INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICSRep. Prog. Phys. 70 (2007) 187 doi:10.1088/0034-4885/70/1/R01Theory of surface plasmons and surface-plasmonpolaritonsJ M Pitarke1,2, V M Silkin2, E V Chulkov2,3and P M Echenique2,31Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea,644 Posta kutxatila, E-48080 Bilbo, Basque Country, Spain2Donostia International Physics Center (DIPC) and Unidad de Fsica de MaterialesCSIC-UPV/EHU, Manuel de Lardizabal Pasealekua, E-20018 Donostia, Basque Country,Spain3Materialen Fisika Saila, Kimika Fakultatea, Euskal Herriko Unibertsitatea, 1072 Postakutxatila, E-20080 Donostia, Basque Country, SpainReceived 2 August 2006, in nal form 9 October 2006Published 7 December 2006Online at stacks.iop.org/RoPP/70/1AbstractCollective electronic excitations at metal surfaces are well known to play a key role in a widespectrum of science, ranging from physics and materials science to biology. Here we focuson a theoretical description of the many-body dynamical electronic response of solids, whichunderlines the existence of various collective electronic excitations at metal surfaces, suchas theconventional surface plasmon, multipole plasmons and the recently predicted acoustic surfaceplasmon. We also review existing calculations, experimental measurements and applications.This article was invited by Professor M Finnis.0034-4885/07/010001+87$90.00 2007 IOP Publishing Ltd Printed in the UK 12 J M Pitarke et alContentsPage1. Introduction 42. Surface-plasmon polariton: classical approach 82.1. Semi-innite system 82.1.1. The surface-plasmon condition. 82.1.2. Energy dispersion. 102.1.3. Skin depth. 112.2. Thin lms 123. Nonretarded surface plasmon: simplied models 133.1. Planar surface plasmon 143.1.1. Classical model. 143.1.2. Nonlocal corrections. 143.1.3. Hydrodynamic approximation. 153.2. Localized surface plasmons: classical approach 173.2.1. Simple geometries. 173.2.2. Boundary-charge method. 183.2.3. Composite systems: effective-medium approach. 193.2.4. Periodic structures. 213.2.5. Sum rules. 214. Dynamical structure factor 235. Density-response function 235.1. Random-phase approximation (RPA) 245.2. Time-dependent density-functional theory 255.2.1. The XC kernel. 256. Inverse dielectric function 277. Screened interaction 287.1. Classical model 287.1.1. Planar surface. 297.1.2. Spheres. 297.1.3. Cylinders. 317.2. Nonlocal models: planar surface 337.2.1. Hydrodynamic model. 337.2.2. Specular-reection model (SRM). 357.2.3. Self-consistent scheme. 388. Surface-response function 428.1. Generation-rate of electronic excitations 428.2. Inelastic electron scattering 438.3. Surface plasmons: jellium surface 448.3.1. Simple models. 448.3.2. Self-consistent calculations: long wavelengths. 458.3.3. Self-consistent calculations: arbitrary wavelengths. 47Theory of surface plasmons and surface-plasmon polaritons 38.3.4. Surface-plasmon linewidth. 498.3.5. Multipole surface plasmons. 518.4. Surface plasmons: real surfaces 548.4.1. Stabilized jellium model. 548.4.2. Occupied d-bands: simple models. 558.4.3. First-principles calculations. 608.5. Acoustic surface plasmons 638.5.1. A simple model. 638.5.2. 1D model calculations. 658.5.3. First-principles calculations. 698.5.4. Excitation of acoustic surface plasmons. 699. Applications 709.1. Particlesurface interactions: energy loss 709.1.1. Planar surface. 729.2. STEM: valence EELS 759.2.1. Planar surface. 769.2.2. Spheres. 779.2.3. Cylinders. 789.3. Plasmonics 78Acknowledgments 80References 804 J M Pitarke et al1. IntroductionIn his pioneering treatment of characteristic energy losses of fast electrons passing throughthin metal lms, Ritchie predicted the existence of self-sustained collective excitations at metalsurfaces [1]. It had already been pointed out by Pines and Bohm [2, 3] that the long-rangenature of the Coulomb interaction between valence electrons in metals yields collective plasmaoscillations similar to the electron-density oscillations observed by Tonks and Langmuir inelectrical discharges in gases [4], thereby explaining early experiments by Ruthemann [5] andLang [6] on the bombardment of thin metallic lms by fast electrons. Ritchie investigatedthe impact of the lm boundaries on the production of collective excitations and found thatthe boundary effect is to cause the appearance of a newlowered loss due to the excitationofsurface collective oscillations [1]. Two years later, in a series of electron energy-lossexperiments 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 signicant advance in both theoretical and experimentalinvestigations of surface plasmons, which for researches in the eld of condensed matter andsurface physics have played a key role in the interpretation of a great variety of experimentsand the understanding of various fundamental properties of solids. These include the nature ofVan der Waals forces [911], the classical image potential acting between a point classicalchargeandametal surface[1215], theenergytransferingassurfaceinteractions[16],surface energies [1719], the damping of surface vibrational modes [20, 21], the energy lossof charged particles moving outside a metal surface [22, 23] and the de-excitation of adsorbedmolecules [24]. Surface plasmons have also been employed in a wide spectrum of studiesranging fromelectrochemistry [25], wetting [26] and biosensing [2729] to scanning tunnellingmicroscopy [30], the ejection of ions fromsurfaces [31], nanoparticle growth [32, 33], surface-plasmon microscopy [34, 35] and surface-plasmon resonance technology [3642]. Renewedinterest insurfaceplasmonshascomefromrecent advancesintheinvestigationof theelectromagnetic properties of nanostructured materials [43, 44], one of the most attractiveaspects of these collective excitations nowbeing their use to concentrate light in subwavelengthstructures and to enhance transmission through periodic arrays of subwavelength holes inoptically thick metallic lms [45, 46].The so-called eld of plasmonics represents an exciting new area for the application ofsurface and interface plasmons, an area in which surface-plasmon based circuits merge theelds of photonics and electronics at the nanoscale [47]. Indeed, surface-plasmon polaritonscan serve as a basis for constructing nanoscale photonic circuits that will be able to carryoptical signals and electric currents [48, 49]. Surface plasmons can also serve as a basis for thedesign, fabrication and characterization of subwavelength waveguide components [5064]. Inthe framework of plasmonics, modulators and switches have also been investigated [65, 66],as well as the use of surface plasmons as mediators in the transfer of energy from donor toacceptors molecules on opposite sides of metal lms [67].Accordingtotheworkof Pines andBohm, thequantumenergycollectiveplasmaoscillationsinafreeelectrongaswithequilibriumdensitynis hp= h(4ne2/me)1/2,p being the so-called plasmon frequency4. In the presence of a planar boundary, there isa new mode (the surface plasmon), the frequency of which equals in the nonretarded region(where the speed of light can be taken to be innitely large) Ritchies frequency s = p/2at wave vectors q in the ranges/c q qF (qF being the magnitude of the Fermi wave4The electron densityn is usually characterized by the density parameterrs =(3/4n)1/3/a0,a0 being the Bohrradius, a0 = 0.529 . In metals the electron-density parameter of valence electrons is typically in the range 2< rs< 6,which corresponds to plasmon energies on the order of 10 eV and frequencies that lie in the optical regime.Theory of surface plasmons and surface-plasmon polaritons 5vector) and exhibits some dispersion as the wave vector is increased. In the retarded region,where the phase velocitys/q of the surface plasmon is comparable to the velocity of light,surface plasmons couple with the free electromagnetic eld. These surface-plasmon polaritonspropagate along the metal surface with frequencies ranging from zero (at q = 0) towards theasymptotic value s = p/2, the dispersion relation (q) lying to the right of the light lineand the propagating vector being, therefore, larger than that of bare light waves of the sameenergy. Hence, surface-plasmon polaritons in an ideal semi-innite medium are nonradiativein nature, i.e. cannot decay by emitting a photon and, conversely, light incident on an idealsurface cannot excite surface plasmons.In the case of thin lms, the electric elds of both surfaces interact. As a result, thereare (i) tangential oscillations characterized by a symmetric disposition of charge deciency orexcess at opposing points on the two surfaces and (ii) normal oscillations in which an excessof charge density at a point on one surface is accompanied by a deciency at the point directlyacross the thin lm. The phase velocity of the tangential surface plasmon is always less thanthe speed of light, as it occurs in the case of a semi-innite electron system. However, thephase velocity of normal oscillations may surpass that of light, thereby becoming a radiativesurface plasmon that should be responsible for the emission of light [68]. This radiation wasdetected using electron beam bombardment of thin lms of Ag, Mg and Al with thicknessesranging between 500 and 1000 [69, 70]. More recently, light emission was observed in theultraviolet from a metaloxidemetal tunnel diode and was attributed to the excitation of theradiative surface plasmon [71].Nonradiative surface plasmons in both thin and thick lms can couple to electromagneticradiation in the presence of surface roughness or a grating, as suggested by Teng and Stern [72].Alternatively, prism coupling can be used to enhance the momentum of incident light, asdemonstrated by Otto [73] and by Kretchmann and Raether [74]. Since then, this so-calledattenuated reection (ATR) method and variations upon it have been used by several workersin a large variety of applications [7581].During the last decades, there has also been a signicant advance in our understanding ofsurface plasmons in the nonretarded regime. Ritchie [82] and Kanazawa [83] were the rstto attack the problem of determining the dispersion (q) of the nonretarded surface plasmon.Bennett [84] used a hydrodynamical model with a continuous decrease of the electron densityat the metal surface and found that a continuous electron-density variation yields two collectiveelectronic excitations: Ritchies surface plasmon at s, with a negative energy dispersionat low wave vectors, and an upper surface plasmon at higher energies. In the direction normalto the surface, the distribution of Ritchies surface plasmon consists of a single peak, i.e. it hasa monopole character; however, the charge distribution of the upper mode has a node, i.e. ithas a dipole character and is usually called multipole surface plasmon.Bennetts qualitative conclusions were generally conrmed by microscopic descriptionsof theelectrongas. Ontheonehand, Feibelmanshowedthat inthelong-wavelengthlimit the classical results =p/2 is correct for a semi-innite plane-bounded electrongas, irrespective of the exact variation of the electron density in the neighbourhood of thesurface [85]. On the other hand, explicit expressions for the linear momentum dispersion ofthe conventional monopole surface plasmon that are sensitive to the actual formof the electron-density uctuation at the surface were derived by Harris and Grifn [86] using the equationof motion for the Wigner distribution function in the random-phase approximation (RPA)and by Flores and Garca-Moliner [87] solving Maxwells equations in combination with anintegration of the eld components over the surface region. Quantitative RPA calculationsof the linear dispersion of the monopole surface plasmon were carried out by several authorsusing the innite-barrier model (IBM) of the surface [88], a step potential [89, 90], and the6 J M Pitarke et almore realistic LangKohn [91] self-consistent surface potential [92]. Feibelmans calculationsshowed that for the typical electron densities in metals (2 0.Solutionsofequations(2.1)(2.4)cangenerallybeclassiedintos-polarizedandp-polarized electromagnetic modes, the electric eld Eand the magnetic eld Hbeing parallel tothe interface, respectively. For an ideal surface, if waves are to be formed that propagate alongthe interface there must necessarily be a component of the electric eld normal to the surface.Hence, s-polarized surface oscillations (whose electric eld E is parallel to the interface) donot exist; instead, we seek conditions under which a travelling wave with the magnetic eld Hparallel to the interface (p-polarized wave) may propagate along the surface (z = 0), with theelds tailing off into the positive (z> 0) and negative (z< 0) directions. Choosing the x-axisalong the propagating direction, we writeEi = (Eix, 0, Eiz) ei|z| ei(qixt )(2.5)andHi = (0, Eiy, 0) ei|z| ei(qixt ), (2.6)where qi represents the magnitude of a wave vector that is parallel to the surface. Introducingequations (2.5) and (2.6) into equations (2.1)(2.4), one ndsi 1H1y = +c

1E1x, (2.7)i 2 H2y = c

2 E2x(2.8)andi =_q2i i2c2 . (2.9)The boundary conditions imply that the component of the electric and magnetic eldsparallel to the surface must be continuous. Using equations (2.7) and (2.8), one writes thefollowing system of equations:1

1H1y +2

2H2y = 0 (2.10)andH1y H2y = 0, (2.11)which has a solution only if the determinant is zero, i.e.

11+

22= 0. (2.12)This is the surface-plasmon condition.Fromthe boundary conditions also follows the continuity of the 2Dwave vector q enteringequation (2.9), i.e. q1 =q2 =q. Hence, the surface-plasmon condition (equation (2.12)) canalso be expressed as follows [170]:q() =c_

1

2

1 + 2, (2.13)where /c represents the magnitude of the light wave vector. For a metaldielectric interfacewith the dielectric characterized by 2, the solution (q) of equation (2.13) has slope equal toc/

2 at the point q = 0 and is a monotonic increasing function of q, which is always smallerthan c q/

2 and for large q is asymptotic to the value given by the solution of

1 + 2 = 0. (2.14)This is the nonretarded surface-plasmon condition (equation (2.12) with 1 = 2 = q), whichis valid as long as the phase velocity /q is much smaller than the speed of light.10 J M Pitarke et al0 0.01 0.02q ( -1)051015202530(eV)Figure 2. The solid lines represent the solutions of equation (2.16) with p = 15 eV: the dispersionof light in the solid (upper line) and the surface-plasmon polariton (lower line). In the retardedregion(q 0.IntroducingFourier transforms, equations (3.22)(3.25) yieldthe basic differentialequation for the plasma normal modes at z0:2(22p + 22)1(r, ) = 0 (z0) (3.26)and Laplaces equation at z> 0:21(r, ) = 0 (z> 0), (3.27)where bothn1(r, ) and1(r, ) vanish. Furthermore, translational invariance in the planeof the surface allows to introduce the 2D Fourier transform1(z; q, ), which according toequation (3.26) must satisfy the following equation at z0:(q2+ d2/dz2)[22p (q2+ d2/dz2)]1(z; q, ) = 0, (3.28)where q represents a 2D wave vector in the plane of the surface.Now we need to specify the boundary conditions. Ruling out exponential increase atz andnotingthat thenormal component of thehydrodynamical velocityshould9While the use of the ThomasFermi functional G[n] of equation (3.19), which assumes static screening, predicts = 1/3 (32n0)1/3, the value = 3/5(32n0)1/3should be more appropriate when high-frequencies of theorder of the plasma frequency are involved; see, e.g. [176].Theory of surface plasmons and surface-plasmon polaritons 17vanishat theinterface, for eachvalueof qonendssolutionstoequation(3.28) withfrequencies [177, 178]2 2p + 2q2(3.29)and2=12_2p + 2q2+ q_22p + 2q2_. (3.30)Equations (3.29) and (3.30) represent a continuum of bulk normal modes and a surface normalmode, respectively. At long wavelengths, where q/p 1 (but still in the nonretardedregime where s/c< q), equation (3.30) yields the surface-plasmon dispersion relation = p/2 + q/2, (3.31)which was rst derived by Ritchie [82] using Blochs equations, and later by Wagner [179] andby Ritchie and Marusak [180] by assuming, within a Boltzmann transport-equation approach,specular reection at the surface.3.2.Localized surface plasmons: classical approachMetaldielectric interfaces of arbitrary geometries also support charge density oscillationssimilar to the surface plasmons characteristic of planar interfaces. In the long-wavelength (orclassical) limit, in which the interface separates two media with local (frequency-dependent)dielectric functions 1 and 2, one writesDi(r, ) = i Ei(r, ), (3.32)where the indexi refers to the media 1 and 2 separated by the interface. In the case ofsimple geometries, such as spherical and cylindrical interfaces, equations (3.1)(3.3) can besolved explicitly with the aid of equation (3.32) to nd explicit expressions for the nonretardedsurface-plasmon condition.3.2.1.Simple geometries.Sphericalinterface. In the case of a sphere of dielectric function1 in a host medium ofdielectric function2, the classical (long-wavelength) planar surface-plasmon condition ofequation (3.7) is easily found to be replaced by [129]l 1 + (l + 1) 2 = 0, l = 1, 2, . . . , (3.33)which in the case of a Drude metal sphere (1 of equation (2.15)) in vacuum (2 = 1) yieldsthe Mie plasmons at frequenciesl = p_l2l + 1. (3.34)Cylindrical interface. Inthe case of aninnitelylongcylinder of dielectric function

1inahost mediumof dielectricfunction 2, theclassical (long-wavelength) surface-plasmon condition depends on the direction of the electric eld. For electromagnetic waveswiththeelectriceldnormal totheinterface(p-polarization), thecorrespondinglong-wavelength(andnonretarded) surface-plasmonconditioncoincideswiththat of aplanarsurface, i.e. [181, 182, 183]

1 + 2 = 0, (3.35)18 J M Pitarke et alwhich for Drude cylinders (1 of equation (2.15)) in vacuum(2 = 1) yields the planar surface-plasmon frequency s = p/2.For electromagnetic waves with the electric eld parallel to the axis of the cylinder (s-polarization), the presence of the interface does not modify the electric eld and one easilynds that only the bulk mode of the host mediumis present, i.e. one nds the plasmon condition

2 = 0. (3.36)In some situations, instead of having one single cylinder in a host medium, an array of parallelcylinders may be present with a lling fraction f . In this case and for electromagnetic wavespolarized along the cylinders (s-polarization), the plasmon condition of equation (3.36) mustbe replaced by [184]10f1 + (1 f ) 2 = 0, (3.37)which for Drude cylinders (1of equation (2.15)) in vacuum (2 =1) yields the reducedplasmon frequency = fp.3.2.2. Boundary-chargemethod. Inthecaseof morecomplexinterfaces, aso-calledboundary-charge method (BCM) has been used by several authors to determine numericallythe classical (long-wavelength) frequencies of localized surface plasmons. In this approach,one rst considers the -component of the time-dependent surface charge density arising fromthe difference between the normal components of the electric elds inside and outside thesurface:s(r, ) =14_E(r, ) n|r=r + E(r, ) n|r=r+_, (3.38)which noting that the normal component of the displacement vector (see equation (3.32)) mustbe continuous yields the following expression:s(r, ) =14

1 2

1E(r, ) n|r=r+ , (3.39)where n represents a unit vector in the direction perpendicular to the interface.An explicit expression for the normal component of the electric eld at a point of medium2 that is innitely close to the interface (r = r+) can be obtained with the use of Gauss theorem.One ndsE(r, ) n|r=r+ = n (r, ) + 2s(r, ), (3.40)where (r, ) represents the scalar potential. In the absence of external sources, this potentialis entirely due to the surface charge density itself:(r, ) =_d2r

s(r

, )|r r

|. (3.41)Combining equations (3.39)(3.41), one nds the following integral equation:2

1 + 2

1 2s(r, ) _d2r

r r

|r r

|3 ns(r

, ) = 0, (3.42)which describes the self-sustained oscillations of the system.The boundary-charge method has been used by several authors to determine the normal-mode frequencies of a cube [185, 186] and of bodies of arbitrary shape [187, 188]. Morerecent applications of this method include investigations of the surface modes of channels cuton planar surfaces [189], the surface modes of coupled parallel wires [190] and the electronenergy loss near inhomogeneous dielectrics [191, 192]. A generalization of this procedure thatincludes relativistic corrections has been reported as well [193].10It is not necessary that the cylinders are circular and the same result is found in the case of plane parallel layersaligned along the electric eld.Theory of surface plasmons and surface-plasmon polaritons 193.2.3. Compositesystems:effective-mediumapproach. Composite systems with a largenumber of interfaces can often be replaced by an effective homogeneous medium that inthelong-wavelengthlimitischaracterizedbyalocaleffectivedielectricfunctioneff().Bergman [194] and Milton [195] showed that in the case of a two-component system withlocal (frequency-dependent) dielectric functions 1 and 2 and volume fractions fand 1 f ,respectively, the long-wavelength effective dielectric function of the system can be expressedas a sum of simple poles that only depend on the microgeometry of the composite materialand not on the dielectric functions of the components:

eff() = 2_1 f

Bu m_, (3.43)where u is the spectral variableu = [1 1/2]1, (3.44)mare depolarization factors andBare the strengths of the corresponding normal modes,which all add up to unity:

B = 1. (3.45)Similarly,

1eff () = 12_1 + f

Cu n_, (3.46)with

C = 1. (3.47)The optical absorption and the long-wavelength energy loss of moving charged particles areknown to be dictated by the poles of the local effective dielectric function eff() and inversedielectric function 1eff (), respectively. If there is one single interface, these poles are knownto coincide.In particular, in the case of a two-component isotropic system composed ofidenticalinclusions of dielectric function1 in a host medium of dielectric function2, the effectivedielectric function eff() can be obtained from the following relation:(eff 2) E = f (1 2) Ein, (3.48)where E is the macroscopic electric eld averaged over the composite:E = f Ein + (1 f )Eout, (3.49)Einand Eoutrepresenting the average electric eld inside and outside the inclusions,respectively11.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 cylinder in a host medium,equations (3.43) and (3.46) yield

eff() = 2_1 f1u m_(3.50)11In the case of identical inclusions composed of an anisotropic material, as occurs in the case of an array of fullerenesor carbon nanotubes, equation (3.48) still holds, as long as the scalar dielectric function 1 is replaced by its tensorialcounterpart. In this case, the electric elds E and Ein would have, in general, different directions.20 J M Pitarke et aland

1eff () = 12_1 + f1u n_, (3.51)normal modes occurring, therefore, at the frequencies dictated by the following conditions:m1 + (1 m) 2 = 0 (3.52)andn 1 + (1 n) 2 = 0. (3.53)For Drude particles (1 of equation (2.15)) in vacuum (2 = 1), these frequencies are easilyfound to be = mp and = n p, respectively.Indeed, for a single 3Dspherical or 2Dcircular12inclusionina host medium, anelementaryanalysis shows that the electric eld Ein in the interior of the inclusion isEin =uu mE, (3.54)where m = 1/D, D representing the dimensionality of the inclusions, i.e. D = 3 for spheresandD =2 for cylinders. Introduction of equation (3.54) into equation (3.48) leads to aneffective dielectric function of the form of equation (3.50) with m = 1/D, which yields (seeequation (3.52)) the surface-plasmon condition dictated by equation (3.33) with l = 1 in thecase of spheres (D = 3) and the surface-plasmon condition of equation (3.35) in the case ofcylinders (D = 2). This result indicates that in the nonretarded long-wavelength limit (whichholds for wave vectors q such that sa/c< q a1, a being the radius of the inclusions) boththe absorption of light and the energy-loss spectrum of a single 3D spherical or 2D circularinclusion exhibit one single strong maximum at the dipole resonance where 1 + 22 = 0 and

1 + 2 =0, respectively, which for a Drude sphere and cylinder (1 of equation (2.15)) invacuum (2 = 1) yield = p/3 and = p/2.In the case of electromagnetic waves polarized along one single cylinder or array ofparallel cylinders (s-polarization), the effective dielectric function of the composite is simplythe average of the dielectric functions of its constituents, i.e.

eff = f1 + (1 f ) 2, (3.55)which can also be written in the form of equations (3.50) and (3.51), but now withm =0andn =f , respectively. Hence, for this polarization the absorption of light exhibits nomaxima (in the case of a dielectric host medium with constant dielectric function2) andthe long-wavelength energy-loss spectrum exhibits a strong maximum at frequencies dictatedby the plasmon condition of equation (3.37), which in the case of Drude cylinders (1ofequation (2.15)) in vacuum (2 = 1) yields the reduced plasmon frequency = fp.MaxwellGarnett approximation. The interaction among spherical (or circular) inclusions ina host mediumcan be introduced approximately in the framework of the well-known MaxwellGarnett (MG) approximation [129].The basic assumption of this approach is that the average electric eld Ein within a particlelocated in a systemof identical particles is related to the average eld Eout in the mediumoutsideas in the case of a single isolated (noninteracting) particle, thereby only dipole interactionsbeing taken into account. Hence, in this approach the electric eld Ein is taken to be of the formof equation (3.54) but with the macroscopic electric eld E replaced by the electric eld Eout12In the case of p-polarized electromagnetic waves, innitely long cylinders can be represented by 2D circularinclusions.Theory of surface plasmons and surface-plasmon polaritons 21Figure 6. Complementary systems in which the regions of plasma and vacuum are interchanged.The top panel represents the general situation. The bottom panel represents a half-space lled withmetal and interfaced with vacuum. The surface-mode frequenciess1ands2of these systemsfull the sum rule of equation (3.58).outside, which together with equations (3.48) and (3.49) yields the effective dielectric functionand effective inverse dielectric function of equations (3.50) and (3.51) with the depolarizationfactors m = n = 1/D (corresponding to the dilute limit, where f 0) replaced bym =1D(1 f ) (3.56)andn =1D[1 + (D 1)f ] . (3.57)3.2.4. Periodic structures. Over the years, theoretical studies of the normal modes of complexcomposite systems had been generally restricted to mean-eld theories of the MaxwellGarnetttype, which approximately account for the behaviour of localized dipole plasmons [129].Nevertheless, anumber of methodshavebeendevelopedrecentlyfor afull solutionofMaxwells equations in periodic structures [196201]. The transfer matrix method has beenused to determine the normal-mode frequencies of a lattice of metallic cylinders [202] androds [203], a so-called on-shell method has been employed by Yannopapas et al to investigatethe plasmon modes of a lattice of metallic spheres in the low lling fraction regime [204]and a nite difference time domain (FDTD) scheme has been adapted to extract the effectiveresponse of metallic structures [201].Most recently, anembeddingmethod[199] hasbeenemployedtosolveMaxwellsequations, whichhasallowedtocalculatethephotonicbandstructureofthree-andtwo-dimensional latticesofnanoscalemetal spheresandcylindersinthefrequencyrangeofthe Mie plasmons [143]. For small lling fractions, there is a surface-plasmon polaritonwhichinthenonretardedregionyields thenondispersiveMieplasmonwithfrequencyp/D. As the lling fraction increases, a continuum of plasmon modes is found to existbetween zero frequency and the bulk metal plasmon frequency [143], which yield strongabsorption of incident light and whose energies can be tuned according to the particleparticleseparation [205].3.2.5. Sum rules. Sum rules have played a key role in providing insight in the investigationof a variety of physical situations. A useful sum rule for the surface modes in complementary22 J M Pitarke et almedia with arbitrary geometry was introduced by Apell et al [206], which in the special caseof a metal/vacuum interface implies that [207]2s1 + 2s2 = 2p, (3.58)wheres1is the surface-mode frequency of a given system ands2represents the surfacemode of a second complementary system in which the regions of plasma and vacuum areinterchanged (see gure 6).For example, a half-space lled with a metal of bulk plasma frequency p and interfacedwith vacuum maps into itself (see bottom panel of gure 6), and therefore equation (3.58)yieldss1 = s2 = p/2, (3.59)which is Ritchies frequency of plasma oscillations at a metal/vacuum planar interface.OtherexamplesareaDrudemetal sphereinvacuum, whichsustainslocalizedMieplasmons at frequencies given by equation (3.34), and a spherical void in a Drude metal,which shows Mie plasmons at frequenciesl = p_l + 12l + 1. (3.60)The squared surface-mode frequencies of the sphere (equation (3.34)) and the void(equation (3.60)) add up to 2p for all l, as required by equation (3.58).Thesplittingof surfacemodesthat occursinthinlmsduetothecouplingof theelectromagnetic elds in the two surfaces (see equation (2.26)) also occurs in the case oflocalized modes. Apell et al [206] proved a second sum rule, which relates the surface modescorresponding to the in-phase and out-of-phase linear combinations of the screening chargedensities at the interfaces. In the case of metal/vacuum interfaces this sum rule takes the formof equation (3.58) but nows1ands2being in-phase and out-of-phase modes of the samesystem.For a Drude metal lm with equal and abrupt planar surfaces, the actual values of thenonretarded s1 and s2 are those given by equation (2.27), which full the sum rule dictatedby equation (3.58). For a spherical fullerene molecule described by assigning a Drude dielectricfunction to every point between the inner and outer surfaces of radiir1 andr2, one nds thefollowing frequencies for the in-phase and out-of-phase surface modes [208]:2s =2p2_1 12l + 1_1 + 4l(l + 1)(r1/r2)2l+1_, (3.61)also fullling the sum rule of equation (3.58).Another sum rule has been reported recently [202, 203], which relates the frequenciesof the modes that can be excited by light (as dictated by the poles of the effective dielectricfunctionof equation(3.43)) andthose modes that canbe excitedbymovingchargedparticles (asdictated by the poles of the effective inverse dielectric function of equation (3.46)). Numericalcalculations for various geometries have shown that the depolarization factorsmandnentering equations (3.43) and (3.46) satisfy the relation [202, 203]n = 1 (D 1) m, (3.62)where D represents the dimensionality of the inclusions.Furthermore, combining equations (3.50) and (3.51) (and assuming, therefore, that onlydipole interactions are present) with the sum rule of equation (3.62) yields equations (3.56)and (3.57), i.e. the MG approximation. Conversely, as long as multipolar modes contributeto the spectral representation of the effective response (see equations (3.43) and (3.46)), theTheory of surface plasmons and surface-plasmon polaritons 23strength of the dipolar modes decreases (see equations (3.45) and (3.47)) and a combinationof equations (3.43) and (3.46) with equation (3.62) leads to the conclusion that the dipolarresonances must necessarily deviate from their MG counterparts dictated by equations (3.56)and (3.57). That a nonvanishing contribution from multipolar modes appears together with adeviation of the frequencies of the dipolar modes with respect to their MG counterparts wasshown explicitly in [202].4. Dynamical structure factorThe dynamical structure factor S(r, r

; ) represents a key quantity in the description of bothsingle-particle and collective electronic excitations in a many-electron system [209]. The ratefor the generation of electronic excitations by an external potential, the inelastic differentialcross section for external particles to scatter in a given direction, the inelastic lifetime of excitedhot electrons, the so-called stopping power of a many-electron system for moving chargedparticles and the ground-state energy of an arbitrary many-electron system (which is involvedin, for example the surface energy and the understanding of Van der Waals interactions) areall related to the dynamical structure factor of the system.The dynamical structure factor, which accounts for the particle-density uctuations of thesystem, is dened as follows:S(r, r

; ) =

n 0n(r1) n0(r2) ( En + E0). (4.1)Here, n0(r) represent matrix elements, taken between the many-particle ground state |0 ofenergy E0 and the many-particle excited state |n of energy En, of the operator (r) n0(r),where (r) is the electron-density operator [210]: (r) =N

i=1(r ri), (4.2)with and ri describing the Dirac-delta operator and electron coordinates, respectively, andn0(r) represents the ground-state electron density, i.e.n0(r) =< 0| (r)|0> . (4.3)The many-body ground and excited states of a many-electron system are unknown andthe dynamical structure factor is, therefore, difcult to calculate. Nevertheless, one can usethezero-temperaturelimitoftheuctuationdissipationtheorem[211], whichrelatesthedynamical structure factorS(r, r

; ) to the dynamical density-response function(r, r

; )of linear-response theory. One writes,S(r, r

; ) =

Im(r, r

; ) (), (4.4)whererepresents the normalization volume and (x) is the Heaviside step function.5. Density-response functionTake a system of N interacting electrons exposed to a frequency-dependent external potentialext(r, ). Keeping terms of rst order in the external perturbation and neglecting retardationeffects, time-dependent perturbation theory yields the following expression for the inducedelectron density [210]:n(r, ) =_dr

(r, r

; ) ext(r

, ), (5.1)24 J M Pitarke et alwhere(r, r

; )representstheso-calleddensity-responsefunctionofthemany-electronsystem:(r, r

; ) =

nn0(r)n0(r

)_1E0 En + h( + i)1E0 + En + h( + i)_, (5.2) being a positive innitesimal.The imaginary part of the true density-response function of equation (5.2), which accountsfor the creation of both collective and single-particle excitations in the many-electron system,is known to satisfy the so-called f -sum rule:_dIm(r, r

; ) =

_n0(r)(r, r

)_, (5.3)with n0(r) being the unperturbed ground-state electron density of equation (4.3).5.1.Random-phase approximation (RPA)In the so-called random-phase or, equivalently, time-dependent Hartree approximation, theelectron density n(r, ) induced in an interacting electron systemby a small external potentialext(r, ) is obtained as the electron density induced in a noninteracting Hartree system (ofelectrons moving in a self-consistent Hartree potential) by both the external potential ext(r, )and the induced potentialH(r, ) =_dr

v(r, r

) n(r

, ), (5.4)withv(r, r

) representing the bare Coulomb interaction. Hence, in this approximation onewritesn(r, ) =_dr

0(r, r

; )_ext(r

, ) +_dr

v(r

, r

) n(r

, )_, (5.5)which together with equation (5.1) yields the following Dyson-type equation for the interactingdensity-response function:(r, r

; ) = (r, r

; ) +_dr1_dr2 0(r, r1; )v(r1, r2) (r2, r

; ), (5.6)where 0(r, r

; ) denotes the density-response function of noninteracting Hartree electrons:0(r, r

; ) =2

i,j(fi fj)i(r)j(r)j(r

)i(r

) j + i + i. (5.7)Here, fiareFermiDiracoccupationfactors, whichat zerotemperaturetaketheformfi =(F i),F being the Fermi energy and the single-particle states and energiesi(r)and i are the eigenfunctions and eigenvalues of a Hartree Hamiltonian, i.e._122+ vH[n0](r)_i(r) = i i(r), (5.8)Theory of surface plasmons and surface-plasmon polaritons 25wherevH[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 electrondensity:n0(r) =N

i=1|i(r)|2. (5.10)5.2.Time-dependent density-functional theoryIntheframeworkoftime-dependent density-functional theory(TDDFT)[212], theexactdensity-response functionof aninteractingmany-electronsystemis foundtoobeythe followingDyson-type equation:(r, r

; ) = 0(r, r

; ) +_dr1_dr2 0(r, r1; ){v(r1, r2) + fxc[n0](r1, r2; )} (r2, r

; ). (5.11)Here, the noninteracting density-response function 0(r, r

; ) is of the form of equation (5.7)but with the single-particle states and energies i(r) and i being now the eigenfunctions andeigenvalues of the KohnSham DFT, i.e:_122+ vKS[n0](r)_i(r) = i i(r), (5.12)wherevKS[n0](r) = vH[n0](r) + vxc[n0](r) (5.13)withvxc[n0](r) =Exc[n]n(r)n=n0. (5.14)Exc[n] represents the unknown XC energy functional and n0(r) denotes the exact unperturbedelectron density of equation (4.3), which the DFTshows to coincide with that of equation (5.10)but with the Hartree eigenfunctions i(r) of equation (5.8) being replaced by their KohnShamcounterparts of equation (5.12). The XC kernel fxc[n0](r, r

; ) denotes the Fourier transformoffxc[n0](r, t ; r

, t

) =vxc[n](r, t )n(r

, t

)n=n0, (5.15)with vxc[n](r, t ) being the exact time-dependent XC potential of TDDFT.If short-range XC effects are ignored altogether by setting theunknown XC potentialvxc[n0](r) and XCkernel fxc[n0](r, r

; ) equal to zero, the TDDFT density-response functionof equation (5.11) reduces to the RPA equation (5.6).5.2.1. The XC kernel. Along the years, several approximations have been used to evaluatethe unknown XC kernel of equation (5.15).26 J M Pitarke et alRandom-phase approximation (RPA). Nowadays, one usually refers to the RPA as the resultof simply setting the XC kernel fxc[n0](r, r

; ) equal to zero:fRPAxc[n0](r, r

; ) = 0, (5.16)but still using in equations (5.7) and (5.10) the full single-particle states and energies i(r) andi of DFT (i.e. the solutions of equation (5.12)) withvxc[n0](r) set different from zero. Thisis sometimes called the DFT-based RPA.Adiabaticlocal-densityapproximation(ALDA). Inthisapproximation, alsocalledtime-dependent local-density approximation (TDLDA) [213], one assumes that both the unperturbedn0(r) and the induced n(r, ) 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 XCkernel of a homogeneous electron gas at the local density:fALDAxc[n0](r, r

; ) =d2[nxc(n)]dn2n=n0(r)(r r

), (5.17)where xc(n) is the XC energy per particle of a homogeneous electron gas of density n.PGG and BPG. In the spirit of the optimized effective-potential method [214], Petersilka,Gossmann and Gross (PGG) [215] derived the following frequency-independent exchange-only approximation for inhomogeneous systems:fPGGx[n0](r, r

; ) = 2|r r

|

ifi i(r) i(r

)2n0(r)n0(r

), (5.18)where i(r) denote the solutions of the KohnSham equation (5.12).More recently, Burke, Petersilka, and Gross (BPG) [216] devised a hybrid formula for theXC kernel, which combines expressions for symmetric and antisymmetric spin orientationsfromthe exchange-onlyPGGscheme andthe ALDA. For anunpolarizedmany-electronsystem,one writes [216]fBPGxc[n0](r, r

; ) =12_f,PGGxc+ f,ALDAxc_, (5.19)where fxcand fxcrepresent the XC kernel for electrons with parallel and antiparallel spin,respectively.Averageapproximation. Theinvestigationofshort-rangeXCeffectsinsolidshasbeenfocusedtoagreat extent ontothesimplest possiblemany-electronsystem, whichisthehomogeneous electron gas. Hence, recent attempts to account for XCeffects in inhomogeneoussystems have adopted the following approximation [217, 218]:favxc [n0](r, r

; ) = fhomxc( n; |r r

|; ), (5.20)where nrepresentsafunctionof theelectrondensitiesat points randr

, typicallythearithmetical average n =12_n0(r) + n0(r

)_, (5.21)andfhomxc( n; |r r

|; ) denotes the XC kernel of a homogeneous electron gas of density n,whose 3D Fourier transformfhomxc( n; Q, ) is directly connected to the so-called local-eldfactor G( n; Q, ):fhomxc( n; Q, ) = 4Q2G( n; Q, ). (5.22)Theory of surface plasmons and surface-plasmon polaritons 27In the ALDA, one writesGALDA( n; Q, ) = G( n; Q 0, = 0)= Q24d2[nxc(n)]dn2n= n, (5.23)which in combination with equations (5.20)(5.22) yields the ALDAXCkernel ofequation (5.17). However, more accurate nonlocal dynamical expressions for the local-eldfactor G( n; Q, ) are available nowdays, which together with equations (5.20)(5.22) shouldyield an accurate (beyond the ALDA) representation of the XC kernel of inhomogeneoussystems.During the last decades, much effort has gone into the determination of the static local-eldfactor Gstatic( n; Q) =G( n; Q, = 0) [219226], the most recent works including diffusionMonte Carlo (DMC) calculations [227, 228] and the parametrization of the DMC data of [228]given by Corradini et al [229]:Gstatic( n; Q) = C Q2+ B Q2/(g +Q2) + Q4eQ2, (5.24)whereQ =Q/qF, and the parameters B, C, g, and are the dimensionless functions of nlisted in [229].Calculations of the frequency dependence of the local-eld factor G( n; Q, ) have beencarried out mainly in the limit of long wavelengths (Q 0) [230235], but work has alsobeen done for nite wave vectors [236239].6. Inverse dielectric functionIn the presence of a many-electron system, the total potential(r, ) of a unit test charge atpoint r that is exposed to the external potentialext(r, ) can be expressed in the followingform:(r, ) = ext(r, ) + H(r, ), (6.1)where H(r, ) represents the induced potential of equation (5.4). Using equations (5.1) and(5.4), the total potential (r, ) of equation (6.1) is easily found to take the following form:(r, ) =_dr

1(r, r

; ) ext(r

, ), (6.2)where

1(r, r

; ) = (r r

) +_dr

v(r r

) (r

, r

; ). (6.3)This is the so-calledinverse longitudinal dielectric functionof the many-electronsystem, whosepoles dictate the occurrence of collective electronic excitations and which can be evaluated inthe RPA or in the framework of TDDFT from the knowledge of the density-response functionof equations (5.6) and (5.11), respectively.Some quantities, such as the optical absorption and the electron energy loss of chargedparticles moving in arbitrary inhomogeneous media, can be described by the so-called effectiveinverse dielectric function 1eff (Q, ), which is dened as a 3DFourier transformof the inversedielectric function 1(r, r

; ):

1eff (Q, ) =1

_dr_dr

eiQ(rr

)

1(r, r

; ) (6.4)28 J M Pitarke et aland which at long wavelengths (Q 0) should take the form of equation (3.46)13.In particular, in the case of a homogeneous system and in the classical long-wavelengthlimit, where the total potential(r, ) of a unit test charge at point r only depends on theexternal potentialext(r, ) at that point, the inverse dielectric function takes the followingform:

1(r, r

; ) = 1() (r r

), (6.5)which in combination with equation (6.2) yields the classical formula(r, ) = ext(r, )/(), (6.6)() representing the so-called local dielectric function of the medium.7. Screened interactionAnother key quantity in the description of electronic excitations in a many-electron system,whichalsodictatestheoccurrenceof collectiveelectronicexcitations, isthefrequency-dependent complex screened interactionW(r, r

; ). This quantity yields the total potential(r, ) of a unit test charge at point r in the presence of an external test charge of densitynext(r

, ) at point r

:(r, ) =_dr

W(r, r

; ) next(r

, ). (7.1)The potential ext(r, ) due to the external test charge density next(r, ) is simplyext(r, ) =_dr

v(r, r

) next(r

, ). (7.2)Hence, a comparison of equations (6.2) and (7.1) yieldsW(r, r

; ) =_dr

1(r, r

; ) v(r

, r

), (7.3)and using equation (6.3):W(r, r

; ) = v(r, r

) +_dr1_dr2v(r, r1) (r1, r2; ) v(r2, r

). (7.4)Fromequations (6.4) and (7.3) one easily nds the following representation of the effectiveinverse dielectric function:

1eff (Q, ) =1vQ_dr_dr

eiQ(rr

)W(r, r

; ), (7.5)where vQ = 4/Q2denotes the 3DFourier transformof the bare Coulomb interaction v(r, r

).7.1.Classical modelIn a classical model consisting of two homogeneous media characterized by local (frequency-dependent) dielectric functions 1 and 2 and separated by an interface of arbitrary geometry,the total potential at each mediumis simply given by equation (6.6) and is, therefore, a solutionof Poissons equation2(r, ) = 4

i()next(r, ), (7.6)13In the long-wavelength (Q 0) limit, longitudinal and transverse dielectric functions with the same polarizationcoincide.Theory of surface plasmons and surface-plasmon polaritons 29

i() being 1 or 2 depending on whether the point r is located in medium 1 or in medium 2,respectively. Hence, the screened interaction W(r, r

; ) entering equation (7.1) is a solutionof the following equation:2W(r, r

; ) = 4

i()(r r

). (7.7)For simple geometries, such as the planar, spherical and cylindrical interfaces,equation(7.7)canbesolvedexplicitlybyimposingtheordinaryboundaryconditionsofcontinuity of the potential and the normal component of the displacement vector at the interface.7.1.1. Planar surface. Inthecaseof twosemi-innitemediawithlocal (frequency-dependent) dielectric functions 1 (at z< 0) and 2 (at z> 0) separated by a planar interfaceat z = 0 (see gure 1), there is translational invariance in two directions, which we take to benormal to the z axis. Hence, one can dene the Fourier transform W(z, z

; q, ), q being themagnitude of a 2Dwave vector in the plane of the interface, andimposingthe ordinaryboundaryconditions of continuity of the potential and the normal component of the displacement vectorat the interface, one nds:W(z, z

; q, ) = 2q____eq|zz

| + g eq(|z|+|z

|)_/1, z< 0, z

< 0,2 g eq|zz

|/(1 2), z> 0,_eq|zz

| g eq(|z|+|z

|)_/2, z> 0, z

> 0,(7.8)wherez) is the smallest (largest) ofz andz

, andgis the classical surface-responsefunction:g() =

1() 2()

1() + 2(), (7.9)or, equivalently,g() = nu n, (7.10)where u is the spectral variable of equation (3.44) and n = 1/2.An inspection of equations (7.8) and (7.9) shows that the screened interactionW(z, z

; q, ) has poles at theclassical bulk- and surface-plasmon conditions dictated by

i = 0 and by equation (3.7), respectively.7.1.2.Spheres. In the case of a sphere of radius a and local (frequency-dependent) dielectricfunction 1 embedded in a host medium of local (frequency-dependent) dielectric function 2,we rst expand the screened interaction W(r, r

; ) in spherical harmonics:W(r, r

; ) =

l,m42l + 1Wl(r, r

; ) Yl,m() Yl,m(

), (7.11)30 J M Pitarke et aland we then derive the coefcients of this expansion by imposing the boundary conditions.One nds [240]14:Wl(r, r

; ) =____(r)l+1 + (l + 1) gl(r r

)la2l+1_/1, r, r

< a,(l + 1) gl(r)l+1/(1 2), r> a,_(r)l+1 l gla2l+1(r r

)l+1_/2, r, r

> a,(7.12)where r) is the smallest (largest) of r and r

, andgl() =

1() 2()l 1() + (l + 1) 2(), (7.13)or, equivalently,gl() = nlu nl, (7.14)with u being the spectral variable of equation (3.44) andnl =l2l + 1. (7.15)As in the case of the planar surface, the screened interaction of equations (7.11)(7.13)has poles at the classical bulk- and surface-plasmon conditions, which in the case of a singlesphere in a host medium are dictated by i = 0 and by equation (3.33), respectively.Introducing equations (7.11)(7.13) into equation (7.5), one nds the following expressionfor the effective inverse dielectric function [182]:

1eff (Q, ) = 12+ f (1112)_1 + 3x

l=0(2l + 1) gl jl(x) l(x)_, (7.16)where

l(x) =l jl1(x) 1 (l + 1) jl+1(x) 2

1 2(7.17)and x = Qa. Here, f represents the volume fraction lled by the sphere and jl(x) are sphericalBessel functions of the rst kind [241]. This equation represents the dilute (f 0) limit ofthe effective inverse dielectric function derived by Barrera and Fuchs for a system composedof identical interacting spheres in a host medium [242].In the limit as Qa1, an expansion of equation (7.16) yields

1eff (Q, ) = 12_1 3f

1 2

1 + 22_, (7.18)whichis preciselythelong-wavelengtheffectiveinversedielectricfunctionobtainedinsection 3.2.3 from equations (3.48) and (3.54) withD =3 and which admits the spectralrepresentation of equation (3.51) with n = 1/3. This result demonstrates the expected result14The direct contribution to the screened interaction is missing in the rst line of equation (5) of this reference, aspointed out in [182]. This contribution, together with part of the term represented in the second line of equation (5),would give the bulk contribution 11 to equation (7) of the same reference. Hence, equation (5) of this referencemust be replaced by equations (7.11)(7.13) of the present manuscript with 2 = 1.Theory of surface plasmons and surface-plasmon polaritons 31that in the limit as Qa1 a broad beam of charged particles interacting with a single sphereof dielectric function 1 in a host medium of dielectric function 2 can only create collectiveexcitations at the dipole resonance where 1 + 22 = 0 (equation (3.33) with l = 1), which fora Drude sphere in vacuum yields = p/3.7.1.3. Cylinders. In the case of an innitely long cylinder of radius a and local (frequency-dependent) dielectric function 1 embedded in a host medium of local (frequency-dependent)dielectric function 2, we expand the screened interaction W(r, r

; ) in terms of the modiedBessel functions Im(x) and Km(x) [241], as follows:W(r, r

; ) =2_0dqz cos_qz(z z

)_

m=0mWm(,

; ) cos_m(

)_, (7.19)wherez and represent the projections of the position vector along the axis of the cylinderand in a plane perpendicular to the cylinder, respectively, qz denotes the magnitude of a wavevector along the axis of the cylinder and mm are Neumann numbersm =___1, m = 0,2, m1.(7.20)The coefcientsWm(,

; ) are then derived by imposing the boundary conditions, i.e. byrequiring that the total scalar potential and the normal component of the displacement vectorsbe continuous at the interface. One nds:Wm(,

; )=___Im(qz) K

m(x) gmIm(qz>)/I

m(x)_/1, ,

< a,_x I

m(x) Km(x)_1gmIm(qz)/(1 2), > a,_Im(qz a,(7.21)where x = qza andgm(x, ) =I

m(x) Km(x) [1() 2()]I

m(x) Km(x) 1() Im(x) K

m(x) 2(), (7.22)or, equivalently,gm(x, ) = nmu nm, (7.23)with u being the spectral variable of equation (3.44) andnm = x I

m(x) Km(x). (7.24)In the limit as x 0 (p-polarization)15, the depolarization factors of equation (7.24) areeasily found to ben0 = 0 (corresponding to the plasmon condition2 = 0)16and nm = 1/215As we are considering longitudinal elds, where the electric eld E and the wave vector q have the same direction,in the limit as x = qza 0 the electric eld lies in the plane perpendicular to the axis of the cylinder.16This mode, however, does not contribute to the effective inverse dielectric function and, therefore, to the energyloss of moving charged particles, since when x =qza = 0 (p-polarization) the strength of this mode is equal to zerofor all values of the total wave vector Qa (see [183]).32 J M Pitarke et al0 2 4 6 8 10qza00.10.20.30.40.5nmm=0m=1m=2Figure 7. Depolarizationfactors nm= x I

m(x) Km(x), as a functionof x = qza, form = 0, 1, 2, 3, 4, 5 (thin solid lines), and m = 10 (thick solid line). As m , the depolarizationfactor nm equals the planar surface-plasmon value nm = 1/2 for all values of x = qza.(corresponding to the planar surface-plasmon condition 1 +2 = 0) for all m = 0; in the limitas x , equation (7.24) yields nm = 1/2 for all m. For the behaviour of the depolarizationfactorsnmof equation (7.24) as a function of xsee gure 7. This gure shows that theenergies of all modes are rather close to the planar surface-plasmon energy (correspondingtonm = 1/2), except form = 0. Them = 0 mode, which corresponds to a homogeneouscharge distribution around the cylindrical surface, shifts downwards from the planar surface-plasmon energy (n0 = 1/2) as the adimensional quantityx =qza decreases, as occurs withthe symmetric low-energy mode in thin lms (see equations (2.26) and (2.27)).Introducing equations (7.19)(7.22) into equation (7.5), one nds the following expressionfor the effective inverse dielectric function [182]:

1eff (Q, ) = 12+ f(1112)_1 +2x2+ y2

m=0mJm(y)gm

m(x, y)_, (7.25)where

m(x, y) =I

m(x) f(1)m(x, y) 1 + K

m(x) f(2)m(x, y) 2I

m(x) Km(x) [1 2], (7.26)f(1)m(x, y) = x Jm(y) Km1(x) + y Jm1(y) Km(x), (7.27)f(2)m(x, y) = x Jm(y) Im1(x) y Jm1(y) Im(x), (7.28)x = qza, and y = qa, qz and q representing the components of the total wave vector Q alongthe axis of the cylinder and in a plane perpendicular to the cylinder, respectively. The volumefraction lled by the cylinder is denoted byf , andJm(x) are cylindrical Bessel functions ofthe rst kind [241]. A spectral representation of the effective inverse dielectric function ofequations (7.25)(7.28) was reported in [183].Theory of surface plasmons and surface-plasmon polaritons 33In the limit as Qa1, an expansion of equations (7.25)(7.28) yields

1eff (Q, ) = 12_1 f

1 2x2+ y2_x2

2+ 2y2

1 + 2__, (7.29)which admits the spectral representation of equation (3.46) with two nonvanishing spectrastrengths: C0 = x2/(x2+y2) and C1 = y2/(x2+y2), the corresponding depolarization factorsbeing n0 = 0 and n1 = 1/2, respectively.Equation (7.29) demonstrates that in the limit as Qa1 and for a wave vector normalto the cylinder (x = 0), moving charged particles can only create collective excitations at thedipole resonance wheren1 =1/2, i.e.1 + 2 =0, which for a Drude cylinder in vacuumyields Ritchies frequencys =p/2. Conversely, still in the limit asQa1 but for awave vector along the axis of the cylinder (y = 0), moving charged particles can only excitethe bulk mode of the host medium dictated by the condition u = 0 (corresponding to n0 = 0),i.e. 2 = 0, in agreement with the discussion of section 3.2.3 (Simple geometries).7.2.Nonlocal models: planar surfaceNonlocal effects that are absent in the classical model described above can be incorporated ina variety of semiclassical and quantal approaches, which we here only describe for a planarsurface.7.2.1.Hydrodynamic model.Semiclassical hydrodynamic approach. Withinasemiclassical hydrodynamicapproach,thescreenedinteractionW(r, r

; )(asdenedinequation(7.1)) canbeobtainedfromthe linearized hydrodynamic equations (3.22)(3.24). For a semi-innite metal in vacuumconsisting of an abrupt step of the unperturbed electron densityn0(z) (see equation (3.25)),we can assume translational invariance in the plane of the surface, and noting that the normalcomponent of the hydrodynamical velocity should vanish at the interface equations (3.22)(3.24) yield the following expression for the 2D Fourier transform W(z, z

; q, ):W(z, z

; q, ) = 2q___

s(z z

) + s(z + z

) 2 g

s(z) s(z

)1 0s, z< 0, z

< 0,2 g

s(z, z> 0,eq|zz

| g eq(z+z

), z> 0, z

> 0,(7.30)where z) is the smallest (largest) of z and z

,

s(z; q, ) =( + i) eq|z| q 2p e|z|

_( + i) 2p_ , (7.31)g(q, ) =2p22( + q) 2p, (7.32) =1_2p + 2q2( + i), (7.33)34 J M Pitarke et al = 1/3(32n0)1/3(as in equations (3.29) and (3.30)) and

0s(q, ) = s(z = 0; q, ). (7.34)An inspection of equations (7.30)(7.34) shows that the hydrodynamic surface-responsefunction g(q, ) and, therefore, the hydrodynamic screened interaction W(z, z

; q, ) becomesingular at the hydrodynamic surface-plasmon condition dictated by equation (3.30). We alsonote that the second moment of the imaginary part of the hydrodynamic surface-responsefunction g(q, ) is found to be_dImg(q, ) = 22 n, (7.35)where n represents the electron density: n = 2p/4.Finally, we note that in the long-wavelength (q 0) limit the hydrodynamic screenedinteraction of equations (7.30)(7.34) reduces to the classical screened interaction ofequations (7.8) and (7.9) with the dielectric functions 1 and2 being replaced by the Drudedielectric function (equation (2.15)) and unity, respectively. The same result is also obtainedby simply assuming that the electron gas is nondispersive, i.e. by taking the hydrodynamicspeed equal to zero.Quantumhydrodynamicapproach. Withinaquantizedhydrodynamicmodelofamany-electron system, one rst linearizes the hydrodynamic Hamiltonian with respect to the inducedelectron density and then quantizes this Hamiltonian on the basis of the normal modes ofoscillation (bulk and surface plasmons) corresponding to equations (3.29) and (3.30). OnendsH = HG + HB0+ HS0 , (7.36)where HGrepresentstheThomasFermi groundstateof thestaticunperturbedelectronsystem [176], and HB0and HS0are free bulk and surface plasmon Hamiltonians, respectively:HB0=1

q,qz_1/2 + BQ_aQ(t )aQ(t ) (7.37)andHS0 =1A

q_1/2 + Sq_bq(t )bq(t ). (7.38)Here,and A represent the normalization volume and the normalization area of the surface,respectively, aQ(t ) andbq(t ) are BoseEinstein operators that annihilate bulk and surfaceplasmons with wave vectors Q =(q, qz) and q, respectively, andBQ andSqrepresent thedispersion of bulk and surface plasmons:_BQ_2= 2p + 2Q2(7.39)and_Sq_2=12_2p + 2q2+ q_22p + 2q2_. (7.40)Hence, within this approach one can distinguish the separate contributions to the imaginarypart of the hydrodynamic surface-response functiong(q, ) of equation (7.32) coming fromthe excitation of either bulk or surface plasmons. One nds [243]Img(q, ) = ImgB(q, ) + ImgS(q, ), (7.41)Theory of surface plasmons and surface-plasmon polaritons 35whereImgB(q, ) =12q_0dqz ( BQ)(2p/BQ) q2zq4z + q2z(q2+ 2p/2) + 4p/(44)(7.42)andImgS(q, ) =2qq + 2q2pSq( Sq), (7.43)with Q =_q2+ q2zandq =12_q +_22p + 2q2_. (7.44)For the second moments of ImgB(q, ) and ImgS(q, ), one nds_0dImgB(q, ) =4qq + 2q2p(7.45)and_0dImgS(q, ) =42qq + 2q2p, (7.46)which add up to the second moment of equation (7.35).In the limit as q 0 the bulk contribution to the so-called energy-loss function Img(q, )(see equations (7.41)(7.44)) vanishes, and the imaginary part of both equations (7.32) and(7.43) yields the classical result:Img(q, ) 2s( s), (7.47)which can also be obtained fromequation (7.9) with 1 replaced by the Drude dielectric functionof equation (2.15) and 2 set equal to unity. Equation (7.47) shows that in the classical (long-wavelength) limit the energy loss is dominated by the excitation of surface plasmons of energys = p/2, as predicted by Ritchie.7.2.2. Specular-reectionmodel (SRM). Analternativeschemetoincorporatenonlocaleffects, which has the virtue of expressing the screened interaction W(z, z

; q, ) in terms ofthe dielectric function (Q, ) of a homogeneous electron gas representing the bulk material,is the so-called specular-reection model reported independently by Wagner [179] and byRitchie and Marusak [180]. In this model, the medium is described by an electron gas inwhich all electrons are considered to be specularly reected at the surface, thereby the electrondensity vanishing outside.For a semi-innite metal in vacuum, the unperturbed electron densityn0(z) is taken tobe of the form of equation (3.25), and the SRM yields a screened interaction of the form ofequation (7.30) but with the quantitiess(z; q, ) andg(q, ) being replaced by the moregeneral expressions:

s(z; q, ) =q_+dqzQ2eiqzz

1(Q, ) (7.48)36 J M Pitarke et alandg(q, ) = 1 0s(q, )1 + 0s(q, ), (7.49)with 0s(q, ) dened as in equation (7.34), and Q =_q2+ q2z.Theinversedielectricfunction1(Q, )enteringequation(7.48)representsthe3DFourier transform of the inverse dielectric function1(r, r

; ) of a homogeneous electrongas. From equation (6.3), one nds

1(Q, ) = 1 + vQ(Q, ), (7.50)where (Q, ) represents the 3DFourier transformof the density-response function(r, r

; ).In the framework of TDDFT, one uses equation (5.11) to nd(Q, ) = 0(Q, ) + 0(Q, )_vQ + fxc( n; Q, )_(Q, ), (7.51)with 0(Q, ) and fxc( n; Q, ) being the 3DFourier transforms of the noninteracting density-responsefunctionandtheXCkernel ofequations(5.7)and(5.15), respectively. Forahomogeneouselectrongas, theeigenfunctions i(r)enteringequation(5.7)areall planewaves; thus, the integrations can be carried out analytically to yield the well-known Lindhardfunction0(Q, ) [244]. If one sets the XC kernelfxc( n; Q, ) equal to zero, introductionof equation (7.51) into equation (7.50) yields the RPA dielectric function

RPA(Q, ) = 1 vQ0(Q, ), (7.52)which is easy to evaluate.The RPA dielectric functionRPA(Q, ) of a homogeneous electron gas can be furtherapproximated in the framework of the hydrodynamic scheme described in section 3.1.3. Onends

hydro(Q, ) = 1 +2p2Q2( + i), (7.53)which in the classical (long-wavelength) limit yields thelocal Drude dielectric function ofequation (2.15). Introduction of equation (7.53) into equation (7.48) yields the hydrodynamicscreened interaction of equations (7.30)(7.34).We know from equation (7.1) that collective excitations are dictated by singularities in thescreened interaction or, equivalently, maxima in the imaginary part of this quantity. For z andz

coordinates well inside the solid (z, z

), one ndsWin(z, z

; q, ) =_dqz2eiqz(zz

)vQ

1(Q, ), (7.54)which in the case of the Drude dielectric function (Q, ) of equation (2.15) and for positivefrequencies (> 0) yieldsImWin(z, z

; q, ) 2qp( p) eq|zz

|. (7.55)For z and z

coordinates both outside the solid (z, z

> 0), one ndsWout(z, z

; q, ) = 2q_eq|zz

| g(q, ) eq(z+z

)_, (7.56)which in the classical (q 0) limit and for positive frequencies (> 0) yieldsImWout(z, z

; q, ) 2qs( s) eq(z+z

). (7.57)Theory of surface plasmons and surface-plasmon polaritons 370 0.5 1 (a.u.)02550Im[-W(z,z;q,)](a.u.)Figure 8. The solid line represents the energy-loss function, Im[W(z, z

; q, )], versus, asobtained at z = z

, q = 0.4qF, and rs = 2.07 from equation (7.54) by using the full RPA dielectricfunction RPA(Q, ). The thick dashed and dotted lines represent separate contributions from theexcitation of bulk collective modes and eh pairs occurring at energies BQ=q 0. (8.39)Hence, in the presence of a 3D substrate that is spatially separated from the 2D sheet(zd> 0), introduction of I (zd) = 4 zd into equation (8.38) yields at zd1: =_2zd, (8.40)Theory of surface plasmons and surface-plasmon polaritons 65which is the result rst obtained by Chaplik in his study of charge-carrier crystallization inlow-density inversion layers [290].If the 2D sheet is located inside the 3D substrate (zd0), I (zd) =0, which meansthat the effective dielectric function of equation (8.34) has no zero at low energies ( 1). When the 2D sheet is located far inside the metal surface, the sound velocity isfound to approach the Fermi velocity ( 1). When the 2D sheet is located far outside themetal surface, the coefcient approaches the classical limit ( 2zd) of equation (8.40).Finally, we note that apart from the limiting casezd =Fandq =0.01 a10(whichyields a plasmon linewidth negligibly small) the small width of the plasmon peak exhibited ingure 25 is entirely due to plasmon decay into eh pairs of the 3D substrate.8.5.2. 1Dmodel calculations. Foramorerealistic(but still simplied)descriptionofelectronic excitations at the (0001) surface of Be and the (111) surface of the noble metals Cu,Ag, and Au, the 1D model potentialvMP(z) of [248] was employed in [125, 126]. The useof this model potential allows to assume translational invariance in the plane of the surfaceand to trace, therefore, the presence of collective excitations to the peaks of the imaginarypart of the jellium-like surface response function of equation (7.73). The important differencebetween the screened interaction W(z, z

; q, ) used here to evaluate g(q, ) and the screenedinteraction used in equation (8.34) in the framework of the simple model described abovelies in the fact that the single-particle orbitalsi(z) and energiesiare now obtained bysolving equation (7.65) with the jellium KohnSham potential vKS[n0](z) replaced by the66 J M Pitarke et al0 0.1 0.2 0.3 0.4 0.5 (eV)-2502550(q,)zd=F(a)zd=-Fzd=0zd=F/2x 0.2 x 0.20 1 2 3 4 5 (eV)-1001020(q,)(b)zd=Fzd=-Fzd=F/2zd=0Figure 25. Effective dielectric function of a 2Dsheet that is located at the jelliumedge (zd = 0), asobtained from equation (8.34) and self-consistent RPA calculations of the 3D screened interactionW(zd, zd; q, )andthe2Ddensity-responsefunction2D(q, )with(a) q =0.01and(b)q = 0.1 [127]. The real and imaginary parts of eff(q, ) are represented by thick and thin solidlines, respectively. The dotted lines represent the effective 2Denergy-loss function Im[1eff (q, )]forzd= F, zd=F/2 andzd=F. The vertical dashed line represents the upper edgeu=vFq +q2/2m of the 2D eh pair continuum, where 2D eh pairs can be excited. Thecalculations presented here for zd =F and q = 0.01 a10have been carried out by replacing theenergy by a complex quantity + i with = 0.05 eV. All remaining calculations have beencarried out for real frequencies, i.e. with = 0. The 2D and 3D electron-density parameters havebeen taken to be r2Ds= 3.14 and rs = 1.87, corresponding to Be(0001).model potential vMP(z). Since this model potential has been shown to reproduce the keyfeatures of the surface band structure and, in particular, the presence of a Shockley surfacestate within an energy gap around the Fermi level of the materials under study, it provides arealistic description of surface-state electrons moving in the presence of the 3D substrate.Figure 26 shows the imaginary part of the surface-response function of equation (7.73), asobtained for Be(0001) and for increasing values of q by using the 1D model potential vMP(z)Theory of surface plasmons and surface-plasmon polaritons 670 2 4 6 8 10 12 14 16Img(q,)/(eV-1)Energy (eV)00.20.400.20.400.20.40.6Figure26. EnergylossfunctionIm[g(q, )]/ofBe(0001)versustheexcitationenergy,obtained from equation (7.73) with the use of the 1D model potential of [248], as reported in [125].The magnitude of the wave vector q has been taken to be 0.05 (top panel), 0.1 (middle panel), and0.15 (bottom panel), in units of the inverse Bohr radius a10. In the long-wavelength limit (q 0),g(q, ) is simply the total electron density induced by the potential of equation (7.75).Figure 27. Energyloss functionIm[g(q, )]/of Be(0001) versus the excitationenergyobtainedfromequation(7.73) withthe use of the 1Dmodel potential of [248] andfor various values of q [293].of [248]. As follows fromthe gure, the excitation spectra is clearly dominated by two distinctfeatures: (i) the conventional surface plasmon at hs 13 eV, which can be traced to thecharacteristic pole that the surface-response functiong(q, ) of a bounded 3D free electrongas withr3Ds= 1.87 exhibits at this energy23, and (ii) a well-dened low-energy peak withlinear dispersion.That the low-frequency mode that is visible in gure 26 has linear dispersion is clearlyshown in gure 27, where the imaginary part of the surface-response functiong(q, ) ofBe(0001) is displayed at low energies for increasing values of the magnitude of the wavevector in the rangeq = 0.01 0.12 a10. The excitation spectra is indeed dominated at lowenergies by a well-dened acoustic peak at energies of the form of equation (8.37) with an coefcient that is close to unity, i.e. the sound velocity being at long wavelengths very closeto the 2D Fermi velocity v2DF(see table 5).The energy-loss function Img(q, ) of the (111) surfaces of the noble metals Cu, Ag,23This surface plasmon would also be present in the absence of surface states, i.e. if the 1D model potentialvMPof [248] were replaced by the jellium KohnSham potential of equation (5.13).68 J M Pitarke et alFigure 28. Energy-loss function Im[g(q, )]/ of the (111) surfaces of the noble metals Cu, Agand Au [126], shown by solid, dashed, and dashed-dotted lines, respectively, versus the excitationenergy, as obtained from equation (7.73) with the use of the 1D model potential of [248] andfor q = 0.01 a10and = 1 meV. The vertical solid lines are located at the energies =v2DFq,which would correspond to equation (8.37) with = 1.Figure 29. The solid line shows the energy of the acoustic surface plasmon of Be(0001) [125], asobtained from the maxima of the calculated surface-loss function Img(q, ) shown in gure 27.Thethickdottedlineandtheopencirclesrepresent themaximaoftheenergy-lossfunctionIm[1/eff(q, )] obtained from equation (8.34) withzdfar inside the solid (thick dotted line)and with zd = 0 (open circles). The dashed line is the plasmon dispersion of a 2D electron gas inthe absence of the 3D system. The grey area indicates the region of the (q,) plane (with the upperlimit at up2D =v2DFq + q2/2m2D) where eh pairs can be created within the 2D Shockley band ofBe(0001). The area below the thick solid line corresponds to the region of momentum space wheretransitions between 3D and 2D states cannot occur. The quantities mininter and qminare determinedfrom the surface band structure of Be(0001). 2D and 3D electron densities have been taken to bethose corresponding to the Wigner radii r2Ds= 3.14 and r3Ds= 1.87, respectively.and Au is displayed in gure 28 for q = 0.01 a10. This gure shows again the presence of awell-dened low-energy collective excitation whose energy is of the form of equation (8.37)with 1.Figure 29 shows the energy of the acoustic surface plasmon of Be(0001) versus q (solidline), as derived from the maxima of the calculated Img(q, ) of gure 27 (solid line) andTheory of surface plasmons and surface-plasmon polaritons 69Figure30. Aperiodicgratingofconstant L. Thegratingperiodicstructurecanprovideanimpinging free electromagnetic radiation with additional momentum 2/L.fromthemaximaoftheeffectiveenergy-lossfunctionIm[1/eff(q, )](seegure25)obtained from equation (8.34) forzd = 0 (dashed line). Little discrepancies between thesetwo calculations should be originated in (i) the absence in the simplied model leading toequation (8.34) of transitions between 2D and 3D states, and (ii) the nature of the decayand penetration of the surface-state orbitals, which in the framework of the model leading toequation (8.34) are assumed to be fully localized in a 2D sheet at z = zd.8.5.3.First-principles calculations. First-principles calculations of the imaginary part of thesurface-response functiongg=0,g=0(q, ) of Be(0001) have been carried out recently [128],and it has been found that this metal surface is indeed expected to support an acoustic surfaceplasmon whose energy dispersion agrees with the solid line represented in gure 29 (if thedispersion of gure 29 calculated for surface state effective mass m = 1 is scaled according tothe ab initio value m = 1.2 [292]). Furthermore, these calculations have been found to agreeclosely with recent high-resolution EELS measurements on the (0001) surface of Be [128](undergrazingincidence), whichrepresenttherstevidenceoftheexistenceofacousticsurface plasmons.8.5.4. Excitation of acoustic surface plasmons. As in the case of the conventional surfaceplasmonat theRitchiesfrequencys, acousticsurfaceplasmonsshouldbeexpectedtobeexcitednot onlybymovingelectrons (as occurs intheEELSexperiments reportedrecently [128]) but also by light. Now we focus on a possible mechanism that would leadtotheexcitationofacousticsurfaceplasmonsbylightin, e.g. vicinalsurfaceswithhighindices [293].At long wavelengths (q 0), the acoustic surface-plasmon dispersion curve is of theform of equation (8.37) with 1. As the 2D Fermi velocityv2DFis typically about threeorders of magnitude smaller than the velocity of light, there is, in principle, no way that incidentlight can provide an ideal surface with the correct amount of momentum and energy for theexcitation of an acoustic surface plasmon to occur. As in the case of conventional surfaceplasmons, however, a periodic corrugation or grating in the metal surface should be able toprovide the missing momentum.Let us consider a periodic grating of constant L (see gure 30). If light hits such asurface, the grating periodic structure can provide the impinging free electromagnetic waveswith additional momentum arising from the grating periodic structure. If free electromagneticradiation hits the grating at an angle , its wave vector along the grating surface has magnitudeq =csin 2Ln, (8.41)where L represents the grating constant, and n = 1, 2, . . .. Hence, the linear (nearly vertical)dispersion relation of free light changes into a set of parallel straight lines, which can matchthe acousticplasmon dispersion relation as shown in gure 31.For a well-denedacoustic surface plasmoninBe(0001) tobe observed, the wave number qneeds to be smaller than q 0.06 a10(see gure 29)24. For q = 0.05 a10, equation (8.41) with24Acoustic surface plasmons in Be(0001) have been observed for energies up to 2 eV [128].70 J M Pitarke et alFigure 31. A schematic representation of the dispersion relation of acoustic surface plasmons(solid line) and free light impinging on a periodic grating of constant L (essentially vertical dottedlines).n = 1 yields a grating constant L = 66 . Acoustic surface plasmons of energy 0.6 eVcould be excited in this way. Although a grating period of a few nanometres sounds unrealisticwith present technology, the possible control of vicinal surfaces with high indices could provideappropriate grating periods in the near future.9. ApplicationsSurface plasmons have been employed over the years in a wide spectrum of studies rangingfrom condensed matter and surface physics [924] to electrochemistry [25], wetting [26],biosensing[27, 28, 29], scanningtunnellingmicroscopy[30], theejectionof ionsfromsurfaces [31], nanoparticle growth [32, 33], surface-plasmon microscopy [34, 35] andsurface-plasmon resonance technology [3642]. Renewed interest in surface plasmons hascome from recent advances in the investigation of the optical properties of nanostructuredmaterials [43, 44], one of the most attractive aspects of these collective excitations now beingtheir use to concentrate light in subwavelength structures and to enhance transmission throughperiodic arrays of subwavelength holes in optically thick metallic lms [45, 46], as well as thepossible fabrication of nanoscale photonic circuits operating at optical frequencies [48] andtheir use as mediators in the transfer of energy from donor to acceptor molecules on oppositesides of metal lms [67].Here we focus on two distinct applications of collective electronic excitations at metalsurfaces:therolethatsurfaceplasmonsplayinparticlesurfaceinteractionsandthenewemerging eld called plasmonics.9.1.Particlesurface interactions: energy lossLet us consider a recoilless fast point particle of charge Z1moving in an arbitraryinhomogeneous many-electron system at a given impact vector b with nonrelativistic velocityv, for which retardation effects and radiation losses can be neglected25. Using Fermis goldenrule of time-dependent perturbation theory, the lowest-order probability for the probe particle25Thisapproximationisvalidforheavychargedparticles, e.g. ionsandalsoforswift electronsmovingwithnonrelativistic velocities that are large compared to the velocity of target electrons. In the case of electrons, Z1 = 1.Theory of surface plasmons and surface-plasmon polaritons 71to transfer momentum q to the medium is given by the following expression [294]:Pq = 4LAZ21_0d_dq

(2)3 eib(q+q

)ImW(q, q

; ) ( q v) ( + q

v), (9.1)whereL andA represent the normalization length and area, respectively, andW(q, q

; ) isthe double Fourier transform of the screened interaction W(r, r

; ) of equation (7.4):W(q, q

; ) =_dr_dr

ei(qr+q

r

)W(r, r

; ). (9.2)Alternatively, the total decay rate1of the probe particle can be obtained from theknowledge of the imaginary part of the self-energy. In the GW approximation of many-bodytheory [295], and replacing the probe-particle Green function by that of a non-interactingrecoilless particle, one nds [250]:1= 2 Z21

f_dr_dr

i (r) f (r

)ImW(r, r

; i f) i(r

) f(r), (9.3)wherei(r) represents the probe-particle initial state of energyi, and the sum is extendedover a complete set of nal states f(r) of energy f. Describing the probe-particle initial andnal states by plane waves in the direction of motion and a Dirac function in the transversedirection, i.e.(r) =1Aeivr_(r b), (9.4)where r represents the position vector perpendicular to the projectile velocity, one nds thatthe decayrate of equation(9.3) reduces indeedtoa sumover the probabilityPqof equation(9.1),i.e.1=1T

qPq, (9.5)Tbeing a normalization time.For a description of the total energyEthat the moving probe particle loses due toelectronic excitations in the medium, one can rst dene the time-dependent probe-particlecharge densityext(r, t ) = Z1(r b v t ), (9.6)and one then obtains the energy that this classical particle loses per unit time as follows [87] dEdt= _dr ext(r, t )Vind(r, t )t, (9.7)where Vind(r, t ) is the potential induced by the probe particle at position r and time t , whichto rst order in the external perturbation yields (see equation (7.1)):Vind(r, t ) =_dr

_+dt

_+d2ei(t t

)Wind(r, r

; ) ext(r

, t

) (9.8)withW(r, r

; ) = W(r, r

; ) v(r, r

), (9.9)72 J M Pitarke et alFigure 32. Particle of chargeZ1moving with constant velocity at a xed distancez from thesurface of a plane-bounded electron gas. Inside the solid, the presence of the surface causes (i) adecrease of loss at the bulk-plasmon frequencyp varying withz as K0(2pz/v) and (ii) anadditional loss at the surface-plasmon frequency s varying with z as K0(2sz/v). Outside thesolid, energy losses are dominated by a surface-plasmon excitation at s.Finally, one writes:E =_+dt_dEdt_. (9.10)Introducing equations (9.6) and (9.8) into equation (9.7), and equation (9.7) intoequation (9.10), one nds that the total energy lossE can indeed be written as a sum overthe probability Pq of equation (9.1), i.e.E =

q(q v) Pq, (9.11)where q v is simply the energy transferred by our recoilless probe particle to the medium.9.1.1. Planarsurface. In the case of a plane-bounded electron gas that is translationallyinvariant in two directions, which we take to be normal to thez axis, equations (9.6)(9.8)yield the following expression for the energy that the probe particle loses per unit time: dEdt= iZ21_d2q(2)2_+dt

_0dei(qv

)(t t

)W[z(t ), z(t

); q, ], (9.12)where q is a 2D wave vector in the plane of the surface, v

represents the component of thevelocity that is parallel to the surface, z(t ) represents the position of the projectile relative tothe surface, andW(z, z

; q, ) is the 2D Fourier transform ofW(r, r

; ).In the simplest possible model of a bounded semi-innite electron gas in vacuum, in whichthe screened interaction W(z, z

; q, ) is given by the classical expression equation (7.8) with

1 being the Drude dielectric function of equation (2.15) and 2 = 1, explicit expressions canbe found for the energy lost per unit path length by probe particles that move along a trajectorythat is either parallel or normal to the surface.Parallel trajectory. In the case of a probe particle moving with constant velocity at a xeddistance z from the surface (see gure 32), introduction of equation (7.8) into equation (9.12)Theory of surface plasmons and surface-plasmon polaritons 73yields [293] dEdx=Z21v2___2p_ln(kcv/p) K0(2pz/v)_+ 2sK0(2sz/v), z< 0,2sK0(2s|z|/v), z> 0,(9.13)where K0() is the zero-order modied Bessel function [241], and kc denotes the magnitudeof a wave vector above which long-lived bulk plasmons are not sustainable.For particle trajectories outside the solid (z> 0), equation (9.13) reproduces the classicalexpressionofEcheniqueandPendry[22], whichwasfoundtodescribecorrectlyEELSexperiments[131]andwhichwasextendedtoincluderelativisticcorrections[296]. Forparticle trajectories inside the solid (z> 0), equation (9.13) reproduces the result rst obtainedby Nu nez et al [297]. Outside the solid, the energy loss is dominated by the excitation ofsurface plasmons at s. When the particle moves inside the solid, the effect of the boundaryis to cause (i) a decrease in loss at the bulk plasma frequency p, which in an innite electrongas would be dE/dx = Z212pln(kcv/p)/v2and (ii) an additional loss at the surface-plasmafrequency s.Nonlocal effects that areabsent intheclassical equation(9.13) wereincorporatedapproximately by several authors in the framework of the hydrodynamic approach and thespecular-reection model described in sections 7.2.1 and 7.2.2 [298303]. More recently,extensive RPA and ALDA calculations of the energy-loss spectra of charged particles movingnear a jellium surface were carried out [245] within the self-consistent scheme described insection 7.2.3. At high velocities (of a fewBohr units) and for charged particles moving far fromthe surface into the vacuum, the actual energy loss was found to converge with the classicallimit dictated by the rst line of equation (9.13). However, at low and intermediate velocitiessubstantial changes in the energy loss were observed as a realistic description of the surfaceresponse was considered.Corrections to the energy loss of charged particles (moving far from the surface into thevacuum) due tothe nite widthof the surface-plasmonresonance that is not present, inprinciple,in jellium self-consistent calculations, have been discussed recently [304]. These correctionshave been included to investigate the energy loss of highly charged ions undergoing distantcollisions at grazing incidence angles with the internal surface of microcapillary materials, andit has been suggested that the correlation between the angular distribution and the energy lossof transmitted ions can be used to probe the dielectric properties of the capillary material.ForamorerealisticdescriptionoftheenergylossofchargedparticlesmovingnearaCu(111) surface, theKohnShampotential vKS(z) usedintheself-consistent jelliumcalculations of [245] was replaced in [305] by the 1D model potentialvMP of [248]. It wasshown, however, that although the Cu(111) surface exhibits a wide band gap around the Fermilevel and a well-dened Shockley surface state the energy loss expected from this model doesnot differ signicantly from its jellium counterpart. This is due to the fact that the presence ofthe surface state compensates the reduction of the energy loss due to the band gap.Existing rst-principles calculations of the interaction of charged particles with solidsinvoke periodicity of the solid in all directions and neglect, therefore, surface effects and,in particular, the excitation of surfaces plasmons [306, 307]. An exception is a recent rst-principles calculation of the energy loss of ions moving parallel with a Mg(0001) surface [308],which accounts naturally for the nite width of the surface-plasmon resonance that is presentneither in the self-consistent jellium calculations of [245] nor in the 1D model calculationsof [305].Atypical situation in which charged particles can be approximately assumed to move alonga trajectory that is parallel to a solid surface occurs in the glancing-incidence geometry, where74 J M Pitarke et alions penetrate into the solid, they skim the outermost layer of the solid, and are then specularlyrepelled by a repulsive, screened Coulomb potential, as discussed by Gemell [309]. By rstcalculating the ion trajectory under the combined inuence of the repulsive planar potential andthe attractive image potential, the total energy loss can be obtained approximately as followsE = 2v_ztpdEdx(z)_vz(z)_1dz, (9.14)ztp and vz(z) denoting the turning point and the value of the component of the velocity normalto the surface, respectively, which both depend on the angle of incidence.Accurate measurements of the energy loss of ions being reected from a variety of solidsurfaces at grazing incidence have been reported by several authors [310314]. In particular,Winter et al [312] carried out measurements of the energy loss of protons being reected fromAl(111). From the analysis of their data at 120 keV, these authors deduced the energy lossdE/dx(z) and found that at large distances from the surface the energy loss follows closelythe energy loss expected from the excitation of surface plasmons. Later on, RPA jelliumcalculations of the energy loss from the excitation of valence electrons were combined with arst-Born calculation of the energy loss due to the excitation of the inner shells and reasonablygood agreement with the experimental data was obtained for all angles of incidence [315].Normal trajectory. Let us now consider a situation in which the probe particle moves along anormal trajectory fromthe vacuumside of the surface (z> 0) and enters the solid at z = t = 0.The position of the projectile relative to the surface is thenz(t ) = vt . Assuming that theelectron gas at z0 can be described by the Drude dielectric function of equation (2.15) andintroducing equation (7.8) into equation (3.8) yields [293]: dEdz=Z21v2___2p_ln(kcv/p) h(pz/v)_+ 2sh(sz/v), z< 02sf (2s|z|/v), z> 0,(9.15)whereh() = 2 cos() f () f (2), (9.16)with f () being given by the following expression:f () =_0x ex1 + x2dx. (9.17)Equation (9.15) shows that (i) when the probe particle is moving outside the solid the effectof the boundary is to cause energy loss at the surface-plasmon energys, and (ii) when theprobe particle is moving inside the solid the effect of the boundary is to cause both a decreasein loss at the bulk-plasmon energy p and an additional loss at the surface-plasmon energy s,as predicted by Ritchie [1].Now we consider the real situation in which a fast charged particle passes through a nitefoil of thickness a (see gure 33). Assuming that the foil is thick enough for the effect of eachboundary to be the same as in the case of a semi-innite medium, and integrating along thewhole trajectory from minus to plus innity, one nds the total energy that the probe particleloses to collective excitations:E =Z21v2_a 2p lnkcvp2 p + s_. (9.18)This is the result rst derived by Ritchie in a different way [1], which brought him tothe realization that surface collective excitations exist at the lowered frequencys. The rstTheory of surface plasmons and surface-plasmon polaritons 75Figure 33. Particle of charge Z1 passing perpendicularly through a nite foil of thickness a. Thepresence of the boundaries leads to a decrease in the energy loss at the bulk-plasmon frequencyp and an additional loss at the surface-plasmon frequencys, the net boundary effect being anincrease in the total energy loss in comparison to the case of a particle moving in an innite mediumwith no boundaries.term of equation (9.18), which is proportional to the thickness of the lm represents the bulkcontribution, which would also be present in the absence of the boundaries. The second andthird terms, which are both due to the presence of the boundaries and become more importantas the foil thickness decreases, represent the decrease in the energy loss at the plasma frequencyp and the energy loss at the lowered frequency s, respectively. Equation (9.18) also showsthat the net boundary effect is an increase in the total energy loss above the value which wouldexist in its absence, as noted by Ritchie [1]. Amore accurate jelliumself-consistent descriptionof the energy loss of charged particles passing through thin foils has been performed recentlyin the RPA and ALDA [316].9.2.STEM: valence EELSThe excitation of both surface plasmons on solid surfaces and localized Mie plasmons onsmall particles has attracted great interest over the years in the elds of scanning transmissionelectron microscopy [130135] and near-eld optical spectroscopy [136].EELS of fast electrons in STEM shows two types of losses, depending on the nature ofthe excitations that are produced in the sample: atomically dened core-electron excitationsat energies>100 eV and valence-electron (mainly collective) excitations at energies upto 50 eV. Core-electron excitations occur when the probe moves across the target, andprovidechemical informationabout atomic-sizeregionsofthetarget [317]. Conversely,valence-electron excitations provide information about the surface structure with a resolutionof the order of several nanometres. One advantage of valence EELS is that it provides astrongsignal, evenfornon-penetratingtrajectories(theso-calledaloo