LETTER pubs.acs.org/NanoLett
Ultrafast Vibrations of Gold Nanorings T. A. Kelf,† Y. Tanaka,† O. Matsuda,† E. M. Larsson,‡ D. S. Sutherland,§ and O. B. Wright*,† †
Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo, 060-8628, Japan Department of Applied Physics, Chalmers University of Technology, S-41296, G€oteborg, Sweden § iNANO Center, University of Aarhus, Aarhus 8000, Denmark ‡
bS Supporting Information ABSTRACT: We investigate the vibrational modes of gold nanorings on a silica substrate with an ultrafast optical technique. By comparison with numerical simulations, we identify several resonances in the gigahertz range associated with axially symmetric deformations of the nanoring and substrate. We elucidate the corresponding mode shapes and find that the substrate plays an important role in determining the mode damping. This study demonstrates the need for a plasmonic nano-optics approach to understand the optical excitation and detection mechanisms for the vibrations of plasmonic nanostructures. KEYWORDS: Nanostructure, vibrations, gigahertz, ultrafast, ring, plasmonics
A
wide variety of experiments have been carried out on the vibrational properties of metal, semiconductor, and insulating nanoparticles and nano-objects. Research involving ultrafast or Raman spectroscopy has been reported for both regular arrays18 and irregular ensembles,919 for example. Shapes such as wires,1,19 tubes,20 rods,12,13,16,18 spirals,7 spheres or ellipsoids,6,10,11,21 shells, boxes or cages,4,15,22 cubes,2,14 triangles,5 disks or dots,3,8 prisms,17,23 and pyramids5 have been investigated with frequencies in the gigahertz range. In addition, experiments have been performed on single nano-objects such as spheres,2426 prisms,23 cubes,27 or tubes,20 allowing a more accurate measurement of intrinsic damping. Research in this area is fuelled by fundamental interest in the dependence of vibrational resonance frequencies on shape, in the damping or dephasing mechanisms, or in periodic nanostructures with tailored acoustic dispersion relations. To our knowledge no such investigations have been performed on nanoring structures. Research on metallic nanorings, for example, has concentrated on their fascinating plasmonic properties.2833 Light confined in such nanorings leads to surface plasmon modes and enhanced electric fields that can be tailored by the choice of the ring dimensions, and such rings have been proposed for applications in chemical sensing,3439 waveguiding,40 data storage,41 quantum information processing,42 optical antennas,43 optical confinement,4447 and lasing.48 Here we study experimentally the vibrations of suboptical-wavelength gold nanorings on a silica substrate using an ultrafast optical technique. We report a number of gigahertz resonances and compare them with the results of numerical simulations. The corresponding mode shapes are elucidated, and we find that the substrate plays an important role in determining the damping of the modes. The nanorings are produced using a colloidal lithographic approach.28 Briefly, sulfate functionalized polystyrene (PS) nanospheres are electrostatically self-assembled onto a silica r 2011 American Chemical Society
substrate of thickness 1 mm, and a thin gold film is then evaporated at normal incidence. An argon ion beam is then used to sputter the gold until it is removed from the interstitial regions between the spheres. During the sputtering process some of the gold is redeposited underneath the PS spheres where it is protected from the Ar beam, forming gold nanorings. Finally the PS spheres are removed by an oxidative ultravioletozone treatment leaving the free-standing gold nanorings. Sample characterization is performed using scanning electron microscopy (SEM) and atomic force microscopy (AFM), as shown in Figure 1. (These images were taken from a sample grown under the same conditions as the present sample with the same geometry.) These images indicate that the nanorings are uniformly distributed over the surface with an density of ∼13 μm2. The nanorings have an outer diameter of 120 nm, a height of 35 nm, and a wall thickness of 10 nm (with approximately (15% variations in each of these). To investigate the vibrational properties of the nanorings, we use a two-color pumpprobe technique. Light from a modelocked Ti:Sapphire laser with a central wavelength of 804 nm, a pulse duration of 200 fs, and a repetition rate of 80 MHz excites the samples. The pump beam passes through an acousto-optic modulator working at frequency 1.1 MHz before being focused onto the sample with linear polarization at normal incidence through a 50 objective (with a numerical aperture (NA) of 0.45) to a 2 μm diameter (full width at half-maximum intensity (fwhm)) spot with an average fluence ∼20 mJ/cm2. The probe beam, which is frequency-doubled to a wavelength of 402 nm, passes through a delay line and is then split into two beams, probe and reference. The probe is then focused with linear polarization Received: June 18, 2011 Revised: August 18, 2011 Published: August 23, 2011 3893
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Figure 1. (a) AFM image of a single gold nanoring on a silica substrate. The data has been clipped to show the cross section. (b,c) Normal and oblique angle SEM images of the sample.
(in the same plane as the pump) through the same objective to a 1 μm diameter (fwhm) spot, and the reflected probe light is detected using a balanced photodiode where it is compared with the intensity of the reference beam. With these spot sizes ∼12 nanorings are effectively sampled. A lock-in amplifier is used to detect the variations in probe reflectivity R (δR/R ∼ 104) of the probe light at the acousto-optic modulation frequency as a function of the delay time between the pump and the probe pulses. The optical absorption depths of gold at the pump and probe wavelengths are 13 and 16 nm, respectively,49 indicating that the nanorings are thick enough for efficient optical excitation and probing. The inset in Figure 2 shows a representative time scan for the relative reflectivity change (solid line). At t = 0, defined as the point when the pump pulse arrives, a spike in the relative intensity variation is observed. This is caused by the excitation of hot electrons within the gold. This effect may be enhanced owing to the excitation of surface plasmons around the nanoring, allowing a greater coupling to the incident optical field. The use of linearly polarized pump light in the present study leads to a nonaxially-symmetric optical energy absorption profile.28 The optical excitation wavelength in this study is in fact off the main nanoring plasmon resonance arising from a dipolar excitation, that was measured to be at 1270 nm by white light spectroscopy (see Supporting Information). The hot electrons diffuse ∼100 nm, and thermalize with the gold lattice in ∼500 fs50,51 by excitation of lattice phonons, causing a transient temperature rise ∼100 K. (The average surface temperature rise is also ∼100 K, as determined by the ring distribution, the laser repetition rate, the optical spot size and the substrate thermal conductivity (see ref 52).) This ultrafast electron diffusion and subsequent thermal diffusion in gold are expected to produce an approximately axially symmetric transient thermal stress distribution on the time scales (∼100 ps) of interest. This causes the nanoring to vibrate in an axially symmetric vibrational mode and strain pulses to be launched into the substrate.53 Strains of amplitude ∼104 are excited, leading to displacements in the picometer range. These strain pulses and ring vibrations lead to oscillatory timedependent reflectivity changes, visible in the inset of Figure 2 over the measured time interval of 1.4 ns. The modulus of the
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Figure 2. Modulus of the temporal Fourier transform of the experimental reflectivity variation δR(t) for gold nanorings on a silica substrate. The shaded left-hand set of peaks represent the gold nanoring vibrational modes, whereas the shaded right-hand peak is a Brillouin peak. Inset: the solid red line shows the experimental relative reflectivity change as a function of delay time and the dashed blue line a fit based on a sum of exponentially decaying sinusoidal waves (including the background variation).
temporal Fourier transform (FT), shown in Figure 2, is evaluated after subtracting the thermal background variation. (The background is removed by subtracting a fitted double-exponential function. The spike at t = 0 is removed by setting data points in its vicinity to zero. In addition, the data before t = 0 is padded with zeros so as to make zero time the center point in the data sequence.) Our measured time interval allows frequencies above ∼1 GHz to be resolved. The Fourier spectrum shows a high frequency peak as well as lower frequency peaks. The high frequency peak at 43.6 GHz is a Brillouin peak and depends on the interference between light reflected from the substrate or ring surface and from propagating longitudinal strain pulses in the substrate.53 Since a number (∼10) of nanorings are excited and probed, one can assume that a single plane longitudinal strain pulse effectively propagates normal to the substrate surface. In that case the oscillation frequency of the probe reflectivity is given fB ¼
2vl ðn2 sin2 θÞ1=2 λ
ð1Þ
where vl and n are the longitudinal sound velocity and refractive index of the substrate, respectively, λ is the probe wavelength, and θ is the angle of probe incidence. For the normal incidence measurements presented here, and making use of the literature values of vl = 5970 m/s and n = 1.47 (refs 53 and 49, respectively), the predicted oscillation frequency is fB = 43.7 GHz, which is in good agreement with experiment. Experiments at varying θ also showed agreement with eq 1, suggesting that the single plane acoustic wave approximation is reasonable. The damping associated with this Brillouin peak depends mainly on the ultrasonic attenuation in the silica substrate.54 The lower frequency peaks in the 515 GHz range in Figure 2 arise from the vibrations of the nanorings. The solid line in Figure 3a shows the modulus of the temporal Fourier transform in this frequency range, characterized by four main peaks. In order to extract the experimental frequencies, Q-factors, and phases for each resonance, the experimental time-domain data was fitted to a set of five damped sinusoidal waves with four frequencies close to the four main lower-frequency peaks and one close to the Brillouin peak, but with arbitrary phase, amplitude, and damping, and these five parameters were adjusted for each resonance so as to minimize the mean absolute 3894
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Figure 3. (a) Experimental (solid line) and simulated (dashed line) modulus of the temporal Fourier transform (FT) of δR(t) for the nanoring oscillations plotted vs frequency up to 20 GHz. The theoretical curve is obtained from an average of the modulus of the FT of the vertical displacement Uz over the top of the nanoring. The bracketed numbers represent the modes identified in the numerical simulation. Downward pointing arrows are fitted resonant frequencies. The scales of the two curves are normalized. The scales are arbitrarily adjusted. (b) Dependence of the five simulated peak frequencies identified below ∼20 GHz as a function of ring outer diameter D for a constant ring height and thickness. (c) The same as (b) but plotted against 1/D.
Table 1. Values for the Parameters Used to Fit the Experimental Data for the Reflectivity Variation to a Sum of Damped Sinusoidal Waves: (iiv) Refer to the Four Main LowerFrequency Peaks, whereas “Bulk” Refers to the Brillouin Peaka mode
(i)
(ii)
(iii)
(iv)
bulk
12.4
13.6
43.6
frequency (GHz)
5.5
8.5
relative amplitude
0.36
0.09
0.11
1.0
2.8 0.11
11.6 0.18
9.3 0.33
1.1 0.74
Q-factor phase (ψ/π)
0.17 128 0.91
a
The Q-factor for this latter peak depends on the attenuation of bulk longitudinal waves in the substrate.
difference between the data and the fitting function δRðtÞ ¼
5
∑ Ai cosð2πfi t þ ψi Þexpð γi tÞ i¼1
ð2Þ
where γi = 2πfi/Qi and Qi is the corresponding Q-factor. Table 1 shows the resulting fitted parameters obtained for each of the experimentally detected modes. The fitted frequencies, indicated by the downward-pointing arrows in Figure 3a, correspond closely to the experimentally measured peak positions, and the fit in the time domain, shown by the dashed line in the inset of Figure 2, gives good agreement with that observed. This fitting procedure is subject to errors of up to ∼10% (with the phase compared to 2π) for the extracted parameters for a given data set, but variations between different data sets were approximately twice as large presumably owing to variations in the nanoring geometry. The observed Q-factors are not only determined by the damping of the nanoring vibration but also by inhomogeneous broadening owing to the distributions in nanoring dimensions. Simulations, described below, suggest that the latter contribution significantly affects the peak widths of the observed vibrational modes. The phase lead ψ for each of the modes is shown in Table 1 in units of π. If the excitation of thermal stress is considered instantaneous on a time scale compared to the oscillation period, the nanorings should start oscillating from a point of maximum displacement, yielding a cosine variation in displacement.55,56 Varying degrees of phase lead or lag are in fact observed. It has been noted for nanoparticles that the phase depends on the electronlattice energy transfer time,55,10
related to the time it takes for the particle to heat up and reach the point of maximum expansion. However these time scales are of the order of a few picoseconds, and so should have little influence over the nanoring vibrational modes we observe with oscillation periods ∼100 ps. Another mechanism must be involved, such as the extra time taken for thermal diffusion in the substrate, since in our case the nanorings are coupled thermally and mechanically to the substrate. Deviations from a cosine variation were also observed in ultrafast vibrations of metal nanoshells.15 The sign of ψ also depends on the details of the optomechanical detection mechanism, such as, for example the sign and magnitude of the photoelastic constants involved. In order to elucidate the experimental results, we conducted a set of finite-difference time-domain simulations57 with rings of rectangular cross section. An initial temporal Gaussian impulsive force with temporal variation exp(π2Δ2t2)cos(2πf0t), where f0 = 60 GHz and Δ = 30 GHz, was applied uniformly and vertically downward over the top surface of a single ring, preserving circular symmetry. This function results in a frequency spectrum of the nanoring vibrations in approximate agreement with experiment (see Figure 2), although the exact effect of the optical excitation is not reproduced. (The impulsive force function of duration ∼20 ps is short compared to the period ∼100 ps of the structural vibrations of the nanorings observed. Therefore the precise form of this function does not significantly influence the predicted frequency spectrum in the range 515 GHz.) Because of this approximation we will not discuss the predictions for the phase ψ. The silica substrate was chosen to be a 300 300 300 nm3 cube, and absorbing boundary conditions were used at the sides and on the bottom surface.58 The mesh size was 1.3 nm3, the time step was 0.104 ps, and the total time T of the simulation was either 2.65 or 6.8 ns for full displacement field or for ring top displacement acquisition, respectively. The dimensions of the polycrystalline gold nanoring are outer diameter 120 nm, height 35 nm, and annular thickness 10 nm, matching the estimated dimensions in experiment. Literature values of the densities and elastic constants were used. (For Au, density F = 19320 kg/m3, Young’s modulus E = 81.2 GPa, Poisson’s ratio ν = 0.24. For silica F = 2220 kg/m3, E = 72.2 GPa, Poisson’s ratio ν = 0.17 (see ref 59).) The modulus of the temporal Fourier transform of the simulated average vertical (z-directed) displacement Uz of the ring top surface (obtained for T = 6.8 ns) is shown by the dashed line in Figure 3a. Five main resonances could be resolved, labeled (15) in Figure 3a, over the same range as the peaks observed in experiment. (The simulation predicts peaks above 20 GHz, but 3895
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Figure 4. (a) Vertical (xz) cross-sectional maps for the modulus of the temporal Fourier transform of the simulated displacement Uz in the z-direction and of Ux in the x-direction for modes (15). (b) Corresponding maps for the phase of the temporal Fourier transform. (c) Mesh images showing the motion of the nanoring cross section for each mode, scaled arbitrarily. The two images at each frequency correspond to deformations at the times of maximum displacement.
these were not resolved in experiment, as discussed later in the paper and in the Supporting Information.) Peak (2) dominates the simulated Fourier spectrum for Uz, and this results in the other resonances showing up as peaks superimposed on a background composed of the wings of peak (2). (This peak does not appear in the Fourier spectrum for the radial displacement Ur of the ring. Subsidiary peaks not evident in Uz were found in Ur at 5.8 and 15.8 GHz, very close to modes (1) and (5) (see Supporting Information).) The simulated mode frequencies obtained from Uz at 6.1, 8.4, 11.7, 14.0, and 16.2 GHz are in the same range as the experimental peaks. Discrepancies with the experimentally observed frequencies are presumably partly due to the more complex geometry of the nanoring cross section, evident in Figure 1c, that stems from the use of spheres in the templating step. As discussed later, the relative heights of the peaks in the experiment are expected to be different from that of the spectrum of the vertical displacement Uz. Figure 4a shows xz plane maps of the vertical and horizontal (x-directed) displacements Uz and Ux for frequencies close to the five main resonances of the simulation (obtained for T = 2.65 ns). Figure 4b shows the corresponding phase, and (c) shows the resulting deformations. The two images for each frequency in (c) correspond to the times of maximum displacement for which the oscillation phase differs by 180°. The mode shapes are best seen in animation in the Supporting Information. Mode (1) is the lowest-order ring flexural mode, basically a breathing-like ring mode, characterized by the nanoring expanding and contracting radially. Relatively little substrate motion is associated with this mode (as is evident in the animation). Mode (2) is the mode associated with bodily motion of the ring in the vertical direction, and this engenders a significant amount of vertical substrate motion (involving the ηzz strain component), both inside and outside the ring. (The relatively large damping of mode (2) implies that some of the deformation of this mode must be present at the frequencies corresponding to the other modes.) This mode is heavily damped by acoustic radiation to the bulk. Acoustic radiation is best seen in the animations of Uz, also included in the Supporting Information. The frequency of mode (2) is in accord with simple estimates based on the mass of the ring and the stiffness of the substrate. (Taking the stiffness of the substrate to be of the order of EA1/2 (see ref 60), where E is the Young’s modulus of the substrate and A is the cross-sectional area of the ring in a horizontal plane, the estimated frequency is ∼7 GHz.) Mode (3) is the second flexural mode. It engenders a significant amount of lateral strain (ηxx and ηyy) in the substrate both inside and outside the ring. The flexural modes (1) and (3)
show frequencies not dissimilar to that of a gold cantilever of rectangular cross section with one end fixed and of the same width and height as the nanoring. (The first two frequencies for the fixed-end gold cantilever system are 2.7 and 16.9 GHz (see ref 61) compared to 6.1 and 11.7 GHz for the ring modes (1) and (3)). Modes (4) and (5), showing up as much smaller resonances in Figure 3, are formed from a combination of the second order flexural mode and mainly z-directed substrate motion. Further insight into the nature of the modes was obtained by varying the ring diameter in the simulation. The variations of the five identified modes as a function of the outer ring diameter D and its reciprocal 1/D are shown in Figure 3b and c, respectively, for 50 nm < D < 120 nm. (Further details of these simulations are given in the Supporting Information.) It is immediately obvious that the mode (2) frequency is independent of the ring diameter. This is not surprising because this mode depends on the vertical motion of the ring as a whole, and so on reducing the diameter the mass and effective spring constant from the substrate decrease by approximately the same factor provided that D . w, where w is the 10 nm wall thickness. The variation with frequency of mode (1), approximately proportional to 1/D, is indicative of a breathing mode.12,19 Modes (35) show a somewhat less pronounced D dependence. Because of the mechanical coupling to the substrate, one expects significant losses by acoustic radiation of both bulk and surface waves. Indeed, these effects are clearly visible in the cross sections in Figure 4. Apart from mode (2), all modes were found in the simulation to have Q-factors greater than 20. In contrast, Mode (2) is strongly damped with an amplitude decay time of ∼70 ps (Q ∼ 2). This mode is particularly heavily damped because of the significant amount of vertical substrate motion that couples the ring motion to the acoustic far field. (A very rough estimate of the Q-factor can be made by comparison with the theory of radiation from an acoustic piston to a fluid (see ref 62). For mode (2), ka ∼ 1, where a = D/2 and k is the acoustic wavenumber for longitudinal waves in the substrate, and in this case the acoustic power P radiated to the silica substrate can be approximated to P ∼ FvlAZ2ω2, where F is the substrate density, vl is its longitudinal sound velocity, A = πa2, Z is the vertical vibrational amplitude of the ring (treated as incompressible), and ω is the resonance angular frequency. For an average stored energy per cycle of mω2Z2/2, where m is the ring mass, this leads to Q = mω/(2πFvla2) ∼ 0.5 of the same order of magnitude as found in the simulation. This simple model also predicts an approximate Q µ 1/D dependence for 50 nm < D < 120 nm for 3896
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Nano Letters this resonance, which was also found in the simulation (see Supporting Information).) Modes (13) may correspond to the first three largest peaks observed in experiment. If this is the case, it is not clear why the second experimental peak is sharper than in the simulation, that is, why this mode is not so strongly damped in experiment. As with the discrepancies in some mode frequencies, this could be caused by the nonideal experimental ring geometry or by imperfect ring-substrate adhesion. Another possibility is that a broad peak similar to mode (2) is present in experiment at a similar frequency but with a lower amplitude, for example, ∼2 times smaller compared to the height of this peak in Figure 3a. The influence of inhomogeneous broadening is presumably responsible for the absence of high Q-factors in experiment. The acoustic radiation for all 5 modes is particularly evident in animations of the z component of the displacement when plotted on a color scale (see Supporting Information). In order to understand the relative amplitudes and absolute phases of the detected oscillation peaks, the coupling of both the pump and probe light to the nanoring vibrations should be known in detail. Concerning the excitation, the detailed spatiotemporal absorption of the pump light energy in the nanoring, governed by plasmonic interactions, together with the resulting electron diffusion, thermal diffusion, and thermal expansion should be calculated. Concerning the detection, the coupling of the reflected probe light optical mode to the oscillating nanorings, also governed by plasmonic interactions, and to the substrate should be calculated; this coupling comes in principle from two sources in this experiment, the photoelastic effect that couples normally incident probe light to strain components ηxx, ηyy and ηzz (arising from the dependence of the refractive indices on these components), and the effect of the nanoring or substrate motion (that can lead to light being scattered out of the probe collection optics). (Non-normally incident light arising from scattering by the ring and further scattered back to the detector may also couple to shear strain components.) One could, for example, speculate that vibrational mode (1), mainly involving ring flexing, might have a significant contribution to the modulation of the probe light from the ring motion, whereas mode (2), with a strong substrate deformation, might have a significant contribution from the substrate photoelastic effect. One clue to the nature of the detection process is the absence of detected structural vibrations above ∼20 GHz. If the photoelastic effect in gold were a dominant mechanism for detection, we would expect to see higher-order modes, for example, at ∼50 GHz or above, corresponding to, for example, longitudinal acoustic resonances in the gold in the z direction (see Supporting Information). It therefore appears likely that the photoelastic effect in the silica substrate plays an important role in detection; the negligible coupling of the probe light to these higher-order modes of acoustic wavelength smaller than ∼100 nm (significantly shorter than the optical probe wavelength) might stem from a canceling of the effect of positive and negative strains on the probe light within the region of the substrate illuminated by the probe beam. The detailed three-dimensional plasmonic nano-optics involved in these excitation and detection mechanisms would be an interesting project for numerical simulation but is beyond the scope of this paper. In conclusion, we have investigated the vibrational properties of gold nanorings and compared the experimental frequency spectrum to that of simulations. These simulations show that all the oscillations are damped by acoustic radiation to the substrate,
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as expected from this type of geometry of a structure attached to a substrate. The modulation of the reflected probe light comes from a combination of the photoelastic effect and sample motion. Even for a nanostructure with a generic shape like a sphere, a rigorous calculation of the effects on the probe light field of the photoelastic modulation and nanostructure deformation during vibrational motion has not been reported. A detailed understanding of the optical detection mechanism for nanorings or for yet more complex structures such a combined phononic and photonic or plasmonic crystal will require a careful consideration of the three-dimensional deformed geometry. Experiments with different ring sizes, materials, or angles of optical incidence63 should make it easier to experimentally pin down the plasmonic contributions to the observed probe light modulation and would allow our predictions for the resonance frequency variations with ring size to be tested. In addition, experiments on single nanorings would also be useful to allow the damping mechanisms to be more accurately probed. The current samples have random ring spacings, and so the mechanical coupling between adjacent nanorings should not produce sharp features in the vibrational spectrum. However, for periodic arrays, or arrangements of nearly touching nanostructures, coupling between nanorings should become important.13,36,37,40 With judicious choice of geometry, including the possible use of concentric ring or disk structures,64 it may prove feasible to greatly enhance the acousto-plasmonic interaction, thereby opening the way to the development of novel ultrafast modulators based on this concept.
’ ASSOCIATED CONTENT
bS
Supporting Information. We include movies of the simulated sample deformations and Uz components of the displacement at the five lowest resonance frequencies. Experimental results for white light spectroscopy and further details of the simulations are also provided. 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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