Keywords - Ruđer Bošković InstituteExcitonic precursor states in ultrafast pump–probe...

Phys. Status Solidi B 247, No. 8, 1907–1919 (2010) / DOI 10.1002/pssb.200983944 p s sb

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basic solid state physics

Excitonic precursor states in ultrafast
pump–probe spectroscopies of surface bands
Branko Gumhalter*,1, Predrag Lazic2, and Nadja Doslic2

1 Institute of Physics, 10000 Zagreb, Croatia2Rudjer Boskovic Institute, 10000 Zagreb, Croatia

Received 15 October 2009, revised 12 March 2010, accepted 16 March 2010

Published online 22 June 2010

Keywords electronic states, photoelectron spectra, pump–probe spectroscopy, surfaces and interfaces

*Corresponding author: e-mail [email protected], Phone: þ385 1 4698 805, Fax: þ385 1 4698 889

We discuss the formation and evolution of transient coherent

excitonic states induced by ultrashort laser irradiation of metal

surfaces supporting the surface and image potential bands

(typically the low index surfaces of Cu and Ag). These states,

which evolve into the image potential states in the course of

screening of primary optically excited electron–hole pair, may

play the role of early intermediate states in pump–probe

spectroscopies of surfaces (e.g. two-photon-photoemission or

sum-frequency generation) if the formation of image charge

density proceeds on the time scale of the order of or longer than

the pump–probe pulse duration and delay. In this regime a

pump–probe experiment may yield information on the

characteristics of such states rather than the states in relaxed

image potential bands. Time scales of the various stages of these

processes are estimated using an exactly solvable model of

surface screening.

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Early time interactions of quasiparticlesexcited at surfaces The interpretation of excitation anddetection of quasiparticles at surfaces in pump–probeexperiments utilizing ultrashort laser pulses requires theassessment of quasiparticle dynamics and system relaxationfrom a true early evolution standpoint. While the non-adiabatic dynamics has been discussed in connection withthe studies of surfaces and surface reactions by pulsed laserbeams (for review see Refs. [1, 2]), the non-Markovianrelaxation effects [3, 4] following optical excitation ofsurfaces have only recently become the subject of interestand theoretical modelling [5]. Here we shall discuss theseeffects on the example of two-photon-photoemission (2PPE)from paradigmatic systems whose initial ground stateaccommodates partly occupied quasi-two-dimensionalsurface state bands (Q2D SS-bands) on top of the bulkvalence and conduction band structure [typically on (111)surfaces of Cu and Ag]. The discussion will be focused onelectronic screening processes as the dominant relaxationmechanism and in this context on the dynamics of formationof the much discussed Q2D image potential band states(IS-states) that are observed in the surface projected bandgaps between the vacuum level EV and the Fermi energy EF

of the substrate [6]. These bands are not incorpotated in the

initial equilibrium electronic structure because they arise as aresult of the response of the system to introduction ofelectronic charge outside the surface.

In the initial step of 2PPE from surface bands [7] anelectron excited out of the SS-band state by absorption of apump pulse photon of energy vpu � EV � EF will first feelthe initial state effective substrate potential and theinteraction with unscreened SS-hole left behind. This leadsto formation of a coherent primary exciton whose boundstate discrete energy levels El < EV may form a Rydberg-like series characterized by the quantum numbers l thatreflect the specificities of constrained electronic motion[8–10]. Subsequently, the two photocreated quasiparticles(SS-hole and excited electron) start interacting with thedynamical surface electronic response which will affect theamplitudes, energies and phases of primary excitonic states.As a result of these many-body interactions the primarystates of excited quasiparticles will evolve in the course oftime into a relaxed set of intermediate 2PPE states fjnig thatincorporate the effects of image screening in the form ofdeveloped image potential bands. The sequence of excitationand relaxation processes in the first stage of a 2PPE eventinduced by ultrashort pump pulse is illustrated schematicallyin the left-side panel of Fig. 1a.


1908 B. Gumhalter et al.: Excitonic precursor states in ultrafast spectroscopy of surface bandsp

hys

ica ssp st

atu

s

solid

i b

Edir= 1+ 2 +ESS

|ESS> , SS

|EIS>, IS

EV

1

22

Eind= EIS + 2

a)

surface state hole

intermediate state electron

pump pulseat t

probe pulseat t+ 2PPE

photo-emittedelectron

b)

transient excitonic levelsevolve into IS-levels

Figure 1 (online colour at: www.pss-b.com) (a) Illustration of therole of transient excitonic states as intermediate states in 2PPE fromsurface state band. The relaxation of transient excitonic states intoimage potential state(s) is denoted symbolically by a wiggly arrow.Variation of delay t between two phase locked pulsesmay probe therelaxation of the physical basis set of intermediate states. (b) Dia-grammatic illustration of the pump–probe photoexcitation processsketched in (a). Dashed and full lines denote the SS-hole and excitedelectron propagators, respectively. Dashed and full wavy linesdenote the dynamically screened interactions involving bosonizedelectronic charge density fluctuations that contribute to the vertexand self-energy renormalizations of the skeleton diagram (cf. dis-cussion in Section 4). Open end boson propagators describingexcitation of real surface charge density fluctuations are not shownin the picture.

In the second stage of a 2PPE experiment the probe pulsephotons eject electrons from the intermediate set ofdeveloped physical states (transient or relaxed) and therebyprovide information on the excited state dynamics ofthe system (see right-side panel of Fig. 1a). The outgoingphotoelectron states are due to their delocalizationmuch lessaffected by the interaction with surface localized SS-holethan the intermediate states.

In order to understand the motivation for discussingrelaxation processes in ultrafast 2PPE experiments it isimportant to observe that the many-body interactions whichaccompany evolution of the system from the initial excitedstate, over the transient excitonic states, to the relaxed statesfjnig that include the states of image potential bands, are notswitched on adiabatically but rather instantaneously with thephotoexcitation of primary electron–hole pair (e–h pair).


From the theoretical point of view this prevents the tracing oftime-development operator back to early times (t ! �1)by using the adiabatic hypothesis on which the standardpropagator techniques for descriptions of evolution ofquasiparticles are based. A clear discussion of this principalissue was presented in Ref. [14].

In this paper we outline a model for description oftemporal evolution and coherence of excited electron statesat surfaces that reveals the most salient features ofquasiparticle dynamics on the time scale typical of ultrafastpump–probe experiments. The model does not assume pre-existence of stationary image potential bands in the initialstate but recovers them as a result of evolution of the excitedsystem towards the screened state. On the basis of earlierstudies of quasiparticle dynamics in surface bands [15–17]we expect that the electronic many-body relaxation pro-cesses will take place on the few femtosecond (fs) timescale. During this interval the dynamics of quasiparticleswillbe dominantly governed by coherent energy and amplituderelaxation because the irreversible decay of quasiparticlesfrom quasistationary states may proceed and be traceableonly after the formation of the latter.

2 Formulation of the model of unscreenedsurface exciton We start from a model in which prior tothe photoexcitation of an e–h pair the many-electron systemis described by one-electron eigenstates [18–21, 23] of theeffective initial state (subscript ‘i’) one-particle Hamiltonian

Hiðp; rÞ ¼p2

2mþ UiðrÞ; (1)

with the particle momentum p ¼ ðP; pzÞ and radiusvectorr ¼ ðR; zÞ. Here bold capitals denote components of theoperators and vectors in the lateral direction (i.e. parallel tothe surface), and lower case italics the components in the z-direction perpendicular to the surface. The effective one-electron potential UiðrÞ is periodic in the lateral directionsbut for the sake of simplicity we shall assume in thefollowing UiðrÞ ¼ UiðzÞ. This means that the effect oflateral periodicity will be accounted for through differenteffective electron (m ¼ me) and hole mass ðm ¼ mhÞ in therespective bands. Hence

Hiðp; rÞ ¼bP2

2mþ

p2z2m

þ UiðzÞ; (2)

and the equation

HijK; ji ¼ EK;jjK; ji ¼ �h2K2=2mþ ej� �

jK; ji; (3)

yields the eigenenergies EK;j and eigenstates jK; ji of Hi,with K and j denoting a 2D electron wavevector parallel tothe surface and a quantum number describing electronmotion perpendicular to the surface, respectively. In thefollowing the energies EK;j will be taken to describe theunperturbed initial state one-electron band structure.

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In the forthcoming discussions we shall also refer to thefinal (subscript ‘f’) or relaxed one-particle Hamiltonian

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Hfðp; rÞ ¼p2

2mþ UfðrÞ; (4)

whose set of eigenstates includes also the image potentialband states produced by the relaxation of effective surfacepotential when an electron is adiabatically brought to thesurface [24]. Here we shall also assume UfðrÞ ¼ UfðzÞ. Theset of eigenstates that diagonalize Hf is denoted by fjK; nigand their calculation was presented in Ref. [24], and in a slabmodel in Ref. [16]. Hence, the sets of one-electron statesjK; ji and jK; ni that diagonalizeHi andHf , respectively, arenon-equivalent since they correspond to Hamiltonians withdifferent effective one-particle potentials UiðzÞ and UfðzÞ.An overview of the studies of many-body excitonicinteractions in the models with UiðrÞ ¼ UfðrÞ can beobtained from Refs. [10–13].

2.1 Primary surface exciton Hamiltonian For theconvenience of forthcoming discussion we shall assumethat the ground state of the system incorporates the statesjK; j ¼ SSi of the SS-band because they represent aninitial state feature pre-existent relative to the interactionsswitched on with optical excitation of the system. Togetherwith the bulk states jK; j 6¼ SSi they constitute a complete setof eigenstates that diagonalize Hi. It is this physical set ofeigenstates occupied up to EF in which individual electronsare ‘experimentally prepared’ before being exposed to theinteraction with pulsed laser fields. In this simplified picturean excited e–h pair can be viewed as a two-body systemdescribed by the Hamiltonian [18, 20–22]

He�h2body ¼ Heðpe; reÞ þ Hhðph; rhÞ þ Vðre � rhÞ; (5)

where re;h ¼ ðRe;h; ze;hÞ and pe;h ¼ ðPe;h; pz;e;hÞ denote theelectron (e) and hole (h) radius vectors and momenta,respectively. Here He and Hh are, respectively, one-particleHamiltonians of the form (2) for the excited electron andhole which propagate in different bands, and

Vðre � rhÞ ¼ � e2

jre � rhj(6)

is the yet unscreened two-body e–h Coulomb interactionthat depends only on the relative coordinate jre � rhj.In view of the below discussed relaxation processesand time scales on which they proceed, the pair interaction(6) becomes effective only after the excitation of e–h pair.In other words, in the present problem the two-bodyinteraction (6) does not obey the adiabatic boundaryconditions but rather is switched on suddenly with thecreation of primary e–h pair over the initial groundstate [25].

Exploiting the symmetry of the problem, the one-particle wavefunctions that diagonalize He and Hh on theright-hand-side (RHS) of Eq. (5) can be, respectively, given

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in the form

cKe;jeðreÞ ¼ hrejKe; jei ¼ eiKeRefjeðzeÞ; (7)

cKh;jhðrhÞ ¼ hrhjKh; jhi ¼ eiKhRhfjhðzhÞ: (8)

These wavefunctions constitute a complete set ofelectron and hole states that may be used in construction ofpair states for description of early exciton dynamics.

We assume that prior to photoexcitation the electronis in a surface state described by the wavefunctioncKe;SSðreÞ ¼ hrejKe; SSi. In optically pumped transitionsabsorption of a photon lifts an electron to a higherunoccupied state jK; ji, leaving a hole in the initialstate jK; SSi. From that instant onward the dynamics ofexcited e–h pair is in the absence of other interactionsdescribed by the primary exciton Hamiltonian (5). However,the creation of two uncompensated charges at the surface,viz. an electron in the state jKe; ji and a hole in thestate jKh; SSi ¼ j � Ke; SSi, also switches on the quasipar-ticle interactions with the charge density fluctuations inthe metal. In the course of time this renormalizes the barequasiparticle energies and dynamically screens out theirmutual Coulomb interaction (6). This is illustrated schema-tically in Fig. 1b. To study temporal evolution of this processwe first need a description of the primary excitonic states.

2.2 Primary exciton wavefunctions andenergies To obtain the surface exciton wavefunctions atthe instant 0þ after its excitation, we observe that if the e–hinteraction Vðre � rhÞ in (5) were absent, the problem wouldbe separable and the total e–h wavefunction given by a linearcombination of the products of electron and hole wavefunc-tions (7) and (8). Since only the lowest lying levels ofelectronic bands are important for formation of excitonicstates [20], we may for practical purposes assume a singleexcited state band produced by effective UiðzeÞ above thesurface projected bulk band gap aroundEV. Then, the formofHamiltonian (5) suggests in the first approximation therepresentation of early exciton wavefunction as described inRefs. [18–21] and adjusted to the present problem charac-terized by the parallel to the surface (lateral) translationalinvariance. By introducing the e–h pair lateral relativecoordinates

r ¼ Re � Rh; RR ¼ 1

2ðRe þ RhÞ; (9)

and their conjugate 2D momentum operators

QQ ¼ 1

2ðPe � PhÞ; PP ¼ Pe þ Ph; (10)



hys

ica ssp st

atu

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solid

i b

and substituting them in (5), one finds following Ref. [18]the transformed Hamiltonian:

H

� 20

¼ � �h2

2Mr2

RR4

��h2r2

r

2M � �h2

2

1

me

� 1

mh

� �rrrRR

� �h2

2m�e

@2

@z2eþ UiðzeÞ �

�h2

2m�h

@2

@z2h

þ UiðzhÞ �e2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 þ ðze � zhÞ2q ;

(11)

where me and mh are the effective masses of electron andhole for motion in lateral direction, respectively, andM ¼ memh=ðme þ mhÞ is the corresponding reduced massof the pair. Now, since the total lateral momentum operatorfor e–h pair PP ¼ �i�hrRR commutes with (11), itseigenvector P ¼ ðKe þ KhÞ is a constant of motion. Hence,the exciton eigenstate jP; li with lateral momentum �hP andexciton quantum number l (or a set of quantum numbers)can be expanded in unperturbed e–h pair wavefunctionscomposed of (7) and (8), viz.

jP; li ¼X

Ke;Kh;j

AP;lKe;j;Kh;SS

jKe; j;Kh; SSi; (12)

where j ¼ je ranges over the set of all empty eigenstateshzejji of the initial Hamiltonian Hi. For reasons explainedearlier the final relaxed states of image potential bands donot enter this summation because they do not constitute thebasis of the initial ground state of the system. The energiesof exciton bound states of interest, el, should be sought asthe eigenvalues of a two-body Hamiltonian (5). Now, in thetransformed coordinates representation (11) the lateralmomentum operator PP can be replaced by its eigenvalue Pand hence

He�h2bodyjP; li ¼ EjP; li: (13)

Following the analogy with Refs. [18, 20] the solutionof (13) should yield the exciton energies and wavefunctions.We note that in the case of 3D-spherical bands discussedin Refs. [18, 20] the exciton bound state energies acquire theRydberg-like form�ER=l

2 (l � 1) where ER ¼ Me4=2�h2 isexpressed through the reduced e–h mass M appropriate to3D bands. A restricted version of Hamiltonian (11) withP ¼ 0 and appropriate to description of excitons in narrowquantum wells was studied in Ref. [26]. In the presentcase and in the effective mass approximation for non-degenerate Q2D bands the exciton bound state energiesshould appear in a similar form [cf. Eqs. (10) and (30) inRef. [18]], viz.

EP;l ¼P2

2b� el; (14)

where b is a function of effective masses of electrons andholes excited in unoccupied 2D jbands and SS-band,respectively, and el denote bound state exciton energies. In a

10 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

two-band model the first term on the RHS of (14) reduces toP2=2ðme þ mhÞ.

In optical transitions the momentum of the absorbedphoton is negligible and hence the total excitonmomentum iszero, viz.P ¼ Ke þ Kh ¼ 0. Then thewavefunction describ-ing a primary exciton produced by photoexcitation of anelectron from the SS-band reads

hre; rhjP; li ¼ CP;lðre; rhÞ

¼Xj

fjðzeÞfSSðzhÞX

KeþKh¼0

eiðKeReþKhRhÞAP;lKe;j;Kh;SS

:

(15)

Prior to the restriction P ¼ 0, the second sum on the RHS of(15) represents a 2D Fourier transform of the wavepacketamplitude in the lateral momentum space of an excitonwith total momentum �hP, i.e. it gives the correspondingwavepacket in the ðx; yÞ-coordinate space[19, 20]. In thespecial case of optically induced transitions for which P ¼ 0this wavepacket reduces to the form

FP¼0;lj;SS ðrÞ ¼

XKe

eiKe rAP¼0;lKe;j;�Ke;SS

: (16)

In this limit the appropriate Hamiltonian for description ofoptically induced excitons is obtained from (11) with PPreplaced by P ¼ 0, and reads

HðexcÞP¼0 ¼ �

�h2r2r

2Mþ He

z ðzeÞ

þ Hhz ðzhÞ þ Ve�hðr ; ze; zhÞ;

(17)

where

Hez ðzeÞ ¼ � �h2

2m�e

@2

@z2eþ UiðzeÞ; (18)

Hhz ðzhÞ ¼ � �h2

2m�h

@2

@z2hþ UiðzhÞ; (19)

Ve�hðr ; ze; zhÞ ¼ � e2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðze � zhÞ2

q : (20)

For non-zero and small P the mixed third term onthe RHS of (11) containing rrrRR can under certainconditions be treated by perturbation theory followingRef. [18].

2.3 Approximate calculation of primary excitonwavefunctions An approximate solution to Eq. (17) maybe obtained following the arguments of Refs. [27, 28]employed in the description of motion of electrons and holesin perturbed periodic fields. We now make use of theassumption that even in the presence of e–h interaction (6)

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the effective potential UiðzhÞ is strong enough to localizethe hole in pre-existent SS-band within a narrow region(of the order of few atomic radii) at the surface, i.e. within theextension of surface state electron density jfSSðzÞj

2where

fSSðzÞ ¼ hzjj ¼ SSi is the eigenstate of (19). This isanalogous to the adiabatic limit Ansatz of Kohn andLuttinger [28] in which the z-component of hole motion issolved first. For the currently studied problem the pertinentsolution was presented in Refs. [16, 17]. The adiabatic limitAnsatz suggests the introduction of an effective excitedelectron Hamiltonian

www

HðeffÞP¼0 ¼ �

�h2r2r

2M þ Hez ðzeÞ þ V

ðeffÞe�h ðr ; zeÞ

¼ H0ðr ; zeÞ þ VðeffÞe�h ðr ; zeÞ;

(21)

in which

VðeffÞe�h ðr ; zeÞ ¼ �e2

Zdzh

jfSSðzhÞj2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 þ ðze � zhÞ2q ; (22)

plays the role of ‘impurity potential’ exerted by the hole in apre-existent SS-state on the excited electron located atðr ; zeÞ. The signature of the integrated out hole coordinatezh appears in the anisotropy of V

ðeffÞe�h ðr ; zeÞ which

asymptotically behaves as;

limr!1

VðeffÞe�h ðr ; zeÞ ¼ � e2

r� e2zezh

r3þOð1=r4Þ; (23)

Figure 2 (online colour at: www.pss-b.com) Three dimensionalcontour plot illustrating the anisotropy of the modulus of effectivepotential jV ðeffÞ

e�h ðr ; zeÞj [cf. Eq. (22)] in the case of Cu(111) surface.Positive direction of the z-axis (i.e. exterior of the metal) is on the

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2e

pand zh ¼

RzhjcSSðzhÞj

2dzh (note that

a similar argument was employed in Ref. [26] to obtain aneffective potential governing exciton dynamics in narrowquantum wells). Thereby the Schrodinger equation withHamiltonian (21) expressed in the coordinates ðr ; zeÞbecomes equivalent to the donor state wavefunctionequation (3.1) of Ref. [28]. In the extreme limit in whichjfSSðzhÞj

2is replaced by the density of a fixed point charge,

the formation of the screening cloud on ultrashort time scalewas studied in Ref. [29].

At this stage it is convenient to invoke the effective masstheory of Ref. [28] and apply it to (21) to obtain quantitativeestimates of the primary exciton energy levels andwavefunctions. The solution follows the well knownvariation-of-constant method. We first assume that theunperturbed z-component He

zðzeÞ of (17) gives rise toelectron band gap extending several eV below EV and to aband (or bands) above the gap whose states near the bottomare non-degenerate. Next we assume that the effective e–hpotential (22) gives rise to bound states that have dimensionslarge in comparison with the lattice period in the lateraldirections so as that the conditions are fulfilled forwriting thewavefunction of (21) in the form analogous to that discussedin Ref. [28] and appropriate to the present boundaryconditions

LHSof the picture. Light shaded sheet denotes thefirst surface planeof the crystal occupying the region of the viewer.
Cðr ; zeÞ ¼ Fðr ; zeÞcjðK0; r ; zeÞ: (24)
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Here cjðK0; r ; zeÞ ¼ uðK0; rÞfjðzeÞ is the unperturbed
wavefunction describing electron at the bottom (K ¼ K0)of the jth band of H0ðr ; zeÞ of Eq. (21). The role of z-component of the band bottom wavefunction fjðzeÞ is takenup by the lowest energy wavefunction ofHe
z ðzeÞ, Eq. (18), inthe jth band. If the surface projected band gap straddles thevacuum level, then far outside the surface this eigenstatetends asympotically to the vacuum energy wavefunction.

Fðr ; zeÞ satisfies the effective mass equation

��h2r2

r

2M � �h2

2m�e

@2

@z2eþ V

ðeffÞe�h ðr ; zeÞ

" #Fðr ; zeÞ

¼ eFðr ; zeÞ; (25)

where m�e is the effective mass for electron motion in the

z-direction near the jth band bottom of H0ðr ; zeÞ defined in(21) which fixes the zero of e. Thereby the effect ofUiðzeÞ onthe electron motion is absorbed in m�

e and the form ofwavefunction cjðk0; r ; zeÞ. In the above described approxi-mation the total exciton wavefunction corresponding toHamiltonian (17) is obtained as a product of (24) andfSSðzhÞ.

The effective potential VðeffÞe�h ðr ; zeÞ is generally aniso-

tropic as signified by expression (23) and illustrated in Fig. 2.Equation (25) is analogous to Eq. (3.4) of Ref. [28] and itssolution may be attempted by several methods. Here wepresent the results of numerical solutions of Eq. (25) forCu(111) surface obtained for few lowest bound state energylevels el by using the DVR MCTDH and Fourier GridHamiltonian methods. The parameters used are M ¼ 0:33,



hys

ica ssp st

atu

s

solid

i b

m�e ¼ 1, and el’s are measured from the relevant jth band

bottom. We find:

� 20

e0ðt ¼ 0þÞ ¼ �3:1961 eV

e1ðt ¼ 0þÞ ¼ �1:9516 eV

e2ðt ¼ 0þÞ ¼ �1:2928 eV

e3ðt ¼ 0þÞ ¼ �1:1261 eV

e4ðt ¼ 0þÞ ¼ �0:9634 eV

e5ðt ¼ 0þÞ ¼ �0:7730 eV

e6ðt ¼ 0þÞ ¼ �0:7271 eV; etc :

(26)

- 4 -2 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

time [fs]

Population probability

Figure 3 (online colour at: www.pss-b.com) Temporal variationof the relative population probabilities jaP¼0

l;SS ðt0Þj2(normalized to

respective jDj;SSCP¼0;lj;SS ðr ¼ 0Þj2) in electronic transitions from the

SS-band on Cu(111) to four lowest stationary surface excitoniclevels (26) detached from a single generic jth band (dotted, shortdashed, long dashed and full curves, respectively), and to a resonantlevel eres ¼ vpu þ eSS (topmost dotted curve). The bottom of the jthband is taken to coincide with the vacuum level. Transitions areinduced by ultrashort Gaussian pump pulse Gðt1Þ centred aroundt1 ¼ 0, with carrier frequency vpu ¼ 4:2 eV and FWHM¼ 3 fs.

This substantiates the possibility of existence ofexcitonic precursor states in the intermeadiate stage of a2PPE process. These el are also in good semiquantitativeagreement with the results obtained for transient excitons inbulk Cu [10]. Of course, for t > 0 the excitonic energy levelswill vary in time following to the dynamics of screening ofthe bare e–h potential and the formation of quasiparticle-surface image potential (see Section 4). Here it is interestingto note that final bound state energy levels of the imagepotential [24] derived from (4) are nearly degenerate withhigher el listed in (26).

3 Population of primary excitonic levels inducedby ultrashort pump pulses The wavefunction prerequi-sites established in the preceding section enable us toestimate the evolution of optically induced populations of theprimary states of transient surface excitons. The desiredinformation is contained in quantum mechanical amplitudesaPl;SSðt0Þ describing these processes, i.e. the coefficients of thecomponents of exciton wavefunctions obtained from firstorder time-dependent perturbation theory applied to electroninteraction with the pump laser field [32]. Hence, thetransition amplitudes appear linear and the populationsquadratic in the field strength E. In the considered process anelectronmakes a transition from the initial state jKe; SSi intoone of the jKe; ji states that constitute the ðP; lÞ-th excitonicstate (15). Making use of the above expressions we find thatthe probability amplitude pertaining to optically inducedtransition into excitonic state jP ¼ 0; li is given by a coherentsum over j-bands:

aP¼0l;SS ðt0Þ ¼

Xj

Dj;SSe�iEP¼0;lt

0F

P¼0;lj;SS ðr ¼ 0Þ

� 1

i

Z t0

�1dt1Epuðt1ÞeiðEP¼0;l�eSSÞt1 ;

ð27Þ

where Dj;SS ¼ mmm j;SSE ¼ hKe; jjreEjKe; SSi is the one-

electron dipole matrix element, FP¼0;lj;SS ðr ¼ 0Þ is obtained

from (16), and Epuðt1Þ is the modulation of pump pulseelectromagnetic field. Assuming rotating wave approxi-mation and Gaussian profile Gðt1Þ of the pump pulse withcarrier frequency vpu, viz. Epuðt1Þ ¼ Gðt1Þe�ivput1 , we findthat the integration on the RHS of (27) can be readily carriedout. The conservation of energy in a single optically inducedtransition (i.e. in the absence of the probe pulse) isestablished in the limit t0 ! 1 taken in the integral on the


RHS of (27). On the other hand, in pump–probe inducedtransitions typical of 2PPE the conservation of energy sets inonly after the completion of interactions with both laserfields. Mathematically this means after additional integ-ration over the duration of interaction with the probe fieldEprðt0Þ, viz.

R t

�1 dt0Eprðt0Þ . . ., and letting t ! 1.Temporal variation of the probability of electronic

transitions induced by an ultrashort Gaussian pulse fromthe occupied part of SS-band into l-th excitonic level is givenby the absolute square of expression (27). This is illustratedin Fig. 3 for the case of Cu(111) surface (cf. this dependencewith the ones shown Figs. 2a and 6a in Ref. [11]). It is seenthat the pumping of a single excitonic level is a smoothprocess largely determined by the pulse duration. However,the population of final 2PPE states proceeding fromexcitonic levels may exhibit interference effects arising inthe coherent sum of coefficients (27) over the manifold fjligof excitonic states.

4 Surface exciton in the presence of dynamicalimage screening We shall describe the effects ofdynamical screening at the surface within linear responseformalism. This is equivalent to bosonization of theinteraction of individual charges with the electroniccharge density fluctuations in the system. The procedure isbased on the observation that within linear responseformalism the propagator of electronic charge densityfluctuations can be conveniently represented by an equival-ent boson propagator with the same excitation spectrum[16, 33]. The quantum numbers describing bosonized chargedensity excitations are their momentum Q parallel to thesurface and the excitation energy v0. Thereby the dynamicalscreening effects are modelled by the interaction of chargesof excited electron and hole with the boson field whose

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fluctuations are described by the linear electronic responsefunction of the system. This passage was elaborated inRef. [16] for the interaction of quasiparticles in surfacebands with the substrate electronic response, and is shownschematically in Fig. 4 for the surface mediated componentof the total e–h interaction discussed below.

Wefirst outline a general framework for the description ofcoherent relaxation processes during which the energies ofquasiparticles in non-stationary states are modified by theirinteractions with the developing electronic response of thesystem. Incoherent relaxation is usually associated withthe quasiparticle recoil and irreversible decay out of thestationary states [16] and therefore can be well defined onlyupon the formation of the latter.We shall demonstrate that theaction of coherent response can be modelled by an effectivetime-dependent potentialwhich gives the leading contributionto surface mediated e–h interaction energy and is manifestlyindependent of the individual states of excited quasiparticles[34]. Our point of departure is the picture of screening of e–hpotential (22) that is illustrated diagrammatically in Fig. 4. Inthis formulation the screening process is completely describedby the full response function of the system, xðr2; r1; t � t1Þelaborated in Ref. [16] and adapted to the present problembelow. From this we may find the form of effective screenedpotential in the direct space as a function of the quasiparticlecoordinates re and rh or, by exploiting the translationalsymmetry along the surface, in the mixed space as afunction of ðQ; ze;hÞ. Thereby we avoid preassumed statesjii in the formulation of early quasiparticle dynamics. In thefield-theoretical description this corresponds to accounting forthe renormalization of bare e–h interaction which becomeseffective with the photoexcitation of the pair (cf. Fig. 1b).

Going over from space and time to the mixed ðQ;vÞ-representation we may write for the total e–h interaction

V tote�hðQ; ze; zh;vÞ ¼ Ve�hðQ; ze; zh;vÞ

þVðindÞe�h ðQ; ze; zh;vÞ:

ð28Þ

(z1,z2,Q,t-t1)z1 z2

(ze,t)

(zh,t1)z=0

t

Figure 4 (online colour at: www.pss-b.com) Diagrammatic illus-tration of the screened component ~Ve�hðQ; ze; zh; tÞ of e–h interac-tion (31) for the caseofhole chargedensityoverlapping the substratesurface at z ¼ 0.

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Here Ve�hðQ; ze; zh;vÞ is the Fourier transform withrespect to r and t of the suddenly switched on barecomponent of e–h potential Ve�hðr ; ze; zhÞQðtÞ withVe�hðr ; ze; zhÞ < 0 given by (20). In accord with this, thescreened or polarization induced component of the total e–hinteraction reads

VðindÞe�h ðQ; ze; zh;vÞ ¼ � i

vþ id

Zdz2

Zdz1VQe

�Qjze�z2j

� xQðz1; z2;vÞVQe�Qjz1�zhj:

ð29Þ

The prefector i=ðvþ idÞ is the Fourier transform ofHeaviside step function, the minus sign in front arisesfrom the attractive interaction between the positive holecharge and the electronic charge density fluctuations in themetal, and VQ ¼ 2pe2=Q is a 2D Fourier transform ofthe Coulomb interaction e2=jRj. We have omitted fromexpression (28) the unscreened e–h exchange interaction[12, 13] because its role in the discussed situation of surfacebandswith small spatial overlapwas estimated to be ofminorimportance.

In the case when both the excited electron and the holeare localized well outside the surface and their chargeshave negligible overlap with the remaining electron densityin the system, the exponential functions in the integrandof (29) factorize. This leads to the quantum analog of theclassical image theorem [30, 16] that allows a simpledefinition of the surface response function in the exterior ofthe system (see next section). However, in a general situationof the holes excited in surface bands which overlap withthe bulk electronic density and the image plane this is notthe case. In this situationwemust examine expression (29) inits general form to assess temporal evolution of theeffective e–h potential. Since within the Kohn–LuttingerAnsatz [28] the hole motion is solved first, it is the temporalevolution of the hole density that causes the substrate chargedensity fluctuations which produce the induced potential(29). This imposes the use of retarded xQðz1; z2;vÞ in theevaluation of (29). To this end we introduce the spectralrepresentation

xQðz1; z2;vÞ ¼Z 1

0

dv0DQðz1; z2;v0Þ

� 1

v� v0 þ id� 1

vþ v0 þ id

� �;

ð30Þ

and use it to calculate the Fourier transform of (29) whichincorporates the time evolution of screened e–h interaction.Such a procedure yields a convolution of the suddenlyswitched on bare e–h potential [hence the appearance ofthe Fourier transform of the step function in (29)] withthe response function x, as it should within the linearresponse theory. This is shown schematically in Fig. 4 and inpractice amounts to integratingV

ðindÞe�h ðQ; ze; zh; t � t1Þ over t1

constrained to the interaction interval ð0; tÞ [10]. This



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produces a reactive (i.e. real or non-dissipative) retardedpotential acting on the excited electron

~VðindÞe�h ðQ; ze; zh; tÞ ¼ QðtÞ

Z t

0

dt1VðindÞe�h ðQ; ze; zh; t � t1Þ

¼ QðtÞZ

dz2

Zdz1V

2Qe

�Qjze�z2je�Qjz1�zhj

�Z 1

0

dv0N Qðz1; z2;v0Þð1� cosv0tÞ;

ð31Þ

where N Qðz1; z2;v0Þ ¼ 2DQðz1; z2;v0Þ=v0. To furtherobtain the effective potential which the hole imparts on theelectron in the approximation of frozen up hole wavefunc-tion [28], the RHS of (29) is multiplied by the hole densityjfSSðzhÞj

2and then integrated over zh.

Several important general features of the dynamicallyscreened e–h interaction can be readily deduced fromexpression (31). First, the induced potential (31) starts fromzero at t ¼ 0 and saturates for t ! 1 at a finite valuedetermined by

~VðindÞe�h ðQ; ze; zh;1Þ ¼ V2

Q

Zdz1

Zdz2e

�Qjze�z2j

�Z 1

0

dv0N Qðz1; z2;v0Þe�Qjz1�zhj:

ð32Þ

This expression is analogous to and plays the role of thecompressibility sum rule [31] or perfect screening sum rule[33] valid for probe charges residing in the interior or exteriorrelative to the surface, respectively. Its Fourier inversion intor -space yields the repulsive electron interaction with theelectronic polarization cloud induced by the hole. Second,any prominent peak of non-negligible weight in the spectraldensity N Qðz1; z2;v0Þ gives rise to attenuated oscillationsof ~V

ðindÞe�h ðQ; ze; zh; tÞ around the saturation value, irrespective

of the detailed structure of N Qðz1; z2;v0Þ. This will beillustrated in the next section in an exactly solvable model ofscreened surface exciton.

Expression (31) also enables the assessment of formationof the screening or image potential ~U

ðimÞe ðQ; ze; tÞ felt by an

electron that is promoted at instant t ¼ 0 to a distance ze inthe surface region. Following the above notation we find~UðimÞe ðQ; ze; tÞ ¼ � 1

2~VðindÞe�h ðQ; ze; ze; tÞ. This implies that the

screening and attenuation of primary e–h excitonic potentialand the formation of electron image potential proceed on thesame pace because they are controlled by the same excitationspectrum N Qðz1; z2;v0Þ. This point will be further elabo-rated in the next section.

5 Exact model solution of the dynamics ofexciton screening

5.1 Dynamical surface screening of e–hinteraction In order to learn about and understand betterthe physics of complex relaxation processes it is desirableto establish their descriptions within exactly solvablemodels whose solutions can be easily interpreted. A simple


illustration of the evolution of screening of e–h pairinteractions at surfaces can be given for excited quasipar-ticles that have negligibly small overlap with the chargedensity inside the metal substrate (external exciton). Atypical example of this case is photoexcitation from the statesand/or orbitals localized outside the substrate image planewhose position is determined by the centroid of the inducedcharge density. This model is of great relevance despite itsrelative simplicity because linear response formalismprovides exact description of screening of external pertur-bations owing to the validity of perfect screening sum rule[33] outside the image plane which defines the physicalsurface for electronic excitations. This situation alsomakes the application of Kohn–Luttinger approximation ofquasistatic hole charge [28] well justified.

The above described conditions appear to be approxi-mately satisfied on Ag(111) surfaces which support thesurface state- and image potential state-bands. According toexperimental evidence [35–37] all three low index surfacesof silver exhibit linear and positive (upward) surfaceplasmon dispersion, in contrast to the free-electron likemetal surfaces for which it is negative and well described injellium (Je) based models [38]. If, following the suggestionsof Rocca et al. [37] and analyses of Feibelman [39], the bulkand surface plasmons in Ag are thought of as collectiveexcitations split off the bottom of 4d–5s band of electron–hole excitations in the interior of the metal, then the positionof the effective image plane will shift to the region of 4dorbitals, i.e. inward relative to the region of occupied surfaceand unoccupied image potential states that are localizedoutside the surface by the bulk band gaps. On the otherhand, since small charge in SS-bands makes very smallcontribution to the response function (30), the relevantreference plane for SS-hole screening may be taken to bethe same image potential plane that is shifted towards theinterior by the presence of 4d-bands. Analogous argumentscould also be extended to the Cu(111) surface (cf. Figs. 1 and3 in Ref. [16] for the localization of SS- and IS-wavefunc-tions and for positive surface plasmon dispersion,respectively).

Within these boundary conditions the limit of (28) for zeand zh lying outside the image plane yields a selfconsistente–h potentialVðeffÞ

e�h which in the ðQ; z;vÞ representation takesthe form [30, 33]

VðeffÞe�h ðQ; ze;vÞ ¼ V

ðeffÞe�h ðQ; ze;vÞ þ V

ðimÞe�h ðQ; ze;vÞRQðvÞ:

ð33Þ

Here VðeffÞe�h ðQ; ze;vÞ is a 2D Fourier transform of e–h

potential VðeffÞe�h ðr ; ze; tÞ of the type appearing on the LHS of

Eq. (25). This is a definite approximation close to the imageplane, but the one enabling good insight into underlyingphysics before the large scale computations based onexpression (28) that describes a general situation areperformed [47].

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To derive the total potential (33) we first determine thebare e–h effective potential V

ðeffÞe�h ðQ; ze;vÞ and the polariz-

ation interaction VðimÞe�h ðQ; ze;vÞ from the ðQ;vÞ-Fourier

transforms of VðeffÞe�h ðr ; ze; tÞ ¼ V

ðeffÞe�h ðr ; zeÞQðtÞ and of

VðimÞe�h ðr ; ze; tÞ ¼ V

ðimÞe�h ðr ; zeÞQðtÞ, respectively (a more

complicated time dependence of the interaction is brieflydiscussed at the end of this subsection). Note thatVðeffÞe�h ðr ; zeÞ is the same interaction that enters Eq. (25),

and the interaction of the hole with the electron image isgiven by the image plane reflected expression [30]

www

VðimÞe�h ðr ; zeÞ ¼ �e2

Zdzh

jfhðzhÞj2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 þ ðze þ zhÞ2q ; (34)

where fhðzhÞ is here assumed to have negligible overlapwith the substrate bulk wavefunctions. The remainingingredient is the surface response function RQðvÞ obtainedfrom xQðz1; z2;vÞ (cf. Appendix B in Ref. [16]) andexpressed through its spectral representation [33]

RQðvÞ ¼Z 1

0

dv0SQðv0Þ

� 1

v� v0 þ id� 1

vþ v0 þ id

� �; (35)

where SQðv0Þ is the spectral weight of surface electronicexcitations of wavevector Q and energy v0.

Transforming the whole expression (33) back to realtime representation and obeying the retarded character of thescreening process as in Section 4, we find

VðeffÞe�h ðQ; ze; tÞ ¼ V

ðeffÞe�h ðQ; zeÞQðtÞ � V

ðimÞe�h ðQ; zeÞQðtÞ

�Z 1

0

dv0NQðv0Þ½1� cosðv0tÞ�:

ð36Þ

Here NQðv0Þ ¼ 2SQðv0Þ=v0 plays the role of surfaceexcitation spectrum that satisfies perfect screening sum ruleR10

dv0NQ¼0ðv0Þ ¼ 1 [33].The second term on the RHS of (36) can be visualized as

a FT of the (retarded) dynamic image mediated interactionbetween the photoexcited electron and hole at lateralseparation r . This is in full analogy with the screening ofexcitonic potentials in bulk systems [10] and it is gratifyingthat equivalent expressions (in terms of appropriate quantumnumbers) are obtained in the interior and exterior regions ofthe dielectric. Hence, expression (31) intrapolates betweenthe two limits across the atomically narrow interface. In theexterior the peculiar transient behaviour of the inducedpotential is governed by a transient factor in the second termon the RHS of (36), viz.

RQðtÞ ¼Z 1

0

dv0NQðv0Þ½1� cosðv0tÞ�; (37)

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and analogously so in the interior of bulk systems (cf.Eq. (13) in Ref. [10]). Thereby the initial many-bodyproblem of the reactive component of screening isformulated as an effective time-dependent one-electronpotential problem. Transformation of (36) back intor -space and substitution into (25) in place of V

ðeffÞe�h leads

to a highly non-stationary one-particle time-dependentSchrodinger equation governing the evolution of excitonicwavefunction Fðr ; ze; tÞ. It should be noted that such asimple form of VðeffÞðQ; ze; tÞ here follows from theassumption of sudden switching on of V

ðeffÞe�h ðQ; ze; tÞ and

VðimÞe�h ðQ; ze; tÞ that both exhibit the time dependence/ QðtÞ,

and the form of RQðtÞ employed in going from (33) to (36)prior to the treatment of quasiparticle recoil and decay fromstationary states [16, 17, 34].

In order to demonstrate temporal evolution of the(dominant) long wavelength contribution to inducedpotential in (36) at real metal surfaces we shall use in (37)semi-empirical forms of the surface excitation spectraSQðv0Þ obtained earlier (cf. Fig. 1 in Ref. [40, 41] for Cusurface and Fig. 1 in Ref. [42] for Ag surface). The behaviourof RQ¼0ðtÞ for the case of electron promotion in front ofAg(111) and Cu(111) surfaces is illustrated in Fig. 5. It isseen that upon the excitation at t ¼ 0 the electron issolely affected by the bare potential V ðeffÞðQ; zeÞ becauseRQðt ¼ 0Þ ¼ 0, whereas after a sufficiently long interval[t > 15 fs for Ag(111) and t > 3 fs for Cu(111)] the totalselfconsistent potential saturates and reaches the limit of therelaxed or screened potential

VðrelaxÞe�h ðQ; zeÞ ¼ V

ðeffÞe�h ðQ; zeÞ � V

ðimÞe�h ðQ; zeÞ: (38)

One of the important messages conveyed by Fig. 5 is thatdue to the different electronic excitation spectra typifying theAg(111) andCu(111) surfaces the transient characteristics ofimage screening of e–h pair interactions at these surfaces arealso notably different. In the case of Ag surface the imagescreening is dominantly governed by virtual excitations ofsurface plasmons of energy �hvs ’ 3:8 eV [42], whereas onCu surface the screening process or the saturation of imagecharge is a much faster process because it is dominantlygoverned by higher energy interband excitations [40, 41].

A comparison of the time dependence in Fig. 3 with theones shown in Fig. 5 indicates that on Cu(111) surface thepopulation saturation of excitonic levels is an equally fastprocess as the renormalization of the e–h pair interaction bysurface screening. On the other hand, on Ag(111) surface thesaturation of population is a notably faster process than thescreening of bare e–h interaction.

Wemay summarize the above findings by noting that theprimary excitonic energy levels and wavefunctions will inthe absence of other dynamical interactions evolve from theeigenvalues and eigenfunctions of (25) to less binding andmore delocalized levels and wavefunctions determined bythe screened e–h potential (38). It may also be observed thatat large e–h separations the total static potential (38) acquires



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2 4 6 8 10 12 14t [fs]

0.2

0.4

0.6

0.8

1.0

1.2

Transient factor (Ag)

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2

t [fs]

Transient factor (Cu)

Figure 5 (online colour at: www.pss-b.com) Top panel: transientfactor RQ¼0ðtÞ appearing in the second term on the RHS of (36) thatdeterminesevolutionof the imagecomponentofe–hpotential actingon the electron upon its promotion in front of Ag(111) surface. Thetransient oscillatory behaviour of image screening is caused byvirtual excitation of surface plasmon which is a well definedexcitation in this system (cf. surface excitation spectrum SQðv0ÞofAg shown in Fig. 1 of Ref. [42]). Bottompanel: same for Cu(111)surface.Here the screening ismuch faster because it is dominated byhigher energy interband transitions (cf. surface excitation spectrumSQðv0Þ of Cu shown in Fig. 1 of Ref. [41]).

a dipolar form:

� 20

limr!1

VðrelaxÞe�h ðr ; zeÞ ¼ � 2e2zhze

r3; (39)

with the strength determined by the image of hole dipole�2ezh depending on the separation of hole density centre ofmass zh from its image at �zh [note the difference withrespect to (23)].

Lastly, we observe that expression (33) offers a moregeneral description of the screening of bare excitonicinteraction. Since in the pump pulse-induced excitation ofa primary e–h pair the hole is created in a pre-existent state itssubsequent decay will cause a time dependent depletion ofthe initial hole charge density appearing in the integrand onthe RHS of (22). This depletion process can be described bythe hole survival probability LhðtÞ which for SS-band holeswas elaborated in Refs. [16, 17]. Hence, replacing the Kohn–Luttinger frozen up hole density with the one described bythe hole survival probability, the screening of e–h interactionmay be described by a version of expression (33) inwhich thev-dependence of thematrix elementsV

ðeffÞe�h andV

ðimÞe�h is given

by the Fourier transform of LhðtÞQðtÞ instead of the potentialswitching step function QðtÞ only. However, both


approaches require the solution of the same primary excitonproblem.

The validity of the above arguments for the screening ofe–h interaction outside the image plane should be alsoapplicable without resorting to the effective e–h barepotential (22) in which the hole wavefunction is frozen up(i.e. treated adiabatically after the hole has been photo-excited). Following the analogy with expression (36) andwithout resorting to preassumed states, the time-dependentinteraction between electron and hole point charges locatedat ze and zh outside the surface, respectively, can be writtendirectly in the ðQ; z; tÞ-space in terms of the Coulombmatrixelements VQ as

Ve�hðQ; ze; zh; tÞ ¼ Ve�hðQ; ze; zhÞQðtÞ

þ ~VðimÞe�h ðQ; ze; zh; tÞ;

(40)

with

Ve�hðQ; ze; zhÞ ¼ �VQe�Qðjze�zhjÞ; (41)

and

~VðimÞe�h ðQ; ze; zh; tÞ ¼ VQe

�QðjzejþjzhjÞQðtÞ

�Z 1

0

dv0NQðv0Þ½1� cosðv0tÞ�;ð42Þ

and then transformed back into the ðr ; zÞ-space and theresult substituted on the RHS of expression (17) to replacethe bare (unscreened) e–h interaction. Thereby the problemof screened external exciton is modelled by a two-particleHamiltonian involving a time-dependent interparticlepotential.

5.2 Dynamical image screening of the excitedelectron and hole Photoexcition of an electron–hole pairin front of the surface also switches on individualinteractions of constituent quasiparticles with the electronicresponse of the system. This induces dynamical imagescreening of each quasiparticle. In the present formulation ofexternal excitons such processes can be treated on the samefooting as the screening of e–h interaction discussed inSection 5.1. In the field-theoretical description these effectsmanifest themselves through the quasiparticle self-energycorrections [34]. Specifically, an electron excited in theregion outside the surface image plane will experience itsown quantum image potential which in the mixed ðQ; ze;vÞ-representation is given by [33]

UðimÞe ðQ; ze;vÞ ¼ UðQ;�ze;vÞRQðvÞ; (43)

where UðQ;�ze;vÞ is the ðQ;vÞ-Fourier transform of thesuddenly switched on Coulomb image potential UðQ;�zeÞfor the electron located at ðr ; zeÞ. Following the argumentsof Section 5.1, we find that the evolution of image potentialis described by

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4 6 8 10ze (a.u.)

0.8

0.6

0.4

0.2

V exctot ze,t a.u.

Figure 6 (online colour at: www.pss-b.com) Illustration of thetemporal development of total time-dependent electron potentialenergy Vexc

tot ðr ; ze; tÞ ¼ ~VðimÞe�h ðr ; ze; zh; tÞ þ 1=2U

ðimÞe ðr ; ze; tÞ ap-

pearing on the RHS of expression (46) for the hole at 1 a.u. outsidethe image plane on Ag(111) surface and mutual e–h separationr ¼ 1 a:u: Full line: t ¼ 0 (primary excitonic potential). Longdashed line: t ¼ 2:25 fs (combination of partly screened e–h andincompletely developed e-image potential energy contributions).Short dashed line: t ¼ 1 (fully relaxed electron potential energywith prevailing electron image contribution). The short dashed linereaches asymptotically (beyond ze ¼ 10 a:u: measured from theimage plane) the classical electron image energy �e2=4ze shownby the uppermost thin full line.

UðimÞe ðQ; ze; tÞ ¼ �UðQ;�zeÞQðtÞ

�Z 1

0

dv0NQðv0Þ½1� cosðv0tÞ�:ð44Þ

Note that this interaction is responsible for the evolutionof unrelaxed initial set of eigenstates jK; jiofHi, over into thefinal set of relaxed eigenstates jK; ni of Hf discussed inSection 2.1. A completely analogous procedure can befollowed to obtain dynamical image potential U

ðimÞh ðQ; zh; tÞ

of the hole photoexcited outside the image plane.We note in passing that in the present linear screening

theory the long time limits of polarization mediatedquasiparticle interactions derived in Sections 4 and 5 donot exhibit any Debye–Waller factor (DWF) type ofrenormalizations [44, 45]. This is in contrast to thequasiparticle propagators which exhibit the DWF effectsupon renormalization by the interactions with the samebosonized polarization fields [5, 16].

5.3 Representation of exciton screening byan effect ive t ime-dependent two-bodyHamiltonian The screening dynamics of e–h potential(36), as well as of the image potentials of external individualquasiparticles, is factorized from thematrix elements of bareinteractions. These expressions may now be used in a simplemodelling of the screening induced energy relaxation of ane–h pair photoexcited outside the surface image plane. Inthis picture the e–h interaction as well as the individualquasiparticle interactions with the surface response arerepresented by effective time-dependent potentials whosemagnitudes are varying on the ultrashort time scale. Hence,to assess the temporal variation of primary excitonicenergy levels we can formulate an effective two-bodytime-dependent excitonic Hamiltonian describing relaxationdynamics of an e–h pair subsequent to its optical excitationat, say, t ¼ 0. This effective Hamiltonian takes the form

www

HexcP¼0ðtÞ ¼ Hexc

P¼0 þ VðimÞtot ðtÞ (45)

with HexcP¼0 given by expression (17), and

VðimÞtot ðtÞ ¼ ~

VðimÞe�h ðr ; ze; zh; tÞ

þ 1

2UðimÞ

e ðr ; ze; tÞ þ1

2U

ðimÞh ðr ; zh; tÞ;

ð46Þ

where the terms on the RHS of (46) are given by ther -Fourier transforms of the respective expressions in theQ-space. At t ¼ 0 expression (45) describes the primaryexciton because V

ðimÞtot ðt ¼ 0Þ ¼ 0, and in the limit of full

relaxation, t ! 1, the Hamiltonian (45) tends to theexpression given by Eq. (2) of Ref. [43]. This is illustrated inFig. 6. It should be noted that the thus formulated time-dependent Hamiltonian (45) does not incorporate dissipa-tive effects. The latter may be incorporated a posteriori

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following the methods of Ref. [16] once the quasistationarystate is reached.

The e–h dynamics governed by the effective Hamiltonian(45) can be visualized as the evolution of excited e–h pair thatis initially dominated by interband e–h interaction whichrapidly weakens due to dynamic screening. This gives way tothe domination of individual interactions of quasiparticleswith their developing screening charges as described by thelast two terms on the RHS of (46). When the asymptotic limitof fully screened total interaction is reached (given by the sumof non-binding or very weakly binding e–h potential and fullydeveloped electron and hole image potentials), the individualquasiparticle states will be well described by the final stateone-particle Hamiltonian (4). In other words, the dynamics ofexcited e–h pairs evolves from propagation in early excitonic,strongly correlated states, over into the regime of electron andhole propagation in the states of developed IS-band andrelaxed SS-band, respectively. In this context the relaxedelectron image potential energies may be viewed as excitoniclevels for which the role of ‘exciton hole’ has in the course ofscreening been taken from the SS-hole and transferred to theemergent electron image screening charge induced at thesurface (i.e. to the excited electron’s exchange correlationhole). Once the stationary state of the electronic systemdescribed by the eigenstates of (4) has been attained, theelectron decoherence and decay from stationary statesmay bedescribed by the formalisms of Refs. [16, 17, 46].

6 Discussion of the dynamics of transientexcitonic states The above developed model description



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of screening of quasiparticles at surfaces provides basicinsight into the early evolution of energetics of intermediateelectronic states probed in time and energy resolved pump–probe investigations of the surface electronic structure anddynamics. It is argued that in the case of pump–probespectroscopies of occupied surface bands on Cu(111) andAg(111) surfaces the pump-induced primary electronictransitions can be viewed as creation of transient excitonicstates whose energy levels lie in the surface projected bandgaps. The subsequent evolution of primary excitonic levels isgoverned by the dynamics of screening processes at thesurface. Reactive (i.e. non-dissipative) interactions ofphotoexcited quasiparticles with the developing screeningcharge renormalize the initial bare e–h interaction and giverise to the formation of relaxed image potential bands inconcurrent processes.

This approach offers a dynamical picture of theintermediate states and energy levels in pump–probespectroscopies of surface bands. These levels arise as boundstates of the effective time dependent potential (46) whichdue to dynamical screening evolves from the initial excitonicpotential dominated by the unscreened Coulomb attractionof photoexcited SS-hole over into the relaxed or saturatedimage potential caused by the screening charge developed atthe surface. In that sense both the primary excitonic statesand the relaxed image potential states can be viewed asbound excitonic states of a potential that is initially governedby the SS-hole and finally by the exchange-correlation hole(i.e. image charge) of individual quasiparticles. In theKohn–Luttinger approximation [28] employed in Section 2.3 tocalculations of primary excitonic levels this implies diabatictransitions elðt ! 1Þ ! en where en are energies of imagepotential states calculated inRef. [24].Ageneral formulationof the relaxation dynamics of excitonic states based on thesurface response outlined in Section 4 will be furtherelaborated elsewhere [47].

To facilitate discussions of the characteristics oftransient excitonic states we introduced in Section 5 asimplified, yet highly illustrative model of screening of e–hpairs excited outside the substrate surface. This constraint onquasiparticle charge distributions allows closed formrepresentations of the interactions which drive the primaryexcitonic states into the states of relaxed image potential.Special merit of this approach lies in easy visualisation of thedynamics of transient excitonic states at surfaces, andthereby of the role these states may play in ultrafast pump–probe experiments. Although the validity of solutions of sucha simplifiedmodel is restricted to exterior of the image plane,the general physical implications deriving thereof shouldbe extendable beyond the regions of formal applicability ofthe model itself. Thus we find that the screening of bare e–hpair interaction outside the image plane and the formation ofindividual quasiparticle image potentials proceed at thecommon pace. On Ag(111) the relaxation during whichtransient excitons fade and the image potential bands arebeing formed is completed within an interval of the order of�15 fs, whereas on the Cu(111) this time is much shorter, of


the order of �3 fs (cf. Fig. 5). This strong difference arisesfrom the markedly different structures of surface electronicexcitation spectra which govern the dynamics of screening atrespective substrate surfaces. In the spirit of Section 2 thiscan be rephrased in that the evolution of effective one-particle Hamiltonians Hi into Hf is a much faster process onCu(111) than on Ag(111) surface.

We have also investigated the role of ultrashort pumppulse duration on the dynamics of population of primaryexcitonic states. For the parameters characteristic of thestudied systems and the currently available ultrashort laserpulses of FWHM �3 fs which may still provide requiredresolution in interferometric experiments [1, 48], thesaturation of population of a primary excitonic level occursin the interval of�4–5 fs. This is comparable to the durationof formation of image potential bands on Cu(111) surfaceand means that for pump–probe delays exceeding thistime the probe pulse would measure occupation of thestates in the already relaxed image potential bands, with allthe implications regarding the existence of well definedintra- and inter-band transitions among the stationarystates deriving from the relaxed Hexc

P¼0ðt ! 1Þ, the ensuingnon-adiabatic and dissipative processes governing thedecay rates and lifetimes of excited states, etc. On theother hand, in the case of Ag(111) surface the requiredinterval may exceed �15–18 fs because of a much slowerscreening of excitonic interaction and saturation of the imagepotential. That altogether permits longer duration oftransient excitonic states and slower formation of imagepotential bands.

The third factor which limits direct observation of the‘standard’ features of quasiparticles in ultrafast experiments,viz. their energy and exponential decay governed by lifetimeeffects, is the early evolution of quasiparticles after beingpromoted into the established or pre-existent quasistationarystates. According to our recent calculations [16, 17, 46] theMarkovian regime of exponential decay and phase saturationof quasiparticles in surface bands on Cu(111) is reachedwithin first 6–7 fs after the photoexcitation. This imposespump–probe delays which exceed that interval, otherwisethe transient, early excited particle features are likely to bedetected.

The effects of ultrafast excitation and relaxationprocesses outlined above will effectively appear convolutedin the amplitudes of 2PPE events, viz. integrated over allintermediate times in the diagrams of the form illustrated inFig. 1b. Therefore, in order to assess quasiparticle energiesand lifetimes in relaxed quasistationary states, the delaytime between the applied pump and probe pulses shouldexceed the three discussed temporal intervals. On the otherhand, for shorter delays the experiment may detect non-relaxed or transient intermediate states, in particular on theAg(111) surface. The latter system poses a challenge topossible detection of transient excitonic and early quasi-particle states as intermediate states in time resolved 2PPEand interferometric 2PPEþ 1PPE experiments [1, 48]. Oneof the goals of the present discussion is to motivate such

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Original

Paper

experiments and the development of a unified theory for theirinterpretation.

Note added in proof: Recent measurements haveprovided evidence for positive (upward) surface plasmondispersion also onAu(111) surface [49]. Hence, themodel ofdynamical screening of ‘‘external exciton’’ outlined inSection 5 should be applicable to this surface as well.

Acknowledgements This work has been supported in partby theMinistry of Science, Education and Sports of the Republic ofCroatia through theResearchProjects nos. 035-0352828-2839, 098-0352828-2863 and 098-0352851-2921.

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