MULTICOMPONENT RESONANT NANOSTRUCTURES: PLASMONIC AND PHOTOTHERMAL EFFECTS 9785763842593

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MULTICOMPONENT RESONANT NANOSTRUCTURES: PLASMONIC AND PHOTOTHERMAL EFFECTS

Vadim Zakomirnyi

Institute of Engineering Physics and Radio Electronics Siberian Federal University Krasnoyarsk, Russia 2019

© Vadim Zakomirnyi, 2019 ISBN 978-5-7638-4259-3 Printed by Siberian Federal University, Krasnoyarsk, Russia

Abstract In recent decades, plasmonic nanoparticles have attracted considerable attention due to their ability to localize electromagnetic energy at a scale much smaller than the wavelength of optical radiation. The study of optical plasmon waveguides (OPWs) in the form of chains of nanoparticles is important for modern photonics. However, the widespread use of OPWs is limited due to the suppression of the resonance properties of classical plasmon materials under laser irradiation. The study of the influence of nanoparticle heating on the optical properties of waveguides and the search for new materials capable of stable functioning at high temperatures is an important task. In this thesis, the processes occurring during heating of plasmon nanoparticles and OPWs are studied. For this purpose, a model was developed that takes into account the heat transfer between the particles of an OPW and the environment. The calculations used temperature-dependent optical constants. As one of possible ways to avoid thermal destabilization of plasmon resonanses, new materials for OPWs formed by nanoparticles were proposed. I show that titanium nitride is a promising thermally stable material, that might be useful for manufacturing of OPWs and that works in high intensity laser radiation. Another hot topic at present is the study of periodic structures of resonant nanoparticles. Periodic arrays of nanoparticles have a unique feature: the manifestation of collective modes, which are formed due to the hybridization of a localized surface plasmon resonance or a Mie resonance and the Rayleigh lattice anomaly. Such a pronounced hybridization leads to the appearance of narrow surface lattice resonances, the quality factor of which is hundreds of times higher than the quality factor of the localized surface plasmon resonance alone. Structures that can support not only electric, but also magnetic dipole resonances becomes extremely important for modern photonics on chip systems. An example of a material of such particles is silicon. Using the method of generalized coupled dipoles, I studied the optical response of arrays of silicon nanoparticles. It is shown that under certain conditions, selective hybridization of only one of the dipole moments with the Rayleigh anomaly occurs.

To analyze optical properties of intermediate sized particles with N = 103 − 105 atoms and diameter of particle d < 12 nm an atomistic approach, where the polarizabilities can be obtained from the atoms of the particle, could fill an important gap in the description of nanoparticle plasmons between the quantum and classical extremes. For this purpose I introduced an extended discrete interaction model where every atom makes a difference in the formation optical properties of nanoparticles within this size range. In this range are first principal approaches not applicable due to the high number of atoms and classical models based on bulk material dielectric constants are not available due to high influence from

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quantum size effects and corrections to the dielectric constant. To parametrize this semiempirical model I proposed a method based on the concept of plasmon length. To evaluate the accuracy of the model, I performed calculations of optical properties of nanoparticles with different shapes: regular nanospheres, nanocubes and nanorods. Subsequently, the model was used to calculate hollow nanoparticles (nano-bubbles).

List of goals of current research: 1. Investigate the influence of limitation caused by thermal effects arising from the excitation of an optical plasmon waveguide in the form of a linear chain of spherical nanoparticles in high energy laser radiation. 2. Show the effect of heating and subsequent melting of the first irradiated particle in the chain on the efficiency of the transmission of an optical signal through an optical plasmon waveguide. 3. Investigate the possibility of using titanium nitride as an alternative material with high thermal stability for optical plasmonic waveguides from spherical and spheroidal nanoparticles. 4. Obtain information on the effect of imperfections (various types of defects) that may appear in two-dimensional arrays of silicon nanoparticles where collective optical effects associated with the manifestation of high-quality lattice resonances. 5. Develop a model for describing the optical properties of plasmon nanoparticles based on a discrete atomic interaction model using a plasmon length based parametrization. 6. Demonstrate the size, shape and aspect ratio dependence of surface plasmon resonances for small (2 − 12 nm in diameter) silver spherical- and cubical-clusters and nanorods. 7. Demonstrate the resonance properties of hollow nanoparticles and compare classic electrodynamics simulations of optical properties of small nanoparticles using Mie theory with my discrete atomic interaction model.

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Preface The work presented in this thesis has been carried out at the Department of Theoretical Chemistry and Biology, Royal Institute of Technology, Stockholm, Sweden, and at the Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University (SibFU), Krasnoyarsk, Russia in the framework of the double doctorate cooperation between KTH and SibFU.

List of papers included in the Thesis Paper I V. S. Gerasimov, A. E. Ershov, S. V. Karpov, A. P. Gavrilyuk, V. I. Zakomirnyi, I. L. Rasskazov, H. ˚ Agren, S. P. Polyutov, ”Thermal effects in systems of colloidal plasmonic nanoparticles in high-intensity pulsed laser fields”, Optical Materials Express, 7(2), 555 (2017). Paper II V. I. Zakomirnyi, I. L. Rasskazov, V. S. Gerasimov, A. E. Ershov, S. P. Polyutov, S. V. Karpov, H. ˚ Agren, ”Titanium nitride nanoparticles as an alternative platform for plasmonic waveguides in the visible and telecommunication wavelength ranges”, Photonics and Nanostructures - Fundamentals and Applications 30, 50-56 (2018). Paper III V. I. Zakomirnyi, S. V. Karpov, H. ˚ Agren, I. L. Rasskazov, ”Collective lattice resonances in disordered and quasi-random all-dielectric metasurfaces”, Journal of the Optical Society of America B 36(7), E21 (2019). Paper IV A. D. Utyushev, I. L. Isaev, V. S. Gerasimov, A. E. Ershov, V. I. Zakomirnyi, I. L. Rasskazov, S. P. Polyutov, H. ˚ Agren, S. V. Karpov, ”Engineering novel tunable optical high-Q nanoparticle array filters for a wide range of wavelengths”, submitted in Optics Express (2019). Paper V V. I. Zakomirnyi, Z. Rinkevicius, G. V. Baryshnikov, L. K. Sørensen, H. ˚ Agren, ”The Extended Discrete Interaction Model: Plasmonic Excitations of Silver Nanoparticles”, accepted in The Journal of Physical Chemistry C (2019). ˚gren, I. Paper VI V. I. Zakomirnyi, A. E. Ershov, V. S. Gerasimov, S. V. Karpov, H. A L. Rasskazov, ”Collective lattice resonances in arrays of dielectric nanoparticles: a matter of size”, Optics Letters 44(23), 5743–5746 (2019). Paper VII V. I. Zakomirnyi, I. L. Rasskazov, L. K. Sørensen, P. S. Carney, Z. Rinkevicius, H. ˚ Agren, ”Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models”, manuscript (2019).

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Comments on my contributions to the papers included • I was responsible for a part of the simulations and participated in discussions of theory and results of Papers I, IV, VII. • I was responsible for the major part of calculations and simulations in Papers II, III, V and VI. I also contributed to figures preparation, discussion of the results and writing of the manuscript.

List of other papers not included in the Thesis Paper I V. I. Zakomirnyi, I. L. Rasskazov, S. V. Karpov, S. P. Polyutov, ”New ideally absorbing Au plasmonic nanostructures for biomedical applications”, Journal of Quantitative Spectroscopy and Radiative Transfer 187, 54-61 (2017). Paper II A. E. Ershov, V. S. Gerasimov, A. P. Gavrilyuk, S. V. Karpov, V. I. Zakomirnyi, I. L. Rasskazov, S. P. Polyutov, ”Thermal limiting effects in optical plasmonic waveguides”, Journal of Quantitative Spectroscopy and Radiative Transfer 191, 1-6 (2017). Paper III V. I. Zakomirnyi, I. L. Rasskazov, V. S. Gerasimov, A. E. Ershov, S. P. Polyutov, S. V. Karpov, ”Refractory titanium nitride two-dimensional structures with extremely narrow surface lattice resonances at telecommunication wavelengths”, Applied Physics Letters 111(12), 123107 (2017).

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Acknowledgments ˚gren and Professor Zilvinas I would like to express my deep gratitude to Professor Hans A Rinkevicius, my research supervisors, for their patient guidance, enthusiastic encouragement and useful critiques of this research work. I wish to thank Professor Sergey V. Karpov, who was my first supervisor and who provided me an opportunity to participate in ongoing research on plasmonics project in the group at Kirensky Institute of Physics (Krasnoyarsk, Russia). Thanks to my colleague Dr. Lasse Kragh Sørensen for the fantastic patience and great participation in my work. Many thanks to Dr. Sergey P. Polyutov for great contribution in my scientific career and to the Siberian Federal University (Krasnoyarsk, Russia) for ample institutional resources. The Scholarship of President of Russian Federation for education abroad is gratefully acknowledged for financial support. I truly appreciate lectures given by Prof. Olav Vahtras, Prof. Yaoquan Tu, Prof. M˚ arten Ahlquist and Dr. Victor Kimberg. Special thanks are given to Prof. Faris Gel‘mukhanov for some nice talks and discussions together, and Prof. Lars Thyl`en for his interest in my work and inspiring discussions about future of plasmonics and photonics. The administrative staff at Theochem Department, in particular Nina Bauer, is acknowledged for all administrative help. Special thanks to Nina for her help with my visa and for teaching me how to understand Swedish humor. My special appreciate to my colleagues and friends for their support in providing relevant assistant and help to complete my study. Thanks to Valeriy Gerasimov, Alexander Ershov, Karan Ahmadzadeh, Iulia-Emilia Brumboiu, Haofan Sun, Lucia Labrador Paez, Michal Biler, Nicolas Rolland, Nina Ignatova, Qingyun Liu, Rafael Carvalho Couto and Vin´ıcius Vaz da Cruz. Special thanks to Glib Baryshnikov for tons of coffee and endless discussions about chemistry. No one was more important to me in the implementation of this project than Ilia Rasskazov. Thank you for valuable guidance, timely suggestions and support throughout my research project work. Last but not least, I want to thank my wife Anna, whose love and care support me and provide endless inspiration. Vadim I. Zakomirnyi Stockholm, 2019-12-19

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Contents 1 Introduction

1

2 Thermal and optical effects in plasmonic nanoparticle waveguides

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2.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1.1

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1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Thermal

Dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Optical properties of meltednanoparticle nanoparticles .waveguides . . . . . . . . .. .. .. .. .. . . . . . and 2.1.2 optical effects in plasmonic 2.1.3

Thermodynamic properties of nanoparticles . . . . . . . . . . . . . . .

2.2.2

Optical properties of optical plasmonic waveguides from titanium ni-

. . .6 . . . 17 7

2.1 Model. . . .2.1.4 . . . . .Thermodynamic . . . . . . . . . . . properties . . . . . . . .of . optical . . . . . plasmonic . . . . . . . .waveguides . . . . . . . .. .. .. .. . . . . . . .10 . . . . 2.1.1 Dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Transmission and dispersion properties of optical plasmonic waveguides 11 2.1.2 Optical properties of melted nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . properties . . . . . . . .of . .nanoparticles. . . . . . . . . . . . . . . . .. .. .. .. . . . . . . . . . .. .. .. .. .. . . . . . . .13 2.1.3 2.2 Thermodynamic . . . . 2.2.1 Thermal properties and optical properties optical plasmonic waveguides 2.1.4 Thermodynamic of opticalofplasmonic waveguides . . . . from . . . . . . . . . . silverand nanoparticles . . . . . . . of . . optical . . . . . .plasmonic . . . . . . . waveguides . . . . . . 13. . . . 2.1.5 Transmission dispersion. .properties

18 18 18 19 22 23

2.2 Results. . . . . . . .tride . . . nanoparticles . . . . . . . . . . .. .. . . . . . . . . . .. .. .. .. .. . . . . . . . . . .. .. .. .. . . . . . . . . . .. .. .. .. .. . . . . . . .17 . . . . 25 2.2.1 2.3 Thermal and optical properties of optical plasmonic waveguides Conclusions for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 from silver nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Optical properties optical plasmonic 3 Collective effects in of structures of resonantwaveguides nanoparticles 23 from titanium nitride nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1

2.3

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1forExtended Conclusions Chaptercoupled 2. . . . . dipole . . . . .approximation . . . . . . . . . . . . . . . . .. .. .. .. . . . . . . . . . .. .. .. .. .. . . . . . 3.1.2

Types of imperfections in arrays on silicon nanoparticles . . . . . . . .

23

. .23 . . . . 33 25

Results . . . . . . . .of . .resonant . . . . . . .nanoparticles . . . . . . . . . .. .. .. .. . . . . . . . . . .. .. .. .. .. . . . . . . .27 3 Collective3.2effects in .structures . . . . 35 3.2.1

Periodic arrays of silicon nanoparticles . . . . . . . . . . . . . . . . . .

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3.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Extended coupled dipole approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Types of imperfections in arrays on silicon nanoparticles. . . . . . . . . . . . . . . . . 37 3.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Periodic arrays of silicon nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Arrays of silicon nanoparticles with imperfections. . . . . . . . . . . . . . . . . . . . . . 3.2.3 Finite size effects in arrays of silicon nanoparticles . . . . . . . . . . . . . . . . . . . . . 3.2.4 Optical filters based on arrays of plasmonic nanoparticles. . . . . . . . . . . . . . . .

39 39 41 45 47

3.3 Conclusions for Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.1 Extended discrete interaction model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.2 Parametrization of extended discrete interaction model for silver . . . . . . . . . . 55 4.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Polarizability of spherical silver nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Polarizability of silver nanoparticles with complicated geometry: nanocubes and nanorods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Plasmon resonances of hollow nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Polarizability of hollow nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60 60 63 65 67

4.3 Conclusions for Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Paper I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Paper III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Paper IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Paper V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Paper VI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Paper VII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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List of abbreviations NP - Nanoparticle DA - Dipole Approximation OPW - Optical Plasmonic Waveguide CLR - Collective Lattice Resonance IR - Infrared (wavelength region) ED - Electric Dipole MD - Magnetic Dipole FDTD - Finite-Difference Time-Domain SPP - Surface Plasmon Polariton SPR - Surface Plasmon Resonance TLSPR - SPR localized along the transverse axis (short axis in spheroids) LLSPR - SPR localized along the longitudinal axis (long axis in spheroids) DIM - Discrete Interaction Model ex-DIM - Extended DIM cd-DIM - Coordination Dependent DIM

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Chapter 1 Introduction Currently, plasmonics constitutes one of the most interesting areas of development in photonics. With ”plasmonics”, we, of course, mean the manifestation of the so-called plasmon resonances which are produced by group oscillations of conduction electrons that lead to an increase in the absorption of electromagnetic radiation at certain wavelengths. The plasmon resonance in small nanoparticles strongly depends on their geometric shape, size, and material. Homogeneous spherical nanoparticles of silver and gold are well-studied objects in nanoplasmonics. However, due to technical limitations associated with the complexity of the experimental manufacturing of silver and gold nanoparticles, attempts have been made to use alternative plasmon materials. One looks for materials that can have advantages compared to the classical materials in terms of, for example, increased heat resistance and chemical stability. Obviously, one can find advantages in combining various materials, for example, for the manufacturing of nanoparticles with a core-shell structure, where the shell performs a protective function. In addition to using various materials for nanoparticles, the influence of the geometry of nanoparticles is of interest. It is well known that the appearance of additional plasmon resonances is observed for nanoparticles in the form of prolate or oblate spheroids. The nature of such resonances are rather well studied theoretically and these resonance particles have been repeatedly used in various applications. There are also studies of more complex geometries of nanoparticles, such as pyramids, cubes, nanostars, nanorods, nanobubbles. It is worthwhile to consider the reasons for the interest in plasmonic nanoparticles and structures of plasmonic nanoparticles from the point of view of possible applications in modern nanophotonics. The potential applications of plasmonic nanoparticles are based on their unique feature of supporting plasmon resonances that enhance the local field near the nanoparticle. An electromagnetic field near a nanoparticle at a plasmon resonance

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Chapter 1 Introduction

wavelength can be two or more orders of magnitude stronger than at other frequencies. In the simplest case of a single nanoparticle in laser illumination, this inevitably leads to heating of the nanoparticle and possibly also to a change in their phase state, in other words, melting. If such a nanoparticle is placed near a living cell, the super hot nanoparticle can burn the cell membrane and destroy the cell. This technique is known and widely used in so-called plasmon photothermal therapy of cancer cells. In the case of a pair of closely spaced nanoparticles, the field between them can be enhanced by more than four orders of magnitude. By placing the molecule there, and controlling the external radiation, one can ”highlight” the molecule and obtain an amplified Raman spectrum. Chains of equidistant nanoparticles of various shapes attract attention of researchers due to the ability to transmit plasmon excitation, which is nothing but a plasmon waveguide. Despite the obvious advantages of using such waveguides in modern nanophotonics, there are a number of effects that limit the widespread use of such waveguides and which is the source of much current research. In recent years, two-dimensional arrays of nanoparticles have begun to attract great attention. Periodic arrays of plasmonic nanoparticles have a unique feature: the manifestation of collective modes, which are formed due to the hybridization of surface plasmon resonances and the Wood-Rayleigh lattice anomalies. This hybridization leads to the appearance of a narrow collective lattice resonance with a quality factor many times higher than the quality factor of the surface plasmon resonance. Collective lattice resonances have attracted attention over the past decade, starting with pioneering theoretical research [1–3] and applied experimental work in vibrational spectroscopy [4], ultra-narrowband absorption [5], sensors [6, 7], lasers [8], and enhanced fluorescence [9, 10]. Collective lattice resonances have been studied in a wide range of periodic nanostructures with various types of unit cells: single [11] or double layers [12] nanodisks, cylinders and nanoshells [13], nanoparticle dimers [14, 15], complex nanoparticles [16], split ring resonators [17], oligomers [18] and other complex configurations [19–22]. The position and shape of a collective plasmon resonance is affected not only by the size, material, and shape of the single nanoparticles, but also by the geometry of the array itself. Thus, an ordered equidistant lattice of nanoparticles will differ from the structure in which the distances between the particles differ along the X and Y axis, or from the structure in which the nanoparticles are ordered in something resembling a honeycomb and other geometries. Currently, attention is focused on lattices from classical plasmon materials [11, 23–25] (Au and Ag) with an surface plasmon resonance peak of individual nanoparticles located in the visible or near infrared region. Although nanoplasmonics since long has constituted a research branch that has received strong attention as a versatile nanotechnology and by now turned into mature research with significant applications in areas like bioimaging, photonics and energy harvesting, there is

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1 Introduction

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still a lag between experiment and theoretical capability to design nanostructures with particular plasmonic properties. Among a number of classical models, Mie theory has been instrumental in predicting light scattering and plasmonic resonances in metallic nanoparticles. Since Mie theory relies on the concept of a dielectric constant it is, however, restricted to a size comprising larger nanoparticles where the classical bulk dielectric constant remains valid and frequency ranges where experimental results are available. At the other hand, pure quantum approaches, like time-dependent density functional theory, are applicable only for the very small particles. This leaves the 1−15 nm size region unattainable by either classical (n.b. Mie) and quantum theory, which is unfortunate considering the wide applicability of small plasmonic nanoparticles within that size range, e.g. for cell imaging [26, 27]. The first part of my thesis concerns the study of the interaction of metal nanoparticles with laser radiation as one of the main directions in nanoplasmonics. My work touches on many possible applications in fields such as nanosensors, biomedicine [28–45], biotechnology, laser excitation of plasmon polaritons in waveguides from chains of plasmon nanoparticles, photochromic reactions induced by laser excitation of resonant domains in disordered colloidal aggregates of nanoparticles and various nonlinear optical processes. Monographs and reviews [46–50] cover a large number of recent works and applications. There are a number of earlier works that form the basis for the research development reported in my thesis: In [51– 53] studies of heating nanoparticles with laser radiation were presented and in [54–56] the effects of pulsed laser radiation on aggregates of plasmon nanoparticles have been analyzed taking into account the effects of the melting of nanoparticles. However, the most part of current research does not take into account the change in optical properties with the change in temperature of the nanoparticles, and vice versa. Thus, there is interest in developing thermodynamic models that take into account heat transfer between nanoparticles inside the optical plasmonic waveguide, the environment, and the substrate, and that also take into account the temperature dependencies of the optical properties of the nanoparticles. Also, I thoroughly addressed the problem of diffractive behavior of electric dipole and magnetic dipole resonances in imperfect arrays of spherical silicon nanoparticles. A comprehensive analysis of various types of disorder revealed the effect on the hybridization scenario of the electric dipole and magnetic dipole modes with lattice modes. Among other things, it is not obvious how collective lattice resonances in arrays of finite sizes of dielectric nanoparticles with strong electric dipole and magnetic dipole resonances differ from collective lattice resonances in infinite arrays. Thus, due to strong self-interactions between the electric dipole and magnetic dipole modes, I studied the problem of the validity of the infinite array approximation when working with collective lattice resonances in arrays of nanoparticles with electric dipole and magnetic dipole resonances. Motivated by the wide applicability of small plasmonic nanoparticles and by the need to

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Chapter 1 Introduction

find a ”bridge” in the length gap between classical and quantum theory to describe plasmon generation, I introduced in the last part of my thesis an extended discrete interaction model to simulate optical properties of nanoparticles with different geometry in size range between 1 and 15 nm. Despite years of research efforts in nanoplasmonic, the area remains wide open for further development of theory and modelling with ramifications for applications in many technological areas, like biomiaging, photonics, energy harvesting and other current front-edge technologies. It is my hope that my thesis makes a contribution to that endeavour.

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides Changes in the optical properties of the particle material due to melting is an important factor in the process of interaction of laser radiation with nanoparticles (NPs). This is due to the fact that with an increase in the temperature of the NPs, the intensity of phonon vibrations increases. As a result of this, the frequency of electron scattering by phonons increases, which leads to an increase in the electron relaxation constant (above the Debye electron relaxation temperature it increases in proportion to the temperature) [57]. During melting, the gradual destruction of the periodic structure in the crystal leads to the scattering of conduction electrons by lattice defects (mainly vacancies and dislocations) up to complete amorphization. The melting process is accompanied by a sharp increase in relaxation constants. In addition to phonons, the contribution to the electronic relaxation of a metal is determined by the scattering of electrons by point defects, dislocations, particle boundaries, and electrons [57]. It was shown that heating of nanoparticles and their subsequent melting significantly affects their resonance properties. A theoretical approach that describes the heat transfer between nanoparticles and the environment [54, 56, 58] is also applicable in the case of a single laser pulse with a duration much shorter than the time to establish a thermodynamic equilibrium. However, such models do not take into account the effect of the substrate, which can play a crutial role of a cooling device.

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2.1 2.1.1

Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

Model Dipole approximation

The electrodynamics part of plasmonic model is based on the dipole approximation [59], which allows us to calculate the electromagnetic interaction between NP and the incident radiation. Let us consider an NP in a medium with a dielectric constant εh which is irradiated by a plane electromagnetic wave E (r) = E0 exp (ik ⋅ r). Here E(r) is electric field at √ location r, ∣k∣ = 2π εh /λ is a wave vector, λ is a wavelength, E0 is amplitude of the electric component of the electromagnetic field. In a general case, the dipole moment d induced at the NP can be described by the following equation [59]: d = εh αe E(r),

(2.1)

where αe is the electric dipole polarizability of the NP [59, 60]: 1 i 1 = − ∣k∣3 , αe α(0) 6π

(2.2)

where α(0) is a quasistatic polarizability of the NP [61]: α(0) = 4πR3

ε − εh . ε + 2εh

(2.3)

Here R is a radius and ε is the dielectric permittivity of the NP.

2.1.2

Optical properties of melted nanoparticles

When the NP absorbs electromagnetic radiation, it heats up until it is completely melted (liquid). This factor may be accompanied by cyclically repeated rises and drops of temperature of resonantly excited NP. These cycles appear due to the termination of the resonant interaction between incident optical radiation and liquid of particles. Therefore, it is necessary to take into account the fact that NP also can be in a molten state. In other words, NP can be represented as layered nanoparticle with a solid core and a liquid shell. In this case, we can apply the concept of a nanoshell to such NP, the materials of the core and shell of which are the same, but in different aggregate states. The quasistatic polarizability for the n-th NPs from eq. (2.3) will be changed in following way [62]:

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2. Thermal and optical effects in plasmonic nanoparticle waveguides

2.1 Model

7

αn = 4πRn3 (0)

(εln − εh )(εsn + 2εln ) + fn (εsn − εln )(εh + 2εln ) , (εln + 2εh )(εsn + 2εln ) + 2fn (εsn − εln )(εln − εh )

(2.4)

where εsn and εln are the dielectric constants of the particle material in the solid and liquid state, respectively. The value fn is the fraction of the solid part of the whole nanoparticle. Equation (2.4) also takes into account the cases when the NP is a complete liquid (fn = 0) or a complete solid (fn = 1). Due to high surface tension forces, the NPs will keep their spherical shape even when they are complete liquids, which makes eq. (2.4) applicable for any values of fn . The dielectric constants εsn and εln also take into account finite size effects: s,l εs,l n → εtab +

ωp2 ωp2 − , ω 2 + iγbulk ω ω 2 + iγfin ω

(2.5)

where ω = 2πc/λ is the frequency of incident radiation, εstab and εltab are the tabulated experimental values of the dielectric constant for solid bulk material at temperature 300 K [63] and dielectric constant for fully melted liquid material [64] respectively, ωp is a plasma frequency, γbulk and γfin are relaxation constants [65]: γfin = γbulk + AL

υF , Leff

(2.6)

where υF is the Fermi velocity; Leff is the electron effective mean free path [66],and AL is a dimensionless parameter, which is close to 1 in our studied cases [62]. It should be noticed that γbulk is a parameter depending on the temperature Tion , and can be approximated by the following expression [57]: γbulk (Tion ) = bT + c,

(2.7)

where b and c are coefficients obtained from a linear approximation of experimental data [67].

2.1.3

Thermodynamic properties of nanoparticles

The absorption of laser radiation leads primarily to the heating of conductivity electrons (electronic subsystem) in nanoparticles and their crystal lattice (ionic subsystem). The changes in temperature of the electronic subsystem Tnel , caused by the absorption of electromagnetic energy and heat transfer with the ionic subsystem (with temperature Tnion ), is described by the equation [68, 69]:

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

Cnel

Wn dTnel , = −g[Tnel − Tnion ] + dt Vn

(2.8)

where Cnel is the volumetric heat capacity of the electronic subsystems: Cnel = 68 Tnion J⋅m−3 ⋅K−1 , Tnion is temperature of the ionic subsystem, Vn is particle volume, g = 4 ⋅ 1016 J⋅m−3 ⋅K−1 ⋅s−1 is the temperature-independent energy exchange rate between the electron and ion subsystems, that should be higher than the Debye temperature [70–72], Wn is the power of energy absorbed by the nanoparticle. In the dipole approximation, Wn is defined by the following expression [56, 73]: Wn =

ω∣dn ∣2 1 Im ( ∗ ) . 2εh αn

(2.9)

where ω is the frequency of incident radiation, dn is a vector of the dipole moment of the n-th NP, asterisk ∗ means complex conjugate value of the polarizability from eq. (2.2).

The changes in the temperature of the ionic subsystem Tnion of the NP is mainly determined by the heat exchange between the electronic subsystem and the environment. Taking into account the solid-to-liquid phase transition in NPs, we use the equation for the thermal ion : energy of the NPs lattice Qion n instead of the T dQion n = gVn [Tnel − Tnion ] + qnl Vn , dt

(2.10)

where qnl is a heat flow per unit volume explaining heat losses [68]: qnl = −

3 (T ion − T0 ) 2Rn n

√

χm cm0 ρm , t

(2.11)

where χm is thermal conductivity of interparticle medium, cm0 is heat capacity, ρm is density, t is time of a laser pulse. The temperature of the ionic subsystem, taking into account the melting process, is expressed in terms of Qion n : Tnion =

Qion Qion (1) (2) (1) n n − Qn m ion H(Qn − Qion H(Qion n )+ n − Qn ) + T (Rn )H(Qn − Qn ), (2.12) Cn V n Cn V n (2)

(1)

(2)

where Qn and Qn are the thermal energies of the particle corresponding to the beginning and to the end of melting process respectively, Cn is the volumetric heat capacity of the

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9

ionic subsystem of NPs, T m (Rn ) is the melting point depending on particle size [74], H(x) is the Heaviside function.

The temperature of the ionic subsystem of a nanoparticle during melting might also be determined as follows [56]: ⎧ Qion (1) ⎪ n ⎪ , where Qion ⎪ n < Qn ⎪ ⎪ Cnion Vn ⎪ ⎪ ⎪ (1) (2) Tnion = ⎨ TnL , where Qn ≤ Qion n ≤ Qn ⎪ ⎪ ⎪ Qion (2) ⎪ n − LVn ⎪ , where Qion ⎪ n > Qn . ⎪ ⎪ ⎩ Cnion Vn

(2.13)

where L is the volumetric heat of fusion, Cnion is the specific heat of the ion subsystem of the n-th nanoparticle, TnL = T L (Rn ) is the melting temperature, taking into account the size of (1) the nanoparticle [74], Qn is heat corresponding to the beginning of melting particles:

(2)

Qn = Cnion Vn TnL , (1)

(2.14)

and Qn corresponds to the heat at the end of the melting of the particle: Qn = Qn + LVn . (2)

(1)

(2.15)

Thus, it becomes possible to determine the mass fraction fn of the liquid phase from eq. (2.4): (1) ⎧ 0, where Qion ⎪ n < Qn ⎪ ⎪ ⎪ ⎪ ⎪ Qion − Q(1) n (1) (2) fn = ⎨ n , where Qn ≤ Qion n ≤ Qn ion V ⎪ C ⎪ n n ⎪ ⎪ (2) ⎪ ⎪ where Qion ⎩ 1, n > Qn

(2.16)

The heat transfer rate between the particle and the environment can be determined from the following expression [75]: υn = −κ ∫ ∇T (r, t) ⋅ ndS,

(2.17)

Sn

where κ and T (r, t) are the thermal conductivity coefficient and ambient temperature, respectively, n is the vector perpendicular to the surface of the NP. Integration of the expression is performed over the entire surface Sn of the NP. The heat transfer rate due to radiation is many times lower than the heat transfer rate due to heat conduction, therefore, in this model, the contribution of the former can be neglected.

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Further, from the heat equation for the environment, we can determine the values T (r, t): ∂T (r, t) = adiff ∆T (r, t), ∂t

(2.18)

where adiff is a diffusion coefficient of the environment. We use the following boundary conditions to solve these equations: • particle and ambient temperatures were taken equal at the initial time t = 0: Tnion = Tnel = T (r, t = 0) = T0 = 300 K;

• ambient temperature on the surface of the substrate does not depend on time: T (rsub , t) = T0 = 300 K;

• ambient temperature is constantly at a distance infinitely remote from the system (∣r∣ ≫ max(Rn )): T (x = ±∞, y, z, t) = T (x, y = +∞, z, t) = T (x, y, z = ±∞, t) = 300 K; n

• the ambient temperature on the surface of the particles is equivalent to the temperature of the ionic subsystem of the NP: T (∣r − rn ∣ = Rn , t) = Tnion .

2.1.4

Thermodynamic properties of optical plasmonic waveguides

In this section I apply thermodynamic model for OPW. In general case, OPW is a chain of NPs with center-to-center distance bigger than diameter if NP. In most experimental setups the OPW is located on the substrate. The excitation of the OPW can be implemented in practice, for example, using a probe near-field optical microscope. The external field En = E(rn ) incident on the n-th NP located at the point rn can be described as: En = E0 exp(ikrn ).

(2.19)

Often only first (n = 1) nanoparticle of the waveguide is considered to be excited by the external radiation [76–80], as a result of which En = 0 for n ≠ 1. In this case, the dipole moment dn induced on the n-th nanoparticle can be found by solving the equations of coupled dipoles (2.1), which will take the following form when taking into account the influence from the substrate and the interaction between all the particles: ˆ dn = εh αne [En δn1 + ∑ G(ω; rn , rm )dm ] , N

m=1

22

(2.20)

2.1 Model

2. Thermal and optical effects in plasmonic nanoparticle waveguides

11

ˆ δn1 is the Kronecker symbol, G(ω; rn , rm ) is a 3 × 3 the interparticle interaction tensor (Green’s tensor) which describes the electromagnetic field at the point rn induced by an electric dipole located at the point rm and oscillating with a frequency of ω. In the general case, the Green’s tensor has the following form: ˆ ˆ free (ω; rn , rm ) + G ˆ refl (ω; rn , rm ), G(ω; rn , rm ) = G

(2.21)

ˆ refl (ω; rn , rm ) are Green’s tensors describing the electric field ˆ free (ω; rn , rm ) and G where G in a homogeneous medium and the electric field reflected from the substrate, respectively. ˆ free (ω; rn , rm ) and G ˆ refl (ω; rn , rm ) are shown explicitly the in articles [81– Expressions for G 83]. It should be noted that the summation in the expression (2.20) is performed over all ˆ free (ω; rn , rn ) = 0, which means that each NP does not interact with indices. However, G itself.

2.1.5

Transmission and dispersion properties of optical plasmonic waveguides

Dispersion relations are one of the most important concepts that quantitatively determine the ability of a linear chain of plasmonic nanoparticles to support SPPs. There are various approaches to estimate dispersion relations of finite [84–87] and infinite [88–95] chains of NPs. In my thesis I use the eigendecomposition method of Ref. [96]. In the general case, according to the Bloch theorem, the dipole moment and the incident ext ⋅ exp(iqnh), where q is the Bloch field can be described as dn = d ⋅ exp(iqnh) and Eext n =E eigenvector. So, it is possible to rewrite (2.20) for a infinite chain of particles: ∞ 1 ˆ nm eiqnh ] d = Eext . [ Iˆ − ∑ G α n=−∞

(2.22)

One can note that the expression in square brackets has the same dimension as the inverse dipole polarizability of the NP. Thus, according to the method of eigenvector decomposition, it is convenient to characterize the electromagnetic response of the OPW with the so-called effective polarizability α ˜ [96] such that 1/˜ α is an eigenvalue of the following equation: ∞ 1 ˆ nm eiqnh ] . [ Iˆ − ∑ G α n=−∞

(2.23)

The maxima of Im[˜ α] = F (ω, q) correspond to the resonances of the OPW, which represent the passband of the OPW, or, in other words, its dispersion relation. A significant advantage

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Figure 2.1: Schematic representation of the OPW from NPs of spherical shape (a), prolate spheroids (b) and oblate spheroids (c). Here we consider an OPW located along the X axis with a center-to-center distance h. The radius of the spherical particles is b as well as for the minor semiaxis for spheroids, while for the major semiaxis it is a. Image taken from Paper II. Copyright 2017 Elsevier. of the eigenvector expansion method is the possibility to simultaneously estimate the eigenmodes of the OPW and their Q-factor [96, 97]. Thus, the function Im[˜ α] = F (ω, q) provides a complete physical representation of the dispersion relations for the OPW, which, generally speaking, is impossible to obtain using other methods considered in literature [84, 86, 88–90]. The calculation of the transmission spectrum of OPW is the most effective way to obtain actual damping of the SPP at the end of the waveguide. Suppose that the external field Eext excites only the first nanoparticle in the OPW in the form of a linear chain of N identical NPs fig. 2.1. The solution of eq. (2.20) on the right side provides the dipole moments dn induced on each nanoparticle in the OPW. Experimentally, the electric field strength at the 2 end of the OPW is IN ∝ ∥dN ∥ which characterizes the SPP attenuation. Therefore, the SPPs propagation efficiency can be described by the following quantity [79]: Qtr =

∥dN ∥2 . ∥d1 ∥2

(2.24)

Thus, one can refer the spectral dependence of Qtr as the transmission spectrum of the OPW. Three different shapes of NPs are considered in my thesis: spheres, oblate and prolate spheroids. The quasistatic polarizability eq. (2.3) of NPs with such shapes is determined by the following expression: αn = (0)

V ε − εh , 4π εh + Dstat (ε − εh )

(2.25)

where V is the volume of the NP, Dstat is the static depolarization coefficient [62]. For NPs with dimensions much smaller than the wavelength of the incident light, retarda-

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13

Figure 2.2: Extinction spectra of a single Ag NP with radius R = 8 nm for various values of temperature: 1 – room temperature; 2 – melting point, solid state; 3 – melting point, liquid state. Image taken from my article [100]. Copyright 2017 Elsevier.

Figure 2.3: Transmission spectra of OPW for various polarizations of exciting laser radiation (see legend) at the initial moment of time t = 0 and room temperature T = 300 K. Image taken from my article [100]. Copyright 2017 Elsevier.

tion effects should be taken into account [60, 98]. Therefore, to adequately describe the electromagnetic properties, it is necessary to introduce the so-called dynamic correction [99] of the polarizability of spheroidal NPs. The polarizability αn of the n-th NP in the OPW then takes the form: αn = αn [1 − (0)

(0)

−1 k2 2k 3 Ddyn α0 − i α0 ] , lE 3

(2.26)

where αn is defined by eq. (2.25), Ddyn is the dynamic geometric factor [99], lE is the length of the NP semiaxis along which the electric field is directed. The static Dstat and dynamic Ddyn depolarization factors for oblate and prolate spheroids can be found using well-known expressions [62, 99]. For spherical NPs: Dstat = 1/3 and Ddyn = 1.

2.2 2.2.1

Results Thermal and optical properties of optical plasmonic waveguides from silver nanoparticles

In my thesis I consider single Ag NPs in water and its extinction efficiency Qext as shown in eq. (2.27). Extinction efficiency is determined by extinction cross section σext , which is a

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

sum of the cross sections due to absorption and scattering. Qext =

σext , πR2 4π∣k∣ Im(d ⋅ E(r)). σext = ∣E0 ∣2

(2.27)

From fig. 2.2 it can be seen that the extinction efficiency of single spherical Ag NP with radius R = 8 nm decreases by 1.5 times for various temperatures and state of aggregation when the temperature of NP reaches the melting temperature (T ≈ 1080 K for Ag). In addition, the maximum of the extinction spectrum is strongly shifted to the short-wavelength region at the end of melting, when the nanoparticle completely passes into the liquid state. In this case, the value of Qext decreases by half compared with a solid nanoparticle at room temperature. Thus, it becomes obvious that temperature effects will significantly affect the transmission properties of OPWs, especially if only the n = 1 nanoparticle is excited by the external radiation.

Figure 2.3 represents the frequency-dependent transmission (eq. (2.24)) for three different polarizations of the external field, which coincide with the Cartesian coordinate axes, at the initial moment of time t = 0, when temperature is equal to room temperature (T = 300 K) for the OPW from N = 11 Ag particles with the geometry described in fig. 2.1(a). The maximum value of Qtr is reached at a wavelength of λ = 402 nm with the polarization directed along the X axis (see fig. 2.1(a)). Thus, only X polarization is of interest for the cases studied in my thesis, while for other polarizations I do not see any promising applications. Next, I turn to the discussion of temperature kinetics in OPWs. Obviously, thermal effects directly depend on the intensity of the exciting laser radiation. For small values of intensity of the laser pulse, none of the nanoparticles reach the melting temperature. However, in this case it will be practically impossible to register an optical signal at the end of the chain due to the strong attenuation of the SPP. As can be seen from fig. 2.3 the amplitude of the SPP at the end of the waveguide decreases very much (here by 70 times), even in the case of the best transmission. For high values of intensity of the laser pulse, substantial heating of the nanoparticles will be observed. In this case, their resonance properties will be almost completely suppressed. Thus, I consider exciting laser radiation with an intermediate intensity located between these two cases. The results of numerical simulations show that for the intensity of a laser pulse being I = 1.57 × 108 W/cm2 , only the first nanoparticle reaches the melting temperature. In this case, the transmission properties of the OPW will not decrease significantly. This value of the laser pulse intensity is standard for many lasers.

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15

Figure 2.4: The temperature of the ionic subsystem Tnion of the first, second, and third particles in an OPW (with geometry from fig. 2.1(a)) after t = 1 ns excitation of a laser pulse with λ = 402 nm and I = 1.57 × 108 W/cm2 intensity. The inset shows the temperature T2ion in the time interval from t = 250 ps to t = 1000 ps. Image taken from my article [100]. Copyright 2017 Elsevier. Figure 2.4 shows the time dependence of the temperature of the ionic subsystem for the first three NPs in an OPW, when the first nanoparticle is excited by a laser pulse at a wavelength of λ = 402 nm. It can be seen that the n = 1 nanoparticle reaches the melting temperature at t = 37 ps. Moreover, the maximum temperature reached by n = 2 and n = 3 nanoparticles at the same time: T2ion ≈ 480 K and T3ion ≈ 350 K, respectively. After t = 37 ps, the melting process of the n = 1 nanoparticle continues for about 134 ps and then its temperature increases slightly. After t = 200 ps, the temperature of the first three nanoparticles becomes constant: T1ion ≈ 1140 K, T2ion ≈ 375 K and T3ion ≈ 310 K. However, the heat from n = 1 nanoparticles reaches n = 2 nanoparticles at t ≈ 420 ps moment in time and T2ion slightly increases during the second half of the pulse (see insert in fig. 2.4). From fig. 2.4 it can be seen that the temperature of the second nanoparticle increases slightly over an extended period of time. The temperature Tnion of the n ≥ 4 NPs also remains unchanged for the chosen parameters of laser radiation. Figure 2.5 shows the transmission spectra (eq. (2.24)) of the OPW from spherical NP for various stages of melting of n = 1 nanoparticle. The transmission spectra of the OPW slightly changes when the n = 1 nanoparticle reaches its melting temperature (t = 37 ps, T ≈ 1080 K - dashed red line). hanges in the transmission spectra are induced by changes in the dielectric constant of the particle. It should be noted that the NP is still in the solid state at t = 37 ps. However, a substantial suppression of the resonance properties of the n = 1 nanoparticle occurs at t = 171 ps, when its become completely liquid (right after the end of melting process). It is seen that the efficiency of the transmission of OPW decreases three times in this case (dash-dotted green line). As a result, the transmitted energy also

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

Figure 2.6: The temperature distribution T at time t = 1 ns for the first three nanoparticles (plane XOY , z = 0) in the OPW (with geometry from fig. 2.1(a)). The first nanoparticle is excited by a laser pulse with intensity I = 1.57×108 W/cm2 . We draw the attention of readers that the color scale is presented in a non-linear scale for clarity. Image taken from Paper I. Copyright 2017 Optical Society of America.

Figure 2.5: Transmission spectra of an OPW (with geometry from fig. 2.1(a)) excited by laser radiation with an intensity of I = 1.57×108 W/cm2 at different points in time: initial time point t = 0 (solid line); start of melting of n = 1 nanoparticle, t = 37 ps (dashed line); end of melting of n = 1 nanoparticle, t = 171 ps (dashdotted line). Image taken from Paper I. Copyright 2017 Optical Society of America.

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17

decreases. In turn, this leads to a decrease in the temperature of the second and subsequent particles (see fig. 2.4). A further increase in the temperature of the nanoparticles in the chain occurs due to heat exchange between them (see the inset in fig. 2.4).

I plot the temperature distribution in the XOY plane (at Z = 0) for the OPW at t = 576 ps on fig. 2.6. The technological substrate is depicted schematically (grey color). Clearly, the temperature substantially increases for only the first three NPs. Despite the fact that the temperature of the n = 1 NP reaches the melting point (according to fig. 2.4), heat transfer from more heated ones nanoparticles to less heated occurs through the interparticle environment.

2.2.2

Optical properties of optical plasmonic waveguides from titanium nitride nanoparticles

Refractory materials are often considered to be effective in avoiding the negative influence of thermal effects on the OPW functionality. It is important to mention that even though TiN does not melt, its permittivity is temperature-dependent, which is taken into account in my work. Therefore, in this section, I will consider the dispersion and transmission properties of OPWs from TiN nanoparticles. I assumed that the chain of nanoparticles is located in a homogeneous medium and only the first (n = 1) nanoparticle is excited. In this section, the discussion will be based on the fact that TiN NPs do not heat up to the melting temperature and do not reach a phase transition. So, I will move directly to the dispersion properties of OPW from TiN NPs.

Figures 2.7 and 2.8 represent values of log[Im(˜ α)] from eq. (2.23) as a function of the frequency of SPP ω and eigenmodes of the wave vector q. According to the eigenvector expansion method, high values of Im(˜ α) correspond to high values of the Q eigenmode factor. While the summation in the eq. (2.23) occurs at infinity, we consider the OPW with a finite but sufficiently large number of particles, namely, N = 1000 NPs. I start with the OPW from spherical NPs whose dispersion relations are shown in the first column of figs. 2.7 and 2.8. It can be seen that SPPs efficiently propagate both for longitudinal (X) and transverse (Y ) polarizations at frequencies ω ≈ 2.5 − 3.5 rad/fsec. However, the branch corresponding to the highest values of Im(˜ α) has a rather slight slope, which corresponds to a low group SPP velocity in this spectral range.

It was previously shown that the use of non-spherical NPs in OPWs significantly increases the group SPP velocity [86, 89] and at the same time minimizes the SPP suppression [101]. Thus, it is of interest to consider the dispersion relations in the OPW from prolate and oblate spheroids with different values of the semiaxes ratio b/a. From fig. 2.7 and 2.8 it

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

Figure 2.7: Dispersion relations for an OPW from spherical and spheroidal particles at various temperatures for the longitudinal (X) polarization of the SPP. White dashed line represents light line ω = q/c. Image taken from Paper II. Copyright 2017 Elsevier. can be seen that in the case of longitudinal polarization, the general shape of the dispersion curve is almost the same as for the OPW from spherical NPs. This behavior is explained by an insignificant difference in the depolarization coefficients L for different geometries of the NPs with the same values of the short semi-axis b parallel to the polarization of the SPP. However, for oblate spheroids with b/a = 0.4, the values of Q near the light line ω = q/c are significantly larger.

In the case of transverse polarization, the values of log[Im(˜ α)] increase significantly, especially for the spectral range ω ≈ 1.5−2.5 rad/fsec for OPWs from prolate and oblate spheroids with b/a = 0.4. The dispersion dependencies for OPWs from oblate spheroids have an even greater slope compared to OPWs from prolate spheroids with the same values of b/a. In addition, the frequency of the eigenmodes decreases to ω ≈ 1 − 2 rad/fsec, which corresponds to the telecommunication wavelength range. In addition, the throughput of an OPW is increased in this case. Finally, the dispersion branch acquires a significant negative slope, which leads to an increase in the group velocity of the SPP and antiparallel propagation of the group and phase velocities of the SPP. The propagation of transversely polarized SPP with antiparallel group and phase velocities is described in detail in [89, 97, 102]. However, it should be noticed that the negative slope of the dispersion curve is not a direct evidence that OPWs are negative refractive metamaterials.

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19

Figure 2.8: Dispersion relations for OPW from spherical and spheroidal particles at various temperatures for longitudinal (Y ) polarization of the SPP. White dashed line denotes the light line ω = q/c. Image taken from Paper II. Copyright 2017 Elsevier. Despite the fact that TiN is thermally stable material [103], suppression of an SPP due to overheating of the OPW [100, 104] was describe in Section 2.1.3 is a crucial factor. From figs. 2.7 and 2.8 it can be seen that the dispersion relations of the OPW from TiN NPs remain almost unchanged even at T = 800○ C. The magnitude of the eigenmodes inevitably decreases at high temperature, but the suppression of the SPP is much lower than might be expected for ordinary plasmonic materials. Note that in my work the heating of the OPW is uniform, which is the most extreme case of overheating. In practice, only three neighboring NPs experience the highest heating in the case of local excitation of the SPP [100]. One of the interesting features that can be observed with a careful analysis of figs. 2.7 and 2.8 is that the SPP band (the spectral range corresponding to high-Q proper modes) varies significantly from longitudinal to transverse polarization for OPWs made from spheroids with b/a = 0.4. For longitudinal polarization, the frequency bandwidth of the OPW corresponds to the visible wavelength range, while for transverse polarization it lies in the telecommunication wavelength range. Thus, OPWs from prolate or oblate NPs can simultaneously operate in these two important wavelength ranges, which allows the use of such waveguides as hybrid photonic interconnectors. Next, I turn to the transmission properties of OPW, considering the propagation of an SPP in short chains from N = 20 NPs. From fig. 2.9 it can be seen that the bandpass 31

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

Figure 2.9: Transmission spectra of OPW from TiN NPs with various shapes for polarization X (a)-(c) and for polarization Y (d)-(f) SPP at various temperatures: (a), (d) T = 23○ C; (b), (e) T = 400○ C; (c), (f) T = 800○ C. Image taken from Paper II. Copyright 2017 Elsevier. of the OPW can be adjusted by switching the polarization of the SPP from longitudinal to transverse. Moreover, Qtr slightly decreases at high temperatures, which is crucial for waveguide applications of OPWs. As expected, the most effective SPP propagation occurs in OPWs from oblate spheroids with small aspect ratios (here b/a = 0.4), which is consistent with the results presented in [86, 101].

Finally, another attractive property of OPWs is their ability to confine electromagnetic energy at scales much shorter than the wavelength of the propagating excitation. This feature distinguishes OPWs from the classical [105] strip waveguides, whose transverse dimensions are usually comparable or several times larger then the wavelength of the propagating signal. Localization of the electromagnetic field near the OPW allows to locate several OPWs in close proximity to each other without the risk of an overlapping SPP propagating in a neighboring OPW, something which cannot be achieved in strip waveguides.

Figure 2.10 shows the temperature-dependent intensity distribution of ∣E∣2 /∣E0 ∣2 for an OPW from TiN NPs at a distance 10 nm from the upper surface of the NPs. The frequencies ω were chosen to correspond to the maximum values of Qtr for Y -polarization of the SPP from fig. 2.9 (d-f). It is shown that in the case of spherical NPs, the electric field is densely localized near the first excited NP and rapidly decays along the OPW. The most effective localization of the electric field is observed in the case of prolate spheroids. Distribution of ∣E∣2 /∣E0 ∣2 looks completely different for OPWs from the oblate spheroids due to the high local field at the tips of oblate spheroids. Finally, due to the refractory behavior, the

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2. Thermal and optical effects in plasmonic nanoparticle waveguides

21

Figure 2.10: Localization of the electric field ∣E∣2 /∣E0 ∣2 OPW from spherical, prolate and oblate nanoparticles (with b/a = 0.4 for both types of spheroids) TiN NPs for the Y polarization of SPP at different temperatures. The frequencies were taken in accordance with the maximum values of Qtr from the fig. 2.9. In all cases, the first particle on the left is excited. Image taken from Paper II. Copyright 2017 Elsevier. confinement of the electric field for OPWs from TiN NPs remains almost unchanged at high temperatures.

2.3

Conclusions for Chapter 2

In this chapter, an original theoretical model of my thesis was reviewed and summarized. It describes light-induced dipole interaction between nanoparticles and environment in highintensity optical fields, taking into account thermal effects. The proposed model includes taking into account the temperature dependence of the dielectric constant of the particle material, as well as the heat exchange of nanoparticles with the environment. The model developed in this thesis was used to study the thermal effects that occur during the propagation of surface plasmon polaritons excited by pulsed laser radiation in an optical plasmonic waveguide (OPW). It was shown that thermal effects significantly decrease the efficiency of OPW transmission due to the suppression of the plasmon resonance of the nanoparticles. In this work, the optimal conditions for transmitting information with the optical waveguide were determined. It was shown that TiN is a promising alternative material that can be used in OPWs in the form of chains of nanoparticles (NPs) that can support the effective propagation of

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Chapter 2 Thermal and optical effects in plasmonic nanoparticle waveguides

SPPs [106]. The bandwidth of linear periodic chains of titanium nitride (TiN) NPs can be adapted to both the visible range and the telecommunication wavelength range by changing the shape of the nanoparticles and polarization of the surface plasmon polariton (SPP). Despite the inevitable ohmic losses and overheating of nanoparticles, the SPP attenuation remains almost unchanged even at extremely high temperatures due to the pronounced refractory properties of TiN. Along with cheap methods of large-scale production of TiN nanoparticles, all these features make it a promising plasmonic material for waveguide applications using linear periodic chains of NPs. The results obtained allow me to offer new applications of OPWs associated with their high sensitivity to the intensity of exciting radiation. Further development of the proposed model and the study of thermal effects in OPWs or other periodic nanostructures is promising and will open up new practical applications of plasmonic nanosystems [107].

34

Chapter 3 Collective effects in structures of resonant nanoparticles Dielectric nanoparticles have attracted an increased interest in photonics due to their ability to preserve not only electric, but also magnetic dipole moments. In this chapter, based on the extended coupled dipole approximation, I review three types of defects in two-dimensional arrays of spherical Si nanoparticles that are studied in my thesis: disorder in the positions of Si nanospheres of the same size; size disorder of nanospheres located in an ordered twodimensional lattice; and quasi-ordered two-dimensional arrays of nanospheres with the same size. A comprehensive analysis in my thesis of these scenarios reveals various effects of disorder on the coupling of electric dipole and magnetic dipole resonances with lattice modes. Next, I have demonstrated ED ↔ MD cross-interactions in sufficiently large NPs arrays, where such interactions are usually considered to be negligible.

3.1 3.1.1

Model Extended coupled dipole approximation

Lets consider an array of N spherical NPs embedded in vacuum which is irradiated by electromagnetic plane waves with electric E0 and magnetic H0 components. The n-th particle located at rn acquires electric dn and magnetic mi dipole moments which are coupled to other dipoles and to an external electromagnetic filed via the extended coupled dipole equations [108–110]:

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Chapter 3 Collective effects in structures of resonant nanoparticles

dn = αe (Einc (rn ) + ∑ Gnj dj − ∑ gnj × mj ) , Ntot

Ntot

j≠n

j≠n

mn = αm (Hinc (rn ) + ∑ Gnj mj + ∑ gnj × dj ) , Ntot

Ntot

j≠n

j≠n

(3.1) (3.2)

where αne and αnm are electric and magnetic dipole polarizabilities [110] of the n-th particle, respectively, ε0 and µ0 are the dielectric constant and magnetic permeability of vacuum, E0n = E0 (rn ), H0n = H0 (rn ), and ˆ nj = Anj Iˆ + Bnj ( rnj ⊗ rnj ) , G 2 rnj

rnj Cˆnj = Dnj ×, rnj

(3.3)

where Iˆ is a 3 × 3 unit tensor, ⊗ denotes a tensor product, and Anj , Bnj and Dnj are defined as follows: Anj =

exp(ikrnj ) 2 1 ik (k − 2 + ) , rnj rnj rnj

(3.4)

Dnj =

exp(ikrnj ) 2 ik (k + ) , rnj rnj

(3.6)

Bnj =

exp(ikrnj ) 3 3ik (−k 2 + 2 − ) , rnj rnj rnj

(3.5)

Electric and magnetic dipole polarizabilities are explicitly defined as [62]: 3i mψ1 (mkRn )ψ1′ (kRn ) − ψ1 (kRn )ψ1′ (mkRn ) , 2k 3 mψ1 (mkRn )ξ1′ (kRn ) − ξ1 (kRn )ψ1′ (mkRn ) 3i ψ1 (mkRn )ψ1′ (kRn ) − mψ1 (kRn )ψ1′ (mkRn ) , αnm = 3 2k ψ1 (mkRn )ξ1′ (kRn ) − mξ1 (kRn )ψ1′ (mkRn ) αne =

(3.7) (3.8)

where m is the refractive index of the NP material, Rn is the radius of the n-th particle, ψ1 (x) and ξ1 (x) are Riccati-Bessel functions, and prime denotes the derivation with respect to the argument in parentheses.

The essence of CLRs can be understood from a closed-form analytical solution of eq. (3.1) and eq. (3.2) obtained for an infinite array [3, 110]. In this case, dn = d ∥ E0 and mn = 36

3. Collective effects in structures of resonant nanoparticles

3.1 Model

25

m ∥ H0 for each NP [110], therefore, the last terms in eq. (3.1) and eq. (3.2) vanish, since E0 ⊥ H0 . Thus, for a special case of a regular 2D lattice illuminated with a normally impinging wave with ∣E0 ∣ = E0x and ∣H0 ∣ = H0y , the non-zero components of d and m are dx =

H0y E0x , my = , 1/αe − G0xx 1/αm − G0yy

(3.9)

e,m − where G0xx and G0yy are diagonal elements of the 3 × 3 tensor G0 = ∑∞ j=2 G1j , and (1/α 0 −1 Gxx,yy ) are effective electric and magnetic polarizabilities which capture the features of the NP’s surrounding [110–112].

The electric dn and magnetic mn dipoles induced on each NP can be found from the solution of eq. (3.1). In this work, I describe the optical response of a finite array of NPs with the extinction efficiency: Qe =

N 4k 0∗ Im ∑ (dn ⋅ E0∗ n + mn ⋅ H j ) , 2 I0 N R n=1

(3.10)

where I0 is the intensity of the incident field, and the ∗ denotes a complex conjugate. In the general case of polydisperse array with Rn ≠ R, the average radius ⟨R⟩ = ∑N n=1 Rn /N is used to define Qe . For an infinite array, after substituting eq. (3.9) in eq. (3.10), one gets: Qinf e =

4k Im [(1/αe − G0xx )−1 + (1/αm − G0yy )−1 ] . R2

(3.11)

Though, higher-order multipoles in all-dielectric NPs are pronounced, for example, in large [113– 115] and nonspherical [116] single Si NPs, or in structures of closely packed Si NPs [117], full-wave simulations and mode analysis [118] show that ED and MD are predominant in arrays of spherical Si NPs with R = 65 nm, and high order electric and magnetic field oscillations can be ignored in this case. Thus, the extended coupled dipole approximation accurately describes optical properties of arrays from relatively small Si NPs.

3.1.2

Types of imperfections in arrays on silicon nanoparticles

From eq. (3.1) and eq. (3.2) I can conclude that two types of disorder can be achieved [76]: off-diagonal and diagonal. These types affect either off-diagonal or diagonal elements of the interaction matrix in eq. (3.1) and eq. (3.2). The first type of disorder (off-diagonal) affects ˆ nj and Cˆnj which are the functions of the NPs positions, while the second only tensors G type of disorder (diagonal) affects only αne,m which are functions of the shape and size of the NPs.

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Figure 3.2: (a) Refractive index m of Si from Ref. [119]; (b) Extinction spectra for a single Si NP of various radii R taking into account high-order multipoles. Spectral positions of electric and magnetic dipole resonances are denoted as ’ed’ and ’md’, respectively. Image taken from Paper III. Copyright 2019 Optical Society of America.

Figure 3.1: Schematic representation of different types of disorder considered in this work: (a) x-disorder, (b) y-disorder, (c) size disorder, and (d) quasi-random array. Image taken from Paper III. Copyright 2019 Optical Society of America.

For ordered arrays of NPs it is shown in fig. 3.3 that two types of couplings can be distinguished. For fixed wavelengths, the optical response of the lattices strongly depends on variations of either hx and hy . Thus, to get more insight, I introduce an off-diagonal disorder in the following manner. I study the positional disorder along the x axis keeping the y coordinates constant, and vice versa, as shown in fig. 3.1(a) and 3.1(b), respectively. I refer to these two types of positional disorders as x-disorder and y-disorder, correspondingly. For both cases, I introduce the deviation σx,y which characterizes the degree of disorder. For each n-th particle with initial (xn , yn ) coordinates, I randomly set new coordinates as dis (xdis n , yn ) for x-disorder and (xn , yn ) for y-disorder within the following limits: xn − σx ≤ xdis n ≤ x n + σx ,

and yn − σy ≤ yndis ≤ yn + σy .

(3.12)

dis Both xdis n and yn are randomly generated using a uniform distribution for each n-th NP and for each lattice with given (hx , hy ). Thus, the effects of positional disorder are uncorrelated.

The schematics of the lattice with diagonal (size) disorder is shown in fig. 3.1(c). In this specific case, I keep the original coordinates of each NP, and randomly change the radius Rn of each n-th NP within the following limits using a uniform distribution:

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27

Rn − σR ≤ Rndis ≤ Rn + σR .

(3.13)

Again, as in the case of off-diagonal disorder, Rndis is introduced randomly for each NP and for each lattice configuration, which provides uncorrelated results. Finally, fig. 3.1(d) shows the last type of considered defects - a special combination of diagonal and off-diagonal disorders, which attract specific interest [120, 121]. It is a well-known fact that the coupling between a SPR of single NP and lattice modes strongly depends on the number of NPs in the array [112, 122]. Nevertheless, periodic lattices of strictly spaced NPs are usually considered studies of this effect of finite size. In this thesis, I fix the initial coordinates and NP sizes in the array and randomly remove the NP from the initial lattice, leaving the other NPs untouched. This type of imperfections is somewhat similar to vacancies in crystal structures. I refer to lattices shown in fig. 3.1(d) as quasi-random arrays. I emphasize that each lattice configuration for each type of disorder with given σx , σy and σR or number of NPs removed from the lattice in the case of quasi-random arrays reviewed here, has been simulated only once, without computing ensemble averages. The closeness to a statistical average has been guaranteed by simulating a reasonably large number of NPs.

3.2 3.2.1

Results Periodic arrays of silicon nanoparticles

I start to shortly review the optical properties of a single Si nanosphere. Figure 3.2(a) shows the refractive index of Si used in my calculations [119], while fig. 3.2(b) shows the extinction efficiency Qe for a single Si nanosphere of various radii R. For a single sphere, and only in this case, I calculate Qe taking into account high-order harmonics [62] required for the convergence of the electromagnetic light scattering problem [123]. It can be seen from fig. 3.2(b) that indeed, for given sizes, the Si nanospheres have distinct and predominant ED and MD resonances in the visible wavelength range. This is a general reason to consider arrays from Si nanospheres with R = 65 nm radius. However, in the special case of a size disorder, all possible radii of NPs will fall into the range shown in fig. 3.2(b), i.e. 50 nm ≤ Rn ≤ 80 nm. Therefore, the coupled dipole approximation can be used with strong confidence. Next, it is insightful to discuss optical properties of ordered Si nanostructures. Figures 3.3(a) and 3.3(b) show two different types of lattices which have been studied in this thesis: with fixed period along the x axis, hx , and varying period along the y axis, hy , and with fixed hy and varying hx . Such variations of interparticle distances make it possible to get ED or

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Chapter 3 Collective effects in structures of resonant nanoparticles

Figure 3.3: (a) and (b) Schematic representation, and (c) and (d) extinction spectra Qe of ordered 2D lattices from N = 20 × 20 Si NPs with R = 65 nm. Two configurations are considered: (left) fixed hx = 540 nm and varying hy , and (right) fixed hy = 450 nm and varying hx . Spectral positions of ED and MD resonances are denoted as ’ed’ and ’md’, respectively. Dashed RAx and RAy lines denote Rayleigh anomalies λ = hx and λ = hy , correspondingly. Image taken from Paper III. Copyright 2019 Optical Society of America.

MD coupling with lattice modes [124]. In both cases, the incident electric E0 and magnetic H0 fields are aligned along the x and y axes, correspondingly. Lattices from N = 20 × 20 Si NPs have been considered here.

In the first case, as it is clearly seen from fig. 3.3(c), ED strongly couples to lattice modes which leads to the emergence of quite sharp collective lattice resonances. The position of the MD resonance slightly shifts to shorter wavelengths for large hy . Note that Qe for MD increases near the Rayleigh anomaly λ = hy . In the second case, according to fig. 3.3(d), the same strong coupling with lattice modes occurs for MD, while the position of ED gradually shifts to shorter wavelengths and the corresponding Qe decreases with increasing hx . Thus, the coupling occurs for the incident field (electric or magnetic) perpendicular to the axis along which the interparticle distance changes. In other words, for the particular case considered, EDs (E0 is parallel to x axis) couple to RAy , and vice versa, MDs (H0 is parallel to y axis) couple to RAx .

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3. Collective effects in structures of resonant nanoparticles

29

Figure 3.4: Extinction spectra Qe for various degrees of positional disorder σx along the x axis, as shown in Fig 3.1(a). Image taken from Paper III. Copyright 2019 Optical Society of America.

3.2.2

Figure 3.5: Extinction spectra Qe for various degrees of positional disorder σy along the y axis, as shown in Fig 3.1(b). Image taken from Paper III. Copyright 2019 Optical Society of America.

Arrays of silicon nanoparticles with imperfections

Figures 3.4 and 3.5 show extinction spectra for arrays of NPs with different degrees of xand y-disorders. It can be seen that these two types of positional disorders affect the optical properties of the NPs in a different way, depending on the coupling regime. As it might be expected from the analysis of Fig. 3.3(d), the x-disorder significantly affects the MD, since the latter strongly couples to the Rayleigh anomaly RAx . Clearly, from Fig. 3.4, one may observe slight suppression of the MD with the increasing of the degree of disorder, σx , both for ED and MD coupling scenarios. It can also be noticed that the coupling of MD and RAx remains observable even for sufficiently large σx in Fig. 3.4(f), where MD is suppressed. ED remains almost the same for each case shown in Fig. 3.4. Figure 3.5 shows an expected trend: since ED couples to RAy , y-disorder affects only the former, keeping MD almost the same for various σy . However, Figs. 3.5(e)-3.5(f) show almost total suppression of ED for σy = 150 nm, while in the case of strong x-disorder shown in Figs. 3.4(e)-3.4(f), MD is quite pronounced. It might be explained by the fact that the

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Chapter 3 Collective effects in structures of resonant nanoparticles

Figure 3.6: Extinction spectra Qe of NPs arrays with ED coupling (left), and MD coupling (right) for various degrees of positional disorder σx (c,d), σy (e,f), and σxy (g,h). Corresponding values of hx and hy are shown in legends. Dashed vertical lines denote positions of Rayleigh anomalies RAy at λ = 500 nm (left), and RAx at λ = 540 nm (right). Image taken from Paper III. Copyright 2019 Optical Society of America. MD response is stronger than the ED resonance in Si NPs of the considered sizes, according to Fig. 3.2(b). Thus, it is easier to suppress ED than MD for the same degree of positional disorders σy and σx , respectively. Figure 3.6 shows the detailed comparison of the extinction spectra for arrays with ED or MD couplings. Indeed, x-disorder strongly suppresses the MD, while y-disorder suppresses the ED resonance. Since the ED is generally weaker than the MD, the former is almost completely disappears for high degrees of y-disorder. For the completeness, Figs. 3.6(g)3.6(h) show the spectra for arrays with xy-disorder, which has been introduced in the same way as the x- and y-disorders, but with simultaneous randomization of both xn and yn coordinates of each NP. It can be seen that in the general case, such a combined disorder gives a superposition of both x - and y -disorders, which suppresses both ED and MD resonances. Next, I move to the diagonal type of disordering. Figure 3.7 shows extinction spectra for arrays with various degrees of size disorder, σR . It is clearly seen that random variations of NP sizes strongly suppress both ED and MD resonances. However, MD remains observable only for σR = 5 nm, while for larger σR it almost completely disappears. Contrary, the ED resonance is preserved in all cases, and, of note, EDs strongly couple with Rayleigh anomalies, RAy , even for high degrees of diagonal disorder, as shown in Fig. 3.7(e). This

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3. Collective effects in structures of resonant nanoparticles

31

Figure 3.8: Extinction spectra Qe of NPs arrays with ED coupling (left), and MD coupling (right) for various degrees of size disorder σR . Corresponding values of hx and hy are shown in legends. Dashed vertical lines denote positions of Rayleigh anomalies RAy at λ = 500 nm (left), and RAx at λ = 540 nm (right). Image taken from Paper III. Copyright 2019 Optical Society of America.

Figure 3.7: Extinction spectra Qe for the same 2D lattices as in Figs. 3.3(c) and 3.3(d), but for various degrees of size disorder σR , as shown in Fig 3.1(c). Image taken from Paper III. Copyright 2019 Optical Society of America.

effect might be explained by the different behavior of polarizabilities αie and αim [110] which yields different impact of size disorder on ED and MD resonances. To get a deeper insight, I plot Qe for arrays with fixed hx and hy , as shown in Fig. 3.8. Indeed, Figs. 3.8(c), 3.8(e), 3.8(g) show that size disorder has a surprisingly weak effect on the ED resonance of arrays with strong ED coupling. It can be seen from Fig. 3.8(g) that the maximum Qe for the ED resonance drops by no more than 10% for σR = 15 nm compared to the ordered array shown in Fig. 3.8(a). For arrays with MD coupling, Qe for ED resonance drops stronger, by the factor of 2 for σR = 15 nm, as shown in Fig. 3.8(h). As for the MD resonance, in both the ED and MD coupling cases, the extinction efficiency for MD sharply drops for σR = 5 nm. For larger σR , the MD resonance becomes almost indistinguishable. Based on the previous discussion of diagonal and off-diagonal types of disorders, I can

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Chapter 3 Collective effects in structures of resonant nanoparticles

conclude that the simultaneous use of positional and dimensional disorders can lead to a superposition of effects as shown in Figs. 3.4, 3.5 and 3.7. Thus, I do not consider arrays of randomly located NPs of different size. Instead, I introduce a specific combination of positional and size disorders as shown in Fig. 3.1(d). These quasi-random arrays are fundamentally different from the ones shown in Figs. 3.1(a)-3.1(c) since random elements of the interaction matrix in (3.1) are strictly set to zero in the case of quasi-random arrays, while in the previously considered scenarios, off-diagonal or diagonal elements have acquired random deviations according to σx , σy or σR .

Figure 3.10: Extinction spectra Qe of NPs arrays with ED coupling (left), and MD coupling (right) for (a)-(b) N = 30 × 30 array, and for its various quasi-random modifications (solid lines): (c)-(d) 81% = 729, (e)-(f) 49% = 441, and (g)-(h) 16% = 144 NPs kept untouched. For comparison, Qe of strictly periodic (dashed lines) arrays of the same number of NPs are shown: (c)(d) N = 27 × 27 = 729, (e)-(f) N = 21 × 21 = 441, and (g)-(h) N = 12 × 12 = 144, grey dash-dot lines show Qe of a single Si NP with R = 65 nm. Image taken from Paper III. Copyright 2019 Optical Society of America.

Figure 3.9: Extinction spectra Qe for quasirandom 2D lattices, as shown in Fig 3.1(d), for different number of NPs: (a)-(b) 81% = 729, (c)-(d) 49% = 441, and (e)-(f) 16% = 144 kept untouched in N = 30 × 30 arrays of NPs with R = 65 nm. Note the different color scale in the last row (e)-(f). Image taken from Paper III. Copyright 2019 Optical Society of America.

Next, I review NPs with the same size, R = 65 nm, but increase their number to N = 30 × 30 (while previously discussed arrays had N = 20 × 20 NPs). Here, I randomly remove NPs, 44

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3. Collective effects in structures of resonant nanoparticles

33

leaving the rest (here 81%, 49% or 16%) of NPs untouched. I note that the consideration of larger arrays is preferable for this type of disorder, since coupling effects may be totally suppressed in arrays from a small number of NPs left in the lattice [112]. Nevertheless, in the smallest array considered here, I keep 144 quasi-randomly located NPs, which is sufficient for coupling effects to occur. Intuitively, one can expect suppression of the ED and MD resonances with an increase in the number of NPs removed from the ordered array. Indeed, Fig. 3.9 confirms such an expectation. However, it can be seen that lattices which contains 81% of the initial NPs have almost the same optical properties as the original periodic arrays. Moreover, Figs. 3.9(e)-3.9(f) show that ED and MD are coupled to Rayleigh anomalies (though quite weakly) in the arrays with only 16% NPs left, and extinction spectra of such arrays tend to become closer to the Qe of a single NP. For comparison, Fig. 3.10 shows spectra of ordered arrays (as in Figs. 3.3(a)-3.3(b)) from exactly the same number of NPs as in the quasi-random arrays, i.e. 27 × 27, 21 × 21, and 12 × 12, and with the same hx and hy . It can be seen from Figs. 3.10(c)-3.10(d) that Qe of the quasi-random array from 729 NPs is also almost the same as Qe for the periodic 27 × 27 array. Moreover, even with the increasing number of NPs removed from the array, Qe of the quasi-random lattices is quite close to strictly ordered arrays with the same number of NPs. However, in the most extreme cases of quasi-random arrays shown in Fig. 3.10(g)3.10(h), the collective ED resonances are almost suppressed, while the MD coupling remains observable, though, the corresponding peak of the MD resonance is blue-shifted compared with the ordered arrays.

3.2.3

Finite size effects in arrays of silicon nanoparticles

From the analysis of (3.9), one could expect to observe resonances if Re (1/αe,m − G0xx,yy ) vanishes for either ED or MD resonances. Indeed, Fig. 3.11(a) shows that the dimensionless representation of the above parameter becomes zero near λ ≈ hy and λ ≈ hx for dx and my , respectively, which corresponds to (0, ±1) and (±1, 0) Wood-Rayleigh anomalies. Note that in the general case of hx ≠ hy considered here, a simple rotation of the incident field polarization, e.g. (E0x , 0, 0) → (0, E0y , 0), does not yield the interchange between ED and MD CLRs spectral positions, since it only implies the interchange G0xx ↔ G0yy in (3.9), which will likely violate the Re (1/αe,m − G0xx,yy ) = 0 condition due to G0xx ≠ G0yy and non-trivial wavelength dependence of polarizabilities αe,m (λ) (Fig.(4) [110]).

Figure 3.11(b) shows that extinction spectra for finite-size arrays gradually approach the spectrum for the infinite lattice as N increases, which is consistent with reported trends for arrays of plasmonic NPs [112, 122]. Indeed, the ED CLR at λ ≈ 490 nm for arrays with Ntot > 50 × 50 becomes almost indistinguishable from one for the infinite array, as it 45

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Chapter 3 Collective effects in structures of resonant nanoparticles

Figure 3.11: (a) Real parts of normalized denominators of (3.9) which correspond to dx (ED) and my (MD); (b) Extinction efficiency for infinite (∞) and for N × N finitesize arrays; (c) and (d) zoomed-in spectra for ED and MD CLRs, respectively. Dashed vertical lines indicate the position of the CLR peak for the infinite array. Image taken from Paper VI. Copyright 2019 Optical Society of America.

Figure 3.12: Normalized intensities of electric field induced by MDs (left) and of magnetic field induced by EDs (right) for N ×N arrays at wavelengths: (a) 493 nm, (b) 588 nm, (c) 490 nm, (d) 586.5 nm. Each dot represents the NP, and the actual sizes of arrays vary for different N × N . Image taken from Paper VI. Copyright 2019 Optical Society of America.

is clearly seen from Fig. 3.11(c). Of note, for plasmonic NPs arrays, the corresponding threshold, when Qext becomes almost the same for finite and infinite lattices is ≈ 20 × 20 NPs [112]. Analogously, Qext for an MD CLR at λ ≈ 586 nm in finite-size arrays becomes similar to the infinite case if N grows, as it is shown in Fig. 3.11(d). However, what is really surprising and unexpected is that Qext of finite-size arrays is noticeably different even for the Ntot = 100 × 100 case.

Figure 3.12 shows the corresponding intensities, i.e. ∣Emag ∣ and ∣Hel ∣ , for each NP in the array. It can be seen that the normalized intensity of the electric field induced by MDs is quite small compared to the incident field, and increases only at the boundaries of the array, which again agrees well with results for plasmonic NPs [122]. The maximum value 2 2 of ∣Emag ∣ / ∣E0 ∣ , which is already quite small for 30 × 30 arrays in Fig. 3.12(a), gradually decreases for larger arrays, and almost vanishes for the 70 × 70 array in Fig. 3.12(c), thus 2

46

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3. Collective effects in structures of resonant nanoparticles

35

providing negligible difference for ED CLRs of infinite and sufficiently large finite-size arrays in Fig. 3.11(c). On the contrary, the maximum intensity of the magnetic field induced by 2 2 electric dipoles, i.e. ∣Hel ∣ / ∣H0 ∣ , increases for larger arrays, and again a divergence takes places near the boundaries of the array. Although the overall contribution of the crossinteraction between EDs and MDs to Qfin ext gradually decreases as N grows, the “boundary effect” is pronounced even for sufficiently large arrays, and thus can not be completely ignored in this case.

3.2.4

Optical filters based on arrays of plasmonic nanoparticles

In the previous section I showed that geometric parameters of an array of NPs, such as the interparticle distance, size and shape of the nanoparticles, significantly affect the position of electric and magnetic dipole resonances and may lead to suppression of the MD resonance, as well as the ED resonances. However, with proper parameters of the array it is also possible to prevent the appearance of several close lines in the spectrum instead of one line, which might be important for selective narrow-band filters in the tunable spectral range. To illustrate this I consider 2D arrays of nanodisks with height H and radius R arranged in a regular square lattice with period h. The arrays are embedded in a homogeneous environment with refractive index nm = 1.45, which corresponds to quartz in the spectral range under study. Such structures can be fabricated using a lithography technique on a quartz substrate and subsequent sputtering a layer of quartz on top of the array. A homogeneous environment is an important factor in the model, because the Q-factor of the CLR drops dramatically in the case of a half-space geometry, where the substrate and the superstrate have different refractive indices [125]. The reflection spectra of such structures are calculated with a commercial FDTD method software. FDTD is a widely used computational method of electrodynamics, which in general shows excellent agreement with experimental results for CLRs [11, 24, 126–128]. The optical response of the infinite array is simulated by considering a single particle unit cell with periodic boundary conditions applied at the lateral boundaries of the simulation box and perfectly matched layers used at the remaining top and bottom sides. Arrays are illuminated from the top by plane waves with normal incidence. The reflection has been calculated at the top of the simulation box using a discrete Fourier transform monitor which is placed above the plane-wave source. An adaptive mesh has been used to accurately reproduce the nanodisk shape. Finally, extensive convergence tests for each set of parameters have been performed to avoid undesired reflections on the perfectly matched layers. The suppression of surface plasmon resonances under extreme conditions was studied in the papers [104, 129, 130] in my thesis. It was shown that heating of nanoparticles by pulsed

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Chapter 3 Collective effects in structures of resonant nanoparticles

Figure 3.13: Reflection spectra for TiN (left) and Au (right) nanodisks arrays with: (a) fixed h = 1100 nm, and for different R = H as shown in legend; (b) fixed h = 650 nm, and for different R = H as shown in legend; (c) and (d) corresponding quality factors of CLRs; (e) fixed R = H = 90 nm and for different h as shown in legend; (f) fixed R = H = 55 nm and for different h as shown in legend; (g) and (h) corresponding quality factors of CLRs. Image taken from Paper IV. Copyright 2019 Optical Society of America.

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laser radiation results in a reduction of the Q-factor and CLR amplitude. In particular, for the CLR at λ = 1100 nm in Fig 3.13, the Q-factor is 1.5 ⋅ 103 at T = 23○ C, Q = 1.1 ⋅ 103 at T = 400○ C, and Q = 0.7 ⋅ 103 at T = 900○ C. Thus, the high radiation resistance of TiN can be an additional advantage when using arrays exhibiting CLRs at high temperatures [131]. The use of TiN as a plasmonic material with high radiation resistance provides an extreme stability at high temperatures compared to conventional plasmonic materials (Au and Ag). Au nanodisks arrays (Fig. 3.13) demonstrate CLRs in the long-wavelength part of the visible and near IR ranges.

3.3

Conclusions for Chapter 3

The effect of various types of disordering on the optical response of 2D arrays of spherical Si nanoparticles was theoretically analyzed in my thesis. Electric and magnetic dipole resonances dominate in spherical Si nanoparticles (NPs) in the considered range of 50 nm ≤ R ≤ 80 nm, so I used the extended coupled dipole approximation, which adequately describes the electromagnetic properties of arrays of Si NPs [118, 132].

First, I showed the existence of two types of collective resonances in 2D arrays arising from the strong coupling of electrical or magnetic dipole resonances of one NP with the lattice modes (Rayleigh anomalies) of the 2D array. Such a connection arises when the corresponding component of the incident field (electric or magnetic) is orthogonal to the variable period (hy or hx ) of the lattice, and the other period (hx or hy ) is constant [124]. Second, I showed that the electric and magnetic response is affected by positional disorder only when the low frequencies are displaced along an axis orthogonal to the corresponding component of the incident electromagnetic illumination. In my case, for E0 ∥ x and H0 ∥ y, the electric and magnetic dipole resonances are strongly suppressed only for y or x disordering, respectively. Obviously, both resonances change when the nanoparticles are shifted along the x and y axes at the same time. Next, I showed that the collective magnetic dipole response almost completely disappears in the case of diagonal (dimensional) disordering with σR > 5 nm. However, the electric dipole moment remains quite stable, especially in the case of strong collective coupling between the electric dipole resonance and lattice modes, even for strongly polydisperse arrays with σR = 15 nm. I considered quasi-random arrays as a special combination of diagonal and diagonal disorders. Instead of simultaneously displacing the nanoparticles and changing their sizes, I arbitrarily removed the nanoparticles from the lattice, leaving other nanoparticles at the starting points with the original sizes. It was noted that in the lattice where only 16% of the nanoparticles remained, collective electric and magnetic resonances

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Chapter 3 Collective effects in structures of resonant nanoparticles

are observed. However, the extinction spectra of such arrays are, as a rule, similar to the spectra of a single nanoparticle. Furthermore, I have shown that the finite size of arrays of dielectric NPs plays an important role for the emergence of both electric dipole (ED) and magnetic dipole (MD) collective lattice resonances (CLR). While ED CLRs in finite-size arrays converge to the infinite-array model for ≈ 50 × 50 NPs, MD CLRs in finite-size arrays are quite different from the ones of infinite arrays even for 100 × 100 NPs, thus the common use of numerical and theoretical models for infinite arrays should be handled with great caution. The reviewed results provide a comprehensive analysis and understanding of the effect that disordering has on collective resonances as well as effect of finite size in 2D arrays of dielectric nanoparticles. Although I examined the special case of spherical nanoparticles of Si embedded in vacuum, similar trends can be expected for dielectric arrays of other forms or materials, see [133], if higher-order multipoles can be neglected. In this section I showed that the geometric parameters of arrays of dielectric nanoparticles, such as the period, size and shape of nanoparticles, significantly affect the position of the electric dipole resonance in the spectrum and can suppress the magnetic dipole resonance. A clever choice of geometric parameters can also prevent the appearance of several close lines or their splitting in the reflection spectrum instead of one line, which is unacceptable for selective narrow-band filters in the tunable spectral range. Arrays of nanoparticles in the reflection mode demonstrate the effect of optical filtering with fine tuning of the spectral position of the resonance line to the required wavelength by tilting the grating with respect to the incident radiation.

50

Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles Finally, I present a new atomistic model for plasmonic excitations and optical properties of metallic nanoparticles (NPs). This model collectively describes the atomic complete response in terms of fluctuating dipoles and charges that depend on the local environment and on the morphology of NPs. This could be single element metal NPs as well as composite or alloy structure. Being atomic dependent, the model describes the total optical properties, the complex polarizability and the plasmonic excitation of a cluster and can refer these properties to a detailed level where geometric characteristics of the cluster plays a role, making it possible to explore the role of material, alloy mixing, size, form shape, aspect ratios, and other geometric factors, down to the atomic level. My conviction is that it will be useful for the design of plasmonic NPs with particular strength and field distributions, and can have wide ramifications in bioimaging, where small plasmonic particles often are desired. The model is parameterized from experimental data and is at present practically implementable for NPs up to more than 12 nm, for nanorods even more, thus covering a significant part of the gap between the scales where quantum calculations and classical models based on the bulk dielectric constant. I have applied the method to both spherical and cubical clusters along with nanorods and hollow NPs and have demonstrated the size and shape dependence of the plasmonic excitations and connected this to the geometry of the NPs using the plasmon length.

51

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.1

Model

4.1.1

Extended discrete interaction model

Similarly to the original DIM suggested by Jensen et al. [134, 135] the ex-DIM [136] aims to describe the polarizability and optical properties of metallic nanoparticles by representing the nanoparticle as a collection of interacting atomistic charges and dipoles. The starting point of both models is a Lagrangian with an energy expression for interacting fluctuating charges and dipoles in an external electric field subject to a charge equilibration constraint: L[{µ, q}, λ] =E[{µ, q}] − λ(q tot − ∑ qi ) N i

1 N 1 N N 1 N (0) = ∑ qi c−1 ∑ ∑ qi Tij qj + ∑ µi α−1 ii qi + ii µi 2 i 2 i j≠i 2 i −

N N N 1 N N (2) (1) ∑ ∑ µi Tij µj − ∑ ∑ µi Tij qj + ∑ qi Vext 2 i j≠i i j≠i i

− ∑ µi Eext − λ(q tot − ∑ qi ). N

N

i

i

(4.1)

In Eq. (4.1) the first term is the self-interaction energy of fluctuating charges, the second term is the interaction energy between fluctuating charges, the third term is the self-interaction energy of fluctuating dipoles, the fourth term is the interaction energy between fluctuating charges and dipoles, the fifth term is the interaction energy between fluctuating dipoles, the sixth term is the interaction energy between fluctuating charges and the external potential, the sixth term is the interaction energy between fluctuating dipoles and the external field, and the last term is a charge equilibration condition expressed via the Lagrangian multiplier λ. Here, the qi is the fluctuating charge assigned to the i-th atom, µi is the fluctuating dipole assigned to the i-th atom, the cii is the i-th charge self-interaction tensor, the αii is the (0) (1) (2) i-th dipole self-interaction tensor, the Tij , Tij , and Tij are the electrostatic interaction tensors, the Vext is the external potential, the Eext is the external electric field, q tot is the total charge of the NP, and N is the total number of atoms in a NP. Similarly to DIM and cd-DIM, our ex-DIM uses Gaussian electrostatics to describe the interaction of fluctuating charges and dipoles. However, in our model normalized Gaussian charge distributions are explicitly dependent on the coordination number of the atom with which it is associated (0) (1) (2) (see Eq. (4.10)), and thus the electrostatic interaction tensors, Tij , Tij , and Tij , have more complex form compared to the ones used in DIM or cd-DIM. Assuming we have two ′ , D) centred on the i-th and j-th Gaussian charge distributions, G(r; fcn , C) and G(r′ ; fcn

52

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.1 Model

41

(0)

atoms with position vectors C and D, the electrostatic interaction tensor Tij between these charges can be computed as ′ , D) erf (γrij ) G(r; fcn , C)G(r′ ; fcn (0) Tij = ∫ ∫ dr′ dr = ′ ∣r − r ∣ rij √ ′ acn acn γ= and rij = ∣C − D∣. acn + a′cn

(4.2)

(1)

(2)

Following A. Mayer [137] the higher order electrostatic interaction tensors, Tij , and Tij , (0) can be obtained by taking the derivatives of Tij with respect to i-th atom coordinates i.e. (1)

Tij

(2)

Tij

= −∇ri Tij = (0)

2γrij rij [erf (γrij ) − √ exp(−γ 2 rij2 )] , 3 rij π

= −∇ri ⊗ ∇rj Tij =

−

(0)

rij ⊗ rij − rij2 I 5 rij

4γ 3 rij ⊗ rij 2 √ 2 exp(−γ 2 rij ). πrij

2γrij [erf (γrij ) − √ exp(−γ 2 rij2 )] π

(4.3) (4.4)

Above given expressions for interaction tensors can be easily reduced to the ones used in DIM if one replaces coordination number dependent Gaussian exponents, acn and a′cn , with appropriate effective radii (see Eqs. (11) − (13) in Jensens work [135]). The fluctuating charges and dipoles are determined by minimizing the energy E[{µ, q}]. According to Jensen et al. [135] this minimization problem can be recast into a problem of solving a set of linear equations: −M 0⎞ ⎛µ⎞ ⎛ Eext ⎞ ⎛ A ⎜−MT −C 1⎟ ⎜ q ⎟ = ⎜Vext ⎟ , ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎝ 0 1 0 ⎠ ⎝ λ ⎠ ⎝ q tot ⎠

(4.5)

where the column vector µ is the collection of µi dipoles, the column vector q is the collection of qi charges, λ is a Lagrangian multiplier associated with charge equilibration condition. The matrix elements of A, C, and M matrices are defined as Aij = δij α−1 ij − (1 − δij )Tij , (2)

Cij = δij c−1 ii + (1 − δij )Tij ,

Mij =

(1) (1 − δij )Tij

53

.

(0)

(4.6)

Vadim Zakomirnyi

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

Equation (4.5) can be solved by inversion of the left-hand side matrix for small- and mediumsize nanoparticles or by the iterative approach, like the conjugate gradient method for large size nanoparticles in an external field and potential. In the calculations presented here I solve the linear equations by inversion for each frequency since I do not apply an external field. After that the fluctuating charges and dipoles are determined the polarizability of the NP can be directly obtained by computing the second derivative of E[{µ, q}] with respect to external field Eext . According to Jensen et al. [135, 138] the polarizability of a NP can be defined as αnp = ∑ N i

∂µi . ∂Eext

(4.7)

The above described scheme for determination of the polarizability of a nanoparticle is generic and has been employed in the original, coordination dependent and extended discrete interaction models [134, 139–143]. The differences between these models originate from the functional form used to describe the fluctuating charges and dipoles and from the parameterization of the self-interaction and electrostatic interaction tensors. To lay the foundation for our extended discrete interaction model I first consider the parameterization of DIM and cd-DIM. In the original DIM model the self-interaction tensors (cii and αii ) are parameterized using atomistic capacitance and polarizability derived from bulk material (0) (1) (2) properties, and the electrostatic interaction tensors (Tij , Tij , and Tij ) are computed using normalized Gaussian charges and dipoles with parametrization using TDDFT. In the cd-DIM, the fluctuating charges are excluded from the energy expression E[{µ, q}], the self-interaction tensor (αii ) between dipoles is parameterized using a coordination number dependent atomistic polarizability derived via the Clausius-Mossotti relation [144], and the (2) electrostatic interaction tensor (Tij ) is computed the same way as in DIM. In order to extend these models and achieve a description of more complex surface topologies, I spatially spread in our model the Gaussian dipoles and charges in a way that they explicitly depend on their local chemical environment. Here, I use the scheme of Grimme [145], originally proposed for the computation of dispersion corrections in DFT calculations, for evaluating i is then computed as atomic coordination numbers. The atomic coordination number fcn i = ∑ ∑ [1 + e−k1 (k2 (Ri fcn N N

cov +Rcov )/r −1) ij j

i j≠i

−1

] ,

(4.8)

where Ricov and Rjcov are the scaled covalent radius of the i-th and j-th atoms, respectively, rij is the distance between the i-th and j-th atoms, k1 and k2 are empirical parameters equal to 16.0 and 4.0/3.0, respectively [146].

54

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.1 Model

43

In the case of fluctuating charges and dipoles, the normalized Gaussian charge distribution 3/2

a G(r; C) = ( ) π

exp[−a(r − C)2 ]

(4.9)

used in the DIM and cd-DIM models can be replaced with the coordination number dependent Gaussian charge distribution G(r; fcn , C) = (

3/2

acn ) π

exp[−acn (r − C)2 ] with acn = a(1 + bfcn ) .

(4.10)

The coordination number dependent dipoles are obtained from coordination dependent Gaussian charges by taking its gradient i.e. µ(r; fcn , C) = −∇r G(r; fcn , C). Here, a is the fixed exponent of Gaussian charge distribution centred on atom with position vector C, b is the coordination number scaling factor, which defines the coordination number dependent spread of the Gaussian charge distribution.

4.1.2

Parametrization of extended discrete interaction model for silver

In paper V, I adopt a scheme based on the concept of plasmon length [147]. The parameterization of the self-interaction tensors, cii and αii , in ex-DIM is central since these tensors play the dominant role in defining the behavior of the polarizability of the NPs. Furthermore, in the case of dynamic polarizabilities, the frequency dependence is solely defined by these tensors. Similarly to DIM and cd-DIM, I use in ex-DIM a diagonal isotropic form for the self-interaction tensors, i.e. cii,kl = δkl c and αii,kl = δkl α for k, l = x, y, z. Here, I employed a different strategy based on the plasmon length [147] to parametrize the cii and αii tensors. Starting from the self-interaction tensor via the Clausius-Mossotti relationship for a spherical NP: αii,kl (ω) = δkl fα with fα =

6 3 ε(ω) − ε0 , R π i ε(ω) + 2ε0

(4.11)

where Ri is the radius of the i-th atom, ε(ω) is the frequency dependent dielectric constant of the material, and ε0 is the dielectric constant of the environment. In DIM [148] Eq. (4.11) is approximated as

55

Vadim Zakomirnyi

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles αii,kl (ω = 0) = αi,s,kl ,

αii,kl (ω > 0) = αi,s,kl (L1 (ω) + L2 (ω, N )),

(4.12) (4.13)

where αi,s,kl is the static polarizability and L1 (ω) and L2 (ω, N ) are two separately normalized frequency dependent Lorentzians. The resonance frequency ωi,2 (N ) in L2 (ω, N ) is size-dependent ωi,2 (N ) = ωi,2 (1 + A/N 1/3 ),

(4.14)

where N is the total number of atoms and ωi,2 and A two fitted parameters. In this way the size-dependent frequency is inversely proportional to the radius for a spherical NP. The problems here are the discontinuity going from the static to the dynamic case due to the separately normalized Lorentzians and that the size dependent resonance frequency in L2 (ω, N ) does not take into account the geometry of the NP. The cd-DIM [138] modifies the radius of Eq. (4.11) to a coordination number dependent radius Ri (fcn ) and dielectric constant ε(ω, fcn , r) αii,kl (ω) = δkl fα with fα =

6 3 ε(ω, fcn , r) − ε0 . R (fcn ) π i ε(ω, fcn , r) + 2ε0

(4.15)

Here ε(ω, fcn , r) is described by the sum of a the experimental dielectric constant εexp , a size-dependent Drude equation minus the Drude function for spherical NP: ε(ω, fcn , r) = εexp + εsize Drude (ω, fcn , r) − εDrude (ω),

(4.16)

where the the plasma frequency in the size dependent Drude function is modified by the coordination number. By using an effective coordination number there is a smooth transition from the inside to the outside of the coordination sphere. Both the DIM and cd-DIM should be able to describe the size dependence of spherical and spherical-like NP if properly parametrized. For shapes far from spherical symmetry, like nanorods with a large aspect ratio, the functional shape in the DIM and cd-DIM does not appear to be appropriate. This motivated me to develop a method which can take into account both the surface effects and geometry effects of nano clusters. In Ex-DIM αii,kl (ω) = (

Ri (fcn ) ) αi,s,kl L(ω, P) Ri,bulk 3

56

(4.17)

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.1 Model

45

is the static polarizabilty αi,s,kl [149] multiplied by a normalized Lorentzian L(ω, P) and the relative shift in radius from the bulk radius is determined by the coordination number. In this parameterization scheme, the chemical environment enters the definition of the αii tensor via Ri (fcn ) defined as Ri (fcn ) = r1 (1 −

fcn fcn ) + r2 , 12 12

(4.18)

which regulates the radii of the atom depending on the coordination number. For Ag I use A and r2 = Ri,bulk = 1.56 ˚ A which are the surface and bulk radii, respectively [138]. r1 = 1.65 ˚ L(ω, P) regulates the geometric dependence via the size dependent resonance frequencies of three size-dependent Lorentzian oscillators L(ω, P) = N (Lx (ω, Px ) + Ly (ω, Py ) + Lz (ω, Pz )),

(4.19)

where each Li (ω, Pi ) depends on the plasmon length Pi in the i-th direction and the frequency ω with the common normalization factor N ensures that the Lorentzian oscillators are normalized in the static limit of ω = 0. With a size-dependent Lorentzian oscillator in each direction it is possible to describe more complicated geometric structures with multiple plasmon resonances without having a new functional dependence for each distinct geometry and thereby make the ex-DIM more universal. The Lorentzian oscillator is chosen as Li =

1 , ωi2 (Pi ) − ω 2 − iγω

(4.20)

where γ describes the broadening of the spectra and ωi (Pi ) is the size-dependent resonance frequency which enables the geometric description of the plasmon excitations. With the choice of Lorentzian oscillator in Eq. (4.20) the normalization constant becomes N =(

−1

1 1 1 + + ) . ωx2 (Px ) ωy2 (Py ) ωz2 (Pz )

(4.21)

The choice of Lorentzian oscillator in Eq. (4.20) and the common normalization in Eq. (4.21) will in this way give the higher peak for the lower incident frequency which, for nanorods, corresponds to the long side. The size-dependent resonance frequency ωi (Pi ) can be written as ωi (Pi ) = ωa (1 + A ⋅ f (N, i)), 57

(4.22)

Vadim Zakomirnyi

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

where ωa and A are atom specific fitted parameters for the bulk resonance and size-dependence while f (N, i) is a function of the number of atoms and dimension along the i-th direction measured in units of atom i. f (N, i) must then in the bulk and atomic limits fulfil lim f (N, i)) = 0

lim f (N, i)) = 1,

N,i→∞

N,i→1

(4.23)

which can easily be accomplished using a single parameter namely the the plasmon length Pi f (N, i) =

1 , Pi

(4.24)

where the plasmon length Pi is defined as the maximum distance between any atoms along the i-th direction plus the radius of each of the end point atoms. This use of the plasmon length is consistent with the experimental work from Tiggesb¨aumker et al. [150]. One can notice that the SPR cannot be directly proportional to the plasmon length as defined by Ringe et al. [147] since in the bulk limit the SPR would tend to minus infinity. Performing a Taylor expansion of Eq. (4.24) the first order is linear in the plasmon length and therefore the linear dependence on the plasmon length as observed by Ringe et al. [147] is consistent with a sample of clusters of a limited size range. For spherical clusters Eq. (4.24) reduces to the usual size dependence for classical models, also seen in the DIM and cd-DIM, but for rods, discs and other shapes far from spherical there is a distinct difference where the ex-DIM can have up to three distinct plasmon resonances. The cii tensor responsible for the self interaction energy in charge transfer processes is in the DIM modelled as ci (ω) = ci,s L1 (ω)

(4.25)

using the same size-independent Lorentzian as in Eq. (4.13) for the polarizability in the DIM and a fitted parameter ci,s for the ”static atomic capacitance”. In the cd-DIM the charges and hence charge transfer and capacitance is completely removed. In the ex-DIM model I adopt a simplified two parameter scheme: cii,kl = δkl fc with fc = ci,s [1 + d

Ri (fcn ) ] L(ω, P) , Ri (12)

(4.26)

where c is the ”static atomic capacitance” parameter, similar by its physical origin to the capacitance used in DIM, d is a scaling factor for the coordination number dependence of

58

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.1 Model

47

the capacitance, set to 0.1, and L(ω, P) is the Lorentzian oscillator defined in Eq. (4.19). Here, I stress that in our parameterization of the cii tensor the frequency dependence is exactly the same as for the polarizability. To reduce the number of parameters needed to be fitted and make the method easier to extend to other elements I make use of experimental or theoretical literature values or make argumented choices for parameters which affect the peak position of the SPRs. The polarizabilities α are taken from the Schwerdtfeger and Nagle collection [149] which for silver is 55 au. The value of the capacitance parameter c, as I will show later, has very little influence on the overall polarizability and peak position as long as c is outside areas of numerical instability. For the optimization of spherical like clusters I have fixed the value at 0.0001 au since all systems appear to be numerically stable with this choice. The Lorentzian broadenings γ should be small compared to the incident frequency and, not surprisingly, they show no significant influence on the position of the SPR. During the optimization γ has been fixed at 0.016 au which gives a reasonable broadening of the peak(s) with an full width at half maximum compared to that extrapolated from the Ringe et al. data. [147] While the SPR(s) does not shift with γ, except when two close lying double peaks merge, the width and height of the SPR(s) are significantly influenced thereby making it difficult to get a good set of parameters when optimized together with α for a small set of small clusters. Despite being optimized with γ = 0.016 it is no problem to adjust this parameter later or to make γ size dependent to obtain different peak heights since the placement of the SPR(s) is not affected by small changes in γ. The only parameters that need to be fitted are therefore the size-dependent resonance frequency ωa and the the size-dependence factor A. Systematic investigations, like the one performed by Scholl et al. [151], are therefore essential for an accurate fit of ωa and A. By plotting the energy of the SPR as a function of the inverse plasmon length I can fit the a simple linear function as shown in Fig. 4.1. From the fit in Fig. 4.1 the bulk limit for the SPR for Ag will be 3.25eV in our model and show a slow variation of the SPR as a function of the inverse plasmon length. With the definition of the plasmon length in Eq. (4.24) the inverse plasmon length cannot exceed the inverse diameter of an atom and the SPR is therefore finite. By choosing a representative set of spherical clusters with a plasmon length of 1.4 − 3.8 nm an optimal resonance frequency, ωi (Pi ) in Eq. (4.22), which exactly reproduces the SPR from the fit in Fig. 4.1 for every cluster can be found. The optimal resonance frequency is here reproduced with a deviation of 10−6 − 10−5 of the SPR compared to experiment. Here I use several spherical cluster with the same plasmon length but with different surface topology to simulate slightly different surfaces. So while the radius in the N = 459, 555 and

59

Vadim Zakomirnyi

48

Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles 0.16

pruned data ft to pruned data set data excluded ft all data

0.15

Energy [au]

0.14 0.13 0.12 0.11 0.1

0

0.005

0.01 0.015 0.02 Inverse plasmon length [au-1]

0.025

0.03

Figure 4.1: Linear fit (a ⋅ x + b) of experimental data with error bars from Scholl et al. [151]. The purple points are the pruned data and green line the fit of the pruned data with coefficients a = 0.671 ± 0.059 and b = 0.119 ± 0.001. The blue points are data excluded from the pruning and the orange line a fit of all data with coefficients a = 0.822 ± 0.106 and b = 0.119 ± 0.001. Image taken from Paper V. Copyright 2019 American Chemical Society. 603 atoms clusters are the same the number of atoms and the surface topology are not. I find ωa = 0.0794 au and A = 9.41 au.

Inserting the fitted ωa - and A-values and recalculating the clusters from the fit along with a test set of larger clusters with 276 − 11849 atoms and 2 − 7 nm radius make it possible to reproduce the SPR from the fit of the experimental values as seen in Fig. 4.3.

As seen from Fig. 4.3 I can reproduce the SPR of any spherical like cluster irrespectively of size with an error limited by the experimental error. To ensure that the behaviour of the polarizabilty is correct for all frequencies I calculated the polarizability dependent frequency for 200 points in the 3.0 − 4.6 eV region.

4.2 4.2.1

Results Polarizability of spherical silver nanoparticles

Since DIM, cd-DIM ad ex-DIM models have been applied to bare spherical-like silver clusters it would be natural to compare them to the experimental data since cd-DIM has been compared to the same data before [138] and ex-DIM is parameterized from the experimental data. The extracted data from the DIM and cd-DIM models have therefore been plotted

60

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.2 Results

49

Figure 4.2: The optimum (ωi (Pi )) which reproduces the plasmon peak at the fitted experimental values from Fig. 4.1 for a given cluster. Fitting the optimum (ωi (Pi )) to Eq. (4.22) I find ωa = 0.0794 au and A = 9.41 au. The 1409 and 1433 atom clusters, also included in the fit, are located between or underneath the 1481 and 1505 atom clusters. Image taken from Paper V. Copyright 2019 American Chemical Society.

Figure 4.3: The plasmon peak as as function of the inverse plasmon length for the clusters used for the fit in Fig. 4.2 and a test set with larger clusters calculated with the fitted ωa - and A-values compared to the experimental fit and pruned data. The 1481 and 1505 atoms recalculated clusters are located between or underneath the 1401 and 1433 atoms clusters. Image taken from Paper V. Copyright 2019 American Chemical Society.

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

Figure 4.4: Comparison between the ex-DIM, DIM [148], cd-DIM [138] models and Mie theory and experiment for bare silver clusters[151]. For the DIM the TO clusters are truncated octahedrons and the Ih are icosahedral clusters. The diameter for the DIM clusters are estimated from the clusters used in the ex-DIM. The 1481 and 1505 atom recalculated clusters are located between or underneath the 1401 and 1433 atom clusters. Image taken from Paper V. Copyright 2019 American Chemical Society.

against experimental data and ex-DIM calculations as shown in Fig. 4.4. From the plotted data it is evident that for the truncated octahedrons the DIM model shows no discernible trend while for the icosahedra there is a red shift of around 0.2 eV with size but only for the range 147 − 1415 atoms (1.8 − 3.4 nm) thereafter there is no shift. The cd-DIM model does show a red shift in the plasmon length with increasing size but only by around 0.097 eV for the 2 − 10 nm clusters while the experimental data gives a red shift of 0.38 eV in that region. The limit of cd-DIM therefore deviates significantly from the experimental results and the results of ex-DIM. While Chen et al. [138] gives an arbitrary shift of 0.2 eV to the experimental data to compensate for solvent effects this does not change the fact that the shift in the SPR in the cd-DIM is only around a quarter of what it should be according to experiment [151]. The poor performance of the DIM and cd-DIM for spherical-like clusters is probably not due to methodological issues but rather due to the parameterization. This can be understood since ex-DIM and DIM in the spherical cases are very similar, except for the surface atoms, and a better fit of parameters should therefore be possible. Mie theory [152] is known to be in good agreement with experiment for medium and large NPs, but not so for small NPs. As seen in Fig. 4.4 Mie theory underestimates the size

62

4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.2 Results

51

Figure 4.5: The scheme of longitudinal and transverse plasmon length and coordination numbers for the Ag nanorod with N = 8743 atoms which is 12.09 nm x 4.20 nm. Image taken from Paper V. Copyright 2019 American Chemical Society. dependence of small silver clusters when compared to experiment even when Mie theory is size corrected based on the electron effective mean free path [66].

4.2.2

Polarizability of silver nanoparticles with complicated geometry: nanocubes and nanorods

As seen from Fig. 4.6 ex-DIM predict that the more acute size-dependence translate into a larger shape-dependence of the SPR as the Ag cubes are red shifted around 0.6 eV in comparison to the Ag spheres, in the region examined here, which is in line with the findings of Gonz´alez et al. [153]. The size-dependence of the cubes and spheres are here shown to be reasonably similar. While a small red shift in the SPR with increasing cluster size is seen for spherical-like NP very significant red shifts can be observed for nanorods depending on the aspect ratio. This very large red shift can be used to tune the SPR to a given region, thereby making nanorods versatile sensors. The SPR for nanorods is, however, split into two due to the cylindrical symmetry and excitation of collective oscillations of conduction electrons of nanorods and two peaks are seen in the UV−vis spectrum. The TLSPR is typically very slightly blue shifted in comparison to the a spherical cluster with the same plasmon length while the LLSPR can be red shifted much below what can be done by increasing the size of a spherical cluster. Furthermore, the polarizability for the red shifted peak is also greatly enhanced with increasing aspect ratio, here defined as the ratio between the plasmon length in the longitudinal and transverse directions. Since tunable nanorods are of great application interest I have examined a series of nanorods

63

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

Figure 4.6: Comparison of the SPR for sphere and cubes with different plasmon lengths. The very red and blue shift seen for the 665 and 1687 atoms cubical clusters are due to double peaks where the most red and blue shifted peak, respectively has the highest polarizability. With a larger γ-value both outliers will be shifted more in line with the rest of the cubes. The 1481 and 1505 atom recalculated clusters are located between or underneath the 1401 and 1433 atom clusters. Image taken from Paper V. Copyright 2019 American Chemical Society.

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4. Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.2 Results

53

to elucidate the interplay between the aspect ratio and diameter with respect to the SPR. I have constructed a series of nanorods where each end is a half sphere connected by a cylinder. The nanorods are designated as Ag(x, y) where x is the plasmon length of the longitudinal axis and y is the plasmon length of the transverse axis in nm as shown in Fig. 4.5. Ag(y, y) is, with this definition, a sphere with an effective diameter of y. Here I use nanorods with a diameter from 2.23 − 6.18 nm, a length up 14.06 nm, aspect ratio up to 5.4 and containing up to 16567 atoms. For all figures I calculate the polarizabilty at 400 different frequencies. In Fig. 4.5 it is clearly that only the top layer of atoms has a coordination number below 11 − 12 and, as expected, the atoms with the lowest coordination number are on the edges. This means that only the surface atoms are directly affected by the changes introduced by the coordination numbers. The red shift of the LLSPRs is clearly visible from Fig. 4.7 and furthermore the shift is directly proportional to the aspect ratio. The dependence on the diameter of the nanorod can also be seen. The increasing slope of the LLSPR with diameter is also observed for gold nanorods [154]. The slight blue shift of the TLSPR is observed as approximately linear. The experimental results et al. [155], in which the average width of the nanorods varies from 55 − 59 nm, indicate that the red shift is directly proportional to the aspect ratio and with a slight increase in slope compared to our results, which may be assigned to the refractive index in the surrounding medium. Here our results refers to nanorods on an ultra thin carbon film [151] while the experimental results were obtained in a 0.1M KNO 3 aqueous solution. The relative polarizability and peak width between the LLSPR and TLSPR in Fig. 4.7 is seen to increase significantly with increasing aspect ratio. The polarizability per atom will also increase linearly with the aspect ratio. Both the LLSPR and the absorbance can in this way be controlled by the aspect ratio and the diameter of the nanorods. The polarizability thus depends substantially on the geometry of NP.

4.2.3

Plasmon resonances of hollow nanoparticles

I have applied the derived ex-DIM model also to hollow metallic nanospheres or nanoshells. These support plasmon resonances with frequencies that are sensitive functions of both the inner and outer radius of the metallic shell. For the specific case of nanoshells, the highly geometry-dependent plasmonic response can be seen as an interaction between the essentially fixed-frequency plasmon response of a nanosphere and that of a nanocavity. The sphere and cavity plasmons are electromagnetic excitations that induce surface charges at the inner and outer interfaces of the metal shell as shown on Fig. 4.9. Because of the finite thickness of the shell layer, the sphere and cavity plasmons interact with each other. This interaction results in a splitting of the plasmon resonances into two new resonances:

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54 1.2×106

Ag(x,2.23) LLSPR Ag(x,2.23) TLSPR Ag(x,4.20) LLSPR Ag(x,4.20) TLSPR Ag(x,6.18) LLSPR Ag(x,6.18) TLSPR ft Ag(x,2.23) LLSPR ft Ag(x,4.20) LLSPR ft Ag(x,6.18) LLSPR exp

1000 900 Energy [nm]

800000 Polarizability

1100

Ag(8.15,4.20) Ag(8.94,4.20) Ag(10.12,4.20) Ag(10.91,4.20) Ag(12.09,4.20) Ag(12.88,4.20) Ag(14.06,4.20)

1×106

600000 400000

800 700 600 500

200000 0 200

400 400

600

800 Energy [nm]

1000

1200

300

1400

1

1.5

2

2.5

3 3.5 Aspect ratio

4

4.5

5

5.5

Figure 4.7: The polarizability as a function of the incident energy for Ag(x, 4.20) nanorods with different longitudinal plasmon lengths. Image taken from Paper V. Copyright 2019 American Chemical Society.

Figure 4.8: The LLSPR, TLSPR and fit of the LLSPR as a function of the aspect ratio for different nanorods. For the Ag(x, 2.23) nanorods the TLSPR becomes a double peak and here only the right TLSPR is included. These are compared to the experimental LLSPR in a 0.1 M KNO3 aqueous solution [155]. Image taken from Paper V. Copyright 2019 American Chemical Society.

Figure 4.9: Scheme of existing modes in nanobubbles: symmetrical ω− and antisymmetrical ω+ bondings. Image taken from Paper VII.

Figure 4.10: Logarithm of imaginary part of polarizability for set of nano-bubbles with fixed thickness of the shell (0.8 nm) and various total radius from 0.8 nm up to 8.6 nm. Image taken from Paper VII.

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4.2 Results

55

the lower energy symmetric or bonding plasmon and the higher energy antisymmetric or antibonding plasmon. The position of splitted plasmon modes with frequencies ω− and ω+ can be defined by following equation using the Drude model [156]:

ω2 ω±2 = B 2

 ⎡ 2l+1 ⎤ ⎥ ⎢ rcore 1  ⎥ ⎢   ⎢1 ± 1 + 4l(l + 1)( ) ⎥, ⎥ ⎢ 2l + 1 r total ⎥ ⎢ ⎦ ⎣

(4.27)

where rcore and rtotal are radius of core and total radius of particle, ωsphere = ω√B3 correspond to position of resonance for solid spherical particle with same total radius, l is is order of spherical harmonics (l = 1 in our case).

4.2.4

Polarizability of hollow nanoparticles

The studied nano-bubbles are presented as core-shell nanoparticles with an empty core inside. According to eq. (4.27) I can observe splitting of plasmon resonance into the two above mentioned resonances - the symmetrical bonding with ω− and anti-symmetrical with ω+ and can be expected to show red- and blue shifts correspondingly. The shift is dependent on both the total radius of the particle and the radius of the hole. On Fig. 4.10 I show the spectral dependence for the imaginary part of the polarizability for the nano-bubbles with fixed thickness of the shell (1.6 nm) and different total radius rtotal . Starting from solid (with rhole = 0) spherical NP with rtotal = 0.8 , I increase the total size and size of hole simultaneously but keeping the shell thickness fixed at ≈ 0.8 nm, which corresponds to 3 atomic layers. When the size of the hole in the NP becomes bigger, the SPR of the solid NP becomes split into two resonances. One resonance shows a significant red shift with increasing rtotal - from λ = 400 nm for rtotal = 2.5 nm up to λ = 900 nm for rtotal = 8.4 nm. This resonance corresponds to symmetrical ”bonding”, and its frequency is well described in eq. (4.27) as ω− . It is clear that with increasing rtotal I can see almost linear dependence for the position of the symmetric resonance vs. rtotal . At the same time, as it was predicted in eq. (4.27), I‘m able to show the second resonance at λ = 218 nm, corresponding to the anti-symmetrical mode with frequency ω+ from eq. (4.27). This second resonance starts to appear at rtotal ≈ 2.5 nm. In contrast to the symmetrically bonded resonance, the anti-symmetrical one shows a very weak blue shift with increasing rtotal .

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Chapter 4 Extended discrete interaction model for calculating optical properties of plasmonic nanoparticles

4.3

Conclusions for Chapter 4

In this chapter I reviewed the extended discrete interaction model (ex-DIM) presented in my thesis, and demonstrated some simulations of the geometric and environmental dependence of plasmon nanoparticles (NPs). The frequency dependent dielectric function in this model is obtained from the Clausius-Mossotti relation as a sum of three Lorentzian oscillators and with Gaussian charge distributions and atomic radii that vary with the coordination number.The three frequency dependent Lorentzian oscillators depend on the plasmon length in the x, y, z− directions, with the so-called plasmon length[147]. I show both theoretically and numerically that the surface plasmon resonance (SPR) is inversely proportional to the plasmon length. I also show that the model can be parameterized from experiment with numerical accuracy of the same order as the experimental accuracy. In addition, I show that some parameters, such as broadening and capacitance, do not affect the position of the SPR peak to any noticeable extent, and that acceptable values for these parameters can be selected without adjustment. Having parameterized the model for a set of spherical clusters, I used the model to predict the position of the plasmon resonance for a set of spherical clusters. To further demonstrate the capabilities of the ex-DIM model, I also performed a series of calculations on cubic and nanorod nanoparticles. For cubes, the SPRs turn up with a red shift compared to a spherical cluster with the same plasmon length. Nanorods show a significant red shift for the longitudinal resonance and a very weak blue shift for the transverse resonance with increasing aspect ratio. Having calculated several series of nanorods with different diameters and aspect radii, I could show that the red shift is directly proportional to the aspect ratio. Next, I considered core-shell nanoparticles with an empty hole inside. I could demonstrate that our model can not only predict the shift of SPRs in such kind of particles, but also make it possible to get the correct positions of resonances in spectral ranges that are beyond the region where the model has been parametrized.

68

Chapter 5 Summary The first part of the dissertation is devoted to the study of thermodynamic properties of plasmon nanostructures. The importance of taking into account the influence of temperature and the state of aggregation of a nanoparticle on its optical characteristics and vice versa was demonstrated. Subsequently, the method was used to model the optical and thermodynamic properties of chains of plasmon nanoparticles that can act as waveguides. It was shown that taking into account the thermal effects leads to a significant suppression in the waveguiding properties of such chains. One effective way to avoid the limitations associated with the deterioration of the waveguide properties of nanoparticles is to use heat-resistant materials. Titanium nitride was proposed as a refractory material. However, alternative plasmonic materials such as AZO, GZO and ITO can also be considered as promising materials for optical plasmon waveguides. Another possible way to prevent the negative influence of thermal suppression of plasmon resonances in chains of nanoparticles is to use a technological substrate that removes heat from the waveguide and save the nanoparticles from melting. These and many other technological solutions indicate that waveguides made of plasmonic nanoparticles are excellent candidates as a new generation of elements for integrated circuits. In addition to the interest in microelectronics, plasmonics has great potential in many other fields. So, two-dimensional structures in the form of arrays of resonant nanoparticles are successfully used as filters and sensors with high sensitivity in biology and chemistry. The next part of the dissertation was devoted to the study of the optical properties of such two-dimensional structures from silicon nanoparticles. As being one of the most accessible and well-studied materials we have available, silicon opens up prospects for the production of high-tech devices with good reproducibility. However, defects in such devices can significantly reduce their versatility and increase the manufacturing costs. In this work, I simulated defects arising in two-dimensional structures from resonant silicon nanoparticles.

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Chapter 5 Summary

From the obtained results I was able to make the conclusion that two-dimensional structures might have high resistance to defects. In addition, such structures can be used as variable ultra-narrow-band filters. Theoretical research in this area is currently focused on using evolutionary mechanisms to create structures with predicted properties. Defects in two-dimensional structures can be used, if controlled, as a flexible way to create devices with completely new properties, such as focusing light. As highlighted in the previous chapter, new materials are one of the potentially interesting directions in the development of this area. I consider all above reviewed applications to have a great technological value. However, research in these areas is limited to the methods used sometimes. In addition to the requirements for computing resources, most methods have a great fundamental limitation the limited size and/or geometry of the structure. Here we are talking about the traditional difficulties of accurate modeling of properties of nanoparticles with sizes less than 10 nm and/or nanoparticles with an exotic (non-spherical) shape. In addition, most models rely on experimental data of the optical properties of nanoparticles and cannot predict their optical properties outside of existing experimental data. In this thesis, the model of discrete atomic interaction was expanded to remedy this situation. This model takes into account the contribution to the optical properties of nanoparticles from each atom. The obtained results for spherical nanoparticles, nanocubes and nanorods are in good general agreement with some theoretical predictions on larger particles made by other methods, for example, from the Mie theory. However, for the precise predictions of small nanoparticles can only be made by atomic discrete interaction models, of which our extended model constitutes the present state-of-art. Calculations were also performed for hollow nanoparticles. Since the model takes into account the contribution of each atom, it might be useful in studies of atomically thin nanostructures. Universal miniaturization invariably leads to the growing interest in modeling of nanoparticles with a certain number of atomic layers. The classical electrodynamics models do not allow to describe the optical properties of such structures. Also, the development of a discrete interaction model should consider new materials. Thus I believe the future is wide open both for applications of the current model and for its further development, with wide ramifications for applications already reviewed in this thesis, and for much more.

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85

Paper I

Thermal effects in systems of colloidal plasmonic nanoparticles in high-intensity pulsed laser fields V. S. Gerasimov, A. E. Ershov, S. V. Karpov, A. P. Gavrilyuk, V. I. Zakomirnyi, L. Rasskazov, H. Agren, S. P. Polyutov Optical Materials Express, 7(2), pp. 555-568 (2017)

Paper I Vol. 7, No. 2 | 1 Feb 2017 | OPTICAL MATERIALS EXPRESS 555

Thermal effects in systems of colloidal plasmonic nanoparticles in high-intensity pulsed laser fields [Invited]

Vol. 7, No. 2 | 1 Feb 2017 | OPTICAL MATERIALS EXPRESS 555

Thermal effects in systems of colloidal

V. S. G ERASIMOV, 1 A. E. E RSHOV, 1,2,3 S. V. K ARPOV, 1,3,4,* plasmonic nanoparticles in high-intensity A. P. G AVRILYUK , 1,2 V. I. Z AKOMIRNYI , 1,5 I. L. R ASSKAZOV, 1,6 laser fields [Invited] H. Åpulsed GREN , 5 AND S. P. P OLYUTOV 1 1 A. E. E RSHOV 1,2,3 S. 1 Siberian Federal University,,Krasnoyarsk 660041,, Russia V. S. G ERASIMOV

V. K ARPOV, 1,3,4,* 1,5 I. KSC Research ,Center RAS, Krasnoyarsk A. of P.Computational G AVRILYUKModeling, , 1,2 V. I.Federal Z AKOMIRNYI L. RSB ASSKAZOV , 1,6 660036, RussiaH. ÅGREN , 5 AND S. P. P OLYUTOV 1 3 2 Institute

Siberian State Aerospace University, Krasnoyarsk 660014, Russia 1 SB RAS, Krasnoyarsk 660036, Russia 2

Siberian Federal University, Krasnoyarsk 660041, Russia 4 Kirensky Institute of Physics, Federal Research Center KSC

of Computational Modeling, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, 5 Royal Institute Institute of Technology, Stockholm SE-100 44, Sweden Russia 6 The Beckman 3

for Advanced Science and Technology, University of Illinois at Siberian Institute State Aerospace University, Krasnoyarsk 660014, Russia 4 Kirensky Institute Urbana-Champaign, Urbana, IL 61801, USA of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, Russia

* [email protected] 5

Royal Institute of Technology, Stockholm SE-100 44, Sweden Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana,light IL 61801, USA processes in nanocolloids and composite Abstract: We have studied induced 6 The

materials * [email protected] containing ordered and disordered aggregates of plasmonic nanoparticles accompanied by their strongAbstract: heating . A universal comprehensive physical model that combines mechanical, electroWe have studied light induced processes in nanocolloids and composite materials dynamical, and ordered thermaland interactions nanoscale has been developed a tool for investigations . containing disordered at aggregates of plasmonic nanoparticlesasaccompanied by their This model was used to gain deep insight onphysical phenomena placemechanical, in nanoparticle strong heating . A universal comprehensive model that that take combines electro-aggregates dynamical, under high-intensity pulsed laser radiationhas resulting in the assuppression of nanoparticle and thermal interactions at nanoscale been developed a tool for investigations . Thisproperties. model was used to gain deep insight on phenomena place in nanoparticle aggreresonant Verification of the model was carriedthat outtake with single colloidal Au and Ag gates under pulsed laser radiation resulting in the suppression of nanoparticle nanoparticles andhigh-intensity their aggregates . resonant properties. Verification of the model was carried out with single colloidal Au and Ag c 2017  Optical Society of America nanoparticles and their aggregates .

c 2017 OCIS codes: (160 .4236) Nanomaterials; (250 .5403) Plasmonics; (240 .6680) Surface plasmons .  Optical Society of America OCIS codes: (160 .4236) Nanomaterials; (250 .5403) Plasmonics; (240 .6680) Surface plasmons .

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#279883 Journal © 2017

http://dx.doi.org/10.1364/OME.7.000555 Received 1 Nov 2016; revised 12 Dec 2016; accepted 15 Dec 2016; published 25 Jan 2017

http://dx.doi.org/10.1364/OME.7.000555 89 Received 1 Nov 2016; revised 12 Dec 2016; accepted 15 Dec 2016; published 25 Jan 2017

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12 . L . Hirsch, R . Stafford, J . Bankson, S . Sershen, B . Rivera, R . Price, J . Hazle, N . J . Halas, and J . West, “Nanoshellmediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proceedings of the National Academy of Sciences 100, 13549–13554 (2003) . 13 . S . Hashimoto, D . Werner, and T . Uwada, “Studies on the interaction of pulsed lasers with plasmonic gold nanoparticles toward light manipulation, heat management, and nanofabrication,” Journal of Photochemistry and Photobiology C: Photochemistry Reviews 13, 28 – 54 (2012) . 14 . E . Y . Lukianova-Hleb, X . Ren, R . R . Sawant, X . Wu, V . P . Torchilin, and D . O . Lapotko, “On-demand intracellular amplification of chemoradiation with cancer-specific plasmonic nanobubbles,” Nature medicine 20, 778–784 (2014) . 15 . A . Lenert, D . M . Bierman, Y . Nam, W . R . Chan, I . Celanovi´c, M . Soljaˇci´c, and E . N . 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Richardson, “Generating heat with metal nanoparticles,” Nano Today 2, 30–38 (2007) . 22 . A . P . Gavrilyuk and S . V . Karpov, “Processes in resonant domains of metal nanoparticle aggregates and optical nonlinearity of aggregates in pulsed laser fields,” Appl. Phys. B 97, 163–173 (2009) . 23 . B . S . Luk’yanchuk, A . E . Miroshnichenko, M . I . Tribelsky, Y . S . Kivshar, and A . R . Khokhlov, “Paradoxes in laser heating of plasmonic nanoparticles,” New Journal of Physics 14, 093022 (2012) . 24 . A . E . Ershov, A . P . Gavrilyuk, S . V . Karpov, and P . N . Semina, “Optodynamic phenomena in aggregates of polydisperse plasmonic nanoparticles,” Appl . Phys . B 115, 547–560 (2014) . 25 . V . S . Gerasimov, A . E . Ershov, A . P . Gavrilyuk, S . V . Karpov, H . Ågren, and S . P . Polyutov, “Suppression of surface plasmon resonance in au nanoparticles upon transition to the liquid state,” Opt . Express 24, 26851–26856 (2016) . 26 . A . Alabastri, S . Tuccio, A . Giugni, A . Toma, C . Liberale, G . Das, F . Angelis, E . Fabrizio, and R . Zaccaria, “Molding of Plasmonic Resonances in Metallic Nanostructures: Dependence of the Non-Linear Electric Permittivity on System Size and Temperature,” Materials 6, 4879–4910 (2013) . 27 . H . Reddy, U . Guler, A . V . Kildishev, A . Boltasseva, and V . M . Shalaev, “Temperature-dependent optical properties of gold thin films,” Optical Materials Express 6, 2776 (2016) . 28 . V . P . Drachev, S . V . Perminov, S . G . Rautian, V . P . Safonov, and E . N . Khaliullin, “Polarization effects in silver nanoaggregates caused by local and nonlocal nonlinear-optical responses,” J . Exp . Theor . Phys . 95, 901–915 (2002) . 29 . A . V . Butenko, Y . E . Danilova, P . A . Chubakov, S . V . Karpov, A . K . Popov, S . G . Rautian, V . P . Safonov, V . V . Slabko, V . M . Shalaev, and M . I . Stockman, “Nonlinear optics of metal fractal clusters,” Z . Phys . 17, 283–289 (1990) . 30. A. E. Ershov, A. P. Gavrilyuk, and S. V. Karpov, “Plasmonic nanoparticle aggregates in high-intensity laser fields: Effect of pulse duration,” Plasmonics 11, 403–410 (2015) . 31 . A . E . Ershov, A . P . Gavrilyuk, S . V . Karpov, and P . N . Semina, “Effect of local environment in resonant domains of polydisperse plasmonic nanoparticle aggregates on optodynamic processes in pulsed laser fields,” Chin. Phys. B 24, 47804 (2015) . 32 . A . E . Ershov, A . P . Gavrilyuk, S . V . Karpov, and S . P . Polyutov, “Restructuring of plasmonic nanoparticle aggregates with arbitrary particle size distribution in pulsed laser fields,” Chin. Phys. B 25, 117806 (2016) . 33 . Y . E . Danilova, N . N . Lepeshkin, S . G . Rautian, and V . P . Safonov, “Excitation localization and nonlinear optical processes in colloidal silver aggregates,” Physica A 241, 231–235 (1997) . 34 . R . A . Ganeev, A . I . Ryasnyansky, S . R . Kamalov, M . K . Kodirov, and T . J . Usmanov, “Nonlinear susceptibilities, absorption coefficients and refractive indices of colloidal metals,” Physica D 34, 1602 (2001) . 35 . S . V . Karpov, M . K . Kodirov, A . I . Ryasiyanskiy, and V . V . Slabko, “Nonlinear refraction of silver hydrosoles during their aggregation,” Quantum Electronics 31, 904–908 (2001) . 36 . N . N . Lepeshkin, W . Kim, V . P . Safonov, J . G . Zhu, R . L . Armstrong, C . W . White, R . A . Zhur, and V . M . Shalaev, “Optical nonlinearities of metal-dielectric composites,” J . Nonlinear Opt . Phys . & Materials 8, 191 (1999) . 37 . F . A . Zhuravlev, N . A . Orlova, V . V . Shelkovnikov, A . I . Plekhanov, S . G . Rautian, and V . P . Safonov, “Giant nonlinear susceptibility of thin films with (molecular j-aggregate)-(metal cluster) complexes,” JETP Lett. 56, 264– 267 (1992) . 38 . S . V . Karpov, A . K . Popov, S . G . Rautian, V . P . Safonov, V . V . Slabko, V . M . Shalaev, and M . I . Shtokman, “Observation of a wavelength- and polarization-selective photomodification of silver clusters,” JETP Lett. 48, 571–573 (1988) . 39 . V . P . Safonov, V . M . Shalaev, V . A . Markel, Y . E . Danilova, N . N . Lepeshkin, W . Kim, S . G . Rautian, and R . L . Armstrong, “Spectral dependence of selective photomodification in fractal aggregates of colloidal particles,” Phys. Rev . Lett . 80, 1102–1105 (1998) .

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40 . S . V . Karpov, A . K . Popov, and V . V . Slabko, “Photochromic reactions in silver nanocomposites with a fractal structure and their comparative characteristics,” Technical Physics 48, 749–756 (2003) . 41 . Y . S . Barash, Van der Waals Forces (Nauka, Moscow, 1988) . 42 . J . N . Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1992) . 43 . L . D . Landau and E . Lifshitz, Theory of Elasticity (Butterworth-Heinemann, Oxford England Burlington, MA, 1986) . 44 . S . V . Karpov, I . L . Isaev, A . P . Gavrilyuk, V . S . Gerasimov, and A . S . Grachev, “General principles of the crystallization of nanostructured disperse systems,” Colloid J . 71, 313–328 (2009) . 45. F. Claro and R. Rojas, “Novel laser induced interaction profiles in clusters of mesoscopic particles,” Appl. Phys. Lett . 65, 2743–2745 (1994) . 46 . V . A . Markel, L . S . Muratov, M . I . Stockman, and T . F . George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys . Rev . B 43, 8183–8195 (1991) . 47 . G . Y . Panasyuk, J . C . Schotland, and V . A . Markel, “Short-distance expansion for the electromagnetic half-space green’s tensor: general results and an application to radiative lifetime computations,” J . Phys . A 42, 275203 (2009) . 48 . W . C . Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990) . 49 . B . T . Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys . J . 333, 848–872 (1988) . 50 . V . A . Markel, L . S . Muratov, M . I . Stockman, and T . F . George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys . Rev . B 43, 8183–8195 (1991) . 51 . V . A . Markel, V . M . Shalaev, E . B . Stechel, W . Kim, and R . L . Armstrong, “Small-particle composites . i . linear optical properties,” Phys . Rev . B 53, 2425–2436 (1996) . 52 . C . F . Bohren and D . R . Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998) . 53 . V . A . Markel, V . M . Shalaev, E . B . Stechel, W . Kim, and R . L . Armstrong, “Small-particle composites . i . linear optical properties,” Phys . Rev . B 53, 2425–2436 (1996) . 54 . P . B . Johnson and R . W . Christy, “Optical constants of the noble metals,” Phys . Rev . B 6, 4370–4379 (1972) . 55 . J . C . Miller, “Optical properties of liquid metals at high temperatures,” Philosophical Magazine 20, 1115–1132 (1969) . 56 . C . Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc ., New York, 1986), 6th ed . 57 . M . Otter, “Temperaturabhängigkeit der optischen konstanten massiver metalle,” Z . Phys . 161, 539–549 (1961) . 58 . T . Castro, R . Reifenberger, E . Choi, and R . P . Andres, “Size-dependent melting temperature of individual nanometersized metallic clusters,” Phys . Rev . B 42, 8548–8556 (1990) . 59 . T . Castro, R . Reifenberger, E . Choi, and R . P . Andres, “Size-dependent melting temperature of individual nanometersized metallic clusters,” Phys . Rev . B 42, 8548–8556 (1990) . 60 . Y . A . Frenkel, The Kinetic Theory of Liquids (Nauka, Moscow, 1975) . 61 . O . Yeshchenko, I . Bondarchuk, V . Gurin, I . Dmitruk, and A . Kotko, “Temperature dependence of the surface plasmon resonance in gold nanoparticles,” Surface Science 608, 275 – 281 (2013) . 62 . D . Dalacu and L . Martinu, “Temperature dependence of the surface plasmon resonance of au/sio2 nanocomposite films,” Appl. Phys. Lett. 77, 4283–4285 (2000) . 63 . J . C . Miller, “Optical properties of liquid metals at high temperatures,” Philosophical Magazine 20, 1115–1132 (1969) . 64 . P . B . Johnson and R . W . Christy, “Optical constants of the noble metals,” Phys . Rev . B 6, 4370–4379 (1972) . 65 . V . A . Markel and A . K . Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys . Rev . B 75, 085426 (2007) . 66 . I . L . Rasskazov, S . V . Karpov, and V . A . Markel, “Surface plasmon polaritons in curved chains of metal nanoparticles,” Phys . Rev . B 90, 075405 (2014) . 67 . I . L . Rasskazov, S . V . Karpov, G . Panasyuk, and V . A . Markel, “Overcoming the adverse effects of substrate on the waveguiding properties of plasmonic nanoparticle chains,” J . Appl . Phys . 119, 043101 (2016) . 68 . I . L . Rasskazov, S . V . Karpov, and V . A . Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt . Lett . 38, 4743–4746 (2013) .

1.

Introduction

Electromagnetic interactions in metal nanoparticles (NPs) are among the most intriguing phenomena in plasmonics [1] which are used in numerous breakthrough applications: optoelectronics [2–5], sensing [6–9], biomedicine [10] . Despite the fact that fundamental processes in plasmonics are well understood nowadays, there are some specific interactions where this understanding has not been reached yet and which give rise to emerging applications [11] . Interaction of NPs with high-intensity pulsed laser fields [12–14], plasmonic heating [15–17] and near-field heat transfer [18, 19] are among them . In the paper [20], authors investigate theoretically, numerically and experimentally the tem-

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perature distribution in metal NPs excited by a modulated incoming light, the response in amplitude and phase of the temperature variations . Photothermal effects in colloidal NPs, including energy absorption by NPs, heat exchange with the environment and phase transitions of the particles and the environment are discussed in Ref . [21] . All of these processes are related to strong heating and even melting of NPs [22–24] . As a consequence, Q-factor of the NP surface plasmon resonance (SPR) considerably drops [25], which may dramatically change the physical pattern of interaction between NPs (single or aggregated) and electromagnetic radiation [26, 27] . Moreover, high-intensity light may cause mechanical effects due to interparticle forces . Therefore, the development of physical model which enables to describe interrelation between electromagnetic, thermodynamic and mechanical interactions in systems of NPs is of topical interest . Note that in alternative models the factor of suppression of SPR in heated NPs was not taken into account (see in particular [28]) . However to get deep insight on the nature of processes in plasmonic colloidal aggregates experimentally studied earlier in [29] and subsequent publications (see below) we employed the optodynamical model proposed in [24] (its earlier version was published in [22]) . We use the optodynamical model to study the interaction of pulsed laser radiation of different duration with mono- and polydisperse dimers and trimers composed of plasmonic nanoparticles as resonant domains of colloid Ag multiparticle aggregates [30] . In paper [31] we have analyzed how the local environment and the associated local field enhancement produced by surrounding particles affect the optodynamic processes in resonant domains of NP aggregates, including their optical properties . In [32] we have studied processes of interaction of pulsed laser radiation with resonant domains in Ag colloidal aggregates with different interparticle gaps and particle size distributions . Nonlinear optical effects observed in nanocolloids with silver and gold NP aggregates included four-wave mixing of pulsed laser radiation [29], nonlinear optical activity (nonlinear gyrotropy), spontaneous rotation of the polarization ellipse [28], non-linear absorption and refraction [33–35], the reverse Faraday effect [28], the optical Kerr effect [36], harmonics generation [34], enhancement of nonlinear responses of organic molecules adsorbed on nanoparticle surfaces [37] and optical memory [38–40] . In the paper [39] authors report on a record in composite medium containing large colloidal Ag aggregates of pulsed laser radiation with five different wavelengths of visible and near IR range in one irradiated spot . In this paper, we summarize our previous and new results regarding thermal effects in single and aggregated NPs in high-intensity pulsed laser fields. We show how thermal effect may lead to new phenomena changing the usual pattern of light interaction with systems of bound plasmonic nanoparticles . 2.

Model

We have developed combined physical model which thoroughly describes interaction between pulsed laser radiation and metal NP aggregates . Basic interactions in such systems can be divided into three major parts: mechanical, electrodynamical and thermodynamical . It is obvious that each of these parts is well known and developed, however implementation of above mentioned interactions into one combined model opens the opportunity to describe vast majority of various phenomena in aggregates of NPs . 2.1.

Mechanical interactions

Let us start with simple mechanical part which is based on the motion equations . Consider the aggregate of N spherical NPs. In general case, each NP in aggregate moves under the influence of van der Waals Fvdw , viscous Fv and light-induced optical forces Fopt . Moreover, we consider interparticle friction Ff and elastic Fel forces to take into account interaction between touching NPs . The motion of i-th NP (where i = 1, 2, ..., N) with mass mi can be characterized by the

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following expression: � � mi r¨ i = (Fvdw ) i + (Fel ) i + Fopt + (Fv ) i + (Ff ) i , i

(1)

where ri denotes the position of i-th NP center of mass and double dot over the vector denotes the time derivative . In this paper, we provide general expressions and brief explanation of these interactions . More comprehensive study can be found in Ref . [24] . Now, let us describe each force from eq. (1) more specifically. The pair potential energy of the van der Waals interaction between i-th and j-th NPs is described by well-known expression [41]: ⎛ 2Ri R j 2Ri R j AH ⎜⎜⎜⎜ + + (Uvdw ) i j = − ⎜ 6 ⎝ h2i j + 2Ri hi j + 2R j hi j h2i j + 2Ri hi j + 2R j hi j + 4Ri R j ⎞ (2) h2i j + 2Ri hi j + 2R j hi j ⎟⎟⎟⎟ + ln 2 ⎟⎠ , hi j + 2Ri hi j + 2R j hi j + 4Ri R j � � where R is the radius of spherical NP; hi j = ��ri j �� − (Ri + R j ) is the interparticle gap, ri j = ri −r j ; AH is the Hamaker constant [42] . Therefore, one should take into account the interaction of i-th � with the rest of aggregate to get (Fvdw )i = − N j=1, ji ∂(Uvdw )i j /∂ri . The solution of Hertz problem for deformation of two touching spheres [43] can be utilized to determine potential energy of elastic interaction between two touching NPs with adsorption layers (neglecting the Poisson’s ratio) [43, 44]: (Uel )i j =

� 5/2 � (Ri + hi )(R j + h j ) � 1/2 (Eel ) i · (Eel ) j 8 � H (hi + h j − hi j ). (3) hi + h j − hi j 15 Ri + hi + R j + h j (Eel ) i + (Eel ) j

Here h is the thicknesses of unstrained NP’s adsorption layer; Eel is the elasticity modulus of NP adlayer; H (x) is the Heaviside function . Thus, taking into account interaction of i-th NP with other NPs contacting therewith, one can obtain total elastic force (Fel )i = � − N j=1, ji ∂(Uel )i j /∂ri . Next, assume that laser pulse with duration τ irradiates NP aggregate . Then, the potential energy of such optical interaction between NP aggregate and incident excitation can be found as follows [45]: � � �∗ N � di 1 � 1 di · E∗ (ri ) + di · − E(ri ) − ε 0 α i |E0 | 2 · H (τ − t) . Uopt = − Re 4 2 ε0 α i i=1

(4)

Here di is the dipole moment which is induced on i-th NP; E(ri ) is the strength of external electromagnetic field at ri point, E0 is the amplitude of the field; α i is the dipole polarizability of the i-th particle and symbol (∗) denotes complex conjugate . Optical force (Fopt )i which acts on i-th NP can be found by simple derivation of eq . (4): (Fopt )i = −∂Uopt /∂ri . Viscous friction forces acting on i-th NP moving in a viscous medium with the velocity vi obeys the Stokes law: (5) (Fv )i = −6πη(Ri + hi )vi ,

where η is the dynamical viscosity of the interparticle medium . Finally, we consider tangential friction force which occurs when NPs in aggregate move transversally relative to each other . Such interaction can be described by analogy with the forces of dry friction (although they are not the ones) . The coefficient of friction is an effective parameter that characterizes the degree of interaction between the touching NP’s adlayers . The effective friction coefficient can significantly exceed unity due to strong intermolecular interaction of NP’s adlayers and their non-uniform deformation . The direction of tangential friction force

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is opposite to the projection of the relative velocity of particles (vi − v j ) on the plane of adlayers contact . Thus, the tangential friction force exerted on the i-th NP by the j-th NP is determined by the following expression: (Ff )i = −μ

N � � �� (F ) ��� q . el i j i j

(6)

j =1 j i

Here μ is the effective friction coefficient; qi j is the normalized vector of the projection of particles relative velocity on the plane of contact of particle adlayers: � � (v j − vi ) − ni j (v j − vi ) · ni j qi j = �� � �� , �� (v j − vi ) − ni j (v j − vi ) · ni j ���

where ni j = ri j /|ri j | . 2.2.

Electrodynamical interactions

Electrodynamical part is based on the coupled dipole approximation [46], which allows one to calculate the electromagnetic interactions between NPs, incident radiation and a substrate [47] . Let us consider the aggregate of NPs in host medium with permittivity ε√M which is illuminated by electromagnetic plane wave E (r) = E0 exp (ik · r) . Here |k| = 2π ε M /λ is the wave vector, E0 is the amplitude of the electric component of the electromagnetic field. In general case, incident plane wave does not excite all NPs in the aggregate . We introduce coefficient κ i to take this fact into account, where κ i = 1 for i-th nanoparticle which is excited by incident field and κ i = 0 otherwise . In this case, the dipole moment di induced on the i-th NP can be described by following equation [46]: ⎡ ⎤ N ⎢⎢⎢ ⎥⎥ � ⎢ ˆ i , r j )d j ⎥⎥⎥⎥ , (7) G(r di = ε 0 α i ⎢⎢⎣E (ri ) κ i + ⎦ ji

ˆ i , r j ) is the 3 × 3 Green’s interaction tenwhere α i is the dipole polarizability of the i-th NP, G(r sor that describes the electric field produced at ri by electric dipole d j located at r j . Expressions ˆ i , r j ) can be found elsewhere [48] . for G(r Finally, the dipole polarizability of i-th NP which takes into account the radiation reaction correction [49], has the form [50]: α i−1 = [α i(0) ] −1 −

i |k| 3 , 6π

(8)

where α i(0) is so-called bare or Lorenz-Lorentz quasistatic polarizability of the particle . For the extinction efficiency of a particle ensemble surrounded by other particles we have the formula [51] (for a single particle, averaging over ensemble is omitted): Qe =

σe . N � πRi2

(9)

i=1

Here the extinction cross section is given by the equation σe = 4π |k| Im

N � (di · E∗ (ri )) i=1

94

|E0 | 2

.

(10)

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2.3.

Thermal effects

The key effect which should be taken into account to describe the impact of thermal effects on optical response of NP is variation of NPs permittivity under conditions of their strong heating, e.g. in high-intensity pulsed laser field. In our work, we take into consideration the fact that a particle can be in the melting process . Thus, the heated NP is considered as a coreshell nanosphere with a solid core and a liquid shell . Therefore, the α i(0) can be described as follows [52]:       ε Li − ε M ε Si + 2ε Li + f i ε Si − ε Li ε M + 2ε Li (0) 3     . (11) α i = 4πRi ×  ε Li + 2ε M ε Si + 2ε Li + 2 f i ε Si − ε Li ε Li − ε M Here ε Si and ε Li are the temperature and size dependent permittivities of the particle material in the solid and liquid state, respectively, f i is the mass fraction of solid material in the particle . The expression (11) takes into account the extreme cases of completely solid ( f i = 0) and fully liquid particles ( f i = 1) . The full description of f i will be given below . It should be noticed that metal NP keeps its spherical shape even in the fully liquid state due to extremely high values of surface tension . Therefore, the expression (11) is valid for any values of f i . Permittivities ε Si and ε Li take into account finite size effects (FSE) [53] and temperature dependence of optical constants: = ε S,L ε S,L i tab +

2 ωpl

ω (ω + iΓ0 )

−

2 ωpl

ω (ω + iΓi )

,

(12)

εS,L i

is the permittivity of the solid or liquid where ω is the electromagnetic field frequency, material of the particles, ε S,L is the corresponding tabulated experimental values for bulk at the tab temperature of 300 K for solid material [54] and at the melting point for liquid one [55], Γ0 is the time dependent bulk electron relaxation constant for corresponding temperature, ωpl is the plasma frequency of the particle material, Γi is the temperature and size dependent electron relaxation constant of a particle:   vF (13) Γi = Γ0 Tiion + A . Ri

Here vF is the Fermi velocity and Tiion is the temperature of ion component . The value of A is taken to be unit in most of cases [52] . However, the relaxation processes depend on the state of particle surface and on other factors . The dependence of the relaxation constant for solid bulk material Γ0 (T ) on the temperature can be approximated by the following expression [56]: Γ0 (T ) = bT + c,

(14)

where b and c are the linear dependence coefficients obtained by linear approximation of the experimental data [57] (detailed discussion will be published elsewhere) . Description of thermodynamical processes is based on a two-component model, that considers the temperature variations of the conduction electron and ion (crystal lattice) components of each NP . The model takes into account heating of the electron component by incident optical radiation, heat transfer between the electron and ion components, heat exchange between the ion component and the environment [22, 24], as well as size dependence of NP’s melting point [58] . The temperature of the electron subsystem Tie which depends on the absorption of radiation energy by nanoparticle and on heat exchange with the crystal lattice (ion subsystem) is described by the following expression [22]: Cie

  Wi dTie = −g Tie − Tiion + , dt Vi

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(15)

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where Cie = 68Tie J · m −3 · K −1 is the volumetric heat capacity, Vi is the volume of a spherical particle, g = 4 × 1016 J · m −3 · K −1 is the energy exchange rate between electron and ion subsystems, Wi is the radiation power absorbed by the particle . In dipole approximation, Wi can be found using the equation [24]: � � ω |di | 2 1 (16) Wi = Im ∗ , 2ε 0 αi

where asterisk denotes the complex conjugate . Developed model takes into account the phase transition at the melting point of the particle . Therefore, the heat exchange between ion component of the particle with its electronic component and the environment is described in terms of the amount of heat Qion i which is absorbed by a particle [22]:

� � dQion i = gVi Tie − Tiion + υi , (17) dt where υi is the rate of heat transfer between particle and its environment . The temperature of the ion subsystem of a particle during the melting process is described as follows [24]:

Tiion

⎧ Qion ⎪ ⎪ i ⎪ ⎪ , ⎪ ion V ⎪ ⎪ C i ⎪ ⎪ ⎨ Li =⎪ Ti , ⎪ ⎪ ⎪ ⎪ Qion ⎪ i − LVi ⎪ ⎪ , ⎪ ⎩ Ciion Vi

(1) when Qion i < Qi (2) when Q i(1) ≤ Qion i ≤ Qn

(18)

(2) when Qion i > Qi .

Here L is the volumetric heat of fusion, Ciion is volumetric heat capacity of the ion subsystem of the i-th particle material, TiL = T L (Ri ) is the size dependent melting temperature [59], Q i(1) is the heat that corresponds to the onset of the particle melting: Q i(1) = Ciion Vi TiL , and

Q i(2)

(19)

is the heat that corresponds to the end of particle’s melting process: Q i(2) = Q i(1) + LVi .

(20)

Thus, we can determine the mass fraction f i of the molten material in equation (11) as follows: ⎧ (1) ⎪ 0, when Qion ⎪ ⎪ n < Qi ⎪ ⎪ (1) ion ⎪ ⎪ ⎨ Qi − Qi (2) fi = ⎪ . (21) , when Q i(1) ≤ Qion ⎪ i ≤ Qi ⎪ Ciion Vi ⎪ ⎪ ⎪ ⎪ ⎩ 1, when Qion > Q (2) i

i

The rate of heat transfer from a particle to the environment can be found using the following expression [43]: � υi = −κ ∇T (r, t) · ndS. (22) Si

Here κ and T (r, t) are coefficients of thermal conductivity and the temperature of host medium, correspondingly, and n is the normal vector to the particle surface . The integration is performed

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Qe

Qe

20 ◦ C h, nm 0 4 12 20

20

10

(a)

0 360

380

400

1 .0

0 .5

λ, nm 420

0 .0

(b) 500

550

600

λ, nm

Fig . 1 . Extinction spectra of Ag — (a) and Au — (b) NPs with 20 nm radius at room (20 ◦ C, solid line) and melting (≈ 1064 ◦ C) temperatures for different values of liquid shell thicknesses h . Dielectric constant of the environment is ε h = 1.78 . The spectral range on graphs is limited by experimental data for optical constants of liquid metals [57] .

over the surface Si of the i-th particle . The rate of heat exchange due to the radiation is much less than due to the thermal conductivity so that the first one is omitted in this model. The heat equation for the environment surrounding a particle is solved to determine the values of T (r, t): ∂T (r, t) = aΔT (r, t), (23) ∂t where a is the thermal diffusivity of the environment material . The particle and the interparticle medium are heated when laser radiation is absorbed that results in a change of the adlayer elasticity modulus due to destruction of molecular bonds in the polymer grid of the adlayer. This occurs over finite relaxation time τr and is described by the expression [6, 44]:   Uf . (24) τr (T ) = τ0 exp k BT The value of τ0 is assumed to be 10−12 s which is characteristic vibration period of atoms in molecules [22,60], Uf is the energy of chemical bonds in the polymer adlayer (taken to be about 1 eV in our calculations, which is typical for the chemical bond energy of polymers) . Taking into account the finite relaxation time, the temperature dependence of the elasticity modulus is described by the following equation [22]: (Eel ) i d(Eel ) i =− , dt τr ((Tm ) i )

(25)

where (Tm )i is the average temperature of the heated area near the i-th particle . 3. 3.1.

Results Thermal effects in single spherical nanoparticle

Let us start with the temperature dependence of SPR in isolated metal NP . As it was shown previuosly, the heating process in Au and Ag NPs [61] or thin films [27, 62] can considerably affect both the peak magnitude of SPR and position of its maximum . Here we present an extensive consideration of the melting process in single NP . Figure 1 shows the extinction spectra Q e (λ) of liquid-shell/solid-core Ag and Au NPs with different core-to-shell ratio . Values of

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liquid metal permittivity at the melting temperature were taken for gold in Ref . [57] and for silver — from Ref . [63]; at room temperature for gold — from [57] and [64], and for silver — from [64] . It is clearly seen that SPR of NPs is strongly suppressed during the melting process . Strong heating of NPs is accompanied by increasing scattering of electrons at the crystal lattice defects (vacancies, dislocations, grain boundaries). This explains the significant increase of the relaxation constant for conduction electrons at the melting point . Our experiment on manifestation of heating and melting of Au NPs is described in detail in [25] . Compared to silver, gold NPs are resistant to oxidation even in the temperature range slightly above the melting point . Therefore, in our work [25] we studied annealed Au NPs deposited on quartz substrate . These NPs do not undergo further changes during several heating cycles up to the melting point . In Ref . [61], Au NPs were heated to temperatures below the melting point which leads to a broadening of SPR and a decrease of its amplitude . In [62], the same effect was observed at the melting point of Au NPs with 2 nm, 8 nm and 15 nm radii . However considerable contribution of FSE can overcome the suppression of SPR due to heating in these cases . To get rid of undesirable manifestation of FSE, we experimentally study NPs with much larger size, namely ≈ 56 nm . In this case FSE is nearly negligible . Experimentally obtained data in [25] shows evolution of the extinction spectrum of Au NPs during temperature growth beyond the melting point up to 1120 ◦ C . It was shown that the NP temperature growth results in the gradual decrease of a SPR amplitude . The complete suppression of SPR corresponds to the temperature 1120 ◦ C . 3.2.

Photochromic effects in disordered nanoparticle aggregates

Now we turn to practical applications of pulsed laser interaction with disordered colloidal NP aggregates . We discuss how suppression of SPR in melting NPs can change interactions of radiation with the systems of bound NPs . Colloidal plasmonic aggregates formed in disperse systems are certainly one of the most popular objects for experimental research . For better understanding of such processes, it is worthwhile to consider light interaction with resonant domains of such aggregates . Resonant domain represents a group of several bound NPs . Such groups are randomly distributed over a large aggregate which in turn is composed of several thousands or more NPs [24] . Each frequency of inhomogeneously broadened plasmonic spectrum of such aggregate corresponds to resonant frequency of individual set of domains comprising the aggregate . Interaction of disordered NP aggregates with monochromatic high-intensity pulsed laser radiation can be accompanied either by solid-liquid state transition of NPs and subsequent sharp decrease of the SPR Q-factor or by structural changes in different resonant domains [39, 40] . Both these changes result in instant formation of narrow dips in absorption spectrum . The latter means that disordered colloidal NP aggregates manifest photochromic effects . Figure 2 shows numerical results based on the model [24] for modification processes in multiparticle Ag aggregates exposed to monochromatic pulsed laser radiation, which strictly speaking for the first time reproduce experimental data on this effect [39, 40] . The physical model proposed in our work provides us with opportunity to obtain further insight into the physical nature of the processes underlying photochromic effects . We show that these effects can be divided into dynamic and long-term types, which have different nature . Figure 2(a) shows the dynamic effect which occurs during the pulse duration and caused by melting of NPs in resonant domains without changes of interparticle distance . Positions of resonant domains in disordered aggregate composed of 3000 Ag NPs and excited by incident radiation with different wavelengths 450 and 650 nm are shown in Fig. 2(c),(d). These figures show that we have different set of domains spatially separated from each other (in 3D aggregate some of domain particles are partially obscured by other NPs) . The kinetics of dynamic effects under picosecond pulsed radiation was studied in [24] and

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Qe

Qe

Initial λ, nm 500 600 700

0 .4 0 .2

0 .4 0 .2

0 .0 ΔQ e

0 .0 (a) 400

(c)

λ, nm 600

800

ΔQ e

1000

(b) 400

(d)

E

λ, nm 600

800

1000

E

Fig . 2 . (a), (b) — changes in extinction spectra Q e (λ) and differential spectra ΔQ e (λ) = (Q e )in − (Q e )a of monodisperse Ag NP aggregate excited by pulsed radiation with 20 ps pulse duration and I = 2.4 × 108 W·cm −2 intensity (here (Q e )in is initial extinction, (Q e )a is extinction after irradiation); vertical dashed lines indicate excitation wavelengths . (a) — at the end of a pulse (dynamic effect), (b) — in 40 ns after a pulse (long-term effect); changes in resonant domain are shown in inset (b) (see the description in the text) . NP aggregates were simulated with molecular dynamics method described in [24] . (c),(d) — positions of resonant domains in 3D aggregate consisting of N = 3000 Ag NPs with resonant wavelengths 450 nm (c) and 650 nm (d); polarization of incident field is indicated by arrows (in both cases 200 nanoparticles are highlighted) .

characteristic time was about 20 ps . After the pulse is over, NP becomes solid again . In picosecond laser pulses, due to inertia of NPs, they do not have enough time to change their positions in domains during the laser pulse neither under the action of the van der Waals forces nor under the light-induced interaction . At such time scales, changes in the extinction spectra caused only by the suppression of resonant properties of melted NPs which give rise to fast nonlinear optical response of the NP aggregate [24] . Figure 2(b) shows the long-term photochromic effect . It is related to restructuring of resonant domains caused by melting of NP polymer adsorption layers (1–2 nm thick) and degradation of their elasticity due to contact with hot metal NP core . It results in consequent shift of adjacent NPs in domains to each other up to contact of their metal cores . Kinetics of such process under nanosecond pulsed radiation was investigated in [24] and characteristic time exceeded 1–2 ns . Therefore, in the case of nanosecond pulsed laser radiation, the nonlinear response is associated with restructuring of resonant domains . The inset in Figure 2(b) shows the fragment of aggregate with a two-particle resonant domain (marked dark) before (left) and after (right) its photo-induced modification. This modification is accompanied by approaching the particles to each other due to deformation of adlayers . Note, that in nanosecond time scale the decreased absorption in plasmonic spectrum within a dip is accompanied by increased absorption in the long wavelength region adjacent to the dip . After the action of nanosecond pulses minor spectral changes may continue in the later stages . Thus, after a pulsed laser irradiation with different

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wavelengths, plasmonic absorption spectrum of disordered colloidal NP aggregates may acquire several dips exactly at the radiation wavelengths . These dips clearly demonstrates the spectral selectivity of the modification process. Visual effect after radiation is that the region of the sample exposed to laser pulse acquires the color and polarization of the incident radiation . This effect demonstrates a polarization selectivity because radiation with different linear polarizations interacts with individual set of domains even at the same wavelengths [40] . 3.3.

Thermal effects in ordered nanostructures

Now we consider interaction of pulsed laser radiation with ordered nanostructures . To this end, we study optical plasmonic waveguides (OPW) in the form of plasmonic NP chains [65, 66] which attract significant attention due to their applicability for transmission of modulated optical radiation by means of surface plasmon polaritons (SPP) . A great number of publications underlines the interest in this objects . But never before the effect of strong heating of nanoparticles in the OPW was studied . In the meanwhile, practical employment of OPWs without providing them with cooling device will be limited . From practical point of view, OPW is considered to be fabricated on some substrate instead of free space . Thus, Green’s tensor in eq . (7) will be described as follows: ˆ free (ri , r j ) + G ˆ refl (ri , r j ), ˆ i , rj ) = G G(r

(26)

ˆ free (ri , r j ) and G ˆ refl (ri , r j ) are Green’s tensors that describe electric field in a homowhere G geneous environment and reflected from the substrate, correspondingly. More comprehensive discussion of eq . (26) can be found in [67] . Consider the simplest case of a linear chain consisting of N = 11 identical spherical Ag NPs with 8 nm radius . NPs are arranged equidistantly in the host medium with permittivity 1.78 over the surface of the quartz substrate (see Fig . 3) . Here we use spherical NPs just to demonstrate thermal effects . The papers [65, 68] show that the use of NPs in the form of prolate or oblate spheroids can dramatically increase the transmission efficiency . The substrate keeps constant temperature 20 ◦ C . The distance between the centers of the neighboring particles is 24 nm. We assume that only the first (i = 1) locally excited NP interacts with an incident radiation which is conventional condition for numerical study of OPW in publications (see in particular [65, 66]) . Experimentally it can be implemented by using, for example, a nearfield scanning optical microscope (NSOM) tip. Thus, transmission properties of OPW can be described by the transmission coefficient [65, 66, 68]: |d N (t)| , (27) |d1 (t = 0)| where d N (t) and d1 (t = 0) are oscillation amplitudes of dipole moments induced on the last (at arbitrary moment of time) and the first (at initial moment of time) particles, correspondingly. The impact of thermal effects on SPP propagation is very important factor for such type of OPW excitation . However temperature dependence of NP’s dielectric constants is usually overlooked in most studies. In our work we used a finite element method in original realization for calculations of temperature distribution around OPW . It is obvious that thermal effects in OPW should strongly depend on the intensity I of incident laser radiation . For low I magnitudes, none of NPs will reach the melting point . However in this case, it will be nearly impossible to register optical signal at the end of OPW due to strong SPP attenuation . For high values of I, we will observe extreme heating of OPW NPs . In this case, their resonant properties will be dramatically suppressed . Therefore, we consider the case of intermediate magnitude of intensity between these two extreme cases . Our calculations show that for I = 1.57 × 108 W · cm −2 , only first NP reaches the melting point. We expect that Qtr will not drop dramatically in this case . Qtr =

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excitation 16nm

Host medium

Y

Substrate

X

Z

24nm

Fig . 3 . The OPW geometry used in numerical simulations .

Q tr

Y , nm

20

315

.0

10

40

0 .

(a)

500 .0

600

-20 -20

.0

80 0 .0

0

0

350 .0

0 ps 37 ps 171 ps (b)

0 0

20

X, nm

360

400

440

λ, nm 480

Fig . 4 . The temperature T distribution at t = 1 ns for first three OPW Ag NPs. Wavelength λ = 402 nm and polarization along the X axis — (a) . Transmission spectra of the OPW at different moments of time: the initial moment of time (solid line); the beginning of i = 1 NP melting (dashed line); the end of i = 1 NP melting (dotted line) — (b) .

Figure 4(a) represents XOY -plane (z = 0) temperature distribution in the OPW at the end of the 1 ns pulse irradiation. We can see that only first three particles are significantly heated, and the temperature of the i = 1 particle reaches the melting point . We take into account heat exchange between NPs through environment and between a NP chain and thermally stable substrate . Transmission spectra of OPW at different stages of i = 1 particle melting process are shown on Fig . 4(b) . The OPW transmission spectrum slightly changes when the i = 1 NP reaches the melting temperature (T ≈ 1080 K for NP with 8 nm radius, dash line) . This change is associated with the change of the dielectric constant of the particle material . It should be noticed that NP is still in the solid state at 37 ps. However significant suppression of SPR in the i = 1 NP occurs when it becomes fully liquid at 171 ps moment of time (the end of the first NP melting process). It is clearly seen that OPW transmission efficiency drops threefold in this case (dotted line) . As a consequence, the transmitted energy is also reduced . In turn, it results in a decrease of absorbed energy and the temperature of the second and subsequent NPs . The further increase of NP temperature in a chain occurs due to heat exchange between them, between NPs and host medium including substrate . 4.

Conclusion

To conclude, we have developed experimentally verified combined theoretical model that describes light-induced dipole interactions between NPs and surrounding medium in highintensity optical fields. This model takes into account vast majority of various physical phenomena that take place in different systems of bound NPs . The interactions in NP aggregates

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cover three major group of the processes: mechanical, electrodynamical, thermal . The melting processes in Au and Ag NPs have been studied numerically and experimentally [25] . Our experimental results evidence the validity of the proposed model . It was shown that the melting of plasmonic NP sufficiently suppresses its SPR . The developed model was used to reproduce numerically photochromic effects in disordered colloidal Ag NP aggregates observed experimentally . Strong heating of NPs in resonant domains of aggregates explains dynamic and long-term laser photochromic effects which have application potential in data storage devices . Finally, we have shown that thermal effects significantly impair the transmission efficiency of OPW due to suppression of the NP resonant properties and deterioration of SPR Q-factor . Optimal conditions for stable waveguiding through OPW are limited by an intensity and wavelength of incident pulsed laser radiation . We also conclude that technological substrates for OPWs must be provided with cooling devices . 5.

Acknowledgments

This work was performed within the State contract of the RF Ministry of Education and Science for Siberian Federal University for scientific research in 2017–2019 and SB RAS Program No II .2P (0358-2015-0010) . The calculations were performed using the MVS-1000 M cluster at the Institute of Computational Modeling, Federal Research Center KSC SB RAS .

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PAPER II

Titanium nitride nanoparticles as an alternative platform for plasmonic waveguides in the visible and telecommunication wavelength ranges V. I. Zakomirnyi, I. L. Rasskazov, V. S. Gerasimov, A. E. Ershov, S. P. Polyutov,S. V. Karpov, H. Agren Photonics and Nanostructures - Fundamentals and Applications 30, pp. 50-56 (2018).

Paper II Photonics and Nanostructures – Fundamentals and Applications 30 (2018) 50–56

Contents lists available at ScienceDirect

Photonics and Nanostructures – Fundamentals and Applications journal homepage: www.elsevier.com/locate/photonics

Invited Paper

Titanium nitride nanoparticles as an alternative platform for plasmonic waveguides in the visible and telecommunication wavelength ranges V.I. Zakomirnyi a,b,∗ , I.L. Rasskazov c , V.S. Gerasimov a,d , A.E. Ershov a,d,e , S.P. Polyutov a , a,b ˚ S.V. Karpov a,e,f , H. Agren a

Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University, Krasnoyarsk 660041, Russia Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, KTH Royal Institute of Technology, 10691 Stockholm, Sweden c The Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA d Institute of Computational Modeling SB RAS, 660036 Krasnoyarsk, Russia e Siberian State University of Science and Technology, Krasnoyarsk 660014, Russia f Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036 Krasnoyarsk, Russia b

a r t i c l e

i n f o

Article history: Received 15 November 2017 Received in revised form 12 March 2018 Accepted 12 April 2018 Available online 17 April 2018 Keywords: Nanoparticle Titanium nitride Surface plasmon polariton Plasmon waveguide Refractory plasmonics

a b s t r a c t We propose to utilize titanium nitride (TiN) as an alternative material for linear periodic chains (LPCs) of nanoparticles (NPs) which support surface plasmon polariton (SPP) propagation. Dispersion and transmission properties of LPCs have been examined within the framework of the dipole approximation for NPs with various shapes: spheres, prolate and oblate spheroids. It is shown that LPCs of TiN NPs support high-Q eigenmodes for an SPP attenuation that is comparable with LPCs from conventional plasmonic materials such as Au or Ag, with the advantage that the refractory properties and cheap fabrication of TiN nanostructures are more preferable in practical implementations compared to Au and Ag. We show that the SPP decay in TiN LPCs remains almost the same even at extremely high temperatures which is impossible to reach with conventional plasmonic materials. Finally, we show that the bandwidth of TiN LPCs from non-spherical particles can be tuned from the visible to the telecommunication wavelength range by switching the SPP polarization, which is an attractive feature for integrating these structures into modern photonic devices. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Plasmonic nanoparticles (NPs) are one of the cornerstones of modern science and technology due to an almost uncountable number of applications [1]. The uniqueness of the NPs originates from their ability to support localized surface plasmons (LSPs) which enables strong confinement of electric fields at scales much smaller than the wavelength of the incident radiation. Strong enhancement of local fields is vital for a wide variety of possible applications such as surface enhanced Raman spectroscopy [2], upconversion [3], biomedicine [4] and solar energy harvesting [5]. Arrays of non-touching NPs enable nanoscale electromagnetic energy transfer via propagation of so-called surface plasmon

∗ Corresponding author at: Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University, Krasnoyarsk 660041, Russia. E-mail address: [email protected] (V.I. Zakomirnyi).

polaritons (SPPs) [6,7] which paves the way for utilization of nanostructures in chemical sensing [8–11], nanoantennas [12,13] and nanosized waveguides [14–19]. Propagation of SPPs is usually considered in 1D or 2D arrays of NPs from conventional plasmonic materials: Ag [6,15,16,20–24] and Au [17,18,25,26]. Various types of technological aspects such as disorder [15,20], polydispersity [15] impact of the substrate [21,27,28], thermal effects [29] have been thoroughly considered so far. However, high Ohmic losses and consequent overheating of metal NPs lead to strong suppression of the LSPs [30] and, as a result, significant attenuation of the SPP. Moreover, the SPP frequencies lie mostly in the visible or near infrared ranges, while integration with CMOS-compatible devices requires bandwidth of LPCs to lie within the telecommunication wavelength range. These drawbacks make it unlikely that LPCs of plasmonic NPs can be utilized as a tool for efficient guiding of electromagnetic energy over distances of several hundred nanometers for applications in highly integrated optical devices operating below the diffraction limit of light [31].

https://doi.org/10.1016/j.photonics.2018.04.005 1569-4410/© 2018 Elsevier B.V. All rights reserved.

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There are some strategies to deal with strong attenuation of SPPs. For example, utilization of non-spherical NPs can lead to nondecaying propagation of SPPs in LPCs in homogeneous media [32] or to minimization of SPP suppression due to interaction with the dielectric substrate [27]. However, inevitable Ohmic losses in metals and consequent overheating of NPs represent a more complicated problem to deal with. The temperature dependent optical properties of noble metals [30,33–36] imply strong suppression of LSP resonances at high temperatures, which is of crucial importance for SPP propagation. Finally, chemical instability, inability to dynamically adjust optical properties [37] limit successful implementations of Ag and Au LPCs of NPs in photonic devices. Recent progress in plasmonics, however, indicates a way to overcome drawbacks of conventional noble metals by using socalled alternative plasmonic materials [38,39]. Numerous materials have been benchmarked against conventional Ag and Au in terms of local field enhancement [40] and refractory behavior [41,39,42]. Metal transition nitrides, particularly, titanium nitride (TiN) have been considered as most promising material from this perspective [43–46]. TiN thin films and NPs have a great potential in photodetection [47], solar energy harvesting [48], sustainable energetics [49], nonlinear optics [50,51], and biomedicine [52]. The position of the LSP resonance peak of a single TiN NP lies in the near infrared region [46] which is a vital feature for utilization of TiN NP arrays in photonic devices operating at telecommunication wavelengths [53,54]. Finally, low-cost large area fabrication [55], thermal stability [56] are important advantages of TiN which immediately enable its wide practical implementation. In this paper, we propose to utilize TiN for waveguiding applications in LPCs of spherical and spheroidal NPs. TiN optical properties provide an opportunity to shift the SPP frequency to telecommunication wavelengths, while the TiN refractory behavior could potentially resolve issues with strong SPP suppression at high temperatures.

2. Model

Table 1 Temperature-dependent optical constants for TiN [70].

ε∞ ωp , rad/fs �D , rad/fs ωL,1 , rad/fs � 1 , rad/fs ω0,1 , rad/fs ωL,2 , rad/fs � 2 , rad/fs ω0,2 , rad/fs

1 dn − ˛

N 

m= / n

106

800 ◦ C

6.50246 11.37931 0.56213 11.76819 1.80793 6.47208 4.03393 3.08411 2.93219

4.87685 11.47047 0.80521 15.9342 2.78026 7.17094 3.52391 2.76507 2.79545

ˆ nm dm = Eext G n ,

(1)

ˆ nm is the Green’s tensor Here ˛ is the polarizability of the NP, G which describes the electric field at xn point created by a point dipole located at the xm point. Explicit expressions for the Green’s function can be found elsewhere [15,16]. In this paper, we consider three different shapes of NPs: spheres, prolates and oblate spheroids. The quasistatic polarizability of NPs with these shapes is defined by the following expression: ˛0 =

εp − ε h 4� , V εh + L(εp − εh )

(2)

where V is the volume of the NP, εp is the dielectric permittivity of the NP material, L is a static depolarization factor [66]. For NPs with dimensions much smaller than the wavelength of the incident illumination, retardation effects should be taken into account [67,68]. Moreover, a so-called dynamic correction [69] to the polarizability of spheroidal NPs has to be introduced for an adequate description of its electromagnetic properties. Thus, the polarizability ˛ takes the form:



A substantial part of theoretical and experimental studies of electromagnetic properties of metal nanostructures are based on the dipole approximation (DA) [15,16,21,23,57–62]. This approximation is quite simple yet efficient for accurate description and prediction of optical properties of plasmonic nanostructures. The applicability of DA is, however, strictly limited by the geometry of the structure [63,64]: the distance between the NPs in a periodic array should be significantly larger than the size of the NPs. For example, in the case of nanospheres, the center-to-center distance between neighboring NPs should exceed the NP diameter by the factor of 1.2 and 1.4 for transverse and longitudinal polarization of incident irradiation, respectively, as shown in Ref. [65]. Otherwise, it is necessary to take into account quadrupole [65] or higher order interaction for an adequate description of the electromagnetic properties of such structures. Although the dipole approximation serves as a well-known approach, extensively described in literature, in this section we provide a general formalism for the convenience of the Readers. Let us consider an LPC of N identical NPs whose centers are located at points xn = (n − 1)h, where h is the center-to-center distance, and n = 1, . . ., N. We consider LPCs located in a homogeneous environment with dielectric permittivity εh . Assume that the LPC is excited ext by an external monochromatic electric field Eext (xn ) (time n =E dependence exp(−iωt) is omitted in all expressions, here ω is the frequency of the incident field). In this case, the dipole moments dn

400 ◦ C

7.86981 11.21219 0.39501 9.86939 2.15736 6.18342 2.28396 1.32176 3.06892

induced in each NP are coupled to each other and to the external field via the following coupled dipole equations:

˛ = ˛0 1 −

2.1. Dipole approximation

23 ◦ C

k2 2k3 D˛0 − i ˛0 3 lE

−1

(3)

,

where ˛0 is defined by Eq. (2), D is a dynamic geometrical factor √ [69], k = εh ω/c is the wave number, c is the speed of light in a vacuum, lE is the length of the semiaxis of the NP along which the electric field is applied. Static L and dynamic D depolarization factors for prolate and oblate spheroids can be found with wellknown expressions [66,69] which are not provided in this paper. For spherical NPs: L = 1/3 and D = 1. Finally, the TiN dielectric permittivity can be described by the Lorentz oscillator model [70]: εp = ε∞ −

ωp2 ω2 + i�D ω

+

2  j=1

2 ωL,j 2 − ω2 − i� ω ω0,j j

,

(4)

where ε∞ describes high energy interband transitions outside the probed energy spectrum, ωp is the plasma frequency, �D 2 , ω , and � describe the is the Drude relaxation constant, ωL,j 0,j j Lorentz oscillator strength, energy, and damping, respectively. Temperature-dependent constants entering Eq. (4) are represented in Table 1. 2.2. Dispersion properties Dispersion relations are one of the most important properties which quantify the ability of LPCs from metal NPs to support SPP propagation. There are various approaches to estimate dispersion

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Fig. 1. Schematic representation of LPCs from (a) spheres, (b) prolate spheroids, and (c) oblate spheroids.

relations of finite [57,71,72] and infinite [16,73–79] LPCs, however, in this work, we use an efficient eigendecomposition method [80]. In accordance with the Bloch theorem, the dipole moment and external field in Eq. (1) can be expressed as dn = d · exp(iqnh) and ext Eext · exp (iqnh), where q is the eigenvector of the Bloch n =E mode. Thus, Eq. (1) can be written as follows:



1 ˆI − ˛

∞ 



ˆ nm eiqnh d = Eext . G

n=−∞

(5)

We note that the expression in square brackets has the same dimension as the inverse dipole polarizability ˛−1 of a NP. Thus, according to the eigendecomposition method, it is convenient to characterize the electromagnetic response of the LPC by so-called effective polarizability ˛ ˜ [80] such that 1/˛ ˜ is the eigenvalue of the following operator:



1 ˆI − ˛

∞ 



ˆ nm eiqnh . G

n=−∞

(6)

Maxima of Im[˛] ˜ = F(ω, q) correspond to LPC resonances, which represent the bandwidth of the LPC, or, in other words, the dispersion relation of the LPC. An essential advantage of the eigendecomposition method is the simultaneous estimation of the LPC eigenmodes and their quality factor [23,80]. Thus, the functionIm[˛] ˜ = F(ω, q) provides a complete physical insight into the dispersion relations for LPCs, which is, generally speaking, impossible with other methods considered in the literature [16,57,72–74].

2.3. Transmission properties While dispersion properties quantify the bandwidth of LPCs, it is useful to calculate the actual damping of the SPP at the end of the waveguide. The most convenient way to do so is to calculate the transmission spectrum of the LPC. Let us assume that the external field Eext excites only the first NP in the LPC. This kind of excitation has been used in other works as well [15,21]. Experimentally, it corresponds to an excitation by a near-field optical microscope tip. ext In this case, the incident field is described as Eext ı1n , where n =E ımn is the Kronecker delta. Solution of Eq. (1) with this right-hand side provides the dipole moments dn induced on each NP in the LPC. Experimentally, the intensity of the electric field at the end of the LPC IN ∝ � dN � 2 characterizes the attenuation of the SPP. Therefore, the efficiency of the SPP propagation can be described by the following quantity [21]:

QN =

�dN �2 �d1 �2

.

(7)

Thus, we will refer to spectral dependence of QN as to the transmission spectrum of the LPC.

3. Results 3.1. LPC geometry In this work, we study LPCs of NPs with different shapes, as shown in Fig. 1. For spherical NPs, there is no difference in NP orientation with respect to the LPC symmetry axis, however, for spheroidal NPs it is no longer so. We fix the orientation of the spheroids to the following configuration: the shorter semi-axis b of each spheroid is aligned along the X axis, while the longer semiaxis a is orthogonal to the X axis. It is notable, however, that LPCs with spiral-like orientation of spheroidal NPs is a subject for a oneway propagation and a Faraday rotation [81,82], something that is not considered in this work. We set the geometrical parameters of the LPCs to the following constant values: shorter semi-axis b = 25 nm and center-to-center distance h = 75 nm. These parameters are widely used in numerical simulations of SPP propagation in LPCs from Ag NPs [6,7,14,21,23,57,63]. The longer semi-axis a has been changed for spheroids with different aspect ratios b/a. DA is adequate for LPCs with these geometric parameters [63]. Finally, the dielectric permittivity of the host medium was set to εh = 2.25. In a real experiment, this kind of surrounding can be achieved by depositing NPs on a quartz substrate and subsequently covering them by a poly(methyl methacrylate) superstrate [83]. Although it is obvious that the optical properties of surrounding medium are temperature-dependent, we assume that εh is constant for the considered frequencies and temperatures [84]. 3.2. Dispersion properties Let us start with the LPC dispersion properties. We plot log[Im(˛)] ˜ vs. the SPP frequency ω and the eigenmode wavevector q. According to the eigendecomposition method, high values of Im(˛) ˜ correspond to high Q-factor eigenmodes. While the summation in Eq. (6) runs to infinity, we consider finite, but sufficiently long, LPCs namely, N = 1000 NPs. We begin with LPCs from spherical NPs, the dispersion relations of which are depicted in the first column of Figs. 2 and 3. It can be seen that the SPPs effectively propagate for both longitudinal (X) and transverse (Y) polarizations at ω≈2.5–3.5 rad/fs frequencies. However, the branch which corresponds to highest values of Im(˛) ˜ has quite small slope, which corresponds to a low group velocity of SPPs at this spectral range. It is well known, however, that the utilization of non-spherical NPs in LPCs significantly increases SPP group velocity [16,72] and simultaneously minimizes the suppression of the SPP [32]. Thus, it is of interest to consider dispersion relations of LPCs from prolate and oblate spheroids with different values of the aspect ratio b/a. It can be seen from Figs. 2 and 3, that in the case of longitudinal polarization, the overall shape of the dispersion branches is almost the same as for LPCs from spherical NPs. This kind of behavior is explained by a negligible difference of the depolarization factors L for different shapes of the NPs with the same values of the shorter semi-axis b which is parallel to polarization of SPP. However, for oblate spheroids with b/a = 0.4, the Q-factor of the eigenmodes near the light line ω = q/c is significantly larger. In the case of transverse polarization, values of log[Im(˛)] ˜ substantially increase, especially for ω≈1.5–2.5 rad/fs spectral range

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53

Fig. 2. Dispersion relations for LPCs from spherical and spheroidal particles at different temperatures for longitudinal (X) polarization of SPPs. White dashed line represents the light line ω = q/c.

Fig. 3. The same as in Fig. 2, but for transverse (Y) polarization of SPPs.

for LPCs from prolate and oblate spheroids with b/a = 0.4. Dispersion branches for LPCs from oblate spheroids have even greater slopes as compared with LPCs from prolate spheroids with the same values of b/a. In addition, the frequency of the eigenmodes decreases to ω≈1–2 rad/fs, which basically corresponds to the telecommunication wavelength range. Moreover, the bandwidth of the LPC significantly increases in this case. Finally, the dispersion branch acquires a significant negative slope, which leads to an increase of the SPP group velocity and to anti-parallel propagation of the group and phase velocities of the SPP. Propagation of transversely polarized SPPs with anti-parallel group and phase velocities has been explained in detail in Refs [7,16,23]. Nonetheless, it is worthwhile

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to mention that the negative slope of the dispersion curve is not a direct proof that LPCs are metamaterials with negative refraction. Suppression of SPP due to overheating of LPCs is a crucial drawback [29] which significantly limits the widespread use of silver and gold NPs for waveguiding applications. In this sense, TiN is advantageous due to its refractory properties [41]. It can be seen from Figs. 2 and 3 that the dispersion relations of the LPCs of TiN NPs remain almost intact even at T = 800 ◦ C. The eigenmode quality factor inevitably decreases at high temperature, however, the suppression of SPPs is much lower than one could expect for a conventional plasmonic materials. We note that in this work LPCs are assumed to be uniformly heated, which represents the most

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Fig. 4. Transmission spectra of LPCs from TiN NPs of various shapes for X polarization (a)–(c) and for Y polarization (d)–(f) of the SPPs, and at different temperatures: (a) and (d) T = 23 ◦ C; (b) and (e) T = 400 ◦ C; (c) and (f) T = 800 ◦ C.

Fig. 5. Electric field localization |E|2 /|E0 |2 of LPCs from spherical, prolate and oblate (with b/a = 0.4 for both type of spheroids) TiN NPs for Y polarization of the SPP at different temperatures. The frequencies were taken in accordance with the maximum values of QN from Fig. 4. Excitation is at the NP on the left.

extreme case of overheating. In practice, only three neighboring NPs experience the most pronounced heating in the case of local SPP excitation [29]. One of the exciting features that can be observed from a careful analysis of Figs. 2 and 3 is that the SPP bandwidth (spectral range which corresponds to high-Q eigenmodes) significantly changes from longitudinal to transverse polarization for LPCs made of spheroids with b/a = 0.4. For longitudinal polarization, the LPC bandwidth corresponds to the visible wavelength range, while for transverse polarization it lies in telecom. Thus, TiN LPCs can simultaneously operate at two important wavelength ranges, which makes it possible to utilize such LPCs as a hybrid photonic interconnector. 3.3. Transmission properties We turn to the transmission properties of the LPCs by considering SPP propagation in short LPCs from N = 20 NPs. It can be seen from Fig. 4 that, indeed, the LPC bandwidth can be adjusted by switching the SPP polarization from longitudinal to transverse

value. Moreover, QN slightly decrease at high temperatures, which is of crucial importance for waveguiding applications of LPCs. As it may be expected, the most efficient propagation of SPPs takes place in LPCs of oblate spheroids with small aspect ratios e.g. b/a = 0.4, which is consistent with results reported in Refs [32,72]. 3.4. Electric field localization Finally, get another attractive property of LPCs is the ability to confine the electromagnetic energy at scales much smaller than the wavelength of the propagating excitation. This feature differs the LPCs from classic strip waveguides [85], the transverse dimensions of which are usually comparable or several times larger that the wavelength of propagating signal. A tight localization of the electromagnetic field near the LPCs makes it possible to arrange several LPCs in a close vicinity to each other without a risk of overlapping of SPPs propagating in neighboring LPCs, which cannot be achieved in strip waveguides. Fig. 5 shows temperature-dependent intensity plots |E|2 /|E0 |2 for LPCs from TiN NPs at 10 nm distance from top-most points of the

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NPs. The frequencies ω of SPPs were chosen to match the maximum values of QN for Y polarization of the SPPs from Fig. 4(d–f). It can be seen that in the case of spherical NPs, the electric field is tightly localized near the first excited NP and rapidly attenuates along the LPC. The most efficient localization of the electric field is observed in the case of prolate spheroids, moreover, as it can be seen from Fig. 5, the field is confined within ≈500 nm spatial region near the LPC, which is several times smaller than the wavelength of the SPP. The intensity |E|2 /|E0 |2 pattern looks completely different for LPCs from oblate spheroids due to the high values of the local field at the tips of the oblate spheroids. Finally, due to the refractory behavior, electric field confinement of LPCs from TiN NPs remains almost the same even at elevated temperatures. 4. Conclusion To conclude, we have shown that TiN is a promising alternative material that can be used in periodic arrays of nanoparticles to support efficient propagation of surface plasmon polaritons. The bandwidth of linear periodic chains from TiN nanoparticles can be tailored from the visible range to telecom through varying the nanoparticle shape and polarization of the surface plasmon polaritons. Despite inevitable Ohmic losses and overheating of the nanoparticles, the attenuation of the surface plasmon polaritons remains almost the same even at high temperatures due to advantageous TiN refractory properties. Along with cheap methods of large-area fabrication of TiN nanoparticles, these features make TiN a promising plasmonic material for waveguiding applications using linear periodic chains of the nanoparticles. Acknowledgments This work was supported by the RF Ministry of Education and Science, the State contract with Siberian Federal University for scientific research in 2017–2019 and SB RAS Program No II.2P (0358-2015-0010). References [1] S.A. Maier, Plasmonics: Fundamentals and Applications, Springer US, Boston, MA, 2007. [2] J.-F. Li, Y.-J. Zhang, S.-Y. Ding, R. Panneerselvam, Z.-Q. Tian, Core–shell nanoparticle-enhanced Raman spectroscopy, Chem. Rev. 117 (2017) 5002–5069. [3] D.M. Wu, A. García-Etxarri, A. Salleo, J.A. Dionne, Plasmon-enhanced upconversion, J. Phys. Chem. Lett. 5 (2014) 4020–4031. [4] N.S. Abadeer, C.J. Murphy, Recent progress in cancer thermal therapy using gold nanoparticles, J. Phys. Chem. C 120 (2016) 4691–4716. [5] J. Li, S.K. Cushing, F. Meng, T.R. Senty, A.D. Bristow, N. Wu, Plasmon-induced resonance energy transfer for solar energy conversion, Nat. Photonics 9 (2015) 601–607. [6] M. Quinten, A. Leitner, J.R. Krenn, F.R. Aussenegg, Electromagnetic energy transport via linear chains of silver nanoparticles, Opt. Lett. 23 (1998) 1331. [7] S.A. Maier, P.G. Kik, H.A. Atwater, S. Meltzer, E. Harel, B.E. Koel, A.A. Requicha, Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2 (2003) 229–232. [8] M.E. Stewart, C.R. Anderton, L.B. Thompson, J. Maria, S.K. Gray, J.A. Rogers, R.G. Nuzzo, Nanostructured plasmonic sensors, Chem. Rev. 108 (2008) 494–521. [9] A.B. Evlyukhin, S.I. Bozhevolnyi, A. Pors, M.G. Nielsen, I.P. Radko, M. Willatzen, O. Albrektsen, Detuned electrical dipoles for plasmonic sensing, Nano Lett. 10 (2010) 4571–4577. [10] G. Li, X. Li, M. Yang, M.-M. Chen, L.-C. Chen, X.-L. Xiong, A gold nanoparticles enhanced surface plasmon resonance immunosensor for highly sensitive detection of ischemia-modified albumin, Sensors 13 (2013) 12794–12803. [11] V.G. Kravets, F. Schedin, R. Jalil, L. Britnell, R.V. Gorbachev, D. Ansell, B. Thackray, K.S. Novoselov, A.K. Geim, A.V. Kabashin, A.N. Grigorenko, Singular phase nano-optics in plasmonic metamaterials for label-free single-molecule detection, Nat. Mater. 12 (2013) 304–309. [12] A.F. Koenderink, Plasmon nanoparticle array waveguides for single photon and single plasmon sources, Nano Lett. 9 (2009) 4228–4233. [13] J. Munárriz, A.V. Malyshev, V.A. Malyshev, J. Knoester, Optical nanoantennas with tunable radiation patterns, Nano Lett. 13 (2013) 444–450.

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PAPER III

Collective lattice resonances in disordered and quasi-random all-dielectric metasurfaces V. I. Zakomirnyi, S. V. Karpov, H. Agren, I. L. Rasskazov Journal of the Optical Society of America B 36(7), E21-E29 (2019)

Paper III Research Article Research Article

Vol. 36, No. 7 / July 2019 / Journal of the Optical Society of America B Vol. 36, No. 7 / July 2019 / Journal of the Optical Society of America B

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Collective Collective lattice lattice resonances resonances in in disordered disordered and and quasi-random all-dielectric metasurfaces quasi-random all-dielectric metasurfaces VADIM I. ZAKOMIRNYI,1,2,3 VADIM I. ZAKOMIRNYI,1,2,3

SERGEI V. KARPOV,3,4,5 HANS ÅGREN,1,2 AND ILIA L. RASSKAZOV6,* SERGEI V. KARPOV, HANS ÅGREN, AND ILIA L. RASSKAZOV * Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, 1 Department Chemistry and SE-10691, Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal InstituteofofTheoretical Technology, Stockholm, Sweden 2 Royal Institute of Research Technology, Stockholm, Sweden Federal Siberian Clinical CentreSE-10691, under FMBA of Russia, Krasnoyarsk, 660037, Russia 2 3Federal Siberian Research Clinical Centre under FMBA of Russia, Krasnoyarsk, 660037, Russia Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University, Krasnoyarsk 660041, Russia 3 4Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University, Krasnoyarsk 660041, Russia Siberian State University of Science and Technology, Krasnoyarsk, 660014, Russia 4 5Siberian State University of Science and Technology, Krasnoyarsk, 660014, Russia Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia 5 6Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia The Institute of Optics, University of Rochester, Rochester, New York 14627, USA 6 The Institute of author: Optics, [email protected] University of Rochester, Rochester, New York 14627, USA *Corresponding *Corresponding author: [email protected] Received 18 December 2018; revised 7 March 2019; accepted 13 March 2019; posted 14 March 2019 (Doc. ID 355522); published 16 April 2019 Received 18 December 2018; revised 7 March 2019; accepted 13 March 2019; posted 14 March 2019 (Doc. ID 355522); published 16 April 2019 3,4,5

1,2

6,

1

Collective lattice resonances in disordered 2D arrays of spherical Si nanoparticles (NPs) have been thoroughly Collective lattice 2Ddipole arraysapproximation. of spherical Si nanoparticles have studied within theresonances frameworkinofdisordered the coupled Three types of(NPs) defects havebeen beenthoroughly analyzed: studied within the framework of the coupled dipole approximation. Threethat types defects have been analyzed: positional disorder, size disorder, and quasi-random disorder. We show theofpositional disorder strongly positional disorder, quasi-random disorder. We showcoupling, that the depending positional on disorder strongly suppresses disorder, either thesize electric dipoleand (ED) or the magnetic dipole (MD) the axis along suppresses either electric dipole (ED) or the magnetic dipole only (MD) depending along which the NPs arethe shifted. Contrarily, size disorder strongly affects thecoupling, MD response, while on the the ED axis resonance which the NPs are shifted. Contrarily, size disorder strongly affects only the MD response, while the ED resonance can be almost intact, depending on the lattice configuration. Finally, random removing of NPs from an ordered can be almost intact, depending on the lattice configuration. Finally, random NPs lattice from an ordered 2D lattice reveals a quite surprising result: hybridization of the ED and MD removing resonancesofwith modes re2D lattice reveals even a quite hybridization thetoED and resonances with lattice array. modesThe remains observable in surprising the case ofresult: random removing ofofup 84% ofMD the NPs from the ordered mains observable evenbe in important the case offor random of up to 84% of ofmetasurfaces, the NPs fromsolar the cells, ordered reported results could rationalremoving design and utilization andarray. otherThe allreported could be important for rational design and utilization of metasurfaces, solar cells, and other all© 2019 Optical Society of America dielectric results photonic devices. dielectric photonic devices. © 2019 Optical Society of America https://doi.org/10.1364/JOSAB.36.000E21 https://doi.org/10.1364/JOSAB.36.000E21

interest. Arrays of all-dielectric NPs have already found a numinterest. Arrays of all-dielectric have already found a number of excellent applications in NPs photonics and nanotechnology ber of excellent applications in photonics and nanotechnology spanning light-guiding [40–42], metamaterials [43], metasurspanning light-guiding [40–42], metamaterials metasurfaces [44–48], mid-infrared filters [49], and[43], others [50]. faces [44–48], mid-infrared [49], and others [50]. However, the coupling betweenfilters localized oscillations in a single However, NP the coupling in a single dielectric and latticebetween modes localized has beenoscillations addressed only quite dielectric NP andwith lattice modes has been addressed recently [51–54], particular attention to ED andonly MDquite courecently [51–54], with particular ED and MD coupling [55] and overlapping [56] attention in 2D Sitonanodisk arrays. pling [55] and overlapping [56] in 2D Si nanodisk arrays. While various aspects of diffractive behavior of ED and MD While various aspects of diffractive behavior ED andinMD resonances in arrays of NPs have been heavilyofstudied the resonances in arrays of NPs have been heavily studied in the recent decade [57–60], just a few works have addressed the efrecent [57–60], justdisorders a few works have addressed effects ofdecade positional and size [61–63], and only the for arfects of positional and size disorders [61–63], and only for arrays with pure ED coupling. Quite interesting results have also rays pure ED Quiteand interesting resultsharvesting have also beenwith reported for coupling. lasing [64,65] solar energy been reported lasingtypes [64,65] and solar energy [62,66–68] in for various of quasi-periodic and harvesting aperiodic [62,66–68] in variousit is types of quasi-periodic structures. However, a well-known fact thatand the aperiodic presence structures. However, it ischains a well-known fact that2D thestructures presence of imperfections in 1D of NPs [69,70], of imperfections in 1D chains of NPs [69,70], 2D structures [71–76], 3D metamaterials [77], and fractal aggregates [78] [71–76], metamaterials [77], and fractal aggregates [78] may lead 3D to various intriguing effects. mayInlead various intriguing address effects. this problem within the this to work, we thoroughly In thisdipole work,approximation, we thoroughly and address thisthree problem the coupled study typeswithin of impercoupled dipole approximation, and study types imperfections in 2D arrays of Si nanospheres: (i)three disorder in of positions fections in 2D arrays nanospheres: disorderininsizes positions of Si nanospheres of oftheSi same size; (ii)(i)disorder of Si of Si nanospheres of the same size; (ii) disorder in sizes of Si

1. INTRODUCTION 1. INTRODUCTION Strong coupling between lattice modes in arrays of nanoparStrong(NPs) coupling between lattice modes in arrays of nanoparticles and Mie-type oscillations localized within a single ticles (NPs) and Mie-type oscillations within single NP has attracted significant attentionlocalized over the last adecade. NP has attracted significant attention overarrays the last decade. Pioneering theoretical predictions for 1D of Ag NPs Pioneering theoretical experimental predictions for 1D arraysforof2D Agarrays NPs [1–3] and consequent verification [1–3] and consequent experimental verification for 2D arrays of Au NPs [4–6] have given a momentum to a great number of of Au NPsapplications [4–6] have given a momentum a great number of excellent of collective lattice to resonances in lasers excellent applications of collective lattice resonances [19–22], in lasers [7–12], biosensors [13–18], emission enhancement [7–12], [13–18], emission enhancement [19–22], and colorbiosensors printing [23–26]. andTo color printing [23–26]. date, most of the studies have considered diffractive To date, most electric of thedipole studies(ED) haveoscillations consideredand diffractive coupling between Wood– coupling between electric dipole (ED)ofoscillations and Wood– Rayleigh anomalies [27,28] in arrays classic plasmonic NPs Rayleigh [27,28] quite in arrays of classic plasmonic NPs such as Auanomalies or Ag. However, recently significant attention such as Auturned or Ag. However, quiteplasmonic recently significant has been to alternative materialsattention such as has been turned [29], to alternative plasmonic as indium-tin-oxide aluminum [30–33],materials transitionsuch metal indium-tin-oxide metal nitrides [22,34,35],[29], and aluminum nickel [36]. [30–33], The use oftransition these materials nitrides nickel [36]. The use these materials makes it[22,34,35], possible toand tailor a wavelength of ofcollective lattice makes within it possible to spectral tailor a range, wavelength of collective modes a wide from UV [31] to IRlattice [35], modes within a wide spectralactivity range, [37,38] from UV [31] to IR or enable a magneto-optical that paves the[35], way or magneto-optical activity [37,38]applications. that paves the way forenable a rich avariety of novel and promising for In a rich variety of novel and promising applications. this context, dielectric NPs with both electric and magIndipole this context, dielectric NPs both electric magnetic (MD) resonances [39]with represent a case ofand specific netic dipole (MD) resonances [39] represent a case of specific 0740-3224/19/070E21-09 Journal © 2019 Optical Society of America 0740-3224/19/070E21-09 Journal © 2019 Optical Society of America

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The electric di and magnetic mi dipoles induced on each NP can be found from the solution of Eq. (1). In this work, we describe the optical response of the arrays of NPs with the extinction efficiency � N � X 4k μ0 0� 0� Im · E � m · H d , (8) Qe � i i i j I 0 N hRi2 ε0 i�1

nanospheres arranged in an ordered 2D lattice; and (iii) quasiordered 2D arrays of Si nanospheres of the same size. A comprehensive analysis of these scenarios reveals different impact of disorder on ED and MD coupling with lattice modes. The paper is organized as follows. In Section 2, we provide a theoretical background for the coupled dipole approximation; next, in Section 3 we discuss general features of ED and MD coupling in ordered lattices; then, the impact of positional and size disorder on optical properties of 2D lattices of Si NPs, as well as their quasi-random modifications, are discussed in Section 4; Finally, we draw general conclusions in Section 5.

where I 0 is the intensity of the incident field, and the asterisk denotes a complex conjugate. Note that in the general case of polydisperse array with R i ≠ R, the mean radius hRi � P N i�1 R i ∕N is used to define Q e . The coupled dipole approximation quite accurately describes optical properties of arrays from relatively small Si NPs. Full-wave simulations [82] show that ED and MD are predominant in arrays of Si NPs with R � 65 nm, and high-order electric and magnetic field oscillations can be ignored in this case, though higher-order multipoles in all-dielectric NPs are pronounced, for example, in large [83–85] and nonspherical [86] single Si NPs, or closely packed arrays of Si NPs [40].

2. COUPLED DIPOLE APPROXIMATION Consider an array of N spherical NPs embedded in vacuum. Under the incident plane-wave illumination with electric E0 and magnetic H0 components, the ith particle located at ri acquires electric di and magnetic mi dipole moments that are coupled to other dipoles and to an external electromagnetic field via the coupled dipole equations [39,79,80]: ! rffiffiffiffiffi X N X μ0 N ˆ Gˆ ij dj − C ij mj , (1a) di � αei E0i � ε0 j≠i j≠i H0i � mi � αm i

N X j≠i

! rffiffiffiffiffi X ε0 N ˆ Gˆ ij mj � C ij dj , μ0 j≠i

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3. ORDERED ARRAYS We start with the optical properties of a single Si nanosphere. Though it has been widely discussed in the literature, for instance, in Ref. [39], we plot these results for the reader’s convenience. Figure 1(a) shows the refractive index of Si used in calculations [87], while Fig. 1(b) shows the extinction efficiency Q e for a single Si nanosphere of various radii R. For a single sphere, and only in this case, we calculate Q e , taking into account high-order harmonics [81] required for the convergence of the electromagnetic light scattering problem [88]. It can be seen from Fig. 1(b) that indeed, for given sizes, Si nanospheres have distinct and predominant ED and MD resonances in the visible wavelength range. In what follows, we consider arrays from Si nanospheres with R � 65 nm radius. However, in the special case of a size disorder, all possible radii of NPs will fall into the range shown in Fig. 1(b), i.e., 50 nm ≤ R i ≤ 80 nm. Therefore, the coupled dipole approximation can be used with strong confidence. Next, it is insightful to discuss optical properties of ordered Si nanostructures. Figures 2(a) and 2(b) show two different types of lattices that have been studied in this work: (i) with fixed period along the x axis, hx , and varying period along

(1b)

where αei and αm i are ED and MD polarizabilities of the ith particle, respectively, ε0 and μ0 are the dielectric constant and magnetic permeability of vacuum, E0i � E0 �ri �, H0i � H0 �ri �, and � � rij ⊗ rij rij (2) , Cˆ ij � Dij × , Gˆ ij � Aij Iˆ � B ij r 2ij r ij where Iˆ is a 3 × 3 unit tensor, ⊗ denotes a tensor product, and Aij , B ij , and Dij are defined as follows: � � exp�ikr ij � 2 1 ik Aij � k − 2� , (3) r ij r ij r ij � � exp�ikr ij � 3 3ik −k2 � 2 − , (4) B ij � r ij r ij r ij � � exp�ikr ij � 2 ik k � , (5) Dij � r ij r ij

where r ij � jrij j � jri − rj j is the center-to-center distance between the ith and jth particles, k � 2π∕λ is a wavenumber, and λ is a wavelength of external illumination. ED and MD polarizabilities are explicitly defined as [81] αei �

3i nψ 1 �nkR i �ψ 10 �kR i � − ψ 1 �kR i �ψ 10 �nkR i � , 2k3 nψ 1 �nkR i �ξ10 �kR i � − ξ1 �kR i �ψ 10 �nkR i �

(6)

αm i �

3i ψ 1 �nkR i �ψ 10 �kR i � − nψ 1 �kR i �ψ 10 �nkR i � , 2k3 ψ 1 �nkR i �ξ10 �kR i � − nξ1 �kR i �ψ 10 �nkR i �

(7)

where n is the refractive index of the NP material, R i is the radius of the ith particle, ψ 1 �x� and ξ1 �x� are Riccati–Bessel functions, and prime denotes the derivation with respect to the argument in parentheses.

Fig. 1. (a) Refractive index n of Si from Ref. [87]; (b) extinction spectra for a single Si NP of various radii R, taking into account high-order multipoles. Spectral positions of ED and MD resonances are denoted as “ed” and “md,” respectively.

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while the second type of disorder affects only αe,m i , which are the functions of NPs’ shape and size. As was shown for ordered arrays of NPs in Fig. 2, two types of coupling can be distinguished. For fixed illumination, optical response of lattices strongly depends on variations of either hx and hy . Thus, to get more insight, we introduce the offdiagonal disorder in the following manner. We study positional disorder along the x axis, keeping y coordinates constant, and vice versa, as shown in Figs. 3(a) and 3(b), respectively. We will refer to these two types of positional disorder as x disorder and y disorder, correspondingly. For both cases, we introduce deviations σ x,y , which characterize a degree of disorder. For each ith particle with initial x i , y i  coordinates, we randomly set new dis coordinates as x dis i , y i  for x disorder and x i , y i  for y disorder within the following limits: x i − σ x ≤ x dis i ≤ xi  σx ,

and

y i − σ y ≤ y dis i ≤ yi  σ y :

(9)

Both x dis and ydis are randomly generated using a uniform i i distribution for each ith NP and for each lattice with given hx , hy . Thus, the effects of positional disorder are uncorrelated. The schematics of the lattice with diagonal (size) disorder is shown in Fig. 3(c). In this case, we keep original coordinates of each NP, and randomly change the radius R i of each ith NP within the following limits using a uniform distribution:

Fig. 2. (a) and (b) Schematic representation, and (c) and (d) extinction spectra Q e of ordered 2D lattices from N  20 × 20 Si NPs with R  65 nm. Two configurations are considered: (left) fixed hx  540 nm and varying hy , and (right) fixed hy  450 nm and varying hx . Spectral positions of ED and MD resonances are denoted as “ed” and “md,” respectively. Dashed RAx and RA y lines denote Rayleigh anomalies λ  hx and λ  hy , correspondingly.

R i − σ R ≤ R dis i ≤ Ri  σR :

(10)

Again, as in the case of off-diagonal disorder, R dis i is introduced randomly for each NP and for each lattice configuration, which provides uncorrelated results. Finally, Fig. 3(d) shows a special combination of diagonal and off-diagonal disorders, which attracts specific interest [64,65]. It is a well-known fact that the coupling between a single NP resonance and lattice modes strongly depends on the number of NPs in the array [89,90]. However, periodic lattices

the y axis, hy , and (ii) with fixed hy and varying hx . Such variations of interparticle distances make it possible to get ED or MD coupling with lattice modes [55]. In both cases, the incident electric E0 and magnetic H0 fields are aligned along the x and y axes, correspondingly. Lattices from N  20 × 20 Si NPs have been considered. In the first case, as is clearly seen from Fig. 2(c), ED strongly couples to lattice modes, which leads to the emergence of quite sharp collective lattice resonances. The position of the MD resonance slightly shifts to shorter wavelengths for large hy . Note that Q e for MD increases near the Rayleigh anomaly λ  hy. In the second case, according to Fig. 2(d), the same strong coupling with lattice modes occurs for MD, while the position of ED gradually shifts to shorter wavelengths, and the corresponding Q e decreases with increasing hx . Thus, the coupling occurs for the incident field (electric or magnetic) perpendicular to the axis along which the interparticle distance is changed. In other words, for the particular case considered in this work, EDs (E0 is parallel to x axis) couple to RA y , and, vice versa, MDs (H0 is parallel to y axis) couple to RA x . 4. DISORDERED ARRAYS A. Types of Disorder

According to Eq. (1), two types of disorder can be distinguished [69]: (i) off-diagonal and (ii) diagonal. These types affect either off-diagonal or diagonal elements of the interaction matrix in Eq. (1), respectively. The first type of disorder affects only tensors Gˆ ij and Cˆ ij , which are the functions of the NPs’ positions,

Fig. 3. Schematic representation of different types of disorder considered in this work: (a) x disorder, (b) y disorder, (c) size disorder, and (d) quasi-random array.

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of strictly arranged NPs are usually considered to study this finite-sized effect. In our work, we fix the initial coordinates and the sizes of NPs, and randomly remove NPs from the lattice, keeping other NPs untouched. This type of imperfection is somewhat similar to vacancies in crystal structures. In what follows, we will refer to lattices shown in Fig. 3(d) as quasi-random arrays. We emphasize that each lattice configuration for each type of disorder with given σ x , σ y , and σ R , or number of NPs removed from the lattice in the case of quasi-random arrays, has been simulated only once, without computing ensemble averages. A reasonable closeness to statistical average has been granted by simulating a large enough number of NPs. B. Off-Diagonal (Positional) Disorder

Figures 4 and 5 show extinction spectra for arrays of NPs with different degrees of x and y disorders. It can be seen that these two types of positional disorder affect the optical properties of NPs in a different way, depending on the coupling regime. As might be expected from the analysis of Fig. 2(d), the x disorder significantly affects MD, since the latter strongly couples to the Rayleigh anomaly RA x. Clearly, from Fig. 4, one may observe a slight suppression of the MD with the increasing of the degree of disorder, σ x , both for ED and MD coupling scenarios. It also has to be noted that the coupling of MD and RA x remains observable even for sufficiently large σ x in Fig. 4(f ), where MD is suppressed. The ED remains almost the same for each case shown in Fig. 4.

Fig. 5. Same as in Fig. 4, but for various degrees of positional disorder σ y along the y axis, as shown in Fig. 3(b).

Figure 5 shows an expected trend: since ED couples to RA y , the y disorder affects only the former, keeping MD almost the same for various σ y . However, Figs. 5(e) and 5(f ) show almost total suppression of ED for σ y  150 nm, while in the case of strong x disorder shown in Figs. 4(e) and 4(f ), MD is quite pronounced. Finally, Fig. 6 shows a detailed comparison of the extinction spectra for arrays with ED or MD couplings. Indeed, the x disorder strongly suppresses the MD, while the y disorder suppresses the ED resonance. Since the ED is generally weaker than the MD, the former almost completely disappears for high degrees of y disorder. For completeness, Figs. 6(g) and 6(h) show spectra for arrays with xy disorder, which has been introduced in the same way as the x and y disorders, but with simultaneous randomization of both x i and y i coordinates of each NP. It can be seen that, in general, such a combined disorder yields a superposition of both x and y disorders suppresses both ED and MD resonances. C. Diagonal (Size) Disorder

Figure 7 shows extinction spectra for arrays with various degrees of size disorder, σ R . It is clearly seen that random variations of NP sizes strongly suppress both ED and MD resonances. However, MD remains observable only for σ R  5 nm, while for larger σ R, it almost completely disappears. Contrarily, the ED resonance is preserved in all cases, and, of note, EDs strongly couple with Rayleigh anomalies,

Fig. 4. Extinction spectra Q e for the same 2D lattices as in Figs. 2(c) and 2(d), but for various degrees of positional disorder σ x along the x axis, as shown in Fig. 3(a).

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To get a deeper insight, we plot Q e for arrays with fixed hx and hy , as shown in Fig. 8. Indeed, Figs. 8(c), 8(e), and 8(g) show that size disorder has a surprisingly weak effect on the ED resonance of arrays with strong ED coupling. It can be seen from Fig. 8(g) that maximum Q e for the ED resonance drops by no more than 10% for σ R  15 nm compared to the ordered array shown in Fig. 8(a). For arrays with MD coupling, Q e for ED resonance drops more strongly, by a factor of 2 for σ R  15 nm, as shown in Fig. 8(h). As for the MD resonance, in both the ED and MD coupling cases, the extinction efficiency for MD sharply drops for σ R  5 nm. For larger σ R, the MD resonance becomes almost indistinguishable. D. Quasi-Random Arrays

From the previous discussion of diagonal and off-diagonal types of disorder, we can conclude that a simultaneous implementation of positional and size disorders should likely result in the superposition of the effects shown in Figs. 4, 5, and 7. Thus, we do not consider arrays of randomly located NPs of different size. Instead, we introduce a specific combination of positional and size disorders, as shown in Fig. 3(d). These quasi-random arrays are fundamentally different from those shown in Figs. 3(a)–3(c), since random elements of the interaction matrix in Eq. (1) are strictly set to zero in the case of quasi-random arrays, while in previously considered scenarios, off-diagonal or diagonal elements have acquired random deviations according to σ x , σ y , or σ R. Here, we consider NPs with the same size, R  65 nm, but increase their number to N  30 × 30 (while previously discussed arrays had N  20 × 20 NPs). Next, we randomly remove 171, 459, or 756 NPs, leaving the rest 81%, 49%, or 16% of NPs untouched, respectively. We note that the consideration of larger arrays is preferable for this type of disorder, since coupling effects may be totally suppressed in arrays from the small number of NPs left in the lattice [89]. However, in the smallest array considered here, we keep 144 quasi-randomly

Fig. 6. Extinction spectra Q e of NPs arrays with ED coupling (left), and MD coupling (right) for various degrees of positional disorder (c) and (d) σ x , (e) and (f) σ y , and (g) and (h) σ xy . Corresponding values of hx and hy are shown in legends. Dashed vertical lines denote positions of Rayleigh anomalies RAy at λ  500 nm (left), and RAx at λ  540 nm (right).

RA y , even for high degrees of diagonal disorder, as shown in Fig. 7(e). This effect might be explained by the different behavior of polarizabilities αei and αm i [39], which yield a different impact of size disorder on ED and MD resonances.

Fig. 8. Extinction spectra Q e of NP arrays with ED coupling (left), and MD coupling (right) for various degrees of size disorder σ R. Corresponding values of hx and hy are shown in legends. Dashed vertical lines denote positions of Rayleigh anomalies RAy at λ  500 nm (left), and RAx at λ  540 nm (right).

Fig. 7. Extinction spectra Q e for the same 2D lattices as in Figs. 2(c) and 2(d), but for various degrees of size disorder σ R , as shown in Fig. 3(c).

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located NPs, which is sufficient for the emergence of coupling effects. Intuitively, one could expect the suppression of ED and MD couplings with the increasing number of NPs removed from the ordered array. Indeed, Fig. 9 confirms such an expectation. However, it can be seen that lattices that contain 81% of the initial NPs have almost the same optical properties as the original periodic arrays. Moreover, Figs. 9(e)–9(f ) show that ED and MD are coupled to Rayleigh anomalies (though quite weakly) in the arrays with only 16% NPs left, and extinction spectra of such arrays tend to become closer to Q e of a single NP. For comparison, Fig. 10 shows spectra of ordered arrays [as in Figs. 2(a) and 2(b)] from exactly the same number of NPs as in quasi-random arrays, i.e., 27 × 27, 21 × 21, and 12 × 12, and with the same hx and hy . It can be seen from Figs. 10(c) and 10(d) that Q e of the quasi-random array from 729 NPs is also almost the same as Q e for the periodic 27 × 27 array. Moreover, even with the increasing number of NPs removed from the array, Q e of the quasi-random lattices is quite close to that for the strictly ordered arrays with the same number of NPs. However, in the most extreme cases of quasi-random arrays shown in Figs. 10(g) and 10(h), the collective ED resonances are almost suppressed, while the MD coupling remains observable, although the corresponding peak of MD resonance is blueshifted compared with the ordered arrays.

Research Article

Fig. 10. Extinction spectra Q e of NPs arrays with ED coupling (left), and MD coupling (right) for (a) and (b) N  30 × 30 array, and for its various quasi-random modifications (solid lines); (c) and (d) 81%  729, (e) and (f) 49%  441; and (g) and (h) 16%  144 (NPs kept untouched). For comparison, Q e of strictly periodic (dashed lines) arrays of the same number of NPs are shown; (c) and (d) N  27 × 27  729; (e) and (f ) N  21 × 21  441; and (g) and (h) N  12 × 12  144; gray dashed-dotted lines show Q e of a single Si NP with R  65 nm. Corresponding values of hx and hy are shown in legends. Dashed vertical lines denote positions of Rayleigh anomalies RA y at λ  500 nm (left), and RA x at λ  540 nm (right).

5. CONCLUSION We have theoretically analyzed the impact of various types of imperfections on the optical response of 2D arrays of spherical Si NPs. ED and MD resonances are dominant in Si nanospheres in the considered range 50 nm ≤ R ≤ 80 nm; thus, we have used the coupled dipole approximation, which adequately describes electromagnetic properties of Si NP arrays [82]. We first have shown the existence of two types of collective resonances in 2D arrays emerging from the strong coupling of either ED or MD resonances of a single NP with lattice modes (Rayleigh anomalies) of the 2D array. Such a coupling occurs when the corresponding component of the incident field (electric or magnetic) is orthogonal to the varied period (hy or hx ) of the lattice, while the other period (hx or hy ) is constant [55]. Second, we have shown that electric or magnetic responses are affected by the positional disorder only when NPs are shifted along the axis that is orthogonal to the corresponding component of incident electromagnetic illumination. In our case, for E0 ∥x and H0 ∥y, ED and MD resonances are strongly suppressed only for y or x disorders, respectively. Obviously, both resonances are affected when NPs are shifted along the x and y axes simultaneously. Next, we have demonstrated that the collective MD response almost completely vanishes in the case of diagonal (size) disorder with σ R > 5 nm. However, the electric counterpart remains quite stable, especially in the case of strong collective coupling between the ED resonance and lattice modes, even for highly polydisperse arrays with σ R  15 nm.

Fig. 9. Extinction spectra Q e for quasi-random 2D lattices, as shown in Fig. 3(d), for different numbers of NPs. (a) and (b) 81%  729, (c) and (d) 49%  441, and (e) and (f ) 16%  144 kept untouched in N  30 × 30 arrays of NPs with R  65 nm. Note the different color scale in the last row, (e) and (f ).

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Finally, we have considered quasi-random arrays as a special combination of off-diagonal and diagonal disorders. Instead of simultaneously shifting the NPs and changing their sizes, we have randomly removed NPs from the lattice, keeping other NPs at original points with original sizes. Surprisingly, arrays with only 16% of NPs left in the lattice exhibit both electric and magnetic collective resonances. However, the extinction spectra of such arrays tend to be similar to spectra of a single NP. The reported results provide a comprehensive analysis and a fundamental understanding of the impact that disorder has on collective resonances in 2D arrays of all-dielectric NPs. While we have considered spherical Si NPs embedded in vacuum, one could expect similar trends for all-dielectric arrays of NPs of other shapes or materials [91] as long as high-order multipoles can be neglected. Thus, we believe that the reported results may pave the way for future applications in all-dielectric nanophotonics. Funding. Russian Foundation for Basic Research (RFBR) (18-42-240013); Russian Science Foundation (RSF) (18-1300363); Siberian Federal University (SibFU) (3.8896.2017). Acknowledgment. The reported study was funded by Russian Foundation for Basic Research, Government of Krasnoyarsk Territory, Krasnoyarsk Regional Fund of Science, the research project No 18-42-240013; the State contract with Siberian Federal University for scientific research in 2017–2019 (Grant No.3.8896.2017). H. A. and V. Z. acknowledge the support of the Russian Science Foundation (Project No.18-13-00363) (numerical calculations of spectral properties of planar dielectric nanostructures). REFERENCES 1. S. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. 121, 12606–12612 (2004). 2. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. 120, 10871–10875 (2004). 3. V. A. Markel, “Divergence of dipole sums and the nature of nonLorentzian exponentially narrow resonances in one-dimensional periodic arrays of nanospheres,” J. Phys. B 38, L115–L121 (2005). 4. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. 101, 143902 (2008). 5. Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008). 6. V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101, 087403 (2008). 7. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013). 8. W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, “Lasing action in strongly coupled plasmonic nanocavity arrays,” Nat. Nanotechnol. 8, 506–511 (2013). 9. M. Dridi and G. C. Schatz, “Model for describing plasmon-enhanced lasers that combines rate equations with finite-difference time-domain,” J. Opt. Soc. Am. B 30, 2791–2797 (2013). 10. A. H. Schokker and A. F. Koenderink, “Lasing at the band edges of plasmonic lattices,” Phys. Rev. B 90, 155452 (2014).

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PAPER IV

Engineering novel tunable optical high-Q nanoparticle array filters for a wide range of wavelengths A. D. Utyushev, I. L. Isaev, V. S. Gerasimov, A. E. Ershov, V. I. Zakomirnyi, L. Rasskazov, S. P. Polyutov, H. Agren, S. V. Karpov submitted in Optics Express (2019).

Paper IV

Engineering novel tunable optical high-Q nanoparticle array filters for a wide range of wavelengths A. D. U TYUSHEV, 1,2 I. L. I SAEV, 3 V. S. G ERASIMOV, 3,1,4 A. E. E RSHOV, 3,1,2,4 V. I. Z AKOMIRNYI , 1,4,5,7 I. L. R ASSKAZOV, 6 S. P. P OLYUTOV 1,4,7 H. ÅGREN 4,5 S. V. K ARPOV, 7,1,2,4,* 1 Siberian

Federal University, Krasnoyarsk, 660041, Russia State University of Science and Technology, 660014, Krasnoyarsk, Russia 3 Institute of Computational Modeling, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia 4 Federal Siberian Research Clinical Center under FMBA of Russia, Krasnoyarsk, 660037, Russia 5 Division of Theoretical Chemistry and Biology, Royal Institute of Technology, SE-100 44, Stockholm, Sweden 6 The Institute of Optics, University of Rochester, Rochester, NY 14627, USA 7 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia 2 Siberian

* [email protected]

Abstract: The interaction of non-monochromatic radiation with arrays comprising plasmonic and dielectric nanoparticles has been studied using the Finite-Difference Time-Domain electrodynamics method. It is shown that LiNbO3 , TiO2 , GaAs, Si, and Ge all-dielectric nanoparticle arrays can provide a complete selective reflection of an incident plane wave within a narrow spectral line of collective lattice resonance with a Q-factor of 103 or larger at various spectral ranges, while plasmonic refractory TiN and chemically stable Au nanoparticle arrays provide high-Q resonances with moderate reflectivity. Arrays with fixed dimensional parameters make it possible to fine tune the position of a selected resonant spectral line by tilting the array relative to the direction of the incident radiation. These effects provide grounds for engineering novel selective tunable optical high-Q filters in a wide range of wavelengths, from visible to middle IR. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1.

Introduction

Design and fabrication of new compact optical elements with high-Q response in the visible, near infrared (IR), and middle IR wavelength ranges is a research topic in applied optics with high priority and wide ramifications. Much attention has been focused on devices in the form of periodic one-dimensional (1D) or two-dimensional (2D) arrays composed of plasmonic or all-dielectric nanoparticles (NPs) with Mie resonance. New ideas underlying such devices stem from the effect first predicted in theoretical studies of regular plasmonic structures by Schatz and Markel [1–3]. According to their predictions, periodic arrays of NPs with strong electromagnetic coupling are capable to support high-Q collective lattice resonances (CLRs) with the Fano type profile in extinction spectra due to the interference of fields from individual particles and the Wood-Rayleigh anomaly [4, 5]. In the general case, the lattice resonance arises in periodic arrays in which the phase of the external field of a plane wave in the vicinity of an individual array element coincides with the phase of the field produced by neighboring elements. If such coherence takes place within the entire array at a given wavelength, a resonance excitation occurs at this wavelength. Thus, resonance excitation is produced by the hybrid coupling of localized low Q-factor resonances of NPs and their non-localized interactions covering the entire array. The position and Q-factor of these resonances depend on the geometry of the array lattice, the material composition and the shape of the NPs [6]. Under particular conditions, the Q-factor of

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Vadim Zakomirnyi such resonances can exceed the Q-factor of a single NP by 102 –103 times. CLRs in periodic arrays of plasmonic [7–14] and all-dielectric [15–22] NPs have been extensively discussed during the recent decade owing to the great number of potential applications in color printing [23–26], biosensing [27–30], lasing [31, 32], fluorescence enhancement [33] and other applications [6, 34]. Light transmission through 2D subwavelength hole arrays in perfect-conductor films and the reflection on same array of nonabsorbing scatterers was shown to be complete at same wavelength of CLR in the dipole approximation in Refs. [35, 36]. In Refs. [37–40], CLRs were observed in the arrays of Si NPs with Q-factor values from 20 to several hundreds. Note that in all-dielectric systems mainly silicon is used for fabricating nanoparticle arrays. In this regard, the relevance of the analysis of materials suitable for fabricating such arrays is obvious, since a certain material with its optical constants is best suited for each spectral range. In addition, an analysis and optimization of the dimensional characteristics of arrays are needed for a significant increase of the CLR Q-factors. The goal of our paper is to verify the relevance of the concept of CLRs for solving applied problems and the advantages they can offer. Within the frame of this goal we propose the design of a device that makes it possible to select radiation in the reflection mode from the spectral continuum within a tunable ultra-narrow spectral line and to control its position with a high Q-factor. Data on the optimal structure of the device – particle size and shape – lattice period and a particle material are obtained to achieve this goal. 2.

Methods

We consider 2D arrays of nanodisks (NDs) with height H and radius R arranged in a regular square lattice with period h, as shown in Fig. 1(a). The arrays are embedded in a homogeneous environment with refractive index nm = 1.45, which corresponds to quartz in the spectral range under study. Such structures can be fabricated using a lithography technique on a quartz substrate and subsequent sputtering a layer of quartz on top of the array. A homogeneous environment is an important factor in the model, because the Q-factor of CLR drops dramatically in the case of the half-space geometry, where the substrate and the superstrate have different refractive indices [41]. The reflection spectra of such structures are calculated with a commercial Finite-Difference Time-Domain (FDTD) method software [42]. FDTD is a widely used computational method of electrodynamics, which in general shows excellent agreement with experimental results for CLRs [8, 13, 40, 43]. The optical response of the infinite array is simulated by considering a single particle unit cell with periodic boundary conditions applied at the lateral boundaries of the simulation box and perfectly matched layers (PML) used at the remaining top and bottom sides, as shown in Fig. 1(b). Arrays are illuminated from the top by plane waves with normal incidence along the Z axis and polarization along the Y axis. The reflection has been calculated at the top of the simulation box using a discrete Fourier transform monitor which is placed above the plane-wave source. The angular dependencies were obtained using the broadband fixed angle source technique [44]. An adaptive mesh has been used to accurately reproduce the nanodisk shape. Finally, extensive convergence tests for each set of parameters have been performed to avoid undesired reflections on the PMLs. We study arrays of both plasmonic (TiN and Au) and all-dielectric (LiNbO3 , TiO2 , Si, GaAs, Ge) nanoparticles. Figure 2 shows tabulated experimental data for the real and imaginary parts of the complex refractive index n which have been used for each material. 3.

Results

The lattice period h varies in accordance with Rayleigh anomalies (±1, 0) and (0, ±1), the positions of which for the case of normal incidence and homogeneous environment with refractive index nm are defined as

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Paper IV (a)

(b)

nm=1.45

NP

periodic BC

periodic BC

PML reflectance monitor plane wave source

PML

Fig. 1. (a) Sketch of the NDs array under consideration; (b) FDTD simulation setup.

Fig. 2. Real and imaginary parts of complex refractive index n for (a) and (b) plasmonic TiN [45] and Au [46]; (c) and (d) all-dielectric LiNbO3 [47] (here we consider only ordinary refractive index), TiO2 [48], GaAs [49], Si [50] and Ge [51].

 λ p,q = hnm / p2 + q2 ,

(1)

where λ is the vacuum wavelength, p and q are integers which represent the phase difference (in 2π units) between waves scattered by two adjacent elements of the array and incident wave in the x and y directions. Eq. (1) describes the condition of constructive interference for particles within the XOY plane [52]. Before discussing CLRs in NP arrays we note that the shape of the NPs is an important parameter that affects the Q-factor of the CLRs. In the calculations we examined two types of the NP shapes: nanodisks and nanoparallelepipeds. Both shapes can be simply experimentally fabricated [38, 40]. We found that nanodisks demonstrate slightly higher value of Q-factors compared to nanoparallelepipeds, so that the further studies were carried out only with nanodisks. 3.1.

Reflection spectra of plasmonic nanoparticle arrays

Plasmonic nanoparticle arrays were the first type of structures used for observation of CLRs with the Fano-type profile [1–3]. Au is a widely used material in these arrays for which the CLRs are observed in the red range of the visible spectrum [7–9]. The use of TiN ND arrays (Fig 3) provides moderate reflectivity with a high Q-factor of CLRs in the telecommunication spectral range. The optimal TiN ND radius is 90 nm. A larger particle size results in a decrease of the reflectivity and Q-factor.

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Fig. 3. Reflection spectra for TiN (left) and Au (right) ND arrays with: (a) fixed h = 1100 nm, and for different R = H as shown in legend; (b) fixed h = 650 nm, and for different R = H as shown in legend; (c) and (d) corresponding quality factors of CLRs; (e) fixed R = H = 90 nm and for different h as shown in legend; (f) fixed R = H = 55 nm and for different h as shown in legend; (g) and (h) corresponding quality factors of CLRs.

The suppression of surface plasmon resonances under extreme conditions was studied in the papers [53–55]. It was shown that heating of nanoparticles by pulsed laser radiation results in a reduction of the Q-factor and CLR amplitude. In particular, for the CLR at λ = 1100 nm in Fig 3, the Q-factor is 1.5 · 103 at T = 23◦ C, Q = 1.1 · 103 at T = 400◦ C, and Q = 0.7 · 103 at T = 900◦ C. Thus, the high radiation resistance of TiN can be an additional advantage when using arrays exhibiting CLRs at high temperatures [14]. The use of TiN as a plasmonic material with high radiation resistance provides an extreme stability at high temperatures compared to conventional plasmonic materials (Au and Ag). Au ND arrays (Fig. 3) demonstrate CLRs in the long-wavelength part of the visible and near IR ranges. Reflectivity and Q-factor of the Au ND array are somewhat higher compared to TiN. 3.2. 3.2.1.

Dielectric nanoparticle arrays Reflection spectra

In the case of dielectric nanoparticle structures the main intrinsic property which makes them attractive refers to the combination of high real part of the refractive index Re(n) and a low imaginary part Im(n) for different spectral ranges which ensures low absorption inside the

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Paper IV particles. Figures 2(c),(d) show, in particular, the refractive indices of the following suitable lossless materials satisfying these requirements: LiNbO3 , TiO2 , Si, GaAs, Ge. In this paper we have selected materials for each spectral range to obtain high-Q CLR in each given case. Two materials were chosen for the visible spectral range: LiNbO3 and TiO2 . First of all, we begin by discussing the general features regarding CLRs in arrays of dielectric nanoparticles. We should note that in arrays with large dielectric particles with a size of tens of nanometers and above, the conditions arise for the appearance of a magnetic dipole resonance along with an electric dipole excitation. The geometric parameters of the array such as period, size and shape of nanoparticles significantly affect the position of electric and magnetic dipole resonances and enable to suppress the magnetic dipole resonance as well as to prevent the appearance of several close lines in the spectrum instead of a single line which is unacceptable for selective narrow-band filters in the tuning spectral range.

Fig. 4. (a) Reflection spectra of the LiNbO3 ND arrays with R = H = 60 nm for different lattice period h (shown in legend), (b), (e) configurations of the electric and magnetic fields at λ = 365 nm (the period h = 250 nm); (c), (f) configurations for λ = 370 nm (the period h = 250 nm); (d), (g) configurations for period h = 325 nm at λ = 474 nm.

Studies of the radiation reflection from NP arrays call for a clarification of what type of resonance excitation is associated with reflection and what is its origin. Fig. 4(a) shows two overlapping spectral lines in the reflection spectrum of the LiNbO3 ND array associated with the excitation of both electric dipole and magnetic dipole CLRs. Different values of the lattice

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Fig. 5. Dependencies of the reflection spectra of the LiNbO3 ND array with period h = 400 nm on a ND aspect ratio: (a) variation of the ND height H at R = 60 nm, (b) variation of the ND radius R at H = 60 nm.

periods used in the calculations make it possible to detect the appearance of a magnetic dipole resonance in the reflection spectrum separately from the electric dipole excitation band. For instance, Figs. 4(b)–(g) show the configuration of electric and magnetic fields in orthogonal planes inside a unit cell for the wavelength λ = 365 nm (b,e), for λ = 370 nm (c,f) and for λ = 474 nm (d,g). We can see that at λ = 365 nm, the field configuration corresponds to the magnetic dipole along X axis while at λ = 370 nm and λ = 474 nm it is typical for electric dipole along Y axis (see also [38, 39, 56]). Calculations for all subsequent spectra in the paper were also preceded by an analysis of CLRs to confirm its electric dipole nature. The effect of the aspect ratio (R/H) of the NDs on the obtained dependencies is of particular interest. We look for maximum values of the reflection coefficient in combination with high Q-factors of the CLRs provided by an optimal value of the particle aspect ratio R/H which in our case is close to 1. The results in Fig. 5 demonstrate that for a fixed value of the particle radius R = 60 nm and a decrease in R/H < 1 due to the growing particle height H, the Q-factor starts to decrease while the reflection coefficient remains equal to 1. This is accompanied by a red CLR shift. When R/H > 1 by reducing the height, the Q-factor is growing, but at the same time the reflection coefficient decreases. In this case we observe a blue CLR shift. At a fixed height H = 60 nm, an increase in the radius in aspect ratio R/H > 1 results in a drop in the Q-factor and a red shift of CLR while maintaining the reflection coefficient equal to 1. A decrease in the radius in R/H < 1 is accompanied by an increase in the Q-factor and a simultaneous fall in the reflection coefficient with a blue CLR shift. Figures 6(a)-(d) with reflection spectra of LiNbO3 and TiO2 ND arrays show that the larger the particle size, the lower the Q-factor, but at the same time the higher the reflection coefficient. So the optimal combination of these factor gives a particle radius of 60 nm with a Q-factor equal to 103 . Figures 6(c),(d),(g),(h) show that employing these materials in ND arrays provides ultra-narrowband resonances in the entire visible range with a Q-factor over 103 and high reflection. The utilization of TiO2 in ND arrays also provides high reflectivity and Q-factor, however, with smaller size of the particle – 50 nm compared to LiNbO3 that results in narrowing the spectral range with high reflectivity (Fig. 6). The next step is to vary the lattice period with the given optimal radius. Calculations show that the Q-factor of the CLR increases with wavelength. The best materials for the near-IR range are Si and GaAs. These materials in the IR range demonstrate the properties of dielectrics (Fig. 2) with a near-zero imaginary part of the refractive index and its high real part, which makes it possible to excite Mie resonances in such particles with radius below 100 nm – much smaller than the wavelength. Fig. 7 shows the reflection spectra of Si ND arrays with ND radii R = 100, 110, and 120 nm and radius/height ratio R/H = 1. Besides

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Fig. 6. Reflection spectra for LiNbO3 (left) and TiO2 (right) ND arrays with: (a) and (b) fixed h = 400 nm and different R = H as shown in legend; (c) and (d) corresponding quality factors of CLRs; (e) fixed R = H = 60 nm and for different h as shown in legend; (f) fixed R = H = 50 nm and for different h as shown in legend; (g) and (h) corresponding quality factors of CLRs.

that, Fig. 7 shows that CLRs with reflected radiation appear in the entire telecommunication wavelength range by varying the array period. In Ref. [40] authors experimentally studied CLRs in near IR range in the reflection mode in Si ND arrays with Q-factor equals 300 – 10 for ND aspect ratio 2.4 – 3.78 and H = 100 nm. As follows from Fig. 7 the ND parameters used in Ref. [40] could not provide higher Q-factor because of too large particle sizes and non-optimal aspect ratio R/H over 3 instead of 1. Figure 7 shows that Q-factor can be much higher for the parameters of arrays given in this figure. Reflection spectra for GaAs ND arrays have optimal characteristics in the range between visible and telecom wavelengths (Fig. 7). The optimal ND radius equals 70 nm and period 600–700 nm with an ultrahigh Q-factor. The use of the Ge ND arrays provides high reflectivity and Q-factor in the middle IR range with optimal ND radius 165 nm. A smaller size of the particle results in a decrease of reflectivity, larger ones are accompanied by lower Q-factors (Fig. 8). Besides, we note one more important feature of the CLRs in the studied arrays, namely that minor array defects, that may occur during their experimental fabrication, do not significantly affect the Q-factor of the CLRs [57].

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Fig. 7. Reflection spectra for Si (left) and GaAs (right) ND arrays with: (a) fixed h = 900 nm, and for different R = H as shown in legend, (b) fixed h = 700 nm, and for different R = H as shown in legend; (c) and (d) corresponding quality factors of CLRs; (e) fixed R = H = 100 nm and for different h as shown in legend; (f) fixed R = H = 80 nm and for different h as shown in legend; (g) and (h) corresponding quality factors of CLRs.

Fig. 8. Reflection spectra for Ge ND arrays with: (a) fixed h = 1600 nm, and for different R = H as shown in legend, (b) fixed R = H = 165 nm and for different h as shown in legend; (c) and (d) corresponding quality factors of CLRs.

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Fig. 9. Reflection spectra for (a) TiO2 and (b) Si ND arrays with different sizes but with fixed R/H and h/R ratios. 3.2.2.

Scale invariance of collective lattice resonances

Arrays of all-dielectric NPs demonstrate an important feature in conditions of utilizing materials with zero dispersion: the spectral position of CLR in NP arrays as all-dielectric systems can be predicted by multiplying all dimensional parameters of the array (particle radius, height and lattice period) by the same number K. The new resonance position will correspond to Kλ (where λ is the previous wavelength value, see Fig. 9). This feature is a consequence of the scale-invariance of Maxwell’s equations in the case of non-absorbing and non-dispersive materials. This is the easiest way to predict reflection at a specific wavelength. So if we determine the optimal parameters of the structure (with maximum reflectivity and the CLR Q-factor) by applying the multiplier K to all lattice parameters we can predict the resonance position at any required wavelength. Figs. 9(a),(b) show a twofold (for TiO2 ND array) and 1.5 (for Si ND array) increase in the parameters of the arrays with corresponding shift of the resonance lines, which demonstrates the scale invariance of CLRs. Scale invariance allows to design and to fabricate optical filters for operation in an arbitrary spectral range from near to far IR. 3.3.

Fine tuning the resonance line using the angular dependence of reflection

The possibility of employing NP arrays for spectral selection of radiation with desired wavelengths in the reflection mode requires deflection of the selected monochromatic radiation away from the incidence direction of the non-monochromatic radiation by tilting the array at least a few degrees. The results of an investigation of this possibility are shown in Fig. 10. It was found that the slope of the array in the range of α = 1◦ − 4◦ is accompanied by a slight decrease in Q-factor of resonance and reflection coefficient. However, the more important effect of this tilting is the shift of the resonance line to the short-wavelength range by about 3 nm when tilting in one plane, and to the long-wavelength range by over 100 nm in another plane. Obviously, the found property of arrays makes it possible to use them for spectral selection and to fine-tune the spectral position of the resonance line to the required wavelength. Our calculations indicate that at larger angles (α > 4◦ ), the resonance shift continues to grow. However, with the increase in the angle up to α > 20◦ , additional resonances with growing amplitude start to emerge at the short-wavelength wing of the CLR, which along with a decrease of the CLR Q-factor impair the selectivity of filtration. To prevent such effects, one should limit the angle to 8–10◦ or less. Similar results have been obtained for other materials. Figs. 10(c),(d) demonstrate a slight decrease in the CLR Q-factor from Q = 3.6 · 103 for α = 0◦ to Q = 2.9 · 103 and Q = 103 for α = 4◦ for TM polarizations, and its increase for TE polarizations with subsequent decrease, respectively. Fig. 10 also demonstrates different effects of tilting the array around the Y axis for different polarization of incident radiation. Rotations of the array around the Y axis for TE polarization turn up to be more sensitive to positions of the CLRs than for TM polarization. The explanation of this feature is given considering that

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Fig. 10. (a) Sketch of the vector configuration for TM and TE polarization, (b) spectral positions of (+1,0) and (0,±1) Wood-Rayleigh anomalies as a functions of the incidence angle (solid lines) and corresponding positions of reflectance maxima (partially filled circles): The incidence angle dependence of the reflection spectra for Si ND array with h = 900 nm and R = H = 100 nm for (c) TM and (d) TE polarizations.

the Wood-Rayleigh anomaly in this case follows simple rules. The condition for constructive interference for the case of oblique incidence and square unit cell (the wave vector is in XOZ plane) reads as k x h = 2πp + k h sin α,

k y h = 2πq .

(2)

Here k x , k y are x and y components of the scattered wave vector, α is the angle between the wave vector of the incident wave and the Z axis, and k is the absolute value of the wave vector of the incident wave:  2πnm = k x2 + k y2 . (3) k= λ For the case of p = 0, which corresponds to the TM polarization, we have: λ=±

nm cos α h. q

(4)

For q = 0, which corresponds to the TE polarization: λ=

nm (1 ± sin α) h. ±p

(5)

Thus, the angular dependence of the spectral shift for the TM polarization obeys Eq. (4), while for TE polarization it is described by Eq. (5). These fast and slow dependencies are shown in Fig. 10(d) and do fully correspond to the spectral features in Fig. 10(a),(b). Effect of tilting 1D metal-dielectric grating on the position of the CLR spectral line with demonstration of its red shift as the angle of incidence increases was shown in particular in [58], whereas in [40], a blue shift is observed. Our Eqs. (2)-(5) explain this difference. Note that in 1D particle arrays (gratings) [58], the control of position of the extracted spectral line from spectral continuum in the reflection mode with adjustable accuracy by tilting the array in two planes is impossible compared to 2D arrays.

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Conclusion

Based on the results obtained in this work we can make the following conclusions. Periodic structures consisting of dielectric highly refractive nanoparticles with low absorption demonstrating collective lattice resonances can be used in the reflection mode as selective ultra-narrowband spectral filters. The position of the spectral lines can be adjusted using the lattice period. Arrays with nanoparticles of various shapes (nanodisks and nano-parallelepipeds) demonstrate similar optical properties and can be synthesized by different available experimental techniques. Dielectric nanoparticle arrays are preferable structures for lossless narrowband reflection compared to plasmonic ones with low refraction and significant absorption of highly conductive materials in the range of the collective lattice resonances. The geometric parameters of dielectric nanoparticle arrays such as period, size and shape of nanoparticles significantly affect the position of electric dipole resonance in a spectrum and makes it possible to suppress the magnetic dipole resonance as well as to prevent the appearance of several close lines or their splitting in the reflection spectrum instead of a single line that is unacceptable for selective narrow-band filters in the tuning spectral range. The use of arrays of plasmonic nanoparticles makes it possible to achieve a high-Q CLR response in conditions of strong NP heating (TiN), chemically aggressive media or biological environments (Au, TiN) in spite of a low value of the reflection coefficient. The results obtained indicate that the most suitable material can be determined by taking into account the optical characteristics as well as operating conditions. Scale invariance of dielectric NPs arrays makes it possible to design and fabricate filters for operation in arbitrary spectral ranges with low dispersion materials from near to far IR. The nanoparticle arrays in the reflection mode demonstrate an optical filtering effect with fine tuning of the spectral position of the resonance line to the required wavelength by means of tilting the array with respect to the incident radiation. The proposed model provides a quantitative interpretation of the angular dependence of the characteristics of a collective lattice resonance for different geometries of the radiation incidence onto the array. The findings presented in the this work lay the ground for the design and engineering of novel selective tunable optical high-Q filters in a wide range of wavelengths, all the way from the visible down to mid IR, which are useful for a multitude of applications within applied optics. Various common highly refractive dielectric materials can be used for selecting specific spectral ranges. The criteria for precision in the production of the filters are largely met by contemporary fabrication technology [40, 59], making it possible to validate our models and predictions for further fine tuning of the design for special application purposes. Funding Information The reported study was funded by the Russian Science Foundation (Project No.18-13-00363) (the reflection spectra of plasmonic NPs arrays); the RF Ministry of Science and Higher Education, the State contract with Siberian Federal University for scientific research in 2017–2019 (Grant No.3.8896.2017)(the reflection spectra of all-dielectric NPs arrays); A.E. thanks the grant of the President of Russian Federation (agreement 075-15-2019-676). Disclosures The authors declare no conflicts of interest.

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Express 24, 26851–26856 (2016). 54. A. E. Ershov, V. S. Gerasimov, A. P. Gavrilyuk, and S. V. Karpov, “Surface plasmon resonances in liquid metal nanoparticles,” Appl. Phys. B 123, 182 (2017). 55. V. S. Gerasimov, A. E. Ershov, S. V. Karpov, A. P. Gavrilyuk, V. I. Zakomirnyi, I. L. Rasskazov, H. Ågren, and S. P. Polyutov, “Thermal effects in systems of colloidal plasmonic nanoparticles in high-intensity pulsed laser fields [Invited],” Opt. Mater. Express 7, 555–568 (2017). 56. I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013). 57. V. I. Zakomirnyi, S. V. Karpov, H. Ågren, and I. L. Rasskazov, “Collective lattice resonances in disordered and quasi-random all-dielectric metasurfaces,” J. Opt. Soc. Am. B 36, E21–E29 (2019). 58. Z. Wang, R. Zhang, and J. Guo, “Quadrupole mode plasmon resonance enabled subwavelength metal-dielectric grating optical reflection filters,” Opt. Express 26, 496–504 (2018). 59. D. G. Baranov, D. A. Zuev, S. I. Lepeshov, O. V. Kotov, A. E. Krasnok, A. B. Evlyukhin, and B. N. Chichkov, “All-dielectric nanophotonics: the quest for better materials and fabrication techniques,” Optica 4, 814–825 (2017).

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PAPER V

The Extended Discrete Interaction Model: Plasmonic Excitations of Silver Nanoparticles V. I. Zakomirnyi, Z. Rinkevicius, G. V. Baryshnikov, L. K. Sorensen, H. Agren The Journal of Physical Chemistry C, accepted (2019)

Paper V

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

pubs.acs.org/JPCC

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Extended Discrete Interaction Model: Plasmonic Excitations of Silver Nanoparticles Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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†,‡,§ †,∥ † † Vadim I. Zakomirnyi, Zilvinas Rinkevicius, Glib V. Baryshnikov, Lasse K.of Sørensen, Extended Discrete Interaction Model: Plasmonic Excitations Silver ,‡,†,⊥ and Hans Ågren*

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Nanoparticles

Department of Theoretical Chemistry and Biology, School †,∥ of Engineering Sciences in Chemistry, Biotechnology and Health, Royal †,‡,§ Vadim I. Zakomirnyi, Glib V. Baryshnikov,† Lasse K. Sørensen,† Institute of Technology, Stockholm Zilvinas SE-10691,Rinkevicius, Sweden ,‡,†,⊥ ‡ andSiberian Hans Ågren* Federal Research Clinical Centre under FMBA of Russia, Kolomenskaya 26, Krasnoyarsk 660037, Russia § † Kirensky Institute of Physics,Chemistry Federal Research Center SB RAS, Krasnoyarsk 660036,Biotechnology Russia Department of Theoretical and Biology, SchoolKSC of Engineering Sciences in Chemistry, and Health, Royal ∥ Institute of Stockholm SE-10691, Sweden Department of Technology, Physics, Kaunas University of Technology, Kaunas LT-51368, Lithuania ‡ ⊥ Federal Siberian Research Clinical Centre under FMBA of University, Russia, Kolomenskaya 26, Krasnoyarsk Russia College of Chemistry and Chemical Engineering, Henan Kaifeng 475004, Henan,660037, P. R. China § Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, Russia ∥ Department of Physics, Kaunas University of Technology, Kaunas LT-51368, Lithuania ABSTRACT: We present a new atomistic model for plasmonic ⊥ College ofand Chemistry Chemical Engineering, University, Kaifeng 475004, Henan, P. R. China excitations opticaland properties of metallic Henan nanoparticles, which collectively describes their complete response in terms of fluctuating dipolesWeand charges that depend on plasmonic the local ABSTRACT: present a new atomistic model for environment on theproperties morphology of the composite excitations and and optical of metallic nanoparticles, nanoparticles. Being describes atomically the total optical which collectively theirdependent, complete response in terms of properties, thedipoles complex and the plasmonic fluctuating and polarizability, charges that depend on the local excitation of a cluster refer the detailedof composition and environment and on thetomorphology the composite geometric characteristics of the cluster, making possible nanoparticles. Being atomically dependent, the ittotal opticalto explore the role of the material, alloy mixing, size, form shape, properties, the complex polarizability, and the plasmonic aspect ratios, and factors down to the atomic excitation of a other clustergeometric refer to the detailed composition and level geometric and making it useful for plasmonic particles characteristics of the design cluster, of making it possible to with explore particular strength fieldalloy distribution. the role of the and material, mixing, size,The formmodel shape, is aspect ratios, andexperimental other geometric to thepractically atomic parameterized from datafactors and, atdown present, level and making it useful up for to themore designthan of plasmonic particles implementable for particles 10 nm (for nanorods even more), thus covering a significant part of the gap with the particular strengthpure and quantum field distribution. The are model is between scales where calculations possible and where pure classical models based on the bulk dielectric parameterized experimental data and, at present, practically constant apply. Wefrom utilized the method to both spherical and cubical clusters along with nanorods where we demonstrate both implementable for particles up to more than 10 nm (for nanorods more), covering a significant part of the gapusing the the size, shape, and ratio dependence of plasmonic excitations and even connect thisthus to the geometry of the nanoparticles between the scales where pure quantum calculations are possible and where pure classical models based on the bulk dielectric plasmon length. constant apply. We utilized the method to both spherical and cubical clusters along with nanorods where we demonstrate both the size, shape, and ratio dependence of plasmonic excitations and connect this to the geometry of the nanoparticles using the plasmon length. sufficiently large proving that the concept of a classical INTRODUCTION dielectric constant remains valid. Other classical electroPlasmonic excitations of metal elements can generate very dynamics methods, such as the discrete dipole approximation2 strong electric fields in their vicinity through the interaction sufficiently large proving that 3the concept of a classical INTRODUCTION and theconstant T-matrix method, alsoclassical been widely with Plasmonic electromagnetic radiation, to can the generate possibility dielectric remains valid.have Other electro- used for excitations of metalleading elements veryto 2 calculation of optical properties of nanospheres, dynamics methods, such as the discrete dipole approximationnanodisks, detectstrong signals of fields single molecules. There the is interaction a delicate electric in their vicinity through 3 and method, other complex geometrical configurations. The andnanorods, the T-matrix have also been widely used for requirement for matchingradiation, the frequency with electromagnetic leading of to the the incident possibility light to efficiencyof ofoptical these properties methods of hasnanospheres, been confirmed many times, calculation nanodisks, with detect that ofsignals the oscillating electrons, depends of single surface molecules. There which is a delicate and nevertheless, they tend to lose accuracy for small nanorods, and other complex geometrical configurations. The particles ultimately on electronic structure and indirectly also on the requirement for matching the frequency of the incident light with a ofdiameter below has 10 been nm. confirmed Corrections for times, these models efficiency these methods many size, with shape, material of nanostructured It still thatand of the oscillating surface electrons,particles. which depends 7 improve the take into quantum effectsparticles andthat nevertheless, theyaccount tend to lose accuracysize for small remains a great challenge to design active also plasmonic ultimately on electronic structure and indirectly on the 8,9 however, use dielectric constants results; with a diameter below 10they nm.still Corrections for these models of bulk nanomaterials composition, and structure size, shape,with and arbitrary material ofsize, nanostructured particles. It still 10 7 and do not consider the materials, empirically, improve the that take intoobtained account quantum size effects whereremains the structures the dimension of aactive few nanometers. a greathave challenge to design plasmonic 8,9 discrete atomicthey structure the nanoparticles. It is thus however, still use of dielectric constants of bulk results; Classical electrodynamics methods serve in general as viable nanomaterials with arbitrary size, composition, and structure 10 do notwill consider the size and materials, empirically, obviousobtained that the dielectric and constant vary in where the haveofthe of a few nanometers. approaches forstructures prediction thedimension optical properties, including 1−3 discrete atomic structure offorthe nanoparticles. It isbroad thus plasmon become unpredictable small clusters with Classical electrodynamics serve inwhere generala as viableof for methods larger particles variety plasmonic generation obvious that the dielectric will complex vary in shapes. size andHere, the resonances or for clustersconstant with more approaches for prediction the optical properties, approaches, including the offinite difference timeincluding domain 1−3 4,5 become unpredictable for small clusters with broad plasmon largerelement particles where a variety of 6 plasmonic generation and theforfinite method (FEM), (FDTD) method resonances or for clusters with more complex shapes. Here, the approaches, timefrequencydomain frequently have including been usedtheforfinite bothdifference time- and Received: August 3, 2019 1 and the finiteMie element method (FEM),6 (FDTD) method4,5Furthermore, is frequently domain calculations. theory Revised: September 24, 2019 have problems been usedinvolving for both nanoparticles time- and frequencyReceived: August 3, 201916, 2019 Published: October used frequently for scattering that are domain calculations. Furthermore, Mie theory1 is frequently Revised: September 24, 2019 Published: October 16, 2019 used for scattering problems involving nanoparticles that are © XXXX American Chemical Society A DOI: 10.1021/acs.jpcc.9b07410

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© XXXX American Chemical Society

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the strengths and weaknesses of the models. We then discuss a different parameterization scheme based on experimental results and not on TDDFT as has otherwise been done with similar atomistic models. The interplay of parameters and how these parameters should be optimized and chosen are also discussed. We demonstrate the size, shape, and aspect ratio dependence of surface plasmon resonances (SPRs) for silver spherical and cubical clusters and nanorods. We, here, theoretically and numerically show the inverse proportionality between the plasmon length and the SPR and not direct proportionality as proposed by Ringe et al.26 Furthermore, we show that, for nanorods with the same diameter, the longitudinal localized surface plasmon resonance (LLSPR) and transverse localized surface plasmon resonance (TLSPR), measured in nanometers, are proportional to the aspect ratio and red and blue shifts, respectively. Furthermore, we show that the slope of the SPRs depends on the diameter of the nanorod and the polarizability per atom increases linearly with the aspect ratio in nanorods. Finally, we conclude and give an outlook of future developments and applications.

particle size can be smaller than the mean free path of the conduction electrons, and the surface to volume ratio can become so large that a significant deviation from a nonlocal bulk-value description of the dielectric constant can be expected. It is clear that there is a breaking point in size where quantum mechanics-based descriptions, accounting also for electronic structure, are necessary and where classical electrodynamics becomes too crude. Full quantum methods, such as the time-dependent density functional theory (TDDFT), can be useful for calculating the absorption spectra of a small cluster of noble metals with the number of atoms N ≈ 10−300 using TDDFT when the cluster displays some higher symmetry.11−16 However, due to obvious scaling limitations, such calculations cannot be performed for a large number of atoms. For calculations of optical properties of intermediate-sized particles with 102−105 atoms and a diameter of a particle of d < 10 nm, an atomistic approach where the polarizabilities can be obtained from the atoms of the particle could fill an important gap in the description of nanoparticle plasmons between the quantum and classical extremes. Here, the classical dipole approximation can be applicable to construct the total polarizability (or the dielectric constant) from a set of interacting complex polarizable atomic dipoles. However, still for the polarizabilities as such, fixed constants are typically used for each type of element, obtained either from bulk measurements or electronic structure calculations. This neglects, for example, any charge rearrangement that takes place in the real cluster on formation or by molecular sensitization. An elaborate model yet very simple compared to quantum chemical calculations to deal with this is the so-called interaction model.17 In the simplest form, it is a set of atomic polarizabilities that interact in accordance with classical electrostatics without an external electric field. One of the significant drawbacks of an interaction model is, however, that the atoms on the surface and inside of the cluster do not differ very much. The model has been significantly expanded by inclusion of a damping term of the internal electric field18,19 and has been extended also to compute for the frequency dependence of the dipole polarizability20,21 and to an atomistic discrete interaction model (DIM).22,23 The discrete interaction models have been successfully applied to study the polarizability of organic molecules and metallic clusters, to model electrostatic interactions in molecular dynamics simulations, and to describe a heterogeneous environment in hybrid quantum mechanics/classical mechanics calculations; see, for example, ref 23. However, despite the atomistic nature, the DIM has limited capabilities to describe the dependence of the polarizability of the surface topology of the metallic nanoparticles, and furthermore, it cannot be used to study the polarizability of composite nanoparticles. Recently, Chen et al.24 developed a coordination-dependent discrete interaction model (cd-DIM), which attempts to overcome these limitations of the original DIM. The new cd-DIM has been successfully applied to study optical properties of ligand-coated silver nanoparticles. Here, we propose an extended discrete interaction model (ex-DIM), which goes beyond the cd-DIM and enables a robust description of the polarizability of nanoparticles with different geometries. In the following theory section, we will describe our new extended discrete interaction method and compare it with the discrete interaction model and coordination-dependent discrete interaction model by Jensen et al.22,25 and Chen et al.,24 respectively, emphasizing

■

THEORY

We split the description of our extended discrete interaction model into two parts: a theoretical foundation and detailed description of the ex-DIM presented in this section and a parameterization scheme of the ex-DIM along with applications in the following section. Extended Discrete Interaction Model. Similar to the original DIM suggested by Jensen et al.,20,22 our extended discrete interaction model aims to describe the polarizability and optical properties of metallic nanoparticles by representing the nanoparticle as a collection of interacting atomistic charges and dipoles. The starting point of both models is a Lagrangian function with an energy expression for interacting fluctuating charges and dipoles in an external electric field subject to a charge equilibration constraint. ij L[{μ , q},λ] = E[{μ , q}] − λjjjjq tot − j k N

1 1 = ∑ qi cii−1qi + 2 i 2 1 + 2 − −

N

∑ i

N

N

i

j≠i

∑∑ N

∑ i

N

{

i

N

N

i

j≠i

∑ ∑ qi T(0) ij qj

1 μi αii−1μi − 2 μi T(1) ij qj +

μi Eext

yz

∑ qizzzzz

N

N

i

j≠i

∑∑

μi T(2) ij μj

N

∑ qi V ext

ji − λjjjjq tot − j k i

zy

∑ qizzzzz N i

{

(1)

In eq 1, the first term is the self-interaction energy of fluctuating charges, the second term is the interaction energy between fluctuating charges, the third term is the selfinteraction energy of fluctuating dipoles, the fourth term is the interaction energy between fluctuating charges and dipoles, the fifth term is the interaction energy between fluctuating dipoles, the sixth term is the interaction energy between fluctuating charges and the external potential, the seventh term is the interaction energy between fluctuating dipoles and the B

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number-dependent atomistic polarizability derived via the Clausius−Mossotti relation,29 and the electrostatic interaction tensor (T(2) ij ) is computed by the same way as in the DIM. In order to extend these models and achieve a description of more complex surface topologies, we spatially spread in our model the Gaussian dipoles and charges in a way that they explicitly depend on their local chemical environment. Here, suggestively, the scheme of Grimme,11 originally proposed for the computation of dispersion corrections in DFT calculations, can be used for evaluating atomic coordination numbers. The atomic coordination number ficn is then computed as

external field, and the last term is a charge equilibration condition expressed via the Lagrangian multiplier λ. Here, qi is the fluctuating charge assigned to the ith atom, μi is the fluctuating dipole assigned to the ith atom, cii is the ith charge self-interaction tensor, αii is the ith dipole self-interaction (1) (2) are the electrostatic interaction tensor, T(0) ij , Tij , and Tij tensors, Vext is the external potential, the Eext is the external electric field, qtot is the total charge of the nanoparticle, and N (1) (2) is the number of atoms in a nanoparticle. T(0) ij , Tij , and Tij are for the ex-DIM, shown in more detail in Appendix A. The fluctuating charges and dipoles are determined by minimizing the energy E[{μ, q}]. According to Jensen et al.,22 this minimization problem can be recast into a problem of solving a set of linear equations i ext y jij A − M 0 zyzijj μ yzz jjjj E zzzz z jj T jj− M − C 1 zzzjjjj q zzzz = jjjj V ext zzzz zzjj zz jj jj zz zj z jj 1 0 z{k λ { jj qtot zz k 0 { k

i f cn =

Rcov i

(2)

N i

(3)

]−1 (5)

Rcov j

3/2

exp[−a(r − C)2 ]

3/2

(6)

exp[− acn(r − C)2 ] with acn

= a(1 + bfcn )

(7)

The coordination number-dependent dipoles are obtained from coordination-dependent Gaussian charges by taking their gradient, that is, μ(r; fcn, C) = − ∇rG(r; fcn, C). Here, a is the fixed exponent of the Gaussian charge distribution centered on the atom with the position vector C, and b is the coordination number scaling factor, which defines the coordination numberdependent spread of the Gaussian charge distribution. This constitutes the general form of the extended DIM. However, before taking on its full implementation, it is necessary to scrutinize the calculation of its most important parameters, namely, the self-interaction tensors cii and αii. Here, we adopt a scheme based on the concept of plasmon length26 as described in the section below. Modeling Capacitance and Polarizability. The parameterization of the self-interaction tensors, cii and αii, in the exDIM is central since these tensors play the dominant role in defining the behavior of the nanoparticle polarizability. Furthermore, in the case of dynamic polarizabilities, the frequency dependence is solely defined by these tensors. Similar to the DIM and cd-DIM, we use in the ex-DIM a diagonal isotropic form for the self-interaction tensors, that is, cii, kl = δklc and αii, kl = δklα for k, l = x, y, z. Here, we will employ a different strategy based on the plasmon length26 to parameterize the cii and αii tensors. Starting from the self-interaction tensor via the Clausius− Mossotti relationship for a spherical particle

∂μi

∂Eext

cov cov i + R j )/ rij − 1)

j≠i

ia y G(r; fcn , C) = jjj cn zzz kπ {

Equation 2 can be solved by inversion of the left-hand side matrix for small- and medium-size nanoparticles or by the iterative approach, such as the conjugate gradient method for large-size nanoparticles in an external field and potential. In the calculations presented here, we solve the linear equations by inversion for each frequency since we do not apply an external field. After that, the fluctuating charges and dipoles determined by the polarizability of the nanoparticle can be directly obtained by computing the second derivative of E[{μ, q}] with respect to external field Eext. According to Jensen et al.,22,24 the polarizability of a nanoparticle can be defined as

∑

i

2

used in the DIM and cd-DIM can be replaced with the coordination number-dependent Gaussian charge distribution

Cij = δijcii−1 + (1 − δij)T(0) ij

α np =

1

iay G(r; C) = jjj zzz kπ {

A ij = δijαij−1 − (1 − δij)T(2) ij

Mij = (1 −

N

where and are the scaled covalent radius of the ith and jth atoms, respectively, rij is the distance between the ith and jth atoms, and k1 and k2 are empirical parameters equal to 16.0 and 4.0/3.0, respectively.30 In the case of fluctuating charges and dipoles, the normalized Gaussian charge distribution

where the column vector μ is the collection of μi dipoles, the column vector q is the collection of qi charges, and λ is a Lagrangian multiplier associated with charge equilibration condition. The matrix elements of A, C, and M are defined as

δij)T(1) ij

N

∑ ∑ [1 + e−k (k (R

(4)

The above-described scheme for determination of the polarizability of a nanoparticle is generic and has been employed in the original, coordination-dependent, and extended discrete interaction models.18−21,27,28 The differences between these models originate from the functional form used to describe the fluctuating charges and dipoles and from the parameterization of the self-interaction and electrostatic interaction tensors. To lay the foundation for our extended discrete interaction model, we first consider the parameterization of the DIM and cd-DIM. In the original DIM, the selfinteraction tensors (cii and αii) are parameterized using atomistic capacitance and polarizability derived from bulkmaterial properties, and the electrostatic interaction tensors (1) (2) (T(0) ij , Tij , and Tij ) are computed using normalized Gaussian charges and dipoles with parameterization using TDDFT. In the cd-DIM, the fluctuating charges are excluded from the energy expression E[{μ, q}], the self-interaction tensor (αii) between dipoles is parameterized using a coordination

αii , kl(ω) = δklfα with fα =

6 3 ε(ω) − ε0 Ri π ε(ω) + 2ε0

(8)

where Ri is the radius of the ith atom, ε(ω) is the frequencydependent dielectric constant of the material, and ε0 is the C

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dielectric constant of the environment. In the DIM, approximated as

24

respectively. L(ω, P) regulates the geometric dependence via the size-dependent resonance frequencies of three sizedependent Lorentzian oscillators

eq 8 is

αii , kl(ω = 0) = αi , s , kl

(9)

αii , kl(ω > 0) = αi , s , kl(L1(ω) + L 2(ω , N ))

L(ω , P) = N (Lx(ω , Px) + Ly(ω , Py) + Lz(ω , Pz))

(10)

where αi, s, kl is the static polarizability and L1(ω) and L2(ω, N) are two separately normalized frequency-dependent Lorentzian functions. The resonance frequency ωi,2(N) in L2(ω, N) is size-dependent ωi ,2(N ) = ωi ,2(1 + A /N1/3)

(11)

where N is the number of particles and ωi,2 and A are two fitted parameters. In this way, the size-dependent frequency is inversely proportional to the radius for spherical particles. The problems here are the discontinuity going from the static to the dynamic case due to the separately normalized Lorentzian functions and that the size-dependent resonance frequency in L2(ω, N) does not take into account the geometry of the particle. The cd-DIM24 modifies the radius of eq 8 to a coordination number-dependent radius Ri( fcn) and dielectric constant ε(ω, fcn, r)

Li =

−1

(18)

The choice of the Lorentzian oscillator in eq 17 and the common normalization in eq 18 will in this way give the higher peak for the lower incident frequency, which, for nanorods, corresponds to the long side. The size-dependent resonance frequency ωi(Pi) can be written as

Here, ε(ω, fcn, r) is described by the sum of the experimental dielectric constant εexp and the size-dependent Drude equation minus the Drude function for spherical particles (13)

ωi(Pi) = ωa(1 + A ·f (N , i))

where the plasma frequency in the size-dependent Drude function is modified by the coordination number. By using an effective coordination number, there is a smooth transition from the inside to the outside of the coordination sphere. Both the DIM and cd-DIM should be able to describe the size dependence of spherical and sphere-like particles if properly parameterized. For shapes far from a spherical symmetry, such as nanorods with a large aspect ratio, the functional shape in the DIM and cd-DIM does not appear to be appropriate. We are therefore interested in developing a method that can take into account both the surface effects and geometry effects of nanoclusters. We here extend the DIM where

(19)

where ωa and A are atom-specific fitted parameters for the bulk resonance and size dependence, while f(N, i) is a function of the number of atoms and dimension along the ith direction measured in units of atom i. f(N, i) must then in the bulk and atomic limits fulfill lim f (N , i) = 0

N , i →∞

lim f (N , i) = 1

N ,i→1

(20)

which can easily be accomplished using a single parameter, namely, the plasmon length Pi f (N , i) =

3

1 Pi

(21)

where the plasmon length Pi is defined as the maximum distance between any atoms along the ith direction plus the radius of each of the endpoint atoms. This use of the plasmon length is consistent with the experimental work from Tiggesbäumker et al.32 We here notice that the SPR cannot be directly proportional to the plasmon length as defined by Ringe et al.26 since, in the bulk limit, the SPR would incline to minus infinity. Performing a Taylor expansion of eq 21, the first order is linear in the plasmon length, and therefore the linear dependence on the plasmon length as observed by Ringe et al.26 is consistent with a sample of clusters of a limited size range. For spherical clusters, eq 21 reduces to the usual size dependence for classical models, also seen in the DIM and cdDIM, but for rods, discs, and other shapes far from spherical, there is a distinct difference where the ex-DIM can have up to three distinct plasmon resonances.

(14)

is the static polarizabilty αi, s, kl31 multiplied by a normalized Lorentzian function of L(ω, P) and the relative shift in radius from the bulk radius is determined by the coordination number. In this parameterization scheme, the chemical environment enters the definition of the αii tensor via Ri(fcn) defined as f yz f ji R i(fcn ) = r1jjj1 − cn zzz + r2 cn z j 12 12 { k

(17)

ij 1 1 1 yzzz + 2 + 2 N = jjjj 2 z j ωx (Px) ωy (Py) ωz (Pz) zz { k

(12)

ij R i(f ) yz cn z zz αi , s , klL(ω , P) αii , kl(ω) = jjjj z R i ,bulk { k

1 ωi 2(Pi) − ω 2 − iγω

where γ describes the broadening of the spectra and ωi(Pi) is the size-dependent resonance frequency, which enables the geometric description of the plasmon excitations. With the choice of the Lorentzian oscillator in eq 17, the normalization constant becomes

ε(ω , fcn , r ) − ε0 6 αii , kl(ω) = δklfα with fα = R i 3(fcn ) π ε(ω , fcn , r ) + 2ε0

size ε(ω , fcn , r ) = εexp + εDrude (ω , fcn , r ) − εDrude(ω)

(16)

where each Li(ω, Pi) depends on the plasmon length Pi in the ith direction and the frequency ω with the common normalization factor N ensuring that the Lorentzian oscillators are normalized in the static limit of ω = 0. With a sizedependent Lorentzian oscillator in each direction, it is possible to describe more complicated geometric structures with multiple plasmon resonances without having a new functional dependence for each distinct geometry and thereby make the ex-DIM more universal. The Lorentzian oscillator is chosen as

(15)

which regulates the radii of the atom depending on the coordination number. For Ag, we use r1 = 1.65 Å and r2 = Ri, bulk = 1.56 Å, which are the surface and bulk radii, D

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The cii tensor responsible for the self-interaction energy in charge transfer processes is in the DIM modeled as ci(ω) = ci , sL1(ω)

Scholl et al. The aim therefore is not to reproduce the absolute polarizability of small TDDFT calculations but to predict the SPR of larger clusters with different geometry. Even though there have been significant advances in geometry optimization of metal clusters, the problem still remains to be hard and not applicable for larger metal clusters due to a multitude of local minima close-lying in energy.34−36 For all clusters and nanorods, we therefore start from a perfect lattice where all atoms have the same distance to their nearest neighbor and then cut out the nanoclusters with the desired structure. From the 310 Ag-atom cluster, the average distance to the closest neighbor is 5.28 au with a minimum and maximum distance of 5.17 and 5.41 au. Similar distances are seen for the other optimized clusters,35 and we have therefore fixed the closest neighbor at 5.27 au in our lattice. Since the surface topologies of all measured nanoparticles with the same radius are slightly different, we try to capture this effect by having several clusters with the same plasmon length but a different number of particles in our optimization. This is possible since there are only small variations in the SPR with respect to small changes in the surface topology for sphere-like clusters. In the optimization, we do not use faceted clusters nor clusters with surface defects such as bumps or holes. In order to reduce the number of parameters needed to be fitted and make the method easier to extend to other elements, we make use of experimental or theoretical literature values or make argued choices for parameters that affect the peak position of SPRs. The polarizabilities (α) are taken from Schwerdtfeger and Nagle,31 which for silver are 55 au. The value of the capacitance parameter c, as shown in Appendix C, has very little influence on the overall polarizability and peak position as long as c is outside areas of numerical instability. For the optimization of sphere-like clusters, we have fixed the value at 0.0001 au since all systems appear to be numerically stable with this choice. The Lorentzian broadening γ should be small compared to the incident frequency and, not surprisingly, should show no significant influence on the position of the SPR as seen in Appendix D. During optimization, γ has been fixed at 0.016 au, which gives what we deemed a reasonable broadening of the peak(s) with FWHM compared to that extrapolated from data by Ringe et al..26 While the SPR(s) does not shift with γ except when two close-lying double peaks merge, the width and height of the SPR(s) are significantly influenced thereby making it difficult to get a good set of parameters when optimized together with α for a small set of small clusters. Despite being optimized with γ = 0.016, there is no problem adjusting this parameter later or making γ size-dependent to obtain different peak heights since the placement of the SPR(s) is not affected by small changes in γ. The only parameters that need to be fitted are therefore the size-dependent resonance frequency ωa and the size-dependence factor A. These two parameters are the decisive parameters in determining the SPR. Systematic investigations, like the one performed by Scholl et al.,33 are therefore essential for an accurate fit of ωa and A. Due to the scarce amount of data and because of what appear to be outliers in the data, we decided to perform some data pruning and base our parameter fit on the pruned data. By plotting the energy of the SPR as a function of the inverse plasmon length, we can fit a simple linear function as shown in Figure 1. From the fit in Figure 1, the bulk limit for the SPR for Ag will be 3.25 eV in our model and will show a slow variation of

(22)

using the same size-independent Lorentzian function as in eq 10 for the polarizability in the DIM and a fitted parameter ci, s for the “static atomic capacitance”. In the cd-DIM, the charges and hence the charge transfer and capacitance are completely removed. In the ex-DIM, we adopt a simplified two-parameter parameterization scheme É ÄÅ ÅÅ R i(fcn ) ÑÑÑÑ ÑÑL(ω , P) cii , kl = δklfc with fc = ci , sÅÅÅÅ1 + d ÅÅ R i(12) ÑÑÑÑÖ (23) ÅÇ where c is the “static atomic capacitance” parameter, similar by its physical origin to the capacitance used in the DIM, d is a scaling factor for the coordination number dependence of the capacitance, set to 0.1, and L(ω, P) is the Lorentzian oscillator defined in eq 16. Here, we stress that, in our parameterization of the cii tensor, the frequency dependence is exactly the same as for the polarizability. The outlined parameterization scheme of our ex-DIM not only satisfies the above given principal conditions for our model, physical limits, and geometric dependencies but also enables rapid reparameterization of the ex-DIM for new types of composite nanoparticles or/and their environments.

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PARAMETERIZATION OF THE EXTENDED DISCRETE INTERACTION MODEL AND APPLICATIONS Similar to its predecessors, the extended discrete interaction model is an empirical approach, and thus its accuracy and applicability are defined by the quality of its parameterization. Here, we outline the basic ideas behind the parameterization of the ex-DIM and discuss the optimization of the parameters for nanoparticles. As the ex-DIM aims to describe static and dynamic polarizabilities of metallic nanoparticles accurately, the goal of the parameterization is to obtain a set of parameters with which the SPRs are computed using eq 4 as close as possible to established benchmark values for a selected set of nanoparticles. We here optimize the model using a training set of spherical clusters and compare the results to both the experimental values and a validation set of larger clusters to ensure that the model gives reliable results for all cluster sizes. Afterward, we investigate if the redshift observed in the Au clusters by Ringe et al.26 when going from more spherical clusters to cubical clusters also is present for Ag clusters. Finally, we turn our attention toward silver nanorods to show that both the longitudinal and transverse SPRs can be accurately predicted using the ex-DIM. Parameterization of the ex-DIM Model. As discussed in detail in Appendix B, parameterizing the ex-DIM using TDDFT data from small silver clusters encounters problems with the magic number of atoms, limited size of clusters, and the more physical problem of not having real SPR. Therefore, trying to optimize the parameters minimizing the difference in the polarizability between a small set of clusters calculated using TDDFT and a given parameter range does not appear to be a viable approach. We have therefore instead opted to optimize the parameters in the ex-DIM directly from experimental results. For this, we have used the systematic size-dependence investigation of silver clusters performed by E

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Figure 1. Linear fit (ax + b) of experimental data with error bars from Scholl et al.33 The purple points are the pruned data, and the green line is the fit of the pruned data with coefficients of a = 0.670796 ± 0.05917 and b = 0.119511 ± 0.0006792. The blue points are data excluded from the pruning, and the yellow-orange line is a fit of all data with coefficients of a = 0.822497 ± 0.1057 and b = 0.119488 ± 0.001297.

Figure 3. Plasmon peak as a function of the inverse plasmon length for the clusters used for the fit in Figure 2 and a test set with larger clusters calculated with the fitted ωa and A values compared to the experimental fit and pruned data. The 1481 and 1505 atom recalculated clusters are located between or underneath the 1409 and 1433 atom clusters.

the SPR as a function of the inverse plasmon length. With the definition of the plasmon length in eq 21, the inverse plasmon length cannot exceed the inverse diameter of an atom, and the SPR is therefore finite. By choosing a representative set of spherical clusters with a plasmon length of 1.4−3.8 nm, an optimal resonance frequency, ωiPi in eq 19, which exactly reproduces the SPR from the fit in Figure 1, for every cluster can be found (Figure 2). The optimal resonance frequency is here reproduced with a

As seen from Figure 3, we are able to reproduce the SPR of any sphere-like cluster irrespective of size with an error limited by the experimental error. To ensure that the behavior of the polarizabilty is correct for all frequencies, we calculated the polarizability-dependent frequency for 200 points in the 3.0− 4.6 eV region. Comparison between the ex-DIM, DIM, and cd-DIM with Mie Theory. Since all three models have been applied to bare sphere-like silver clusters, it would be natural to compare them to the experimental data since the cd-DIM has been compared to the same data before24 and the ex-DIM is parameterized from the experimental data. The extracted data from the DIM and cd-DIM have therefore been plotted against the experimental data and ex-DIM calculations as shown in Figure 4. From the plotted data, it is evident that, for the truncated octahedrons, the DIM shows no discernible trend, while for the icosahedra, there is a redshift of approximately 0.2 eV with size but only for the range of 147−1415 atoms (1.8− 3.4 nm); thereafter, there is no shift. The cd-DIM does show a redshift in the plasmon length with increasing size but only by

Figure 2. Optimum ωi(Pi), which reproduces the plasmon peak at the fitted experimental values from Figure 1 for a given cluster. Fitting the optimum ωi(Pi) to eq 19, we find that ωa = 0.0794 au and A = 9.41 au. The 1409 and 1433 atom clusters, also included in the fit, are located between or underneath the 1481 and 1505 atom clusters.

deviation of 10−6−10−5 of the SPR compared to experiments. We here use several spherical clusters with the same plasmon length but with different surface topologies to simulate slightly different surfaces. So while the radius in the 459, 555, and 603 clusters is the same, the number of atoms and the surface topology are not. We here find that ωa = 0.0794 au and A = 9.41 au. Inserting the fitted ωa and A values and recalculating the clusters from the fit along with a test set of larger clusters with 276−11,849 atoms and a 2−7 nm diameter, we are able to reproduce the SPR from the fit of the experimental values as seen in Figure 3.

Figure 4. Comparison between the ex-DIM, DIM,25 and cd-DIM,24 Mie theory, and experiment for bare silver clusters.33 For the DIM, the TO clusters are truncated octahedrons and Ih are icosahedral clusters. The diameter for the DIM clusters are estimated from the clusters used in the ex-DIM. The 1481 and 1505 atom recalculated clusters are located between or underneath the 1401 and 1433 atom clusters. F

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shape dependence of the SPR as the Ag cubes are redshifted at approximately 0.6 eV in comparison to the Ag spheres in the region examined here, which is in line with the findings of González et al.42 The size dependence of the cubes and spheres is here shown to be reasonably similar. Silver Nanorods. While a small red shift in the SPR with increasing cluster size is seen for sphere-like particles, very significant red shifts can be observed for nanorods depending on the aspect ratio. This very large red shift can be used to tune the SPR to a given region, thereby making nanorods versatile sensors. The SPR for nanorods is, however, split into two due to the cylindrical symmetry and excitation of collective oscillations of conduction electrons of nanorods, and two peaks are seen in the UV−vis spectrum. The SPR along the short transverse axis (TLSPR) is typically very slightly blueshifted in comparison to that of a spherical cluster with the same plasmon length, while the SPR along the long longitudinal axis (LLSPR) can be redshifted much below what can be done by increasing the size of a spherical cluster. Furthermore, the polarizability for the redshifted peak is also greatly enhanced with an increasing aspect ratio, here defined as the ratio between the plasmon length in the longitudinal and transverse directions. Since tunable nanorods are of great application interest, we have examined a series of nanorods to elucidate the interplay between the aspect ratio and diameter with respect to the SPR. We have constructed a series of nanorods where each end is a half-sphere connected by a cylinder. The nanorods are designated as Ag(x, y) where x is the plasmon length of the longitudinal axis and y is the plasmon length of the transverse axis in nanometers as shown in Figure 6. Ag(y, y) is, with this

approximately 0.097 eV for the 2−10 nm clusters, while the experimental data give a redshift of 0.38 eV in that region. The limit of the cd-DIM therefore deviates significantly from the experimental results and the results of the ex-DIM. While Chen et al.24 give an arbitrary shift of 0.2 eV to the experimental data to compensate for solvent effects, this does not change the fact that the shift in the SPR in the cd-DIM is only approximately a quarter of what it should be according to experiments.33 The poor performance of the DIM and cd-DIM for spherelike clusters is most likely not due to methodological issues but rather due to the parameterization. This can be understood since the ex-DIM and DIM in the spherical cases are very similar except for the surface atoms, and a better fit of parameters should therefore be possible. For the DIM,25 Jensen and Jensen reported puzzling parameters. While the resonance frequency ωi,1 = 0.0747 is similar to ωa = 0.0794 in the ex-DIM, the broadening γi,1 = 0.0604 is of the same size as the resonance frequency ωi,1, which is very unusual in Lorentzian and other oscillator models and a sign of something that has gone wrong in the parameterization. We here notice that Jensen and Jensen22 use a capacitance parameter, c = 3.45, which is right in the region of numerical instability in the exDIM as shown in Appendix C. In cd-DIM, the size-dependent Drude function does not appear to be optimized for silver at all even though Karimi et al.37 had no problems fitting a similar function for gold. Mie theory1 is known to be in good agreement with experiments for medium and large particles but not so for small particles. As seen in Figure 4, Mie theory underestimates the size dependence of small silver clusters when compared to experiments even when Mie theory is size-corrected based on the electron effective mean free path.38 Silver Cubes. Ringe et al.26 showed that Au cubes are redshifted up to 0.2 eV in comparison to more sphere-like clusters with the same plasmon length. Because of differences in dielectric constants39 between Au and Ag, the SPR in Ag nanoparticles has a more acute size dependence than that of their Au counterparts.40,41 As seen from Figure 5, we predict that the more acute size dependence translates into a larger

Figure 6. Longitudinal and transverse plasmon length and coordination numbers for the Ag(12.09, 4.20) nanorod, which contains 8743 atoms.

definition, a sphere with an effective diameter of y. We here use nanorods with a diameter from 2.23 to 6.18 nm, a length of up to 14.06 nm, and an aspect ratio of up to 5.4 and containing up to 16,567 atoms. For all figures, we calculate the polarizabilty at 400 different frequencies. In Figure 6, we clearly see that only the top layer of atoms has a coordination number below 11−12, and, as expected, the atoms with the lowest coordination number are on the edges. This means that only the surface atoms are directly affected by the changes introduced by the coordination numbers. The red shift of the LLSPR for the Ag(x, 2.23) and Ag(x, 4.20) nanorods calculated with the ex-DIM is clearly visible from Figures 7 and 8. Due to the emergence of a double peak in Figure 7, the slight blue shift of the TLSPR is not as visible as in Figure 8. As seen in Figure 9, the red shift of the LLSPR in nanometers is directly proportional to the aspect ratio and the difference in the LLSPR changes with the diameter of the

Figure 5. Comparison of the SPR for sphere and cubes with different plasmon lengths. The greater red and blue shifts seen for the 665 and 1687 atom cubical clusters, respectively, are due to double peaks where the most red- and blueshifted peaks have the highest polarizability. With a larger γ value, both outliers will be shifted to be more in line with the rest of the cubes. The 1481 and 1505 atom recalculated clusters are located between or underneath the 1401 and 1433 atom clusters. G

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Figure 7. Polarizability as a function of the incident energy for Ag(x, 2.23) nanorods with different longitudinal plasmon lengths.

Figure 8. Polarizability as a function of the incident energy for Ag(x, 4.20) nanorods with different longitudinal plasmon lengths.

slight blue shift of the TLSPR is best seen for the Ag(x, 4.20) and Ag(x, 6.18) nanorods and is also approximately linear. The experimental results of Jakab et al.45 in which the average width of the nanorods varies from 55 to 59 nm indicate that the red shift is directly proportional to the aspect ratio and with a slight increase in the slope compared to our results. The large red shift in comparison to our results is due to the differences in the refractive index in the surrounding medium. Here, our results refer to nanorods on an ultrathin carbon film,33 while the experimental results were obtained in a 0.1 M KNO3 aqueous solution. The relative polarizability and peak width between the LLSPR and TLSPR in Figures 7 and 8 are seen to increase significantly with an increasing aspect ratio. The polarizability per atom in Figure 10 is seen to increase linearly with the aspect ratio, and only minor changes are seen with respect to the diameter of the nanorod. Both the LLSPR and the absorbance can in this way be controlled by the aspect ratio and the diameter of the nanorods. The polarizability thus depends substantially on the geometry. For applications of nanorods, the refractive index of the surrounding medium must also be carefully considered.

Figure 9. LLSPR, TLSPR, and fit of the LLSPR as a function of the aspect ratio for different nanorods. For the Ag(x, 2.23) nanorods, the TLSPR becomes a double peak (see Figure 7), and here, only the right TLSPR is included. These are compared to the experimental LLSPR in a 0.1 M KNO3 aqueous solution.45

nanorod. The dependence on the diameter of the nanorod can also be seen from the slope of the fit for the Ag(x, 2.23), Ag(x, 4.20), and Ag(x, 6.18) nanorods, which are 139 ± 4, 150 ± 3, and 162 ± 7, respectively. The increasing slope of the LLSPR with the diameter is also observed for gold nanorods.43,44 The H

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radii, we show that the red shift is directly proportional to the aspect ratio and that the slope in all series of nanorods of different diameters shows a slight dependence on their plasmon length in the transverse direction similar to that seen for Au nanorods.43,44 We compare these results to experiments and find that differences in the refractive index of the surroundings only appear to give a constant shift for the LLSPR.45 Furthermore, we show that the polarizability per atom increases linearly with the aspect ratio, thereby making it possible to control both the peak position and the polarizability of the SPR for nanorods. Our ex-DIM is flexible and versatile and brings wide ramifications, for instance, in the design of small plasmonic nanoparticles in mixed or alloyed systems with particular geometries in metal particle organic hybrids where the organic part receives comparable parameterization as the metal part in heterogeneous environments and in external fields. Work is ongoing to capitalize on these expectations.

Figure 10. Polarizability per atom as a function of the aspect ratio.

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OUTLOOK AND DISCUSSION Motivated by the wide applicability of small plasmonic nanoparticles with a size between 1 and 15 nm and by the need to bridge this length gap between the classical and quantum theories to describe plasmon generation, we have presented in this work an extended discrete interaction model (ex-DIM) to simulate the geometric and environmental dependence of plasmons of this size. The frequency-dependent dielectric function is obtained from the Clausius−Mossotti relation as a sum of three Lorentzian oscillators and with Gaussian charge distributions and atomic radii that vary with the coordination number. The three frequency-dependent Lorentzian oscillators depend on the plasmon length in the x, y, and z directions with the plasmon length defined as in the work of Ringe et al.26 We here show both theoretically and numerically that the SPR is inversely proportional to the plasmon length and not proportional. We also show that, due to the limitations of applicable quantum calculations (N.B. TDDFT) and due to the interplay of parameters that fit an atomistic model for metal clusters, the use of absolute polarizabilities from TDDFT is not a viable approach. Instead, we show that the model can be parameterized from experiments33 with a numerical accuracy of the same order as the experimental accuracy. Furthermore, we show that certain parameters such as the broadening and capacitance do not influence the peak position of the SPR to any appreciable extent and that reasonable values for these parameters can be chosen without fitting. We see almost no effect of the capacitance below a given value, while above that value, the system may become numerically unstable in contrast to some earlier work on fitting atomic capacitance parameters. Having parameterized the model to a set of spherical clusters, we show that not only the training set but also the validation set with larger clusters are all very close to the experimental results unlike the DIM22,25 and cd-DIM24 results. To demonstrate the capabilities of the ex-DIM, we also performed a set of calculations on cubes and nanorods. For the cubes, we show that the SPR is redshifted in comparison to a spherical cluster with the same plasmon length, which is in line with the findings of González et al.42 and the experimental findings of Ringe et al.26 for Au clusters with the difference that the geometric dependence for Ag appears to be larger than that of Au. For the nanorods, we show a significant red shift for the longitudinal resonance and a very slight blue shift for the transverse resonance with the aspect ratio. By calculating several series of nanorods with different diameters and aspect

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APPENDIX A

Electrostatic Interaction Tensors

Similarly to the DIM and cd-DIM, our ex-DIM uses Gaussian electrostatics to describe the interaction of fluctuating charges and dipoles. However, in our model, normalized Gaussian charge distributions are explicitly dependent on the coordination number of the atom with which it is associated (see eq 7), (1) and thus the electrostatic interaction tensors, T(0) ij , Tij , and T(2) ij , have more complex forms compared to the ones used in the DIM or cd-DIM. Assuming that we have two Gaussian charge distributions, G(r; fcn, C) and G(r ’ ; f ’cn, D) centered on the ith and jth atoms with position vectors C and D, the electrostatic interaction tensor T(0) ij between these charges can be computed as T(0) ij =

∫∫

G(r; fcn , C)G(r’; f ’cn , D) |r − r’|

dr’dr =

erf(γrij) (24)

acna’cn and rij = |C − D| acn + a’cn

γ=

rij

(25)

Following Mayer,46 the higher-order electrostatic interaction (2) tensors, T(1) ij and Tij , can be obtained by taking the derivatives of T(0) ij with respect to the ith atom coordinates, that is Ä ÑÉÑ rij ÅÅÅ 2γrij Ñ (0) ÅÅerf(γr ) − T(1) exp(− γ 2rij 2)ÑÑÑÑ Å ij ij = −∇ri T ij = 3Å ÑÑÖ (26) π rij ÅÅÇ Ä ÑÉÑ rij ⊗ rij − rij 2 I ÅÅÅ 2γrij Ñ ÅÅerf(γr ) − exp( − γ 2rij 2)ÑÑÑÑ ÅÅ ij 5 ÅÅÇ ÑÑÖ π rij

(0) T(2) ij = −∇ri ⊗ ∇rj T ij

=

−

4γ 3rij ⊗ rij π rij 2

exp( − γ 2rij 2)

(27)

The above given expressions for interaction tensors can be easily reduced to the ones used in the DIM if one replaces the coordination number-dependent Gaussian exponents, acn and a′cn, with appropriate effective radii (see eqs 11−13 in Jensen’s work22). I

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APPENDIX B

Surface versus Bulk Plasmon Resonance in Small Silver Clusters

The accurate estimation of the electronic and spectral properties, such as static dipole polarizability, ionization potential, and electron affinity, and atomization energy and absorption and emission cross sections, is still a major challenge for small silver clusters.15,47−49 The important property of silver clusters is that they can be considered at the same time as bulk metal particles and surface-active systems depending on the size of these species. Indeed, the electronic properties and spectra of small silver clusters (till 120 atoms that approximately correspond to the diameter of a spherical nanoparticle approximately 1.6 nm) are well described by the so-called shell model50 that explains the strongly nonlinear behavior of these species with respect to the position of absorption maxima, static polarizability, ionization potential, and electron affinity parameters.47,51 Particularly, it has been found that silver clusters with a magic number of atoms (8, 18, 34, 58, and 92) show a localized maximum value of the plasmon-like absorption energies at approximately 4.1 eV.51,52 These numbers actually correspond to the fully filled states of 1s, 1p, 1d, 1f, 1g, and 1h electronic shells. Some previously published DFT calculations confirm such behavior51,15 demonstrating that s valence electrons are distributed in delocalized orbitals in the following sequence of electronic shells: 1s2, 1p6, 1d10, 2s2, 1f14, 2p6,1g18, 2d10, 3s2, 1h22, etc. It means that small silver clusters absorb light due to excitations of s electrons delocalized over the whole volume of the cluster, which explains that this band can be assigned as plasmon-like resonance or so-called bulk resonance, different from the commonly known SPR by a physical meaning. It is reasonable to suggest that the position of the bulk resonance should be weakly sensitive to the size and shape of the nanoparticle. Even for large silver nanoparticles, the bulk core (if we consider a cage model of the particle) still possesses similar localization and energy of s electrons as a small cluster, while the energy of the SPR expectantly depends significantly on the shape, curvature, type, and defects of the surface. Scholl et al.33 clearly defined both types of resonances (bulk and surface) for spherical silver nanoparticles of the size of 2−23 nm from electron energy-loss spectroscopy by directing the electron beam to different zones of the particles from the edge through the bulk. Based on such focused excitation, surface and bulk resonances can be selectively observed in the range of 4.1−3.8 and 3.8−3.2 eV, respectively. One can see that the energy variation is more pronounced for the surface resonance, a tendency which is clearly size-dependent, while the bulk resonance is less size-dependent. The energy values are in a good agreement with our TDDFT/cam-B3LYP/Lanl2DZ53−55 calculations for small metal clusters of Agn (n = 18−34) and with other previously published results. An applicability of long-range corrected (LC) functionals for the correct simulation of optical transitions for small silver clusters in the gas phase was shown in few recent publications.51,56 Particularly, LC hybrid functionals significantly reduce the occurrence of spurious states in the optical absorption spectra while maintaining the intensity of plasmonlike features of the spectra for larger silver clusters.56 The final spectra calculated using the long-range corrected cam-B3LYP functional for the nine Agn (n = 18−34) clusters (Figure 11, black curves) all exhibit an absorption peak at

Figure 11. Absorption spectra of the Agn (n = 18−34) clusters calculated by the TDDFT/cam-B3LYP/Lanl2DZ method in the gas phase (black curves and black peaks correspond to the vertical electronic transitions) compared with experimental data (red curves were taken from refs 52 and 57). The Gaussian line shape with a full width at half maximum of 0.2 eV was used for the calculated spectra convolutions.

approximately 4 eV in good agreement with the experimental data for Ag18, Ag20, and Ag30 clusters (Figure 11, red curves).52,57 For these three cases, the main computed absorption peak is blueshifted at approximately 0.2 eV relative to the experimental curves, which can be attributed to the matrix effect16,51 (experimental spectra were measured for the clusters isolated in the noble gas matrix). Such a blue shift accords with that published in ref 51 (0.17 eV, estimated from direct comparison between the experimental spectra measured in a neon matrix at 6 K and TDDFT/ωB97x gas phase calculations). Visualization of Kohn−Sham molecular orbitals on Figure 12 responsible for the main most intensive electronic transition (black peaks under the spectral curves, Figure 11) clearly shows the delocalization of the valence and conductive s electrons mixed with the localized d functions over the whole volume of the studied clusters. It proves the “bulk” nature of the predicted resonance peak. Increasing of the cluster size greatly complicates the calculations of the bulk resonance position and its parameters because the number of excited states that lie lower than 4 eV significantly increases with the cluster that grows without additional symmetry constraints. For instance, for the Ag34 cluster, at least 250 excited states should be calculated for correctly reproducing the bulk plasmon resonance peak (here using Casida’s transitionbased approach).58 One alternative way to solve this limitation and to simulate the optical absorption spectra of Ag clusters up to 561 atoms is a TDDFT time-evolution formalism proposed by Yabana et al.59 and realized by Weissker et al. using the realspace code Octopus.13,60,61 However, it does not principally solve the size limitation of TDDFT approximation with respect to silver clusters even using the less computationally expensive LDA approximation or applying symmetry constraints. Moreover, the small silver clusters are very unstable even in the presence of a coverage shell of organic ligands, something that complicates the investigation of their optical properties and J

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Figure 13. Polarizabilty of a 1481 atom silver cluster with different values for the capacitance parameter c. Since all curves with c = 0.001 or smaller are all on top of each other, not all curves are directly visible. For the no capacitance curve, the interaction energy between fluctuating charges has been omitted. All curves have been calculated with the fitted parameters ωa and A.

from c = 0.001 and below (see Figure 14). While the capacitance term in eq 1 does not seem to be important for

Figure 12. Doubly occupied (Occ.) and unoccupied (Unocc.) molecular orbitals pairs that correspond to the main electronic configuration of the most intensive electronic transition in the absorption spectra of Agn (n = 18−34) clusters calculated by the camB3LYP/Lanl2DZ method (the contour value of the isosurface is 0.015 au).

comparison between the theoretical end experimental spectra.48,62−64 In this context, the different models for the frequency-dependent permittivity of silver particles look more promising for the explanation of surface and bulk plasmon resonances of real-size systems (up to 20,000 atoms within our coordination-dependent discrete interaction model that correspond to the approximate size of the spherical nanoparticles of approximately 12 nm).

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APPENDIX C

Figure 14. Polarizabilty of the Ag(6.18, 2.23) nanorod with different values for the capacitance parameter c. The values 6.18 and 2.23 are designated to the longitudinal and transverse plasmon lengths in nm, respectively. For the no capacitance curve, the interaction energy between fluctuating charges have been omitted. All curves have been calculated with the fitted parameters ωa and A. All curves calculated with the capacitance parameter c below 0.001 or omitted are overlapping each other. The range of the polarizability for c = 1 has been cropped in order to better present the rest of the curves.

Capacitance Parameter

Since there is no known connection between the capacitance parameter c and any measurable atomic quantity, this parameter should in principle be fitted as attempted by others.20,22,27,28 We, however, found that the capacitance parameter could not be fitted from a set of spherical clusters since there was no discernible difference in the frequencydependent polarizabilty for values in the range of 105 to 10−8 except in the region of 10 to 10−2 where the method showed numerical instability as shown for a 1481 atom silver cluster in Figure 13. Due to the numerical instability from 10 to 10−2 and the otherwise numerical insensitivity outside this region, we have chosen to use 0.0001 au as the value of the capacitance parameter c in the fitting. It is, here, a bit surprising that the value of the capacitance parameter c does not seem to matter for clusters with only one type of atom and the exclusion of the interaction energy between fluctuating charges does not alter the polarizabilty either as seen in Figure 13. Due to the numerical instability, the polarizability can take negative values as seen for c = 1 in Figure 13, which looks more like the real part of a complex resonance. For nanorods, we observe a similar picture though the region with numerical instability is larger and stability is only seen

nanoclusters consisting of only one type of element, we expect it to be more important for composite materials, different close-lying metallic clusters, and clusters in external electromagnetic fields or with coordination number-dependent Gaussian charge distributions though further investigations along those lines are needed for definite conclusions.

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APPENDIX D

Gamma Parameter

While the γ parameter in the Lorentzian function in eq 17 does not affect the position of the SPR, provided that γ is small in comparison to ω, the smoothing effect of γ, however, influences the absolute polarizability significantly. Since we do not fit our model to the absolute polarizability from TDDFT calculations but instead the SPR to experimental values, the γ value is instead chosen. K

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the position of the SPR, it is possible to later fit either a γ value or size-dependent function for γ if experimental data is found without having to fit ωa and A again.

In Figure 15, the polarizability of a 1481 atom spherical cluster with different γ values in the 0.002−0.030 au range. In

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Vadim I. Zakomirnyi: 0000-0002-2049-7259 Zilvinas Rinkevicius: 0000-0003-2729-0290 Glib V. Baryshnikov: 0000-0002-0716-3385 Hans Ågren: 0000-0002-1763-9383 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS H.Å. and V.I.Z. acknowledge the support of the Russian Science Foundation (project no. 18-13-00363). L.K.S. acknowledges the support of Carl Tryggers Stifetelse, project no. CTS 18-441.

Figure 15. Polarizability of a 1481 atom spherical silver cluster with different γ values.

the 0.002−0.006 au range, several peaks are visible, and the SPR is clearly a double peak, but from approximately 0.010 au and above, the double-peak SPR becomes a smooth single peak. For the very small γ values, the SPR is therefore shifted slightly but remains constant at 0.010 au and above. Furthermore, the absolute polarizability increases more than fourfold from γ at 0.030 to 0.002 showing that γ would be a very sensitive parameter if the absolute polarizability was fitted to TDDFT calculations. While the polarizability drops, the FWHM, of course, increases significantly. For nanorods, the effect of γ is similar to that of spheres. As seen for the Ag(6.18, 2.23) nanorod in Figure 16, the TLSPR

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REFERENCES

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Figure 16. Polarizabilty of the Ag(6.18, 2.23) nanorod with different γ values. The values 6.18 and 2.23 are designated to the longitudinal and transverse plasmon lengths in nm, respectively.

for lower γ values is a clear double peak, while larger γ values smooth the two peaks. In order to have a γ value that still can show some structure and have a FWHM that is reasonably consistent with the experimental values from Ringe et al.,26 we have chosen γ = 0.016 au. While we here show absolute and not normalized polarizabilities, it is obvious that, because the γ parameter is not fitted but chosen, only the relative or normalized polarizabilities should be interpreted. We here show the absolute values in order to demonstrate how the polarizability increases with the number of atoms and the geometry of the cluster. Since the polarizability in the range tested here does not affect L

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PAPER VI

Collective lattice resonances in arrays of dielectric nanoparticles: a matter of size V. I. Zakomirnyi, A. E. Ershov, V. S. Gerasimov, S. V. Karpov, H. Agren, I. L. Rasskazov Optics Letters 44(23), 5743-5746 (2019)

Paper VI Letter

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Collective lattice resonances in arrays of dielectric nanoparticles: a matter of size V. I. Zakomirnyi,1,2,3 A. E. Ershov,4,5,6 H. Ågren,1,2 AND I. L. Rasskazov7, *

V. S. Gerasimov,4,5

S. V. Karpov,3,5,6

Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal Institute of Technology, Stockholm SE-10691, Sweden Federal Siberian Research Clinical Centre under FMBA of Russia, Krasnoyarsk 660037, Russia 3 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, Russia 4 Institute of Computational Modeling SB RAS, Krasnoyarsk 660036, Russia 5 Siberian Federal University, Krasnoyarsk 660041, Russia 6 Siberian State University of Science and Technology, Krasnoyarsk 660014, Russia 7 The Institute of Optics, University of Rochester, Rochester, New York 14627, USA *Corresponding author: [email protected] 1

2

Received 9 October 2019; accepted 28 October 2019; posted 31 October 2019 (Doc. ID 379884); published 26 November 2019

Collective lattice resonances (CLRs) in finite-sized 2 D arrays of dielectric nanospheres have been studied via the coupled dipole approximation. We show that even for sufficiently large arrays, up to 100 × 100 nanoparticles (NPs), electric or magnetic dipole CLRs may differ significantly from the ones calculated for infinite arrays with the same NP sizes and interparticle distances. The discrepancy is explained by the existence of a sufficiently strong crossinteraction between electric and magnetic dipoles induced at NPs in finite-sized lattices, which is ignored for infinite arrays. We support this claim numerically and propose an analytic model to estimate a spectral width of CLRs for finite-sized arrays. Given that most of the current theoretical and numerical researches on collective effects in arrays of dielectric NPs rely on modeling infinite structures, the reported findings may contribute to thoughtful and optimal design of inherently finite-sized photonic devices. © 2019

phenomenon has been extensively discussed for plasmonic NPs [27–33] with strong electric dipole (ED) resonances, and for all-dielectric NPs with ED and magnetic dipole (MD) optical resonances [34–37]. To date, the overwhelming majority of studies on CLRs deal with infinitely large arrays of NPs and ignore the presence of physical boundaries of arrays, either in full-field simulations [38–40] or dipole [41–44] and higher-order [45,46] semi-analytic approximations. It is commonly assumed that boundary effects are negligible for large arrays synthesized in experimental studies; thus, an infinite-array model is often considered as a satisfactory approximation. Nonetheless, the effects of the finite size have been thoroughly discussed for CLRs in regular nanostructures with Au [47–49], Ag [49–52], and graphene [49] constituents via the dipole approximation. Generally, one could expect the quality factor of CLRs in 20 × 20 and larger arrays of plasmonic NPs to be close to that of infinite arrays. However, in Refs. [47–51], NPs are considered as purely EDs, since ED oscillations predominate in plasmonic NPs. Thus, it is not obvious a priori, how CLRs in finite-sized arrays of dielectric NPs with strong ED and MD resonances differ from CLRs in infinite arrays, though brief discussions of up to 21 × 21 [17] and 30 × 30 [37] Si NP arrays have been reported recently. In this Letter, we address this problem and find regimes where the “infinite array” approximation is no longer reliable for CLRs in arrays of dielectric NPs with both ED and MD resonances. Figure 1 shows a 2D array of Ntot = N × N identical spherical NPs embedded in a vacuum and illuminated by a plane wave with Einc (r) = E0 exp(ik · r) and Hinc (r) = H0 exp(ik · r), where E0 = (E 0x , 0, 0) and H0 = (0, H0y , 0) are amplitudes of the electric and magnetic fields, respectively, and k is a wave vector. The time dependence exp(−iωt) is assumed and suppressed. Each NP is considered as a point dipole, so ED and MD moments di and mi induced on the i-th particle are coupled to dipoles on other j  = i particles and to the incident

Optical Society of America https://doi.org/10.1364/OL.44.005743

All-dielectric nanophotonics is a rapidly growing field in modern physics [1] that provides a low-loss platform for an impressive number of applications such as color printing [2], biosensing [3–5], lasing [6], waveguiding [7–9], and flat [10,11] and nonlinear [12–14] optics. While even a single dielectric nanoparticle (NP) may exhibit extraordinary electromagnetic response [15,16], its periodic array possesses a richer variety of properties, extensively discussed recently [17–21]. Such interest in regular arrays of dielectric NPs is justified (among other factors) by the emergence of a tunable high-quality-factor lattice-mediated electromagnetic response that gives rise to collective lattice resonances (CLRs) [22–24]. CLRs originate from the strong electromagnetic coupling between NPs comprising the lattice, which usually occurs at wavelengths close to the Wood–Rayleigh anomalies [25,26] of the lattice. This 0146-9592/19/235743-04 Journal © 2019 Optical Society of America

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Fig. 1. Sketch of considered 2D array from N × N spherical NPs with radius R and center-to-center distances h x and h y along X and Y directions. Fig. 2. (a) Real parts of normalized denominators of Eq. (2), which correspond to d x (ED) and m y (MD); (b) extinction efficiency for infinite (∞) and for N × N finite-sized arrays. Dashed vertical lines show the spectral positions of (0, ±1) and (±1, 0) Wood–Rayleigh anomalies; (c), (d) zoomed-in spectra for ED and MD CLRs, respectively. Dashed vertical lines indicate the position of the CLR peak for the infinite array. Arrays from Si NPs (refractive index from [58]) with R = 65 nm, h x = 580 nm, and h y = 480 nm are considered.

field [41,53,54]: e



di = α Einc (ri ) + 

Ntot 

mi = α m Hinc (ri ) +

j =i

Gi j d j −

Ntot  j =i

Ntot  j  =i

Gi j m j +



gi j × m j  ,

Ntot  j  =i



gi j × d j  ,

(1a)

(1b)

due to G 0x x  = G 0y y and non-trivial wavelength dependence of polarizabilities α e ,m (λ) (Fig. 4 in [41]). To characterize the electromagnetic response of the finite-sized array, we use the extinction efficiency [41,54]

where α e ,m are ED and MD polarizabilities [55], and × denotes a vector product. Tensor G i j and vector gi j describe the interaction between i-th and j -th dipoles [41,53,54]. Note that G i j is responsible for interaction between dipoles of the same kind (ED ↔ ED or MD ↔ MD), while gi j stands for ED ↔ MD cross-interaction. The essence of CLRs can be understood from a closed-form analytical solution of Eq. (1) obtained for the infinite array [29,41]. In this case, di = d  E0 and mi = m  H0 for each NP [41]; therefore, the last terms in Eq. (1) vanish, since E0 ⊥H0 . Thus, for a special case of a regular 2D lattice illuminated with a normally impinging wave with |E0 | = E 0x and |H0 | = H0y , the non-zero components of d and m are     d x = E 0x / 1/α e − G 0x x , m y = H0y / 1/α m − G 0y y , (2) where G 0x x and G 0y y are diagonal elements of 3 × 3 tensor  e ,m G0 = ∞ − G 0x x ,y y )−1 are effective electric j =2 G 1 j , and (1/α and magnetic polarizabilities that capture the features of the NP’s surrounding [33,41,47]. The summation in G 0 implies the use of the non-trivial Ewald method well known for 1D [56] and 2D [57] lattices, which thus has been implemented in this work. From the analysis of Eq. (2), one could expect to observe resonances if Re(1/α e ,m − G 0x x ,y y ) vanishes for either ED or MD moments. Indeed, Fig. 2(a) shows that the dimensionless representation of the above parameter becomes zero near λ ≈ h y and λ ≈ h x for d x and m y , respectively, which corresponds to (0, ±1) and (±1, 0) Wood–Rayleigh anomalies. Note that in the general case of h x  = h y considered here, a simple rotation of the incident field polarization, e.g., (E 0x , 0, 0) → (0, E 0y , 0), does not yield the interchange between ED and MD CLR spectral positions, since it implies only the interchange G 0x x ↔ G 0y y in Eq. (2), which will likely violate the Re(1/α e ,m − G 0x x ,y y ) = 0 condition

Q fin ext =

4k |E0 |2 Ntot R 2

Im

Ntot   i=1

 di · E∗inc (ri ) + mi · H∗inc (ri ) ,

(3) where the asterisk denotes a complex conjugate, and di and mi are defined from the solution of Eq. (1). For an infinite array, after substituting Eq. (2) in Eq. (3), one gets  −1  −1  4k . (4) Im 1/α e − G 0x x + 1/α m − G 0y y Q inf ext = 2 R

We are now ready to consider Q ext for infinite and finite-sized arrays. Figure 2(b) shows that extinction spectra for finite-sized arrays gradually approach the spectrum for the infinite lattice as N increases, which is consistent with reported trends for arrays of plasmonic NPs [47,49]. Indeed, ED CLR at λ ≈ 490 nm for arrays with Ntot > 50 × 50 becomes almost indistinguishable from one for the infinite array, as it is clearly seen in Fig. 2(c). Of note, for plasmonic NP arrays, the corresponding “threshold,” when Q ext becomes almost the same for finite and infinite lattices, is ≈ 20 × 20 NPs [47]. Analogously, Q ext for an MD CLR at λ ≈ 586 nm in finite-sized arrays becomes similar to the infinite case if N grows, as shown in Fig. 2(d). However, what is really surprising and unexpected is that Q ext of finite-sized arrays is noticeably different even for the Ntot = 100 × 100 case. Moreover, the Fano-type profile for CLRs [30] in Fig. 2(d) is significantly different for infinite and finite-sized arrays near λ ≈ h x , which implies the existence of non-negligible electromagnetic interaction emerging in finite-sized arrays. To understand and explain these trends, we recall the difference between Eqs. (1) and (2), i.e., the last terms of Eqs. (1a) and (1b), which, respectively, provide the electric field at the i-th NP mediated by MDs on other j  = i NPs and, vice versa, the

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Fig. 4. (a) ED and (b) MD CLRs spectral width λ calculated with Eq. (5) (analytical) and from data in Fig. 2(b) (numerical).

λr 

h x ,y 2π N

(5)

for CLRs coupled with the (±1, 0) or (0, ±1) Wood–Rayleigh anomaly with λ ≈ h x and λ ≈ h y , respectively. Therefore, for a given spectral width λinf of the CLR for the infinite array [which is readily obtained via Eq. (4)], one can immediately get its size-dependent counterpart with λ = λinf + λr without extensive simulations of finite-sized arrays. Figure 4 compares analytical estimates of λ with the corresponding numerical data from Fig. 2(b). One can see that, indeed, Eq. (5) provides reliable estimates of λ for sufficiently large arrays (Ntot > 30 × 30 and Ntot > 70 × 70 for ED and MD, respectively). It is worthwhile to emphasize that the pronounced discrepancy between analytic approximation and numerical calculations for smaller arrays supports the claim that cross-interaction terms, ignored in Eq. (5), indeed matter for CLRs in finite-sized arrays, especially for the MD case. Here, we have considered arrays of NPs embedded in a homogeneous environment under normal illumination via the coupled dipole approximation, which is a quite insightful approach valid for experimentally feasible setups [11]. Qualitatively similar effects are expected in homogeneous media with refractive index  = 1 or under oblique incidence, though the position of the CLRs will be shifted due to the change in G 0 and/or α e ,m [11,30]. We also anticipate that reported finitesized effects will likely emerge in a more sophisticated manner for higher-order electromagnetic interactions [46,60,61] or in non-homogeneous environments [62–65]. Even though we have limited the discussion to Si NPs, the obtained results are qualitatively valid for appropriately scaled arrays of dielectric particles from other materials [66] at corresponding frequencies, as long as CLRs are emerged. For instance, it is a well-known practice to verify concepts of all-dielectric nanophotonics with millimeter-sized ceramic particles under microwave illumination [7,21]. To conclude, we have shown that the finite size of arrays of dielectric NPs plays an important role for the emergence of both ED and MD CLRs. We have demonstrated that ED ↔ MD cross-interactions significantly contribute to both types of CLRs, even in sufficiently large NP arrays, where such interaction is usually considered to be negligible. While ED CLRs in finite-sized arrays converge to the infinite-array model for ≈ 50 × 50 NPs, MD CLRs in finite-sized arrays are quite different from the ones of infinite arrays even for 100 × 100 NPs; thus, the common use of numerical and theoretical models for infinite arrays should be handled with great caution. Given that a significant number of works on CLRs in all-dielectric nanostructures deal with numerical or theoretical considerations of infinite arrays, we believe that the reported results may lead to deeper understanding and more thoughtful research of

Fig. 3. Normalized intensities of electric field induced by MDs (left) and of magnetic field induced by EDs (right) for N × N arrays at wavelengths that correspond to peaks of ED (left) and MD (right) CLRs [see Figs. 2(b)–2(d) for details]: (a) 493 nm, (b) 588 nm, (c) 490 nm, (d) 586.5 nm. Each dot represents the NP, and the actual sizes of arrays vary for different N × N.

magnetic field at the i-th NP mediated by EDs. Figure 3 shows the corresponding intensities, i.e., |Emag |2 and |Hel |2 , for each NP in the array. It can be seen that the normalized intensity of the electric field induced by MDs is quite small compared to the incident field, and increases only at the boundaries of the array, which again agrees well with results for plasmonic NPs [49]. The maximum value of |Emag |2 /|E0 |2 , which is already quite small for 30 × 30 arrays in Fig. 3(a), gradually decreases for larger arrays, and almost vanishes for the 70 × 70 array in Fig. 3(c), thus providing a negligible difference for ED CLRs of infinite and sufficiently large finite-sized arrays in Fig. 2(c). On the contrary, the maximum intensity of the magnetic field induced by electric dipoles, i.e., |Hel |2 /|H0 |2 , increases for larger arrays, and again a divergence takes places near the boundaries of the array. Although the overall contribution of cross-interaction between EDs and MDs to Q fin ext gradually decreases as N grows, the “boundary effect” is pronounced even for sufficiently large arrays, and thus cannot be completely ignored in this case. In other words, for MD CLR, the last term in Eq. (1b) has to be taken into account to get a reliable estimate of the extinction efficiency, which is clearly justified by the discrepancy between Q ext for infinite and finite arrays in Fig. 2(d). Finally, to get even deeper insight, we provide the following analytical considerations. For the electromagnetic field confined in a finite volume, the corresponding wave vector is also distributed in a finite volume of the reciprocal space: r kr  1 [59]. In our case, the finiteness of the lattice mode in space is determined by the lattice size r ≈ Nh x ,y , where N and h x ,y correspond to the direction perpendicular to the polarization of the respective component of the electromagnetic field (i.e., Nh x for H0 , and Nh y for E0 ). Thus, the confinement of the wave vector in the reciprocal space is kr ≈ 2π λr /λ2 , which yields

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Vadim Zakomirnyi 5746

Letter

Vol. 44, No. 23 / 1 December 2019 / Optics Letters

electromagnetic phenomena in this rapidly developing field of nanophotonics.

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Funding. Russian Science Foundation (19-72-00066). Acknowledgment. We thank Alexander Moroz for fruitful and valuable discussions. Disclosures. The authors declare no conflicts of interest. REFERENCES 1. I. Staude, T. Pertsch, and Y. S. Kivshar, ACS Photon. 6, 802 (2019). 2. A. B. Evlyukhin and S. I. Bozhevolnyi, Opt. Mater. Express 9, 151 (2019). 3. O. Yavas, M. Svedendahl, P. Dobosz, V. Sanz, and R. Quidant, Nano Lett. 17, 4421 (2017). 4. A. Krasnok, M. Caldarola, N. Bonod, and A. Alú, Adv. Opt. Mater. 6, 1701094 (2018). 5. F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, Nat. Photonics 13, 390 (2019). 6. K. S. Daskalakis, P. S. Eldridge, G. Christmann, E. Trichas, R. Murray, E. Iliopoulos, E. Monroy, N. T. Pelekanos, J. J. Baumberg, and P. G. Savvidis, Appl. Phys. Lett. 102, 101113 (2013). 7. R. S. Savelev, A. P. Slobozhanyuk, A. E. Miroshnichenko, Y. S. Kivshar, and P. A. Belov, Phys. Rev. B 89, 035435 (2014). 8. E. N. Bulgakov and D. N. Maksimov, Opt. Lett. 41, 3888 (2016). 9. R. M. Bakker, Y. F. Yu, R. Paniagua-Domínguez, B. Luk’yanchuk, and A. I. Kuznetsov, Nano Lett. 17, 3458 (2017). 10. R. C. Devlin, M. Khorasaninejad, W. T. Chen, J. Oh, and F. Capasso, Proc. Natl. Acad. Sci. 113, 10473 (2016). 11. R. C. Ng, J. C. Garcia, J. R. Greer, and K. T. Fountaine, ACS Photon. 6, 265 (2019). 12. V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, Opt. Express 24, 15965 (2016). 13. S. Liu, G. A. Keeler, J. L. Reno, M. B. Sinclair, and I. Brener, Adv. Opt. Mater. 4, 1457 (2016). 14. D. Smirnova, A. I. Smirnov, and Y. S. Kivshar, Phys. Rev. A 97, 013807 (2018). 15. M. V. Rybin, K. L. Koshelev, Z. F. Sadrieva, K. B. Samusev, A. A. Bogdanov, M. F. Limonov, and Y. S. Kivshar, Phys. Rev. Lett. 119, 243901 (2017). 16. R. Colom, R. McPhedran, B. Stout, and N. Bonod, J. Opt. Soc. Am. B 36, 2052 (2019). 17. V. E. Babicheva and A. B. Evlyukhin, Laser Photon. Rev. 11, 1700132 (2017). 18. E. N. Bulgakov and D. N. Maksimov, Phys. Rev. Lett. 118, 267401 (2017). 19. E. N. Bulgakov and D. N. Maksimov, J. Opt. Soc. Am. B 36, 2221 (2019). 20. A. Rahimzadegan, D. Arslan, R. N. S. Suryadharma, S. Fasold, M. Falkner, T. Pertsch, I. Staude, and C. Rockstuhl, Phys. Rev. Lett. 122, 015702 (2019). 21. H. K. Shamkhi, K. V. Baryshnikova, A. Sayanskiy, P. Kapitanova, P. D. Terekhov, P. Belov, A. Karabchevsky, A. B. Evlyukhin, Y. Kivshar, and A. S. Shalin, Phys. Rev. Lett. 122, 193905 (2019). 22. M. B. Ross, C. A. Mirkin, and G. C. Schatz, J. Phys. Chem. C 120, 816 (2016). 23. V. G. Kravets, A. V. Kabashin, W. L. Barnes, and A. N. Grigorenko, Chem. Rev. 118, 5912 (2018). 24. W. Wang, M. Ramezani, A. I. Väkeväinen, P. Törmä, J. G. Rivas, and T. W. Odom, Mater. Today 21(3), 303 (2018). 25. R. W. Wood, Proc. Phys. Soc. London 18, 269 (1902). 26. L. Rayleigh, Proc. R. Soc. A 79, 399 (1907). 27. S. Zou and G. C. Schatz, J. Chem. Phys. 121, 12606 (2004). 28. S. Zou, N. Janel, and G. C. Schatz, J. Chem. Phys. 120, 10871 (2004).

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PAPER VII

Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models V. I. Zakomirnyi, I. L. Rasskazov, L. K. Sorensen, P. S. Carney, Z. Rinkevicius, H. Agren in manuscript (2019)

Paper VII Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models

Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models

Vadim I. Zakomirnyi,1, 2 Ilia L. Rasskazov,3 Lasse K. Sørensen,1 P. Scott Carney,3 Zilvinas Rinkevicius,1 and Hans Ågren4, 1, 5

1) Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal Institute of Technology, Stockholm, SE-10691, Sweden 2) Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia. 3) The Institute of Optics, University of Rochester, Rochester, NY 14627, USA 4) Federal Siberian Research Clinical Centre under FMBA of Russia, 660037, Kolomenskaya, 26 Krasnoyarsk, Russia. 5) College of Chemistry and Chemical Engineering, Henan University, Kaifeng, Henan 475004 P. R. China

(Dated: 25 November 2019)

Using the extended discrete interaction model and Mie theory we investigate the tunability of the optical polarizability of small metallic nano-bubbles. We show that the energies of the two dipolar plasmon resonances vary with the ratio of particle radius to hole radius in a manner similar to what has been predicted for clusters using a semiclassical approach of two coupled harmonic oscillators for uniform metallic shell structures. We show that, according to the extended discrete interaction model, the dipolar plasmon resonances also are present in the 2 − 13 nm size region and show the same functional dependence as for the larger nanobubbles. Using previously fitted data from experiment we can predict the size-dependence of the plasma frequency for nano-bubbles in the 1 − 15 nm size region. We find that sizecorrected Mie theory is not able to reproduce the same functional from of the dipolar modes for the nano-bubbles in same size region. Keywords: ex-DIM, nano-bubbles, plasmonics, Mie theory I.

INTRODUCTION

Although nanoplasmonics since long has constituted a research branch that has received strong attention as a versatile nanotechnology and by now turned into mature research with significant applications in areas like bioimaging, photonics and energy harvesting, there is still a lag between experiment and theoretical capability to design nanostructures with particular plasmonic properties. Among a number of classical models, Mie theory has been instrumental in predicting light scattering and plasmonic resonances in nanosized metallic particles. Since Mie theory relies on the concept of a dielectric constant it is, however, restricted to a size comprising larger nanoparticles where the classical bulk dielectric constant remains valid and frequency ranges where experimental results are available. Although corrections for Mie theory that take into account quantum size effects have been proposed and shown to improve results1 , the use of dielectric constants of bulk materials makes it impossible to take into account structural differentiation for the dielectric response and in particular properties that more precisely relate to the discrete atomic structure of the nanoparticles. As the surface to volume ratio becomes large in the sub-15 nm region, and the particle size then becomes smaller than the mean free path of the conduction electrons, there are limits to its use towards smaller nanoparticles. These limits are yet there to uncover. At the other end, pure quantum approaches, like timedependent density functional theory (TD-DFT), applicable for the very small particles, are constantly developed

towards larger size. However, inherent scaling restrictions limit their current applicability to about 1 nm.2–5 This leaves the 1 − 15 nm size region unattainable by either classical (n.b. Mie) and quantum theory, which is unfortunate considering the wide applicability of small plasmonic nanoparticles within that size range, e.g. for cell imaging.6 Discretization of the interactions down to the atomic level is thus called for, something that indeed has been explored more recently by discrete interaction models7–10 . The simplest variant of this set of models considers a set of atoms with fixed polarizabilities to interact in accordance with classical electrostatics. However, despite the atomistic nature, there are limited capabilities inherent in this approach to describe the dependence of the polarizability of the surface topology or structure of the metallic nanoparticles as the atom polarizabilities themselves are assumed constant throughout the particle. The proposition of Chen et al. 10 to expand the discrete interaction model by making the atom polarizabilities coordination dependent (cd-DIM), forms a way to overcome limitations of the original DIM. In a recent work we presented an extended discrete interaction model (ex-DIM) to simulate the geometric dependence of plasmons in the "missing" size range of 1 − 15 nm where the frequencydependent dielectric function from the Clausius-Mossotti relation is replaced by a the static atomic polarizability times the sum of three size-dependent Lorentzian oscillators and, furthermore, with Gaussian charge distributions and atomic radii that vary with the coordination number11 . The frequency-dependent Lorentzian oscillators depend on the plasmon length along the three Carte-

165

Vadim Zakomirnyi Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models sian directions using the concept of the plasmon length as defined in the work of Ringe et al. 12 . As shown in this previous paper the ex-DIM model, which is parametrized versus experimental data, enables a robust description of the polarizability of nanoparticles with different geometries11 . The purpose of the present work is three-fold, on one hand to test more precisely how far towards small size Mie theory is applicable by contesting results against exDIM and on the other to provide a bridge between the two theories (Mie and ex-DIM) so to cover the full nanoscale for plasmonic generation with a working approach for numerical simulations. Thirdly, using nano-bubbles as test beds, we also explore the role of cavity versus volume size of spherical particles as an additional design criterion for plasmonics generation along with the intrinsic size dependence of the particles.

II.

THEORY

We will here sketch the physical problem and the prediction for the red- and blue-shifts of the hybridized plasmon modes seen for nano-bubbles in Fig. 1 and then in the following subsections describe the physical models we use to tackle the problem. The surface charges on the nano-bubbles can be considered as a hybridization of the plasmon modes from a sphere and a cavity and in a semiclassical approach (SCA) be described as two coupled harmonic oscillators.13,14

2 = ω±

The position of hybridized plasmon modes seen in Fig. 1 with frequencies ω− and ω+ are in SCA defined from the following equation15–17 :

   2l+1  2 ωB Rcore 1 ± 1 , 1 + 4l(l + 1) 2 2l + 1 Rtot

(1) where Rcore and Rtot are radii of core and total radius of particle, ωB is the bulk plasma frequency, and l is the order of spherical harmonics (l = 1 in our case). √ The surface plasmon resonance frequency ωsph = ωB / 3 corresponds to the dipolar (l = 1) resonance of a solid spherical particle with radius rtot . Prodan et al. have shown that the SCA not only gives a good intuitive picture but also provides an effective way to predict the different surface plasmon resonance frequencies for both small and large nano-bubbles.16,17 Measurements of the dielectric function of Ag in the frequency range that we here examine gives ωB = 8.5 − 9.6 eV18–22 though greater variances of ωB have seen throughout time in literature. These measurement have all been on systems which are large enough to be considered as bulk systems. For very small spherical clusters in the 1 − 15 nm size range Scholl et al.23 showed that the surface plasmon resonance ωsph varied up to 0.5 eV depending on the size of the particle. We recently showed11 that fitting the experimental data to the inverse plasmon length12 one could predict ωsph of solid particles in the 1 − 15 nm size range. Since ωsph is size-dependent and directly proportional to ωB in Eq. 1 we would also expect that ωB should show similar size dependence. Since we expect that ωsph will be in the 3.3 − 3.6 eV frequency range for solid particles in the size we here will use we would expect that ωB should be in the 5.7 − 6.2 eV frequency range provided Eq. 1 gives a valid description of the physical problem. A.

FIG. 1. Scheme of hybridized modes in nano-bubbles: symmetric ω− and anti-symmetric ω+ modes.

2

ex-DIM

We recently derived a new atomistic model, known as the extended Discrete Interaction Model (ex-DIM)11 , which is an further development of the DIM model8,9 where we introduced significant improvements in the description of the surface topology and geometric dependence of the SPR(s). The aim of the ex-DIM is to model systems in the 1 − 15 nm size range which is out of the reach of quantum mechanical models due to the scaling of these methods and where classical models fail since the concept of a bulk dielectric constant in this region is no longer valid. Unlike previous interaction models we have not parameterized the ex-DIM from TD-DFT calculations but instead directly from the experimental data 23 since this is the only way that accurate data for the size range of 1 − 15 nm can be obtained.11 In the ex-DIM model every atom is represented by a Gaussian charge distribution and endowed with a polarizability and a capacitance which governs the atomic interaction subject to a charge equilibration constraint as seen from the Lagrangian

166

Paper VII Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models L[{µ, q}, λ] = E[{µ, q}] − λ(q tot − N

−

N

N 

qi ) =

i

N

N

1 2

N 

qi c−1 ii qi +

i

1 2

N

N  N  i

(0)

qi Tij qj +

j=i

N

1 2

3

N 

µi α−1 ii µi

i

N

    1  (2) (1) µi Tij µj − µi Tij qj + qi Vext − µi Eext − λ(q tot − qi ) . 2 i i i i i j=i

j=i

In Eq. 2 the Lagrangian is written in the usual form as the interaction energy minus the constraint. Here, the E is the energy, q tot is the total charge of the nanoparticle, N is the number of atoms in a nanoparticle qi is the fluctuating charge assigned to the i-th atom, µi is the fluctuating dipole assigned to the i-th atom, the cii is the i-th charge self-interaction tensor, the αii is the i-th (0) (1) (2) dipole self-interaction tensor, the Tij , Tij , and Tij are the electrostatic interaction tensors, the Vext is the external potential and the Eext is the external electric field. The first term in the energy is the self-interaction energy of fluctuating charges, the second term is the interaction energy between fluctuating charges, the third term is the self-interaction energy of fluctuating dipoles, the fourth term is the interaction energy between fluctuating charges and dipoles, the fifth term is the interaction energy between fluctuating dipoles, the sixth term is the interaction energy between fluctuating charges and the external potential, the seventh term is the interaction energy between fluctuating dipoles and the external field, and the last term is a charge equilibration condition expressed via the Lagrangian multiplier λ. In order to capture complex surface topologies a coordination number, as defined by Grimme24 , is assigned to each atom. The coordination number fcn modifies the polarizabilty through the scaling of the radius  3 Ri (fcn ) αi,s,kl L(ω, P) (3) αii,kl (ω) = Ri,bulk and likewise for the capacitance   Ri (fcn ) L(ω, P). (4) cii,kl = δkl fc with fc = ci,s 1 + d Ri (12)

In Eqs. 3 and 4 Ri,bulk is the bulk radius of the atom, Ri (fcn ) the coordination number scaled radius11 , αi,s,kl the static atomic polarizability25 , d = 0.1 a scaling factor and a size-dependent Lorentzian L(ω, P). We here notice that for the nano-bubbles, shown in Fig. 1, we in this way modify the polarizabilty and capacitance of all atoms on both the outer and inner surface. The geometric dependence of the SPR is determined by the size-dependent Lorentzian L(ω, P) L(ω, P) = N (Lx (ω, Px ) + Ly (ω, Py ) + Lz (ω, Pz )), (5) where we have tried to chose a more universal functionality form by having a Lorentzian for every spatial direction. Each Lorentzian depends on the plasmon length Pi 12 in the given direction ωi (Pi ) = ωa (1 + A/Pi ),

(2)

(6)

be simulated for solid particles. ωa and A are the only fitted parameters in the ex-DIM model.11 While the ex-DIM is parameterized to spherical silver clusters in the 1 − 15 nm size and 3.35 − 3.75 eV frequency range we have shown by calculations on nanorods and nanocubes that the ex-DIM model remains valid in a much broader frequency range. The increased range is due to the interaction between the atoms in the exDIM model where the SPRs can be red or blue shifted far beyond the parametric range as we have shown for nanorods.11 The ex-DIM model does not appear to be limited in the frequency range by the parameterization range and unlike classical models which are limited by the experimental range for which the dielectric constant have been measured. B.

Mie Theory

Within the framework of Mie theory26 extended by Aden and Kerker for two concentric spheres27 , scalar dipolar polarizability of a nano-bubble can be defined as follows:

αMie =

3i ψ1 (xout )a1 − nψ1 (xout )b1 , 2k 3 ξ1 (xout )a1 − nξ1 (xout )b1

(7)

provided that refractive indices of inner sphere and surrounding medium are both unity. Here

A1 =

a1 = ψ1 (nxout ) − A1 χ1 (nxout ) , b1 = ψ1 (nxout ) − A1 χ1 (nxout ) ,

(8)

nψ1 (nxin )ψ1 (xin ) − ψ1 (nxin )ψ1 (xin ) , nχ1 (nxin )ψ1 (xin ) − χ1 (nxin )ψ1 (xin )

(9)

where xin,out = 2πωrin,out /c are dimensionless size parameters, n is the refractive index of the shell, ξ1 (z) = ψ1 (z) − iχ1 (z), ψ1 (z) and χ1 (z) are Ricatti-Bessel functions of the first order, and k is a wave-vector in a free space. In classic electrodynamics, the description of the metal permittivity relies on the concept that electrons are drifting as a cloud (which is essentially a Drude model). However, for finite-size shells, collisions of the electrons on the boundaries of the particle yield in a modification of the permitivity, commonly described as follows:

and in this way cluster size dependence and complicated geometrical shapes, with up to at least three SPRs, can

167

ε = εbulk +

2 2 ωB ωB − , ω 2 + iΓ0 ω ω 2 + iΓω

(10)

Vadim Zakomirnyi Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models where εbulk is experimentally measured data for bulk sample28 , and29

Γ = Γ0 +

υF , Leff

Leff =

4 r23 − r13 . 3 r22 + r12

ex-DIM heat maps we also plot the predicted mode hybridization from Eq. 1 as fitted for ex-DIM. Due to Mie theory predicting a different functional form of the SPRs this is not plotted on the heat maps from Mie theory.

(11)

For Ag, one has21,22 Γ0 = 1.034 eV, ωB = 9.6 eV, and υF = 1.36 × 106 m/s. III.

4

RESULTS

According to Eq. 1 the red- and blue shift of the two SPR’s is only dependent on the relative radius between the hole and the particle and not the radius of the particle as seen for solid particles.11,23 In order to analyse and illustrate these effects we will initially show that the hybridized plasmon modes also are present for small clusters and examine if the SPRs have the same functional form but with a different size-dependent plasma frequency ωB and then present the data in three different ways. For the data presentation we first show the effect of a hole growing inside a particle of a fixed radius. This will be followed by the opposite process where the radius of the hole is fixed and the radius of the particle is growing. Finally we fix the thickness of the particle shell, defined as the difference between the radius of the particle and the hole, and increase the radius of the particle. Here we define the total radius of the particle as the distance from the center of the cluster to the center of the outermost atom plus the effective radius of the outermost atom. The hole is likewise defined as the distance from the center of the cluster to the nearest atom minus the effective radius of the innermost atom. In this way the radii of the cluster and hole are the effective radius of the cluster and hole. It is well known that higher order modes of the SPR becomes increasingly important for particles of sufficiently large size30 , but in order to better compare the smaller clusters calculated with the ex-DIM to some of the larger clusters calculated using Mie theory we have restricted the Mie theory calculations to the electric dipole mode in these comparisons. This means restricting the resonance of the plasmon modes in Eq. 1 to l = 1. In all the heat map plots we have expressed the relative radius between the hole and particle (Rhole /Rtot ) along the x-axis and the frequency in eV along the y-axis. The polarizabilty is plotted as the logarithm of the normalized polarizabilty per volume so that the maximum is one. In this way the colorbar indicates the exponent relative to the maximum and the polarizabilty per atom or volume can be visualized. For the ex-DIM the volume is proportional to the number of atoms in the particle. Since the heat maps are interpolated there can be local dips in the polarizabilty due to the distance between the interpolated points and large gradients along the x- and y-direction for the polarizabilty. We have tried to mitigate this effect by plotting the logarithm of the polarizabilty. Secondly, in order to display the anti-symmetric mode a logarithmic plot was necessary. On top of the

A.

Hybridization of small particles

By plotting the peak position of the symmetric- and anti-symmetric modes from many clusters in the 1 − 13 nm size range with varied hole sizes, as shown in Figure 2, the red- and blue-shifting of the two SPRs from Ex-DIM are obvious. Fitting the symmetric- and anti-symmetric modes separately to Eq. 1 we can find ωBex−DIM for each mode separately and in this way check if our results are in line with the predicted positions from SCA.15–17 From our fit we find that the resonance frequency of the sym− = 5.83 eV and anti-symmetric metric mode ωB ex−DIM + = 5.77 eV for all clusters calculated mode ωB ex−DIM are in excellent agreement with each other and that the ex-DIM model in excellent functional agreement with the predicted development of the SPRs from Eq. 1. We will here use the average for the plasma frequency ωBex−DIM = 5.8 eV in the ex-DIM heat map plots. We here see that ωBex−DIM = 5.8 eV matches the predicted ωB for the larger end of the small clusters and not the bulk plasma frequency ωB as it should be since the bulk plasma frequency is not well defined for these small clusters. By plotting and fitting ωB for clusters of different sizes, as seen in Fig. 3, we are able to observe a shift in ωBex−DIM with respect to the size of the cluster similar to that seen for small clusters in the 1 − 15nm size range.11,23 From the fit of the size dependence of the plasma frequency in Fig. 3 we see that the size dependence of ωBex−DIM is 25.05x + 5.6 eV and close to the size-dependence seen for filled clusters11,23 and the predicted range from Sec. II. Since the SPR varies with the inverse of the plasmon length and in order to have a decent size hole the nano-bubbles become so large that the SPR for the solid particles vary very little and we therefore use the average ωBex−DIM = 5.8 eV in the heat map plots. For small Au nano-bubbles Prodan et al. 16 used the time-dependent local density approximation implemented with the jellium model and using following values for the surface plasmon 6.4 eV and the plasma frequency 9.0 eV. By simply using the bulk values on systems with a size of a few nanometers size effects become difficult to capture and does not appear to be present in the calculations presented by Prodan et al. .16 The average value for the plasma frequency ωBex−DIM = 5.8 eV we find for small nano-bubbles is therefore also significantly different than the bulk plasma frequency in Ag which is 8.5 − 9.6 eV18–22 It is clear from Fig. 2 that the calculated points for the dominant symmetric mode follow the fit very closely along the entire curve whereas the anti-symmetric mode is not as regular. The resonance frequency for the antisymmetric mode for clusters with small ratios could not be clearly separated from the broad symmetric peak and

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FIG. 2. The SPRs of the symmetric and anti-symmetric modes as a function of the relative radius of the core and particle along with curves of the fit of Eq. 1. From the fit we find − = the resonance frequency of the symmetric mode ωB ex−DIM + = 5.77 eV. 5.83 eV and the anti-symmetric mode ωB ex−DIM

5

FIG. 4. Fit of the symmetric mode from Mie theory to Eq. 1  6 and a function where rrcore is used in Eq. 1. The plasma tot frequency is in this case ωB = 6.82 eV and ωBr6 = 5.97 eV

Mie theory does not appear to give the same functional form of the symmetric mode as shown by Prodan et al. .17 We have therefore not included the curve of the fitted symmetric mode in the heat map plots for Mie theory. Due to the different functional form of the SPR in Mie theory it is not possible to compare the fitted plasma frequency ωB to the experimental values or to ex-DIM. Even if the data from Mie theory appears to form three different closely bordering lines we are not able to observe any size dependence for the symmetric mode since we do not know the functional form to fit to. We would expect that the size-corrected Mie theory used here will show some size-dependence of ωB . FIG. 3. The plasma frequency, fitted from the symmetric mode, as a function of the inverse plasmon length along with a fit of the size dependence of the plasma frequency. The fit values for the size dependence are 25.05x + 5.6 eV.

tended towards zero for Rhole → 0 and have therefore been omitted. For very large ratios the symmetric mode is easily red shifted but the anti-symmetric mode is not as easily blue shifted. Due to the onset of interband transitions we have not been able to identify the anti-symmetric mode in Mie theory since the interband transition contribution is significantly larger than that from the anti-symmetric mode. A plot of the functional form of the symmetric mode from Mie theory can be seen in Fig. 4. From Fig. 4 it is obvious that the functional form of the resonance in Mie theory does not follow the predicted plasmon modes from SCA in Eq. 1.17 Using a guessed functional form shows that Mie theory predicts a higher order power of the ratio and the r6 function fitted to the Mie data in Fig. 4 is     6 2 1 ωB rcore  2  ω± = 1± , (12) 1+8 2 3 rtot

where we see a higher power in the ratio compared to Eq. 1 for l = 1. We here stress that Unlike the ex-DIM,

B.

Fixed radius

First, we turn to small Ag nanoparticles where the radius of the particle is fixed but the hole inside grows. From Eq. 1 the expected behaviour of the symmetric and anti-symmetric modes reflects a significant red shift of the symmetric mode and a slight blue shift of the antisymmetric mode with increasing hole size for a fixed radius. Starting with the small clusters from the ex-DIM calculations in Fig 5 we observe the expected red shift of the symmetric mode with increasing ratio. In the region where the ratio is between 0.3−0.6 one sees a clear broadening towards the blue region of the peak and when the ratio is above 0.6 the SPR of the anti-symmetric mode is clearly visible and shows a small blue shift with increasing ratio. Below a ratio of approximately 0.3 no anti-symmetric peak is visible in the heat map plots nor is easily identifiable from the raw data without further data analysis. Comparing Fig 5 (a) and (b) we see that there is no real difference between clusters of these sizes. From Fig. 5 (a) it is seen that the symmetric mode returns to the filled sphere for Rhole → 0. The antisymmetric mode Eq. 1 is also predicted to be present for Rhole → 0, which seems contrary to the prediction from a filled sphere, but as can be seen from Fig. 5 (a) the polarizabilty of the anti-symmetric mode quickly tends

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Vadim Zakomirnyi Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models

6

to zero for small ratios and is therefore not present for filled spheres.

(a)

(a)

(b) FIG. 6. Ag nanoparticles calculated using Mie theory with fixed total radius Rtotal with cluster sizes of 8 nm (a) and 20 nm (b) and various radius of holes Rcore .

(b) FIG. 5. Ag nanoparticles calculated using ex-DIM with fixed total size of 5.97 nm (Rtotal = 2.99 nm) (a) and 7.94 nm (Rtotal = 3.97 nm)(b) and various radius of hole Rcore . The slightly different look of the two figures is due to different range of the x-axis.

For the 5.97 nm cluster shown in Fig 5 (a) the absolute polarizabilty is approximately proportional to the number of atoms in the cluster irrespective of the relative radius. For even smaller clusters we see the same trend but for the larger 7.94 nm cluster the polarizability per atom increases up to 1/3 going from the solid particle to a shell thickness of 1.05 nm, giving a relative radius of 0.87. With a shell thickness of 1.05 nm the shell is only a few atoms thick and removing more atoms from such shell does not make sense. Using Mie theory the red shift of the symmetric mode is clearly visible for the 8 nm and 20 nm clusters plotted in Fig. 6. For both Figs. 6 (a) and (b) it is clear that there are two peaks due to the drop in polarizabilty around 4 eV. The peak of the anti-symmetric mode is, however, not easy to identify since the second peak from the interband transitions is very broad, flat and larger than the peak from the anti-symmetric mode. Comparing the ≈ 8 nm clusters in Fig. 5 (b) from

the ex-DIM and Fig. 6 (a) from Mie theory we see that for filled or spheres with holes with ratio below ≈ 0.3 ex-DIM and Mie theory predicts SPRs very close to each other. But for larger ratios the ex-DIM is significantly red shifted in comparison to Mie theory which is due to the different functional form for the peak resonance frequency for the SPR of the symmetric mode. Another clear difference between the ex-DIM and Mie theory is the prediction of the shift of polarizabilty per volume which increases faster for Mie theory than for the ex-DIM.

C.

Fixed hole

By fixing the hole size at 1.77 nm or 3.74 nm as shown in Fig. 7 we see the same trend as for the fixed radius in Fig. 5 where it is only the ratio that matters. For a fixed hole size in Fig. 8 one also sees that the size of the hole does not matter for Mie theory. Comparing the ≈ 3.75 nm hole in Fig 7 (b) from ex-DIM and in Fig. 8 it is seen that ex-DIM clearly red shifts the symmetric mode much faster than using Mie theory.

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(a)

(a)

(b) FIG. 8. 3.75 nm and 19 nm holes - Mie theory

IV.

(b) FIG. 7. Fixed size of hole at 1.77 nm (a) and 3.74 nm (b)

D.

Fixed thickness

By fixing the thickness of the shell and letting the cluster grow the polarizabilty per atom becomes significantly more evident from the heat map plots of the ex-DIM as seen in Fig. 9. For the very thin 0.72 nm shell the symmetric mode does not show as nice behaviour with respect to the increasing ratio as the slightly thicker shell of 1.11 nm. For very large ratios the anti-symmetric mode also becomes increasingly strong though never anywhere near the symmetric mode. For Mie theory we again see that only the symmetric mode is visible though a small dip around 4 eV shows the very broad interband peak in the 5 − 6 eV region. The trend where Mie theory predicts an increasing polarizabilty per volume is also clearly visible and that the polarizabilty per atom for Mie theory increases faster than that for the ex-DIM. Even if the shells in Mie theory can be made in any thickness the 0.5 nm shell is from an atomistic viewpoint not a physically realistic structure.

OUTLOOK AND DISCUSSION

We have shown that symmetric and anti-symmetric plasmon modes of hollow plasmonic nanoparticles predicted for nano-bubbles based on a semi-classical model of interacting harmonic oscillators,15–17 not only are present for small nano-bubbles in the 2 − 13 nm range but also show a similar size-dependence as seen for filled nanoparticles.11,23 For the extended discrete interaction model (ex-DIM) the plasmon modes follow the same functional form though with a different size-dependent plasma frequency ωB . For the symmetric mode we show a significant red shift with increasing ratio between the cluster and hole, from less than 0.1 and all the way to 0.95 and that the symmetric mode ends in the regular surface plasmon resonance of a filled cluster. We here also observe that the anti-symmetric mode disappears for a filled cluster, since the polarizability goes to zero, as would be expected. Applying Mie theory on nano-bubbles in the 1 − 20 nm size range we do not see the same functional behaviour of the symmetric plasmon mode as predicted by Prodan et al. using the model of two coupled harmonic oscillators and found for ex-DIM.17 Due to the different functional form of the plasmon modes in Mie theory we find that ex-DIM here predicts a larger red shift for the symmetric mode than Mie theory. Because the functional form of the plasmon modes for the size-corrected Mie theory is unknown we have not been able the show a size depen-

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Vadim Zakomirnyi Plasmonic nano-bubbles: atomistic discrete interaction versus classic electrodynamics models

(a)

(a)

(b)

(b)

FIG. 9. Spheres with a fixed shell thickness of 0.72 nm (a) and 1.11 nm (b).

FIG. 10. 0.5 nm and 2.5 nm thick shells - Mie theory

dence of the plasma frequency as would also be expected from the size-corrected Mie theory. The results of the present work, focusing on plasmonic nano-bubbles, confirm the flexibility and reliability of the ex-DIM model in studies of small plasmonic particles for a broad range of structures, shapes and compositions. We are currently engaged in widening the scope of these studies to new types of applications as well as in further development of the ex-DIM model itself. Furthermore we are investigating why size-corrected Mie theory differ from the semi-classical picture of two interacting harmonic oscillators as predicted by the ex-DIM and TDLDA calculations using the jellium model on small particles.16

ACKNOWLEDGMENTS

H.Å. and V.Z. acknowledge the support of the Russian Science Foundation (project No. 18-13-00363). L.K.S acknowledges the support of Carl Tryggers Stifetelse, project CTS 18-441. 1 S.

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