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Valence Electron Density-Dependent Pseudo-Permittivity for Nonlocal Effects in Optical Properties of Metallic Nanoparticles Chao Chen, and Shuzhou Li ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00122 • Publication Date (Web): 12 May 2018 Downloaded from http://pubs.acs.org on May 13, 2018
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Valence Electron Density-Dependent Pseudo-Permittivity for Nonlocal Effects in Optical Properties of Metallic Nanoparticles
Chao Chen, and Shuzhou Li*
School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore *E-mail:
[email protected] 1
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ABSTRACT The peak positions of localized surface plasmonic resonance (LSPR) exhibit strongly dependent on sizes of metallic nanoparticles. TDDFT calculations have shown remarkable size effect for metallic nanoparticles smaller than 1 nm, because it could account for fully nonlocal effects. Due to the high resource consumption of TDDFT, several semi-quantum approaches have been proposed to reduce the computation time while addressing nonlocal effects, and it is still desirable to introduce new ideas into this area since physical origins of related fields are not completed known yet. In this work, we took account of both spilling out of s-band electrons and screening effect of d-band electrons in LSPR phenomena, and developed a model using pseudo-permittivity to describe several quantum mechanical effects that contribute to nonlocal effects in LSPR. With incorporation of machine learning, this model is capable to calculate optical response of large nanostructures above nanometer scale. Besides successful prediction for different metallic nanoparticle monomers, the tunneling effect occurring in dimers can also be well described by using the concept of pseudo-permittivity,. The employing of pseudo-permittivity and machine learning is expected to achieve both high accuracy and high efficiency in quantum plasmonics. It provides a new ideology in simulation of wave-matter interactions. KEYWORDS: plasmonics, size effects, semi-classical semi-quantum, electron spilling out, d-band screening, machine learning, dimer structures.
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Localized surface plasmon resonance (LSPR) is a collective oscillation of valence electrons at surface of metallic nanoparticles (MNPs) under external electromagnetic fields1-2. Recent experimental studies have reported that both noble metal nanoparticles and alkali metal nanoparticles show size dependent LSPR peak shift3-6. For silver nanoparticles (AgNPs), a significant blueshift of LSPR energy has been detected with decreasing diameter via Electron Energy Loss Spectroscopy3, while an opposite trend has been observed for sodium nanoparticles (NaNPs), whose LSPR energies have been measured by photo-absorption methods5-7. These size dependent effects are promising phenomena to provide programmable tunability in performance of nano-plasmonic devices at nanometer or sub-nanometric scale. To get accurate predictions of these plasmonic structures, approaches under the frame of quantum mechanics are needed since the validity of classical electromagnetic model (CEM) is limited to large systems3,
8-11
. Recent time-dependent density functional theory
(TDDFT)12-13 simulations have demonstrated optical properties of sub-nanometric MNPs14-15, which also predicted size dependent LSPR energy shift for NaNPs and AgNPs, respectively. However, the scope of applications of fully quantum-mechanical approach are limited to very small MNPs containing no more than a few thousands valence electrons16-17 because of high consumption of computational resources. Therefore, the current computing capacity of traditional TDDFT tools still cannot fulfill the requirement of the fast developing nano-plasmonic devices18-19, which may typically contain even millions of valence electrons and involve sub-nanometric components to enhance performance20. It is generally accepted that the inaccuracy of CEM in quantum plasmonics lies in local response approximations, who intuitively describes light-matter interactions as spatially local responses. Thus, a new family of simulation tools referred as semi-classical model is springing up21-22. These semi-classical models incorporate nonlocal effect into solution of Maxwell equations. Famous examples are hydrodynamics Drude model (HDM)23-25, quantum corrected model (QCM)26-28,
projected dipole model (PDM) and their improved models.
General nonlocal optical response (GNOR)29, curl-free hydrodynamic model, and self-consistent hydrodynamic model (SC-HDM) have made major improvements from 3
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original HDM. The reported artificial fluctuation issue occurred in HDM has been solved by GNOR and curl-free hydrodynamic model. Moreover, SC-HDM also manages to fulfill interbond transitions of d-band electrons into their semi-quantum model. Besides hydrodynamics, another category of semi-classical models uses fictitious materials to solve nonlocal effect in LSPR. The QCM26, 28, a widely known semi-classical model, highlights electron tunneling effect30 in plasmonic systems embodied with sub-nanometric gaps by adding fictitious material with separation-dependent dielectric constant. The PDM approach31 also proposed a fictitious infinitely thin layer of dipoles to account for quantum effects of plasmonic phenomena. PDM can be used to predict quantum plasmonic aspects of dimers with subnanometric gaps. Since nonlocal effect is more remarkable at surface region, especially when immersed in liquid or connected with ligand, some recently works have also put forward core-shell structured models to give more nonlocal treatment to surface regions of MNPs than inner parts32-33. The shell parts of these models are somehow considered as fictitious materials with their dielectric constant dependent on outside chemical environment. The rationality of core-shell treatment lies in the fact that optical properties of surface regions are quite different from core parts. Inspired by achievements of previous works, we focused on build up the valence electron density-dependent pseudo-permittivity (VEDP) model. This new model aims to provide new ideas and alternative approaches in theoretical studies and to achieve both high accuracy and high efficiency in quantum plasmonics. By Runge-Gross theorems, optical properties of MNPs are uniquely determined by their ground state valence electron density functional15, which can be predicted by ground state DFT calculations or machine learning. The concept of pseudo-permittivity, derived from valence electron density, was introduced into VEDP to measure the resistance of valence electrons under external electric field in plasmonic structures. We focused on behavior of s-band and d-band valence electrons since only s-band and d-band valence electrons play roles for most plasmonic metallic elements. Based on control simulations and previous literatures, we found that spilling out effect of s-band valence electrons7,
34-35
and screening effect of d-band electrons7, 4
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36
are critical issues in
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nonlocal behaviors of quantum plasmonics. Models ignoring these two effects may cause incorrectness in calculation of LSPR spectra. In VEDP, each MNP is divided into a shell region and a core region, and different pseudo-permittivity values are assigned to different regions to describe these two effects. The validity domain of VEDP has been benchmark by TDDFT in subnanometric scale for MNPs with closed shell structures and by CEM in 10-nm-scale where quasistatic approximation is still valid via extrapolation method. Within this domain, VEDP is capable to account for unique features of quantum plasmonics such as size dependent LSPR energy shifts of MNP and charge transfer induced plasmons in dimer structures. The calculated near filed spectra inside subnanometric gaps of dimer systems revealed that the gap separation dependent field enhancement could be quite different for different plasmon modes, and no abrupt changes of observable physical quantities when two separated MNPs come into contact, which has reflected the physical reality.
PSEUDO-PERMITTIVITY MODEL The ideology of VEDP has been illustrated in Figure 1. Drude model and quantum jellium model are two widely used models to describe LPSR of MNPs. In classical Drude model, the particle is treated as an object with homogeneous optical properties inside the surface boundary. The free electron density, , remains constant inside particle and drops to zero abruptly at the surface boundary as illustrated in the classical model of Figure1. The frequency dependent permittivity, εω, can be expressed as: ε ω = 1 −
where the plasmon frequency ( ), and damping frequency ( depend on 34:
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=
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= In eq. 3, the parameter is a constant coefficient for calculation of damping frequency ( and this coefficient may change value for different metallic elements34. In this article, the value of for sodium particles is 0.01673 #$ ∗ Å 0.002102 #$ ∗ Å
( ',
( '
, and for silver particles is
respectively. In quantum jellium model, the positively charged ions are
assumed to be uniformly distributed inside surface boundary of a jellium, and free electrons own wave-like properties described by quantum mechanics. The free electron density ( ), has periodic fluctuations inside the jellium, and decrease continuously to zero outside surface boundary (as shown in Figure 1). Jellium model has good performances in describing optical responses dominated by free electrons in conduction band, but poor performances for tightly bounded d-band electrons37. For alkaline metals whose all valence electrons lie in s-band, jellium model could provide accurate results38-39. The VEDP model divides a MNP into core part and shell part, being a combination of classical Drude model and quantum jellium model. In core region, the periodic fluctuational electron density ( has been well averaged to a constant value *+, , which can be well described by Drude model. In the shell region, nonlocal effects become non-negligible thus local electron density (./00 should gradually vanish as that in jellium model. Therefore,
the entire electron density profile in space [ 1] of a MNP is separately represented by
./00 and *+, in VEDP. If 1 is known, other observable physical quantities are all
functionals of 1, according to Hohenberg-Kohn theorems40 and Runge-Gross theorems15. This core-shell treatment of valence electron density and permittivity is similar to MNPs pseudopotentials in DFT, where the unsensitive inner part and sensitive outer part are interpreted separately. Therefore, the calculated permittivity is named as pseudo-permittivity, while this new model is referred as valence electron density-dependent pseudo-permittivity
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model. The shell regions describe the regions where electron spilling out effect occurs. For a single MNP, its shell region slightly overlaps with Wigner–Seitz radius41 of the particle’s outmost layer atoms. The depth of shell regions is usually within several angstroms. Theoretically, permittivity in shell region gradually changes as the value of radical distance (r) increases. Multiple shell layers should be used to increase accuracy of VEDP model. However, for spherical nanostructures, adoption of multilayers in shell region can hardly increase accuracy of simulations numerically, which is shown in Supporting Information with more details. In this study, the whole shell regions of nanospheres were approximately treated as one single layer (as illustrated in Figure 1), and the pseudo-permittivity of this layer was derived from averaged electron density, ./00_456 7 in shell region: :
:
./00_456 7 = 8: ;?@
The cluster size is represented by 7, the number of atoms contained by one MNP. From previous works which visually demonstrated LSPR process36, 42, it has been reported that electrons forming plasmon on particles surfaces are mainly s-band electrons, while d-band electrons stay polarized in the opposite direction within core regions43. Therefore 7 evaluates s-band electrons only. By deriving plasmon frequencies and damping frequencies from 7, pseudo-permittivity of shell region of MNPs can then be calculated from eq. 1. NaNPs with magic numbers (92, 138, 198, 268, 338, 440…) are studied, and these structures are also referred as closed shell clusters44-46. If the pseudo-permittivity of both shell regions and core regions of a MNP determined, its optical properties can then be calculated by solving original Maxwell equations. In this work, FDTD method provided by MEEP (MIT Electromagnetic Equation Propagation) package was used to do the simulations. For the core part of MNPs, their pseudo-permittivity can be treated classically, which equals 7
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to the metals dielectric constant in bulk materials. The pseudo permittivity of NaNPs’ core regions is described by Drude model in eq. 1, and 5.89 eV is the plasmon frequency,
0.38 eV is the damping frequency47. For AgNP, their pseudo-permittivity of core parts can be described by a Drude-Lorentz model that fits the data published by Johnson and Christy48 with Differential Evolution Method : H ω = 1 −
+ ∑ MN
KL
L L
In our calculation, totally N=5 Lorentzian peaks have been adopted, with all the parameters present in Table 1. For the convenience of numerical calculation, the Drude term was rewritten in the form of Lorentz term, and treated as the 0th Lorentz term: H ω = ∑ MP
OL ∗
L L
Table 1. Parameters in fitting a Drude-Lorentz model with 5 Lorentzian items with Johnson and Christy data.
N
#$
#$
Q
0 (Drude term)
0.000
0.048
0.990
1
0.816
3.886
0.065
2
4.481
0.452
0.124
3
8.185
0.065
0.011
4
9.083
0.916
0.840
5
20.29
2.419
5.646
Absorption spectra of closed shell NaNPs are shown in Figure 2a, 2c and 2e, which are calculated by quantum atomistic model, quantum jellium model, and VEDP model, respectively. Both quantum atomistic model and jellium model results show redshift of LSPR
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energy as particle size decreases (Figure 2a, c), and these two models show remarkable agreement quantitatively. This result raised up naturally since these two models describe the orbitals of s-band electrons similarly, and the collective oscillation of s-band free electrons plays the pivotal role in plasmonics of alkaline metals. The VEDP model also exhibits great resemblance with quantum models. By generating pseudo-permittivity of shell region of closed shell NaNPs, VEDP predicted a redshift of 0.15 eV in LSPR energy, as the size of NaNPs decreases from 440 atoms to 92 atoms (Fig 2e), very close to the value of 0.14 eV predicted by jellium model. The absorption spectra of AgNPs with magic numbers (93, 139, 199, 269…) are also studied, which are shown in Fig 2b, 2d and 2f. TDDFT calculations showed that atomistic model and jellium model exhibit quite different physical pictures. Without consideration of d-band electrons, LSPR peaks of AgNPs predicted by jellium models are all around 5.0 eV (Figure 2d), much larger than atomistic model’s simulation results (around 3.5 eV in Figure 2b). This disagreement originates from d-band electrons’ screening effects in LSPR process. When external electric field applied, drift of s-band free electrons would accumulate at surface region and form a dipole against the direction of the external field. The d-band valence electrons, on the other band, would polarize in the opposite direction inside the core region, decreasing the polarity of the dipole moment formed from s-band electrons, which would in turn reduces LSPR energies. Literatures about DFT calculations have also reported that pseudopotentials underestimating d-band screening effect may lead to overestimation of LSPR energy and peak intensity, and vice versa36, 49. The jellium model, totally ignoring d-band electrons, certainly results in very large LSPR energies. To stress this issue, Lorentz terms have been added to pseudo-permittivity of core parts of AgNPs in VEDP model, since ab initio calculations of LSPR of MNPs have proved that d-band screening hardly occur in shell regions7, 36. Absorption spectra of AgNPs obtained from VEDP model show blueshift of LSPR energy with particle size decreasing (Figure 2f). This trend showed good agreement with calculation of quantum atomistic model.
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MACHINE LEARNING ENHANCED VEDP MODEL In VEDP model, core regions are treated classically, while shell regions are treated quantum mechanically. Therefore, information of valence electron densities of shell regions must be achieved from ground state DFT calculations. Although this approach is much more efficient than TDDFT, it still takes more than 5000 CPU hours to calculate absorption spectrum of a MNP containing 2048 sodium atoms. The development of machine learning50 technologies can help to accelerate calculations when quantum effects are to be considered51-54. A neural network based feed-forward back-propagation algorithm named Extreme learning machine (ELM)50 has been adopted to predict valence electron density of MNPs in the shell region. To predict electron densities of MNPs in their shell regions by ELM, raw learning data achieved from DFT calculations is still needed. we firstly calculated electron densities of NaNP and AgNP with magic number atoms via jellium model. Due to the symmetry properties of electron clouds in spherical nanoparticles44-46, the electron density value of each MNP at any point in shell region can be labelled as ./00 7, 9, where 7 is the number of
atoms contained by the MNP and 9 is the radical distance from center of the particle. With
all the data sets (./00 7, 9, 7, 9) from closed shell nanoparticles for 7 ranging from 92 to
832 calculated by ground state DFT, mathematical relationship between ./00 7, 9, 7, and
9 can then be learned by ELM.
It has been reported in literatures that the spilled out free electron density profiles roughly obey Fermi–Dirac distribution function35, 55. By fitting the discretized electron density data in shell regions achieved from DFT calculation and ELM prediction using differential evolution method, we found that ./00 7, 9 obey Fermi-Dirac function multiplied by a cosine function
instead
of
the
pure
Fermi-Dirac
distribution
function.
Therefore,
a
“Fermi-Dirac-Cosine” formula was set up for the convenience of following calculations for spherical MNPs: 10
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./00 7, 9 = *+, + R cosV9 − W*+, 7] ∗
NX
N
Y@Z[\==L]^ _
In eq. 7, the wave number in eq. 7, ω, is determined by periodicity of 1 in core region.
W*+, 7 and W`00a 7 are radius of core region and jellium radius of closed shell particles containing n atoms, respectively. The value of W`00a 7 can be calculated from eq. S1 in
Supporting Information. The definition of W*+, 7 is the position where 1 achieve local maximum near surface boundary, as illustrated in eq. 8. bc?@ = 0
A and B in eq. 7 are two fitting parameters achieved by differential evolution method upon ELM’s prediction data, with values shown in Figure 3a and Figure 3b for NaNPs. By investigating the evolution of parameters, we found that both A and B are approaching fixed values with the increase of particle size. Based on ELM, for large NaNPs containing more than 400 atoms, A and B fluctuate around 0.0037 and 2.005, respectively. Excellent agreements with DFT simulations have been got as is illustrated in Figure 3c. The tolerance of machine learning approximation together with differential evolution was at the level of 10 electrons Åf. Therefore, the accuracy of ELM enhanced VEDP model in optical responses is at the same level as that of DFT based VEDP model. For NaNPs with more than 2000 atoms, A and B can be approximately taken as 0.0037 and 2.005 if DFT level accuracy is not required in rough calculations.The accuracy of VEDP model also depends on the determination of shell region’s outer boundary, W./00 (n), and W./00 (n) locates at the
position where 1 attenuates to less than 10% of *+, , as state in eq. 9. c
