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DISSERTATIONS | DENIS KARPOV | RESONANCE PHENOMENA IN NONLINEAR AND ACTIVE NANOPHOTONICS | N
DENIS KARPOV
RESONANCE PHENOMENA IN NONLINEAR AND ACTIVE NANOPHOTONICS
This work is dedicated to theoretical and experimental investigation of the resonance
optical phenomena occurring in nonlinear and active photonics nanostructures. By
using a wide range of theoretical and experimental techniques we studied glass- metal nanocomposites, whispering gallery
mode semiconductor quantum dots lasers and exciton-polaritons lasing in the bias-
controlled heterostructures.
DENIS KARPOV
Resonance phenomena in nonlinear and active
nanophotonics
Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences
No 251
Academic Dissertation
To be presented by permission of the Faculty of Science and Forestry for public examination in the Auditorium F100 in Futura Building at the University of
Eastern Finland, Joensuu, on December 8, 2016, at 12 o’clock noon.
Department of Physics and Mathematics
Editors: Research Dir. Pertti Pasanen, Pekka Toivanen, Jukka Tuomela, Matti Vornanen
Distribution:
University of Eastern Finland Library / Sales of publications P.O.Box 107, FI-80101 Joensuu, Finland
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ISBN: 978-952-61-2350-9 (Print) ISSNL: 1798-5668
ISSN: 1798-5668 Online
ISBN: 978-952-61-2351-6 (PDF) ISSNL: 1798-5668
ISSN: 1798-5676
70211 KUOPIO FINLAND
email: denis.karpov@uef.fi
Supervisors: Professor Yuri Svirko, Ph.D.
University of Eastern Finland
Department of Physics and Mathematics P.O.Box 111
80100 JOENSUU FINLAND
email: yuri.svirko@uef.fi
Professor Andrei Lipovskii, Ph.D.
St. Petersburg Academic University Department of Physics and Technology of Nanoheterostructures,
194021 ST. PETERSBURG RUSSIA
email: lipovskii@gmail.com
Reviewers: Professor V. A. Makarov, Ph.D Moscow State University International laser center Leninskiye Gory 119991 MOSCOW RUSSIA
email: vamakarov@phys.msu.ru
Professor Erik Vartiainen, Ph.D
Lappeenranta University of Technology School of Engineering Science
P.O.Box 20
FI-53851 LAPPEENRANTA FINLAND
email: Erik.Vartiainen@lut.fi
Opponent: Professor Stefano Pelli, PhD
Institute of Applied Physics nello Carrara Department of Optoelectronics and Photonics Via Madonna del Piano 10
50019 SESTO FIORENTINO ITALY
email: s.pelli@ifac.cnr.it
This work is dedicated to theoretical and experimental investigation of the resonance optical phenomena occurring in nonlinear and active photonics nanostructures. By using a wide range of theoretical and experimental techniques we studied the interaction of intense laser pulses with glass-metal nanocomposites and individual metal nanoparticles deposited on a dielectric surface. In particular, by performing the light-induced transmission measurements we reveal the modification of the metal nanoparticle shape under irradiation with intense femtosecond laser pulses. The numerical simulation allowed us to reveal the contribution of the sharp edges to the polarization and intensity of the second harmonic wave generated by individual metal semispheres deposited onto dielectric substrates. Stimulated emission of InAs quantum dots embedded in the semiconductor ring/disc microcavities with Q-factor as high as 20000 was studied by microphotoluminescence microphotoluminescence
measurements, while silicon carbide 2-dimensional photonic crystals were fabricated and used for development of new nitride growth technology. We also performed theoretical investigation of the exciton polaritons dynamics in a semiconductor microcavity with a saturable absorber. In particular, the role of the dissipative nonlinearity due to emergence of bistability of the polariton condensate was studied. We also develop protocols of soliton formation and destruction in such structures. A microscopic theory of the lasing in the bias-controlled heterostructure was developed. In particular, we simulated the dynamics of the exciton-polariton ensemble and revealed threshold dependence of the number of quasiparticles on the applied bias.
539.122, 620.3
Library of Congress Subject Headings: Photonics; Nanophotonics; Optical resonance; Nanotechnology; Nanostructured materials; Nanocomposites (Materials); Nanoparticles; Metals; Glass; Semiconductors; Quantum dots;
Optoelectronics; Plasmons (Physics); Polaritons; Solitons; Solid state physics;
Second harmonic generation; Nonlinear optics; Bose-Einstein condensation;
Femtosecond lasers; Numerical analysis
I wish to thank my supervisors Prof. Yuri Svirko and Prof. Andrei Lipovskii for support during my PhD studies. I also wish to thank the Head of the Department of Physics and Mathematics Professor Timo Jääskeläinen for opportunity to work in such a pleasant atmosphere.
Special thanks to Dr. Ivan Savenko, my friend and long time collaborator for his guiding in the field of exciton-polaritons. I want to thank Dr. Janne Laukkanen who have taught me everything in the field of fabrication of micro- and nanostructures and good cleanroom practice, Dr. Victor Prokofiev and Dr. Olga Svirko for guiding me through cleanroom facilities. I wish also to thank Dr. Natalia Kryzhanovskay for guiding me throughout micro lasers research. Many thanks to my friends including Dr.
Viatcheslav Vanyukov, Mrs Feruza Tuyakova and Mr Semen Chervinskii for their friendship and help both inside and outside the University.
I very grateful to Mrs Hannele Karppinen, Mrs Katri Mustonen and Dr. Noora Heikkilä for their assistance and backing during my PhD studies.
I wish to express my gratitude to my mother Olga and my wife Elena for their strong and permanent support through all my studies.
Joensuu December 8, 2016 Denis Karpov
ALD atomic layer deposition BEC Bose-Einstein condensate DBR distributed Bragg reflectors DDE drift diffusion equations DS dissipative soliton
EBL electron beam lithography EHR electron hole recombination EMA effective medium approximation EP exciton polariton
ES excited state
FEM finite elements method
FWHM full width at half-maximum GMN glass-metal composite
GPE Gross-Pitaevskii equation GS ground state
ICP inductively coupled plasma MBE molecular beam epitaxy
MG Maxwell Garnet effective medium approximation NP nano particle
Q-factor resonance quality factor QD quantum dot
QW quantum well RIE reactive ion etching SA saturable absorber
SERS surface enhanced Raman scattering
SESAM semiconductor saturable absorber mirror SHG second harmonic generation
SP surface plasmon
SPR surface plasmon resonance TIR total internal reflection of wave WGM whispering gallery mode
This thesis is based on data presented in the following articles, referred to by the Roman numerals I–VI.
I D.V. Karpov and I. G. Savenko, “Operation of a semiconductor microcavity under electric excitation”, Applied Physics Letters 109(6), 061110 (2016)
II D. V. Karpov, I. G. Savenko, H. Flayac, and N. N. Rosanov,
“Dissipative soliton protocols in semiconductor microcavities at finite temperatures”, Physical Review B 92, 075305 (2015) III D.V. Karpov, S. A. Scherbak, Y.P. Svirko and A.A. Lipovskii,
“Second harmonic generation from hemispherical metal nanoparticle covered by dielectric layer”, Journal of Nonlinear Optical Physics & Materials 25, 1650001 (2016)
IV S. Chervinskii, R. Drevinskas, D. V. Karpov, M. Beresna, A.
A. Lipovskii, Yu. P. Svirko & P. G. Kazansky, “Revealing the nanoparticles aspect ratio in the glass-metal nanocomposites irradiated with femtosecond laser”, Scientific Reports 5, 13746 (2015)
V M.V.Maximov, N.V. Kryzhanovskay, A.M.Nadtochiy, E.I.
Moiseev, I.I. Shostak, A.A. Bogdanov, Z.F.Sadrieva, A.E.Zhukov, A.A. Lipovskii, D.V. Karpov, J. Laukkanen, J.
Tommila, “Ultrasmall microdisk and microring lasers based on InAs/InGaAs/GaAs quantum dots”, Nanoscale Research Letters 9:657 (2014)
VI A.E.Zhukov, M.V.Maximov, N.V. Kryzhanovskay, A.M.Nadtochiy, E.I. Moiseev, I.I. Shostak, A.A. Bogdanov, Z.F.Sadrieva, A.A. Lipovskii, D.V. Karpov, J. Laukkanen, J.
Tommila, “Lasing in microdisks of an ultra-small diameter”, Semiconductors 48(12), 1666-1670 (2016)
V.N.Bessolov, D.V.Karpov, E. V. Konenkova , A.А. Lipovskii, A.V. Osipov, A. V. Redkov, I.P. Soshnikov, S.A. Kukushkin,
“Pendeo-epitaxy of stress-free AlN layer on a profiled SiC/Si substrate”, Thin Solid Films 606, 74–79 (2016)
I. Reduto, S. Chervinskii, A. Kamenskii, D. Karpov and A. A.
Lipovskii, “Self-Organized Growth of Small Arrays of Metal Nanoislands on the Surface of Poled Ion-Exchange Glasses”, Technical Physics Letters42(1), (2016)
The publications I-V have been included at the end of this thesis with their copyright holders’ permission.
AUTHOR’S CONTRIBUTION
Author formulated the problem for paper I. In papers I, II, III and IV, the author conducted theoretical analysis and performed numerical simulation. In papers II and III, the author performed parallel computing using supercomputer facilities. The papers I, II and III were written by the author. In papers V, VI as well as two papers not included in the Thesis, author has designed and fabricated the studied micro- and nanostructures and participated in the writing of the parts of the papers related to the fabrication.
Preface ... 7
Contents ... 11
1 Introduction ... 13
2 Surface plasmon resonance in glass-metal composite ... 21
2.1 Linear plasmonics ... 21
2.1.1 Maxwell’s equations ... 22
2.1.2 Localized surface Plasmon: Eigenmode expansion ... 22
2.1.3 Metallic nanoparticles: free electron gas model ... 24
2.1.4 Quasi-static approximation ... 25
Example 1: Sphere ... 26
Example 2: Spheroid ... 27
Example 3: Bisphere ... 29
Example 4: Hemisphere ... 29
2.2 Effective medium approximation ... 30
2.2.1 Composite with spherical inclusions ... 31
2.2.2 Composite with prolate spheroidal inclusions ... 34
2.3 Conclusion of chapter 2 ... 36
3 Nonlinear optics of glass-metal composite ... 38
3.1 Nonlinear response of glass metal composites ... 38
3.2 Second order nonlinearity ... 39
3.3 Hydrodynamic theory of electron gas motion... 40
3.4 Hyperpolarizability of metal particle ... 42
3.5 Third order nonlinearity... 43
3.6 Conclusion of chapter 3 ... 44
4 Disc/ring microcavities with quantum dots ... 47
quantum dots ... 49
4.3 Semiconductor nanostructure fabrication ... 50
4.3.1 Molecular beam epitaxy ... 51
4.3.2 Electron beam lithography ... 52
4.3.3 Atomic layer deposition ... 52
4.3.4 Reactive ion etching ... 53
4.4 Microphotoluminescense measurements ... 54
4.5 Threshold characteristics of QD ring microcavities ... 55
4.6 Temperature dependence ... 56
4.7 Conclusion of chapter 4 ... 58
5 Nonlinear phenomena in exciton-polariton condensate .... 59
5.1 Electric and optical properties of semiconductor nanostructures ... 60
5.2 Exciton-photon strong coupling ... 61
5.3 Exciton-polariton condensation ... 63
5.4 Semiconductor microcavity at the electrical excitation ... 64
5.7 Saturable absorption ... 69
5.8 Dissipative solitons in microcavity ... 70
5.9 Dissipative soliton protocol ... 72
5.10 Conclusion of chapter 5 ... 73
6 Summary ... 75
7 References ... 79
1 Introduction
Linear and nonlinear optical properties of media comprised of metallic and semiconductor nanostructures are of great interest for photonics. This is mainly because the properties of such composite materials are strongly influenced by the resonances associated with their mesoscopic nature. For example, inclusion of metal nanoparticles (NP) into dielectric matrix allows one to excite surface plasmons [1-4], which can be coupled to resonances of molecules or ions in the vicinity of nanoparticles. This effect, which gave birth to the surface enhanced Raman scattering [5-8], also results in the drastic change of the linear and nonlinear absorption [9-14] and can lead to the enhancement of the second harmonic generation in nanocomposites [15-18]. It is worth noting that position and strength of the plasmon resonance in a nanocomposite strongly depend on the shape of nanoparticles.
This makes it possible e.g. to determine the aspect ratio and concentration of spheroidal inclusions by measuring the differential optical density spectra of the glass-metal nanocomposite, see Fig. 1.
The enhanced local electric field in the vicinity of metal inclusion results in the electron ejection from the inclusion to the glass and respectively, the accumulation of the electric charge.
Thus irradiation with intense laser pulses, which results in increasing the local temperature, can give rise to elongation of the nanoparticle by the Coulomb force along the polarization of the incident beam [paper IV, 19-20]. The elongation makes the position of surface plasmon resonance dependent on the light polarization allowing one to deduce the aspect ratio from the transmittance spectrum measured for two orthogonal polarizations. Knowledge of the aspect ratio allows one to define the characteristics of the nanocomposite, as well as to visualize
the metal nanoparticle elongation under external influence (e.g.
stress and bending at the elevated temperature).
Figure 1. (a) Absorbance spectrum of the GMN with spherical inclusion fitted using Maxwell Garnet model. (b) Simulation of differential optical density as a function of the aspect ratio c/a of spheroidal inclusion and wavelength for the GMN modified with an intense laser pulse. Insets show GMN before and after modification. Picture from paper IV.
The plasmon resonance can also have a strong influence on the nonlinear optical response of the glass-metal nanocomposites.
For example, metal hemisphere on the glass surface have a non- zero dipole hyperpolarizability because the inversion symmetry is lifted due to its shape and proximity of the interface. This leads to the second harmonic generation upon irradiation of the hemisphere with intense light pulse. The local field enhancement in the vicinity of the sharp edges of the hemisphere [21-23] is of special importance in this respect. Since the coating with a dielectric shell shifts the plasmon resonance, the dependence of the second harmonic intensity on the shell thickness may reveal the effects of surface plasmon contribution to the second-order nonlinear response. Identification of the hemisphere regions which provide maximum contribution of the SHG intensity is important for potential applications.
Lasing in ring/disc microcavities with different active media is a subject of considerable interest during last three decades [24- 26]. In semiconductor disk and ring microcavities embedded with InAs/InGaAs quantum dots (QD) [paper V, 27-29], the resonant coupling between whispering gallery modes (WGM) and emitted
photons makes it possible to design advanced laser sources due to the high WGM Q-factor. Such structures can be considered as an alternative to lasers based on quantum wells [30] because of their good temperature stability due to 3D carrier confinement in QD. In such microcavities, threshold pump power is as low as 5 𝜇W at room temperature and can be adjusted by changing the geometry of the microcavity.
Figure 2. (a) Scanning electron microscope image of the ring microcavity. (b) Micro photoluminescence spectrum of the ring microcavity with diameter of 2μm and inner diameter of 0.8μm for different pumping powers. Lasing takes place at TE12,1 mode with.
Picture is borrowed from paper V.
Lasers based on ring microcavities (Fig. 2.) are promising for interchip data transfer, modulators [31], switchers [32] and filters [33]. Due to the high Q-factor of WGM [34] they can also be used as a frequency standard in integrated optics. Due to in-plane localization of the WGM modes in ring microcavity [35,36], electrically-pumped ring resonators coupled to the planar waveguide can be can be employed used as light sources and modulators in photonic circuits. Development methods of fabrication of the ultra small resonators is an important task in terms of their integration into optical circuits and minimizing energy consumption. The major role is played by the quality of the wall of the ring/disc. This is because whispering modes propagate along the wall surface, and the roughness of the surface leads to large optical losses thus increasing the lasing threshold. This problem can be resolved by using the electron- beam lithography, which allows one to achieve smooth walls.
Correspondingly, the further advancing in fabrication of the ultra-smooth wall cavities by electron-beam and/or UV lithography is of a strong importance.
Synthesis of new materials (nitrides, organic semiconductors), in which the interband dipole moment is large in comparison with the conventional GaAs, makes possible a condensation of exciton polaritons in microcavities at room temperature [37-50].
This makes the study of the optical properties of such structures extremely important for applications. In particular, the lasing threshold for polaritons is much lower than that for photons thus allowing one to reduce the energy consumption of data transmission devices.
Cavity polaritons ensemble is highly nonequilibrium system due to short exciton lifetime (10-100ps). Therefore description of the spatial-temporal evolution of such a system requires kinetic approach. Introduction of saturable absorber into microcavity leads to additional dissipation that makes this nonequilibrium system nonlinear and leads to formation of solitary waves (solitons). Such nonequlibrium and nonlinear system is stabilized by pumping, which compensates the dissipation losses. This is very different from conventional conservative nonlinear systems, in which soliton formation is possible when dispersion compensates nonlinearity, and we usually have a family of solitons. In contrast, in dissipative nonlinear systems the only soliton can exist if gain compensates losses (see Fig. 3) [51-53].
Semiconductor microcavities under incoherent pump (electrical or optical) can have different fields of application, such as optical routers [54,55], transistors [56], sources of terahertz radiation [57,58], elements of optical circuits [59], high-speed optical switches of polarization [60]. In this context, the study of microcavities under optical pumping is an important task. The introduction of the saturable absorber into a microcavity leads to the dependence of the polariton dissipation rate on the polariton density and can significantly affect the operation of the polariton devices. This can be described via including nonlinear dissipative term in the Gross-Pitaevskii equation and studying of optical solitons at a finite temperature. It is worth mentioning that the
recently reported polariton condensation in WGM microcavity strongly links the parts of this Thesis making it more consistent.
Figure 3. Scheme of nonlinear problem solution for (a) Hamiltonian and (b) dissipative systems
Electrically pumped microcavities with polariton condensate are highly interesting for applications. Description of nonequilibrium Bose-Einstein condensation (BEC) in semiconductor heterostructures under electrical excitation (e.g.
electrically pumped polariton laser) requires solution of the Boltzmann equation for the exciton reservoir supplemented with drift-diffusion equations for charge carriers and the Gross- Pitaevskii equation for polaritons. We consider wide-band-gap semiconductor InAlGaN alloy, which is a promising material for room-temperature BEC and, thus, lasing [61,62]. The large oscillator strength and exciton binding energy and giant Rabi splitting (more than 30meV) lead to robust polariton BEC at 300K (see Fig. 4). The oscillator strength of the InGaN QW excitons is found to be one order of magnitude higher than that of GaAs QW excitons.
Figure 4. Exciton-polariton density in the vicinity of k = 0 as a function of the forward U for the InGaN quantum-well diode. Inset shows color map of the particle distribution in momentum space at (a) U = 2.2 V (under threshold, left) and (b) U = 2.3 V (above threshold, right). Picture was taken from paper I.
The second chapter of the Thesis describes the linear properties of the plasmonic nanostructures and macroscopic properties of media based on metallic inclusions. Plasmon resonances for metallic sphere, bi-sphere, ellipsoid and hemisphere are described using epsilon-method.
The third chapter presents theoretical description of second- harmonic generation in plasmonic nanostructures based on the hydrodynamic theory of the optical nonlinearity of the conduction electrons in a metal. The latter, being combined with the electrostatic approach, made it possible to obtain quasi- analytical expressions for the hyperpolarizability tensor of metal hemisphere. On the basis of the developed approach we predict increase of second harmonic generation at the frequency of the plasmon resonance in hemispherical metal nanoparticles coated with a dielectric layer.
In the fourth chapter of the Thesis we describe the fabrication technology of the ring/disc semiconductor microcavity of diameter as small as 2𝜇m. Here we present the fabrication technique, which provides a small size combined with low roughness of the ring/disc walls. This combination allowed us to obtain cavity with a high quality factor and to achieve lasing at room temperature. Methods of characterization on the basis of micro photoluminescence are presented together with the
experimental results. Developed method of modifying the surface of the silicon carbide by means of electron beam lithography, in order to optimize subsequent growth of aluminum nitride and gallium nitride is also described.
In the fifth part of the Thesis, a model of a semiconductor heterostructure with quantum well (QW) and a saturable absorber (NP) is presented. The effect of the absorption saturation in a microcavity leads to the formation of dissipative solitons in the polariton ensemble. We also consider the effect of acoustic phonon-polariton interaction. We demonstrate protocol (laser pulse consequence and regime of incoherent pumping) of formation and destruction of a dissipative soliton at finite temperatures. We present microscopic theory of polariton laser (for Indium nitride heterostructure), in which we imply the microscopic description of the exciton reservoir.
2 Surface plasmon resonance in glass-metal composite
When a metal nanoparticle is embedded into a dielectric matrix, the electric field strength in its vicinity can be strongly enhanced by collective oscillations conduction electrons in the nanoparticle (surface plasmons). Calculation of the local filed amplitude at the surface plasmon resonance (SPR) is important for various applications such as surface enhanced Raman scattering (SERS) and second harmonic generation (SHG). In this chapter, eigenmode expansion for Maxwell equations is introduced. This technic allows us to compute SPR features for nanoparticles of different shapes. Effective medium approximation based on Maxwell Garnett approach describes the SPR dependence on the nanoparticles concentration. This approach allows one also to describe optical properties of the GMN composed of spheroids.
Such a nanostructure can be produced by irradiating the GMN with intense laser pulse that leads to transformation of spherical nanoparticles to spheroids. Position of the SPR in the spheroid is depends on light polarization.
2.1 LINEAR PLASMONICS
Surface plasmons (SPs) are collective oscillations of conducting electrons on the metallic surfaces coupled to an external electromagnetic field [1-4,63]. Alternatively SPs can be understood as the electromagnetic eigenmodes of the metal- dielectric interfaces. Since in GMN, SPs are localized at the nanoparticle and do not propagate, they often referred to as localized surface plasmons. In this Chapter, we present plasmon
eigenmodes analysis for subwavelength NP. This theory in combination with effective medium approximation was used to describe the linear and nonlinear optical properties of the glass- metal nanocomposites.
2.1.1 Maxwell’s equations
Maxwell’s equations [64] describe relations between the electric field
E
, electric displacement vectorD
, magnetic inductionB
, magnetic field vectorH
, charge density
and current density j:
D (2.1a)
B 0 (2.1b) t
E B (2.1c)
t
H D j (2.1d)
Maxwell’s equations should be supplemented with constitutive relations, which for isotropic linear media can be presented in the following form:
0
D E (2.2a)
0B = H (2.2b)
j E (2.2c)
where ,
and and are permittivity, permeability and conductivity of the medium,
0 and
0are vacuum permittivity and permeability.2.1.2 Localized surface Plasmon: Eigenmode expansion
Consider nanocomposite consisting of metallic inclusions in a dielectric matrix. In presence of the light wave, oscillations of free electrons at the metal-dielectric interface can be resonantly coupled with incident photons and form surface plasmon. At the surface plasmon resonance, the momentum of the coupled electron-photon excitation can be much bigger than photon momentum, i.e. it becomes localized. Resonance frequencies of such localized plasmons depend on the shape of the nanoparticle,
distance between nanoparticles forming ensemble and materials of the inclusion and the host. Most general concept of localized plasmons can be introduced using eigenvalue problem formulation for dielectric constant also referred to as the epsilon method [3, 65].
In the framework of the epsilon method, the solution of the Maxwell equations reduces to finding the eigenvalues and eigenfunctions of the boundary problem for a specified geometry.
Epsilon method permits calculation of the SPR frequencies and electromagnetic field distribution for both individual NPs and their ensembles. The method allows us to describe the properties of localized plasmons and enhancement of the electric field at the plasmon resonance, with a focus on the dependence of the SPR on the nanoparticle shape. In the framework of the epsilon method, the SPR emerges as the frequency corresponding to the eigenvalues
n of the permittivity. In the quasi-static approximation, when the characteristic size of particles much smaller than the incident light wavelength, epsilon method allows one to reduce the solution of Maxwell's equations down to the finding eigenfunctions of the Laplace boundary problem.In spectral representation the eigenfunctions en and hnof the boundary problem satisfy the following equations:
0 0
( ) ( ) 0
( ) ( ) 0
n n n
n n
i i
h r e r
e r h r , (2.3)
where is external light frequency,
n is the eigenvalue. The eigenfunctions are orthogonal,n m nm, n m nm
V V
dV
dV
e e
h h (2.4)and the electric field in the medium can be present as:
0
n n n
AE E e (2.5) where E0is external field and
0
2
( ( ) 1)
( ( )) ( )
n V n
n n
V
dV A
dV
e E
e .
The most important feature of the solutions is the presence of resonant factor in the denominator ,
n
( res)0. That is at the resonance frequency,0
2
( ( ) 1)
( ( ))
n
res V
res n
n res n
V
dV dV
e E
E E e
e (2.6)
It is worth noting that 𝜀𝑛 and 𝑒𝑛do not depend on material and are determined by pure geometrical reasons. In particular, this approach is valid for quite close particles allowing one to understand mechanisms of plasmon hybridization.
2.1.3 Metallic nanoparticles: free electron gas model
Conducting electrons in metals can be considered to move freely.
With this assumption, most of the electronic and optical properties of metals can be described in terms of the Drude- Lorentz-Sommerfeld model [66, 67].
This model allows one to describe motion of conduction electrons along x-axes in terms of the damped harmonic oscillator:
( ) ( ) e x( )
x t x t E t
m
(2.7)
with e as the elementary charge, m as the electron mass, and γ as the damping constant. The x-component of the medium polarization defined as Px enx
0( 1)Ex, here n is electron concentration. By solving Eq. (2.7) one can arrive at the following equation for medium permittivity:
2
1 p
i
, (2.8) where
2
0 p
ne
m
is the plasma frequency.
Drude model does not consider bound electrons contribution to the permittivity that may be important for noble metals. To
account this contribution one need to modify Eq. (2.8) as the following:
2 p
i
, (2.9)
where
is the high frequency permittivity. Plasmon resonance frequencies can be obtained from Eq. (2.9) from the condition
res
n where
nis the eigenvalue of Eqs. (2.3).2.1.4 Quasi-static approximation
Eigenvalue problem for Maxwell equation presented above considers all types of resonance including whispering gallery modes, radiative modes, Mie resonances, etc. However, if the size of the nanoparticles is much smaller than the light wavelength the Maxwell equations can be solved in the framework of the quasi-static approximation, which is valid if the size of the nanoparticle l is
- much smaller than the wavelength of the exciting field 𝜆 and
- much bigger than the mean free path of the electron le, Debye radius rD and the wavelength of an electron 𝜆F
at the Fermi surface.
These requirements can be presented by the following inequality:
e D F
l l r
If the particle size is comparable to or less than the electron mean free path, the important role is played by the scattering of the conduction electrons by the NP surface. This scattering will lead to decrease in the relaxation time of the electron. Specifically, in this case the scattering of the conduction electrons by the surface increases the electron relaxation rate by
vF
A R
(2.10)
where νF, R and A are Fermi velocity, typical size of nanoparticle, while constant A is between 0 and 0.7 depending on the NP shape [68,69]. Fermi velocity for the silver and gold can be estimated as:
F 1.4
v nm/fs. When the particle size is comparable to the Debye radius, the spatial dispersion begins to play an important role.
When the particle size is comparable to the electron’s wavelength, an important role is played by spatial quantization [11,12].
By neglecting effects of the spatial dispersion i.e. assuming
2 1
kl l
one can reduce Eqs.(2.3) down to
0,0.
n n n
e
e (2.11)
That is in this quasi-static approximation, for particle embedded in host with permittivity
host Maxwell equations are reduced to the Laplace equations for potentialn 0,
(2.12)
where en
n . The continuity of the normal component of the electric displacement across the NP/host interface is given the following boundary conditions for the potential:
in
out .n n S host n S
a a (2.13)
Where a is unit vector along the NP surface normal, while superscripts “in” and “out” label potential inside and outside the nanoparticle, respectively.
Example 1: Sphere
Let us consider a sphere with radius R0 embedded in a host medium with permittivity
host . Eigen functions of Laplace operator in spherical coordinates are:0 0
1 0
0
( , ),
( , ),
n nm
nm n
nm
r Y r R
R
R Y r R
r
where Ynm is spherical harmonics, ( , , )r are spherical coordinates. Electrostatic problem is linear, hence the electric potential is written as:
1 0
n nm
n m
Using the boundary conditions (2.13), we can find the eigenvalues of the permittivity as
n 1
host
n n
(2.14)
For example, for Drude dispersion relation (2.10), neglecting damping parameter we obtain well-known formula for plasmon resonances of a sphere:
1
p host
n
n
(2.15)Example 2: Spheroid
In order to solve the Laplace boundary problem for a prolate spheroid it is convenient to use spheroidal coordinates ( , , ) [70]:
xa (1
2)( 21) cos ,
ya (1
2)( 21) sin ,
za
Eigenfunctions of Laplace operator in new coordinates are:
(1) (1)
(2) (2)
( ) ( )( cos sin ),
( ) ( )( cos sin ),
n n
m m nm nm
nm n n
m m nm nm
P Q m m outside
P P m m inside
where Pmn( )
and Qmn( )
associated Legendre polynomials of first and second type respectively. The potential inside spheroid can be presented in the following form:1 0
n nm
n m
Using boundary conditions for continuity of the tangential component of E and the normal component of D we eliminate
, coefficients and obtain relation for plasmon eigenvalues:
0 0
0 0
( )( ( )) ( ( )) ( )
m n n
n m m
n n
host m m
P Q
P Q
where
0 can be expressed using spheroid axes (a<c) as0 2 2
c c a
In dipolar case, when only mode with n=1 is nonzero, the linear polarizability can be expressed from:
0( ( ) host) 0( ( ) host) 0 host 0
V dV V
d E E E
where E can be expressed from (2.5) as
1 0
1 ( )
m host
m m m
E E
and thus polarizability can be expressed as:
2
1 1
4 ( ) 1
3 ( )
m
mm m
hos host
t
ca
(2.16)
For oblate spheroid, using the same approach we have:
0 0
0 0
( )( ( )) ( ( )) ( )
m n n
n m m
n n
host m m
P i Q i P i Q i
and further steps to obtain polarizability are the same as presented above, for more details see [3].
Figure 5. SPR wavelengths for oblate and prolate silver spheroids in the glass matrix as functions of the aspect ratios. Red and black solid lines show SPR wavelength for the light polarized along a- and c-axis, respectively. The following parameters were used for the numerical simulations: ε∞ = 4, λp = 135 nm, γ/ωp = 0.1, εhost = 7.4. Figure borrowed from our paper IV.
Example 3: Bisphere
The eigenvalue problem can be solved using coordinate transformation to the bispherical coordinates ( , , ) [71]:
sin cos sin sin
, ,
cos cos cos
x a y a z a sh
ch ch ch
Continuity of the tangential component of E and the normal component of D allows us to arrive at matrix relation for eigenvalues that could be solved for zero interparticle gap analytically [63,65,72-73].
Example 4: Hemisphere
In case of hemisphere, we have no orthogonality for basic functions and cannot obtain eigenvalues analytically. However, by applying boundary conditions one can arrive at algebraic equations for coefficients in the expansion of the potential in series of Legendre polynomials and first associated Legendre polynomials [70]. For example, when the external electric field is directed normal and parallel to the interface, respectively:
0
1
,
n n n n n
r aE r B P cos
a
(2.17a)
1
|| 0||
1
, ,
n n n n n
r aE r C P cos cos
a
(2.17b)Electric field can be found using equality E
, r
||
, Coefficients B Cn, n in Eqs. (2.17) can be found using the boundary conditions for the potential and electric field [21-23].Calculation of dipole moment and polarizability is quite straightforward. For more details, see Paper III.
Figure 6. Linear absorption spectrum of the silver hemisphere on dielectric substrate. The normal incident light wave is polarized in plane of substrate surface.
Silver permittivity data were taken from [79].
The epsilon method is much better in terms of the numerical burden than the finite elements method (FEM) for hemisphere.
This is because the singularity associated with the sharp edge of the hemisphere greatly complicates the numerical solution. For example, it may take several hours to get results with COMSOL but it takes a few minutes using the epsilon method.
2.2 EFFECTIVE MEDIUM APPROXIMATION
Optical properties of metallic nanoparticles embedded in a transparent host are the subject of considerable experimental and theoretical interest. Much of this attention is due to the possibility to control dielectric function and optical properties of these composite media through the concentration and geometry of the metal inclusions. When the concentration of the inclusions is quite low, the Maxwell Garnett (MG) effective media approximation (EMA) [74-77] is conventionally used for calculation the dielectric function of such media. This approach is based on presentation of the polarizability of the composite
media as the sum of polarizabilities of non-interacting nanoparticles. Generally speaking, the higher metal volume fraction, the higher probability. This is because a small distance between nanoparticles implies strong contributions of the dipole, quadrupole, and higher multipole interactions between them.
The dipole interaction between separated nanoparticles, which has been analyzed in many papers and books [3,65], allows one to obtain first correction of Maxwell Garnett formula as a virial expansion of series of concentration.
Nanoparticle dimers are of considerable importance in this context because of the first step of considering interaction between nanoparticles is two-body approximation. One may expect that nanodimers make a major contribution to macroscopic dielectric constant of the nanocomposite. The plasmonic properties of nanoparticle dimers are briefly discussed in this chapter.
The effective dielectric constant of a GMN with constituent inclusions depends upon the average local field acting in the interior of an inclusion. This average field is not in general equal to the macroscopic field
E
, entering into the macroscopic field equations. Below we present analysis of the averaged electric field in the ensemble of metal nanoparticles.2.2.1 Composite with spherical inclusions
Maxwell Garnett model [75] describe the macroscopic properties of composite materials by averaging the multiple values of the constituents’ dipole moments. This model considers shift of surface plasmon resonance as function of inclusion’s concentration and shape. Maxwell Garnett formula is valid for dilute composites, in which concentration metal volume fraction is low so that the inter-particle interaction does not change plasmon resonance of individual particle and macroscopic field is uniform. In the framework of the Maxwell Garnett model the nanocompsite can be characterized by the effective values of conductivity and permittivity. In the paragraph, we assume that inclusions have spherical shape, are uniformly distributed in the bulk, and have the same size.
Figure 7. Sketch of the glass-metal nanocomposite. Metal spheres with permittivity 𝝐 and polarizabilities α are embedded into glass matrix with permittivity 𝝐𝒉𝒐𝒔𝒕. is the effective permittivity of the GMN, E0 is external electric field.
Following Kirkwood [77], we consider an ensemble of N equal spherical particles embedded in a dielectric matrix. The magnitude of the induced dipole moment of i-th particle is determined by the external field E0 and the interaction with the other particles of the ensemble:
0 ,
i j
j i
G
i jp E r r p , (2.18)
where is the particle polarizability, G
r ri, j is the dyadic Green function describing dipole interaction between i and j particles. In quasi static approximation the Green function reads
30
, 3
4 G I
ij ij
i j
ij
r r
r r r (2.19)
where rij ri ri, denotes tensor multiplication,
I
is identity matrix.Solution of the Eq. (2.18) can be presented in terms of the expansion in series of the polarizability as the following:
2 3
, , , 0
i
j i j i
k j
p G G G E
r ri j
r ri j r rj k(2.20) Polarization of the nanocomposite can be obtained by averaging dipole moment of the individual particle over their space distribution. In an isotropic composite one can arrive at the following equation for the polarization:
2 3
12 1 2 1 2 2
3 3
12 1 2 1 2 2 1 2
3 3 3
123 1 2 3 1 2 2 3 2 3 0
, ,
( , ) , ,
, , , ,
n n n G d
n G G d
n G G d d
P p r r r r r
r r r r r r r
r r r r r r r r r E
(2.21) where n , n12
r r1, 2
and n123
r r r1, ,2 3
are concentration, two- and three-particles distribution functions, respectively.Solving (2.21) for the external electric field and applying statistical averaging we arrive at the following equation:
3
12 1 2 1 2 2 1 2
2 0
3 3
12 1 2 23 2 3
123 1 2 3 1 2 2 3 2 3
2
1 ( , ) , ,
3
( , ) ( , )
, , , , )
n G G d
n n
n n
n G G d d
n n
E r r r r r r r
r r r r
r r r r r r r r r
P
P (2.22) Rewriting Eq. (2.22) in terms of the permittivity of the composite one can arrive at the virial expansion in terms of powers of nanoparticles concentration. In particular, by taking in to account two lowest order terms in the virial expansion one can arrive at:
2
0 0
ε 1
ε 2 3
host host
n
Bn
(2.23)
where
B
is an analog of the second virial coefficient in the statistical theory of the equation of non-ideal gas. It depends only on the interaction of two or more particles and in the framework of the Kirkwood approximation [77] can be presented in the following form:12
2 4
2 n r dr B
n r(2.24) If concentration of the nanoparticles is low, one can neglect B and arrive at the conventional MG equation for the permittivity:
(2 ) 2
ε (2 )
host host
host
host host
f f
(2.25)
2.2.2 Composite with prolate spheroidal inclusions
The polarizability tensor of an isolated spheroid with the radii of a and c (rotation axis) can be presented in the following form:
||,
||, host
N host
(2.26)
where V 4ca2/ 3 is the volume of the spheroid. If z axis is directed along the rotation axis of spheroid, polarization tensor
ij is diagonal,
xx
yy
and
zz
|| . Here N||, are depolarization factors of the spheroid [3],|| ||,
0 0
0 ,
2
0
1
1 ( 1) ln 1
2 1
1 host
N
,
where 0
2 2
c c a
and N
1 N||
/ 2 . Applying Maxwell Garnett approach (MGA) [78] for effective composite permittivity tensor the same way as it was presented above for spherical inclusion, but using the polarizability tensor of the ellipsoids (2.16) we have effective permittivity:,||
,||
||, ||,
(1 )
host 1
f fN
(2.27)
and f is the volume fraction of the spheroidal inclusions. Thus the dielectric constants of the Maxwell-Garnett GMN consisting of metal spheroids depends on both concentration and anisotropy
of the metal inclusions. Then we carry on extremum analysis of previous formula and obtain:
2 3Δ 1 3Δ
1 3Δ 1
SPR p
N f N
N f
(2.28a)
4 3Δ 2 2 3Δ
2 3Δ 1
SPR p
N f N
N f
(2.28b)
where
p 2
c/ p is the plasma wavelength, ΔNN 1/ 3. It is worth to noting that we obtain SPR position as a function of volume fraction, that is differ from formula (2.16) where is no dependence of volume fraction. This is the main motivation of using effective medium theory for real composite in which metal concentration can vary.For
p, range of frequency, we show that SPR has Lorentzian shape,
,
,
2 ,
2 1Im SPR ΔSPR , (2.29) where the linewidth is determined by the electron scattering
rate, Δ , , .
2
SPR SPR
p
Figure 8. SPR wavelengths for light polarized along a-axis (red) and c-axis (blue) (⊥and ∥ respectively) as functions of (a) the aspect ratio at the silver volume fraction f = 0.01(solid lines) and f = 0.1(dashed lines).(b) SPR wavelengths as function of volume fraction at the aspect ratio c/a = 2. Johnson- Christy data for silver were used for simulation [79].
2.3 CONCLUSION OF CHAPTER 2
SPR dominates the optical properties of GMNs. Since the SPR spectral position critically depends on the size, shape and concentration of metal inclusions, one can tailor the linear and nonlinear optical response of the GMN by modifying the shape of the nanoparticles. Vice versa, based on the SPR position one can deduce the shape parameters of ellipsoidal nanoparticles forming the composite, and their concentration. In particular, the modification of the inclusion shape by ultrafast lasers opens the way for optical engineering of GMN.
3 Nonlinear optics of glass- metal composite
The composite media with embedded metal nanoparticles has strong nonlinear response at the plasmon resonance due to enhancement of the local field in the vicinity of the metal nanoparticles. Although in the electric dipole approximation the second-order nonlinear processes including SHG are forbidden in GMN, the electric quadrupole and magnetic dipole mechanisms of the optical nonlinearity still contribute to the SHG in nanocomposites composed of spherical nanoparticles. It is worth noting that the broken inversion symmetry at the metal- dielectric interface may also result in the dipole SHG in GMN. For NPs without inversion symmetry, the SHG in dipole approximation is also allowed.
In this chapter, we consider SHG by metallic NP. Specifically, by using hydrodynamic theory for SHG at metal surfaces [80-85]
we express the hyperpolarizability of a NP through local electric field.
3.1 NONLINEAR RESPONSE OF GLASS METAL COMPOSITES
Optical properties of media with embedded nanoparticles with strong nonlinearity are extensively studied since 80s. Composite materials, in which size of inclusions is much smaller wavelength, show visible nonlinear response due to local field enhancement associated with plasmon resonance. Such media demonstrate various optical phenomena originating from the third-order nonlinearity [11-14] including optical Kerr effect, stimulated Brillouin and Raman scattering, and harmonics generation [15- 18]. Small size of inclusions and large distance between them
