Time-dependent screening of a point charge at a metal surface

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J. Phys.: Condens. Matter 22 (2010) 304013 (9pp) doi:10.1088/0953-8984/22/30/304013

Time-dependent screening of a pointcharge at a metal surfaceV M Silkin1,2,3, A K Kazansky2,4, E V Chulkov1,2,5 andP M Echenique1,2,5

1 Departamento de Fısica de Materiales, Facultad de Quımica, Universidad del Paıs Vasco,Apartado 1072, 20080 San Sebastian, Basque Country, Spain2 Donostia International Physics Center (DIPC), P Manuel Lardizabal 4, 20018 San Sebastian,Basque Country, Spain3 IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain4 Fock Institute of Physics, St Petersburg University, 198904 St Petersburg, Russia5 Centro de Fısica de Materiales CFM, Centro Mixto CSIC-UPV/EHU, 20018 San Sebastian,Basque Country, Spain

E-mail: waxslavs@sc.ehu.es (V M Silkin)

Received 25 February 2010, in final form 24 May 2010Published 13 July 2010Online at stacks.iop.org/JPhysCM/22/304013

AbstractThe space–time evolution of the dynamical screening charge density caused by a suddenlycreated point charge at the Cu(111) surface is investigated in the linear response approximation.Considering a thin slab as a model for the Cu(111) surface, we investigate the confinementeffects on dynamical screening as well. The results have been obtained on the basis ofself-consistent evaluation of the energy–momentum-dependent response function, taking intoaccount the realistic surface band structure of Cu(111). At the initial stage, we observe fastlong-range charge density oscillations due to excitation of the surface plasmon modes. Then weobserve the propagation of the shock wave of the electron–hole excitations along the slab withvelocity determined by the Fermi velocity of bulk Cu. At longer times, we have identified thepropagation along the two slab surfaces of a much slower (with velocity ∼0.3 au, close to theFermi velocity of the Cu(111) surface state) charge disturbance due to acoustic surfaceplasmon. The role of the energy band gap in the direction perpendicular to the surface inestablishing the screening is also addressed.

1. Introduction

In a set of pioneering papers, Manson and Ritchie developed inan elegant way, based on straightforward perturbation theory,a new theoretical representation of the spatial dependence ofthe self-energy of a projectile incident on a many-particletarget [1–4]. This approach, formally equivalent at thesecond-order perturbation level to the well-known and widelyapplied self-energy approach of Hedin and Lundqvist [5]was extended to higher orders of perturbation theory. Thatenabled Manson and Ritchie to evaluate quantum nonlinearcorrections to the image potential experienced by a chargedparticle at a condensed matter surface [6], to the binding energyof the surface polaron [4] and to the excited Rydberg typeelectron states of a solid surface [7], as well to dispersionforces of a general nature [1–3]. In a paper published in

1985 [4] Manson and Ritchie used their formalism to studyrecoil corrections to the van der Waals force between atomsand provided a correction to an expression which had beenuniversally accepted for more than half a century. As a smalltribute, among many others that we could have chosen, tothe deep physical insight of Professor Manson we would liketo quote here the words of the then editor of Nature JohnMaddox, taken from a full page discussion [8] of the paperby Manson and Ritchie ‘The chief interest in this work will bethe neatness of the calculation. From the interaction betweenpositronium atoms, the departure from 1/R6 behavior shouldbe recognizable out to a distance of four atomic units, and maythus be measurable. So should be departures from London’sresult for the interaction between positronium and other atoms.Meanwhile, the calculation is a telling reminder that even thebest-tried algorithms should be constantly reassessed’.

0953-8984/10/304013+09$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA1

J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

A pioneering work of Manson and co-workers on dynamicscreening and interaction of projectiles with many bodysystems was based, out of necessity, on a simplified codifiedHamiltonian to describe this interaction. Stimulated by theirwork, a great deal of other work incorporating a more detaileddescription of this interaction was published, including higher-order corrections to the interaction of charged particles withmatter [9–28]. With the advent of modern computing facilities,the change in the screening properties of surfaces and inparticular the strong non-locality at the surface in generaland specifically the ones [29–32] caused by the presence ofthe surface states in the dynamics of excited states or theenergy loss of charged particles at surfaces became feasible fortheoretical investigation. Nowadays we are able to incorporateinto the problem of dynamic screening and interaction ofprojectiles with many body system effects due to the detailedband structure in the bulk and on the surfaces of the target [33],together with the information on the time required, usually atthe attosecond (as) level (1 as = 10−18 s) for the dynamicscreening response to be established. In this paper, as atribute to the outstanding contributions to the field by ProfessorManson, we analyze in detail the time evolution of thedynamical screening of a point charge in the vicinity of a metalsurface.

The response of an electron system to a localized time-dependent external perturbation enters the description ofvarious processes in metals and at metal surfaces and presentsone of the basic problem in solid state physics. Besidesits pure scientific interest, understanding of the dynamicalscreening processes constitutes a basis for the description ofvarious spectroscopies, low energy electron diffraction, etc.For instance, for a long time this has been a topic of intensivestudies in relation to, e.g. the calculation of x-ray absorptionand emission spectra and of x-ray photoelectron spectra ofmetals [34–45]. With recent developments in ultrashort-pulselaser spectroscopic techniques [46–48] it becomes possible tostudy such processes in solids and their surfaces in real time atfemtosecond (fs) (1 fs = 10−15 s) and sub-fs timescales.

In many situations the screening of an externalperturbation in the host electron system can be considered asbeing instantaneous. Hence theoretically the screening can beconsidered as being stationary. An important question in thisrespect is the spatial extension of a disturbed region around theexternal charge. Normally in three-dimensional (3D) metalsthis screening is very efficient and occurs on a very shortlength of a few au.6 A textbook example is a Thomas–Fermiscreening which decays exponentially with distance from thepoint charge with characteristic decay length determined bya Fermi wavevector of the host electron system. However,the extension of the screening region may be somehow longerdue to the so-called Friedel charge density oscillations whoseamplitude decays with distance R as R−3 [49]. With reductionof the dimensionality of the system, the screening becomesessentially less efficient. In particular, in two-dimensional(2D) electron systems the amplitude of the Friedel oscillations

6 In the present paper we use atomic units (au), i.e. h = e2 = me = 1 unlessotherwise stated. In this system, the unit of time corresponds to 1/41.3 fs andthe unit of distance to 0.529 177 A.

decays like R−2 with distance [50–54]. All these facts hold forthe stationary screening density established around an externalpoint charge (EPC) when all dynamical processes are over.An interesting question in this respect is how the electronsystem has been evolving before reaching this final stationarysituation, i.e. the dynamical aspect in the formation of thescreening. This problem was considered previously for the2D and 3D electron systems. The calculations performedwithin linear response theory [55, 56] and time-dependentdensity functional theory [57, 58] showed that, e.g. in the2D and 3D electron systems with valence charge densitycorresponding to metals the screening charge in the vicinityof the EPC reaches the final stationary distribution on a sub-fs timescale7. However, the charge density imbalance causedby the sudden appearance of the EPC continues to oscillateat longer times at distances much longer from the EPC incomparison with the Thomas–Fermi screening length and canintroduce oscillatory behavior in the preasymptotic decay anddephasing of quasiparticles for a particular case of surface andimage potential states [62–64].

Theoretically the dynamical processes at the sub-fstimescale in an electron system following a sudden creationof a EPC were considered in a variety of publications withthe use of several approaches. First, Canright [55] performedlinear response theory calculations for the case of a 3D electronsystem within a jellium model. Recently similar calculationsfor a 2D jellium system were performed by Alducin et al[56]. The calculations for the same systems within time-dependent density functional theory, i.e. beyond the linearresponse theory, were carried out by Borisov et al [57, 58].The closely related problem of the establishment of an imagepotential experienced by a charged particle suddenly created infront of a metal surface was considered as well [65].

Besides the detailed knowledge gained in the previousstudies about how the screening establishes around an EPC,one important question regarding the role of the band structureof real materials and, in particular, that of the surface electronicstructure, has not yet been addressed. Indeed, many metalsurfaces possess the so-called surface electronic states [66, 67]whose wavefunction is strongly localized at the surface—aninteresting example of the 2D electron system. Hence, at thesemetal surfaces the carriers residing in two electron systems,in the 2D surface state and the 3D bulk ones, do participatein the screening process. Here we study the dynamicalscreening process of a suddenly created EPC at a metal surfaceconsidering its realistic band structure. In the present studywe perform calculations within the framework of the linearresponse theory. It is well known that this theory fails toreproduce the induced charge density in a close vicinity ofthe EPC, however it works well beyond some relatively smalldistance from the EPC [55–58], that is the region of primaryinterest here.

A well-known example of such surfaces is the (111)surface of noble metals like Cu, Ag, and Au with a partlyoccupied electron band of the s–pz surface state at the surface

7 Some interesting questions such as the existence in the initial moments inmetals of unscreened bare Coulomb collisions [59] or of transient excitonicstates [60, 61] are not considered here.

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

Brillouin zone center, � [68–70]. On these surfaces the s–pz surface states have a parabolic-like dispersion with 2Dmomentum k‖ parallel to the surface. Therefore these statesare frequently considered as forming the quasi-2D surfacestate band with the Fermi energy, ESS

F , equal to the surfacestate binding energy at the � point. Hence at these surfacesthe 2D and 3D electronic systems coexist. In this paper we,for the first time, present the calculated results for the casewhen two electronic subsystems of different dimensionalitysimultaneously respond to the sudden creation of the EPC.

The paper is organized as follows. In section 2 we brieflypresent the linear response theory for the calculation of theinduced density in response to the sudden appearance of theEPC in the vicinity of the surface. Section 3 is devoted to thepresentation of the calculated results and discussion of them.The conclusions are presented in section 4.

2. Calculation details

Here we are interested in study of the evolution of the screeningcharge with time in space at a metal surface immediately afterthe sudden introduction of an EPC into the system at somepoint ro at a time t = 0. This can be relevant in problems inwhich a charge is created on a timescale much shorter than thecharacteristic screening times in the host systems. Thus, onecan think about processes like the creation of a core hole in anadsorbate located at the surface or the excitation of an electronfrom an image or a surface state. In this suddenly introducedEPC limit the photoelectron is considered to be decoupled fromthe subsequent transients [55]. Then the external potential Vext

generated by this EPC reads

Vext(r, t) = e

|r − ro|�(t). (1)

Here e is the perturbing charge which is set equal to unity in thefollowing and �(t) is the Heaviside step function, i.e. �(t) =0 for t � 0 and �(t) = 1 for t > 0. Within the frameworkof a linear response theory, the charge density generated by anexternal potential is expressed as

nind(r, t) =∫

dr′∫

dt ′ χ(r, r′, t − t ′)Vext(r′, t ′), (2)

where χ(r, r′, t) is the density response function of aninteracting electron system. Taking the Fourier transform overthe time coordinate in equation (2) and considering Vext in theform of equation (1), one arrives at the next expression [55, 56]

nind(r, t) = 1

π�(t)

∫dr′

|r′ − ro|×

∫ ∞

0

dω

ωIm[χ(r, r′, ω)] (1 − cos ωt). (3)

It is natural to exploit the cylindrical symmetry of a wholesystem and represent the spatial coordinate r in the form {R, z},where R is the cylindrical coordinate in the plane parallel to thesurface measured from an axis through the EPC and z denotesthe coordinate normal to the surface. In this case the EPC is

located at ro = {0, zo}. As a result the expression (3) can berewritten in the following form

nind(R, z, t) = 1

π�(t)

∫ ∞

0

dω

ωIm[nind(R, z, ω)](1 − cos ωt),

(4)where we define nind(R, z, ω) according to

nind(R, z, ω) = 1

2π

∫ ∞

0dq‖ q‖ J0(q‖ R)nind(q‖, z, ω) (5)

with

nind(q‖, z, ω) =∫

dz′ χ(q‖, z, z′, ω)2π

q‖e−q‖|z′−zo|. (6)

Here q‖ is the magnitude of the 2D momentum q‖ in the planeparallel to the surface, and J0 is the zeroth Bessel function.

We calculate the response function of interacting electronsχ(q‖, z, z′, ω) in the random phase approximation [71]using the response function for noninteracting electronsχo(q‖, z, z′, ω)

χ(q‖, z, z′, ω) = χo(q‖, z, z′, ω)

+∫

dz1

∫dz2 χo(q‖, z, z1, ω)

× 2π

q‖e−q‖|z1−z2| χ(q‖, z2, z′, ω) (7)

with

χo(q‖, z, z′, ω) = 2∑i, j

ϕi (z)ϕ∗j (z)ϕ j(z

′)ϕ∗i (z

′)

×∫

dk‖(2π)2

θ(EF − Ei) − θ(EF − E j)

Ei − E j + (ω + iη), (8)

where η is a positive infinitesimal that in the presentcalculations was set to 10 meV, Ei = εi + k2

‖/(2mi), E j =ε j + (k‖ + q‖)2/(2m j), and EF is the Fermi energy ofthe electron system. Here, the one-particle energies εi andwavefunctions ϕi are the solutions of the one-dimensionalSchrodinger equation describing the system in the normaldirection to the surface. The corresponding potential was takenfrom [72]. This potential was designed to reproduce the keyingredients of the Cu(111) surface electronic structure, namelythe width and position of the energy gap at the center of thesurface Brillouin zone and the energy positions of both thesurface and first image states. The effective masses mi andm j associated with the surface and bulk electronic states havebeen taken to reproduce the realistic Cu(111) surface bandstructure. The sums in equation (8) over i and j includeall occupied and unoccupied electronic states up to the cutoffenergy of 150 eV above the Fermi level. In order to evaluateχ and χo we adopt an approach based on the calculation ofthese quantities in the reciprocal space [73, 74] as described indetail in [75]. In the present work we simulate the Cu(111)surface by a slab consisting of 11 Cu atomic layers. Thecorresponding band structure in the vicinity of the Fermi levelis presented in figure 1. As follows from the figure, the useof such relatively thin slabs introduces a quantization of theelectronic states in the direction perpendicular to the surfacewith notable energy splitting between them. However, we

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

EF

Figure 1. Calculated electronic structure of the 11 atomic layerCu(111) slab around the surface Brillouin zone center (k‖ = 0). Twosplit surface state bands, SS+ and SS−, calculated for experimentaleffective mass m∗

SS = 0.42me are shown by thick solid lines. Theother electronic states of the slab are shown by thin solid lines. Thecolored area represents the continuum of the projected bulkelectronic states for the Cu(111) surface. All energies are givenrelative to the Fermi level, EF.

can clearly identify two surface states SS+ and SS−, whichhave symmetric and antisymmetric character with respect tothe reflection symmetry according to the slab center plane.One can observe a splitting of ∼0.1 eV in energy positionsof this pair of surface states. Although this fact introducessome additional structures in the induced charge density, theeffect does not seem dramatic in the present context. Finally,the integrations over ω in equation (4) and q‖ in equation (5)were performed numerically with the use of very fine meshes:1250 and 20 000 points were employed for the summation in ω

and q‖ with the upper limits ωmax = 25 eV and qmax‖ = 10 au,

respectively.

3. Calculation results

In order to facilitate the subsequent discussion, in figure 2we present the evaluated surface loss function which containsinformation on the rate at which the frequency-dependentexternal potential generates electron–hole and collectiveelectronic excitations at surfaces [76]. It is defined as theimaginary part of the surface response function g(q‖, ω)

that is expressed in terms of the density response functionχ(z, z′, q‖, ω) as [77]

g(q‖, ω) = −2π

q‖

∫dz

∫dz′e−q‖(z+z′)χ(q‖, z, z′, ω). (9)

In this figure one can observe several features reflecting, inan integrated way, the singularities in χ(q‖, z, z′, ω). Asχ(q‖, z, z′, ω) directly, through equation (6), determines theinduced charge density, these singularities in χ(q‖, z, z′, ω) aretransmitted to singularities in nind(q‖, z, ω). Subsequently any,even a weak, singularity in nind(q‖, z, ω) produces oscillatory

variations in the induced charge density nind(R, z, ω) in realspace via integration in equation (5). In figure 2(a) one canobserve in the 8–12 eV energy range a clear dominant peakωSP corresponding to the conventional surface plasmon [78].At the momenta q‖ smaller than ∼0.07 au, this peak splitsinto two separate ones which correspond to the symmetricalω−

SP and antisymmetrical ω+SP modes, a well-known effect

for a finite thickness slab [78]. The energy of the lowerdispersing mode ω−

SP goes to zero when q‖ → 0, whereas inthe ω+

SP mode it reaches the bulk plasmon value in the longwavelength limit. Essentially, the new result of this work isthe presence of a complicated peak structure in the surfaceloss function in the low energy part in figure 2(a). Herethe region ‘SS’ of the enhanced Im[g(q‖, ω)] has its originin the intra-band electron–hole transitions within the surfacestate bands as discussed in detail in [79]. At variance withthe previous results for the noble metal surfaces [75, 79], infigure 2 this region has a more complicated structure due tothe presence of two energy split SS+ and SS− surface states.Figure 2(b) shows in more detail the low-energy–low-momentaregion where in the lower-left part of the SS continuum onecan observe, additionally to the prominent ωASP peak relatedto the acoustic surface plasmon (ASP) [80, 81] (correspondingto the out-of-phase collective oscillations of charge densitybetween carriers in the surface state and the bulk states), twoother dispersing peak structures, ω1

SS and ω2SS, that have their

origin in the inter-band electron–hole transitions between theSS+ and SS− surface states. Recently it was demonstrated [82]how the ωASP plasmon produces the long-range oscillations ofthe dynamical charge density in addition to the well-knownstatic Friedel oscillations which are related to the singularityin the static χ(q‖, z, z′, ω = 0) at q‖ = 2kF. The signatureof this singularity is visible in Im[g(q‖, ω = 0)] presentedin figure 2(a) as a feature centered at q‖ = kSS+

F + kSS−F =

0.236 au (where kSS+F = 0.125 au and kSS−

F = 0.111 au arethe Fermi wavevectors of the SS+ and SS− surface state bandsin figure 1, respectively). Additionally, in figure 2 are visibleother features related to the numerous inter-band transitionsbetween the occupied and unoccupied quantum-well statesof the slab. The corresponding singularities in the responsefunction lead to rather complicated patterns in time dependenceof the induced charge density. However, these features inIm[g(q‖, ω)] and the corresponding oscillations in nind(R, z, t)strongly depend on the slab thickness and gradually disappearwith increase in the number of atomic layers in the slab.Therefore we will not concentrate on them anymore.

Here we present results for the EPC placed on the vacuumside at a distance of 1.97 au from the crystal border (i.e. at adistance from the topmost Cu atomic layer corresponding tothe Cu(111) interlayer atomic distance). Figure 3 shows thestatic induced charge density distribution R · nstatic

ind (R, z) inresponse to the EPC, i.e. at t → ∞ when all the dynamicalprocesses are over. Here one can observe that inside the slabthe induced charge density at R smaller than ∼35 au essentiallycoincides with that reported previously for the Cu [82] andAg [83] surfaces. At the same time, along the upper surfacethe Friedel charge density oscillations characteristic of the 2Dsurface state are clearly seen up to R ∼ 150 au. Along the

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

Figure 2. (a) A 2D plot of the normalized surface loss function Im[g(q‖, ω)]/ω for the 11 atomic layer Cu(111) slab versus thetwo-dimensional momentum q‖ and energy ω. The peaks corresponding to the dominating surface plasmon ωSP and its split symmetrical ω+

SPand antisymmetrical ω−

SP modes at small q‖s, and acoustic surface plasmon ωASP are denoted by the corresponding symbols. (b) The same asin (a) at the low-momentum–low-energy region. Here, additionally to the conventional acoustic surface plasmon the ωASP mode can beresolved into two additional branches, ω1

SS and ω2SS, due to the finite thickness slab effect splitting of the Cu(111) surface state (the SS+ and

SS− states seen in figure 1). Eventually upon increase of the slab thickness all three modes converge into a single ωASP mode.

Figure 3. Interpolated plot of the normalized induced charge densityR · nind(R, z) in a static limit (i.e. at t = ∞) produced by an externalpoint charge (EPC) located at the place marked by the dot. Thedensity is shown as a function of lateral distance R from the EPC andperpendicular coordinate z. Two horizontal dashed lines delimit theslab. The thick dotted line shows the main direction of the chargedistribution inside the crystal beneath the EPC.

surface their amplitude decays with distance R from the EPCas R−2. In the subsurface region a complicated chessboard-like pattern [83] can be detected although its shape is stronglyinfluenced by the electron waves reflected from the lowersurface of the slab. Note that in the present case, wherea relatively thin slab is considered, the Friedel oscillationsalong the surface indeed are a result of superposition oftwo density waves with slightly different lengths of 2kSS+

F

and 2kSS−F corresponding to the split surface states SS+ and

SS−, respectively. Additionally, in figure 3 one can see howbeneath the EPC the induced charge density propagates insidethe crystal with some finite angle around 40o relative to thesurface normal due to the presence of the energy gap at theFermi level seen in figure 1, in contrast to predictions of the

jellium model [74, 84–86]. Moreover, one can see how theseoscillations are reflected from the lower surface of the slab atR ∼ 40 au and reach the upper surface at R ∼ 75 au wherethey introduce some distortions of the surface state Friedeloscillations. Indeed, this reflection process is detectable evenat larger Rs.

Figure 4 shows evolution of the induced charge densityat times from 2 au (50 as) up to 10 au (240 as) after theappearance of the EPC at the slab surface. At these smalltimes one can observe how the screening charge evolves in thevicinity of the EPC similar to the pure 2D [56] and 3D [55]cases. Thus a shock wave (SW) corresponding to electron–hole excitations is created immediately after the creation of theEPC. One can infer that the screening hole around the EPC isalso fully developed during a time of few au [57]. Neverthelesssome differences with previous results for the 3D and 2Dsystems can be noted. Thus, in figure 4 one can see that at smallRs the perturbation region propagates almost instantaneouslyin the perpendicular to the surface direction. This reflects thequasi-2D nature of the electronic structure of the slab wherethe electronic states representing bulk Cu are quantized in theperpendicular direction with large energy separation betweenenergy levels, as seen in figure 1. Moreover, in contrast to thestatic case of figure 3, the induced density at initial momentsof time does not feel any effect from the bulk band structureand propagates almost freely at small R in the z direction ashappens in the 3D jellium model [74, 84–86].

Another interesting observation of results shown infigure 4 is the existence even at these initial moments oflong-range charge density oscillations moving along the slabmuch faster than the front of the shock wave created due toelectron–hole excitations and propagating with velocity closeto the bulk Fermi velocity [55, 56]. The amplitude of these

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

Figure 4. Interpolated plot of the normalized induced charge densityR · nind(R, z, t) established in the slab at times t = 2, 4, 6, 8, and10 au after the appearance of the external point charge at the placemarked by the dot. The density is shown as a function of lateraldistance R from the EPC and perpendicular coordinate z. Twohorizontal dashed lines delimit the slab. Dotted lines for t = 6, 8,and 10 au demonstrate the front of the shock wave SW due toexcitation of the electron–hole pairs in the system.

ultrafast oscillations strongly changes with time and reachesits maximum value at the slab surfaces. We relate this effectto the excitations of the ω+

SP and ω−SP surface modes. At initial

moments of time the former dominates, as can be seen in theout-of-phase oscillations at two slab surfaces. With increasingt the lower energy ω−

SP starts to dominate and charge density atboth surfaces starts to oscillate in phase at all Rs.

In figure 5 we present the evolution of the induced chargedensity in the 20–60 au (500–1500 as) time interval. Here onecan see that even at t = 60 au after the appearance of theEPC the induced charge density does not completely feel theenergy gap in the bulk band structure and the flux of chargecontinues to propagate in the z direction at small Rs. Onthe other hand, this figure demonstrates how at time between20 au and 30 au the shock wave formed at the EPC due toelectron–hole excitations reaches the lower surface and formsa wavepacket moving along the slab almost independently of z.The leading component of this shock wave propagates with avelocity of ∼0.6–∼0.8 au which is around 1.5–2 times higherthan the average Fermi velocity of the carriers in the slabquantum states. This correlates with results for the 2D electrongas [56]. It is noteworthy that the ultrafast waves related tothe excitation of the ω−

SP and ω+SP surface modes continue to

propagate along the slab at these relatively long times and caneven be seen at t = 100 au in figure 6. However, one canobserve that the wavelength of these ultrafast charge densityoscillations reduces with time. The explanation for this couldbe that at long times the oscillations mainly correspond to theexcitation of the surface plasmon ωSP mode with large q‖swhile the fastest waves related to the strongly dispersing ω−

SPmode have already gone to larger Rs at these times.

Figure 6 presents the induced charge density distributionat times between 100 and 500 au. Here it is seen how theshock wave propagating along the slab reaches the distanceR ∼ 200 au at t = 200 au. On one hand, behind the trailingedge of this shock wave we do not see the charge densityoscillations due to the bulk plasmon excitation as it occursin the 3D jellium model [55, 57]. On the other hand, fromcomparison with the 2D model [58] we see that along the slabthe spatial spread Lsw of the shock wave is larger by a factorof about 2–3 in the present case: in figure 6 Lsw ∼ 70 andLsw ∼ 100 au for t = 100 au and t = 200 au, respectively, incomparison with Lsw ∼ 40 au at t = 50 au for the 2D electrongas [58]. Eventually upon increase of the slab thickness thevelocity of the trailing edge propagation will be reduced, whichleads to the increase of Lsw. Finally, for a semi-infinite crystalthe oscillating region due to excitation of the 3D bulk plasmonwill occupy almost all space between the EPC and the shockwavefront like in the 3D system [55, 57].

In figure 6 it is seen that once the shock wavefront hasgone away the induced charge density basically reaches itsstationary distribution8. Thus in figure 6 at t � 400 au onecan see the Friedel oscillations in the surface states on theupper surface up to R ∼ 100 au similar to the static caseof figure 3. Nevertheless, the induced charge density stilldoes present some deviations from its stationary distribution,in contrast to what occurs in a 2D electron gas [56, 58]. Thus

8 Notice, that even at t as large as 500 au the induced charge densitydistribution along the dotted line in figure 6 does not completely coincide withthat in figure 3 due to the usage of a finite ωmax value for the upper limit in thenumerical integration in equation (4).

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

Figure 5. Interpolated plot of the normalized induced charge densityR · nind(R, z, t) established in the slab at times t = 20, 30, 40, 50,and 60 au after the appearance of the external point charge at theplace marked by the dot. The density is shown as a function of lateraldistance R from the EPC and perpendicular coordinate z. Twohorizontal dashed lines delimit the slab. Dotted lines demonstrate thefront of the shock wave SW due to excitation of the electron–holepairs in the system.

we observe a wave denoted by ‘ASP’ which propagates alongboth the surfaces with average velocity ∼0.3 au. It correspondsto the excitation of the acoustic surface plasmon ωASP whosepresence is seen in the surface loss function of figure 2. The

Figure 6. Interpolated plot of the normalized induced charge densityR · nind(R, z, t) established in the slab at times t = 100, 200, 300,400, and 500 au after the appearance of the external point charge atthe place marked by the dot. The density is shown as a function oflateral distance R from the EPC and perpendicular coordinate z. Twohorizontal dashed lines delimit the slab. The dotted line fort = 100 au demonstrates the front of the shock wave SW due toexcitation of the electron–hole pairs in the system. For other ts thefront is situated beyond R = 200 au. In the bottom panel the thickdotted line shows the main direction of the charge distribution insidethe crystal beneath the EPC.

formation of this soliton-like charge wave is related to thequasi-linear dispersion of the ASP mode. Some variationswith time of the shape of the ASP waves are related to the

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J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

interference effect between the ωASP mode with the ASP-likeω1

SS and ω2SS modes having their origin in the energy splitting

of ∼0.1 eV between the SS+ and SS− surface states, due to afinite thickness slab effect, as seen in figure 1. This interferencealso produces some non-monotonous propagation of the ASPwaves along both surfaces.

4. Conclusions

We have studied, using linear response theory, the spatial-timeevolution of the induced charge density created by a suddenappearance of an external point charge at the Cu(111) surfacesimulated by the 11 Cu atomic-layer slab. The effect of theenergy gap in the bulk electronic structure and the presenceof the surface state has been investigated. Much larger timeand spatial extensions in comparison with the previous studiesare considered. The formation and propagation with timeof the main shock wave due to electron–hole excitations isdemonstrated. At times shorter than a certain value determinedby the slab thickness (∼30 au in the present case of a 11atomic layer slab) the shock wave propagates from the externalpoint charge like in 3D systems, in both the perpendicular andlateral directions. After reaching the lower surface the shockwave starts to propagate in the whole slab like in 2D systems.Additionally, at t � 150 au we observe ultrafast charge wavespropagating along the slab surfaces due to excitation of thesurface plasmon modes. At t � 200 au it is possible alsoto resolve the soliton-like charge waves corresponding to theexcitation of an acoustic surface plasmon and propagatingalong the surface with velocity close to the group velocity ofthe acoustic surface plasmon.

Acknowledgments

This work has been partially funded by the University ofthe Basque Country UPV/EHU (grant No. GIC07IT36607),the Departamento de Educacion del Gobierno Vasco, and theSpanish Ministerio de Ciencia y Tecnologıa (MCyT) (grantNo. FIS200766711C0101).

References

[1] Manson J R and Ritchie R H 1981 Phys. Rev. B 24 4867[2] Manson J R, Ritchie R H and Ferrell T L 1984 Phys. Rev. B

29 1080[3] Manson J R and Ritchie R H 1984 Phys. Rev. B 29 1084[4] Manson J R and Ritchie R H 1985 Phys. Rev. Lett. 54 785[5] Hedin L and Lundqvist S 1969 Solid State Phys. 23 1[6] Zheng X-Y, Ritchie R H and Manson J R 1989 Phys. Rev. B

39 13510[7] Zheng X-Y, Ritchie R H and Manson J R 1990 Phys. Status

Solidi b 157 K87[8] Maddox J 1985 Nature 314 315[9] Nagy I, Arnau A, Echenique P M and Zaremba E 1989 Phys.

Rev. B 40 11983[10] Echenique P M, Nagy I and Arnau A 1989 Int. J. Quantum

Chem. 36 (Suppl. 23) 521[11] Nagy I, Apagyi B and Ladanyi K 1990 Phys. Rev. A 42 1806[12] Narmann A, Monreal R, Echenique P M, Flores F,

Heiland W and Schubert S 1990 Phys. Rev. Lett. 64 1601[13] Pitarke J M, Ritchie R H, Echenique P M and Zaremba E 1993

Europhys. Lett. 24 613

[14] Nagy I 1995 Phys. Rev. B 51 77[15] Pitarke J M, Ritchie R H and Echenique P M 1995 Phys. Rev. B

52 13883[16] Nagy I and Bergara A 1996 Nucl. Instrum. Methods Phys. Res.

B 115 58[17] Wang N-P and Nagy I 1997 Phys. Rev. A 55 2083[18] Nagy I and Apagyi B 1998 Phys. Rev. A 58 R1653[19] Arista N R and Lifschitz A F 1999 Phys. Rev. A 59 2719[20] Arista N R 2002 Nucl. Instrum. Methods Phys. Res. B 195 91[21] Alducin M, Arnau A and Nagy I 2003 Phys. Rev. A 68 014701[22] Vincent R and Nagy I 2006 Phys. Rev. B 74 073302[23] Bocan G A, Arista N R and Miraglia J E 2007 Phys. Rev. A

75 012902[24] Vincent R, Nagy I and Zaremba E 2007 Phys. Rev. B

76 073301[25] Nazarov V U, Pitarke J M, Takada Y, Vignale G and

Chang Y C 2008 Int. J. Mod. Phys. 22 3813[26] Nagy I and Aldazabal I 2009 Phys. Rev. A 80 064901[27] Villo-Perez I and Arista N R 2009 Surf. Sci. 603 1[28] Nagy I, Glasser M L and March N H 2009 Phys. Lett. A

373 3182[29] Dıez Muino R, Arnau A and Echenique P M 1995 Nucl.

Instrum. Methods Phys. Res. B 98 420[30] Osma J, Sarrıa I, Chulkov E V, Pitarke J M and Echenique P M

1999 Phys. Rev. B 59 10591[31] Sarrıa I, Osma J, Chulkov E V, Pitarke J M and Echenique P M

1999 Phys. Rev. B 60 11795[32] Juaristi J I, Arnau A, Echenique P M, Auth C and

Winter H 1999 Phys. Rev. Lett. 82 1048[33] Vergniory M G, Silkin V M, Gurtubay I G and Pitarke J M

2008 Phys. Rev. B 78 155428[34] Doniach S and Sunjic M 1969 J. Phys. C: Solid State Phys.

3 285[35] Roulet B, Gavoret J and Nozieres P 1969 Phys. Rev. 178 1072[36] Nozieres P, Gavoret J and Roulet B 1969 Phys. Rev. 178 1084[37] Nozieres P and De Dominicis C T 1969 Phys. Rev. 178 1097[38] Langreth D C 1970 Phys. Rev. B 1 471[39] Muller-Hartmann E, Ramakrishnan T V and Toulouse G 1971

Phys. Rev. B 3 1102[40] Minnhagen P 1976 Phys. Lett. A 56 327[41] Mahan G D 1982 Phys. Rev. B 25 5021[42] Gadzuk J W 1987 Phys. Scr. 35 171[43] Kato M 1988 Phys. Rev. B 38 10915[44] Leiro J A and Heinonen M H 1999 Phys. Rev. B 59 3265[45] Despoja V, Sunjic M and Marusic L 2008 Phys. Rev. B

78 035424[46] Baltuska A, Udem T, Uiberacker M, Hentschel M,

Goulielmakis E, Gohle C, Holzwarth R, Yakovlev V S,Scrinzi A, Hansch T W and Krausz F 2003 Nature 421 611

[47] Cavalieri A L, Muller N, Uphues T, Yakovlev V S, Baltuska A,Horvath B, Schmidt B, Blumel L, Holzwarth R, Hendel S,Drescher M, Kleineberg U, Echenique P M, Kienberger R,Krausz F and Heinzmann U 2007 Nature 449 1029

[48] Krausz F and Ivanov M 2009 Rev. Mod. Phys. 81 163[49] Langer J S and Vosko S H 1959 J. Phys. Chem. Solids 12 196[50] Lau K W and Kohn W 1978 Surf. Sci. 75 69[51] Hyldgaard P and Persson M 2000 J. Phys.: Condens. Matter

12 L13[52] Bogicevic A, Ovesson S, Hyldgaard P, Lundqvist B I,

Brune H and Persson M 2000 Phys. Rev. Lett. 85 1910[53] Repp J, Moresco F, Meyer G, Rieder K-H, Hyldgaard P and

Persson M 2000 Phys. Rev. Lett. 85 2981[54] Zaremba E, Nagy I and Echenique P M 2003 Phys. Rev. Lett.

90 046801[55] Canright G S 1988 Phys. Rev. B 38 1647[56] Alducin M, Juaristi J I and Echenique P M 2004 Surf. Sci.

559 233[57] Borisov A, Sanchez-Portal D, Dıez Muino R and

Echenique P M 2004 Chem. Phys. Lett. 387 95[58] Borisov A G, Sanchez-Portal D, Dıez Muino R and

Echenique P M 2004 Chem. Phys. Lett. 393 132

8

J. Phys.: Condens. Matter 22 (2010) 304013 V M Silkin et al

[59] Kwong N-H and Bonitz M 2000 Phys. Rev. Lett. 84 1768[60] Cao J, Gao Y, Miller R J D, Elsayed-Ali H E and Mantell D A

1997 Phys. Rev. B 56 1099[61] Schone W-D and Ekardt W 2000 Phys. Rev. B 62 13464[62] Gumhalter B 2005 Phys. Rev. B 72 165406[63] Lazic P, Silkin V M, Chulkov E V, Echenique P M and

Gumhalter B 2006 Phys. Rev. Lett. 97 086801[64] Lazic P, Silkin V M, Chulkov E V, Echenique P M and

Gumhalter B 2007 Phys. Rev. B 76 045420[65] Alducin M, Dıez Muino R and Juaristi J I 2003 J. Electron

Spectrosc. Relat. Phenom. 129 105[66] Tamm I E 1932 Phys. Z. Sowjet. 1 733[67] Shockley W 1939 Phys. Rev. 56 317[68] Gartland P O and Slagsvold B J 1975 Phys. Rev. B 12 4047[69] Kevan S D (ed) 1992 Angle-Resolved Photoemission (Studies in

Surface Science and Catalysis vol 74) (Amsterdam:Elsevier)

[70] Hufner S 1995 Photoelectron Spectroscopy—Principles andApplications (Springer Series in Solid-State Science vol 74)(Berlin: Springer)

[71] Fetter A L and Walecka J D 1964 Quantum Theory ofMany-Particle Systems (New York: McGraw-Hill)

[72] Chulkov E V, Silkin V M and Echenique P M 1999 Surf. Sci.437 330

[73] Eguiluz A G 1983 Phys. Rev. Lett. 51 1907[74] Eguiluz A G 1985 Phys. Rev. B 31 3303[75] Silkin V M, Pitarke J M, Chulkov E V and Echenique P M

2005 Phys. Rev. B 72 115435[76] Liebsch A 1997 Electronic Excitations at Metal Surfaces

(New York: Plenum)[77] Persson B N J and Zaremba E 1985 Phys. Rev. B 31 1863[78] Ritchie R H 1957 Phys. Rev. 106 874[79] Silkin V M, Alducin M, Juarisi J I, Chulkov E V and

Echenique P M 2008 J. Phys.: Condens. Matter 20 304209[80] Silkin V M, Garcıa-Lekue A, Pitarke J M, Chulkov E V,

Zaremba E and Echenique P M 2004 Europhys. Lett.66 260

[81] Diaconescu B, Pohl K, Vattuone L, Savio L, Hofmann P,Silkin V M, Pitarke J M, Chulkov E V, Echenique P M,Farias D and Rocca M 2007 Nature 448 57

[82] Silkin V M, Nechaev I A, Chulkov E V and Echenique P M2005 Surf. Sci. 588 L239

[83] Silkin V M, Nechaev I A, Chulkov E V and Echenique P M2006 Surf. Sci. 600 3875

[84] Lang N D and Williams A R 1978 Phys. Rev. B 18 616[85] Ishida H and Terakura K 1987 Phys. Rev. B 36 4510[86] Ishida H and Liebsch A 1990 Phys. Rev. B 42 5505

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