Understanding How Acoustic Vibrations Modulate the Optical

Understanding How Acoustic Vibrations Modulate the Optical Response of Plasmonic Metal Nanoparticles Aftab Ahmed,*,†,§ Matthew Pelton,‡,§ and Jeffrey R. Guest*,§ †

Department of Electrical Engineering, California State University Long Beach, Long Beach, California 90840, United States Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, United States § Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States ‡

S Supporting Information *

ABSTRACT: Measurements of acoustic vibrations in nanoparticles provide an opportunity to study mechanical phenomena at nanometer length scales and picosecond time scales. Vibrations in noble-metal nanoparticles have attracted particular attention because they couple to plasmon resonances in the nanoparticles, leading to strong modulation of optical absorption and scattering. There are three mechanisms that transduce the mechanical oscillations into changes in the plasmon resonance: (1) changes in the nanoparticle geometry, (2) changes in electron density due to changes in the nanoparticle volume, and (3) changes in the interband transition energies due to compression/expansion of the nanoparticle (deformation potential). These mechanisms have been studied in the past to explain the origin of the experimental signals; however, a thorough quantitative connection between the coupling of phonon and plasmon modes has not yet been made, and the separate contribution of each coupling mechanism has not yet been quantified. Here, we present a numerical method to quantitatively determine the coupling between vibrational and plasmon modes in noble-metal nanoparticles of arbitrary geometries and apply it to silver and gold spheres, shells, rods, and cubes in the context of time-resolved measurements. We separately determine the parts of the optical response that are due to shape changes, changes in electron density, and changes in deformation potential. We further show that coupling is, in general, strongest when the regions of largest electric field (plasmon mode) and largest displacement (phonon mode) overlap. These results clarify reported experimental results and should help guide future experiments and potential applications. KEYWORDS: nanoparticle, acoustic vibrations, plasmons, transient absorption, phonons, optical response

M

means of detecting the transient phonon oscillations. Energy from an ultrafast pump laser pulse is absorbed by electrons in the nanoparticle and transferred to the lattice within a few picoseconds, resulting in a rapid thermal expansion of the particle. This expansion, in turn, excites coherent acoustic oscillations, or low-frequency phonon modes, that result in a periodic variation in shape and size of the nanoparticle. The strong optical absorption of the plasmonic metal nanoparticles and the rapid heating of the lattice enable efficient excitation of the acoustic modes; high-sensitivity detection of the vibrations is enabled by the sensitivity of the plasmon resonances to changes in the size and shape of the particle, resulting in modulation of the transmission of a probe laser pulse. There are three mechanisms that transduce the mechanical oscillations

easurements of acoustic vibrations in metal nanoparticles have revealed fundamental mechanical properties of the nanoparticles1,2 and have been used as a probe of the mechanical properties of the local fluid environment, revealing phenomena such as the viscoelastic response of simple liquids at high frequencies.3,4 Monitoring the vibrational frequency of the nanoparticles as molecules adsorb on their surface has the potential to enable mass-based sensing in a variety of environments with high sensitivity,5,6 complementing other metal nanoparticle chemical sensing techniques based on enhanced Raman scattering7 or plasmon frequency shifts.8−10 The majority of experimental measurements of nanoparticle vibrations have employed noble-metal nanoparticles because the optical spectra of these particles exhibit strong plasmon resonances whose frequencies depend on the sizes and shapes of the nanoparticles.11 Femtosecond transient absorption (TA) spectroscopy can thus be employed as a sensitive, noncontact © 2017 American Chemical Society

Received: July 8, 2017 Accepted: August 17, 2017 Published: August 17, 2017 9360

DOI: 10.1021/acsnano.7b04789 ACS Nano 2017, 11, 9360−9369

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determined corresponding to the maximum compression and maximum expansions for the various vibrational modes. Finally, the optical response of the deformed geometries are calculated using the FDTD method. We separately calculate the contributions to the overall shift in plasmon resonance due to shape deformation, local changes in ED, and local DP. Calculation details regarding simulating deformed geometries and calculation of permittivity based on local effects are given in the Methods section, with further details in the Supporting Information (SI). Experimentally, the initial expansion of the particle is proportional to the energy in the pump pulse, with greater expansions resulting in greater amplitudes of oscillation for each of the normal modes. It is expected that a larger amplitude of deformation will result in a larger plasmon shift. However, a quantitative relation showing the dependence of plasmon shift on amplitude of deformation has not been reported. Here, we investigate this relation by calculating the optical response corresponding to deformation amplitudes of 1, 0.5, and 0.25 nm for the cases of gold and silver nanospheres as well as gold nanorods. Our calculations show a linear relation between plasmon shift and deformation amplitude, as discussed below. This allows for the use of an arbitrary deformation amplitude in the FDTD calculations; the plasmon shift corresponding to any given vibrational amplitude can then be calculated by linear interpolation of the simulation results. Conversely, the vibrational amplitudes in a TA experiment can be estimated by analyzing the observed shift in plasmon frequency. In our calculations, we use an amplitude of 1 nm for the dominant vibrational mode, allowing the calculations to be performed without requiring very small mesh sizes, and thus reducing the required computational resources. It should be noted that the phonon modes are normalized to the amplitude of the dominant mode, and thus the amplitudes of the remaining modes are smaller than 1 nm. Nanospheres. Vibrational Modes of Nanospheres. Spherical metal nanoparticles have relatively simple plasmonic and vibrational mode structures that can be solved analytically due to their high degree of symmetry and thus serve as a useful test case to validate our numerical simulations. Vibrations of an elastic, isotropic sphere can be calculated by solving Navier’s equation. The eigenmode solutions correspond to the series of spherical harmonics; however, for the uniform initial stress that results from isotropic heating, only spheroidal modes with zero angular momentum are excited.27 The displacement vector u is given by u = ar ̂ d/dr[j0(ωnr/cl)], where j0 is the spherical Bessel function of the first kind, ar ̂ is the radial unit vector, and cl is the longitudinal speed of sound in the particle.39,40 The eigenfrequencies ωn can be approximated analytically as ωn(L,n) = χncl/R, where R is the radius of the sphere and χn is the nth solution of χncot(χn) = 1 − (χncl/2ct)2, where ct is the transverse speed of sound in the particle.39 The mode shapes of the fundamental breathing mode (n = 0) and the first higherorder radial mode (n = 1) of a 60 nm silver nanosphere are shown in the inset of Figure 1a. Figure 1b compares the analytical (solid line) and numerical results (circles) for the absolute displacement as a function of position within the sphere for these two modes (fundamental breathing mode normalized to 1 nm); the results are in excellent agreement. The above equations predict resonance frequencies of 55 and 122 GHz for the two modes of a 60 nm silver sphere, in good agreement with the numerically calculated values of 56 and 122

into changes in the plasmon resonance frequency: (1) changes in shape during oscillation (shape effect); (2) changes in electron density (ED) due to changes in nanoparticle volume (volume effect); and (3) changes in interband transition energies through the deformation potential (DP). Experimental work has mainly focused on the frequency shift1,12−14 and damping15−17 of the vibrations, whereas vibrational amplitudes have received less attention. Spherical nanoparticles have been treated in detail12,13,18 due to the availability of analytical solutions for these geometries. The vibrational modes of spherical particles were first computed by Lamb,19 who found that the period of the symmetric breathing mode is proportional to the radius of the particle. This has been experimentally verified for gold nanoparticles in water with sizes ranging from 8 to 120 nm.20 Crut et al. presented a semianalytical method to study the vibrational modes of core−shell spherical nanoparticles.21 The contribution of various acousto−plasmon coupling mechanisms, particularly DP, to Raman scattering has been studied analytically for spherical nanoparticles.18 Recently, mechanical coupling in gold nanoparticles was studied, revealing ultra-lowfrequency Raman scattering from a dimer configuration when excited near the plasmon resonance frequency.22 Finite-element method calculations have been used to determine the vibrational modes in particles without spherical symmetry.23 In most cases, only mechanical simulations were performed, and plasmon−vibration coupling strengths and mechanisms were not considered. Only a limited number of studies reported both optical and mechanical calculations; however, these studieswhich revealed the role of “hot spots” and focused primarily on Raman scatteringdealt with either only one geometry or only one coupling mechanism.24,25 Raman and TA measurements provide complementary information, as they involve detection of different vibrational modes. As the origins of TA signals remain relatively unexplored, we focus here on acousto−plasmon coupling in TA experiments. Specifically, we perform numerical calculations to quantitatively determine the coupling between vibrational and plasmon modes in noble-metal nanoparticles. We integrate finiteelement (FE) calculations of the mechanical response with finite-difference time-domain (FDTD) calculations of the optical response in order to determine the optical signals that are expected for different vibrational modes. We focus on four commonly investigated nanoparticle geometries: spheres, 1,13,16,20,26−28 rods, 16,17,29−33 shells, 34,35 and cubes.1,36−38 We separately determine the parts of the optical response that are due to shape changes, volume changes (changes in local ED), and changes in interband transition energies (DP). Although we present results that are relevant to time-resolved measurements, this method can be readily extended to include asymmetric vibrational modes that are Raman-active. These results illuminate reported experimental results and will help guide the optimization of future experiments and guide design for potential applications.

RESULTS Calculations. We calculate the frequencies and mode shapes of acoustic vibrations using FE simulations, using the bulk mechanical properties of silver and gold. The initial condition for the oscillations is taken to be an isotropic expansion of the particle, corresponding to uniform heating by an incident laser pulse. Deformed geometries are then 9361


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the nanoparticle vibration is thus a shift of the frequency of the plasmon resonance, as seen in Figure 1c, with the largest contribution coming from DP. The relation between the optical response and deformation amplitude is investigated by calculating the plasmon shift of the fundamental breathing mode of silver and gold nanospheres for varying deformation amplitudes, considering all three coupling mechanisms. Results for a silver nanosphere are shown in Figure 2; results for a gold nanosphere are given in Figure S4 of

Figure 1. Calculation results for a 60 nm silver sphere. (a) Absolute surface displacement as a function of frequency. Isotropic radial vibrational modes (fundamental and higher-order breathing modes) are shown in the inset. The mode shape shows the absolute local displacement of the deformed nanoparticle from the equilibrium position. The vibrational amplitudes are normalized to reduce computational resources and are large compared to experimentally accessible values (see text); however, linear acousto−plasmon coupling allows for the estimation of plasmon shift in the limit of small deformations. (b) Absolute normalized displacement as a function of position within the sphere for the two modes. Solid lines represent the theoretical results, and circles and squares are the simulation results for the fundamental and higherorder modes, respectively. (c,d) Net change in extinction cross section, including combined contributions from shape, electron density (ED), and deformation potential (DP) mechanisms, for the fundamental (c) and higher-order breathing modes (d). The relative contribution of each coupling mechanism is presented in the Supporting Information. Extinction spectra for expanded, nondeformed, and contracted nanospheres are shown in the lower panels of (c,d) by dashed-dotted blue, solid black, and dashed red curves, respectively. The change in extinction cross section (δσext) is plotted in the top panels. The acoustic modes in the insets of (c) and (d) use the same color scales as in (a).

Figure 2. Calculation results for a 60 nm silver sphere showing plasmon resonance shift for varying phonon vibrational amplitudes. (a) Changes in extinction cross section for vibrational amplitude of 1 nm (green dots), 0.5 nm (dashed-dotted red), 0.25 nm (dashed blue), and 0 nm (solid black). (b) Plasmon resonance wavelength as a function of deformation amplitude. Blue squares represent the calculated values, and the dashed red line is a linear fit.

the SI. A simple interpolation algorithm is discussed in the SI along with an example showing the interpolated optical response corresponding to vibrational amplitudes of 125 and 250 pm (see Figure S5). The higher-order radial mode results in a small change in the volume of the sphere, reflected by a surface displacement of about one-fifth of the fundamental mode (Figure 1a), and experimental evidence for this mode has been obtained only through detailed analysis of transient absorption data.26 Further details about the optical response of nanospheres and other geometries studied in this work are provided in the SI, where the contributions from all three coupling mechanisms are separately presented; a summary is presented in Figure 7. A TA experiment measures the difference between the spectrum of the deformed nanoparticle and the undeformed nanoparticle as a function of time as the particle undergoes acoustic vibrations. A TA measurement on a vibrating silver nanosphere is thus predicted to show strong modulation when the probe wavelength is detuned half a line width away from the initial plasmon resonance in either direction, as shown in Figure 1c. The modulation on the blue side of the resonance is 180° out of phase with the modulation on the red side of the resonance, and the modulation passes through zero near the plasmon resonance frequency. Changes in line width result from the different optical absorptivity of silver at the plasmon resonance as the plasmon resonance frequency oscillates and are predicted to be small. All of these numerical predictions are consistent with experimental results.27 The mechanical properties of silver and gold are very similar, resulting in nearly identical vibrational modes, as shown in

GHz, respectively. The analytical solutions thus validate the accuracy of our numerical calculations. Optical Response of Vibrating Nanospheres. The optical response of a spherical metal nanoparticle is dominated by the lowest-order dipolar plasmon mode. In the quasi-static limit, when the sphere diameter is small compared to the optical wavelength, the frequency of this mode is given by the condition11 ϵm(ω) + 2ϵs = 0

(1)

where ϵm and ϵs are the dielectric functions of the metal and the surroundings, respectively. The frequency of the plasmon mode is independent of the size of the particle in this limit, so the radial modes that are excited by heating are not expected to result in a shift in the plasmon frequency due to the shape effect only. We have verified this by solving Maxwell’s equations directly in the FDTD calculations, without employing the quasi-static approximation. The extinction cross section of the particle, on the other hand, is sensitive to the particle volume;11 we see a modulation in the extinction cross section (with negligible peak shift) as the particle becomes larger and smaller. However, the increase in cross section from the size change is almost completely canceled by ED and DP effects. The dominant overall effect of 9362


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Figure 4. Calculation results for a 55 nm long and 26 nm wide gold nanorod (left panels) and a 61 nm long and 22 nm wide nanorod (right panels). (a,d) Displacement of the rod end (solid green) and of the midpoint (dashed magenta) of the nanorod as a function of frequency. Mode shapes are also shown along with a circle and arrow, indicating the point on the rod’s surface where displacement is measured. (b,e) Changes in extinction cross section in response to the extensional vibrational mode. (c,f) Changes in extinction cross section in response to the higher-order vibrational mode.

Figure 3. Calculation results for a 60 nm gold sphere. (a) Displacement of a point on the surface as a function of frequency. Isotropic radial modes (fundamental and higher-order breathing modes) are shown in the inset. The mode shape shows the absolute local displacement of the deformed nanoparticle from the equilibrium position. (b,c) Changes in extinction cross section for the two vibrational modes.

calculations suggest that the lowest-order mode is indeed primarily an extensional mode, the shape of the higher frequency mode does not correspond exclusively to a breathing mode; instead it is a combination of the fundamental (n = 0) breathing and the higher-order (n = 2) extensional modes, due to the proximity of their resonant frequencies.29 For the small aspect ratio of 2.1 shown in Figure 4a, the highfrequency acoustic mode is excited with an amplitude comparable to that of the extensional mode. However, as shown in Figure 4d, this high-frequency mode gets weaker as the aspect ratio increases, consistent with experimental findings.29 Optical Response of Vibrating Nanorods. Similar to the vibrational response, the plasmonic response of the nanorod splits into a lower-frequency (long-wavelength) longitudinal mode and a higher-frequency (short-wavelength) transverse mode. For the longitudinal plasmon, the spatial mode overlaps well with the regions of largest displacement for the extensional acoustic mode, resulting in efficient modulation of the plasmon by this vibration, as shown in Figure 4b,e. This interplay is primarily due to changes in the aspect ratio of the nanorod. The longitudinal plasmon mode can be thought of as a Fabry−Perot (FP) resonance for surface plasmon polaritons propagating along a metal nanowire and reflected from the two ends of the rod. The resulting standing-wave plasmon mode, with a node at the center of the rod, is sensitive to changes in length; the extensional vibrational mode thus produces a significant shift in the longitudinal plasmon resonance.

Figure 3a. On the other hand, gold has stronger optical absorption than silver at wavelengths around 500 nm, corresponding to the blue edge of the spherical nanoparticle plasmon resonance. The result is that the extinction cross section of the particle undergoes significant modulation as the plasmon resonance passes into and out of wavelength regions where gold is strongly absorbing. The calculated change in extinction is significant on the red side of the plasmon resonance, where optical absorption of gold is relatively weak, but is much smaller on the blue side of the resonance, where the absorption of gold is stronger; this is shown in Figure 3b and is consistent with experimental observations.41 The higherorder mode again does not induce large changes in the extinction cross section (Figure 3c). Nanorods. Vibrational Modes of Nanorods. Whereas the high symmetry of nanospheres means that the vibrational and plasmon mode structures of the particles and the coupling between vibrations and plasmons are relatively straightforward, cylindrically symmetric nanorods manifest a more complex coupling.42 Specifically, the two lowest-order vibrational modes of a nanorod are the fundamental extensional mode, which primarily affects the aspect ratio (length/width) of the rod, and a higher-frequency mode that primarily affects the nanorod volume. Figure 4a illustrates these two modes for a 55 nm long and 26 nm wide gold nanorod. The calculated mode frequencies of 19 and 102 GHz agree well with the theoretical predictions of ref 29, which suggests 18.4 and 96 GHz for the extensional and breathing modes, respectively. Whereas our 9363


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ACS Nano For gold and silver nanospheres, the plasmon shift as a function of the amplitude of the extensional mode of the gold nanorod shows a linear relation, as shown in the SI (Figure S6). The effect of the oscillating aspect ratio is a periodic modulation of the plasmon resonance frequency as it is for the breathing modes of nanospheres. However, the longitudinal plasmon frequency occurs at wavelengths where optical absorption in the bulk metal is weak; the optical response is thus predicted to be symmetric (but 180° out of phase) for the red and blue sides of the plasmon resonance, as observed experimentally.29 Because the metal absorption changes little as the plasmon resonance changes, the line width of the plasmon is predicted to be virtually unaffected by the acoustic oscillations; this is again consistent with experiment.30 For the higher-frequency acoustic mode, expansion of the nanorod reduces its aspect ratio, resulting in a blue shift of the plasmon resonance when only shape effects are considered. However, the expansion also significantly increases the volume of the nanorod, modifying the ED and the DP, and thereby shifting the plasmon resonance toward the red (Figures S9 and S10 and Figure 4c,f). The overall optical signal is a balance between these competing effects and is thus sensitive to the size and aspect ratio of the nanoparticles. A non-negligible modulation of the longitudinal plasmon resonance due to the breathing mode is predicted for gold nanorods with lengths of 55 nm and widths of 26 nm.29 Such a modulation has been observed, together with the larger modulation due to the extensional mode, in single-particle transient absorption measurements on gold nanorods.16,17,31 The corresponding signal was absent in ensemble measurements,30 but this may have been a result of rapid dephasing of the breathing mode signal due to inhomogeneities in the sample. Both the extensional and breathing acoustic modes also modulate the transverse plasmon mode of the gold nanorod, as shown in Figures S9 and S10. The effect is negligible, however, for the case of gold, due to strong metal absorption at the transverse plasmon wavelength. In the case of silver, coupling of the extensional vibrational mode to the transverse plasmon mode is calculated to be an order of magnitude weaker than its coupling to the longitudinal plasmon mode. The breathing mode of the silver rod couples to the transverse plasmon mode with about 1/3 of the strength of its coupling to the longitudinal plasmon mode. Optical measurements to date have therefore focused nearly exclusively on the longitudinal plasmon mode. For comparison, Figure S11 shows vibrational as well as optical results for a silver nanorod; because of the large spectral gap between the interband transition frequencies and plasmon resonance of silver nanorods, DP effects are negligible. Nanoshells. Vibrational Modes of Nanoshells. The acoustic vibrational modes of a nanoshell can be understood in terms of coupling between motion of the inner and outer surfaces. The vibrational frequencies thus depend on the outer radius, Rout, and on the radius ratio (Rin/Rout), where Rin is the inner radius. Figure 5 shows calculation results for the two lowest-order vibrational modes for two cases from ref 35: (i) Rout = 40 nm, radius ratio of 0.75 (Figure 5a), (ii) Rout = 25 nm, radius ratio of 0.6 (Figure 5d). In both cases, the initial isotropic stress in the nanoparticle predominantly excites the lowest-order symmetric mode, where the inner and outer surfaces of the shell are displaced in phase. The higher-frequency antisymmetric vibrational mode is excited to a lesser degree, corresponding

Figure 5. Calculation results for a gold nanoshell with inner radius of 30 nm and outer radius of 40 nm (left panels), and a nanoshell with inner radius of 15 nm and outer radius of 25 nm (right panels). (a,d) Radial displacement of the inner surface (solid green) and outer surface (dashed magenta) of the nanoshell as a function of frequency. The isotropic radial phonon modes are also shown along with a circle and arrow indicating the points where displacement is measured. (b,e) Changes in extinction cross section in response to the fundamental and (c,f) higher-order breathing modes.

to out of phase displacement of the two surfaces. The period of the symmetric vibrational mode is longer than that of a solid sphere of the same size and increases with decreasing shell thickness.34 Optical Response of Vibrating Nanoshells. The plasmon modes on the inner and outer surfaces of the shell couple electrostatically, and the resulting optical response is dominated by the symmetric coupled dipolar mode.43 The plasmon resonance frequency shifts as the thickness of the shell varies, providing a widely tunable plasmonic response. The response of the plasmon resonances in nanoshells to the vibrational modes is shown in Figure 5. In both cases, the fundamental breathing (symmetric) mode produces a strong oscillation in plasmon resonance frequency. However, the antisymmetric vibrational mode produces a significant plasmon oscillation only in the case of the smaller shell (Figure 5f); this is consistent with experimental results.35 This occurs for two reasons. First, the amplitude of displacement is larger for the smaller nanoshell; second, in the case of the smaller nanoshell, all three coupling mechanisms produce plasmon shifts toward the red, resulting in a stronger modulation of the probe beam. As the hollow shell expands and contracts, the inner and outer surfaces move by different amounts, changing the radius ratio of the shell. For the smaller nanoshell, the solid shell occupies a larger fraction of the interior of the particle, so that the fractional change in volume of the solid portion is larger than for the larger nanoshell. This means that the smaller nanoshell experiences a larger change in ED and also larger 9364


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Optical Response of Vibrating Nanocubes. Just as only two vibrational modes dominate the mechanical response of a nanocube, only six dominant plasmon modes account for more than 90% of the optical absorption within the visible spectral range.44,45 These dominant plasmon modes are sensitive primarily to the shape of the corners of the cubes, where the electric fields are strongly concentrated.45 They are therefore expected to couple strongly to the fundamental vibrational mode, which involves significant deformation of the corners, and only weakly to the higher-order mode, which does not. This is confirmed by the calculated changes in optical spectra shown in Figure 6b,c. Even though the mechanical amplitude of the higher-order mode is nearly as large as that of the fundamental mode, it results in a much smaller shift in the plasmon peak. This is consistent with TA measurements on single nanocubes, which have shown only a single vibrational frequency for the majority of particles studied.36 A similar importance of field localization has previously been recognized in the context of Raman scattering from metal nanoparticles.24,25 The signal produced by the higher-order mode is further suppressed by the opposing effects of shape and volume changes. For the fundamental vibrational mode, expansion of the cube produces an outward motion of the corners that results in a red shift of the plasmon mode; this adds to the red shift produced by the decreasing ED and a further slight red shift by DP, resulting in a larger overall signal. For the higherorder mode, by contrast, expansion of the cube produces an inward motion of the corners and outward motion of the faces, causing a blue shift of the plasmon mode; this works against the red shift produced by the decreasing ED and DP, resulting in a smaller overall signal. The magnitude and phase of the signal produced by the higher-order mode will depend on the balance between the shape and volume effects, which will, in turn, be determined by the size of the cube. Figure S15 shows results for a larger cube, with edge length of 85 nm. Strong optical scattering from the 85 nm nanocube significantly modifies the extinction spectra as compared to the smaller nanocube, for which scattering is relatively weak. In this case, the optical signal due to the higherorder mode is significantly stronger than for the smaller cube and is comparable to the optical signal due to the fundamental mode. This is consistent with experimental results on ensembles of nanocubes, where smaller cubes have shown signals due only to the fundamental vibrational frequency, whereas cubes with edge lengths larger than 50 nm have shown signals that correspond to both modes.38

effects due to DP, leading to an additional shift of the plasmon resonance frequency. The fact that shape changes dominate the optical response is similar to the case of nanorods and is in contrast to the case of solid nanospheres, where the dominant contribution to the optical signal comes from changes in ED and DP. A similarity between the response of the solid sphere and the smaller nanoshell is that the metal absorption is strong on the blue side of the plasmon resonance, so that a more significant transient signal is expected on the red side of the resonance (Figure 5b,e), consistent with experimental observations.35 For the higher-order radial mode, the relative displacements of the inner and outer surfaces are sensitive to radius ratio. For the larger particle of the two that we simulated, this mode happens to result in nearly equal displacements of the two surfaces. This means a minimal change in the shell thickness, ED, and DP, resulting in a small change in the plasmon resonance (Figure 5c). For the smaller particle, the relative motion of the two surfaces is larger, leading to a larger optical signal (Figure 5f). This is again consistent with the experimental results.35 Nanocubes. Vibrational Modes of Nanocubes. Nanocubes have lower symmetry than either nanorods or nanoshells and exhibit correspondingly more complex optical and mechanical behavior. Of the many acoustic modes that are possible, two have been predicted to dominate the mechanical response.38 Figure 6a shows the calculated deformations corresponding to these two lowest-order acoustic modes for a silver nanocube with 35 nm edge length. These results show that the lowestorder mode primarily involves deformation of the eight corners of the cube, whereas the higher-order mode primarily involves displacement of the centers of the six faces of the cube.

DISCUSSION Table 1 presents a summary of our calculated vibrational frequencies along with experimentally determined values where available. Differences between model and experiment are largely due to the effects of the surrounding medium, which are not included in our mechanical calculations. For most of the particles, the effect of the surroundings is primarily to damp the vibrations.16 This results in a relatively small shift in resonance frequency, so that our calculations are in reasonable agreement with experiment.3 The nanoshells are the exception: motion of the inner surface of real colloidal nanoshells is restricted by the nearly incompressible fluid filling the shell, resulting in significantly higher vibrational frequencies. Accounting for this effect quantitatively will require coupling the finite-element treatment of the nanoparticle vibration, based on the Navier

Figure 6. Calculation results for a 35 nm silver nanocube. (a) Displacement as a function of frequency at the corner (solid green) and face center (dashed magenta) of the nanocube. The fundamental and overtone breathing modes are also shown. (b,c) Changes in extinction cross section for the two modes. 9365


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ACS Nano Table 1. Summary of the Vibrational Frequencies for Different Geometries Investigated in This Work phonon modes (GHz) shape

material

dimensions (nm)

sphere

silver

diameter = 60

55

gold

diameter = 60

50.7

gold

16.1 91 15.3 87.7 15

silver

length = 55 width = 26 length = 61 width = 22 Rin = 30 Rout = 40 Rin = 15 Rout = 25 edge = 35

silver

edge = 85

16.1 23.8

rod

gold shell

gold gold

cube

exptl

37.4 81 52.6

sim (this work) 56 122 54 113 19 102 17 117 28 135 45 171 57 80 23 33

exptl ref 27 13, 20 29 29, 46 35 35 38 38

equation, to a finite-element modeling of the enclosed liquid, based on the Navier−Stokes equation; such a model is beyond the scope of this paper. Nonetheless, the full fluid dynamic treatment is not expected to have a significant effect on the geometry of the vibrational modes, so the coupling between vibrational and plasmonic modes should be mostly unchanged by these effects. Figure 7 summarizes the calculated shifts in plasmon resonance frequency for each vibrational mode of the various nanoparticles due to shape changes, ED, and DP. In the case of spherical nanoparticles, the fundamental breathing mode is the dominant mode, with surface displacement about 5 times larger than that of the next higher-order breathing mode (see Figures 1 and 3), in agreement with experimental results.16,20,27,47,48 The higher-order breathing mode is excited 180° out of phase with respect to the fundamental mode, which would suggest a blue shift of the plasmon resonance; however, the local volume change involves both expansion and compression, as seen in the inset of Figure 1a, modulating the local permittivity in both directions from the equilibrium position. The weighted average permittivity is thus slightly modified from the equilibrium position, producing a negligible red shift in plasmon resonance for a 60 nm sphere. In terms of coupling mechanism, DP plays a significant role in the case of gold nanoparticles as compared to silver because of the spectral location of the interband frequencies. The magnitude of expansion in experiments can be estimated using the absorbed pump energy and the corresponding change in temperature, ΔT = Q/CV, where Q is the absorbed energy and C and V are the heat capacity and volume of the nanoparticle, respectively. Q is calculated as the product of the energy density in the pump pulse and the extinction cross section of the nanoparticle. Finally, the change in volume is calculated using the volumetric thermal expansion of the particle. Using typical values of pump pulse energy (100 nJ) and the calculated extinction cross section of a 60 nm gold sphere (1.4 × 10−14 m2), we estimate a 56 pm change in the radius of the particle. Considering the demonstrated linearity of the plasmon shift (Figure S4b), this corresponds to a plasmon shift of about 1.23 nm, using the results of Figure 3 and Figure 7.

Figure 7. Plasmon resonance shift corresponding to normalized vibrational modes for metal nanoparticles with different geometries (amplitude of the dominant vibrational mode is 1 nm). Contributions to the overall shift as the particle expands are indicated from shape only (red circle), shape + electron density (blue square), and shape + electron density + deformation potential (green diamond). L and H represent the low-frequency and highfrequency vibrational modes, respectively. The dashed vertical line represents the plasmon resonance of the undeformed nanoparticle.

In the case of nanorods, we can perform the reverse calculation and estimate the actual deformations of nanoparticles by comparing transient absorption measurements to the numerically calculated change in plasmon resonance wavelength (δλr). Ruijgrok et al. performed pump−probe measurements on single gold nanorods with an aspect ratio of 2.4 and length of 60 nm.16 From the pump−probe data and using the results of Figure 4e, we estimate a deformation of about 0.3 nm. We can also estimate the expected change in the length of the nanorod using ΔL = LαΔT, where α is the thermal expansion coefficient and L is the initial length of the rod. For the same experiments, this results in ΔL = 0.299 nm, in good agreement with our spectroscopic determination. In the case of nanoshells, a plasmon shift of approximately 2.5 nm has been reported in response to the fundamental breathing mode of a nanoshell with an outer radius of 40 nm.35 Comparing the experimental plasmon resonance shift to our numerical results, the experimental plasmon shift corresponds to a 0.25 nm change in inner radius of the shell and a 0.168 nm change in the outer radius of the shell. On the other hand, using the absorbed pump energy and thermal expansion calculations, an approximate change of 1.4 nm in the inner radius and 0.97 nm in the outer radius is predicted. As this calculation ignores the heat transferred from the metal shell to the liquid within the shell, it is likely that the temperature increase is lower than predicted, and the smaller amplitudes of deformation determined spectroscopically based on our calculations are more accurate. 9366


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ment can slightly shift the vibrational frequencies and damp the oscillations,56 though in many cases, the frequency shifts are negligible;17 an isotropic loss factor has been introduced in the FE calculations to avoid arbitrarily large deformations. Small changes in the vibration frequency and damping due to the environment will not affect conclusions for most of the cases studied in this work, including spheres, rods, and cubes; for nanoshells, however, the calculated mode frequencies and deformations may be affected by the core material. In addition, we do not take into account changes in the plasmon resonance due to heating of the nanoparticle or its environment. Provided that the change in temperature of the particle and its vibrational amplitude are both smalla condition that holds in most experimentsthen the effect of heating on the plasmon resonance will be nearly independent of the effect of vibration, and the two can be separated by appropriate fitting of experimental data.16 We thus focus here exclusively on the coupling between mechanical vibrations and plasmon resonances. Laser-induced heating is modeled by a uniform initial increased temperature, which is accurate for particles larger than a few nanometers, so that the initial heat transfer from the electrons to the lattice is much faster than the vibrational period of the nanoparticle. The dielectric response of a nondeformed nanoparticle is modeled using the Lorentz−Drude (LD) model:

In the case of nanocubes, our results predict the detection of a single vibrational mode (the fundamental breathing mode) for smaller nanocubes with an edge length of 35 nm. The higher-order breathing mode produces a negligible plasmon shift because the small shift in plasmon resonance due to shape modulation is almost completely canceled by ED effects, as seen in Figure S14b. On the other hand, for larger nanocubes with an edge length of 85 nm, optical scattering becomes significant and produces a broad feature on the red side of the resonance. In this case, our results predict the detection of two vibrational modes because the two modes produce comparable plasmons shifts 180° out of phase with one another, as shown in Figure S15. These results are again in agreement with experimental findings.38

CONCLUSIONS We have developed a numerical approach that predicts the optical response due to the mechanical vibrations of plasmonic metal nanoparticles. The method is applicable to arbitrary geometries and provides insight into the mechanisms responsible for coupling mechanical vibrations to plasmon resonances. This allows for studying additional variables in experimental measurements of the nanoparticle vibrations: instead of being limited to analysis of only vibrational frequencies and damping rates, we are now able to quantitatively relate optical signals to mode shapes and vibrational amplitudes. Since the calculations predict the full spectral response due to the vibrating nanoparticles, they can be used to optimize experimental measurements by predicting the probe wavelengths that will be most sensitive to specific vibrational modes. Moreover, they can be used to optimize nanoparticle geometries in order to maximize the optical signal due to a given vibrational amplitude, with potential application to ultrasensitive mass detection.6 Future work will be directed toward applying the method to additional nanoparticle geometries with lower symmetry that have been studied experimentally, such as bipyramids,3,16 nanodisks,49 crosses,50 and core−shell nanoparticles.5,51 In addition, the proposed method can be applied to study vibrational modes that are accessed using Raman spectroscopy and compare the results with experimental observations.18,24,25 Further connection with experiment can be achieved by coupling the Navier equation treatment of nanoparticle vibrations to a Navier−Stokes treatment of surrounding liquid in order to quantify the effects of the nanoparticle environment on the signal transduction. Our method can be extended to separately and quantitatively study the effects of mechanical vibrations on the real and imaginary parts of the dielectric function of a single nanoparticle, enabling comparison to timeresolved interferometric measurements.28,52

M

εr(ω) = εr(∞) + ωp2

∑

ω0,2m m=0

fm − ω2 − jω Γm

where ωp is the plasma frequency, f m is the oscillator strength, ω0,m is the interband transition frequency, and Γm is the line width of the mth Lorentzian term. These parameters are obtained for gold and silver by fitting LD model to the experimental data of refs 57 and 58 and are given in Table 1 of the SI. The bulk plasma frequency ωp = (Ne 2)/(ε0m) is modified by changes in the free-electron density, N, of a deformed nanoparticle: N = N0/(1 + ΔV/V0), where e and m are the charge and effective mass of the electron, respectively, N0 and V0 are the free electron density and volume of the nondeformed nanoparticle, respectively, and ΔV is the change in volume. The local volume change is obtained from the finite-element calculations. Aside from modifying the ED, the deformation of the particle also modifies the interband transition frequencies via the DP. The shifts in ω 0,m are calculated using the DPs for gold and silver: δE0,m = ξm(dV/V), where ξm and δE0,m are the DPs and change in energy of the mth transition, and the interband transition frequency and energy are related by E0,m = ℏω0,m (refs 59 and 60). Details of the calculation of the dielectric constant are provided in the SI. Equations S3 and S4 of the SI describe two possible ways (local and weighted average, respectively) of calculating the optical response of a deformed nanoparticle, based on local or weighted-average dielectric constants. Figures S2 and S3 of the SI show that both methods produce almost identical results, and thus eq S4 can be employed to reduce the required computational resources as well as the simulation time. Extinction cross sections of the particles are calculated using Lumerical FDTD Solutions.61 The deformed geometry is transferred from COMSOL to Lumerical for calculations of optical properties in the time domain; details are provided in the SI. Time-domain results are Fourier transformed to produce plasmon resonance spectra as a function of frequency. The FDTD simulation domain is terminated by a perfectly matched layer. A mesh override region is defined for accurate representation of the nanoparticle, with the size of the mesh determined according to the magnitude of deformations that need to be modeled. A minimum mesh size of 0.15 nm was used, and conformal variant 1 mesh was used. The polarization of the incident light determines which plasmon modes are excited. In the case where the particle has spherical symmetry, the polarization is inconsequential. For the case of rods and cubes, we employed a source with electric

METHODS The vibrational modes of the nanoparticles are calculated by solving the Navier equation with continuity of stress and displacement at the boundary of the nanoparticle. Impulsively excited modes are calculated in the frequency domain using the FE solver COMSOL Multiphysics.50,53 The metal nanoparticles are assumed to be elastically isotropic and homogeneous, bulk elastic constants are used, and the mechanical response is modeled using continuum mechanics.54,55 The COMSOL built-in material database is used to represent gold and silver using their bulk elastic constants and sound velocities. We ignore the mechanical properties of the surrounding medium and perform the FE calculations for metal nanoparticles in vacuum. In solution-based experiments, mechanical interaction with the surrounding environ9367


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ACS Nano field at 45° with respect to the long axis of the rod or the edge of the cube.

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ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b04789. Details of calculations of the dielectric response of the deformed nanoparticles, local vs averaged effects, methods used to transform deformed geometries from COMSOL to Lumerical, interpolation of numerical results, and calculated extinction cross sections for each nanoparticle showing the contribution of each coupling mechanism separately (PDF)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Aftab Ahmed: 0000-0002-9655-634X Matthew Pelton: 0000-0002-6370-8765 Jeffrey R. Guest: 0000-0002-9756-8801 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. This material is based upon work supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC0206CH11357. This material is based in part on work supported by the National Science Foundation under Grant No. DMR1554895. The authors thank Prof. John Sader and Dr. Debadi Chakraborty for helpful discussions. REFERENCES (1) Hartland, G. V. Optical Studies of Dynamics in Noble Metal Nanostructures. Chem. Rev. 2011, 111, 3858−3887. (2) Tang, Y.; Ouyang, M. Tailoring Properties and Functionalities of Metal Nanoparticles through Crystallinity Engineering. Nat. Mater. 2007, 6, 754−759. (3) Pelton, M.; Chakraborty, D.; Malachosky, E.; Guyot-Sionnest, P.; Sader, J. E. Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures. Phys. Rev. Lett. 2013, 111, 244502. (4) Yu, K.; Major, T. A.; Chakraborty, D.; Devadas, M. S.; Sader, J. E.; Hartland, G. V. Compressible Viscoelastic Liquid Effects Generated by the Breathing Modes of Isolated Metal Nanowires. Nano Lett. 2015, 15, 3964−3970. (5) Dacosta Fernandes, B.; Spuch-Calvar, M.; Baida, H.; TréguerDelapierre, M.; Oberlé, J.; Langot, P.; Burgin, J. Acoustic Vibrations of Au Nano-Bipyramids and Their Modification under Ag Deposition: A Perspective for the Development of Nanobalances. ACS Nano 2013, 7, 7630−7639. (6) Arlett, J. L.; Myers, E. B.; Roukes, M. L. Comparative Advantages of Mechanical Biosensors. Nat. Nanotechnol. 2011, 6, 203−215. (7) Stiles, P. L.; Dieringer, J. A.; Shah, N. C.; Van Duyne, R. P. Surface-Enhanced Raman Spectroscopy. Annu. Rev. Anal. Chem. 2008, 1, 601−626. 9368


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