# Theory of surface plasmons and surface-plasmon polaritons

**ABSTRACT** Collective electronic excitations at metal surfaces are well known to play a key role in a wide spectrum of science, ranging from physics and materials science to biology. Here we focus on a theoretical description of the many-body dynamical electronic response of solids, which underlines the existence of various collective electronic excitations at metal surfaces, such as the conventional surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. We also review existing calculations, experimental measurements, and applications. Comment: 54 pages, 33 figures, to appear in Rep. Prog. Phys

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**ABSTRACT:**Semi-classical non-local optics based on the hydrodynamic description of conduction electrons might be an adequate tool to study complex phenomena in the emerging field of nanoplasmonics. With the aim of confirming this idea, we obtain the local and non-local optical absorption spectra in a model nanoplasmonic device in which there are spatial gaps between the components at nanometric and sub-nanometric scales. After a comparison against time-dependent density functional calculations, we conclude that hydrodynamic non-local optics provides absorption spectra exhibiting qualitative agreement but not quantitative accuracy. This lack of accuracy, which is manifest even in the limit where induced electric currents are not established between the constituents of the device, is mainly due to the poor description of induced electron densities.The Journal of Physical Chemistry C 04/2013; · 4.81 Impact Factor - SourceAvailable from: Matthew Foulkes[show abstract] [hide abstract]

**ABSTRACT:**By simulating the passage of heavy ions along open channels in a model crystalline metal using semi-classical Ehrenfest dynamics we directly investigate the nature of non-adiabatic electronic effects. Our time-dependent tight-binding approach incorporates both an explicit quantum mechanical electronic system and an explicit representation of a set of classical ions. The coupled evolution of the ions and electrons allows us to explore phenomena that lie beyond the approximations made in classical molecular dynamics simulations and in theories of electronic stopping.We report a velocity-dependent charge-localization phenomenon not predicted by previous theoretical treatments of channelling. This charge localization can be attributed to the excitation of electrons into defect states highly localized on the channelling ion. These modes of excitation only become active when the frequency at which the channelling ion moves from interstitial point to equivalent interstitial point matches the frequency corresponding to excitations from the Fermi level into the localized states. Examining the stopping force exerted on the channelling ion by the electronic system, we find broad agreement with theories of slow ion stopping (a stopping force proportional to velocity) for a low velocity channelling ion (up to about 0.5 nm fs from our calculations), and a reduction in stopping power attributable to the charge localization effect at higher velocities. By exploiting the simplicity of our electronic structure model we are able to illuminate the physics behind the excitation processes that we observe and present an intuitive picture of electronic stopping from a real-space, chemical perspective.Journal of Physics Condensed Matter 03/2013; 25(12):125501. · 2.36 Impact Factor - Hannah J Joyce, Callum J Docherty, Qiang Gao, H Hoe Tan, Chennupati Jagadish, James Lloyd-Hughes, Laura M Herz, Michael B Johnston[show abstract] [hide abstract]

**ABSTRACT:**We have performed a comparative study of ultrafast charge carrier dynamics in a range of III-V nanowires using optical pump-terahertz probe spectroscopy. This versatile technique allows measurement of important parameters for device applications, including carrier lifetimes, surface recombination velocities, carrier mobilities and donor doping levels. GaAs, InAs and InP nanowires of varying diameters were measured. For all samples, the electronic response was dominated by a pronounced surface plasmon mode. Of the three nanowire materials, InAs nanowires exhibited the highest electron mobilities of 6000 cm(2) V(-1) s(-1), which highlights their potential for high mobility applications, such as field effect transistors. InP nanowires exhibited the longest carrier lifetimes and the lowest surface recombination velocity of 170 cm s(-1). This very low surface recombination velocity makes InP nanowires suitable for applications where carrier lifetime is crucial, such as in photovoltaics. In contrast, the carrier lifetimes in GaAs nanowires were extremely short, of the order of picoseconds, due to the high surface recombination velocity, which was measured as 5.4 × 10(5) cm s(-1). These findings will assist in the choice of nanowires for different applications, and identify the challenges in producing nanowires suitable for future electronic and optoelectronic devices.Nanotechnology 05/2013; 24(21):214006. · 3.84 Impact Factor

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arXiv:cond-mat/0611257v1 [cond-mat.mtrl-sci] 9 Nov 2006

paper1.tex

Theory of surface plasmons and surface-plasmon polaritons

J. M. Pitarke1,2, V. M. Silkin3, E. V. Chulkov2,3, and P. M. Echenique2,3

1Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea,

644 Posta kutxatila, E-48080 Bilbo, Basque Country

2Donostia International Physics Center (DIPC) and Unidad de F´ ısica de Materiales CSIC-UPV/EHU,

Manuel de Lardizabal Pasealekua, E-20018 Donostia, Basque Country

3Materialen Fisika Saila, Kimika Fakultatea, Euskal Herriko Unibertsitatea,

1072 Posta kutxatila, E-20080 Donostia, Basque Country

(Dated: February 6, 2008)

Collective electronic excitations at metal surfaces are well known to play a key role in a wide

spectrum of science, ranging from physics and materials science to biology. Here we focus on a

theoretical description of the many-body dynamical electronic response of solids, which underlines

the existence of various collective electronic excitations at metal surfaces, such as the conventional

surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. We also

review existing calculations, experimental measurements, and applications.

PACS numbers: 71.45.Gm, 78.68.+m, 78.70.-g

Contents

I. Introduction

2

II. Surface-plasmon polariton: Classical

approach

A. Semi-infinite system

1. The surface-plasmon condition

2. Energy dispersion

3. Skin depth

B. Thin films

4

4

4

5

6

7

III. Nonretarded surface plasmon: Simplified

models

A. Planar surface plasmon

1. Classical model

2. Nonlocal corrections

3. Hydrodynamic approximation

B. Localized surface plasmons: classical

approach

1. Simple geometries

2. Boundary-charge method

3. Composite systems: effective-medium

approach

4. Periodic structures

5. Sum rules

7

8

8

8

9

10

10

10

11

12

12

IV. Dynamical structure factor

13

V. Density-response function

A. Random-phase approximation (RPA)

B. Time-dependent density-functional theory

1. The XC kernel

13

14

14

15

VI. Inverse dielectric function

16

VII. Screened interaction

A. Classical model

1. Planar surface

2. Spheres

16

16

17

17

3. Cylinders

B. Nonlocal models: planar surface

1. Hydrodynamic model

2. Specular-reflection model (SRM)

3. Self-consistent scheme

18

19

19

21

22

VIII. Surface-response function

A. Generation-rate of electronic excitations

B. Inelastic electron scattering

C. Surface plasmons: jellium surface

1. Simple models

2. Self-consistent calculations: long

wavelengths

3. Self-consistent calculations: arbitrary

wavelengths

4. Surface-plasmon linewidth

5. Multipole surface plasmons

D. Surface plasmons: real surfaces

1. Stabilized jellium model

2. Occupied d-bands: simple models

3. First-principles calculations

E. Acoustic surface plasmons

1. A simple model

2. 1D model calculations

3. First-principles calculations

4. Excitation of acoustic surface plasmons

25

25

25

26

26

26

27

29

30

32

32

32

35

36

37

39

40

40

IX. Applications

A. Particle-surface interactions: energy loss

1. Planar surface

B. STEM: Valence EELS

1. Planar surface

2. Spheres

3. Cylinders

C. Plasmonics

41

41

42

44

45

45

46

46

X. Acknowledgments

47

References

48

Page 2

2

I.INTRODUCTION

In his pioneering treatment of characteristic energy

losses of fast electrons passing through thin metal films,

Ritchie predicted the existence of self-sustained collec-

tive excitations at metal surfaces [1].

been pointed out by Pines and Bohm [2, 3] that the long-

range nature of the Coulomb interaction between valence

electrons in metals yields collective plasma oscillations

similar to the electron-density oscillations observed by

Tonks and Langmuir in electrical discharges in gases [4],

thereby explaining early experiments by Ruthemann [5]

and Lang [6] on the bombardment of thin metallic films

by fast electrons. Ritchie investigated the impact of the

film boundaries on the production of collective excita-

tions and found that the boundary effect is to cause the

appearance of a new lowered loss due to the excitation

of surface collective oscillations [1]. Two years later, in

a series of electron energy-loss experiments Powell and

Swan [7] demonstrated the existence of these collective

excitations, the quanta of which Stern and Ferrell called

the surface plasmon [8].

Since then, there has been a significant advance in

both theoretical and experimental investigations of sur-

face plasmons, which for researches in the field of con-

densed matter and surface physics have played a key

role in the interpretation of a great variety of exper-

iments and the understanding of various fundamental

properties of solids. These include the nature of Van

der Waals forces [9, 10, 11], the classical image potential

acting between a point classical charge and a metal sur-

face [12, 13, 14, 15], the energy transfer in gas-surface in-

teractions [16], surface energies [17, 18, 19], the damping

of surface vibrational modes [20, 21], the energy loss of

charged particles moving outside a metal surface [22, 23],

and the de-excitation of adsorbed molecules [24]. Sur-

face plasmons have also been employed in a wide spec-

trum of studies ranging from electrochemistry [25], wet-

ting [26], and biosensing [27, 28, 29], to scanning tun-

neling microscopy [30], the ejection of ions from sur-

faces [31], nanoparticle growth [32, 33], surface-plasmon

microscopy [34, 35], and surface-plasmon resonance tech-

nology [36, 37, 38, 39, 40, 41, 42]. Renewed interest in

surface plasmons has come from recent advances in the

investigation of the electromagnetic properties of nanos-

tructured materials [43, 44], one of the most attractive

aspects of these collective excitations now being their

use to concentrate light in subwavelength structures and

to enhance transmission through periodic arrays of sub-

wavelength holes in optically thick metallic films [45, 46].

The so-called field of plasmonics represents an excit-

ing new area for the application of surface and interface

plasmons, and area in which surface-plasmon based cir-

cuits merge the fields of photonics and electronics at the

nanoscale [47]. Indeed, surface-plasmon polaritons can

serve as a basis for constructing nanoscale photonic cir-

cuits that will be able to carry optical signals and electric

currents [48, 49]. Surface plasmons can also serve as a

It had already

basis for the design, fabrication, and characterization of

subwavelength waveguide components [50, 51, 52, 53, 54,

55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In the framework

of plasmonics, modulators and switches have also been

investigated [65, 66], as well as the use of surface plas-

mons as mediators in the transfer of energy from donor to

acceptors molecules on opposite sides of metal films [67].

According to the work of Pines and Bohm, the

quantum energy collective plasma oscillations in a free

electron gas with equilibrium density n is ¯ hωp

¯ h(4πne2/me)1/2, ωp being the so-called plasmon fre-

quency [68]. In the presence of a planar boundary, there

is a new mode (the surface plasmon), the frequency of

which equals in the nonretarded region (where the speed

of light can be taken to be infinitely large) Ritchie’s

frequency ωs = ωp/√2 at wave vectors q in the range

ωs/c << q << qF (qF being the magnitude of the Fermi

wave vector) and exhibits some dispersion as the wave

vector is increased. In the retarded region, where the

phase velocity ωs/q of the surface plasmon is comparable

to the velocity of light, surface plasmons couple with the

free electromagnetic field. These surface-plasmon polari-

tons propagate along the metal surface with frequencies

ranging from zero (at q = 0) towards the asymptotic

value ωs= ωp/√2, the dispersion relation ω(q) lying to

the right of the light line and the propagating vector be-

ing, therefore, larger than that of bare light waves of the

same energy. Hence, surface-plasmon polaritons in an

ideal semi-infinite medium are nonradiative in nature,

i.e., cannot decay by emitting a photon and, conversely,

light incident on an ideal surface cannot excite surface

plasmons.

In the case of thin films, the electric fields of both

surfaces interact. As a result, there are (i) tangential

oscillations characterized by a symmetric disposition of

charge deficiency or excess at opposing points on the two

surfaces and (ii) normal oscillations in which an excess

of charge density at a point on one surface is accompa-

nied by a deficiency at the point directly across the thin

film. The phase velocity of the tangential surface plas-

mon is always less than the speed of light, as occurs in

the case of a semi-infinite electron system. However, the

phase velocity of normal oscillations may surpass that

of light, thereby becoming a radiative surface plasmon

that should be responsible for the emission of light [69].

This radiation was detected using electron beam bom-

bardment of thin films of Ag, Mg, and Al with thick-

nesses ranging between 500 and 1000˚ A [70, 71]. More re-

cently, light emission was observed in the ultraviolet from

a metal-oxide-metal tunnel diode and was attributed to

the excitation of the radiative surface plasmon [72].

Nonradiative surface plasmons in both thin and thick

films can couple to electromagnetic radiation in the pres-

ence of surface roughness or a grating, as suggested by

Teng and Stern [73]. Alternatively, prism coupling can

be used to enhance the momentum of incident light,

as demonstrated by Otto [74] and by Kretchmann and

Raether [75]. Since then, this so-called attenuated reflec-

=

Page 3

3

tion (ATR) method and variations upon it have been

used by several workers in a large variety of applica-

tions [76, 77, 78, 79, 80, 81, 82].

During the last decades, there has also been a signifi-

cant advance in our understanding of surface plasmons in

the nonretarded regime. Ritchie [83] and Kanazawa [84]

were the first to attack the problem of determining the

dispersion ω(q) of the nonretarded surface plasmon. Ben-

nett [85] used a hydrodynamical model with a continu-

ous decrease of the electron density at the metal surface,

and found that a continuous electron-density variation

yields two collective electronic excitations: Ritchie’s sur-

face plasmon at ω ∼ ωs, with a negative energy disper-

sion at low wave vectors, and an upper surface plasmon

at higher energies. In the direction normal to the surface,

the distribution of Ritchie’s surface plasmon consists of

a single peak, i.e., it has a monopole character; however,

the charge distribution of the upper mode has a node, i.e.,

it has a dipole character and is usually called multipole

surface plasmon.

Bennett’s qualitative conclusions were generally con-

firmed by microscopic descriptions of the electron gas.

On the one hand, Feibelman showed that in the long-

wavelength limit the classical result ωs= ωp/√2 is cor-

rect for a semi-infinite plane-bounded electron gas, irre-

spective of the exact variation of the electron density in

the neighborhood of the surface [86]. On the other hand,

explicit expressions for the linear momentum dispersion

of the conventional monopole surface plasmon that are

sensitive to the actual form of the electron-density fluc-

tuation at the surface were derived by Harris and Grif-

fin [87] using the equation of motion for the Wigner dis-

tribution function in the random-phase approximation

(RPA) and by Flores and Garc´ ıa-Moliner [88] solving

Maxwell’s equations in combination with an integration

of the field components over the surface region. Quan-

titative RPA calculations of the linear dispersion of the

monopole surface plasmon were carried out by several

authors by using the infinite-barrier model (IBM) of the

surface [89], a step potential [90, 91], and the more realis-

tic Lang-Kohn [92] self-consistent surface potential [93].

Feibelman’s calculations showed that for the typical elec-

tron densities in metals (2 < rs< 6) the initial slope of

the momentum dispersion of monopole surface plasmons

of jellium surfaces is negative [93], as anticipated by Ben-

nett [85].

Negative values of the momentum dispersion had been

observed by high-energy electron transmission on unchar-

acterized Mg surfaces [94] and later by inelastic low-

energy electron diffraction on the (100) and (111) sur-

faces of Al [95, 96]. Nevertheless, Klos and Raether [97]

and Krane and Raether [98] did not observe a negative

dispersion for Mg and Al films. Conclusive experimental

confirmation of the negative surface plasmon dispersion

of a variety of simple metals (Li, Na, K, Cs, Al, and Mg)

did not come until several years later [99, 100, 101, 102],

in a series of experiments based on angle-resolved low-

energy inelastic electron scattering [103].These ex-

periments showed good agreement with self-consistent

dynamical-response calculations carried out for a jellium

surface [104] in a time-dependent adiabatic extension

of the density-functional theory (DFT) of Hohenberg,

Kohn, and Sham [105]. Furthermore, these experiments

also showed that the multipole surface plasmon was ob-

servable, its energy and dispersion being in quantitative

agreement with the self-consistent jellium calculations

that had been reported by Liebsch [106].

Significant deviations from the dispersion of surface

plasmons at jellium surfaces occur on Ag [107, 108, 109,

110] and Hg [111], due to the presence of filled 4d and

5d bands, respectively, which in the case of Ag yields

an anomalous positive dispersion. In order to describe

the observed features of Ag surface plasmons, various

simplified models for the screening of d electrons have

been developed [112, 113, 114, 115, 116]. Most recently,

calculations have been found to yield a qualitative un-

derstanding of the existing electron energy-loss measure-

ments by combining a self-consistent jellium model for

valence 5s electrons with a so-called dipolium model in

which the occupied 4d bands are represented in terms of

polarizable spheres located at the sites of a semi-infinite

face-cubic-centered (fcc) lattice [117].

Ab initio bulk calculations of the dynamical response

and plasmon dispersions of noble metals with occu-

pied d bands have been carried out recently [118, 119,

120]. However, first-principles calculations of the surface-

plasmon energy and linewidth dispersion of real solids

have been carried out only in the case of the simple-metal

prototype surfaces Mg(0001) and Al(111) [121, 122].

These calculations lead to an accurate description of the

measured surface-plasmon energy dispersion that is supe-

rior to that obtained in the jellium model, and they show

that the band structure is of paramount importance for

a correct description of the surface-plasmon linewidth.

The multipole surface plasmon, which is originated

in the selvage electronic structure at the surface, has

been observed in a variety of simple metals at ω ∼

0.8ωp [99, 100, 101, 102], in agreement with theoreti-

cal predictions. Nevertheless, electron energy-loss spec-

troscopy (EELS) measurements of Ag, Hg, and Li re-

vealed no clear evidence of the multipole surface plas-

mon. In the case of Ag, high-resolution energy-loss spec-

troscopy low-energy electron diffraction (ELS-LEED)

measurements indicated that a peak was obtained at

3.72eV by subtracting the data for two different impact

energies [123], which was interpreted to be the Ag mul-

tipole plasmon. However, Liebsch argued that the fre-

quency of the Ag multipole surface plasmon should be

in the 6 − 8eV range above rather than below the bulk

plasma frequency, and suggested that the observed peak

at 3.72eV might not be associated with a multipole sur-

face plasmon [124].

An alternative spectroscopy technique to investigate

multipole surface plasmons is provided by angle- and

energy-resolved photoyield experiments (AERPY) [125].

In fact, AERPY is more suitable than electron energy-

Page 4

4

loss spectroscopy to identify the multipole surface plas-

mon, since the monopole surface plasmon of clean flat

surfaces (which is the dominant feature in electron-

energy loss spectra) is not excited by photons and thus

the weaker multipole surface mode (which intersects the

radiation line in the retardation regime) can be observed.

A large increase in the surface photoyield was observed

at ω = 0.8ωpfrom Al(100) [125] and Al(111) [126]. Re-

cently, the surface electronic structure and optical re-

sponse of Ag has been studied using this technique [127].

In these experiments, the Ag multipole surface plasmon

is observed at 3.7eV, while no signature of the multipole

surface plasmon is observed above the plasma frequency

(ωp= 3.8eV) in disagreement with the existing theoret-

ical prediction [124]. Hence, further theoretical work is

needed on the surface electronic response of Ag that go

beyond the s-d polarization model described in Ref. [124].

Another collective electronic excitation at metal sur-

faces is the so-called acoustic surface plasmon that has

been predicted to exist at solid surfaces where a par-

tially occupied quasi-two-dimensional surface-state band

coexists with the underlying three-dimensional contin-

uum [128, 129]. This new low-energy collective excitation

exhibits linear dispersion at low wave vectors, and might

therefore affect electron-hole (e-h) and phonon dynamics

near the Fermi level [130]. It has been demonstrated that

it is a combination of the nonlocality of the 3D dynamical

screening and the spill out of the 3D electron density into

the vacuum which allows the formation of 2D electron-

density acoustic oscillations at metal surfaces, since these

oscillations would otherwise be completely screened by

the surrounding 3D substrate [131]. This novel surface-

plasmon mode has been observed recently at the (0001)

surface of Be, showing a linear energy dispersion that

is in very good agreement with first-principles calcula-

tions [132].

Finally, we note that metal-dielectric interfaces of ar-

bitrary geometries also support charge density oscilla-

tions similar to the surface plasmons characteristic of

planar interfaces. These are localized Mie plasmons oc-

curring at frequencies which are characteristic of the

interface geometry [133].

plasmons on small particles has attracted great inter-

est over the years in scanning transmission electron mi-

croscopy [134, 135, 136, 137, 138, 139] and near-field

optical spectroscopy [140].

structuring and manipulating on the nanometer scale

have rekindled interest this field [141]. In nanostructured

metals and carbon-based structures, such as fullerenes

and carbon nanotubes, localized plasmons can be ex-

cited by light and can therefore be easily detected as

pronounced optical resonances [142, 143, 144]. Further-

more, very localized dipole and multipole modes in the

vicinity of highly coupled structures are responsible for

surface-enhanced Raman scattering [145, 146] and other

striking properties like, e.g., the blackness of colloidal

silver [147].

Collective electronic excitations in thin adsorbed over-

The excitation of localized

Recently, new advances in

layers, semiconductor heterostructures, and parabolic

quantum wells have also attracted attention over the last

years. The adsorption of thin films is important, because

of the drastic changes that they produce in the electronic

properties of the substrate and also because of related

phenomena such as catalytic promotion [148]; however,

the understanding of adsorbate-induced collective exci-

tations is still incomplete [149, 150, 151, 152, 153, 154,

155, 156]. The excitation spectrum of collective modes in

semiconductor quantum wells has been described by sev-

eral authors [157, 158, 159, 160, 161]. These systems,

which have been grown in semiconductor heterostruc-

tures with the aid of Molecular Beam Epitaxy [162], form

a nearly ideal free-electron gas and have been, therefore,

a playground on which to test existing many-body theo-

ries [163, 164].

Major reviews on the theory of collective elec-

tronic excitations at metal surfaces have been given by

Ritchie [165], Feibelman [166], and Liebsch [167]. Ex-

perimental reviews are also available, which focus on

high-energy EELS experiments [168], surface plasmons

on smooth and rough surfaces and on gratings [169], and

angle-resolved low-energy EELS investigations [170, 171].

An extensive review on plasmons and magnetoplasmons

in semiconductor heterostructures has been given re-

cently by Kushwaha [172].

This review will focus on a unified theoretical descrip-

tion of the many-body dynamical electronic response of

solids, which underlines the existence of various collective

electronic excitations at metal surfaces, such as the con-

ventional surface plasmon, multipole plasmons, and the

acoustic surface plasmon. We also review existing cal-

culations, experimental measurements, and some of the

most recent applications including particle-solid inter-

actions, scanning transmission electron microscopy, and

surface-plasmon based photonics, i.e., plasmonics.

II.SURFACE-PLASMON POLARITON:

CLASSICAL APPROACH

A.Semi-infinite system

1.The surface-plasmon condition

We consider a classical model consisting of two

semi-infinite nonmagnetic media with local (frequency-

dependent) dielectric functions ǫ1and ǫ2separated by a

planar interface at z = 0 (see Fig. 1). The full set of

Maxwell’s equations in the absence of external sources

can be expressed as follows [173]

∇ × Hi= ǫi1

c

∂

∂tEi,(2.1)

∇ × Ei= −1

c

∂

∂tHi,(2.2)

Page 5

5

ε 1

ε 2

z=0

FIG. 1: Two semi-infinite media with dielectric functions ǫ1

and ǫ2 separated by a planar interface at z = 0.

∇ · (ǫiEi) = 0, (2.3)

and

∇ · Hi= 0,(2.4)

where the index i describes the media: i = 1 at z < 0,

and i = 2 at z > 0.

Solutions of Eqs. (2.1)-(2.4) can generally be classified

into s-polarized and p-polarized electromagnetic modes,

the electric field E and the magnetic field H being paral-

lel to the interface, respectively. For an ideal surface, if

waves are to be formed that propagate along the interface

there must necessarily be a component of the electric field

normal to the surface. Hence, s-polarized surface oscil-

lations (whose electric field E is parallel to the interface)

do not exist; instead, we seek conditions under which a

traveling wave with the magnetic field H parallel to the

interface (p-polarized wave) may propagate along the sur-

face (z = 0), with the fields tailing off into the positive

(z > 0) and negative (z < 0) directions. Choosing the

x-axis along the propagating direction, we write

Ei= (Eix,0,Eiz)e−κi|z|ei(qix−ωt)

(2.5)

and

Hi= (0,Eiy,0)e−κi|z|ei(qix−ωt),(2.6)

where qirepresents the magnitude of a wave vector that

is parallel to the surface. Introducing Eqs. (2.5) and (2.6)

into Eqs. (2.1)-(2.4), one finds

iκ1H1y= +ω

cǫ1E1x,(2.7)

iκ2H2y= −ω

cǫ2E2x, (2.8)

and

κi=

?

q2

i− ǫiω2

c2.(2.9)

The boundary conditions imply that the component

of the electric and magnetic fields parallel to the sur-

face must be continuous. Using Eqs. (2.7) and (2.8), one

writes the following system of equations:

κ1

ǫ1

H1y+κ2

ǫ2

H2y= 0 (2.10)

and

H1y− H2y= 0,(2.11)

which has a solution only if the determinant is zero, i.e.,

ǫ1

κ1

+ǫ2

κ2

= 0.(2.12)

This is the surface-plasmon condition.

From the boundary conditions also follows the conti-

nuity of the 2D wave vector q entering Eq. (2.9), i.e.,

q1 = q2 = q.Hence, the surface-plasmon condition

[Eq. (2.12)] can also be expressed as follows [174],

q(ω) =ω

c

?

ǫ1ǫ2

ǫ1+ ǫ2,(2.13)

where ω/c represents the magnitude of the light wave

vector. For a metal-dielectric interface with the dielectric

characterized by ǫ2, the solution ω(q) of Eq. (2.13) has

slope equal to c/√ǫ2at the point q = 0 and is a mono-

tonic increasing function of q, which is always smaller

than cq/√ǫ2and for large q is asymptotic to the value

given by the solution of

ǫ1+ ǫ2= 0. (2.14)

This is the nonretarded surface-plasmon condition

[Eq. (2.12) with κ1 = κ2 = q], which is valid as long

as the phase velocity ω/q is much smaller than the speed

of light.

2.Energy dispersion

In the case of a Drude semi-infinite metal in vacuum,

one has ǫ2= 1 and [175]

ǫ1= 1 −

ω2

p

ω(ω + iη),(2.15)

η being a positive infinitesimal.

Eq. (2.13) yields

Hence, in this case

q(ω) =ω

c

?

ω2− ω2

2ω2− ω2

p

p

.(2.16)

We have represented in Fig. 2 by solid lines the dis-

persion relation of Eq. (2.16), together with the light line

ω = cq (dotted line). The upper solid line represents the

Page 6

6

00.01

q (A -1)

0.02

0

5

10

15

20

25

30

ω (eV)

FIG. 2: The solid lines represent the solutions of Eq. (2.16)

with ωp = 15eV: the dispersion of light in the solid (upper

line) and the surface-plasmon polariton (lower line). In the

retarded region (q < ωs/c), the surface-plasmon polariton

dispersion curve approaches the light line ω = cq (dotted

line). At short wave lengths (q >> ωs/c), the surface-plasmon

polariton approaches asymptotically the nonretarded surface-

plasmon frequency ωs = ωp/√2 (dashed line).

FIG. 3:

field associated with a surface-plasmon polariton propagating

along a metal-dielectric interface. The field strength Ei [see

Eq. (2.5)] decreases exponentially with the distance |z| from

the surface, the decay constant κi being given by Eq. (2.18).

+ and - represent the regions with lower and higher electron

density, respectively.

Schematic representation of the electromagnetic

dispersion of light in the solid. The lower solid line is the

surface-plasmon polariton

ω2(q) = ω2

p/2 + c2q2−

?

ω4

p/4 + c4q4,(2.17)

which in the retarded region (where q < ωs/c) cou-

ples with the free electromagnetic field and in the non-

retarded limit (q >> ωs/c) yields the classical nondis-

persive surface-plasmon frequency ωs= ωp/√2.

We note that the wave vector q entering the dispersion

relation of Eq. (2.17) (lower solid line of Fig. 2) is a 2D

wave vector in the plane of the surface. Hence, if light

hits the surface in an arbitrary direction the external ra-

diation dispersion line will always lie somewhere between

the light line cq and the vertical line, in such a way that it

will not intersect the surface-plasmon polariton line, i.e.,

light incident on an ideal surface cannot excite surface

plasmons. Nevertheless, there are two mechanisms that

0

0.005

0.01

q (A-1)

0.015

0.02

100

200

300

400

li (A)

metal

vacuum

1/q

FIG. 4: Attenuation length li = 1/κi, versus q, as obtained

from Eq. (2.18) at the surface-plasmon polariton condition

[Eq. (2.17)] for a Drude metal in vacuum. ǫ1 has been taken

to be of the form of Eq. (2.15) with ωp = 15eV and ǫ2 has

been set up to unity. The dotted line represents the large-q

limit of both l1 and l2, i.e., 1/q.

allow external radiation to be coupled to surface-plasmon

polaritons: surface roughness or gratings, which can pro-

vide the requisite momentum via umklapp processes [73],

and attenuated total reflection (ATR) which provides the

external radiation with an imaginary wave vector in the

direction perpendicular to the surface [74, 75].

3.Skin depth

Finally, we look at the spatial extension of the electro-

magnetic field associated with the surface-plasmon po-

lariton (see Fig. 3). Introducing the surface-plasmon con-

dition of Eq. (2.13) into Eq. (2.9) (with q1= q2= q), one

finds the following expression for the surface-plasmon de-

cay constant κiperpendicular to the interface:

κi=ω

c

?

−ǫ2

ǫ1+ ǫ2,

i

(2.18)

which allows to define the attenuation length li= 1/κiat

which the electromagnetic field falls to 1/e. Fig. 4 shows

lias a function of the magnitude q of the surface-plasmon

polariton wave vector for a Drude metal [ǫ1of Eq. (2.15)]

in vacuum (ǫ2= 0). In the vacuum side of the interface,

the attenuation length is over the wavelength involved

(l2> 1/q), whereas the attenuation length into the metal

is determined at long-wavelengths (q → 0) by the so-

called skin-depth.At large q [where the nonretarded

surface-plasmon condition of Eq. (2.14) is fulfilled], the

skin depth is li∼ 1/q thereby leading to a strong concen-

tration of the electromagnetic surface-plasmon field near

the interface.

Page 7

7

B.Thin films

Thin films are also known to support surface collective

oscillations. For this geometry, the electromagnetic fields

of both surfaces interact in such a way that the retarded

surface-plasmon condition of Eq. (2.12) splits into two

new conditions (we only consider nonradiative surface

plasmons), depending on whether electrons in the two

surfaces oscillate in phase or not. In the case of a thin

film of thickness a and dielectric function ǫ1in a medium

of dielectric function ǫ2, one finds [169]:

ǫ1

κ1tanh(κ1a/2)+ǫ2

κ2

= 0(2.19)

and

ǫ1

κ1coth(κ1a/2)+ǫ2

κ2

= 0.(2.20)

Instead, if the film is surrounded by dielectric layers of

dielectric constant ǫ0and equal thickness t on either side,

one finds

ǫ1

κ1ν tanh(κ1a/2)+ǫ0

κ0

= 0(2.21)

and

ǫ1

κ1ν coth(κ1a/2)+ǫ0

κ0

= 0, (2.22)

where

ν =1 − ∆e−2κ0t

1 + ∆e−2κ0t, (2.23)

with

∆ =κ2ǫ0− κ0ǫ2

κ2ǫ0+ κ0ǫ2

(2.24)

and

κ0=

?

q2− ǫ0ω2

c2. (2.25)

Electron spectrometry measurements of the dispersion

of the surface-plasmon polariton in oxidized Al films were

reported by Pettit, Silcox, and Vincent [176], spanning

the energy range from the short-wavelength limit where

ω ∼ ωp/√2 all the way to the long-wavelength limit

where ω ∼ cq. The agreement between the experimen-

tal measurements and the prediction of Eqs. (2.21)-(2.25)

(with a Drude dielectric function for the Al film and a

dielectric constant ǫ0 = 4 for the surrounding oxide) is

found to be very good, as shown in Fig. 5.

In the nonretarded regime (q >> ωs/c), where κ1 =

κ2= q, Eqs. (2.19) and (2.20) take the form

ǫ1+ ǫ2

ǫ1− ǫ2

= ∓e−qa, (2.26)

00.01 0.020.03

q (A -1)

0

10

ω (eV)

FIG. 5: Dispersion ω(q) of the surface-plasmon polariton of

an Al film of thickness a = 120˚ A surrounded by dielectric lay-

ers of equal thickness t = 40˚ A. The solid lines represent the

result obtained from Eqs. (2.21)-(2.25) with ǫ2 = 1, ǫ0 = 4,

and a frequency-dependent Drude dielectric function ǫ1 [see

Eq. (2.15)] with ωp = 15eV and η = 0.75eV [177]. The solid

circles represent the electron spectrometry measurements re-

ported by Petit, Silcox, and Vincent [176]. The dashed line

represents the nonretarded surface-plasmon frequency ωp/√5,

which is the solution of Eq. (2.14) with ǫ2 = 4 and a Drude

dielectric function ǫ1. The dotted line represents the light line

ω = cq.

which for a Drude thin slab [ǫ1of Eq. (2.15)] in vacuum

(ǫ2= 1) yields [1]:

ω =ωp

√2

?1 ± e−qa?1/2.(2.27)

This equation has two limiting cases, as discussed by

Ferrell [69]. At short wavelengths (qa >> 1), the sur-

face waves become decoupled and each surface sustains

independent oscillations at the reduced frequency ωs =

ωp/√2 characteristic of a semi-infinite electron gas with

a single plane boundary. At long wavelengths (qa << 1),

there are normal oscillations at ωpand tangential 2D os-

cillations at

ω2D= (2πnaq)1/2,(2.28)

which were later discussed by Stern [178] and observed

in artificially structured semiconductors [179] and more

recently in a metallic surface-state band on a silicon sur-

face [180].

III. NONRETARDED SURFACE PLASMON:

SIMPLIFIED MODELS

The classical picture leading to the retarded Eq. (2.12)

and nonretarded Eq. (2.14) ignores both the nonlocality

of the electronic response of the system and the micro-

scopic spatial distribution of the electron density near

Page 8

8

the surface. This microscopic effects can generally be ig-

nored at long wavelengths where q << qF; however, as

the excitation wavelength approaches atomic dimensions

nonlocal effects can be important.

As nonlocal effects can generally be ignored in the re-

tarded region where q < ωs/c (since ωs/c << qF), here

we focus our attention on the nonretarded regime where

ωs/c < q. In this regime and in the absence of external

sources, the ω-components of the time-dependent electric

and displacement fields associated with collective oscilla-

tions at a metal surface satisfy the quasi-static Maxwell’s

equations

∇ · E(r,ω) = −4πδn(r,ω), (3.1)

or, equivalently,

∇2φ(r,ω) = 4π δn(r,ω), (3.2)

and

∇ · D(r,ω) = 0,(3.3)

δn(r,ω) being the fluctuating electron density associated

with the surface plasmon, and φ(r,ω) being the ω com-

ponent of the time-dependent scalar potential.

A.Planar surface plasmon

1.Classical model

In the classical limit, we consider two semi-infinite me-

dia with local (frequency-dependent) dielectric functions

ǫ1 and ǫ2 separated by a planar interface at z = 0, as

in Section IIA (see Fig. 1). In this case, the fluctuating

electron density δn(r,ω) corresponds to a delta-function

sheet at z = 0:

δn(r,ω) = δn(r?,ω) δ(z),(3.4)

where r?defines the position vector in the surface plane,

and the displacement field D(r,ω) takes the following

form:

D(r,ω) =

?ǫ1E(r,ω),

ǫ2E(r,ω),

z < 0,

z > 0.

(3.5)

Introducing Eq. (3.4) into Eq. (3.2), one finds that self-

sustained solutions of Poisson’s equation take the form

φ(r,ω) = φ0eq·r?e−q|z|, (3.6)

where q is a 2D wave vector in the plane of the surface,

and q = |q|. A combination of Eqs. (3.3), (3.5), and

(3.6) with E(r,ω) = −∇φ(r,ω) yields the nonretarded

surface-plasmon condition of Eq. (2.14), i.e.,

ǫ1+ ǫ2= 0. (3.7)

2.Nonlocal corrections

Now we consider a more realistic jellium model of

the solid surface consisting of a fixed semi-infinite uni-

form positive background at z ≤ 0 plus a neutralizing

nonuniform cloud of interacting electrons. Within this

model, there is translational invariance in the plane of

the surface; hence, we can define 2D Fourier transforms

E(z;q,ω) and D(z;q,ω), the most general linear relation

between them being

D(z;q,ω) =

?

dz′ǫ(z,z′;q,ω) · E(z′;q,ω),(3.8)

where the tensor ǫ(z,z′;q,ω) represents the dielectric

function of the medium.

In order to avoid an explicit calculation of ǫ(z,z′;q,ω),

one can assume that far from the surface and at low wave

vectors (but still in the nonretarded regime, i.e., ωs/c <

q < qF) Eq. (3.8) reduces to an expression of the form of

Eq. (3.5):

D(z;q,ω) =

?ǫ1E(z;q,ω),

ǫ2E(z;q,ω),

z < z1,

z > z2,

(3.9)

where z1<< 0 and z2>> 0. Eqs. (3.2) and (3.3) with

E(r,ω) = −∇φ(r,ω) then yield the following integration

of the field components Ez and Dx in terms of the po-

tential φ(z) at z1and z2[where it reduces to the classical

potential of Eq. (3.6)]:

?z2

z1

dz Ez(z;q,ω) = φ(z2;q,ω) − φ(z1;q,ω) (3.10)

and

− i

?z2

z1

dz Dx(z;q,ω) = ǫ2φ(z2;q,ω) − ǫ1φ(z1;q,ω).

(3.11)

Neglecting quadratic and higher-order terms in the

wave vector, Eqs. (3.10) and (3.11) are found to be com-

patible under the surface-plasmon condition [88]

ǫ1+ ǫ2

ǫ1− ǫ2

= q?d⊥(ω) − d?(ω)?,(3.12)

d⊥(ω) and d?(ω) being the so-called d-parameters intro-

duced by Feibelman [166]:

d⊥(ω) =

?

?

dz zd

dzEz(z,ω)/

?

dz δn(z,ω)

dz

d

dzEz(z,ω)

=dz z δn(z,ω)/

?

(3.13)

and

d?(ω) =

?

dz zd

dzDx(z,ω)/

?

dz

d

dzDx(z,ω), (3.14)

where Ez(z,ω), Dx(z,ω), and δn(z,ω) represent the

fields and the induced density evaluated in the q → 0

limit.

Page 9

9

For a Drude semi-infinite metal in vacuum [ǫ2 = 1

and Eq. (2.15) for ǫ1], the nonretarded surface-plasmon

condition of Eq. (3.12) yields the nonretarded dispersion

relation

ω = ωs

where ωsis Ritchie’s frequency: ωs= ωp/√2. For neu-

tral jellium surfaces, d?(ω) coincides with the jellium edge

and the linear coefficient of the surface-plasmon disper-

sion ω(q), therefore, only depends on the position d⊥(ωs)

of the centroid of the induced electron density at ωs[see

Eq. (3.13)] with respect to the jellium edge.

?1 − qRe?d⊥(ωs) − d?(ωs)?/2 + ...?, (3.15)

3. Hydrodynamic approximation

In a hydrodynamic model, the collective motion of elec-

trons in an arbitrary inhomogeneous system is expressed

in terms of the electron density n(r,t) and the hydrody-

namical velocity v(r,t), which assuming irrotational flow

we express as the gradient of a velocity potential ψ(r,t)

such that v(r,t) = −∇ψ(r,t). First of all, one writes

the basic hydrodynamic Bloch’s equations (the continu-

ity equation and the Bernoulli’s equation) in the absence

of external sources [181]:

d

dtn(r,t) = ∇ · [n(r,t) ∇ψ(r,t)] (3.16)

and

d

dtψ(r,t) =1

2|∇ψ(r,t)|2+δG[n]

δn

+ φ(r,t),(3.17)

and Poisson’s equation:

∇2φ(r,t) = 4π n(r,t),(3.18)

where G[n] is the internal kinetic energy, which is typi-

cally approximated by the Thomas-Fermi functional

G[n] =

3

10(3π2)2/3[n(r,t)]5/3.(3.19)

The hydrodynamic equations [Eqs. (3.16)-(3.18)] are

nonlinear equations, difficult to solve.

typically uses perturbation theory to expand the electron

density and the velocity potential as follows

Therefore, one

n(r,t) = n0(r) + n1(r,t) + ...(3.20)

and

ψ(r,t) = 0 + ψ1(r,t)...,(3.21)

so that Eqs. (3.16)-(3.18) yield the linearized hydrody-

namic equations

d

dtn1(r,t) = ∇ · [n0(r) ∇ψ1(r,t)], (3.22)

d

dtψ1(r,t) = [β(r)]2n1(r,t)

n0(r)

+ φ1(r,t), (3.23)

and

∇2φ1(r,t) = 4π n1(r,t), (3.24)

where n0(r) is the unperturbed electron density and

β(r) =

?1/3?3π2n0(r)?1/3represents the speed of prop-

tem [182].

We now consider a semi-infinite metal in vacuum con-

sisting of an abrupt step of the unperturbed electron den-

sity at the interface, which we choose to be located at

z = 0:

?¯ n,

0,

agation of hydrodynamic disturbances in the electron sys-

n0(z) =

z ≤ 0,

z > 0.

(3.25)

Hence, within this model n0(r) and β(r) are constant at

z ≤ 0 and vanish at z > 0.

Introducing Fourier transforms, Eqs. (3.22)-(3.25)

yield the basic differential equation for the plasma nor-

mal modes at z ≤ 0:

∇2(ω2− ω2

and Laplace’s equation at z > 0:

∇2φ1(r,ω) = 0

where both n1(r,ω) and ψ1(r,ω) vanish. Furthermore,

translational invariance in the plane of the surface allows

to introduce the 2D Fourier transform ψ1(z;q,ω), which

according to Eq. (3.26) must satisfy the following equa-

tion at z ≤ 0:

(−q2+d2/dz2)?ω2− ω2

where q represents a 2D wave vector in the plane of the

surface.

Now we need to specify the boundary conditions. Rul-

ing out exponential increase at z → ∞ and noting that

the normal component of the hydrodynamical velocity

should vanish at the interface, for each value of q one

finds solutions to Eq. (3.28) with frequencies [183, 184]

ω2≥ ω2

and

ω2=1

2

Eqs. (3.29) and (3.30) represent a continuum of bulk nor-

mal modes and a surface normal mode, respectively. At

long wavelengths, where β q/ωp << 1 (but still in the

nonretarded regime where ωs/c < q), Eq. (3.30) yields

the surface-plasmon dispersion relation

ω = ωp/√2 + β q/2,

p+ β2∇2)ψ1(r,ω)(z ≤ 0), (3.26)

(z > 0),(3.27)

p− β(−q2+ d2/dz2)?ψ1(z;q,ω),

(3.28)

p+ β2q2

(3.29)

?

ω2

p+ β2q2+ βq

?

2ω2

p+ β2q2?

. (3.30)

(3.31)

which was first derived by Ritchie [83] using Bloch’s

equations, and later by Wagner [185] and by Ritchie

and Marusak [186] by assuming, within a Boltzmann

transport-equation approach, specular reflection at the

surface.

Page 10

10

B.Localized surface plasmons: classical approach

Metal-dielectric interfaces of arbitrary geometries also

support charge density oscillations similar to the surface

plasmons characteristic of planar interfaces. In the long-

wavelength (or classical) limit, in which the interface sep-

arates two media with local (frequency-dependent) di-

electric functions ǫ1and ǫ2, one writes

Di(r,ω) = ǫiEi(r,ω),(3.32)

where the index i refers to the media 1 and 2 separated

by the interface. In the case of simple geometries, such as

spherical and cylindrical interfaces, Eqs. (3.1)-(3.3) can

be solved explicitly with the aid of Eq. (3.32) to find

explicit expressions for the nonretarded surface-plasmon

condition.

1.Simple geometries

a.Spherical interface.

electric function ǫ1in a host medium of dielectric func-

tion ǫ2, the classical (long-wavelength) planar surface-

plasmon condition of Eq. (3.7) is easily found to be re-

placed by [133]

In the case of a sphere of di-

lǫ1+ (l + 1)ǫ2= 0,l = 1,2,...,(3.33)

which in the case of a Drude metal sphere [ǫ1 of

Eq. (2.15)] in vacuum (ǫ2= 1) yields the Mie plasmons

at frequencies

ωl= ωp

?

l

2l + 1. (3.34)

b.Cylindrical interface.

long cylinder of dielectric function ǫ1in a host medium

of dielectric function ǫ2, the classical (long-wavelength)

surface-plasmon condition depends on the direction of

the electric field.For electromagnetic waves with the

electric field normal to the interface (p-polarization),

the corresponding long-wavelength (and nonretarded)

surface-plasmon condition coincides with that of a planar

surface, i.e., [187, 188, 189]

In the case of an infinitely

ǫ1+ ǫ2= 0, (3.35)

which for Drude cylinders [ǫ1 of Eq. (2.15)] in vacuum

(ǫ2 = 1) yields the planar surface-plasmon frequency

ωs= ωp/√2.

For electromagnetic waves with the electric field par-

allel to the axis of the cylinder (s-polarization), the pres-

ence of the interface does not modify the electric field

and one easily finds that only the bulk mode of the host

medium is present, i.e., one finds the plasmon condition

ǫ2= 0.(3.36)

In some situations, instead of having one single cylinder

in a host medium, an array of parallel cylinders may be

present with a filling fraction f. In this case and for

electromagnetic waves polarized along the cylinders (s-

polarization), the plasmon condition of Eq. (3.36) must

be replaced by [190]

f ǫ1+ (1 − f)ǫ2= 0, (3.37)

which for Drude cylinders [ǫ1 of Eq. (2.15)] in vac-

uum (ǫ2 = 1) yields the reduced plasmon frequency

ω =√f ωp.

2. Boundary-charge method

In the case of more complex interfaces, a so-called

boundary-charge method (BCM) has been used by sev-

eral authors to determine numerically the classical (long-

wavelength) frequencies of localized surface plasmons. In

this approach, one first considers the ω-component of the

time-dependent surface charge density arising from the

difference between the normal components of the electric

fields inside and outside the surface:

σs(r,ω) =

1

4π[E(r,ω) · n|r=r− + E(r,ω) · n|r=r+],

(3.38)

which noting that the normal component of the displace-

ment vector [see Eq. (3.32)] must be continuous yields

the following expression:

σs(r,ω) =

1

4π

ǫ1− ǫ2

ǫ1

E(r,ω) · n|r=r+,(3.39)

where n represents a unit vector in the direction perpen-

dicular to the interface.

An explicit expression for the normal component of the

electric field at a point of medium 2 that is infinitely close

to the interface (r = r+) can be obtained with the use of

Gauss’ theorem. One finds:

E(r,ω) · n|r=r+ = −n · ∇φ(r,ω) + 2πσs(r,ω),

where φ(r,ω) represents the scalar potential. In the ab-

sence of external sources, this potential is entirely due to

the surface charge density itself:

(3.40)

φ(r,ω) =

?

d2r′σs(r′,ω)

|r − r′|.(3.41)

Combining Eqs. (3.39)-(3.41), one finds the following in-

tegral equation:

2πǫ1+ ǫ2

ǫ1− ǫ2σs(r,ω) −

?

d2r′ r − r′

|r − r′|3· nσs(r′,ω) = 0,

(3.42)

which describes the self-sustained oscillations of the sys-

tem.

The boundary-charge method has been used by sev-

eral authors to determine the normal-mode frequen-

cies of a cube [191, 192] and of bodies of arbitrary

shape [193, 194]. More recent applications of this method

Page 11

11

include investigations of the surface modes of channels

cut on planar surfaces [195], the surface modes of coupled

parallel wires [196], and the electron energy loss near in-

homogeneous dielectrics [197, 198]. A generalization of

this procedure that includes relativistic corrections has

been reported as well [199].

3.Composite systems: effective-medium approach

Composite systems with a large number of inter-

faces can often be replaced by an effective homogeneous

medium that in the long-wavelength limit is charac-

terized by a local effective dielectric function ǫeff(ω).

Bergman [200] and Milton [201] showed that in the

case of a two-component system with local (frequency-

dependent) dielectric functions ǫ1 and ǫ2 and volume

fractions f and 1−f, respectively, the long-wavelengthef-

fective dielectric function of the system can be expressed

as a sum of simple poles that only depend on the mi-

crogeometry of the composite material and not on the

dielectric functions of the components:

ǫeff(ω) = ǫ2

?

1 − f

?

ν

Bν

u − mν

?

,(3.43)

where u is the spectral variable

u = [1 − ǫ1/ǫ2]−1, (3.44)

mν are depolarization factors, and Bν are the strengths

of the corresponding normal modes, which all add up to

unity:

?

ν

Bν= 1.(3.45)

Similarly,

ǫ−1

eff(ω) = ǫ−1

2

?

1 + f

?

ν

Cν

u − nν

?

,(3.46)

with

?

ν

Cν= 1.(3.47)

The optical absorption and the long-wavelength energy

loss of moving charged particles are known to be dic-

tated by the poles of the local effective dielectric function

ǫeff(ω) and inverse dielectric function ǫ−1

tively. If there is one single interface, these poles are

known to coincide.

In particular, in the case of a two-component isotropic

system composed of identical inclusions of dielectric func-

tion ǫ1 in a host medium of dielectric function ǫ2, the

effective dielectric function ǫeff(ω) can be obtained from

the following relation:

eff(ω), respec-

(ǫeff− ǫ2)E = f(ǫ1− ǫ2)Ein, (3.48)

where E is the macroscopic electric field averaged over

the composite:

E = fEin+ (1 − f)Eout,(3.49)

Einand Eoutrepresenting the average electric field inside

and outside the inclusions, respectively [202].

a.Simple geometries.If there is only one mode with

strength different from zero, as occurs (in the long-

wavelength limit) in the case of one single sphere or cylin-

der in a host medium, Eqs. (3.43) and (3.46) yield

ǫeff(ω) = ǫ2

?

1 − f

1

u − m

?

(3.50)

and

ǫ−1

eff(ω) = ǫ−1

2

?

1 + f

1

u − n

?

, (3.51)

normal modes occurring, therefore, at the frequencies

dictated by the following conditions:

mǫ1+ (1 − m)ǫ2= 0 (3.52)

and

nǫ1+ (1 − n)ǫ2= 0.(3.53)

For Drude particles [ǫ1of Eq. (2.15)] in vacuum (ǫ2= 1),

these frequencies are easily found to be ω =√mωpand

ω =√nωp, respectively.

Indeed, for a single 3D spherical or 2D circular [203]

inclusion in a host medium, an elementary analysis shows

that the electric field Einin the interior of the inclusion

is

Ein=

u

u − mE,(3.54)

where m = 1/D, D representing the dimensionality of

the inclusions, i.e., D = 3 for spheres and D = 2 for

cylinders.Introduction of Eq. (3.54) into Eq. (3.48)

leads to an effective dielectric function of the form of

Eq. (3.50) with m = 1/D, which yields [see Eq. (3.52)]

the surface-plasmoncondition dictated by Eq. (3.33) with

l = 1 in the case of spheres (D = 3) and the surface-

plasmon condition of Eq. (3.35) in the case of cylinders

(D = 2). This result indicates that in the nonretarded

long-wavelength limit (which holds for wave vectors q

such that ωsa/c < q a << 1, a being the radius of the in-

clusions) both the absorption of light and the energy-loss

spectrum of a single 3D spherical or 2D circular inclusion

exhibit one single strong maximum at the dipole reso-

nance where ǫ1+ 2ǫ2= 0 and ǫ1+ ǫ2= 0, respectively,

which for a Drude sphere and cylinder [ǫ1of Eq. (2.15)]

in vacuum (ǫ2= 1) yield ω = ωp/√3 and ω = ωp/√2.

In the case of electromagnetic waves polarized along

one single cylinder or array of parallel cylinders (s-

polarization), the effective dielectric function of the com-

posite is simply the average of the dielectric functions of

its constituents, i.e.:

ǫeff= f ǫ1+ (1 − f)ǫ2, (3.55)

Page 12

12

which can also be written in the form of Eqs. (3.50)

and (3.51), but now with m = 0 and n = f, respec-

tively.Hence, for this polarization the absorption of

light exhibits no maxima (in the case of a dielectric host

medium with constant dielectric function ǫ2) and the

long-wavelength energy-loss spectrum exhibits a strong

maximum at frequencies dictated by the plasmon condi-

tion of Eq. (3.37), which in the case of Drude cylinders

[ǫ1of Eq. (2.15)] in vacuum (ǫ2= 1) yields the reduced

plasmon frequency ω =√f ωp.

b.Maxwell-Garnett approximation.

among spherical (or circular) inclusions in a host medium

can be introduced approximately in the framework of the

well-known Maxwell-Garnett (MG) approximation [133].

The basic assumption of this approach is that the av-

erage electric field Einwithin a particle located in a sys-

tem of identical particles is related to the average field

Eout in the medium outside as in the case of a single

isolated (noninteracting) particle, thereby only dipole in-

teractions being taken into account. Hence, in this ap-

proach the electric field Einis taken to be of the form of

Eq. (3.54) but with the macroscopic electric field E re-

placed by the electric field Eoutoutside, which together

with Eqs. (3.48) and (3.49) yields the effective dielec-

tric function and effective inverse dielectric function of

Eqs. (3.50) and (3.51) with the depolarization factors

m = n = 1/D (corresponding to the dilute limit, where

f → 0) replaced by

1

D(1 − f)

The interaction

m =

(3.56)

and

n =1

D[1 + (D − 1)f].(3.57)

4.Periodic structures

Over the years, theoretical studies of the normal modes

of complex composite systems had been generally re-

stricted to mean-field theories of the Maxwell-Garnett

type, which approximately account for the behavior of

localized dipole plasmons [133].

ber of methods have been developed recently for a

full solution of Maxwell’s equations in periodic struc-

tures [204, 205, 206, 207, 208, 209]. The transfer matrix

method has been used to determine the normal-mode

frequencies of a lattice of metallic cylinders [210] and

rods [211], a so-called on-shell method has been employed

by Yannopapas et al. to investigate the plasmon modes

of a lattice of metallic spheres in the low filling fraction

regime [212], and a finite difference time domain (FDTD)

scheme has been adapted to extract the effective response

of metallic structures [209].

Most recently, an embedding method [207] has been

employed to solve Maxwell’s equations, which has al-

lowed to calculate the photonic band structure of three-

and two-dimensional lattices of nanoscale metal spheres

Nevertheless, a num-

FIG. 6:

plasma and vacuum are interchanged. The top panel rep-

resents the general situation. The bottom panel represents a

half-space filled with metal and interfaced with vacuum. The

surface-mode frequencies ωs1and ωs2of these systems fulfill

the sum rule of Eq. (3.58).

Complementary systems in which the regions of

and cylinders in the frequency range of the Mie plas-

mons [147]. For small filling fractions, there is a surface-

plasmon polariton which in the non-retarded region

yields the non-dispersive Mie plasmon with frequency

ωp/√D. As the filling fraction increases, a continuum of

plasmon modes is found to exist between zero frequency

and the bulk metal plasmon frequency [147], which yield

strong absorption of incident light and whose energies

can be tuned according to the particle-particle separa-

tion [213].

5.Sum rules

Sum rules have played a key role in providing insight

in the investigation of a variety of physical situations. A

useful sum rule for the surface modes in complementary

media with arbitrary geometry was introduced by Apell

et al. [214], which in the special case of a metal/vacuum

interface implies that [215]

ω2

s1+ ω2

s2= ω2

p, (3.58)

where ωs1is the surface-mode frequency of a given sys-

tem, and ωs2represents the surface mode of a second

complementary system in which the regions of plasma

and vacuum are interchanged (see Fig. 6).

For example, a half-space filled with a metal of bulk

plasma frequency ωp and interfaced with vacuum maps

into itself (see bottom panel of Fig. 6), and therefore

Eq. (3.58) yields

ωs1= ωs2= ωp/√2, (3.59)

which is Ritchie’s frequency of plasma oscillations at a

metal/vacuum planar interface.

Other examples are a Drude metal sphere in vac-

uum, which sustains localized Mie plasmons at frequen-

cies given by Eq. (3.34), and a spherical void in a Drude

Page 13

13

metal, which shows Mie plasmons at frequencies

ωl= ωp

?

l + 1

2l + 1.(3.60)

The squared surface-mode frequencies of the sphere

[Eq. (3.34)] and the void [(Eq. (3.60)] add up to ω2

all l, as required by Eq. (3.58).

The splitting of surface modes that occurs in thin films

due to the coupling of the electromagnetic fields in the

two surfaces [see Eq. (2.26)] also occurs in the case of

localized modes. Apell et al. [214] proved a second sum

rule, which relates the surface modes corresponding to

the in-phase and out-of-phase linear combinations of the

screening charge densities at the interfaces. In the case

of metal/vacuum interfaces this sum rule takes the form

of Eq. (3.58), but now ωs1and ωs2being in-phase and

out-of-phase modes of the same system.

For a Drude metal film with equal and abrupt planar

surfaces, the actual values of the nonretarded ωs1and ωs2

are those given by Eq. (2.27), which fulfill the sum rule

dictated by Eq. (3.58). For a spherical fullerene molecule

described by assigning a Drude dielectric function to ev-

ery point between the inner and outer surfaces of radii

r1and r2, one finds the following frequencies for the in-

phase and out-of-phase surface modes [216]:

pfor

ω2

s=ω2

p

2

?

1 ±

1

2l + 1

?

1 + 4l(l + 1)(r1/r2)2l+1

?

(3.61)

,

also fulfilling the sum rule of Eq. (3.58).

Another sum rule has been reported recently [210, 211],

which relates the frequencies of the modes that can be

excited by light [as dictated by the poles of the effec-

tive dielectric function of Eq. (3.43)] and those modes

that can be excited by moving charged particles [as dic-

tated by the poles of the effective inverse dielectric func-

tion of Eq. (3.46)]. Numerical calculations for various

geometries have shown that the depolarization factors

mν and nν entering Eqs. (3.43) and (3.46) satisfy the

relation [210, 211]

nν= 1 − (D − 1)mν, (3.62)

where D represents the dimensionality of the inclusions.

Furthermore, combining Eqs. (3.50) and (3.51) (and

assuming, therefore, that only dipole interactions are

present) with the sum rule of Eq. (3.62) yields Eqs. (3.56)

and (3.57), i.e., the MG approximation. Conversely, as

long as multipolar modes contribute to the spectral rep-

resentation of the effective response [see Eqs. (3.43) and

(3.46)], the strength of the dipolar modes decreases [see

Eqs. (3.45) and (3.47)] and a combination of Eqs. (3.43)

and (3.46) with Eq. (3.62) leads to the conclusion that the

dipolar resonances must necessarily deviate from their

MG counterparts dictated by Eqs. (3.56) and (3.57).

That a nonvanishing contribution from multipolar modes

appears together with a deviation of the frequencies of

the dipolar modes with respect to their MG counterparts

was shown explicitly in Ref. [210].

IV.DYNAMICAL STRUCTURE FACTOR

The dynamical structure factor S(r,r′;ω) represents a

key quantity in the description of both single-particle and

collective electronic excitations in a many-electron sys-

tem [217]. The rate for the generation of electronic exci-

tations by an external potential, the inelastic differential

cross section for external particles to scatter in a given di-

rection, the inelastic lifetime of excited hot electrons, the

so-called stopping power of a many-electron system for

moving charged particles, and the ground-state energy

of an arbitrary many-electron system (which is involved

in, e.g., the surface energy and the understanding of Van

der Waals interactions) are all related to the dynamical

structure factor of the system.

The dynamical structure factor, which accounts for the

particle-density fluctuations of the system, is defined as

follows

S(r,r′;ω) =

?

n

δˆ ρ0n(r1)δˆ ρn0(r2)δ(ω−En+E0). (4.1)

Here, δˆ ρn0(r) represent matrix elements, taken between

the many-particle ground state |Ψ0? of energy E0 and

the many-particle excited state |Ψn? of energy En, of the

operator ˆ ρ(r) − n0(r), where ˆ ρ(r) is the electron-density

operator [218]

ˆ ρ(r) =

N

?

i=1

ˆδ(r − ri),(4.2)

withˆδ and ri describing the Dirac-delta operator and

electron coordinates, respectively, and n0(r) represents

the ground-state electron density, i.e.:

n0(r) =< Ψ0|ˆ ρ(r)|Ψ0> . (4.3)

The many-body ground and excited states of a many-

electron system are unknown and the dynamical struc-

ture factor is, therefore, difficult to calculate.

theless, one can use the zero-temperature limit of the

fluctuation-dissipation theorem [219], which relates the

dynamical structure factor S(r,r′;ω) to the dynami-

cal density-response function χ(r,r′;ω) of linear-response

theory. One writes,

Never-

S(r,r′;ω) = −Ω

πImχ(r,r′;ω)θ(ω),(4.4)

where Ω represents the normalization volume and θ(x) is

the Heaviside step function.

V.DENSITY-RESPONSE FUNCTION

Take a system of N interacting electrons exposed to a

frequency-dependent external potential φext(r,ω). Keep-

ing terms of first order in the external perturbation and

Page 14

14

neglecting retardation effects, time-dependent perturba-

tion theory yields the following expression for the induced

electron density [218]:

δn(r,ω) =

?

dr′χ(r,r′;ω)φext(r′,ω),(5.1)

where χ(r,r′;ω) represents the so-called density-response

function of the many-electron system:

χ(r,r′;ω) =

?

−

n

ρ∗

n0(r)ρn0(r′)

?

1

E0− En+ ¯ h(ω + iη)

?

1

E0+ En+ ¯ h(ω + iη)

,(5.2)

η being a positive infinitesimal.

The imaginary part of the true density-response func-

tion of Eq. (5.2), which accounts for the creation of both

collective and single-particle excitations in the many-

electron system, is known to satisfy the so-called f-sum

rule:

?∞

−∞

dωω Imχ(r,r′;ω) = −π∇·∇′[n0(r)δ(r,r′)], (5.3)

with n0(r) being the unperturbed ground-state electron

density of Eq. (4.3).

A. Random-phase approximation (RPA)

In the so-called random-phase or, equivalently, time-

dependent Hartree approximation, the electron density

δn(r,ω) induced in an interacting electron system by a

small external potential φext(r,ω) is obtained as the elec-

tron density induced in a noninteracting Hartree system

(of electrons moving in a self-consistent Hartree poten-

tial) by both the external potential φext(r,ω) and the

induced potential

δφH(r,ω) =

?

dr′v(r,r′) δn(r′,ω),(5.4)

with v(r,r′) representing the bare Coulomb interaction.

Hence, in this approximation one writes

δn(r,ω) =

?

dr′χ0(r,r′;ω)

×

?

φext(r′,ω) +

?

dr′′v(r′,r′′)δn(r′′,ω)

?

, (5.5)

which together with Eq. (5.1) yields the following Dyson-

type equation for the interacting density-response func-

tion:

χ(r,r′;ω) = χ0(r,r′;ω) +

?

dr1

?

dr2χ0(r,r1;ω)

× v(r1,r2)χ(r2,r′;ω),(5.6)

where χ0(r,r′;ω) denotes the density-response function

of noninteracting Hartree electrons:

χ0(r,r′;ω) =

2

Ω

?

i,j

(fi− fj)

×

ψi(r)ψ∗

ω − εj+ εi+ iη

j(r)ψj(r′)ψ∗

i(r′)

.(5.7)

Here, fi are Fermi-Dirac occupation factors, which at

zero temperature take the form fi= θ(εF−εi), εF being

the Fermi energy, and the single-particle states and ener-

gies ψi(r) and εiare the eigenfunctions and eigenvalues

of a Hartree Hamiltonian, i.e.:

?

−1

2∇2+ vH[n0](r)

?

ψi(r) = εiψi(r),(5.8)

where

vH[n0](r) = v0(r) +

?

dr′v(r,r′)n0(r′),(5.9)

with v0(r) denoting a static external potential and n0(r)

being the unperturbed Hartree electron density:

n0(r) =

N

?

i=1

|ψi(r)|2.(5.10)

B.Time-dependent density-functional theory

In the framework of time-dependent density-functional

theory (TDDFT) [220], the exact density-response func-

tion of an interacting many-electron system is found to

obey the following Dyson-type equation:

χ(r,r′;ω) = χ0(r,r′;ω) +

?

dr1

?

dr2χ0(r,r1;ω)

×{v(r1,r2) + fxc[n0](r1,r2;ω)}χ(r2,r′;ω).

Here,the noninteracting

χ0(r,r′;ω) is of the form of Eq. (5.7), but with the single-

particle states and energies ψi(r) and εi being now the

eigenfunctions and eigenvalues of the Kohn-Sham Hamil-

tonian of DFT, i.e:

(5.11)

density-responsefunction

?

−1

2∇2+ vKS[n0](r)

?

ψi(r) = εiψi(r), (5.12)

where

vKS[n0](r) = vH[n0](r) + vxc[n0](r),(5.13)

with

vxc[n0](r) =δExc[n]

δn(r)

????

n=n0

.(5.14)

Exc[n] represents the unknown XC energy functional and

n0(r) denotes the exact unperturbed electron density of

Page 15

15

Eq. (4.3), which DFT shows to coincide with that of

Eq. (5.10) but with the Hartree eigenfunctions ψi(r) of

Eq. (5.8) being replaced by their Kohn-Sham counter-

parts of Eq. (5.12). The xc kernel fxc[n0](r,r′;ω) denotes

the Fourier transform of

fxc[n0](r,t;r′;t′) =δvxc[n](r,t)

δn(r′,t′)

????

n=n0

,(5.15)

with vxc[n](r,t) being the exact time-dependent xc po-

tential of TDDFT.

If short-range XC effects are ignored altogether by set-

ting the unknown XC potential vxc[n0](r) and XC ker-

nel fxc[n0](r,r′;ω) equal to zero, the TDDFT density-

response function of Eq. (5.11) reduces to the RPA

Eq. (5.6).

1.The XC kernel

Along the years, several approximations have been

used to evaluate the unknown XC kernel of Eq. (5.15).

a.Random-phase approximation

days, one usually refers to the RPA as the result of

simply setting the XC kernel fxc[n0](r,r′;ω) equal to

zero:

(RPA).Nowa-

fRPA

xc

[n0](r,r′;ω) = 0, (5.16)

but still using in Eqs. (5.7) and (5.10) the full single-

particle states and energies ψi(r) and εiof DFT [i.e., the

solutions of Eq. (5.12)] with vxc[n0](r) set different from

zero. This is sometimes called DFT-based RPA.

b. Adiabatic local-density approximation (ALDA).

In this approximation, also called time-dependent local-

density approximation (TDLDA) [221], one assumes that

both the unperturbed n0(r) and the induced δn(r,ω)

electron densities vary slowly in space and time and,

therefore, one replaces the dynamical XC kernel by the

long-wavelength (Q → 0) limit of the static XC kernel of

a homogeneous electron gas at the local density:

fALDA

xc

[n0](r,r′;ω) =d2[nεxc(n)]

dn2

????

n=n0(r)

δ(r − r′),

(5.17)

where εxc(n) is the XC energy per particle of a homoge-

neous electron gas of density n.

c. PGG and BPG. In the spirit of the optimized

effective-potential method [222], Petersilka, Gossmann,

and Gross (PGG) [223] derived the following frequency-

independent exchange-only approximation for inhomoge-

neous systems:

fPGG

x

[n0](r,r′;ω) = −

2

|r − r′|

|?

ifiψi(r)ψ∗

n0(r)n0(r′)

i(r′)|2

,

(5.18)

where ψi(r) denote the solutions of the Kohn-Sham

Eq. (5.12).

More

(BPG) [224] devised a hybrid formula for the XC

kernel, which combines expressions for symmetric and

antisymmetric spin orientations from the exchange-only

PGG scheme and the ALDA. For an unpolarized

many-electron system, one writes [224]:

recently, Burke, Petersilka,andGross

fBPG

xc

[n0](r,r′;ω) =1

2

?f↑↑,PGG

xc

+ f↑↓,LDA

xc

?, (5.19)

where f↑↑

with parallel and antiparallel spin, respectively.

d.Averageapproximation

short-range XC effects in solids has been focused in a

great extent onto the simplest possible many-electron

system, which is the homogeneous electron gas. Hence,

recent attempts to account for XC effects in inhomo-

geneous systems have adopted the following approxima-

tion [225, 226]:

xcand f↑↓

xcrepresent the XC kernel for electrons

Theinvestigationof

fav

xc[n0](r,r′;ω) = fhom

xc

(˜ n;|r − r′|;ω), (5.20)

where ˜ n represents a function of the electron densities at

points r and r′, typically the arithmetical average

˜ n =1

2[n0(r) + n0(r′)],(5.21)

and fhom

mogeneous electron gas of density ˜ n, whose 3D Fourier

transform fhom

xc

(˜ n;Q,ω) is directly connected to the so-

called local-field factor G(˜ n;Q,ω):

xc

(˜ n;|r − r′|;ω) denotes the XC kernel of a ho-

fhom

xc

(˜ n;Q,ω) = −4π

Q2G(˜ n;Q,ω).(5.22)

In the ALDA, one writes

GALDA(˜ n;Q,ω) = G(˜ n;Q → 0,ω = 0)

= −Q2

4π

d2[nεxc(n)]

dn2

????

n=˜ n

,(5.23)

which in combination with Eqs. (5.20)-(5.22) yields the

ALDA XC kernel of Eq. (5.17). However, more accu-

rate nonlocal dynamical expressions for the local-field

factor G(˜ n;Q,ω) are available nowdays, which together

with Eqs. (5.20)-(5.22) should yield an accurate (beyond

the ALDA) representation of the XC kernel of inhomo-

geneous systems.

During the last decades,

into the determination of the static local-field factor

Gstatic(˜ n;Q) = G(˜ n;Q,ω = 0) [227, 228, 229, 230, 231,

232, 233, 234], the most recent works including diffu-

sion Monte Carlo (DMC) calculations [235, 236] and the

parametrization of the DMC data of Ref. [236] given by

Corradini et al. [237]:

much effort has gone

Gstatic(˜ n;Q) = CˆQ2+ BˆQ2/(g +ˆQ2) + αˆQ4e−βˆ Q2,

(5.24)

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