Dependence of Plasmon Energies on the Acoustic Normal Modes of

Subscriber access provided by Chicago State University

Article

Dependence of Plasmon Energies on the Acoustic Normal Modes of Ag (n=20, 84 and 120) Clusters n

Clotilde Marjolaine Lethiec, Lindsey Rebecca Madison, and George C. Schatz J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b12497 • Publication Date (Web): 15 Mar 2016 Downloaded from http://pubs.acs.org on March 19, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Dependence of Plasmon Energies on the Acoustic Normal Modes of Agn (n=20, 84 and 120) Clusters Clotilde M. Lethiec, Lindsey R. Madison and George C. Schatz1 Department of Chemistry, Northwestern University, Evanston, IL 60208-3113

ABSTRACT The vibrational and optical properties of Ag20, Ag84 and Ag120 closed shell clusters are investigated through a combination of continuum mechanics and density functional theory approaches. The acoustic vibrational frequencies associated with these tetrahedral silver clusters are found to be in close correspondence for the two theories, demonstrating the ability of finiteelement calculations to reproduce first-principles computational results, even down to few atom structures. TDDFT calculations of the absorption spectra of these clusters indicate a strong plasmon-like mode both for equilibrium structures of the clusters and for structures where the acoustic breathing mode is excited by amounts that are accessible to ultrafast experiments. The plasmon-like mode energy is found to vary linearly with the acoustic mode displacement (for small displacements), with a slope that increases with increasing cluster size. For larger clusters, the TDDFT slope is larger than the FDTD slope, which indicates that there are systematic errors in the continuum theory result for small particles. We also examine the ground and plasmon excited potential energy curves, and show that the displacement in equilibrium geometry between these curves is too small to give breathing mode excitation that is consistent with observations based on vertical excitation alone. This suggests that breathing mode excitation arises during internal conversion after the initial photoexcitation. Keywords: DFT-TDDFT, FEA, electron-phonon coupling, acoustic normal modes, metallic nanoparticles. 1- Introduction Nanometer-sized metallic structures have been of great interest for years because their strong tunable localized plasmon resonance offers many attractive applications in such fields as optoelectronics,1 energy transfer,2 therapeutics,3 chemical sensors,4 solar energy harvesting,5,6 etc. A persistent grey area in understanding is how the molecular-like electronic excitations of small metallic nanoclusters evolve into localized surface plasmon resonances that are observed in larger nanoparticles. Continuing theoretical investigations into nanoscale metallic particles have attempted to elucidate these optical properties in the intermediate size regime.7 In particular, Ag clusters have been the subject of several theoretical studies, leading notably to the conclusion that the commonly assumed 1/R dependence of the plasmon band width on particle size R is incorrect for closed shell tetrahedral Ag clusters.8 This work also demonstrated good agreement of Ag cluster extinction spectra obtained from time-dependent density functional theory (TDDFT) and electrodynamics results from the discrete dipole approximation (DDA) method using an empirical dielectric function for Ag. In other studies it was demonstrated that coupledcluster quantum calculations provide accurate absorption spectra for Ag clusters, although this expensive computational technique is restricted to very small clusters.9 An atomistic electrodynamics model has been proposed to overcome the computational costs associated with 1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 14

ab initio calculations, leading to an accurate description of the optical properties of silver clusters (although involving a fit for the model parametrization).10 Based on earlier work that showed the possibility of producing single-molecule Raman scattering with very small silver clusters,11 firstprinciples simulations have furthermore been applied to small Ag clusters with absorbed pyridine to determine the size dependence and processes involved in enhanced Raman scattering.12 Ultrafast spectroscopy performed on metallic nanoparticles, by impulsively exciting the lattice vibrations, reveals oscillations of the plasmon band and allows the investigation of nonequilibrium transfer processes.13,14 The vibrational response of metallic nanoparticles through time-resolved analysis has been extensively studied in the past, notably for metallic spherical core and core-shell nanoparticles,15-18 nanocubes,19 nanowires,20,21 nanorods,22,23 bipyramids,24,25 and nanoclusters.26 In particular, the fundamental radial “breathing” mode, displaying a radial symmetry, is interesting because it is the only acoustic vibration observable in isotropic excitation time-resolved experiments.27-29 In this mechanism, ultrafast excitation by the pump pulse heats the electrons in the conduction band. The hot electrons subsequently relax and transfer their energy to the lattice, on a time-scale of ~1 ps. This laser-induced heating implies an isotropic expansion of the lattice. Electron/phonon dynamics in semiconductor quantum dots have also been studied, taking advantage of quantum confinement to highlight the electron-hole relaxation process.30 A theoretical study of Ge quantum dots showed good agreement of the vibrational periods when comparing atomistic, semiempirical interaction potentials with Lamb’s model of elasticity for nanoparticles as small as 280 atoms.31 Many analyses have been performed on the acoustic breathing mode of metal nanoparticles, notably focusing on damping25 and dephasing27 processes, lifetime of hot electrons,32 and influence of the environment.28 Some experimental results have, in addition, quantified the expansion of gold spherical nanoparticles associated with acoustic mode excitation as being a 0.4% increase in the radius.33 However, the optical and vibrational properties of the breathing mode have not, to our knowledge, been explored using electronic structure theory for metal particles. Instead, tests of continuum mechanics have involved comparisons of continuum theory with experiment. For example, continuum elastic theory was used to evaluate eigenfrequencies of large (~100 nm) gold bipyramids.24 Also, Vallee and coworkers showed that the frequency of the breathing mode of spherical Pt nanoparticles can be described using a continuum mechanics approach for particles down to 75 atoms.25,34 In addition, vibrational periods of small (up to 4 nm) faceted Au and Pt nanoparticles showed very good agreement with the results from the continuum mechanics description.35 In this paper we consider the fundamental radial mode (breathing mode) properties in tetrahedral36 closed shell Agn clusters, with n = 20, 84 and 120, making detailed comparisons of electronic structure and continuum mechanics calculations of the breathing mode properties, and also examining optical spectra from quantum mechanics and continuum electrodynamics. In the first section, we examine the vibrational eigenfrequency of the three clusters and we show that continuum mechanics (finite-element FEA calculations) leads to frequencies that are very similar to those from the DFT approach, even for the 20 atom cluster. The second part of this paper provides insight into the cluster size dependence of the plasmon energies. Time-dependent density functional theory (TDDFT) calculations were carried out to model the evolution of the photonic absorption in a silver cluster during the expansion and contraction corresponding to the breathing vibration. The evolution of the plasmon resonance in a larger silver tetrahedron has 2 ACS Paragon Plus Environment

Page 3 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


been also studied using finite-difference time-domain (FDTD) simulations. Finally, we address the issue of what fraction of the absorbed photon energy is transmitted to the breathing vibrational mode during photoexcitation through Franck-Condon (vertical) excitation.

2- Computational Details All the DFT calculations presented here (geometry optimization and natural frequency analysis) were performed using the Amsterdam Density Functional (ADF) program.37 The electronic structure of the silver clusters has been computed using the gradient-corrected (GGA) Perdew, Burke and Ernzerhof (PBE) exchange-correlation functional.38 A triple-ζ (TZP) Slater-type basis set with a frozen 4p core (noted TZP/Ag.4p) was employed. Scalar relativistic effects have been taken into account through the zeroth-order regular approximation (ZORA).39 The SCF convergence threshold was set at 10-6. For the geometry optimization, the gradient (energy) convergence criterion was set at 10-4 (10-5). TDDFT was used to calculate the excited states of the system. The first 2000 dipole-allowed transitions of each cluster were considered. The calculated photoabsorption spectra were convoluted with a 0.1 eV full width at half-maximum Lorentzian function in plotting the results. Note that the clusters are assumed to be in vacuum, so no solvation correction has been included in the spectra, or in the vibrational frequency calculation. The vibrational frequencies calculations have been determined both through DFT hessian calculations and by a finite-element analysis (FEA) based on the Abaqus commercial software. The latter simulations have been performed on a continuous tetrahedron whose size was extracted from the DFT geometry optimization (side length of Ag20 is 0.844 nm, Ag84 is 1.746 nm and Ag120 is 2.030 nm). We used the experimental elastic properties of bulk silver (Young’s modulus set at 83 GPa and Poisson’s ratio at 0.37) and a density of 10.49 g.cm-3. The natural frequency extraction procedure has been applied using the Lanczos eigensolver. Finite-difference time-domain (FDTD) simulations have been performed using the Lumerical Solutions commercial software (v. 8.11.318) to calculate the extinction spectra of various silver tetrahedra with a side length L = 4, 6, 8 and 10 nm, respectively. We used a dielectric permittivity tabulated by Palik for silver40 and a refractive index of 1 for the background medium. The simulation region was delimited by perfectly matched layer (PML) absorbing boundaries on all axes. The calculated absorption spectra correspond to the sum of the spectral response under plane-wave irradiation with polarization parallel and perpendicular to the face of the tetrahedron. The small structural details and numerical convergence of the results require a spatial meshgrid of 0.3 nm on the tetrahedron. Extinction cross sections were calculated at wavelengths ranging from 300 to 800 nm.

3- Results and discussion 1. Vibrational Frequencies

3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 14

This section addresses the interesting question of what is the size where the continuum mechanics approach breaks down in the determination of the acoustic frequencies of the Ag120, Ag84 and Ag20 clusters. To do this, the vibrational frequencies of the silver nanoparticles were calculated through both quantum DFT and continuum mechanics calculations. For the latter, the vibrational frequencies have been extracted by using a normal-mode analysis based on natural modes of small vibrations for a finite element model. Two different types of vibrations are extracted from these calculations: spheroidal (i.e., radial) and torsional modes. We note that the terminology used here applies to spherical clusters, but even for tetrahedral clusters the lowest order modes tend to have these symmetries. The lowest order spheroidal mode is the fundamental radial mode of interest, i.e. the breathing mode. DFT calculations, by contrast, provide the complete vibrational energy spectrum. In order to distinguish the breathing mode from other modes in this calculation, we determine the scalar product between the displacement vector of the normal mode and a vector that starts at the center of the tetrahedron and points radially outward toward each atom. The breathing mode is assigned to the mode with the highest scalar product associated with an expanding and contracting motion of the tetrahedron tips. Fig. 1 shows the calculated frequency associated with the breathing mode from DFT (blue stars) and FEA (red, green and purple circles) for the three silver clusters. As already highlighted in previous works, the vibrational frequency decreases as the size of the nanoparticle increases.24,28,35

Fig. 1: Frequency of the breathing mode calculated from both FEA and DFT for the three Agn clusters. Inset: Ag20 clusters modeled for DFT (left) and FEA (right) calculations. We observe in Fig. 1 that the FEA calculations get closer to the DFT results as the cluster size increases. Note that the silver clusters’ elastic constants were taken to be the same as for bulk silver, and therefore are not dependent on the cluster size in making this comparison. This is in agreement with the conclusions drawn in,41 which showed that the elastic constants of spherical silver and gold particles as small as 2 nm are the same as bulk. These conclusions are also con4 ACS Paragon Plus Environment

Page 5 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


sistent with the fact that the speed of sound in such nanoparticles is the same as in the bulk. However the present calculations consider much smaller size particles than studied previously. Different tetrahedral sizes and shapes have been modeled with FEA in Fig. 1. For both bluntedged and sharp-edged tetrahedra, we accounted for the silver van der Waals radius (RvdW = 172 pm) by adding it to the size extracted from DFT geometry optimization (which provides the positions of the centers of the atoms). We see in the figure that those sizes lead, for both geometries, to values of frequencies that are too small compared to DFT results. By not adding the van der Waals radius correction to the blunt-edged tetrahedron, we find reasonable agreement between DFT and FEA calculations, as attested by the very small error between classical continuum and quantum atomistic calculations (9.6% on Ag20, 7.2% on Ag84 and 1.7% on Ag120). These results are promising and interesting since they demonstrate that classical calculations are able to provide accurate eigenfrequency results even for nanoparticles down to a few tens of atoms, avoiding time-consuming ab initio normal mode analysis calculations.

2. Absorption properties We focus in this section on electronic transitions that involve plasmonic excited states of the system, based on TDDFT modeling. Here we follow previous work by Aikens8 in which conduction band transitions which involve a superposition of many particle-hole transitions are considered to be plasmonic excitations. The interest of these calculations is twofold. First, we can address the evolution of the plasmon resonance during the vibrational motion and determine if there is a common trend in the dependence of the plasmon energy on vibrational coordinate for the different clusters. This approach also allows for a quantitative determination of the fraction of the photon energy which is transferred to the vibrational modes. Here, the acoustic frequencies and normal mode shapes have been calculated within the frozen-phonon approach, according to the procedure described below. The total energy of the system, after slightly displacing the atoms by a small amount ∆ , follows the harmonic approximation and can be written as: from their equilibrium position + ∆ ≈ E + ∑, ∅, ( ∆ , ∆ )∆ , ∑, ∆

where , refer to the atoms, is the mass of atom , , ∆ ' is: ∅, &∆ , ∆ ) = ( ∅, (∆ )+

)*

,,- )+.,/

, ! = #, $, % and the hessian matrix

2)3 0(1∆

56 4+

,

2' the potential energy of the system. The hessian is estimated by moving the with 0&1∆ and the forces on all other atom from its equilibrium position by the small displacement ∆ atoms are recorded (frozen-phonon approach). 5 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 14

The hessian matrix ∅, is real and symmetric. Therefore, by choosing a normal coordinate basis in which this matrix is diagonal, the displacement of each atom can be expressed along only one coordinate 7 and the total energy can then be expressed as: + ∆ = E + ∑ 9 89 ∆′9 ,

9 correspond respectively to the acoustic frequency and displacement of normal where 89 and ∆′ mode L. In other words, in the normal coordinate basis, the total energy of the system can be written as a linear combination of harmonic potentials representing the modes. The hessian matrix eigenvectors are normalized in our DFT calculations, in the sense that the total displacement (sum) of the atoms is equal to 1 unit of length (bohr, in our case). We call 0 the geometry corresponding to this global atomic displacement of the acoustic mode. The different cluster geometries considered in the rest of this article are expressed as a multiple (from . The signs “+” and “-“ refer to expansion and contraction of atomic motion, 1 to 6) of 0 respectively. The cluster side lengths L corresponding to the different geometries of the tetrahedral clusters are presented in Table 1.

-6 -4 -2 -1 equilibrium +1 +2 +4 +6

Ag20 0.636 nm 0.706 nm 0.775 nm 0.810 nm 0.844 nm 0.879 nm 0.914 nm 0.983 nm 1.052 nm



Table 1: Side length L of the Agn clusters for various geometries (expressed as multiple of the ), associated with the breathing mode vibration. global displacement 0 The side lengths of the silver clusters calculated in this work are slightly smaller for the equilibrium configuration than those obtained by Aikens.8 This difference can be explained by the 0.204 nm additional length included by Aikens8 to account for the atomic radius. A schematic representation of the Ag20 cluster geometry at the -1, +1 and equilibrium geometries is represented on Fig. 2 a). The observed contraction and expansion features are characteristic of a breathing vibration. Fig. 2 b), c) and d) show the absorption spectra calculated for the different geometries of Ag20, Ag84 and Ag120 clusters respectively. The resonance peak redshifts and becomes sharper as the cluster size increases, showing the same behavior as the localized 6 ACS Paragon Plus Environment

Page 7 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


plasmon resonance (LSPR) in spherical metallic nanoparticles.42 This plasmon-like behavior of the absorption spectra of silver clusters is in agreement with previous results,8 but here we see that this behavior persists away from the equilibrium geometry except for Ag20 at large displacements.

a)

b)

Ag20 expanded

equilibrium

contracted

c)

d) expanded

Ag84

Ag120

expanded

equilibrium

equilibrium

contracted

contracted

Fig. 2: a) Ag20 cluster in contracted (-1), equilibrium and expanded (+1) geometries (left to right). b), c), d) Absorption spectra as a function of photon energy for Ag20, Ag84 and Ag120. Different colors stand for the different atomic displacements from (-4) to (+4). The energies of maximum absorption at equilibrium for the Ag20, Ag84 and Ag120 clusters are 3.40 eV, 2.97 eV and 2.90 eV respectively. We further plot in Fig. 3 the LSPR from TDDFT calculations as a function of the inverse edge length of the clusters in order to compare the evolution of the plasmon energy within the three clusters during the breathing vibrations. We also used FDTD simulations to calculate the absorption cross section of larger silver tetrahedra (L = 4, 6, 8 and 10 nm) and plotted the corresponding plasmon resonances in Fig. 3.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 14

Fig. 3: Energy of the absorption peak as a function of the inverse length of the Ag20 (in red), Ag84 (in blue) and Ag120 (in green) clusters. LSPR from FDTD calculations for silver tetrahedra with L = 4, 6, 8 and 10 nm respectively is also plotted (in purple). The dashed black lines are linear fits. As expected due to normalization of the atomic displacement vector (see above), the Ag20 cluster displays a LSPR shift over a much broader range of energy than the Ag84 and Ag120 clusters. Therefore, for the Ag20 cluster, we consider only expanded and contracted geometries close to the equilibrium geometry (±1, ±2). In this case, the LSPR shift for the Ag20, Ag84 and Ag120 clusters can be fit with a linear function where the slope is 0.83, 1.9 and 3.7 (nm)-1, respectively. The LSPR in a large cluster (in sense of number of atoms) turns out to be at a higher energy than for a smaller cluster, for the same tetrahedral side length. This indicates that a larger number of atoms for a given size results in a higher energy absorption, because the electron density is larger (as might be expected for a plasmonic excitation). For the larger silver tetrahedra simulated by using FDTD, a linear curve is still a good approximation to the L-dependence of the LSPR. In this case, the slope is lower (1.3) and does not follow the trend observed for the different sizes from TDDFT calculations. This result is likely due to the classical approach of these calculations, which assumes that the dielectric function is independent of particle volume. In the TDDFT results, the electron number doesn’t change when the particle is stretched or compressed, so the electron density changes with volume. Accurate classical calculations would therefore require including a correction due to variation of the electronic density, and therefore the dielectric function, with volume of the metallic particle. This point has also been mentioned by Hartland.33 We also note that using FDTD to describe particles with small size deformations that describe acoustic mode oscillations is quite challenging, and provides another source of difference compared to TDDFT. Finally, we focus on the ground and excited state potential energy surfaces of the nanoparticles here examining the dependence of these surfaces on the normal breathing mode. The ground state surface is obtained from DFT for each of the geometries considered in Fig. 3. We then approximate the energy of the plasmon excited state by adding the LSPR energy to the ground


Page 9 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


state energy. The resulting potential curves for both ground and excited states are plotted on Fig. 4.

Ag20

Ag84

Ag120

Fig. 4: Potential energies calculated for the ground state (in blue) and evaluated at the plasmon excited state (in red) for Ag20, Ag84 and Ag120. Also shown is a quadratic fit to the plasmon excited state (green solid line). For all the three clusters the potential curves display approximately harmonic profiles. Fits to quadratic functions that are shown on Fig. 4 can then be used to determine the breathing mode energy associated with a vertical transition from the ground state. Anharmonic effects do not change these results significantly. In particular, we find that the breathing mode excitation energies are 0.006 eV for Ag20, 0.015 eV for Ag84, and 0.023 eV for Ag120, which are a tiny fraction of the photon energy needed to excite the LSPRs. This suggests that breathing mode excitation arises during internal conversion after the initial excitation rather than from initial excitation from the vertical transition. An upper limit to the energy which goes into the acoustic mode would correspond to the case where all the photon energy is converted to breathing excitation: 3.40 eV for Ag20, 2.97 eV for Ag84, and 2.90 eV for Ag120. From the excited state curves in Fig. 4, this corresponds to displacements V equal to +3.2 for Ag20, +6.1 for Ag84, and +6.3 for Ag120, corresponding to edge lengths that increase by around 13% for Ag20, 6.8% for Ag84, and 5.0% for Ag120. Based on the results in Fig. 3, these displacements would correspond to changes in plasmon wavelength of 0.12 eV for Ag20, 0.08 eV for Ag84, and 0.08 eV for Ag120. These fractional displacements and plasmon wavelength changes are higher than has been observed in the experiments, which are of the order of 1% elongation.33,43 On the other hand, if one considers a 1% increase in the tetrahedron side length, the acoustic mode energy is smaller than 0.1 eV for the three considered clusters. A more accurate estimation of the fraction of the photon energy in acoustic excitation would require nonadiabatic dynamics calculations44 that go beyond the scope of this article. 4- Conclusion



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 14

Vibrational properties of normal breathing mode of Ag20, Ag84 and Ag120 closed shell clusters have been investigated by using both quantum and continuum mechanical descriptions. Previous work had established that these small Ag clusters display a resonance in the absorption spectra which can be attributed to a localized plasmon resonance, so these clusters are good model systems for understanding plasmon-phonon interactions. The continuum mechanical acoustic mode frequencies turn out to be in very good agreement with the atomistic approach even for clusters down to several tens of atoms. This result is very promising since it demonstrates that computationally constrained ab initio calculations are not required to describe the acoustic mode behavior of few atom structures. We also found that the variation in plasmon energy with acoustic mode deformation of the cluster is larger for larger clusters. Thus if we compare clusters with the same size but different numbers of atoms, the absorption energy of a cluster made of a larger number of atoms is higher than for an expanded cluster with fewer atoms. In addition, the variation in absorption energy with cluster size is larger than is obtained from FDTD, indicating that the standard continuum theory assumption that the dielectric function does not change when the particle is deformed by acoustic mode excitation is not accurate. Finally, we provided an upper and lower limit of the photon energy that is transferred to the excited acoustic breathing mode by calculating the bond energy change when the cluster is excited. The results indicate that very little of the acoustic mode excitation comes from FranckCondon effects, less than 0.03 eV. The observed oscillations are therefore associated with internal conversion after photoexcitation. However, these observed oscillations are sufficiently small that only a small fraction of the photon energy ends up as acoustic mode excitation. Acknowledgment This research was supported by the Department of Energy, Basic Energy Sciences, under grant DE-FG02-10ER16153. The authors would also like to thank Craig Chapman for his valuable input on finite-element analysis.

References 1.

2.

3.

4.

Choo, H.; Kim, M. K.; Staffaroni, M.; Seok, T. J.; Bokor, J.; Cabrini, P.; Schuck, J.; Wu, M. C.; Yablonovitch, E. Nanofocusing in a Metal-Insulator-Metal Gap Plasmon Waveguide with a Three-Dimensional Linear Taper. Nat. Photonics 2012, 6, 838-844. Wei, W.; Li, S.; Lidong Qin, L.; Xue, C.; Millstone, J. E.; Xu, X.; Schatz, G. C.; Mirkin, C. A. Surface Plasmon-Mediated Energy Transfer in Heterogap Au−Ag Nanowires. Nano Lett. 2008, 8, 34463449. Mocan, L.; Ilie, I.; Tabaran, F. A.; Dana, B.; Zaharie, F.; Zdrehus, C;. Puia, C.; Mocan, T.; Muntean, V.; Teodora, P.; Mosteanu Ofelia and Tantau Marcel and Cornel Iancuet al. Surface Plasmon Resonance-Induced Photoactivation of Gold Nanoparticles as Mitochondria-Targeted Therapeutic Agents for Pancreatic Cancer. Expert Opin. Ther. Targets 2013, 17, 1383-1393. Valsecchi, C.; Brolo, A. G. Periodic Metallic Nanostructures as Plasmonic Chemical Sensors. Langmuir 2013, 29, 5638-5649.


Page 11 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5.

6.

7. 8.

9.

10. 11. 12. 13.

14.

15. 16. 17. 18. 19.

20.

21. 22.

Leenheer, A. J.; Narang, P.; Lewis, N. S.; Atwater, H. A. Solar Energy Conversion Via Hot Electron Internal Photoemission in Metallic Nanostructures: Efficiency Estimates. J. Appl. Phys. 2014, 115, 134301. Sundararaman, R.; Narang, P.; Jermyn, A. S.; Goddard III, W. A.; Atwater, Harry A. Theoretical Predictions for Hot-Carrier Generation from Surface Plasmon Decay. Nat. Commun. 2014, 5, 5788. Guidez, E. B.; Aikens, C. M. Quantum Mechanical Origin of the Plasmon: from Molecular Systems to Nanoparticles. Nanoscale 2014, 6, 11512-11527. Aikens, C. M.; Li, S.; Schatz, G. C. From Discrete Electronic States to Plasmons: TDDFT Optical Absorption Properties of Agn (n = 10, 20, 35, 56, 84, 120) Tetrahedral Clusters. J. Phys. Chem. C 2008, 112, 11272-11279. Bonačić-Koutecký, V.; Pittner, J.; Boiron, M.; Fantucci, P. An Accurate Relativistic Effective Core Potential for Excited States of Ag atom: An Application for Studying the Absorption Spectra of Agn and Agn+ Clusters. J. Phys. Chem. C 1999, 110, 3876-3886. Jensen, L. L.; Jensen, L. Atomistic Electrodynamics Model for Optical Properties of Silver Nanoclusters. J. Phys. Chem. C 2009, 113, 15182-15190. Peyser-Capadona, L.; Zheng, J.; González, J. I.; Lee, T.H.; Patel, S. A.; Dickson, R. M. NanoparticleFree Single Molecule Anti-Stokes Raman Spectroscopy. Phys. Rev. Lett. 2005, 94, 058301. Jensen, L.; Zhao, L. L.; Schatz, G. C. Size-Dependence of the Enhanced Raman Scattering of Pyridine Adsorbed on Agn (n = 2−8, 20) Clusters. J Phys Chem C. 2007, 111, 4756-4764. Wall, S.; Wegkamp, D.; Foglia, L.; Appavoo, K.; Nag, J.; Haglund, R. F.; Stähler, J.; Wolf, M. Ultrafast Changes in Lattice Symmetry Probed by Coherent Phonons. Nat. Commun. 2012, 3, 721. Zamponi, F.; Penfold, T. J.; Nachtegaal, M.; Lübcke, A.; Rittmann, J.; Milne, C. J.; Chergui, M.; van Bokhoven, J. A. Probing the Dynamics of Plasmon-Excited Hexanethiol-Capped Gold Nanoparticles by Picosecond X-ray Absorption Spectroscopy. Phys. Chem. Chem. Phys. 2014, 16, 23157-23163. Hartland, G. V. Coherent Excitation of Vibrational Modes in Metallic Nanoparticles. Annu. Rev. Phys. Chem. 2006, 57, 403-430. Del Fatti, N.; Voisin C.; Christofilos, D.; F. Vallée F.; Flytzanis C. Acoustic Vibration of Metal Films and Nanoparticles. J. Phys. Chem. A. 2000, 104, 4321-4326. Crut, A.; Juvé, V.; Mongin, D.; Maioli, P.; Del Fatti, N.; Vallée, F. Vibrations of Spherical Core-Shell Nanoparticles. Phys. Rev. B 2011, 83, 205430. Hodak, J. H.; Henglein, A.; Hartland, G. V. Electron-Phonon Coupling Dynamics in Very Small (Between 2 and 8 nm Diameter) Au Nanoparticles. J. Chem. Phys. 2000, 112, 5942-5947. Petrova, H.; Lin, C. H.; de Liejer, S.; and Hu, M.; McLellan, J. M.; Siekkinen, A. R.; Wiley, B. J.; Marquez, M.; Xia, Y.; Sader, J. E.; Hartland, G. V. Time-Resolved Spectroscopy of Silver Nanocubes: Observation and Assignment of Coherently Excited Vibrational Modes. J. Chem. Phys. 2007, 126, 094709. Major, T. A.; Crut, A.; Gao, B.; Lo, S. S.; Del Fatti, N.; Vallée, F.; Hartland, G. V. Damping of the acoustic vibrations of a suspended gold nanowire in air and water environments. Phys. Chem. Chem. Phys. 2013, 15, 4169-4176. Major, T. A.; Lo, S. S.; Kuai, Y.; Hartland, G. V. Time-Resolved Studies of the Acoustic Vibrational Modes of Metal and Semiconductor Nano-objects. J. Phys. Chem. Lett. 2014, 5, 866-874. Hu, M.; Wang,X.; Hartland,G. V.; Mulvaney,P.; Juste, J.P.; E. Sader, J. E. Vibrational Response of Nanorods to Ultrafast Laser Induced Heating: Theoretical and Experimental Analysis. J. Am. Chem. Soc. 2003, 125, 14925-14933.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

23. 24.

25. 26. 27. 28. 29.

30.

31. 32.

33. 34. 35.

36. 37. 38. 39. 40. 41.

42.

Page 12 of 14

Zijlstra, P.; Tchebotareva, A.L.; Chon, J. W. M.; Gu, M.; Orrit, M. Acoustic Oscillations and Elastic Moduli of Single Gold Nanorods. Nano Lett. 2008, 8, 3493-3497. Fernandes, B. D.; Spuch-Calvar, M.; Baida, H.; Tréguer-Delapierre, 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. Pelton, M.; Wang, Y.; Gosztola, D.; Sader. J. E.; Mechanical Damping of Longitudinal Acoustic Oscillations of Metal Nanoparticles in Solution. J. Phys. Chem. C 2011, 115, 23732-23740. Qian, H. and Matthew Y. Sfeir, M. Y; Jin, R. Ultrafast Relaxation Dynamics of [Au25(SR)18]q Nanoclusters: Effects of Charge State J. Phys.Chem. C 2010, 114, 19935–19940. Hodak, J. H.; Henglein, A.; Hartland, G. V. Size Dependent Properties of Au Particles: Coherent Excitation and Dephasing of Acoustic Vibrational Modes. J. Chem. Phys. 1999, 111, 8613-8621. Voisin, C.; Del Fatti, N.; Christofilos, D.; Vallée, F. Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles. J. Phys. Chem. B 2001, 105, 2264-2280. Nelet, A.; Crut, A.; Arbouet, A.; Del Fatti, N. D.; Vallée, F.; Portalès, H.; Saviot, L.; Duval E. Acoustic Vibrations of Metal Nanoparticles: High Order Radial Mode Detection. Appl. Surf. Sci. 2004, 226, 209-215. Hannah, D. C.; Brown, K. E.; Young, R. M.; Wasielewski, M. R.; Schatz, G. C.; Co, D. T.; Schaller, R. D. Direct Measurement of Lattice Dynamics and Optical Phonon Excitation in Semiconductor Nanocrystals Using Femtosecond Stimulated Raman Spectroscopy. Phys. Rev. Lett. 2013, 111, 107401. Combe, N.; Huntzinger, J. R.; Mlayah, A. Vibrations of Quantum Dots and Light Scattering Properties: Atomistic Versus Continuous Models. Phys. Rev. B 2007, 76, 205425. Aruda, K. O.; Tagliazucchi, M.; Sweeney, C. M.; Hannah, D. C.; Schatz, G. C.; Weiss, Emily A. Identification of Parameters through which Surface Chemistry Determines the Lifetimes of Hot Electrons in Small Au Nanoparticles. Proc. Natl. Acad. Sci. 2013, 110, 4212-4217. Hartland, G. V. Coherent Vibrational Motion in Metal Particles: Determination of the Vibrational Amplitude and Excitation Mechanism. J. Chem. Phys. 2002, 116, 8048-8055. Hartland, G. V. Optical Studies of Dynamics in Noble Metal Nanostructures. Chem. Rev. 2011, 111, 3858-3887. Sauceda, H. E.; Mongin, D.; Maioli, P.; Crut, A.; Pellarin, M.; Del Fatti, N.; Vallée, F.; Garzón, I. L. Vibrational Properties of Metal Nanoparticles: Atomistic Simulation and Comparison with TimeResolved Investigation. J. Phys. Chem. C 2012, 116, 25147-25156. Fernández, E. M.; Soler, J. M.; Garzón, I. L.;Balbás, L. C. Trends in the Structure and Bonding of Noble Metal Clusters. Phys. Rev. B 2004, 70, 165403. Te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C and van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. Chemistry with ADF. J. Comput. Chem. 2001, 22, 931–967. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. Van Lenthe, E.; Baerends, J.; Snijders, J. G. Relativistic Total Energy Using Regular Approximation. J. Chem. Phys. 1994, 101, 9783-9792. Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: New York, 1998. Sauceda, H. E., Mongin, D., Maioli, P., Crut, A., Pellarin, M., Del Fatti, N., Vallee, F., Garzon, I. L., Vibrational Properties of Metal Nanoparticles : Atomistic Simulation and Comparison with TimeResolved Investigation, J. Phys. Chem. C 2012, 116, 25147-25156. Maier, S. Plasmonics: Fundamentals and Applications. Springer: New York, 2007.


Page 13 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


43.

44.

Plech, A.; Kotaidis, V.; Grésillon, S.; Dahmen, C.; von Plessen, G. Laser-Induced Heating and Melting of Gold Nanoparticles Studied by Time-Resolved X-Ray Scattering. Phys. Rev. B 2004, 70, 195423. Neukirch, A.J., Guo, Z.; Prezhdo, O.V. Time-Domain Ab Initio Study of Phonon-Induced Relaxation of Plasmon Excitations in a Silver Quantum Dot. J. Phys. Chem. C 2012, 116, 15034-15040.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 14

Table of Contents Image: Potential energies calculated (TDDFT) for the ground state and evaluated at the plasmon excited state for the Ag84 cluster during acoustic mode vibration.


Dependence of Plasmon Energies on the Acoustic Normal Modes of

Recommend Documents