Article pubs.acs.org/JPCA
First-Principles Computational Visualization of Localized Surface Plasmon Resonance in Gold Nanoclusters Kenji Iida, Masashi Noda, Kazuya Ishimura, and Katsuyuki Nobusada* Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, 444-8585, Japan ABSTRACT: Cluster-size dependence of localized surface plasmon resonance (LSPR) for Aun nanoclusters (n = 54, 146, 308, 560, 922, 1414) is investigated by using our recently developed computational program of first-principles calculations for photoinduced electron dynamics in nanostructures. The size of Au1414 (3.9 nm in diameter) is unprecedentedly large in comparison with those addressed in previous first-principles calculations of optical response in nanoclusters. These computations enable us to clearly see that LSPR gradually grows and the LSPR peaks red shift with increasing cluster size. The growth of LSPR is visualized in real space, demonstrating that electron charge distributions oscillate in a collective manner around the outermost surface region of the clusters. We further illustrate that the core d electrons screen the collective oscillation of the conduction-like s electrons.
1. INTRODUCTION Localized surface plasmon resonance (LSPR) is an essential optical process in metals, more specifically, noble metal nanoclusters, when subjected to an external laser field. The application of LSPR in stained glass has a lengthy history, dating back to Roman times. In the 20th century, LSPR was intensively investigated in a research field of basic science primarily in the context of understanding of fundamental properties of metal nanoclusters.1−3 LSPR has also recently attracted much attention in applied science, sometimes referred to as plasmonics, extending its potential applicability to functional materials and devices.4 Such valuable functions are strongly related to the characteristic properties of LSPR (i.e., highly resonant response to an applied laser field and strong electric field enhancement in a near field region). For example, light concentration in metal nanoclusters improves the efficiency of photovoltaic cells and metal nanocluster waveguides.5−8 Ultra sensitive response of LSPR to structural details at an interface region is very useful in developing biosensors and extreme measurement tools for single-molecule detection and imaging.9−13 Furthermore, nonlinear optical response due to electric field enhancement is expected to facilitate two or more photons excitation in wide band gap materials. LSPR of metal clusters can be explained conceptually in terms of a classical description of collective oscillation of surface charges. Thus, optical properties of LSPR of relatively large (larger than ∼30 nm) metal clusters have so far been frequently described by Mie scattering theory or a quasi-static approximation.14−16 The finite-difference time-domain method17 is an alternative powerful and efficient computational tool for calculating optical response in nanoclusters with an arbitrary shape. All the approaches mentioned above are apparently not beyond a classical description, in which materials are represented in terms of dielectric constants, and consequently cannot account for electronic structures of metal nanoclusters. © 2014 American Chemical Society
As the size of metal clusters decreases below 20 nm in diameter, LSPR becomes strongly dependent on cluster-size, geometric structure, and constituent atoms, all of which are unarguably related to electronic structures of nanoclusters. It is expected, in this size region, to see the transition from a classical feature of the collective oscillation to quantum mechanical optical response. If the size of clusters further decreases (< 2 nm), a quantum confinement effect is predominant over the classical description. One can carry out first-principles calculations of optical responses of such small clusters,18−20 whereas it is still highly computationally demanding to do so for nanoclusters of more than 2 nm in size. Experimentally, there have been a relatively large number of investigations on size dependence of LSPR in metal clusters.2,21−29 However, it is not easy to perform the experiments for individual (i.e., ligand free and ensemble heterogeneity free) clusters for sizes systematically ranging from a classical region to a quantum one. Recently, optical properties of LSPR were addressed both experimentally and theoretically in the transient size region of ∼1 to ∼20 nm.29−32 Scholl et al. performed an experiment on cluster-size dependence of LSPR for individual silver nanoclusters with diameters from 2 to 20 nm.29 The LSPR peak was found to shift to higher energy (i.e., blue shift) with decreasing cluster size. The theoretical studies on LSPR were conducted by Malola et al. for Au nanoclusters31 and by Xiang et al. for Na nanoclusters.32 Malola et al. carried out large first-principles, time-dependent density functional theory (TDDFT), calculations for Au nanoclusters with different sizes up 2 nm. Their calculations indicated the presence of LSPR in Au314(SH)96 at the maximum size (2 nm). However, since they did not calculate the size-dependence of LSPR for larger clusters, the Received: August 31, 2014 Revised: November 1, 2014 Published: November 4, 2014 11317
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linearly polarized laser field is applied, and a damping factor as a function of T is employed.40 The oscillator strength distribution of eq 4 is obtained by performing Fourier transformation of time-dependent induced dipole in the gold clusters subjected to an external laser field. Thus, this quantity is straightforwardly related to optical absorption spectra38 and describes well the LSPR optical properties. To be precise, df(E)/dE should be given by averaging over all axes of ν = x, y, and z. However, we computationally confirmed that since the clusters considered have approximately spherical symmetry, the difference in df(E)/dE with respect to the polarization direction has no qualitative influence on the following discussions on LSPR. Thus, only the x component (ν = x) is evaluated in this study. To investigate the light-induced electron dynamics, we apply an external laser pulse whose functional form is given as
growth of LSPR was not discussed as a function of cluster size. Xiang et al. investigated, on the basis of first-principles calculations with a practical approximation, the size-dependence of LSPR in Na clusters with diameters ranging from 0.7 to 12.3 nm. They demonstrated a nonmonotonic shift of the LSPR peaks as a function of cluster size and clearly explained the reason for the shift. However, the sodium cluster is a simple electronic structure system and thus the optical features of its LSPR are quite different from those of gold or silver clusters. Furthermore, from the viewpoint of applications of LSPR to functional materials in real systems, an understanding of the LSPR optical properties of gold or silver nanoclusters is highly desirable. Very recently, our group developed a highly efficient firstprinciples TDDFT computational program for electron dynamics solved in real-time and real-space, named GCEED (grid-based coupled electron and electromagnetic field dynamics).33 In this paper, GCEED is applied to the optical response of Aun clusters (n = 54, 146, 308, 560, 922, and 1414). GCEED can carry out full first-principles TDDFT calculations without a practical approximation. To the best of our knowledge, the cluster size of Au1414 (3.9 nm in diameter) consisting of 15554 electrons is unprecedentedly large compared with those addressed in previous first-principles calculations of optical response in nanoclusters. Therefore, the growth of LSPR in the gold clusters is revealed, for the first time, using first-principles calculations. Furthermore, the realspace treatment of GCEED allows us to accurately visualize this growth.
vext(r, t ) = −rνFt (t )cos(ωt )
where rν is one of the Cartesian coordinates (ν = x, y, and z) indicating the laser polarization, Ft is a pulse envelope function, and ω is the laser frequency. The pulse envelope is given by Ft (t ) = F sin 2(πt /τ ) (0 < t < τ )
⎤ ⎡ 1 ∂ ψ (r, t ) = ⎢ − ∇2 + Veff (r, t )⎥ψj(r, t ) ⎦ ⎣ 2 ∂t j
(1)
where Veff is the effective potential given by Veff (r, t ) = vnuc(r) +
∫
ρ(r′, t ) dr′ + vXC(r, t ) |r − r′|
+ vext(r, t )
(2)
Here, vnuc(r), vXC(r), and vext(r) are the nuclear attraction potential, the exchange-correlation potential, and the external potential of an applied laser field, respectively. ρ in eq 2 is the electron density given by N /2
ρ (r , t ) = 2
∑ |ψj(r, t )|2 (3)
j=1
where the factor of 2 indicates that each orbital is fully occupied. For the exchange-correlation potential vXC(r), the adiabatic local density approximation is used. The oscillator strength distribution df(E)/dE is given by39 d f (E ) 2E = Im πF dE
T
∫0 [∫ rν{ρ(r, t ) − ρ(r, 0)}
× {1 − 3(t /T )2 + 2(t /T )3 } dr]eiEt dt
(6)
where τ determines the laser pulse duration. The field strength F is related to the laser intensity by I = 1/2ε0cF2, where ε0 is the permittivity of vacuum and c is the speed of light. To perform the practical calculations, we utilize GCEED wherein the TDKS equation is directly solved at threedimensional Cartesian grid points by a finite-difference formula.41,42 The time-propagation is evaluated by the Taylor expansion method. The computational program is completely free from algorithms involving eigenvalue problems and fastFourier-transformation, which are usually implemented in conventional quantum chemistry or band structure calculations. Because of its excellent parallel efficiency, the program is highly suitable for massively parallel calculations and therefore allows us to carry out first-principles calculations of a large gold cluster system of Au1414. The maximum size of the computation box for Au1414 is set to 64 Å × 64 Å × 64 Å. All electrons other than Au 5d6s electrons are replaced with the effective core potential (ECP) obtained by the Troullier−Martins scheme implemented in the fhi98PP program.43,44 Then, the total number of the electrons of Au1414 is 15554. We employ uniform space grids Δx, Δy, and Δz of 0.25 Å. The real-time propagation is performed by using the constant time step of 1.316 × 10−3 fs. The applied laser pulse duration and intensity are set to 30 fs and 1.0 × 108 W/cm2, respectively. Time propagation takes 6 h for the largest cluster Au1414 by using 62400 CPU cores of the K computer (SPARK64 VIIIfx 2.0 GHz, 8 CPU cores/node). Visualization of the time-dependent induced density is performed by using the Vesta package.45 Using GCEED, we calculate the optical response of the icosahedral Aun clusters (n = 54, 146, 308, 560, 922, and 1414) where one atom at an apex is removed to consider closed-shell electronic structures. In the present computations, we assume the icosahedral structures for all the Au clusters without geometry optimization. Their bond lengths are, on average, less than 3% shorter than those of the bulk gold.46 Although the structure of a gold cluster has been extensively discussed,47,48 previous experimental and theoretical studies show that the icosahedral structure is one of the major compositions at the sizes less than 10 nm.49−51 Furthermore, the optical absorption spectrum of an icosahedral Au147 cluster was shown to be less affected by slight geometry compression.52 We consider that
2. MASSIVELY PARALLEL FIRST-PRINCIPLES DFT CALCULATIONS The present approach is based on the time-dependent Kohn− Sham (TDKS) theory. TDKS equation is written in the form of (in atomic units),34−38 i
(5)
(4)
where F is the strength of a pulse field, T is the total propagation time, rν(ν = x, y, and z) is the axis along which the 11318
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the icosahedral structure is the most reasonable structure for systematically discussing the growth of LSPR. Thus, we focus on such icosahedral structures. The sizes of the clusters are 1.1 nm (Au54), 1.7 nm (Au146), 2.2 nm (Au308), 2.8 nm (Au560), 3.3 nm (Au922), and 3.9 nm (Au1414) in diameter, respectively. Three of these (n = 54, 560, and 1414) are exemplified in Figure 1.
clusters have multiple resonance peak structures and no clear LSPR (see the inset in the figure and also following discussion). This is simply because the cluster size is so small that a quantum confinement effect is predominant and they have discrete electronic energy levels.31,53 The oscillator strengths for the larger clusters rapidly increase and additional spectral features appear in the higher energy range. This is mainly due to the interband transitions of the d electrons.2 It is also noted that the LSPR peaks are not sharp in comparison with those of other metals such as sodium and silver clusters because of the screening effect by the d electrons.2,32,54 We will discuss the screening effect in Section 3.3. As mentioned earlier, the properties of LSPR strongly depend on size, geometric structure, and constituent atoms of a cluster. Since all the clusters considered consist only of gold atoms and have a homogeneous icosahedral structure with near-spherical symmetry, we can reasonably define the renormalized oscillator strength, dividing the original strength by the number of atoms of a cluster (as in the lower panel of Figure 2). The renormalization allows us to focus on the growth of LSPR on a unified scale. The renormalized curves for Au560, Au922, and Au1414 exhibit rather structureless, broad features similar to each other, whereas the curves for the smaller (n ≤ 146) gold clusters show multiple resonance peak structures as mentioned above. This figure clearly demonstrates that LSPR appears in Au308 (2.2 nm) and grows in the larger clusters. Malola et al. also reported the appearance of LSPR in a 2 nm gold cluster.31 The present computed results of the LSPR peak positions and their size dependence, that is, the peaks red shift (blue shift) with increasing (decreasing) cluster size, are in good accordance with experimental observations.21−23,27 The peak position of 2.3 eV for Au1414 is close to the experimental values of 2.34−2.38 eV for gold clusters with the sizes of 3.5− 37 nm in diameter in condensed phase.15,23 These values are also close to the value of Mie plasmon resonance (∼2.4−2.5 eV) evaluated by using bulk gold dielectric function.55 3.2. Growth of LSPR. Here, we carry out further firstprinciples visualization of the growth of LSPR. To this end, we define the quantity of frequency-dependent difference in electron density using the imaginary part of the Fouriertransformed time-dependent induced density as follows:
Figure 1. Icosahedral Aun clusters (n = 54, 560, and 1414).
3. RESULTS AND DISCUSSIONS 3.1. Optical Absorption Spectra. Figure 2 (top) shows the oscillator strength of Aun. It is clear from this figure that
Δρ(r, E) ≡ Im
∫0
T
{ρ(r, t ) − ρ(r, 0)}
× {1 − 3(t /T )2 + 2(t /T )3 }eiEt dt
(7)
where the electron density is calculated by applying an xpolarized laser field. As seen from eqs 4 and 7, Δρ is associated with the spatial component of the oscillator strength. Thus, Δρ yields a clear and detailed insight into the optical response of LSPR in a configuration space. Figure 3 shows Δρ of the Aun clusters at E = 2.7 eV for Au54, 2.7 eV for Au146, 2.6 eV for Au308, 2.5 eV for Au560, and 2.3 eV for Au1414, which correspond to each intense resonance frequency or LSPR frequency. The figures are shown from the direction perpendicular to the x−y plane. The red and blue colors, respectively, indicate increase and decrease in the density from the ground state. The electron density for Au54 is distributed rather irregularly. Relatively positive and negative charge densities for Au146 are distributed around the top and bottom sides along the laser polarization direction. This separation indicates that the beginning of LSPR can be traced to Au146, whereas the corresponding absorption spectrum in Figure 2 still has multiple resonance peak
Figure 2. (Top) Oscillator strengths of Aun (n = 54, 146, 308, 560, 922, and 1414) and (bottom) their renormalized strengths with respect to the number of atoms. Insets in the upper and lower panels show magnifications of the curves for Au54 and Au146 and for Au560, Au922, and Au1414, respectively.
LSPR gradually grows and becomes discernible as the cluster size increases. The LSPR peak positions red shift with increasing size, and those for Au308, Au560, Au922, and Au1414 are found to be 2.6, 2.5, 2.5, and 2.3 eV, respectively. As is predicted, however, the curves for the smaller Au54 and Au146 11319
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frequencies are set to E = 2.7 (Au54) and 2.3 eV (Au1414), respectively. The red and blue colors indicate the increase and decrease in the electron density from the ground state, respectively. The positive (red) and negative (blue) charge densities for Au54 are distributed rather irregularly. The charge densities for Au1414, on the other hand, show regular patterns, indicating that the positive and negative charges are distributed around the upper and lower surface regions, respectively, synchronizing the x-polarized external laser field. This is exactly the same behavior as the surface charge distributions seen in Figure 3. Furthermore, the charge oscillations occur locally around each gold atom inside the cluster in the direction opposite to the surface charge oscillation. This behavior of electron dynamics subject to an applied laser field indicates a typical screening effect, which has been widely recognized, particularly in metal clusters.2,54,56,57 Very recently, Malola et al. also clearly demonstrated the screening effect of d electrons in 2 nm gold clusters by using a linear response TDDFT method.31 The screening effect in the present gold clusters is due to the fact that the conduction (or valence) s-electrons follow the oscillating laser field, whereas the core d-electrons suppress (i.e., screen) the oscillatory motion of the conduction electrons. Here, the importance of the d-electron dynamics in the optical property is definitively demonstrated by freezing the d-electron motion. Figure 5 shows an absorption spectrum of a model
Figure 3. Induced densities of Aun (n = 54, 146, 308, 560, and 1414) seen from the direction perpendicular to the x−y plane. The thresholds of the density maps are set to the same value. The applied laser energies are indicated in the unit of eV.
structures with no clear LSPR peak. As the cluster size further increases, the separation of the charge densities becomes more apparent. These features found in Figure 3 are clear evidence of the growth of LSPR. 3.3. Screening Effect. Although Figure 3 visualizes the charge oscillation of LSPR and its growth depending on cluster size, only the charge distribution on the surface was discussed. To further analyze LSPR, let us next examine the electron dynamics in an inner region of the cluster. We illustrate finitewidth cross sections of Δρ in the x−y plane in Figure 4. These density maps of the cross sections are seen from z < 0 of the disklike moieties obtained by slicing Aun from z = −2 to +2 Å. For convenience, the gold atoms are drawn smaller than those in Figure 3. The incident laser is x-polarized and the
Figure 5. Absorption spectrum of a model Au1414 cluster in which the d electrons are replaced with ECP (top) and a disklike moiety of the induced density of the cluster sliced at z = −2 and +2 Å. The applied laser energies are indicated in the unit of eV.
cluster Au1414 in which all the d electrons are replaced with ECP and only the 1414 s electrons are treated explicitly. The peak intensity becomes significantly strong, and the position largely shifts to the higher energy by ∼3 eV. The d-electron motion has also great influence on the electron dynamics, as shown in the lower panel of Figure 5. This figure is the same as the density map for Au1414 in Figure 4 but for the model Au1414 cluster without the d electrons. The incident laser frequency is set to the resonance peak position of 5.45 eV. The positive and negative charge densities are distributed only around the cluster
Figure 4. Disklike moieties of the induced density of Aun (n = 54 and 1414) sliced at z = −2 and +2 Å. The thresholds of the density maps are set to the same value. The applied laser energies are indicated in the unit of eV. 11320
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(4) Maier, S. A. Plasmonics: Fundamentals and Applications; Springer: New York, 2007. (5) Maier, S. A.; Kik, P. G.; Atwater, H. A.; Meltzer, S.; Harel, E.; Koel, B. E.; Requicha, A. A. G. Local Detection of Electromagnetic Energy Transport Below the Diffraction Limit in Metal Nanoparticle Plasmon Waveguides. Nat. Mater. 2003, 2, 229−232. (6) Lal, S.; Link, S.; Halas, N. J. Nano-Optics from Sensing to Waveguiding. Nat. Photonics 2007, 1, 641−648. (7) Atwater, H. A.; Polman, A. Plasmonics for Improved Photovoltaic Devices. Nat. Mater. 2010, 9, 205−213. (8) Green, M. A.; Pillai, S. Harnessing Plasmonics for Solar Cells. Nat. Photonics 2012, 6, 130−132. (9) Nie, S.; Emory, S. R. Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering. Science 1997, 275, 1102−1106. (10) Kneipp, K.; Wang, Y.; Kneipp, H.; Perelman, L. T.; Itzkan, I.; Dasari, R.; Feld, M. S. Single Molecule Detection Using SurfaceEnhanced Raman Scattering (SERS). Phys. Rev. Lett. 1997, 78, 1667− 1670. (11) Anker, J. N.; Hall, W. P.; Lyandres, O.; Shah, N. C.; Zhao, J.; Van Duyne, R. P. Biosensing with Plasmonic Nanosensors. Nat. Mater. 2008, 7, 442−453. (12) Mayer, K. M.; Hafner, J. H. Localized Surface Plasmon Resonance Sensors. Chem. Rev. 2011, 111, 3828−3857. (13) Zhang, R.; Zhang, Y.; Dong, Z. C.; Jiang, S.; Zhang, C.; Chen, L. G.; Zhang, L.; Liao, Y.; Aizpurua, J.; Luo, Y.; Yang, J. L.; Hou, J. G. Chemical Mapping of a Single Molecule by Plasmon-Enhanced Raman Scattering. Nature 2013, 498, 82−86. (14) Mie, G. Articles on the Optical Characteristics of Turbid Tubes, Especially Colloidal Metal Solutions. Ann. Phys. 1908, 25, 377−445. (15) Kreibig, U.; Zacharias, P. Surface Plasma Resonances in Small Spherical Silver and Gold Particles. Z. Physik 1970, 231, 128−143. (16) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press: Cambridge, 2006. (17) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed.; Artech House, Inc.: Norwood, 2005. (18) Stener, M.; Nardelli, A.; De Francesco, R.; Fronzoni, G. Optical Excitations of Gold Nanoparticles: A Quantum Chemical Scalar Relativistic Time Dependent Density Functional Study. J. Phys. Chem. C 2007, 111, 11862−11871. (19) 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. (20) Burgess, R. W.; Keast, V. J. TDDFT Study of the Optical Absorption Spectra of Bare and Coated Au55 and Au69 Clusters. J. Phys. Chem. C 2011, 115, 21016−21021. (21) Alvarez, M. M.; Khoury, J. T.; Schaaff, T. G.; Shafigullin, M. N.; Vezmar, I.; Whetten, R. L. Optical Absorption Spectra of Nanocrystal Gold Molecules. J. Phys. Chem. B 1997, 101, 3706−3712. (22) Palpant, B.; Prével, B.; Lermé, J.; Cottancin, E.; Pellarin, M.; Treilleux, M.; Perez, A.; Vialle, J. L.; Broyer, M. Optical Properties of Gold Clusters in the Size Range 2−4 nm. Phys. Rev. B 1998, 57, 1963− 1970. (23) Jana, N. R.; Gearheart, L.; Murphy, C. J. Seeding Growth for Size Control of 5−40 nm Diameter Gold Nanoparticles. Langmuir 2001, 17, 6782−6786. (24) Mock, J. J.; Barbic, M.; Smith, D. R.; Schultz, D. A.; Schultz, S. Shape Effects in Plasmon Resonance of Individual Colloidal Silver Nanoparticles. J. Chem. Phys. 2002, 116, 6755−6759. (25) Lindfors, K.; Kalkbrenner, T.; Stoller, P.; Sandoghdar, V. Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy. Phys. Rev. Lett. 2004, 93, 037401. (26) Berciaud, S.; Cognet, L.; Tamarat, P.; Lounis, B. Observation of Intrinsic Size Effects in the Optical Response of Individual Gold Nanoparticles. Nano Lett. 2005, 5, 518−518.
surface regions, and almost no distribution is found in the inner region. This is because only the conduction (or valence) s electrons freely oscillate. Such a surface charge oscillation due to the s electrons without the screening was also found in the study for Na clusters by Xiang et al.32 The present computed results clearly illustrate that the screening by the d-electrons plays a crucial role in determining optical properties.
4. CONCLUDING REMARKS We have studied the cluster-size dependence of LSPR for Aun clusters of up to n = 1414 (3.9 nm in diameter) by performing first-principles photoinduced electron dynamics calculations. The maximum cluster size is unprecedentedly large in comparison with those addressed in the previous, fully quantum mechanical, calculations of optical response in real cluster systems. The computations enable us to see that LSPR gradually grows, and its peak position red shifts (blue shifts) with increasing (decreasing) cluster size. These computed results are in good accordance with experimental observations. The localized surface charge distributions are visualized in real space, vividly illustrating the conduction electrons oscillate in a collective manner. From the visualization, LSPR has proven to be discernible at n ∼ 146, although the optical responses for Aun (n ≤ 146) are still mostly in a quantum regime because a quantum confinement effect remains predominant. The charge oscillation occurs in two regions, the outermost surface region and the inner region. The surface charge responds in synchronization with the applied laser field, whereas the inner charge oscillates locally around each gold atom in a direction opposite to the surface charge distribution. This is clear evidence of a screening effect caused by the d electrons. The present quantum mechanically accurate description of LSPR in gold nanoclusters provides valuable information when utilizing the properties of LSPR in developing plasmonic functional devices.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported by Grants-in-Aid (nos. 25288012 and 25730079) and by the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan. This work was also partly supported by JSPS Core-to-Core Program, A. Advanced Research Networks. The results presented in this paper have been primarily obtained by the K computer at the RIKEN Advanced Institute for Computational Science (Proposal number hp120035 and hp140054). The computation was also partly performed at the Research Center for Computational Science, Okazaki, Japan.
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(27) Cottancin, E.; Celep, G.; Lermé, J.; Pellarin, M.; Huntzinger, J. R.; Vialle, J. L.; Broyer, M. Optical Properties of Noble Metal Clusters as a Function of the Size: Comparison between Experiments and a Semi-Quantal Theory. Theor. Chem. Acc. 2006, 116, 514−523. (28) Peng, S.; McMahon, J. M.; Schatz, G. C.; Gray, S. K.; Sun, Y. Reversing the Size-Dependence of Surface Plasmon Resonances. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 14530−14534. (29) Scholl, J. A.; Koh, A. L.; Dionne, J. A. Quantum Plasmon Resonances of Individual Metallic Nanoparticles. Nature 2012, 483, 421−427. (30) Li, L.-H.; Hayashi, M.; Guo, G.-Y. Plasmonic Excitations in Quantum-Sized Sodium Nanoparticles Studied by Time-Dependent Density Functional Calculations. Phys. Rev. B 2013, 88, 155437. (31) Malola, S.; Lehtovaara, L.; Enkovaara, J.; Häkkinen, H. Birth of the Localized Surface Plasmon Resonance in Monolayer-Protected Gold Nanoclusters. Nano Lett. 2013, 7, 10263−10270. (32) Xiang, H.; Zhang, X.; Neuhauser, D.; Gang, L. Size-Dependent Plasmonic Resonances from Large-Scale Quantum Simulations. J. Phys. Chem. Lett. 2014, 5, 1163−1169. (33) Noda, M.; Ishimura, K.; Nobusada, K.; Yabana, K.; Boku, T. Massively-Parallel Electron Dynamics Calculations in Real-Time and Real-Space: Toward Applications to Nanostructures of more than Ten-Nanometers in Size. J. Comput. Phys. 2014, 265, 145−155. (34) Runge, E.; Gross, E. K. U. Density-Functional Theory for TimeDependent Systems. Phys. Rev. Lett. 1984, 52, 997−1000. (35) Calvayrac, F.; Reinhard, P.-G.; Suraud, E.; Ullrich, C. A. Nonlinear Electron Dynamics in Metal Clusters. Phys. Rep. 2000, 337, 493−580. (36) Giuliani, G. F.; Vignale, G. Quantum Theory of the Electron Liquid; Cambridge University Press: Cambridge, 2005. (37) Fundamentals of Time-Dependent Density Functional Theory; Marques, M. A. L., Maitra, N. T., Nogueira, F. M. S., Gross, E. K. U., Rubio, A., Eds.; Springer: Heidelberg, 2012. (38) Ullrich, C. A. Time-Dependent Density Functional Theory Concepts and Applications; Oxford University Press: New York, 2012, and references therein. (39) Kawashita, Y.; Yabana, K.; Noda, M.; Nobusada, K.; Nakatsukasa, T. Oscillator Strength Distribution of C60 in the TimeDependent Density Functional Theory. J. Mol. Struct.: THEOCHEM 2009, 914, 130−135. (40) Yabana, K.; Nakatsukasa, T.; Iwata, J.-I.; Bertsch, G. F. RealTime, Real-Space Implementation of the Linear Response TimeDependent Density-Functional Theory. Phys. Status Solidi B 2006, 243, 1121−1138. (41) Chelikowsky, J. R.; Troullier, N.; Saad, Y. Finite-DifferencePseudopotential Method: Electronic Structure Calculations without a Basis. Phys. Rev. Lett. 1994, 72, 1240−1243. (42) Chelikowsky, J. R.; Troullier, N.; Wu, K.; Saad, Y. Higher-Order Finite-Difference Pseudopotential Method: An Application to Diatomic Molecules. Phys. Rev. B 1994, 50, 11355−11364. (43) Fuchs, M.; Scheffler, M. Ab Initio Pseudopotentials for Electronic Structure Calculations of Poly-Atomic Systems Using Density-Functional Theory. Comput. Phys. Commun. 1999, 119, 67− 98. (44) Troullier, N.; Martins, J. L. Efficient Pseudopotentials for PlaneWave Calculations. Phys. Rev. B 1991, 43, 1993−2006. (45) Momma, K.; Izumi, F. VESTA 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data. J. Appl. Crystallogr. 2011, 44, 1272−1276. (46) Kittel, C. Introduction to Solid State Physics, 7th ed.; John Wiley & Sons, Inc.: New York, 1996. (47) Baletto, F.; Ferrando, R. Structural Properties of Nanoclusters: Energetic, Thermodynamic, and Kinetic Effects. Rev. Mod. Phys. 2005, 77, 371−423. (48) Barnard, A. S. Modelling of Nanoparticles: Approaches to Morphology and Evolution. Rep. Prog. Phys. 2010, 73, 086502. (49) Häberlen, O. D.; Chung, S.-C.; Stener, M.; Rösch, N. From Clusters to Bulk: A Relativistic Density Functional Investigation on a 11322
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First-Principles Computational Visualization of Localized Surface Plasmon Resonance in Gold Nanoclusters Kenji Iida, Masashi Noda, Kazuya Ishimura, and Katsuyuki Nobusada* Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, 444-8585, Japan ABSTRACT: Cluster-size dependence of localized surface plasmon resonance (LSPR) for Aun nanoclusters (n = 54, 146, 308, 560, 922, 1414) is investigated by using our recently developed computational program of first-principles calculations for photoinduced electron dynamics in nanostructures. The size of Au1414 (3.9 nm in diameter) is unprecedentedly large in comparison with those addressed in previous first-principles calculations of optical response in nanoclusters. These computations enable us to clearly see that LSPR gradually grows and the LSPR peaks red shift with increasing cluster size. The growth of LSPR is visualized in real space, demonstrating that electron charge distributions oscillate in a collective manner around the outermost surface region of the clusters. We further illustrate that the core d electrons screen the collective oscillation of the conduction-like s electrons.
1. INTRODUCTION Localized surface plasmon resonance (LSPR) is an essential optical process in metals, more specifically, noble metal nanoclusters, when subjected to an external laser field. The application of LSPR in stained glass has a lengthy history, dating back to Roman times. In the 20th century, LSPR was intensively investigated in a research field of basic science primarily in the context of understanding of fundamental properties of metal nanoclusters.1−3 LSPR has also recently attracted much attention in applied science, sometimes referred to as plasmonics, extending its potential applicability to functional materials and devices.4 Such valuable functions are strongly related to the characteristic properties of LSPR (i.e., highly resonant response to an applied laser field and strong electric field enhancement in a near field region). For example, light concentration in metal nanoclusters improves the efficiency of photovoltaic cells and metal nanocluster waveguides.5−8 Ultra sensitive response of LSPR to structural details at an interface region is very useful in developing biosensors and extreme measurement tools for single-molecule detection and imaging.9−13 Furthermore, nonlinear optical response due to electric field enhancement is expected to facilitate two or more photons excitation in wide band gap materials. LSPR of metal clusters can be explained conceptually in terms of a classical description of collective oscillation of surface charges. Thus, optical properties of LSPR of relatively large (larger than ∼30 nm) metal clusters have so far been frequently described by Mie scattering theory or a quasi-static approximation.14−16 The finite-difference time-domain method17 is an alternative powerful and efficient computational tool for calculating optical response in nanoclusters with an arbitrary shape. All the approaches mentioned above are apparently not beyond a classical description, in which materials are represented in terms of dielectric constants, and consequently cannot account for electronic structures of metal nanoclusters. © 2014 American Chemical Society
As the size of metal clusters decreases below 20 nm in diameter, LSPR becomes strongly dependent on cluster-size, geometric structure, and constituent atoms, all of which are unarguably related to electronic structures of nanoclusters. It is expected, in this size region, to see the transition from a classical feature of the collective oscillation to quantum mechanical optical response. If the size of clusters further decreases (< 2 nm), a quantum confinement effect is predominant over the classical description. One can carry out first-principles calculations of optical responses of such small clusters,18−20 whereas it is still highly computationally demanding to do so for nanoclusters of more than 2 nm in size. Experimentally, there have been a relatively large number of investigations on size dependence of LSPR in metal clusters.2,21−29 However, it is not easy to perform the experiments for individual (i.e., ligand free and ensemble heterogeneity free) clusters for sizes systematically ranging from a classical region to a quantum one. Recently, optical properties of LSPR were addressed both experimentally and theoretically in the transient size region of ∼1 to ∼20 nm.29−32 Scholl et al. performed an experiment on cluster-size dependence of LSPR for individual silver nanoclusters with diameters from 2 to 20 nm.29 The LSPR peak was found to shift to higher energy (i.e., blue shift) with decreasing cluster size. The theoretical studies on LSPR were conducted by Malola et al. for Au nanoclusters31 and by Xiang et al. for Na nanoclusters.32 Malola et al. carried out large first-principles, time-dependent density functional theory (TDDFT), calculations for Au nanoclusters with different sizes up 2 nm. Their calculations indicated the presence of LSPR in Au314(SH)96 at the maximum size (2 nm). However, since they did not calculate the size-dependence of LSPR for larger clusters, the Received: August 31, 2014 Revised: November 1, 2014 Published: November 4, 2014 11317
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linearly polarized laser field is applied, and a damping factor as a function of T is employed.40 The oscillator strength distribution of eq 4 is obtained by performing Fourier transformation of time-dependent induced dipole in the gold clusters subjected to an external laser field. Thus, this quantity is straightforwardly related to optical absorption spectra38 and describes well the LSPR optical properties. To be precise, df(E)/dE should be given by averaging over all axes of ν = x, y, and z. However, we computationally confirmed that since the clusters considered have approximately spherical symmetry, the difference in df(E)/dE with respect to the polarization direction has no qualitative influence on the following discussions on LSPR. Thus, only the x component (ν = x) is evaluated in this study. To investigate the light-induced electron dynamics, we apply an external laser pulse whose functional form is given as
growth of LSPR was not discussed as a function of cluster size. Xiang et al. investigated, on the basis of first-principles calculations with a practical approximation, the size-dependence of LSPR in Na clusters with diameters ranging from 0.7 to 12.3 nm. They demonstrated a nonmonotonic shift of the LSPR peaks as a function of cluster size and clearly explained the reason for the shift. However, the sodium cluster is a simple electronic structure system and thus the optical features of its LSPR are quite different from those of gold or silver clusters. Furthermore, from the viewpoint of applications of LSPR to functional materials in real systems, an understanding of the LSPR optical properties of gold or silver nanoclusters is highly desirable. Very recently, our group developed a highly efficient firstprinciples TDDFT computational program for electron dynamics solved in real-time and real-space, named GCEED (grid-based coupled electron and electromagnetic field dynamics).33 In this paper, GCEED is applied to the optical response of Aun clusters (n = 54, 146, 308, 560, 922, and 1414). GCEED can carry out full first-principles TDDFT calculations without a practical approximation. To the best of our knowledge, the cluster size of Au1414 (3.9 nm in diameter) consisting of 15554 electrons is unprecedentedly large compared with those addressed in previous first-principles calculations of optical response in nanoclusters. Therefore, the growth of LSPR in the gold clusters is revealed, for the first time, using first-principles calculations. Furthermore, the realspace treatment of GCEED allows us to accurately visualize this growth.
vext(r, t ) = −rνFt (t )cos(ωt )
where rν is one of the Cartesian coordinates (ν = x, y, and z) indicating the laser polarization, Ft is a pulse envelope function, and ω is the laser frequency. The pulse envelope is given by Ft (t ) = F sin 2(πt /τ ) (0 < t < τ )
⎤ ⎡ 1 ∂ ψ (r, t ) = ⎢ − ∇2 + Veff (r, t )⎥ψj(r, t ) ⎦ ⎣ 2 ∂t j
(1)
where Veff is the effective potential given by Veff (r, t ) = vnuc(r) +
∫
ρ(r′, t ) dr′ + vXC(r, t ) |r − r′|
+ vext(r, t )
(2)
Here, vnuc(r), vXC(r), and vext(r) are the nuclear attraction potential, the exchange-correlation potential, and the external potential of an applied laser field, respectively. ρ in eq 2 is the electron density given by N /2
ρ (r , t ) = 2
∑ |ψj(r, t )|2 (3)
j=1
where the factor of 2 indicates that each orbital is fully occupied. For the exchange-correlation potential vXC(r), the adiabatic local density approximation is used. The oscillator strength distribution df(E)/dE is given by39 d f (E ) 2E = Im πF dE
T
∫0 [∫ rν{ρ(r, t ) − ρ(r, 0)}
× {1 − 3(t /T )2 + 2(t /T )3 } dr]eiEt dt
(6)
where τ determines the laser pulse duration. The field strength F is related to the laser intensity by I = 1/2ε0cF2, where ε0 is the permittivity of vacuum and c is the speed of light. To perform the practical calculations, we utilize GCEED wherein the TDKS equation is directly solved at threedimensional Cartesian grid points by a finite-difference formula.41,42 The time-propagation is evaluated by the Taylor expansion method. The computational program is completely free from algorithms involving eigenvalue problems and fastFourier-transformation, which are usually implemented in conventional quantum chemistry or band structure calculations. Because of its excellent parallel efficiency, the program is highly suitable for massively parallel calculations and therefore allows us to carry out first-principles calculations of a large gold cluster system of Au1414. The maximum size of the computation box for Au1414 is set to 64 Å × 64 Å × 64 Å. All electrons other than Au 5d6s electrons are replaced with the effective core potential (ECP) obtained by the Troullier−Martins scheme implemented in the fhi98PP program.43,44 Then, the total number of the electrons of Au1414 is 15554. We employ uniform space grids Δx, Δy, and Δz of 0.25 Å. The real-time propagation is performed by using the constant time step of 1.316 × 10−3 fs. The applied laser pulse duration and intensity are set to 30 fs and 1.0 × 108 W/cm2, respectively. Time propagation takes 6 h for the largest cluster Au1414 by using 62400 CPU cores of the K computer (SPARK64 VIIIfx 2.0 GHz, 8 CPU cores/node). Visualization of the time-dependent induced density is performed by using the Vesta package.45 Using GCEED, we calculate the optical response of the icosahedral Aun clusters (n = 54, 146, 308, 560, 922, and 1414) where one atom at an apex is removed to consider closed-shell electronic structures. In the present computations, we assume the icosahedral structures for all the Au clusters without geometry optimization. Their bond lengths are, on average, less than 3% shorter than those of the bulk gold.46 Although the structure of a gold cluster has been extensively discussed,47,48 previous experimental and theoretical studies show that the icosahedral structure is one of the major compositions at the sizes less than 10 nm.49−51 Furthermore, the optical absorption spectrum of an icosahedral Au147 cluster was shown to be less affected by slight geometry compression.52 We consider that
2. MASSIVELY PARALLEL FIRST-PRINCIPLES DFT CALCULATIONS The present approach is based on the time-dependent Kohn− Sham (TDKS) theory. TDKS equation is written in the form of (in atomic units),34−38 i
(5)
(4)
where F is the strength of a pulse field, T is the total propagation time, rν(ν = x, y, and z) is the axis along which the 11318
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the icosahedral structure is the most reasonable structure for systematically discussing the growth of LSPR. Thus, we focus on such icosahedral structures. The sizes of the clusters are 1.1 nm (Au54), 1.7 nm (Au146), 2.2 nm (Au308), 2.8 nm (Au560), 3.3 nm (Au922), and 3.9 nm (Au1414) in diameter, respectively. Three of these (n = 54, 560, and 1414) are exemplified in Figure 1.
clusters have multiple resonance peak structures and no clear LSPR (see the inset in the figure and also following discussion). This is simply because the cluster size is so small that a quantum confinement effect is predominant and they have discrete electronic energy levels.31,53 The oscillator strengths for the larger clusters rapidly increase and additional spectral features appear in the higher energy range. This is mainly due to the interband transitions of the d electrons.2 It is also noted that the LSPR peaks are not sharp in comparison with those of other metals such as sodium and silver clusters because of the screening effect by the d electrons.2,32,54 We will discuss the screening effect in Section 3.3. As mentioned earlier, the properties of LSPR strongly depend on size, geometric structure, and constituent atoms of a cluster. Since all the clusters considered consist only of gold atoms and have a homogeneous icosahedral structure with near-spherical symmetry, we can reasonably define the renormalized oscillator strength, dividing the original strength by the number of atoms of a cluster (as in the lower panel of Figure 2). The renormalization allows us to focus on the growth of LSPR on a unified scale. The renormalized curves for Au560, Au922, and Au1414 exhibit rather structureless, broad features similar to each other, whereas the curves for the smaller (n ≤ 146) gold clusters show multiple resonance peak structures as mentioned above. This figure clearly demonstrates that LSPR appears in Au308 (2.2 nm) and grows in the larger clusters. Malola et al. also reported the appearance of LSPR in a 2 nm gold cluster.31 The present computed results of the LSPR peak positions and their size dependence, that is, the peaks red shift (blue shift) with increasing (decreasing) cluster size, are in good accordance with experimental observations.21−23,27 The peak position of 2.3 eV for Au1414 is close to the experimental values of 2.34−2.38 eV for gold clusters with the sizes of 3.5− 37 nm in diameter in condensed phase.15,23 These values are also close to the value of Mie plasmon resonance (∼2.4−2.5 eV) evaluated by using bulk gold dielectric function.55 3.2. Growth of LSPR. Here, we carry out further firstprinciples visualization of the growth of LSPR. To this end, we define the quantity of frequency-dependent difference in electron density using the imaginary part of the Fouriertransformed time-dependent induced density as follows:
Figure 1. Icosahedral Aun clusters (n = 54, 560, and 1414).
3. RESULTS AND DISCUSSIONS 3.1. Optical Absorption Spectra. Figure 2 (top) shows the oscillator strength of Aun. It is clear from this figure that
Δρ(r, E) ≡ Im
∫0
T
{ρ(r, t ) − ρ(r, 0)}
× {1 − 3(t /T )2 + 2(t /T )3 }eiEt dt
(7)
where the electron density is calculated by applying an xpolarized laser field. As seen from eqs 4 and 7, Δρ is associated with the spatial component of the oscillator strength. Thus, Δρ yields a clear and detailed insight into the optical response of LSPR in a configuration space. Figure 3 shows Δρ of the Aun clusters at E = 2.7 eV for Au54, 2.7 eV for Au146, 2.6 eV for Au308, 2.5 eV for Au560, and 2.3 eV for Au1414, which correspond to each intense resonance frequency or LSPR frequency. The figures are shown from the direction perpendicular to the x−y plane. The red and blue colors, respectively, indicate increase and decrease in the density from the ground state. The electron density for Au54 is distributed rather irregularly. Relatively positive and negative charge densities for Au146 are distributed around the top and bottom sides along the laser polarization direction. This separation indicates that the beginning of LSPR can be traced to Au146, whereas the corresponding absorption spectrum in Figure 2 still has multiple resonance peak
Figure 2. (Top) Oscillator strengths of Aun (n = 54, 146, 308, 560, 922, and 1414) and (bottom) their renormalized strengths with respect to the number of atoms. Insets in the upper and lower panels show magnifications of the curves for Au54 and Au146 and for Au560, Au922, and Au1414, respectively.
LSPR gradually grows and becomes discernible as the cluster size increases. The LSPR peak positions red shift with increasing size, and those for Au308, Au560, Au922, and Au1414 are found to be 2.6, 2.5, 2.5, and 2.3 eV, respectively. As is predicted, however, the curves for the smaller Au54 and Au146 11319
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frequencies are set to E = 2.7 (Au54) and 2.3 eV (Au1414), respectively. The red and blue colors indicate the increase and decrease in the electron density from the ground state, respectively. The positive (red) and negative (blue) charge densities for Au54 are distributed rather irregularly. The charge densities for Au1414, on the other hand, show regular patterns, indicating that the positive and negative charges are distributed around the upper and lower surface regions, respectively, synchronizing the x-polarized external laser field. This is exactly the same behavior as the surface charge distributions seen in Figure 3. Furthermore, the charge oscillations occur locally around each gold atom inside the cluster in the direction opposite to the surface charge oscillation. This behavior of electron dynamics subject to an applied laser field indicates a typical screening effect, which has been widely recognized, particularly in metal clusters.2,54,56,57 Very recently, Malola et al. also clearly demonstrated the screening effect of d electrons in 2 nm gold clusters by using a linear response TDDFT method.31 The screening effect in the present gold clusters is due to the fact that the conduction (or valence) s-electrons follow the oscillating laser field, whereas the core d-electrons suppress (i.e., screen) the oscillatory motion of the conduction electrons. Here, the importance of the d-electron dynamics in the optical property is definitively demonstrated by freezing the d-electron motion. Figure 5 shows an absorption spectrum of a model
Figure 3. Induced densities of Aun (n = 54, 146, 308, 560, and 1414) seen from the direction perpendicular to the x−y plane. The thresholds of the density maps are set to the same value. The applied laser energies are indicated in the unit of eV.
structures with no clear LSPR peak. As the cluster size further increases, the separation of the charge densities becomes more apparent. These features found in Figure 3 are clear evidence of the growth of LSPR. 3.3. Screening Effect. Although Figure 3 visualizes the charge oscillation of LSPR and its growth depending on cluster size, only the charge distribution on the surface was discussed. To further analyze LSPR, let us next examine the electron dynamics in an inner region of the cluster. We illustrate finitewidth cross sections of Δρ in the x−y plane in Figure 4. These density maps of the cross sections are seen from z < 0 of the disklike moieties obtained by slicing Aun from z = −2 to +2 Å. For convenience, the gold atoms are drawn smaller than those in Figure 3. The incident laser is x-polarized and the
Figure 5. Absorption spectrum of a model Au1414 cluster in which the d electrons are replaced with ECP (top) and a disklike moiety of the induced density of the cluster sliced at z = −2 and +2 Å. The applied laser energies are indicated in the unit of eV.
cluster Au1414 in which all the d electrons are replaced with ECP and only the 1414 s electrons are treated explicitly. The peak intensity becomes significantly strong, and the position largely shifts to the higher energy by ∼3 eV. The d-electron motion has also great influence on the electron dynamics, as shown in the lower panel of Figure 5. This figure is the same as the density map for Au1414 in Figure 4 but for the model Au1414 cluster without the d electrons. The incident laser frequency is set to the resonance peak position of 5.45 eV. The positive and negative charge densities are distributed only around the cluster
Figure 4. Disklike moieties of the induced density of Aun (n = 54 and 1414) sliced at z = −2 and +2 Å. The thresholds of the density maps are set to the same value. The applied laser energies are indicated in the unit of eV. 11320
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surface regions, and almost no distribution is found in the inner region. This is because only the conduction (or valence) s electrons freely oscillate. Such a surface charge oscillation due to the s electrons without the screening was also found in the study for Na clusters by Xiang et al.32 The present computed results clearly illustrate that the screening by the d-electrons plays a crucial role in determining optical properties.
4. CONCLUDING REMARKS We have studied the cluster-size dependence of LSPR for Aun clusters of up to n = 1414 (3.9 nm in diameter) by performing first-principles photoinduced electron dynamics calculations. The maximum cluster size is unprecedentedly large in comparison with those addressed in the previous, fully quantum mechanical, calculations of optical response in real cluster systems. The computations enable us to see that LSPR gradually grows, and its peak position red shifts (blue shifts) with increasing (decreasing) cluster size. These computed results are in good accordance with experimental observations. The localized surface charge distributions are visualized in real space, vividly illustrating the conduction electrons oscillate in a collective manner. From the visualization, LSPR has proven to be discernible at n ∼ 146, although the optical responses for Aun (n ≤ 146) are still mostly in a quantum regime because a quantum confinement effect remains predominant. The charge oscillation occurs in two regions, the outermost surface region and the inner region. The surface charge responds in synchronization with the applied laser field, whereas the inner charge oscillates locally around each gold atom in a direction opposite to the surface charge distribution. This is clear evidence of a screening effect caused by the d electrons. The present quantum mechanically accurate description of LSPR in gold nanoclusters provides valuable information when utilizing the properties of LSPR in developing plasmonic functional devices.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported by Grants-in-Aid (nos. 25288012 and 25730079) and by the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan. This work was also partly supported by JSPS Core-to-Core Program, A. Advanced Research Networks. The results presented in this paper have been primarily obtained by the K computer at the RIKEN Advanced Institute for Computational Science (Proposal number hp120035 and hp140054). The computation was also partly performed at the Research Center for Computational Science, Okazaki, Japan.
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