated Content Supporting Information. Details on modeling and photonic-crystal template design. Additional video clips showing the assembly processes. This material is available free of


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charge via the Internet at http://pubs.acs.org.

Author Information Corresponding Author *E-mail: (M.L.P.) [email protected] Present Address †

Laboratoire Aimé Cotton, ENS, CNRS UMR 9188, Université Paris-Sud, 91405 Orsay,

France. ‡

nanoPrecision products, 411 Coral Circle, El Segundo, California 90245, USA

Competing financial interests The authors declare no competing financial interests.

Acknowledgements This work is funded by the Army Research Office PECASE Award under Grant 56801MS-PCS. Computation was supported by the University of Southern California Center for High Performance Computing and Computation. Imaging capabilities were provided by the Center for Electron Microscopy and Microanalysis.

Table of Content Graphics


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Figure 1 | Pattern formation due to optical trapping and optical binding. (a) A periodic optical potential traps weakly interacting particles at intensity maxima, due to the optical gradient force. (b) Optical binding causes strongly interacting particles to arrange themselves into well-defined patterns, even in a nearly uniform light field. (c) Competition between a periodic optical potential and optical binding can give rise to pattern formation. Figure 2 | Strong interaction-induced pattern formation during optical epitaxial growth. (a) Square (left) and hexagonal (right) lattices, each with lattice constant a. (b) Monte-Carlo simulation results of energy shift per particle as a function of lattice constant. The solid lines are guides to the eye. The energy is in arbitrary unit. (c-e) Particle configurations obtained for square lattices with varying lattice constants. (f-i) Particle configurations for hexagonal lattices with varying lattice constants. Figure 3 | Optical epitaxial growth of a gold particle array. Optical microscope images show 200-nm diameter gold nanoparticles trapped on the photonic crystal slab, visible in the background. (a-f), Snapshots taken with the laser power on; elapsed time is shown below image. (g,h), Snapshots taken with laser power off. The scale bar indicates 5 m. Figure 4 | Energy lowering during the growth process. (a-c) The energy shift of each particle. (d) Average energy shift per particle as a function of time.


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d

10 min.

0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0

20 min.

ACS Paragon Plus Environment 10 20 30

Time (min.)

-1.3

30 min.

40

Optical Epitaxial Growth of Gold Nanoparticle Arrays - Nano Letters

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