Anisotropic Polarizability Induced Plasmon Transfer

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C: Physical Processes in Nanomaterials and Nanostructures

Anisotropic Polarizability Induced Plasmon Transfer Greta Donati, David B Lingerfelt, Christine M. Aikens, and Xiaosong Li J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02425 • Publication Date (Web): 25 Apr 2018 Downloaded from http://pubs.acs.org on May 1, 2018

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The Journal of Physical Chemistry

Anisotropic Polarizability Induced Plasmon Transfer Greta Donati,† David B. Lingerfelt,† Christine M. Aikens,∗,‡ and Xiaosong Li∗,† †Department of Chemistry, University of Washington, Seattle, WA, 98195 ‡Department of Chemistry, Kansas State University, 1212 Mid-Campus Drive North, Manhattan, KS 66506 E-mail: [email protected]; [email protected]

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Abstract Despite the large number of technological applications relying on noble metal nanoparticles collective electron oscillations, or localized surface plasmon resonances (LSPR), a complete understanding of all factors affecting their dynamics has not yet been achieved. In this paper, a non-adiabatic Ehrenfest dynamics approach is employed to investigate the dynamics of a linear chain of silver atoms initialized in the transverse LSPR state. Out-of-chain motions are shown to cause the increase of one specific off-diagonal component of the molecular polarizability, inducing a polarization orthogonal to the direction of the transverse LSPR oscillation and consistent with the molecule’s geometrical orientation. These geometry changes allow the transfer from the initially excited transverse plasmon to a multipolar longitudinal plasmon. This unique plasmon transfer mechanism, allowed only by the symmetry change of the system and never observed before, sheds light on a previously unknown feature of metal nanoparticles.

Introduction

anisms driving their dynamics and decay in order to improve the properties and functionality of LSPR-supporting nanostructures. Indeed, although widely employed in several applications, a complete picture of the main molecular processes dictating the outcome of exciting a dipolar LSPR has not yet been achieved. LSPR decay can take place both radiatively and non-radiatively 24 and the timescales of these processes limit the material’s performance in many applications. When the decay takes place in a radiative way, plasmons decay through the re-emission of photons 25 while in the nonradiative case the decay leads to local heating and/or energetic electrons. 26–28 Recently, these “hot” electrons generated by non-radiative decay of plasmon excitations have shown promise toward a wide range of applications such as photocatalysis, surface imaging, and photodetection/photovoltaic devices. 24 There are still many open questions about the mechanism (or mechanisms) underlying the plasmon decay and other possible relaxation pathways beyond the ones considered previously, and the efficiency of the devices based on the plasmon excitation decay can be considerably improved with greater insights into the mechanism underlying the decay. In this context only a theoreticalcomputational approach from first principles can fully disentangle the complex dynamics taking place after the plasmon formation. In this paper, a mixed quantum-classical Ehrenfest dynamics simulation approach is utilized to investigate the mechanisms of plasmon decay. The role that nuclear motions play in

Localized surface plasmon resonance (LSPR), 1 i.e. a collective oscillation of the conduction electrons taking place in metal nanoparticles after interaction with an electric field, has driven a huge variety of applications such as photovoltaic, molecular sensing, energy transfer, photoemission, nanoscale microscopy, and vibrational spectroscopy. 2–12 Dipolar LSPR’s consist of collective oscillations of conduction electrons across the metal particle, and result from resonant interaction between these conduction electrons and an electromagnetic field. Dipolar LSPR excitations are characterized by large absorption (or scattering, for larger nanoparticles) cross sections giving rise to a high extinction in the UV-vis region 1 that depends on the size, shape and medium surrounding the nanoparticle. 13,14 This feature permits the tuning of these material’s optical properties to obtain most efficient performance for a given application. LSPR’s have been studied satisfactorily using classical electrodynamics (for example, the Drude oscillator model), but this approach is no longer sufficient when an atomistic description of the LSPR is desired, or when these excitations take place in small noble metal particles. 13,15–19 LSPR states have also been appreciated in both “simple” alkali and noble metal particles consisting of only a few atoms, in which case the optical spectra become more complicated. 20–23 It is important to understand the timeevolution of LSPR’s and the molecular mech-


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influencing electronic degrees of freedom and coupling plasmon modes of different symmetries and nodal structures is revealed in the coupled electronic/nuclear dynamics simulation of the dipolar LSPR excitation. Nuclear motions have a strong influence on the plasmon lifetime in this model system, with specific nuclear motions driving the plasmon decay and others allowing the LSPR to survive. 29 In this work it is shown that the spatial orientation of nuclear displacements dictates not only the plasmon decay, but also how it can be coupled to higherorder multipolar LSPR’s that are inaccessible to optical excitation from the far field. A linear silver chain was chosen to study the LSPR decay and transfer phenomena. Since the nanowires belong to the D∞h point group, the delocalized frontier orbitals can be labeled by Greek letters, Σ, Π, etc. This system is a minimal complexity, fully ab initio model for the class of nanorod and nanowire particles, employed in the medical therapeutics field 7,30 The relevance of such minimal systems to study and understand nanoplasmonics has also been underlined by other studies. 31–33 The metal nanostrucutres with cylindrical symmetry exhibit two types of LSPR. The longitudinal ones, resulting from Σ → Σ transitions (see MO energy diagram in Fig. 4), are excited when the external field is applied with polarization along the chain, and the two-fold degenerate transverse ones resulting from Σ → Π transitions are excited when the field is polarized orthogonal to the axis of the chain. 34–39 The transverse plasmon modes have been shown to be strongly influenced by the nuclear motions, 29 and were therefore chosen for this investigation. Our results suggest that there is a strong connection between the static molecular polarizability at a given geometry of the atomic chain and the plasmon decay. In particular, the deformation of the chain causes changes in the molecular polarizability, and as the system is no longer linear, the polarizability is not strictly composed by the diagonal elements only. The increase in magnitude of one specific off-diagonal component of the polarizability tensor is in agreement with the directions along which the transverse plasmon decays and higher multipolar longitu-

dinal plasmon modes will rise. Higher-order multipolar plasmon excitations 40 are known to play an important role in nanophotonic applications, as they are responsible for the waveguiding and local electric field concentration beyond the diffraction limit. The multipolar LSPR are known to be largely dark to photoexcitation from the far field, but are readily excited in the electron energy loss spectroscopy (EELS) experiments. 41 They are characterized by a higher nodal structure than the dipolar plasmon, which is the one associated with a huge absorption cross section for excitation from the electronic ground state. However, we show here that multipolar LSPR can be substantially populated for nontrivial lifetimes under certain deformation conditions of the wire. Specifically, we have identified high frequency phonon modes that couple the transverse dipolar LSPR strongly to multipolar longitudinal plasmons of the distorted atomic chain. For the out-of-chain bending vibrational mode highlighted in the current work, the coupling is particularly strong due to the energetic proximity of the multipolar longitudinal LSPR with nodes in the excited orbitals at the “kinks” induced in the chain by the bending mode.

Methodology x y

z

Figure 1: The linear Ag8 wire model along with Cartesian axes.

The system under study consists of eight Ag atoms and is presented in Fig. 1 (hereafter Ag8 ). The system is initially perturbed by applying an external electric field of 0.001 a.u. in x direction to start the transverse plasmon excitation. Then, the field is turned off and a nonadiabatic Ehrenfest dynamics simulation, 42–46 based on the employment of an efficient triplesplit operator integrator, 42,43 is performed for 500 fs. The initial momenta for the Ehrenfest


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simulation are obtained by selectively activating one particular normal mode. Specifically for this study, a collective bending mode characterized by displacements taking place in the x direction is excited. This mode was chosen because recently it has been shown to lead to transverse plasmon decay in a system very similar to the one under study in this work. 29 The selected mode is schematically shown in Fig. 2 in terms of its displacement vectors.

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Ag-Ag-Ag angles accessed during the dynamics cover the full range presented in Fig. 5.

Results and Discussion Magnitude

x y

z Magnitude

Figure 2: Schematic picture for the activated bending mode of the linear Ag8 wire model in terms of its displacement vectors. Cartesian axes are also shown. In this simulation, a vibrational energy of 2.51 Kcal/mol is initially imparted to the system in order to exaggerate the effect of this vibrational mode on the electronic dynamics, and initial nuclear velocities were assigned accordingly. Ag8 polarizabilities are obtained for geometries representative of the selected normal mode displacements. Optimizations and frequency calculations as well as the Ehrenfest simulations, are performed using the BP86 exchange-correlation functional and LANL2DZ effective core potential and valence basis set as implemented in the development version of the Gaussian software package. 47 The three time steps employed for the Ehrenfest simulation are ∆tN = 0.1 f s, ∆tN e = 0.01 f s, ∆te = 0.001 f s. The time-evolving dipole moment extracted from the Ehrenfest simulation is analyzed by using the short time Fourier transform (STFT), where a moving window of time of 100 fs was employed. Molecular polarizability components were calculated for geometries obtained by gradually displacing along the vibrational mode that was activated in the initial conditions of the dynamics. The average

Figure 3: Top panel: Short time Fourier transform of the x dipole moment component. The initial electronic state at t = 0 f s corresponds to a transverse molecular plasmon. The nuclear initial conditions are generated accordingly to the selective activation of the bending mode shown in Fig. 2. Bottom panel: Short time Fourier transform of the z dipole moment component extracted from the same trajectory.

The STFT of the time-evolving x and z dipole components for the Ag8 chain are shown in Fig. 3. By an inspection of the x dipole component STFT, it is evident that only one main peak characterizes the spectrum. This peak is centered around ∼5 eV and corresponds to the transverse plasmon absorption peak as already demonstrated in previous works. 29,39,48–50 The transverse plasmon peak undergoes a strong intensity decay; indeed the peak intensity is less than half of the initial value already at 200


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fs. After 300 fs it completely disappears. No other peaks are observed in the 3 − 7 eV region or higher (data not shown) on the investigated timescale. The eventual decrease in the total dipole intensity is due to the loss of phase coherence of the electronic transitions contributing to the plasmons, as shown in a previous work. 29 An inspection of the STFT of the z dipole component reveals the rise of peaks centered in the same frequency range of the plasmon excitation since 100 fs. The STFT of the z dipole component is also more complex, showing multiple peaks in the 4.8 − 5.5 eV region. Characterization of the electronic energy eigenstates in this energy range using the linear response (LR) TDDFT revealed that many of these states are higher order multipolar longitudinal plasmon modes, so Σ → Σ in character. Specifically, the states identified as plasmons exhibit a large contribution from electron-electron interactions to their total electronic energies 31,51,52 (see supporting information for methodology employed to discern LSPR states from excitonic ones). In particular, the observed ones in Fig. S1 are: Σ2 → Σ8 (4.48 eV), Σ1 → Σ8 (4.59 eV), Σ4 → Σ9 (4.70 eV), Σ3 → Σ9 (4.93 eV), Σ2 → Σ9 (5.19 eV), and Σ1 → Σ9 (5.31 eV), where the number of nodes in the orbitals involved in the LSPR wavefunction increases with the subscript. These orbitals are shown in Fig. 4. As the chain deformation increases along the x direction during the time, the transition dipole moment between each such longitudinal LSPR and the ground electronic state evolves from initially being purely oriented along z to having nonzero transition dipole strength along x (see Tab. S1 in supporting information for transition dipole strengths from LR-TDDFT at final geometry from Ehrenfest trajectory.) While the transverse dipolar LSPR originally excited for the linear wire also acquires nonzero transition dipole strength along the z direction as the chain is deformed along the chosen vibrational mode, the rate at which the intensity of the peak in the STFT of the dipole’s z component grows in cannot be fully explained by the change in character of the originally excited state. In order to explain the full magnitude of

the z component of the dipole STFT, one must consider the nonadiabaticity of the electronic dynamics along the nuclear trajectory. Specifically, the rise in intensity is consistent with a rise in population of the multipolar longitudinal LSPR’s of the distorted chain. Energy (eV) -0.83

𝚺𝟗

-1.53

𝚺𝟖

-4.59

𝚺𝟓

-5.18

𝚺𝟒

-5.59

𝚺𝟑

-5.86

𝚺𝟐

-6.01

𝚺𝟏

Figure 4: Orbitals involved in dipolar and higher multipolar longitudinal plasmon modes.

So, for the first time the decay of a transverse dipolar LSPR (of an initially linear wire) is observed to be coupled to higher order multipolar longitudinal LSPR’s through the structural evolution of the system. Indeed, the change of the transition dipole moment associated to such electronic transitions seems to be related to (and allowed by) the fact that the symmetry of the states has changed in the new conformation reached by the chain. A closer inspection of the chain time evolution (whose last geometry is the same as the one shown in Fig. 2), clearly shows that the chain is no longer linear and that the silver atoms lie in the xz plane. The symmetry of the system is lowered from D∞h to C2h , where the C2 axis lies along the y direction and the x/z is no longer the sole direction of polarization for the transverse/longitudinal plasmons, allowing the longitudinal plasmons features to be observed even when the transverse


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ones were excited on the initial linear chain. The transverse plasmon decay and the rise of multipolar longitudinal LSPR are all induced by the nuclear dynamics and are entirely dictated by the nature and direction of the nuclear motions. The vibrational modes found to be responsible for the promotion of multipolar longitudinal LSPR’s from the dipolar transverse LSPR in the linear chain are those associated with out-of-chain displacements. The mechanism by which this normal mode affects the plasmon excitation can be understood from the change in molecular polarizability that accompanies the nuclear dynamics. According to the linear symmetry of the initial system, the polarizability tensor for the molecule can be reduced to a scalar and the molecular polarizability can be represented only by the diagonal elements according to the following expression: αiso

1 = (αxx + αyy + αzz ) 3

increasingly non linear. The other off-diagonal components i.e. αxy and αyz retain values identical to zero, in agreement with the fact that no nuclear displacement takes place along the y direction (data not shown).

Isotropic average xz component

1000

-40

2

2

C m J

-1

)

1010

Polarizability (10

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

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990

6 4 2 170 160 150 140 130 Average Ag-Ag-Ag Angle (deg)

(1)

Figure 5: Polarizability isotropic average (top) and particular off-diagonal component, αxz (bottom), shown as function of the silver bond angle (averaged over all the Ag-Ag-Ag bond angles of the chain) while displacing along the activated normal mode.

where for a linear molecule and according to the specific lab-frame orientation of our chain in this work αxx =αyy 6=αzz . However, during the dynamics, the nuclear displacements due to the activated bending mode cause a change in the polarizability that cannot be described in terms of its diagonal elements. The chain deviates from the linear symmetry and for this reason the off-diagonal polarizability contributions have to be included in the molecular polarizability. The anisotropic term, including the off-diagonal terms, can be calculated in the following way: v u u(α − αyy )2 + (αxx − αzz )2 1 u xx +(αyy − αzz )2 αaniso = √ t 2 2 2 2 +6(αxy + αxz + αyz ) (2)

The increase of the αxz component of the static polarizability tensor is in perfect agreement with the transition dipole components characterizing the multipolar longitudinal LSPR activated during the dynamics. The αxz component increases in magnitude at distorted geometries, suggesting that the dipolar LSPR excited along the x direction (which radiates an oscillatory dipolar electric field polarized along x) can excite some resonant state polarized along z. Although direct excitation to higher order multipolar LSPR is generally not a dipole-allowed transition from the ground state, the results presented here suggest that excited state energy transfer pathway can be dipole-allowed between LSPRs. The bond angles shown in Fig. 5 belong to the same range of values explored during the Ehrenfest simulation. These results give the direct connec-

Figure 5 shows the isotropic average and the αxz component of the polarizability as a function of the the bond angles gradually deviating from 180 degrees. The magnitude of the αxz component is zero when the chain is linear, but continuously increases as the geometry becomes


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tion between nuclear and electronic dynamics upon excitation of dipolar, transverse LSPR in monatomic silver nanowires. Indeed, the nature of the molecular polarizability off-diagonal component (in this case αxz ) increasing in magnitude, is in agreement, in an unambiguous way, with the direction of the plasmon transfer. So, restricting the nanoparticle geometry can have nontrivial impact on its performance for particular applications, and recognizing if its symmetry is being lowered can help to understand exactly how the LSPR lifetime, in this case with a decay time constant of only 93.5 fs, will be affected and what decay pathways are available to the LSPR.

sharp rise in transition dipole strength along z. Until now the only recognized way of exciting the multipolar longitudinal LSPR modes was through electron beam spectroscopies. Here, we show that vibrational excitations of particular character can not only break the symmetry and make the multipolar LSPR transitions weakly dipole allowed, but can also effectively transfer population between plasmonic modes of different symmetries for the linear chain. These results increase understanding of plasmon behavior at atomistic level, and can be employed to make predictions about the behavior of larger particles employed in technological applications. The role of nuclear dynamics is a key factor to understanding which molecular parameters can limit the performances of LSPR supporting particles for a given application. In particular, nuclear displacements that lower the symmetry of the nanoparticle can substantially decrease the transverse plasmon lifetimes. The proposed approach, based on the interplay between LSPR and nuclear dynamics, was employed to characterize the evolution of LSPR’s after an external perturbation has been applied to cause alterations in the system geometry; in this way the solid bridge between the nuclear and electronic dynamics is revealed, and a more complete picture of the dynamics of systems in LSPR states that can aid the design of more efficient devices relying on LSPR excitation has been developed.

Conclusion and Perspective In conclusion, the present study shows the influence of nuclear dynamics on the time evolution of dipolar LSPR states in a monoatomic silver chain. In particular, anisotropic LSPR transfer and eventual decay are understood both in terms of a state-resolved nonradiative relaxation pathway and changes in static dipoledipole polarizability during the nuclear dynamics. The spatial orientation of nuclear displacements reveals how the molecular polarizability is changing. In this case, the deviation from linear symmetry along the x direction causes the increase in magnitude of the αxz polarizability off-diagonal component. This induces the loss of population in the dipolar transverse LSPR, and concomitant rise of multipolar longitudinal LSPR populations. The nuclear dynamics removes the clear distinction that exists between “transverse” and “longitudinal” polarization direction of the LSPR’s of the linear wire. By the end of the nuclear trajectory, both the directions lie in the plane of the distorted chain, and polarization is effectively transferred from the transverse to longitudinal directions. The nuclear motions along the particular vibrational mode considered are shown to couple the LSPR’s of different symmetry at the linear geometry, resulting in the transfer of population from the dipolar transverse LSPR to the multipolar longitudinal ones, as indicated by the

Acknowledgement The development of the first-principles electronic dynamics is supported by the US Department of Energy (DESC0006863). The development of linear response TDDFT method for computational spectroscopy was supported by the National Science Foundation (CHE-1565520). C.M.A. is grateful to the US Department of Energy (DE-SC0012273) for financial support. This work was facilitated though the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system and funded by the STF at the University of Washington and the National Science Foundation (MRI-1624430).


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Anisotropic Polarizability Induced Plasmon Transfer

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