Role of the intrinsic surface state in the decay of image states at a metal surface

10 Pages·0.32 MB·English

by  J. Osma

Role of the intrinsic surface state in the decay of image states at a metal surface
J. Osma1, I. Sarr´ıa2, E. V. Chulkov1, J. M. Pitarke2, and P. M. Echenique1,3
1 Departamento de F´ısica de Materiales, Facultad de Ciencias Qu´ımicas, Universidad del Pa´ıs Vasco/Euskal Herriko
Unibertsitatea, Aptdo. 1072, 20080, San Sebasti´an, Basque Country, Spain
2 Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta kutxatila,
48080 Bilbo, Basque Country, Spain
9
9 3 Unidad Asociada al Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cient´ıficas,
9 Cantoblanco, 28049 Madrid, Spain
1 (February 1, 2008)

n
a
J During the last decade the linewidth of image states
1 The role of the intrinsic surface state (n = 0) in the has been measured by inverse photoemission5,6, two-
2 decay of the first image state (n = 1) at the (111) surface photon photoemission7–9, and time-resolved two-photon
of copper is investigated. Inelastic linewidths are evaluated photoemission10–14. Recently, time-resolved two-photon
] fromtheknowledgeoftheimaginarypartoftheelectronself-
i photoemission has been used in combination with the
c energy, which we compute, within the GW approximation of
coherent excitation of several quantum states, and the
s many-body theory, by going beyond a free-electron descrip-
- lifetime of the first six image states on the (100) surface
tion of the metal surface. Single-particle wave functions are
rl obtained bysolving theSchr¨odingerequation with arealistic of copper has been accurately determined15.
t Theoretical calculations of the linewidth of image
m one-dimensionalmodelpotential,anddepartureofthemotion
states were first reported in Refs. 16 and 17, within a
along the surface from free-electron behaviour is considered
.
t through the introduction of the effective mass. The decay many-body free-electron description of the metal sur-
a
m of the first image state of Cu(111) into the intrinsic surface face and with the use of simplified models to approxi-
state is found to result in a linewidth that represents a 40% mate both initial and final electronic states and, also,
-
d of the total linewidth. The dependence of linewidths on the the screened Coulomb interaction. Later on, the decay
n momentum of the image state parallel to the surface is also ofthefirstimagestateonthe(111)surfacesofcopperand
o investigated. nickel metals to the n = 0 crystal-induced surface state
c was calculated18, in terms of Auger transitions,with the
[
use of a three band model to describe the surface band
1 I. INTRODUCTION structure. InRef.18,hydrogenic-likestateswithnopen-
v etration into the solid were used to describe the image-
1 The presence of an electron in front of a solid surface state wavefunctions, a simplifiedparametrisedformwas
2 redistributes the charge in the solid. As a consequence, used for the surface-state wave functions, and screening
2
anattractivepotentialisinduced,whichfarfromthesur- effects were neglected. Self-consistent calculations of the
1
0 face approaches the long-range classical image potential linewidths of image states on copper surfaces have been
9 Vim(z)= gs/4z (z being the distance from the surface, reported recently19, and good agreement with experi-
9 g =(ǫ −1)/(ǫ +1), and ǫ the static bulk dielectric mentallydetermineddecaytimeshasbeenfound. InRef.
s s s s
/ constant−; for a metal, g = 1). If the bulk band struc- 19,thelinewidths ofimagestateswerecomputed,within
t s
a ture projected onto the surface presents an energy gap theGWapproximationofmany-bodytheory20,bygoing
m
near the vacuum level, an electron located in front of beyond a free-electron description of the metal surface.
- the surface cannot propagate into the solid. Therefore, Single-particle wave functions were obtained by solving
d
the electron may be trapped in the vacuum well, and an theSchr¨odingerequationwitharealisticone-dimensional
n
o infinite series of Rydberg-like states appears, which con- modelpotential21,andthescreenedinteractionwaseval-
c verges,for zeroparallelmomentum,towardsthe vacuum uated in the random-phase approximation (RPA)22.
: energy. These so-called image states1–3 are localized in In this paper we focus our attention on the role that
v
i the vacuum region of the surface, the penetration of the the crystal-induced surface state (n = 0) plays in the
X first (n = 1) image wave function into the solid varying relaxation of the first image state at the (111) surface
r typicallybetween4%and22%4. Asaresult,imagestates of copper, which we find to represent a 40% of the to-
a are almost decoupled from bulk electron scattering, and tal linewidth. We present self-consistent calculations
they are much longer lived than bulk excitations: The along the lines of Ref. 19, and we also consider simpli-
lifetime of bulk electrons with energies of 4 eV above fied models for both the electronic wave functions and
the Fermi level is approximately one order of magnitude the screened Coulomb interaction, showing that a de-
smaller than that of the first image state. Furthermore, tailed description of these quantities is of crucial impor-
thelifetimeofhigherorderimagestates(n 1)hasbeen tance in the understanding of the origin and magnitude
predicted1 to scale asymptotically with n3,≥which makes of linewidths of image states. We account for potential
the series to be resolvable. variationparalleltothesurfacethroughtheintroduction
of the effective mass, and we find that the linewidth of
1
thefirstimagestateofCu(111)is20%smallerthaninthe Three different models have been used for the evalua-
case of free-electron behavior along the surface. Finally, tion of the screened interaction, W. First, the specular-
we investigate the dependence of interband23 linewidths reflection model (SRM) of Ritchie and Marusak25 has
on the momentum of the image state parallelto the sur- been considered. In this model, bulk electrons are as-
face, k . Our results indicate that for image state total sumed to be specularly reflected at the surface, the in-
k
energies lying below the top of the gap, the linewidth of terferencebetweentheingoingandoutgoingwavesbeing
the first image state of Cu(111) is increased with k =0 neglected,andtheelectronicchargedensityabruptlyter-
k
6
up to a 20%. minatesatthe surface(z =0),whichwe choosetobe lo-
catedhalf a lattice spacingbeyond the lastatomic layer.
Within this simplified model26, also called semiclassical
II. THEORY infinite barriermodel (SCIBM), the screenedinteraction
is obtained in terms of the wave-vector and frequency
We assume translational invariance in the plane par- dependent bulk dielectric function, which we evaluate in
allel to the surface, which is taken to be normal to the the RPA. Secondly, for the vacuum contribution to the
z-axis,and we evaluate the inelastic linewidth ofthe im- linewidth (z > 0, z′ > 0) the surface response function
agestateφ1(z)eikk·rk withenergyE1 =ǫ1+k2k/(2m1)24 suggestedbyPerssonandZaremba27(PZ)hasbeenused.
Finally,thescreenedinteractionhasbeenevaluated,asin
(we use atomic units throughout, i.e., e2 =h¯ =m =1),
e
Ref.19,bysolvingtheRPAintegralequationfortheden-
as the projection of the imaginary part of the electron
sity response function of inhomogeneous media in terms
self-energy, Σ(r,r′;E ), over the state itself:
1
of the eigenfunctions of the one-electroneffective Hamil-
tonian. These eigenfunctions have been computed by
Γ=−2Z dzdz′φ∗1(z)ImΣ(z,z′,kk;E1)φ1(z′), (1) solving the Schr¨odinger equation with the realistic one-
dimensional model potential suggested in Ref. 21. This
where Σ(z,z′,k ;E ) represents the two-dimensional model potential uses as parameters the width and posi-
k 1
Fourier transform of Σ(r,r′;E1). tion of the energy gap at the Γ¯ point (kk =0) and, also,
In the GW approximation20, the self-energy is ob- the binding energies of both the n = 0 crystal-induced
tained by just keeping the first term of the expansion surface state at Γ¯ and the first image state.
in the screened interaction (W). Then, after replacing Fortheevaluationofn=0andn=1surfacestates,we
the Greenfunctionby the zeroorderapproximation,one have first used simplified models for the wave functions
finds inside and outside the solid. In the vacuum side of the
surface (z > 0), n = 1 and n = 0 states have been
ImΣ(z,z′,k ;E )= d2qk φ∗(z′) approximatedbyaparametrised1s-likehydrogenicwave
k 1 Z (2π)2 f functionandamereexponential,respectively,whichhave
EF≤XEf≤E1 been matched to a decaying wave function in the crystal
ImWind(z,z′,qk;E1 Ef)φf(z), (2) band-gap (z < 0) obtained within a nearly free electron
× −
two-band model17:
where the sum is extended over a complete set of fi-
nal states φf(z)ei(kk+qk)·rk with energies Ef = ǫf + φn(z <0) e∆nzcos(Gz+δn). (5)
∼
(k +q )2/(2m ). m is the effective mass, which ac-
k k f f
Here,n=0andn=1correspondto the crystal-induced
counts for the departure of the motion along the surface
surface state and the first image state, respectively, G is
from free-electronbehavior,E is the Fermi energy, and
F
Wind(z,z′,q ,E)representsthetwo-dimensionalFourier the limit of the Brillouin zone in the direction normalto
k
the surface, and
transformoftheinducedpartofthescreenedinteraction.
In particular, if only transitions into the crystal-
1 1
Eindu=ceǫd+n=(k0+suqrfa)c2e/s(t2amte)φa0(rze)ceoi(nkski+dqekr)e·drk, ownitehfienndesrgy ∆n = Gr4Eg2ap−ǫ¯2n, (6)
0 0 k k 0
where E and ǫ¯ represent the energy gap and the en-
gap n
ImΣ (z,z′,k ;E )= d2qk φ∗(z′) ergy of the n surface state with respect to the midgap,
s k 1 Z (2π)2 0 respectively. The phase shift δn is given by
ImWind(z,z′,q ;E E )φ (z), (3)
ainntdo Einqt.ro(d1u)coinn×egfitnhdiss:contributikon 1to−Im0Σ(z0,z′,kk;E1) δn = 21 ×πta−n−1tahnq−1η1n2hq−η11n2i−, 1i, iiff −10≤≤ηηnn≤≤01,

(7)
d2q
Γ = 2 k dzdz′φ∗(z)φ∗(z′)
s − Z (2π)2 Z 1 0 with ηn =2ǫ¯n/Egap.
Theimagestate onCu(111)is locatedrightatthe top
ImWind(z,z′,q ;E E )φ (z)φ (z′). (4)
× k 1− 0 0 1 ofthegap(δ1 0.9 π/2),boththehydrogenic-likewave
≃ ×
2
function in the vacuum (z > 0) and the decaying s-like structurecalculations. Asforthen=0surfacestate,we
wavefunctioninthebulk(z <0)having,therefore,nodes have used m = 0.42, as in Table I, and for bulk states
0
atthesurface(z =0). Then=0crystal-inducedsurface we have chosen to increase the effective mass from our
stateonCu(111)islocatedatthebottomofthegap(δ computedvalue33 ofm =0.22atthe bottomofthe gap
0 f
≃
0.2 π/2); thus,it is describedby a p-likewavefunction to m =1 at the bottom of the valence band.
f
×
in the bulk. These approximate wave functions (AWF) Our full RPA calculations indicate that the decaying
are exhibited in Fig. 1, together with the corresponding rate of the n = 1 image state into the n = 0 crystal-
wavefunctionsthatweobtainbysolvingtheSchr¨odinger induced surface state results, for m = 1, in a linewidth
0
equation with the one-dimensional potential of Ref. 21 of 16 meV, while use of the more realistic effective mass
(MWF). Bothwavefunctions,AWF andMWF, coincide m =0.42 leads to a linewidth of 12meV. With the use
0
withinthebulk,butthehydrogenic-likewavefunctionfor ofeither the free-electronmassor morerealisticeffective
then=1imagestateappearstobelesslocalizednearthe masses for both bulk and crystal-induced surface states,
surface than our model wave function. The n = 0 and Γ approximatelyrepresentsa40%ofthetotallinewidth,
s
n = 1 surface states on Cu(111) have binding energies Γ = 37meV (m = 1) or Γ = 29meV (m = 1). The
f f
6
(measured with respect to the vacuum level) of 0.83 and morerealisticresultofΓ=29meVforthetotallinewidth
5.32 eV, respectively. The n=1 probability-density has is in good agreement with the experimentally measured
a maximum at 4.3 a.u. outside the crystal edge (z =0). lifetime35of22 3fsat25K13,14. Withinthevacuumside
±
Thepenetrationintothebulkofn=0andn=1surface of the surface the n=1 image state couples dominantly
statesisfoundtobe,attheΓ¯ point,of74.5%and22.1%, to the n = 0 surface state (this coupling approximately
respectively. represents a 90% of the total Γ linewidth); however,
vac
the coupling of image states with all bulk crystal states
occurring through the bulk penetration plays an impor-
III. RESULTS AND DISCUSSION tant role and cannot, therefore, be neglected if one is to
accurately describe the lifetime of image states.
The results of our calculations for the linewidth, Γ , We note that simplified jellium models for the evalua-
s
coming from the decay of the n=1 image state into the tionofthescreenedinteractionleadtounrealisticresults
n = 0 intrinsic surface state on Cu(111) are presented for the contribution of the surface state to the linewidth
in Table I, with the momentum of the image electron ofimage states36. First, we compareour full RPA calcu-
parallel to the surface, k , set equal to zero. Here, the lations(seeTableI)withtheresultsweobtain,alsowith
k
linewidth has been split as follows: useofourmodelinitialandfinalwavefunctions(MWF),
whenourrealisticscreenedinteractionisreplacedbythat
Γ =Γ +Γ +Γ , (8) obtainedwithinthespecularreflectionmodel(SRM)and
s vac sol inter
the model of Perssonand Zaremba (PZ). Bulk contribu-
where Γvac, Γsol and Γinter represent vacuum, bulk and tions to the linewidth are approximately well described
interference contributions, respectively, as obtained by within the specular reflection model, small differences
confining the integrals in Eq. (4) to either vacuum resulting from an approximate description, within this
(z > 0,z′ > 0), bulk (z < 0,z′ < 0) or vacuum-bulk model,oftheso-calledbegrenzungeffects. Astheapprox-
(z>0,z′<0) coordinates. First we show our full RPA imate treatment of Ritchie and Marusak25 ignores the
< >
calculations, in which the screened interaction is ob- quantum mechanical details of the surface, this model
tained on the basis of one-electron eigenfunctions com- fails to describe both vacuum and interference contri-
puted from the realistic one-dimensional model poten- butions to the linewidth. These quantum mechanical
tial of Ref. 21. Within these calculations both n = 0 details of the surface are approximately taken into ac-
and n=1 surface-statewave functions are also obtained count within the jellium model of Ref. 27, thus result-
from the model potential of Ref. 21 (MWF), with either ing in a better approximation for the vacuum contri-
m0 = 1 or m0 = 0.4228–30. Within the specular reflec- bution to the linewidth. Discrepancies between vacuum
tion model31 and the model suggested by Persson and contributions obtained within this model (PZ) and our
Zaremba32 forthescreenedinteraction,wehaveusedthe more realistic full RPA calculations36 appear as a result
MWF n = 0 and n = 1 surface-state wave functions of the jellium model of Ref. 27 being accurate provided
as well as the simplified models (AWF) described in the q /q and ω/E <<1 (q is the Fermi momentum, i.e.,
k F F F
previous section, with m0 =1. EF =qF2/2).
Total linewidths, Γ, as obtained from the decay of the Inordertoinvestigatethe dependenceofΓ onthede-
s
n = 1 image state on Cu(111) into any final state with tails of both n=1 andn=0 wavefunctions, we present
energy Ef, EF Ef E1, (see Eqs. (1) and (2)), are in Table I calculations, within SRM and PZ models for
≤ ≤
presentedinTableII.Here,ourfullRPAcalculationsare thescreenedinteraction,inwhichourrealisticwavefunc-
shown, with all wave functions computed from the one- tions are replaced by the simplified models (AWF) de-
dimensionalmodelpotentialofRef.21(MWF).Realistic scribed in the previous section. As the hydrogenic-like
values ofthe effective massof finalstates havebeen con- wave function used to describe the n=1 image state on
sidered,accordingto the experimentortoab initioband
3
the vacuum side of the surface presents an image-state mentumtransfer,whichmayresultinboth enlargedand
chargegravitycenterlocalizedfurtherawayfromthesur- diminished screened interactions, depending on momen-
face than our more realistic model wave function, both tum and energy transfers. This is illustrated in Fig. 3,
vacuum and interference contributions to the linewidth wheretheimpactoftheintroductionoftheeffectivemass
are largely underestimated within this approximation. on the evaluation of the various contributions (vacuum,
Furthermore, we note that the linewidth is highly sensi- bulk, and interference) to both Γ and Γ (see Eq. (9))
f s
tivetothedetailsoftheimage-statewavefunctions. This is exhibited through the percentage ratio
isaconsequenceofthe criticalbehavioroftheimaginary
partofthe non-localself-energycouplingpoints nearthe Γf(mf =1)
R =100 6 , (10)
surface, as we will discuss below. Γf Γ (m =1)
f f
Now we focus on our full RPA calculation of the total
linewidthofthen=1imagestateonCu(111)(seeTable as a function of the final state energyǫf. If the screened
II),withalleffectivemassessetequaltothefree-electron interaction were independent of the parallel momentum
mass. We show in Fig. 2b separate contributions to the transfer, all ratios would scale as √m, which is repre-
linewidth, Γ, coming from the decay into the various f sented in Fig. 3 as R∆qk = 100√mf. Instead, as the
bulk crystal states, Γ , such that parallel momentum transfer decreases the screened in-
f
teraction is predominantly larger, which results in the
Γ= Γf +Γs. (9) ratio RΓf to be larger than R∆qk, especially in the case
Xf of vacuum contributions to the linewidth which are ex-
pected to be dominated by vertical transitions. When
Fig. 2a exhibits the bulk band structure projected onto the decay from the image state may occur through very
the (111) surface of copper. The arrows indicate the small parallel momentum transfer q , [this is the case of
k
available phase space in the decay of the n = 1 image finalstatesthatarejustbelowthebottomofthegapand
state at the Γ¯ point (kk=0) into the unoccupied portion also the case of the n=0 surface state], a decrease in qk
of the n = 0 surface state and a generic f bulk state, may result in a diminished screenedinteraction(see Fig.
which are represented by their characteristic ǫ0 +qk2/2 4), thus RΓf being slightly smaller than R∆qk for these
and ǫ +q2/2 parabolic dispersions, respectively. Γ , states.
f k vac
Γ and Γ contributions to Γ are represented, to- Now,weanalyzethe behaviorofthe imaginarypartof
sol inter f
gether wi−th the total contribution, Γ , as a function of the image-electron self-energy, which results in vacuum
f
ǫ . andinterferencecontributionstothelinewidthtobecom-
f
The lower edge, at the Γ¯ point, of the energy gap parable in magnitude and opposite in sign (see Table I).
projected onto the Cu(111) surface lies below the Fermi Fig. 5 shows full RPA (solid line) and SRM (dashed-
level (Eg < EF) and, consequently, the decay from the dotted line) calculations of −ImΣs(z,z′,kk =0,E1) (see
(k = 0) image state occurs through finite parallel mo- Eq. (3)) for Cu(111) with all effective masses set equal
k
mentum transfer. Hence, as the coupling of the image tothefree-electronmass,togetherwiththen=1image-
state with the crystaloccurringthroughthe tails ofbulk state wave function, as a function of the z′-coordinate
states outside the crystal is expected to be dominated and for z = 7.4a.u.. We note that the probability for
by verticaltransitions (q 0), vacuum contributions to electron-holepaircreation(thedominantchannelforthe
k
the Γ linewidthareverys≃mall,especiallyforthosebulk decay of these states is provided by this process) is un-
f
states located at the bottom of the valence band (decay derestimated within the SRM.
into these states is only allowed for large values of the Thecouplingbetweenelectronicstatesiswellknownto
momentum transfer; also, their vacuum penetration is bemaximum,withinthebulk,atthepositionoftheelec-
small). Actually, the coupling of the n = 1 image state tron. Nevertheless, as the electron moves into the vac-
with all bulk states taking place at the vacuum side re- uum the maximum of the imaginary part of the electron
sults in a linewidth of only 5 meV, which approximately self-energystays (see Fig. 5) near the surface (z =0), as
represents a 10% of the total Γ linewidth37. demonstrated, within a jellium model of the surface, by
vac
A realistic description of motion along the surface can Deisz et al38. Hence, for any given value of z >0, main
be approximated by introduction of the effective mass, vacuum and interference contributions to the linewidth
as described above. The effective mass of all final states are determined (see Eq. (1)) by the specific shape of
with energies E , E E E , is found to be smaller theimage-statewavefunctioninregionsAandB ofFig.
f f f 1
than the free-electron≤mass≤, thus both Γ and Γ being 5. As the image-state on Cu(111) is located right at
s
about 20% smaller than in the case of free-electron be- the top of the energy gap and the corresponding wave
havior along the surface (see Tables I and II). This is function has, therefore, a node at the surface (z 0),
≃
the result of two competing effects: First, there is the an inspection of Fig. 5 leads us to the conclusion that
effect of the decrease of the available phase space, which vacuum and interference contributions to the linewidth
is easily found to scale as √m. Secondly, as the effective are comparable in magnitude and opposite in sign. On
mass decreases the decay from the image state occurs, Cu(100) the image state is located close to the center
for a given energy transfer, through smaller parallel mo- of the gap, and the corresponding wave function has a
4
node at z 1.3 a.u. On this surface, total vacuum and portanceinthe understandingofthe originoflinewidths
≃
interference contributions to the linewidth are still op- of image states. We have accounted for potential varia-
posite in sign, though interference contributions coming tion parallel to the surface through the introduction of
from the decay into states at the bottom of the valence the effective mass.
band are now positive due to their minor vacuum pen- Wehaveanalyzedtheoriginandmagnitudeofthevari-
etration. If the image state were located at the bottom ouscontributionstothelinewidth. Thoughthedominant
of the gap, matching at the surface would occur at max- contributiontothedecayofthefirstimagestateintothe
imum amplitude and the total interference contribution crystal-inducedsurfacestatecomesfromthecouplingbe-
might be positive, as the sign of the image-state wave tween image and surface states within the vacuum part
function would be the same on both sides just around of the surface, it appears to be approximately canceled
the surface (z =0). out by the contribution from the interference between
Finally, we investigate the dependence of interband23 bulk andvacuum coordinates. For the vacuumcontribu-
linewidths on the momentum of the image electron par- tion to the decaying rate into the intrinsic surface state,
allel to the surface, k . First, we use the MWF image we have found that it approximately represents a 90%
k
state wave function evaluated at the Γ¯ point and intro- of the total vacuum contribution. We also conclude that
duce, within this model, the dependence on k of the the coupling of image states with all bulk crystal states
k
quasiparticle self-energy, as indicated in Eq. (2). We occurring through the bulk penetration plays an impor-
find that for image state total energies lying below the tant role in the determination of lifetimes, and that this
top of the gap (k 0.11a.u.), the linewidth of the first penetration cannot be neglected if one is to accurately
k
≃
imagestate ofCu(111)increasesless than4%. Secondly, describe the lifetime of image states.
we also account for the change of the z-dependent ini- Wehavefound,withinourfullRPAscheme,thatinthe
tial wave function along the dispersion curve of the im- case of kk = 0 the decaying rate of the first image state
age state, by solving the Schr¨odinger equation for a one- on Cu(111) into the intrinsic surface state results, with
dimensionalmodelpotentialthatwebuildfollowingRef. alleffectivemassessetequaltothe free-electronmass,in
21 with various values of k : 0.06, 0.09, and 0.10a.u.. alinewidthof16meV,whileuseofmorerealisticeffective
k
The penetration into the bulk of the n = 1 image state masses leads to a linewidth of 12meV. With the use of
with these values of k is found to vary from 22.1% at either the free-electron mass or more realistic effective
k
k = 0 to 22.6%, 24.1% and 26.2% at k = 0.06 a.u., masses for both bulk and crystal-induced surface states,
k k
0.09 a.u. and 0.10 a.u., respectively. Our results, as Γsapproximatelyrepresentsa40%ofthetotallinewidth,
obtained with all effective masses set equal to the free- Γ = 37meV (mf = 1) or Γ = 29meV (mf = 1). The
6
electron mass, are presented in Table III. Though the morerealisticresultofΓ=29meVforthetotallinewidth
penetration of the image state wave function increases is in good agreement with recent experimental results
with k , the amplitude of this wave function on the bulk reported in Ref. 13.
k
side and near the jellium edge decreases, thus the abso- We have investigated the dependence of interband
lute value of both bulk and interference contributions to linewidths on the momentum of the image electron par-
the linewidth decreasing with k . Nevertheless, the to- alleltothesurface,showingthatforimagestatetotalen-
k
tal overlapbetweenimage state and final wavefunctions ergieslying belowthe topofthe gapthe linewidthofthe
becomes more efficient as kk increases, which results in first image state increaseswith kk up to a 20%. We con-
largervaluesofthetotallinewidth. Also,itisinteresting clude thatthis increaseappearsmainlyasaconsequence
to notice that especially sensitive to the variation of the of the change of the z-dependent initial wave function
momentum of the image state parallel to the surface is withkk,andourresultsindicatethatthecontributionto
the contribution to the linewidth from damping into the the linewidth from damping into the n=0 surface state
n=0 surface state. is responsible for the dependence of the total linewidth
with the momentum parallel to the surface.
IV. SUMMARY
V. ACKNOWLEDGEMENTS
Wehavereportedcalculationsoftheinelasticbroaden-
ingofthefirstimagestateatthe(111)surfaceofcopper, The authors gratefully acknowledge A. Rubio, E.
andwe have investigated,in particular,the rolethat the ZarateandM.A.Cazalillaforfruitfuldiscussionsincon-
intrinsic crystal-induced surface state plays in the decay nection with this research. This project has been sup-
of this image state. We have presented self-consistent portedbyEuskoJaurlaritza(BasqueCountry),theMin-
RPA calculations, by going beyond a free-electron de- isterio de Educaci´ony Cultura (Spain), and Iberdrola S.
scription of the metal surface. We have also considered A..
simplified models for both the electron wave functions
and the screened Coulomb interaction, showing that a
detailed description of these quantities is of crucial im-
5
1P. M. Echenique and J. B. Pendry, J. Phys. C 11, 2065 1971 (1985).
(1978); P.M.EcheniqueandJ.B.Pendry,Prog.Surf.Sci. 29S.D. Kevan,Phys. Rev.Lett. 50, 526 (1983).
32, 111 (1990). 30S. L. Hulbert, P. D. Johnson, N. G. Stoffel, W. A. Royer
2Th. Fauster and W. Steinmann, in Photonic Probes and N. V.Smith, Phys.Rev.B 31, 6815 (1985).
of Surfaces, edited by P. Halevi (Elsevier, Amsterdam, 31The electron density parameter has been taken to be rs =
1995), pp. 347-411. 2.58, so as to reproduce the Fermi energy of 7.53 eV ob-
3R. M. Osgood Jr. and X. Wang, Solid State Phys. 51, 1 tained within our model calculation of theground state.
(1997), and references therein. 32The parameters entering the surface response function of
4E. V. Chulkov, V. M. Silkin, and P. M. Echenique (to be PerssonandZaremba27 havebeentakentobemopt=1.68,
published). h=1.84 and ξ=0.49.
5F. Passek and M. Donath, Phys. Rev. Lett. 69, 1101, 33Abinitiocalculationsofthebulkbandstructurehavebeen
(1992). projected onto the (111) surface of copper, and the dis-
6F. Passek, M. Donath, K. Ertl and V.Dose, Phys. Rev.B persion E(k ) along the Σ¯ : Γ¯ M¯ direction has been
75, 2746, (1995). fitted to E k=k2/(2m) with m=→0.22 and m=3.8 at the
k k
7R.W.Schoenlein, J.G. Fujimoto, G.L.Eesley and T.W. bottom and the top of the gap, respectively. Though the
Capehart, Phys.Rev.Lett. 61, 2596, (1988). effectivemassatthebottomofthegaphasbeenreported34
8S. Schuppler, N. Fischer, Th. Fauster and W. Steinmann, forCu(111) tobem=0.31, ourcalculation of thelifetime
Phys.Rev. B 46, 13539, (1992). isnearlyinsensitivetotheprecisechoiceofeitherm=0.22
9X.Y.Wang,X.J.Shen,R.M.Osgood,Jr.,R.Haightand or m=0.31 of Ref. 34.
F. J. Himpsel, Phys.Rev.B 53, 15738, (1996). 34S. D. Kevan and R. H. Gaylord, Phys. Rev. B 36, 5809
10T. Hertel, E. Knoesel, M. Wolf and G. Ertl, Phys. Rev. (1987).
Lett. 76, 535, (1996). 35The lifetime, τ, is related to the linewidth, Γ, of Eqs. (1)
11M.Wolf,E.KnoeselandT.Hertel,Phys.Rev.B54,5295, and (4) byΓ τ =¯h=660 meV fs.
(1996). 36Wefindthatt·heimpactoftheban·dstructureintheevalua-
12J. D. McNeill, R. L. Lingle, Jr., N.-H. Ge, C. M. Wong, tionofthescreenedinteractionisnotlarge,anddifferences
R. E. Jordan, and C. B. Harris, Phys. Rev. B 79, 4645, between our full RPA calculations, on the one hand, and
(1997). the SRM and PZ results, on the other hand, appear as a
13E.Knoesel,A.HotzelandM.Wolf,J.Electron.Spectrosc. consequenceofsimplifications introduced,within ajellium
Relat. Phenom. 88-91, 577, (1998). model of the surface, in both SRMand PZ approaches.
14I. L. Shumay, U. H¨ofer, Ch. Reuß, U. Thomann, W. Wal- 37Notice that both Γvac and Γinter contributions to the
lauer, and Th. Fauster, Phys.Rev. B 58, (1998). linewidth present a maximum for energies just below the
15U. H¨ofer, I. L. Shumay, Ch. Reuß, U. Thomann, W. Wal- bottom of the energy gap. This is due to the fact that the
lauer and Th. Fauster, Science227, 1480 (1997). vacuumpenetrationofthebulkstatesincreasesasonegoes
16P. M. Echenique, F. Flores and F. Sols, Phys. Rev. Lett. upin energyand thendecreases just below thegap dueto
55, 2348 (1985). thestrongerbulkcharacterofthesestatesreproducingthe
17P. de Andr´es, P. M. Echenique and F. Flores, Phys. Rev. bottom of the gap. This maximum appears here as a cusp
B 35, 4529 (1987). because of the relatively thin slab used in the present cal-
18S. Gao and B. I. Lundqvist, Prog. Theor. Phys. Suppl. culation. This sharp feature will be significantly smoother
106, 405 (1991); S. Gao and B. I. Lundqvist, Solid State for thickerfilms.
Commun. 84, 147 (1992). 38J. J. Deisz, A. Eguiluz, W. Hanke, Phys. Rev. Lett. 71,
19E. V. Chulkov, I. Sarr´ıa, V. M. Silkin, J. M. Pitarke, and 2793 (1993).
P. M. Echenique,Phys. Rev.Lett.80, 4947 (1998).
20L.HedinandS.O.Lundqvist,inSolidStatePhysics,edited
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21E. V. Chulkov, V. M. Silkin and P. M. Echenique, Surf.
Sci. Lett. 391, 1217, (1997).
22Seee.g., D. Pines, Elementary excitations in solids, Chap-
ter 3 (Addison Wesley,New York,1963).
23In these processes the image electron decays into a state
having a different z-dependent (the plane of the surface is
assumed to beperpendicular to thez-axis) wave function.
24The effective mass of the image state is taken to be equal
to thefree-electron mass, i.e., m1=1 (see, e.g., Ref. 2).
25R.H. Ritchieand A.L. Marusak, Surf. Sci. 4, 234 (1966).
26P. M. Echenique, F. J. Garc´ıa de Abajo, V. H. Ponce and
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(1985).
28A. Goldmann, V. Dose and G. Borstel, Phys. Rev. B 32,
6
TABLEI. Calculated linewidth(inmeV)comingfrom the TABLE III. Calculated total linewidth (in meV) of the
decay of the n = 1 image state on Cu(111) into the n = 0 first image state on Cu(111), computed within our full
intrinsicsurfacestate,asobtainedwithinthreedifferentmod- RPA scheme with all wave functions computed from the
els for the description of the surface response and two differ- one-dimensional model potential of Ref. 21, as a function of
ent models for the description of both initial and final wave the momentum of the image electron parallel to the surface,
functions (see text). RPA accounts for our full RPArealistic k (see text). All effective masses have been set equal to the
k
calculation of the screened interaction, SRM for the simpli- free-electronmass. Contributionstothelinewidthfromdecay
fied specular reflection model of Ritchie and Marusak25, and intothen=0crystal-inducedsurfacestate,Γs,aredisplayed
PZ for the vacuum side surface response suggested by Pers- in parenthesis.
son and Zaremba27. MWF accounts for the wave functions
obtained by solving the Schr¨odinger equation with the real- kk Γvac Γsol Γinter Γ
0.0000 47 (42) 44 (16) -54 (-42) 37 (16)
istic one-dimensional model potential of Ref. 21, and AWF
0.0570 48 (44) 40 (13) -49 (-38) 39 (19)
for the approximate model described in the text. The effec-
0.0912 50 (45) 32 (8) -38 (-29) 44 (24)
tive mass of the n=1 image state has been set equal to the
0.1026 50 (44) 28 (6) -31 (-23) 47 (27)
free-electron mass, and for the n = 0 surface state we have
used either m0 = 1 or m0 = 0.42. The momentum of the
image electron parallel to the surface is set equalto zero.
Surf. Res. Wavefunction Γvac Γsol Γinter Γs
RPA (m0=1) MWF 42 16 -42 16
RPA (m0=1) MWF 29 8 -25 12
6
SRM(m0 =1) MWF 11 12 -17 6
SRM(m0 =1) AWF 2 15 -9 8
PZ (m0=1) MWF 55 - - -
PZ (m0=1) AWF 12 - - -
TABLE II. Calculated total linewidth (in meV) coming
from the decay of the n = 1 image state on Cu(111) into
any unoccupied final state with Ef <E1, as obtained within
our full RPA scheme with all wave functions computed from
the one-dimensional model potential of Ref. 21. As in Ta-
ble I, the effective mass and the momentum parallel to the
surface of the n = 1 image state have been set equal to the
free-electron mass and equal to zero, respectively. As for fi-
nal (bulk and intrinsic surface) states, we have used either
m = 1 or the realistic effective masses (m = 1) described
f f
in the text. Contributions to the linewidth from decay into
bulk states, Γ Γs, are displayed in parentheses.
−
Surf. Res. Γvac Γsol Γinter Γ
RPA(m =1) 47(5) 44(28) -54(-12) 37(21)
f
RPA(m =1) 34(5) 32(24) -37(-12) 29(17)
f
6
7
0,4 15
f (z) Cu(111) (a)
n=1 0,3 (a) (b)
0,2
n=1 Vacuum Level
0,1
image state
0 GAP
-0,1 10 n=0 GG S = = S 1 6G ( m+e GV) = 37 (meV)
-0,2 f f S
0,6 V)
f n=0(z) Cu(111) (b) y (e Fermi Level
0,4 g surface state
r
ne Bottom of the Gap
0,2 E
0 f-state
5
-0,2 G
f
e
-0,4 f G -G G
-50 -40 -30 -20 -10 z (a0.u.)10 20 30 40 50 vac inter sol
FIG.1. Wave functions of both (a) n=1 image and (b)
n=0 intrinsic surface states on Cu(111), as obtained within
two different models: MWF (solid line) and AWF (dotted 0
0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 3.5
line) (see text). Full circles represent the atomic positions in
the (111) direction. The geometrical electronic edge (z = 0) q (a.u.) linewidth (meV)
//
has been chosen to be located half an interlayer spacing be-
yond the last atomic layer. Notice the s-like and p-like char- FIG.2. (a)Electronicsurfacebandstructureatthe(111)
actersoftheimageandintrinsic-surfacestatewavefunctions, surface of copper. (b) Vacuum (Γvac), bulk (Γsol), and inter-
respectively. ference ( Γinter) contributions to the linewidth (Γf) coming
−
from the decay into the various f bulk crystal states. The
arrowsdeterminetheavailablephasespaceinthedecayfrom
theΓ¯ pointofthen=1imagestateintotheunoccupiedpor-
tion of both the n = 0 surface state and the generic f bulk
state. Dispersion curves of these final states are depicted.
The energy is measured with respect to the bottom of the
valenceband. Alleffective masses havebeen set equaltothe
free-electron mass.
8
180 0.12
R (%)
G vacuum bulk
160 bottom of
gap 0.1 w =4 eV
surface
140
)
u.
120 interference n=0 n (a. 0.08 w =3 eV
o
cti
n
u
100 solid e F 0.06 w =2 eV
s
n
o
80 total sp
R e
D q R 0.04
60
bottom of
0.02
valence band
40
w =1 eV
20
0
0 1 2 3 4 5 6 7
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Energy (eV)
q (a.u.)
FIG.3. Ratio RΓ of Eq. (10), as a function of the final
f FIG. 4. Imaginary part of bulk (solid lines) and sur-
state energy. The effective mass and the momentum parallel
face (dotted lines) response functions, Im[ 1/ǫ(q,ω)] and
to the surface of the n = 1 image state have been set equal −
to the free-electron mass and equal to zero, respectively. As Im −gs(qk,ω) , which describe the screened interaction far
from(cid:2) the surfac(cid:3)e into the bulk and into the vacuum, respec-
for final (bulk and intrinsic surface) states, we have used the
tively. Bulk response functions have been evaluated in the
realisticeffectivemasses(m =1)describedinthetext. Here,
f
6 RPA,whereassurfaceresponsefunctionshavebeenevaluated
R∆qk =100√mf. within the specular reflection model of the surface (SRM)26
with theRPA for thebulk dielectric function22.
9
1
RPA
) SRM
r
h
o
b n=1
V/ 0.5
e
m
(
}
)
0
w0,
=
Q
z',
z,
(S 0
z
S-
{
m
I jellium edge
_
+
B A
-0.5
-15 -10 -5 0 5 10
z' (a.u.)
FIG. 5. Full RPA (solid line) and SRM (dashed-dotted
line) calculations of −ImΣs(z,z′,kk = 0,E1), as obtained
from Eq. (3) and as a function of the z′ coordinate. The
dotted line represents the n = 1 image state wave function
(MWF). The value of z (z = 7.4 a.u.) is indicated by an
open circle. The sign appearing in regions A and B accounts
forthesign oftheproductbetweentheimaginary partofthe
self-energy and the image state wave function. Full circles
represent the atomic positions in the (111) direction. The
geometrical electronic edge (z=0) has been chosen to belo-
catedhalfaninterlayerspacingbeyondthelast atomiclayer.
All effective masses have been set equal to the free-electron
mass.
10