FDTD Algorithm for Plasmonic Nanoparticles with Spatial Dispersion

Home , Plasmonic nanoparticles, Spatial dispersion

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Li Zhang, B.S.

Graduate Program in Electrical and Computer Engineering

The Ohio State University

2016

Master's Examination Committee:

Prof. Fernando Teixeira, Advisor

Prof. Betty Lise Anderson Copyright by

Li Zhang

2016 ABSTRACT

We develop a finite-difference time-domain (FDTD) algorithm incorporating the hydrodynamic Drude model for the permittivity of noble metals. This model incorporates spatial dispersion (nonlocal) effects on the permittivity. The resulting

FDTD algorithm fully includes eddy current effects and is employed to study the extinction cross section of different metallic nanoparticles: nanospheres, nanoshells, nanospheroids, nanodisks, and nanorings, in the visible spectrum. It is observed that spatial dispersion can have significant effects on the extinction cross section of nanoparticles with characteristic sizes around 20 nm or below. For larger particles, the effect is mostly negligible. It is determined that inclusion of spatial dispersion yields two general trends on the behavior of the extinction cross section versus frequency as the particle size is reduced. The first is a blue-shift on the extinction spectrum and the second is an overall decrease on the extinction cross section as well as on the sharpness of resonance peaks. Also we find the strength of spatial dispersion effects do not have significant dependence on the orientation of nanoparticles.

ii ACKNOWLEDGMENTS

Hereby, I want to give my sincere thanks to Prof. Fernando L. Teixeira, who gave me a lot of help in my academic studies, and who has been alway patient even when

I went through hard times.

I always present a graceful heart to my parents. They are the backbone of my achievement.

iii VITA

2011 ...... B.S. Electrical Engineering

2011-present ...... M.S. Electrical and Computer Engi- neering

FIELDS OF STUDY

Major Field: Electrical and Computer Engineering

iv TABLE OF CONTENTS

Page

Abstract ...... ii

Acknowledgments ...... iii

Vita ...... iv

List of Tables ...... vii

List of Figures ...... viii

Chapters:

1. Introduction ...... 1

1.1 Introduction ...... 1

2. Theoretical Background ...... 3

2.1 Drude model ...... 3 2.2 Nonlocal Permittivity ...... 4 2.3 Hydrodynamic Plasma Equation ...... 5 2.4 Hydrodynamic Drude model for permittivity ...... 9

3. Formulation ...... 13

3.1 Permittivity Data ...... 13 3.2 Nonlocal FDTD Algorithm ...... 16 3.3 Additional Boundary Conditions ...... 19 3.4 Numerical Setup ...... 20 3.4.1 General Aspects ...... 20 3.4.2 Source Excitation ...... 21

v 3.4.3 Extinction Cross Section Calculation ...... 21 3.4.4 Near-to-Far Field Transformation ...... 22 3.4.5 Two-stage PML Formulation ...... 23 3.4.6 Qualitative Illustration ...... 24

4. Results ...... 26

4.1 Au Nanospheres ...... 26 4.2 Au Nanoshells ...... 30 4.3 Au Nanospheroids ...... 34 4.4 Au Nanodisks ...... 42 4.5 Au Nanorings ...... 45

5. Conclusion ...... 47

Bibliography ...... 48

vi LIST OF TABLES

Table Page

3.1 Additional boundary condition for nonlocal models at metal-dielectric interfaces...... 20

vii LIST OF FIGURES

Figure Page

3.1 Real part of the permittivity function function (solid line) for Au compared to experiment data (circle) [1, 2]...... 15

3.2 Imaginary part of the permittivity function function (solid line) for Au compared to experiment data (circle) [1, 2]...... 16

3.3 Qlatitative illustration of the field distribution around Au nanoparticles excited by a plane wave. The top-left plot represent the field around a 4 nm Au nanosphere modeled by the hydrodynamic Drude mode. The top-right plot represent the field around the same 4 nm Au nanosphere modeled by the standard Drude mode. The bottom- left plot represent the field around a Au nanospheroid modeled by the hydrodynamic Drude mode. The bottom-right plot represent the field around the same Au nanospheroid modeled by the standard Drude mode. 25

4.1 Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 30 nm diameter Au nanosphere embedded in silica glass substrate...... 27

4.2 Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 20 nm diameter Au nanosphere embedded in silica glass substrate...... 28

4.3 Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 10 nm diameter Au nanosphere embedded in silica glass substrate...... 29

4.4 Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 5 nm diameter Au nanosphere embedded in silica glass substrate...... 30

viii 4.5 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 60 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate. 32

4.6 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 30 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate. 33

4.7 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 10 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate. 33

4.8 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 5 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate. 34

4.9 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in x direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate...... 36

4.10 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in y direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate...... 37

4.11 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in z direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate...... 38

4.12 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in x direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate. . . . 38

4.13 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in y direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate. . . . 39

ix 4.14 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in z direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate. . . . 40

4.15 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 2 for diﬀerent long axis orientation. The spheroids are embedded in silica glass substrate. . . 41

4.16 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 1/2 for diﬀerent long axis orientation. The spheroids are embedded in silica glass substrate. . . 42

4.17 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 5 nm for diﬀerent heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate. . . . 43

4.18 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 5 nm for diﬀerent heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate. . . . 44

4.19 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 10 nm for diﬀerent heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate. . . . 44

4.20 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 10 nm for diﬀerent heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate. . . . 45

4.21 Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for an Au nanoring with its ring shape laid down on the x−y plane and with its wall height in the z direction. The nanoring is embedded in silica glass substrate...... 46

x CHAPTER 1

Introduction

1.1 Introduction

Nanoscale metallic structures are of interest for a number of applications in optics.

By harnessing plasmon resonances, they can provide subwavelength field confinement at levels otherwise not possible with other types of structures. Nonlocal electromagnetic effects such as spatial dispersion can have important effects in the response of some nanoscale devices made of noble metals [3] such as Au and Ag. In particular, spatial dispersion can play a significant role in affecting the plasmon resonances of metallic nanoparticles such as spheres, disks, and rings. Due to its efficiency and

flexibility, the finite-difference time-domain (FDTD) method is one of the most popular techniques for the analysis of plasmonic structures [4, 5, 6] and optical devices in general [7, 8, 9, 10]. In this thesis, we develop a FDTD algorithm to simulate nonlocal effects in plamonic Au nanoparticles using the hydrodynamic Drude model for the permittivity of noble metals. The latter is an extension of the conventional Drude model where expression for permittivity in the Fourier domain is now an explicit function of both the frequency ω and the wavenumber ~k, reflecting the nonlocal characteristic of the electromagnetic response. The hydrodynamic Drude model can be

1 implemented by using a system of auxiliary partial diﬀerence equations. The auxiliary equations can be discretized in time and space and implemented into a staggered-grid

FDTD algorithm. We employ the proposed nonlocal FDTD algorithm to simulate the optical response of several Au plasmonic metallic nanoparticles of common shapes: nanospheres, nanoshells, nanospheroids, nanodisks, and nanorings. The results are compared against simulations based on a conventional FDTD algorithm that does not account for spatial dispersion effects. Based on those results, some observations are made in regards to the effect of spatial dispersion on the response of the nanoparticles, including size dependence, blue shift of the extinction cross section, and decrease of the sharpness of the resonances. Also we find, for the examples considered, that the strength of spatial dispersion effects does not significantly depend on the orientation of nanoparticles with respect to the direction of propagation of the incidence field.

2 CHAPTER 2

Theoretical Background

2.1 Drude model

The Drude model [11, 12] is commonly used to describe the permittivity of noble

metals at optical frequencies. In this model, metals are treated as a cloud of free elec-

trons immersed in a uniform positive background of charges (stationary ions) based

on the assumption that valence electrons are completely detached from their metal

nuclei. Meanwhile, electron-electron interactions are neglected [13, 14]. Electrons

collisions are assumed to occur only with positive ions, at a given probability. These

collisions yield a frictional force on the electron movement and dissipate their kinetic

energy. The Drude model can aid in the prediction of a host of metal properties

such as the electrical conductivity, thermal conductivity, skin eﬀect, Hall eﬀect, and

others.

Let us consider the motion of a single electron with charge −q and mass m. The position of this electron is denoted as ~r = ~r(t). Suppose an uniform external electrical

ﬁeld E~ present on the region where the electron is located. As noted before, collisions between the electrons and the positive ions cause a frictional force to appear during electron movement. The probability of collisions only depends on the arrangement of

3 metal crystal structure. Thus, the electron mean free path is supposed to be constant.

The frictional force is proportional to the collective mean velocity of electrons: Γ~v, where Γ is damping coeﬃcient. Thus, from Newton’s law, the equation of movement for such an electron can be expressed as

d2~r(t) d~r(t) m + mΓ = −qE~ (t). (2.1) dt2 dt

By taking the Fourier transform of the above, we arrive at

m[−ω2~r(ω) − iωΓ~r(ω)] = −qE~ (ω). (2.2)

The dipole moment of a electron is ~p = −q~r. The polarization of this material is a sum of the dipole moments in per unit volume, that is P~ = N~p, where N is electron number density per volume. Also, the relation between dipole moments and the ~ ~ electric susceptibility function χ is given by P = 0χE. By applying those relations to (2.1), the electric susceptibility of this metal is found as

Nq2 1 χ(ω) = − 2 . (2.3) m0 ω + iωΓ

2 2 We next use the deﬁnition of the plasma frequency ωp ≡ Nq /m0 to rewrite the previous equation as ω2 χ(ω) = − p . (2.4) ω2 + iωΓ As a result, the permittivity of a metal described by Drude model is given as

ω2 (ω) = (∞) − p . (2.5) ω2 + iωΓ

2.2 Nonlocal Permittivity

Most often, it is assumed that permittivity (~r) of a medium depends only on its history of electric ﬁeld at that very point. That means the electric ﬂux density

4 D~ (~r, t) can be written as a function of the history of electric ﬁeld E~ (~r, t) in terms of a convolution integral written as [15]

t D~ (~r, t) = dt0(~r, t − t0)E~ (~r, t0). (2.6) ˆ∞

This corresponds to a linear inhomogeneous, isotropic, time-dispersive medium. How- ever, in more general cases, this assumption may not hold. In particular, the permittivity effects can incorporate the influence of the electric field E~ (~r, ~r0, t) in a nonlocal fashion, for example at neighboring points. This can be expressed mathematically as

t D~ (~r, t) = dV 0 dt0(~r, ~r0, t − t0)E~ (~r0, t0). (2.7) ˆ ˆ∞

This nonlocal eﬀect is called spatial dispersion. Assume further that the medium is translation invariant, i.e.,

(~r, ~r0, t) = (~r − ~r0, t). (2.8)

In this case, the nonlocal dispersion relation can be represented in the Fourier domain as

D~ (ω,~k) = (ω,~k)E~ (ω,~k), (2.9) that is, the permittivity of a spatial dispersive material can be treated as a function of both ~k and ω.

2.3 Hydrodynamic Plasma Equation

The hydrodynamic Drude model, also called warm-plasma model, is an important example of a spatially-dispersive permittivity model. This model is important because it describes well the physics of an electron gas [16, 17]. Compared to the conventional Drude model, the hydrodynamic Drude model is a correction in the sense

5 that electron-electron interactions are now included. This interaction contributes

to a gradient pressure added to the dynamic equation for free-electrons-gas model,

creating a nonlocal term for metal permittivity.

Let us consider the free electron gas model for a metal. As before, we denote

particle number density as N(~r). The particle distribution function is f(~r,~v), where

~v is the velocity of particles. We denote an average value of ~v as

1 ~u = h~vi = d3~v~vf(~r,~v), (2.10) N(~r) ˆ

3 3 where d ~v is a volume element in the velocity space, that is d ~v = dvxdvydvz . As-

suming that no electron is created or destroyed within the electron gas or the cube

element, conservation of the total number of particles on a volume implies

d NdV = − N~u · ndSˆ = − ∇ · (N~u)dV, (2.11) dt ˆ ˛ ˆ

where dV is a volume element in the Cartesian spatial domain, dS is a surface element,

and the surface integral is taken over the boundary of a volume of interest inside the

material. The diﬀerential form of this equation is called continuity equation

∂N + ∇ · (N~u) = 0. (2.12) ∂t

In order to derive the equation for hydrodynamic Drude model, we ﬁrst consider

the Boltzmann equation, which relates the evolution of the particle distribution func-

tion f(~r,~v) under the inﬂuence of internal diﬀusion process, an external force F~ , and

particle collisions. It is written as

∂f F~ ∂f ∂f ∂f ∂f + ~v · ∇f + · (~ex + ~ey + ~ez ) = (2.13) ∂t m ∂vx ∂vy ∂vz ∂t collision

where ~eζ , with ζ = x, y, z, are unit vectors along the Cartesian axes. Applied to our electromagnetic model, we can substitute F~ by the Lorentz force E~ +~v ×B~ [18]. If we

6 assume the distance between the electrons to be suﬃciently large so that the collision

term in the r.h.s. can be neglected, we obtain the collisionless Vlasov equation

∂f q ~ ~ ∂f ∂f ∂f + ~v · ∇f + (E + ~v × B) · (~ex + ~ey + ~ez ) = 0. (2.14) ∂t m ∂vx ∂vy ∂vz

Now, by multiplying this equation by velocity ~v and integrating over entire velocity

(momentum) space, we get

∂f 3 3 q ~ ~ ∂f ∂f ∂f 3 ~v d ~v+ ~v(~v·∇f)d ~v+ ~v (E + ~v × B) · (~ex + ~ey + ~ez ) d ~v = 0. ˆ ∂t ˆ m ˆ ∂vx ∂vy ∂vz (2.15)

By exchanging the order of integration and diﬀerentiation,the ﬁrst term of eq. (2.15)

becomes ∂ ∂ ~vf(~r,~v) d3v = (N~u). (2.16) ∂t ˆ ∂t

The second term of equation 2.15 can be expanded as

X X ∂f d3~v v v eˆ (ζ, ξ = x, y, z) (2.17) ˆ ζ ξ ∂ξ ζ ζ ξ

where spatial derivative ∂/∂ξ can be moved out of the integral. The integral above

evaluates the mean value of vζ vξ. So the previous expression becomes

X X ∂ hv v i Neˆ (2.18) ∂ξ ζ ξ ζ ζ ξ

This expression can be manipulated to be

X X ∂ (hu u i + u hv − u i + u hv − u i + h(v − u )(v − u )i)Neˆ . (2.19) ∂ξ ζ ξ ξ ζ ζ ζ ξ ξ ζ ζ ξ ξ ζ ζ ξ

Since uξ h(vζ − uξ)i = 0, the above is reduced to

X X ∂ (hu u i + h(v − u )(v − u )i)Neˆ . (2.20) ∂ξ ζ ξ ζ ζ ξ ξ ζ ζ ξ

7 We have ∂ ∂ ∂ ∂u hu u i N = (u u N) = u (u N) + u N ζ . (2.21) ∂ξ ζ ξ ∂ξ ζ ξ ζ ∂ξ ξ ξ ∂ξ

Next, deﬁne a pressure tensor P¯ by

Pζξ ≡ m h(vζ − uζ )(vξ − uξ)i eˆζ eˆξ (2.22)

As such, (2.20) becomes

X X ∂ ∂uζ 1 ∂ u (u N) + u N eˆ + P eˆ (2.23) ζ ∂ξ ξ ξ ∂ξ ζ m ∂ξ ζξ ξ ζ ξ

In vector form, the above can be written as

1 ~u [∇ · (N~u)] + N(~u · ∇)~u + ∇ · P¯ (2.24) m

Now, let us consider the third term of (2.15):

X X 3 ~ ~ ∂f d ~vvζ (E + ~u × B)ξ eˆζ = ˆ ∂vξ ζ ξ (2.25) X X ~ ~ 3 ∂ X X ∂vζ ~ ~ (E + ~u × B)ξ d ~v (vζ f)ˆeζ − (E + ~u × B)ξeˆζ ˆ ∂vξ ∂vξ ζ ξ ζ ξ

The integrals in ﬁrst term on the r.h.s.(right hand side) of the above equation can be

evaluted as

X ∂ 3 3 (vζ f) d ~v = ∇ · [vζ f(ˆex +e ˆy +e ˆz)] d ~v. (2.26) ˆ ∂vξ ˆ ξ and, by using Gauss’s theorem,

∇ · [v f(ˆe +e ˆ +e ˆ )] d3~v = fv (ˆe +e ˆ +e ˆ ) · nˆ d2~v. (2.27) ˆ ζ x y z ˛ ζ x y z

If we next extend the closed surface integral to inﬁnity where the momentum distri-

bution approaches zero, the integral can be equated to zero and hence the the ﬁrst

8 term in the r.h.s. of (2.25) vanishes. The second term in the r.h.s. of (2.25) is easily

determined to be

X X ∂vζ ~ ~ X X ~ ~ ~ ~ (E +~u×B)ξeˆζ = δζξN(E +~u×B)ξeˆζ = N(E +~u×B) (2.28) ∂vξ ζ ξ ζ ξ

where δζξ is the Kronecker delta. By substituting these three terms back into (2.15),

we get

∂ 1 q (N~u) + ~u [∇ · (N~u)] + N(~u · ∇)~u + ∇ · P¯ − N(E~ + ~u × B~ ) = 0, (2.29) ∂t m m

and using (2.12) we have

∂ ∂~u ∂N ∂~u (N~u) = N + ~u = N − ~u[∇ · (N~u)]. (2.30) ∂t ∂t ∂t ∂t

Also, by using d ∂~u ~u(~r, t) = + (~u · ∇)~u (2.31) dt ∂t

in (2.30) and multiplying by a factor m/N, the hydrodynamic plasma equation is

obtained d~u 1 m + ∇ · P¯ − q(E~ + ~u × B~ ) = 0. (2.32) dt N

If the plasma medium is near equilibrium (which occurs when high frequency electromagnetic waves excite metals), the ∇ · P¯ term can be treated through a standard

Maxwell-Boltzmann distribution [16]. In this case, ~u = 0 and Pij = δijm hvii so that

∇ · P¯ can be replaced by the gradient of a scalar pressure P , that is ∇P .

2.4 Hydrodynamic Drude model for permittivity

The hydrodynamic Drude model for permittivity is derived from (2.32) by adding a damping force term and setting B~ to zero. In this case, the equation for motion of

9 the electrons in a metal can be written as

d2 d 1 m ~r(t) + mΓ ~r(t) = qE~ (t) − ∇P (t). (2.33) dt2 dt N

By taking the Fourier transform of the above equation

1 −ω2m~r(ω) − iωΓm~r(ω) = qE~ (ω) − ∇P (ω), (2.34) N where ~r(ω) is the Fourier transform of the electron position displacement. Near equilibrium, N can be written as N = N0 + δN, and after inserting the continuity equation ∂N + (N + δN)∇ · ~u + ~u · ∇(N + δN) = 0. (2.35) ∂t 0 0 Neglecting the small terms and taking the Fourier transform, we obtain

−iωN + N0∇ · ~u = 0. (2.36)

We know that ~u = d~r/dt, and in the Fourier domain ~u = −iω~r, so

N = −N0∇ · ~r. (2.37)

The chain rule gives ∂P ∇P = ∇N (2.38) ∂N so that

∇P = β2m∇N, (2.39) where β2 = ∂P/∂ρ = (∂P/∂N)/m is a constant [19, 2]. Using (2.37), the above equation becomes

2 ∇P = −N0β m∇(∇ · ~r), (2.40) so that eq. (2.34) can be written as

−ω2m~r(ω) − iωΓm~r(ω) − β2m∇[∇ · ~r(ω)] = qE~ (ω) (2.41)

10 From the vector identity ∇(∇ · ~r) = ∇2~r + ∇ × ∇ × ~r and assuming that there are

no eddy currents −iω∇ ×~r = q∇ × ~u = ∇ × J~ = 0 present in the metal [3], a Fourier transformation the space domain ∇ → i~k applied to (2.41) gives [2]

−ω2m~r(ω,~k) − iωΓm~r(ω,~k) + β2k2m~r(ω,~k) = qE~ (ω,~k) (2.42) with k2 = |~k|2. By using the same steps as before for the derivation of the local Drude model, the nonlocal hydrodynamic Drude model for the permittivity is obtained as

ω2 (ω,~k) = ˜(∞) − p (2.43) ω2 + iωΓ − β2k2

Although this model has been used in the past [2], neglect of eddy currents is not

justiﬁed in metals and it can produce spurious resonances in the analysis of metal

nanoparticles, as pointed out in [3]. With eddy currents included, (2.41) writes in the

Fourier (ω,~k) domain as

−ω2m~r(ω,~k) − iωΓm~r(ω,~k) + β2m~k[~k · ~r(ω,~k)] = qE~ (ω,~k), (2.44)

which can be decomposed into a longitudinal L and a transversal T parts

2 ~ ~ 2 2 ~ ~ ~ −ω m~rL(ω, k) − iωΓm~rL(ω, k) + β mk ~rL(ω, k) = qEL(ω, k)

2 ~ ~ ~ ~ −ω m~rT (ω, k) − iωΓm~rT (ω, k) = qET (ω, k) (2.45)

ˆ ˆ ˆ ~ ~ where ~rL = k(k · ~r) with k = k/k and ~rT = ~r − ~rL, and similarly for E. As a result,

the permittivity is similarly written in terms of longitudinal and transverse parts:

ω2 (ω,~k) = (∞) − p L ω2 + iωΓ − β2k2 ω2 (ω) = (∞) − p (2.46) T ω2 + iωΓ

so that the ﬁnal expression for the nonlocal Drude permittivity can be written as

~ ˆˆ ~ ¯ ˆˆ ¯(ω, k) = kk L(ω, k) + (I − kk) T (ω). (2.47)

11 Note that this expression for the permittivity is in tensor form, the medium is still

isotropic. This is because the vector ~k is not a preferential direction set by the medium. Rather, it is simply the direction of propagation of the wave itself. As a result, the wave still see the medium as isotropic.

12 CHAPTER 3

Formulation

3.1 Permittivity Data

The analysis in this thesis is focused on obtaining the extinction cross section of Au nanoparticles of diﬀerent shapes embedded in silica glass (amorphous silicon dioxide SiO2) near and at the visible spectrum. In order to perform the analysis, precise data are required for the parameters describing the permittivity function.

Measurement of the permittivity of Au has been extensively performed in the past [20, 21, 1, 22, 23, 24]. Johnson and Christy [1] measured the room temperature complex refractive index n + ik value, from vacuum-evaporated Au thin ﬁlms. The real and imaginary values of the permittivity 0 + i00 can be subsequently simply obtained using the relations

0 = n2 − k2, (3.1)

00 = 2nk. (3.2)

Despite the fact that the conventional Drude free-electron model for Au permittivity can be successfully ﬁtted to incorporate experimental data for wavelength above

600 nm (corresponding to a photon energy of 2 eV), other references suggest that

13 this model deviates signiﬁcantly when comparing data at shorter wavelengths [21].

The discrepancy between measurement data and the Drude function data can be re-

solved by taking interband electron eﬀects (also called bound electron eﬀects) into

consideration. The interband electron eﬀect is basically a result of bounded electron

transitions from d-band to sp-band [25]. The interband electron eﬀect incorporated

into the permittivity gives rise to a frequency response [2] with a sume of terms in

the form of Lorentz oscillators [25, 26, 24], i.e.,

Nt 2 X ∆εjωj . (3.3) ω2 − ω2 − i2δ ω j=1 j j

By combining the above with the spatially dispersive terms in (2.47), the combined

dielectric function is written as

Nt 2 ! 2 2 X ∆εjωj ωp ωp ¯(ω,~k) = I¯ (∞) − −kˆkˆ −(I¯−kˆkˆ) . ω2 − ω2 − i2δ ω ω2 + iωΓ − β2k2 ω2 + iωΓ j=1 j j (3.4)

Observing the experimental data from Johnson and Christy [1], two signiﬁcant peaks

are visible in the imaginary part of the permittivity function. Hence, in most cases,

it is suﬃcient to restrict the number of terms to Nt = 2 in the sum above. The

permittivity function in this case exhibits a total of ten unknown parameters: (∞),

ωp, Γ, ∆ε1, ω1, δ1, ∆ε2, ω2, δ2 and β. In order to incorporate spatial dispersion eﬀects

into the permittivity model, the parameter β needs ﬁrst to be determined. Here, we

assume that β takes the value of 1.0767 × 106m/s for Au as provided in [2].

The expression for the permittivity in eq. (3.4) can be curve ﬁt to experimental

data using a nonlinear least-square solver. By doing so, the nine unknown parameters

near the visible spectrum are found to be: (∞) = 3.8985, ωp = 8.9771eV , Γ = 0.0740 eV, ∆ε1 = 3.3099 eV, ω1 = 4.7617 eV, δ1 = 1.7644 eV, ∆ε2 = 1.2859 eV, ω2 = 3.0908

14 eV, δ2 = 0.5372 eV. These resulting parameters provide a very precise ﬁt to the real part of the permittivity and data a reasonably good ﬁt to the imaginary part of the permittivity. These parameters are in good agreement with those published in [2].

The curve-ﬁt result is shown in Figures 3.1 and 3.2.

Figure 3.1: Real part of the permittivity function function (solid line) for Au compared to experiment data (circle) [1, 2].

15 Figure 3.2: Imaginary part of the permittivity function function (solid line) for Au compared to experiment data (circle) [1, 2].

Within the range of frequencies we are interested in, the relative permittivity of silica glass is purely real and has a nonlinear dependency on frequency, which ranges from 2.07 to 2.50 in the spectrum from 0.6 eV to 6.75 eV [27]. This implies a variation as large as 16%. However, the value is much more stable in visible spectrum, ranging from 2.07 to 2.16. As such, we set treat this permittivity as constant for simplicity, and equal to 2.13 (value at the midpoint of the visible spectrum). Of course, the discrepancy caused by this approximation would become more pronounced at the near infrared and near ultraviolet regions than in the visible region.

3.2 Nonlocal FDTD Algorithm

To integrate the hydrodynamic model of nonlocal permittivity into the ﬁnite- diﬀerence time-domain method (FDTD) algorithm, we extend the conventional FDTD

16 discretization scheme in a staggered-grid [4, 28, 29] by incorporating auxiliary diﬀer-

ential equations and ﬁelds into the algorithm [6]. The equations are ﬁrst obtained in

the (ω,~k) frequency-domain and later Fourier-transformed back to the time-domain to derive the time-stepping algorithm. By incorporating the the Drude-Lorentz nonlocal model for the metal permittivity in the (ω,~k) Fourier domain, Ampere’s equation

becomes [6]

~ ~ ~ ~ h ~ ¯ i ~ ik × H = −iω0E = −iω∞E + iω0 ¯(ω, k) − I∞ · E, (3.5)

~ where ∞ = (∞, k) for notational simplicity. Recall Maxwell’s equation in time

domain [4, 30]

∂D~ ∇ × H~ = + J~ (3.6) ∂t ∂B~ ∇ × E~ = − . (3.7) ∂t

To tackle the nonlocal dielectric function, we deﬁne three auxiliary ﬁelds (currents) ~ ~ ~ denoted as J1, J2, and J3:

ω2 ω2 J~ = −iω kˆkˆ p + iω (I¯− kˆkˆ) p · E~ (3.8) 1 0 ω2 + iωΓ − β2k2 0 ω2 + iωΓ 2 ~ ∆ε1ω1 ~ J2 = −iω0 2 2 E (3.9) ω1 − ω − i2δ1ω 3 ~ ∆ε2ω2 ~ J3 = −iω0 2 2 E. (3.10) ω2 − ω − i2δ2ω

17 These auxiliary equations mimic the eﬀect of induced currents [2, 4, 6] in Maxwell’s equations. By rearranging the equations and transforming them back to the space- time domain, we obtain a set of three diﬀerential equations:

∂2 ∂ ∂ + Γ − β2∇∇· J~ = ω2 E~ (3.11) ∂t2 ∂t 1 0 p ∂t ∂2 ∂ ∂ + 2δ + ω2 J~ = ∆ε ω2 E~ (3.12) ∂t2 1 ∂t 1 2 0 1 1 ∂t ∂2 ∂ ∂ + 2δ + ω2 J~ = ∆ε ω2 E.~ (3.13) ∂t2 2 ∂t 2 3 0 2 2 ∂t

These equations can be discretized in time using a leap-frog scheme

J~n+1 + J~n−1 − 2J~n J~n+1 − J~n−1 E~ n+1 − E~ n 1 1 1 + Γ 1 1 − β2∇(∇ · J~n) = ω2 (3.14) ∆t2 2∆t 1 0 p ∆t J~n+1 + J~n−1 − 2J~n J~n+1 − J~n−1 E~ n+1 − E~ n 2 2 2 + δ 2 2 − ω2J~n = ∆ε ω2 (3.15) ∆t2 1 ∆t 1 2 0 1 1 ∆t J~n+1 + J~n−1 − 2J~n J~n+1 − J~n−1 E~ n+1 − E~ n 3 3 3 + δ 3 3 − ω2J~n = ∆ε ω2 . (3.16) ∆t2 2 ∆t 2 3 0 2 2 ∆t

By rearranging the terms,the semi-discrete update equations for the auxiliary currents ~ ~ ~ J1, J2, and J3 are obtained as follows

E~ n+1 − E~ n J~n+1 = κ J~n + κ J~n−1 + κ (3.17) 1 11 1 21 1 31 ∆t E~ n+1 − E~ n J~n+1 = κ J~n + κ J~n−1 + κ (3.18) 2 12 2 22 2 32 ∆t E~ n+1 − E~ n J~n+1 = κ J~n + κ J~n−1 + κ , (3.19) 3 13 3 23 3 33 ∆t with coeﬃcients

4 + 2β2∆t2∇(∇·) κ = (3.20) 11 2 + Γ∆t Γ∆t − 2 κ = (3.21) 21 2 + Γ∆t 2 2 20ω ∆t κ = p (3.22) 31 2 + Γ∆t

18 2 2 2 − ω1∆t κ12 = (3.23) 1 + δ1∆t δ1∆t − 1 κ22 = (3.24) 1 + δ1∆t 2 2 0∆εω1∆t κ32 = (3.25) 1 + δ1∆t

2 2 2 − ω2∆t κ13 = (3.26) 1 + δ2∆t δ2∆t − 1 κ23 = (3.27) 1 + δ2∆t 2 2 0∆εω2∆t κ33 = . (3.28) 1 + δ2∆t

Furthermore, the update equation for the electric ﬁeld E is obtained from Maxwell’s

equations as

∇ × H~ n+1/2 1 X X E~ = E~ + − [ (κ + 1)J~n + κ J~n−1], (3.29) n+1 n η + η η + η 1i i 2i i 1 2 1 2 i m where

∆t η1 = (3.30) 0∞ κ + κ + κ η = 31 32 33 , (3.31) 2 4∆t

and likewise the updating equation for the magnetic ﬁeld H is

∆t Hn+1/2 = Hn−1/2 − ∇ × E~ n. (3.32) µ0

The remaining spatial derivatives due to the nabla operator present in the equations

above can be discretized in a standard fashion in the staggered FDTD grid [4, 31].

3.3 Additional Boundary Conditions

The nonlocal permittivity model has a diﬃculty dealing with metal-dielectric in- ~ terfaces since the auxiliary current J1 has a spatial term which that is not collocated

19 Nonlocal Term Boundary Condition local 0 nˆ · J~ = 0,n ˆ × J~ 6= 0 nonlocal β2∇(∇·) nˆ · J~ = 0,n ˆ × J~ 6= 0 nonlocal(curl-free) β2∇2 nˆ · J~ = 0,n ˆ × J~ = 0

Table 3.1: Additional boundary condition for nonlocal models at metal-dielectric interfaces.

to E~ at the same point. As a result, an additional boundary condition is needed to supplement the equations. In the nonlocal model, such additional boundary conditions restrict the auxiliary current ﬂow that is moving out of the boundary interface to be zero, based on the assumption that the surface electron density is constant: nˆ · J~ = 0 [32, 33]. In the curl-free variant description of nonlocal term β2∇2, it is also

required to enforce the tangential component of auxiliary current as being curl-free

nˆ ×J~ = 0. This often referred as the Pekar’s additional boundary condition [32]. The

additional boundary conditions are summarized in Table 3.1.

3.4 Numerical Setup

3.4.1 General Aspects

The FDTD simulations are performed on the Oakley Cluster at the Ohio Super-

computing Center(OSC). Each simulation assigns four processors in one node at the

OSC machine. The FDTD grid lattice spacing is set to be identical for all directions,

i.e. ∆x = ∆y = ∆z, in all cases. For better accuracy, the lattice spacing ∆x should

be suﬃcient small. The drawback of utilizing smaller lattice spacing is that longer

computation times are required because the Courant stabiity limit dictates smaller

time steps [34]. If the dimension of lattice is reduced by half, the computational

20 time will increase by approximately 24 = 16 times. The numerical simulations for larger nanoparticles utilize a grid spacing of 1 nm. For smaller particles, grid spac- ings of 0.5 nm, 0.25 nm or 0.125 nm are used, depending on the size of particles.

As many as ten cells of perfectly matched layer (PML) boundary are utilized for the simulation domain. The PML is implemented using complex stretching coordinates scheme [35, 36, 37]. The PML boundary ensures that the residual reﬂection levels are below 0.025% or about -70 dB.

3.4.2 Source Excitation

The excitation used is a uniform plane wave, which is incident from one side of boundaries to the opposite side, along the positive x direction. The electric ﬁeld

in linearly polarized along the y direction. A total-ﬁeld/scattered-ﬁeld technique

(TF/SF) [4, 38] is implemented in FDTD to generate the plane-wave excitation. The

TF/SF interface is placed sufficiently far from the particles so that the influence of evanescent (non-propagating) fields on the surface is small. In our case, we set the

TS/SF interface at 10 cells away from the outermost cell comprising the nanoparticle.

The time-domain waveform of the excitation ﬁeld is a Gaussian pulse modulated by cosine function. The waveform spectrum covers a range from 1 eV to 6 eV. For practical purposes, the Gaussian function is truncated by a rectangular window of

ﬁnite-duration equal to 4.8145 × 10−15.

3.4.3 Extinction Cross Section Calculation

The spectrum of the the extinction cross section is analyzed and compared for the various nanoparticles using the present FDTD algorithm. The extinction cross section is a sum of absorption cross section and scattering cross section. It characterizes the

21 strength of the interaction between the nanoparticles and the incident light. The

extinction cross section equals to the energy loss divided by incident wave energy

ﬂow density Eext . The extinction cross section spectrum thus obtained is normalized Sinc by dividing it the geometrical cross section of the particle facing the incident wave.

The calculation of extinction cross section is based on optical theorem [39]:

4π σ = Im[ˆe · f(kˆ , kˆ ) · eˆ ], (3.33) ext k i i i i

ˆ where ki is the unit wavenumber vector ande î is the unit polarization vector of the ˆ ˆ incident plane wave, and f(ki, ki) is the far-field scattering amplitude tensor in the forward direction. The extinction cross section is also related to the scattered field in the direction of incident wave. In order to calculate the extinction cross section, we need to retrieve the far-field pattern of scattered field.

3.4.4 Near-to-Far Field Transformation

The forward scattering amplitude is calculated using a near-to-far-field (NTFF) transformation [4]. This transformation is based on Huygens principle and it can be expressed by representing the fields on a surface enclosing the scatterer using equivalent current sources. These equivalent currents are obtained, via the equivalence principle, from the tangential electric or magnetic field at the NTFF surface, that is ~ ~ Ms = −nˆs × E (3.34) ~ ~ Js =n ˆs × H,

The NTFF uses the radiation integral to evaluate the far-ﬁeld produced by these

equivalent sources. The expression for the magnetic and electric vector potentials in

22 the far-field are ~0 µ e−jk|~r−r | ~ ~ ~0 0 A(~r) = Js(r ) ds 4π ˛S ~0 ~r − r (3.35) ~0 e−jk|~r−r | ~ ~ ~0 0 M(~r) = Ms(r ) ds , 4π ˛S ~0 ~r − r from which the far-field electrical and magnetic field can be obtained using the fol-

lowing equations 1 1 E~ = −jω[A~ + ∇(∇ · A~)] − × F~ k2 (3.36) 1 1 H~ = −jω[F~ + ∇(∇ · F~ )] + × A.~ k2 µ The forward scattered ﬁeld is derived by taking r → ∞, θ = 0, φ = 0 in the expressions above. We can then apply the optical theorem to obtain the extinction cross section.

3.4.5 Two-stage PML Formulation

In order to reduce spurious reﬂection artifacts coming from the FDTD grid boundaries and improve the accuracy of result, a two-stage formulation of the PML is implemented here [40]. The transitional PML can be regarded as mapping Maxwell’s equations to complex coordinates [37, 41], in which the nabla operator is written as [42] 1 ∂ 1 ∂ 1 ∂ ∇¯ =x ˆ +y ˆ +z ˆ (3.37) sx ∂x sy ∂y sz ∂z with complex stretching variables given by

σζ sζ = κζ + (3.38) jω0

for ζ = x, y, z. In the two-stage formulation, a buﬀer stage is inserted in the PML

before the complex stretched coordinates acquire a non-zero imaginary part. This

allows for the evanescent part of the wavenumber spectrum to exhibit suﬃcient decay

23 before reaching the absorptive part of the PML (with positive imaginary part) [40].

This is done because the PML absorption is mostly effective only for the propagating spectrum. Note that the evanescent spectrum is important in problems involving plasmonic nanoparticles since they are sub-wavelength structures. The formulation of two-stage PML employs a tapered profile for both the real and imaginary parts of sζ in two stages [40]. In the first (inner) stage, only the real part is modified in a tapered fashion. i.e., σ(ζ) = 0 (3.39) 1 ζ κ(ζ) = (2 ∆ζ ) and, in the second (outer) stage, we have

ζ − d1 m σ(ζ) = σmax( ) d2 (3.40)

κ(ζ) = κ0(ζ − ∆ζ), where the parameters σmax, κ0 d1, d2 and m are chosen to minimize the reﬂection coeﬃcient from the PML.

3.4.6 Qualitative Illustration

In Figure 3.3, we present a qualitative illustration of the diﬀerent behavior of nanoparticles simulated using the hydrodynamic Drude model (nonlocal) and the using standard Drude model (local). The incident plane-wave travels from left to right in these pictures. In the two plots at the top, the left one represents the electric ﬁeld intensity around a 4 nm diameter Au nanosphere simulated using the hydrodynamic

Drude model (nonlocal). The right plot represents the ﬁeld magnitude around same nanoparticle simulated using the standard Drude model (local). In the two bottom plots, the left one represent the electric ﬁeld intensity of a Au nanospheroid simulated

24 using the hydrodynamic Drude model. The right plot represents the ﬁeld magnitude around the same nanoparticle simulated using the standard Drude model.

Figure 3.3: Qlatitative illustration of the field distribution around Au nanoparticles excited by a plane wave. The top-left plot represent the field around a 4 nm Au nanosphere modeled by the hydrodynamic Drude mode. The top-right plot represent the field around the same 4 nm Au nanosphere modeled by the standard Drude mode. The bottom-left plot represent the field around a Au nanospheroid modeled by the hydrodynamic Drude mode. The bottom-right plot represent the field around the same Au nanospheroid modeled by the standard Drude mode.

25 CHAPTER 4

Results

4.1 Au Nanospheres

Although from a fabrication viewpoint, nanospheres are not suited to planar techniques, in a geometrical sense nanospheres are the simplest of nanoparticles. Once the optical properties of metal and substrate materials are determined, the optical response of a nanosphere is determined solely by its diameter. In the optical frequency spectrum, surface plasmons can be excited on the surface of nanospheres (as well as in other types of nanoparticles) which create significant localized resonances. The resonance behavior can be modified by the presence of spatial dispersion (nonlocal effects) in the metal. Figure 4.1, 4.2 and 4.3 compare the normalized extinction cross section with and without spatial dispersion for Au nanospheres with three different radii: 30 nm, 20 nm, and 10 nm. From these figures, it can be seen that the 30 nm gold nanosphere result shows a slight blue-shift from the nonlocal case in contrast to the local case by a range of approximately 0.02 eV to 0.035 eV. As the size of the nanoparticles decrease, the impact of nonlocality become more significant. For the 20 nm gold sphere, the blue-shift can be seen from Figure 4.2 to be slightly larger than the 30 nm particle case. The amount of blue-shift in the 20 nm is found to be in the

26 0.04 eV to 0.06 eV range. When the gold nanosphere diameter is decreased further down to 10 nm, the blue-shift eﬀect becomes much greater and a stronger distortion of extinction cross section spectrum shape is also visible. As the diameter of nanosphere goes down to 5 nm, the distortion is so obvious that the two curves are hardly recog- nized as of similar shape. In particular, the resonance peak at approximately 2.4 eV in the local case is virtually absent in the nonlocal case.

Figure 4.1: Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 30 nm diameter Au nanosphere embedded in silica glass substrate.

27 Figure 4.2: Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 20 nm diameter Au nanosphere embedded in silica glass substrate.

28 Figure 4.3: Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 10 nm diameter Au nanosphere embedded in silica glass substrate.

29 Figure 4.4: Comparison between local (solid red) and nonlocal (dashed blue) extinction cross section for a 5 nm diameter Au nanosphere embedded in silica glass substrate.

4.2 Au Nanoshells

Nanoshell particles are highly functional nanoparticles that have been used for many applications [43, 44], including photonics-based imaging [43], Raman spectroscopy biosensing [45], bioimaging [46], chemical catalysis, medical treatments [47], photonics band gap materials [48], among others. A single nanoshell consists of a dielectric core coated by a thin layer of a metallic material (shell). The metal utilized is usually Ag or Au [43]. One attractive feature of nanoshells is that their optical properties can be easily tuned by changing the core constitution, the shell material, and/or the geometry of nanoshells (thickness of metallic shell, diameter of the outer shell, and/or core diameter). Depending on these properties, the optical resonance frequency of nanoshells can typically span a range from visible to near-infrared (NIR).

30 FDTD simulations have been performed to investigate the eﬀect of spatial dispersion on Au nanoshells response. Four diﬀerent shell outer radii are considered:

60 nm, 30 nm, 10 nm, and 5 nm. The results are shown in Figs. 4.5 to 4.8. Akin to the nanosphere case seen before, smaller nanoshell geometries are aﬀected more strongly by the nonlocal behavior. For large shell outer radius, such as 30 nm radius

(Figure 4.5), and when the shell is sufficiently thick, say 20 nm, the nonlocal effect is rather negligible. This is consistent with solid nanospheres results seen before since, as the nanoshell thickness increases, the extinction spectrum of a nanoshell should eventually approach that of a nanosphere. However, when the shell thickness de- creases to 8nm as shown in Figure 4.5, spatial dispersion effects becomes increasingly more important. Similar conclusions can be made in regards to the 30 nm radius nanoshells. As seen before, the response of a 30 nm radius solid Au nanosphere had almost no variation with respect to nonlocal effect. Here instead, it is seen that spatial dispersion has a more visible effect on the extinction cross section when the shell thickness is gradually reduced down to 4 nm. As a rule of thumb, it is observed that the smallest geometrical feature of the nanoparticle basically determines the strength of the spatial dispersion effect on these particles. Another effect contributed by spatial dispersion is a decrease of the peak of the extinction spectrum resonances. This effect is clearly seen in Figures 4.7 and 4.8. Meanwhile, spatial dispersion also reduces the sharpness of the resonance peaks. Similar to the nanosphere cases, a blue-shift on the spetrum can also be observed in all these figures. This suggests, as a trend, that spatial dispersion cause a blue-shift on the extinction spectrum of nanoparticles regardless of their shapes.

31 Figure 4.5: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 60 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate.

32 Figure 4.6: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 30 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate.

Figure 4.7: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 10 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate.

33 Figure 4.8: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for a 5 nm radius Au nanoshell for diﬀerent values of wall thicknesses embedded in a silica glass substrate.

4.3 Au Nanospheroids

Nanospheroids are ellipsoid nanoparticles. The mathematical expression describing the surface of a spheroid is given by

x2 + y2 z2 + = 1 (4.1) a2 b2

The optical properties of a nanospheroid can be tuned by the long axis b and the short axis a. Due to the asymmetry in the nanospheroid shape, the incident wave polarization and incident direction are also important in determining its extinction cross section. As noted before, in the present study, we assume an incident plane- wave travelling in the positive x direction with electric ﬁeld linearly polarized along the y direction.

34 Figures 4.16 to 4.9 show a series of results for the FDTD simulation for Au

nanospheroids. For spheroids with a =5 nm, it can be seen that that spatial dispersion has a significant effect on the extinction cross section. In particular, the shorter the long axis is, the stronger is the effect. By comparing different long axis orientations, we do not find significant evidence that the strength of nonlocal effects depends on long axis orientation. Again, we find that the spatial dispersion causes a blue-shift the extinction cross section spectrum, reduce the peak of resonance, and also reduce the sharpness of the resonance. For the larger spheroid cases with a =10 nm and 16 nm, we observe similar trends: the shorter the long axis is, the stronger the effects of spatial dispersion are. Also, we find the strength of spatial dispersion do not significantly depend on nanospheroids orientation in those cases either. Similarly, spatial dispersion cause blue-shift the optical properties, and reduce the peak and sharpness of the resonances.

35 Figure 4.9: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in x direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

36 Figure 4.10: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in y direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

37 Figure 4.11: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 5 nm and long axis in z direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

Figure 4.12: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in x direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

38 Figure 4.13: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in y direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

39 Figure 4.14: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 10 nm and long axis in z direction for diﬀerent values of short/long axial ratio. The spheroids are embedded in silica glass substrate.

40 Figure 4.15: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 2 for diﬀerent long axis orientation. The spheroids are embedded in silica glass substrate.

41 Figure 4.16: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au spheroids with short axis a = 16 nm and long/short axial ratio equal to 1/2 for diﬀerent long axis orientation. The spheroids are embedded in silica glass substrate.

4.4 Au Nanodisks

Nanodisks comprise a thick layer of metal with circular shape. In contrast to the other nanoparticles considered before, a key advantage of nanodisks is that they are amenable to planar fabrication techniques. Nanodisks can also be viewed as cylindrical rods with relatively small height. Apart from the frequency, material properties, and the relative orientation with respect to the indicent field, two geometrical parameters control the optical response of a nanodisk: its radius and its height. Figures 4.17 to 4.20 show the effect on the extinction cross section of Au nanodisks of various sizes stemming from the presence of spatial dispersion on the metal. We compare the strength of nonlocal effect for different nanodisk orientations as well. From both r = 5 nm and r = 10 nm, no significant dependence on the strength of spatial dispersion

42 eﬀects with respect to orientation has been found. Not surprisingly, we again ﬁnd the same basic trends caused by spatial dispersion seen before for the other nanoparticles: blue-shift of the spectrum and decrease of sharpness of the resonances, both of which becomes relatively stronger as the characteristic dimensions of the nanodisks get smaller.

Figure 4.17: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 5 nm for diﬀerent heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate.

43 Figure 4.18: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 5 nm for diﬀerent heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate.

Figure 4.19: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 10 nm for diﬀerent heights as indicated, with the disk plane facing the y direction. The nanodisks are embedded in silica glass substrate.

44 Figure 4.20: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for Au nanodisks with radius r = 10 nm for diﬀerent heights as indicated, with the disk plane facing the z direction. The nanodisks are embedded in silica glass substrate.

4.5 Au Nanorings

Nanorings can be fabricated using similar planar techniques as nanodisks, but deposited with a hollow core inside. Typically, the diﬀerence between inside and out- side radii (thickness) is small but this is an extra parameter that can be adjusted for tuning of desired resonances. As a result, nanorings have become very popular nanoparticles to extract plasmonic resonances and achieve sub-wavelength function- alities [49, 50, 51, 52, 53]. In our FDTD simulations, we consider a nanoring with the following characteristic dimensions: inner radius Rin = 4.5 nm, outer radius Rout = 7 nm, and height h = 2.5 nm. Again, the incident wave ﬁeld is a plane wave traveling along x direction, while the nanoring is oriented with its ring shape laid down on the xy plane and with its wall height aligned along the z axis. As before, the electric

ﬁeld is polarized in the y direction. Figure 4.21 shows the normalized extinction cross

45 section in this case. The blue-shift is less visible in this case but the decrease on the sharpness of the resonances is once more clearly seen once spatial dispersion eﬀects are included. We also observe a decrease on the resonance peak resulting from spatial dispersion.

Figure 4.21: Comparison between local (solid red) and nonlocal (dashed blue) normalized extinction cross sections for an Au nanoring with its ring shape laid down on the x − y plane and with its wall height in the z direction. The nanoring is embedded in silica glass substrate.

46 CHAPTER 5

Conclusion

In this thesis, we have developed an FDTD algorithm to take into account the eﬀects of spatial dispersion in metals. The FDTD algorithm incorporates such nonlocal eﬀects based on the hydrodynamic Drude model for the permittivity of metals.

We utilize Au as an example, but other noble materials can be similarly modeled.

The FDTD model fully includes eddy current eﬀects and was employed to study the extinction cross section of diﬀerent metallic nanoparticles: nanospheres, nanoshells, nanospheroids, nanodisks, and nanorings [2, 16, 17, 11, 12], in the visible spectrum.

It was observed that spatial dispersion can have a signiﬁcant eﬀect on the extinction cross section of nanoparticles with characteristic sizes around 20 nm or below.

For larger particles, the effect is less important. It was determined that inclusion of spatial dispersion causes two general trends on the behavior of the extinction cross section versus frequency as the particle size is reduced. The first trend is a blue-shift on the extinction spectrum. The second trend is an overall decrease on the extinction cross section together with the sharpness of resonance peaks. For the cases considerd, we have also found the relative orientation of the nanoparticles with respect to the incident plane-wave does not noticeably influence the strength of the effects caused by spatial dispersion on the extiction cross section.

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