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Journal of Electron Spectroscopy and Related Phenomena 127 (2002) 17–28www.elsevier.com/ locate/elspec

O ne, two and three-body channels of the core–valence–valenceAuger photoelectron coincidence spectra of early transition metals

*Andrea Marini , Michele CiniDipartimento di Fisica, Istituto Nazionale di Fisica della Materia, Universita’ di Roma Tor Vergata, Via della Ricerca Scientifica,

1-00133 Rome, Italy

Abstract

We propose a simple model of core–valence–valence Auger and APECS intensities from open-band solids to account forthe alleged negative-U behavior of the spectra of early transition metals. In these systems the maximum of the line shape isshifted by the interaction to lower binding energy, which is the contrary of what happens in closed band materials where the

(2)two-hole Green’s functionG allows to understand the phenomenology of the spectra in terms of theU /W ratio of thev

on-site repulsionU to the band widthW. Actually, for open-band solids, only part of the intensity comes from the decay of(2)unscreened core-holes and is obtained by the two-body Green’s functionG , as in the case of filled bands. The rest of thev

intensity which arises from screened core-holes here is derived using a variational description of the relaxed ground state;this involves the two-holes-one-electron propagatorG , that also contains one-hole contributions. We propose a practicalv

scheme to calculate the three-body Green function by a summation of the perturbation series to all orders. We achieve that byformally rewriting the problem in terms of a fictitious three-body interaction. Our method grants non-negative densities ofstates, explains the apparent negative-U behavior and interpolates well between weak and strong coupling, as wedemonstrate by test model calculations. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Electron impact; Auger emission; Photoemission and photoelectron spectra

PACS: 79.20.Fv; 79.60.-i

1 . Introduction frozen during the ionization so that the spectra iswell described [2] in terms of the two-holes Green’s

(2)The one-step model [1] of the core–valence–val- functionG by the Cini [3,4] and Sawatzky [5]v

ence (CVV) Auger spectra coherently describes the (CS) model.ionization process and the Auger decay; the excita- The CS model explains the phenomenology in-tions due to the core level ionization do, in general, volving band-like, atomic-like and intermediatemix their amplitudes in the Auger decay. For solids situations in terms of theU /W ratio of the on-sitewith closed valence bands no electron–hole excita- repulsionU to the band widthW. For low U /W, thetions are possible and the valence electrons remain line shape is close to the self-convolution on the

local one-hole density of states; with increasingU /*Corresponding author. W, the shape is distorted until, for a critical value of

0368-2048/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PI I : S0368-2048( 02 )00168-8

18 A. Marini, M. Cini / Journal of Electron Spectroscopy and Related Phenomena 127 (2002) 17–28

the ratio, two-hole resonances appear. Detailed the apparent negative-U behavior was proposed byCini and Drchal [17].studies of noble transition metals, like Au [6] and Ag

In this theory, the Auger line shape has two main[7] led to a very good agreement between theory andcontributions, that we call unrelaxed and relaxed,experiment, and also allowed the direct observationrespectively. The unrelaxed contribution is obtainedof off-site interaction effects [8]. In all these cases

(2) assuming that the Auger decay occurs while thethe diagrammatic expansion ofG is just a ladderv

conduction electrons are in their ground stateucl inof successive interactions between the two barethe absence of the core hole. The density of statesholes.which shows up in this contribution is obtained byFor almost completely filled bands, when the

(2)G , like in closed-band systems. However thenumber of holes per quantum staten !1, closed- vhscreening electronic cloud that surrounds the core-band theory can be extrapolated [9]. The core-holehole can participate in the Auger decay; this contri-screening diagrams areO(n ) and one can assumehbution to the total current is described by the Augeragain that, to a first approximation, the valencedecay with the ground stateufl in the presence ofelectrons remain frozen during the core ionization,the core hole as initial state. We show that this lastand in the initial state of the Auger decay the valenceterm is related to the Fourier transform of thet. 0configuration is the same as in the ground state. Thepart of a three-body Green’s function much harder toCVV spectra are still essentially described in terms

(2)(2) calculate thanG . To sum the perturbation series ofvof the two-hole Green’s functionG , computed invthis complex many-body propagator we propose athe absence of the core hole. Using Galitzkii’s lowsimple approach in the spirit of the BLA [18].density approximation [10] (LDA) the dominant

(2) Experimentally, one can single out the relaxeddiagrams of the perturbation expansion of theGvcontribution to the total Auger current by properlyare just the same ladder diagrams which provide thefixing the photoelectron energy in an Auger photo-exact solution forn → 0 [11,12]. This is the barehelectron coincidence spectroscopy (APECS) experi-ladder approximation (BLA) that has been useful toment [19–22], where the Auger electron is detectedinterpret, e.g. the line shape of graphite [13]. Thein coincidence with the photoelectron responsible ofunexpected result was that the BLA is far better thanthe core hole creation.the self-consistent ladder approximation.

The next turn came from experiments on early 3dtransition metals, like Ti and Sc [14], that could not

2 . From the one-step model to the two-bodybe interpreted by the above theory. The maximum ofGreen’s functionsthe line shape results shifted by the interaction to

lower binding energy, which is the contrary of whatIn the one-step model, the APECS current [1] ishappens in closed band materials. Qualitatively the

given by:CS model could work if one admitted thatU , 0,and such an explanation has actually been proposed

J e ,e |Os dp k[15]. j

However, no justification ofU , 0 exists; rather, N N21 2 N21 † N N21 (1)ukC uF lu kF uA d(v 1E 2H(1)2e 2e ) A uF l0 j j k g 0 p k k jU .0 and the theory for almost empty bands is no ]]]]]]]]]]]3N N21 2 2

v 1E 1E 2E 2e 1Gs dsimple extrapolation of the closed-band approach. g 0 c j p

pairing by positiveU is actually possible (see e.g. M. with e , e are the photoelectron, Auger electronp kCini and A. Balzarotti, Phys. Rev. B. 56, 14711 energies,E is the core level energy,v the incom-c g(1997)) but this requires lobes sitting on different ing photon energy,G the Auger width of the coresites. hole andA is the Auger operator. Since the algebrakSarma [16] first suggested that the Auger line is complicated, we seek simplification by physicalshape of Ti looks like some linear combination of the arguments. We write:one-electron density of states and its convolution.Using this hint, and the general framework of the A 5O M a a (2)k cka b a bl l l l

abone-step model, a simplified microscopic theory of

A. Marini, M. Cini / Journal of Electron Spectroscopy and Related Phenomena 127 (2002) 17–28 19

with M Auger matrix elements;a , b are the where:cka b l ll l

local orbitals that belong to the Auger site that is, the 1(2) (2)]9 9 9 9site where the primary core-hole belongs; we will D a b ,b a ;v 5 2 I[FT(u(t)G (a b ,b a ;t))]s dl l l l l l l lpconsider only thelocal scattering at the Auger site,

(5)since in this way we drastically simplify the algebra,and the line shape is little influenced by scattering at

(2) N † † N29 9G a b ,b a ;t 5 2 i kC uT a t a t a a uC ls d s d s ds d h jl l l l 0 a b b 9 a 9 0the other sites [23]. Many Auger line shape calcula- l l l l

tions in solids performed in this way have shown that (6)Nthis is a good approximation. LetuC l be the ground0

N Here, FT stands for Fourier transform and operatorsstate, with energyE , of the system with the core0are in the Heisenberg picture. As above we are usingelectron; the fully interacting valence electrons area special notationa ,b . . . to indicate the set ofdescribed by the Hamiltonian: l l

quantum numbers of thelocal valence spin–orbitalsH n 5H 1 1 12 n dH (3)s ds d s d belonging to the Auger site.c c

If we assume that the Coulomb valence–valencen 5 0,1 is the number of electrons in the core level.c interaction between the orbitals of the Auger site is:N21huF lj are the complete set of eigenstates ofH 0s dj

† †(so in presence of the core hole), with energies H 5 O U a a a a (7)U m n r t m n t rl l l l l l l lN21 m ,n ,r ,tl l l lhE j. N represents the total number of electrons.j

H n contains the electron–electron interaction ands dc the two Auger holes dynamics contained inH 1 iss dthe screening mechanism switched on by the ioniza- solved exactly in the CS model summing all orderstion via thedH potential. then Eq. (1) describes the of the perturbation series of the two-hole GreenN N21

(2)evolution of uC l to each possible eigenstateuF l0 j Function G . The diagrammatic method developsvin presence of the core-hole and the successive Eq. (6) in terms of local non-interacting time orderedAuger decay. The summation on the excited state one-body propagatorS a ,b ;t :s d0 l lN21uF l is the origin of the correlation between thej

h ephotoelectron and the Auger energy measured in S a ,b ;t 5 S a ,b ;t 2 S b ,a ;2 t (8)s d s d s d0 l l 0 l l 0 l lAPECS experiments: tuning the photoelectron

where:energy it’s possible to select (within the Auger widthh †G) the events corresponding to the decay of a few S a ,b ;t 5 2 iu t ka (t)a l (9)s ds d0 l l a bl ldominant intermediate states in the presence of the

core hole [24]. e †S b ,a ;2 t 5 2 iu 2 t ka a (t)l (10)s ds d0 l l b al lThe final two-hole density of states is complicatedN21by the excitations present in the statesuF l. Thesej Here the average is taken over the non-interacting

are due to the screening attractive interactiondH ground stateuc l with energy E . Introducing the0 0between the core hole and the valence electrons.non-interacting two-hole propagator:Only for full bands systems there are no empty states

† †to create hole–electron pairs necessary to screen theg a b ,t r ;t2 t 5 kT a t a t a t a t l (11)s d s ds d s d s dh jl l l l 1 a b t 1 r 1l l l l

core hole; in this case Eq. (1) is greatly simplified(2)the final from ofG within CS theory is:tbecause the valence electrons remain frozen during

the ionization and so the summation over the states (2) 9 9 9 9N21 G a b ,b a ;t 5 g a b ,b a ;t 2 i O Us d s dl l l l l l l l m n r tl l l lhuF lj is well approximated by the ground state m ,n ,r ,tj l l l lN21 N `uF l¯ uC l, with the result:0 0

(2) 9 9E dt g a b ,t r ;t2 t G m n ,b a ;ts d s d1 l l l l 1 l l l l 1J e , es d 2`p k

(12)(2)* 9 9| J e O O M M D a b ,b a ;es d sXPS p cka b cka b l l l l pl l l l

a ,b a ,b9 9l l l l Going to frequency space, this becomes a linearalgebraic system. For almost completely filled bands,1e 2v (4)dk g


when the number of holes per quantum staten ! 1, is, the ground state of the valence electrons in theh

closed-band theory can be extrapolated [9] leading to presence of the core-hole potential.the same Eq. (12). In this scheme, the Auger spectrum has two main

Eq. (12) works also forn ,0.25 and a range of contributions,relaxed and unrelaxed; the latter de-h

U /W, however for open bands systems Eq. (12) is a pends onD v , which is analogous to that calcu-s dcc

partial sum of the perturbation series called bare lated for closed bands materials.ladder approximation (BLA) because it uses un- The relaxed contribution arises from creating thedressed single particle propagators. Cluster calcula- two Auger hole in the stateufl which is totallytions [25] has shown that is a cancellation of self- relaxed around the core-hole (that due to the screen-

relenergy and vertex corrections that permits the exten- ing cloud has a binding energyE #E ). Screeningc c(2)sion of the ladder expansion forG to largern . effects that permit the decay ofucl into ufl arev h

The convolution form of Eq. (12) has further described by the screening-widthd and it’s interest-important consequences. Formally, it is a Dyson ing to note that for long-lived core-hole, whenG /equation in which theU matrix is an instantaneous d ! 1, the total current is dominated by the relaxedself-energy. Therefore, it grants the Herglotz proper- contributionJ e ,e .s drelaxed p k

(2)ty: for any interaction strength,G generates a The electronic structure of the stateufl is calcu-non-negative densities of states. The Herglotz prop- lated in Ref. [17] using ad-times degenerateerty is a basic requirement for a sensible approxi- Hamiltonian of Hubbard type:mation, yet it is not easily obtained by diagrammatic H 0 5H 1H 1H (16)s d 0 e2e c2happroaches.

where:

H 5O E k n (17)s d0 kaka3 . A variational approach to the negative-U

problem is the band term,H is the Hubbard interactione2e

term, and:The above theory, neglecting the summation on

N21 H 5 2W O n (18)the excited statesuf l, works well for full or c2h aj lalnearly full bands but fails for early d transitiondescribes the coupling of the valence electrons to themetals. For these systems a rearrangement of the

†core hole at the origin. Heren 5 a a arevalence electrons after the creation of the core-hole ka ka ka

number operators in the Bloch representation.must be included; in a simplified one-step approachThe stateufl can be obtained fromucl by creatingto this problem [17], Eq. (1) is rewritten as:

an infinite number of electron-hole pairs [26], and itJ e ,e ¯ J e ,e 1J e ,es d s d s dp k relaxed p k unrelaxed p k is orthogonal toucl, kc ufl5 0 (the so-called Ander-

D e 1e 2v D e 1e 2vs d s dff p k g cc p k g (13) son orthogonality catastrophe). On the other hand,]]]] ]]]]]~ G 1rel 2 2F G22v 1E 2e 1G v 1E 2e 1sG1dds d s d the photoemission and Auger spectra can be calcu-g c p g c p

lated with high accuracy, if one includes only thewith: one and two electron–hole excitations [24]. Thus a

simple class of trial states is analyzed, namely:†D v 5 kc uA d v 1H 1 2E A ucl (14)s d s s d dcc k k†ufl5 a1O b a a ucl (19)ka a kaS Dl† kaD v 5 kf uA d v 1H 1 2E A ufl (15)s d s s d dff k k

with the variational parametersb determined soka]The physical idea is that the summation over the that E ;kf uH 0 ufl attains its minimum. As a results dN21states uF l in Eq. (1) is largely exhausted by a simple form forufl is obtained:j

summing over just two orthogonal states, namely, the1 †ground stateucl of H 1 with energy E and thes d ]]ufl5 O a a ucl (20)] a Falan dœ hrelaxed initial state of the Auger transition,ufl, that


where the statesuFal from which the screening the BLA to each two-body interaction [17] (see Fig.electron are removed belong to the Fermi surface, as 1). Together with the all possible two-body interac-one would intuitively expect because this yields the tions we get a single particle diagram correspondinglowest possible energy for the screening electron. to the creation of one of the two Auger holes in the

With the form of Eq. (20) used in the second term screening cloud (the so-called one-body channel forof Eq. (15) the relaxed Auger current due to the the Auger decay).decay from the totally relaxed ground-stateufl is However, this approximate solution of the per-proportional to a three-body Green density of states: turbation series does not treat the two Auger holes

and screening electron on equal footing and theD vs dff resulting density of states is not positive defined.

To avoid this problem recently [18] we have* 9 9 9|O O M M D a b g ,g b a ;v (21)s dcka b cka b l l l l l ll l l l proposed forG an extended bare ladder approxi-a ,b a ,b v9 9l l l l

mation in which all the possible two-body barewith: interactions are allowed, as shown by the second

order diagram of Fig. 2. However, the series cannot19 9 9 ] 9 9 9D a b g ,g b a ;v 52 I FT u t G a b g ,g b a ;ts ds d f s s ddgl l l l l l l l l l l l be summed easily like in Eq. (12), because withp

three bodies involved we meet an extra difficulty.(22)For instance, letH t produce an interaction be-s dU i

tween the two holes in a given term of the expan-9 9 9G a b g ,g b a ;ts dl l l l l l sion; then, the electron line overtakes timet ; there-i

† † † fore the diagram doesnot yield a convolution of a5 kc uT a t a t a t a a a ucl (23)s d s d s dh ja b g g 9 b 9 a 9l l l l l l

function of t2 t times a function oft . This undesir-i iSo we obtain that the Auger-holes/screening-elec- able feature can be removed by using the identities:tron dynamics is described byG ; unfortunately wev

h h hhave no recipes like BLA for the three-body Green’s S a ,b ;t 5 iO S a ,g;t2 t9 S g,b ;t9 (25)s d s d s d0 l l 0 l 0 lgfunction and the problem of the solution of the

2s dDyson equation forG is much harder than forG .v v andIn the next section we will propose a simple recipefor the calculation ofG based on some physicalv e e eS a ,b ;t 5 iO S a ,g;t2 t9 S g,b ;t9 (259)s d s d s d0 l l 0 l 0 lapproximations.

g

where the summations run over all the complete setof spin–orbitals. We note that in the limitt54 . An extended bare ladder approximation

1t9→ 0 , we get the correct normalization onlythanks to the completeness of theg set.The Green’s function (23) yields the expansion:

To see the use of Eqs. (25), consider for instance` `

the application to one of the second-order contribu-n 3n139 9 9G a b g ,g b a ;t 5O 2 1 2 i E dt . . . E dts d s ds dl l l l l l 1 n tions to Eq. (24). Using the standard diagrammatic

n2` 2` rules, we get the l.h.s. of the uppermost equation in

† †kTha t a t a t H ts d s d s d s da b g U 1l l l Fig. 2, while Eqs. (25), that we represent pictorially† as the r.h.s. of the uppermost equation in Fig. 2,. . . H t a a a jls dU n g 9 b 9 a 9 cl l l

(24) permits to write the diagram in a form of a product(in time space) of simple diagrams. In this way,

This describes the propagation of two-holes and one introducing a fictitious3 interaction vertex, alongelectron in the final state, or, if the electron and one with the true interaction vertex (dot), the diagram ishole annihilate, a one-body propagation results. cast in the convolution form. This useful property

In proposing an approximation to Eq. (24), we extends to all the diagrams of the bare-ladder2s dmay proceed by analogy withG using separately approximation.v


Fig. 1. To a first approximation to Eq. (24) each two-body interaction is solved separately using BLA (represented by the box in all thediagrams on the l.h.s.). The first three diagrams of the l.h.s. represent the hole–hole and the two possible hole–electron series of scatteringprocess; the fourth diagram is a non interacting term necessary to avoid double counting while the last diagram describes the dynamics ofthe single Auger hole (see text).

Fig. 2. Second order contribution to the three-body Green’s function. Using Eqs. (25) we cast it in the form of a product of three ‘blocks’.These are easily dealt with by a Fourier transform.


4 .1. Core-approximation Thus, we regard the ansatz (26) as a physicallymotivated approximation, which must be tested

Using Eqs. (25) we got a great simplification for against exact results for its validation.the second order diagram of Fig. 2. However, theinfinite summations (one for each3 interaction) are 4 .2. Summing the three-body laddera high price to pay for that. They arise because Eqs.(25) imply a summation over the complete set ofg. Working out the core approximation (CA) like inOn the other hand, since we use alocal H , the dotU the second-order case showed in Fig. 2 one caninteraction involves only local matrix elements compute all kinds of ladder diagrams, to all orders.between spin–orbitals; so, we are only interested in The partial sum of the series (24) that one obtains inh,ethe local elementsS a ,b ;t . Physically, we mays dl l this way will be referred to as core-ladder-approxi-expect that only the sites which are closest to the mation (CLA). From now on onlylocal indicesAuger site give an important contribution to the appear so we shall dispense ourselves from showingsummations, and we can actually work with a sum this explicitly. At ordern of the perturbation seriesover g drastically limited to the local statesg . Wel for G we get the six contributions represented invhave observed above that summing over the com- Fig. 3; Fourier transforming and summing all orderspleteg set is necessary to get the correct zero-time we obtain a linear system of equations for the CLA

CLAlimit. To preserve normalization using only the local form G abg,g 9b9a9;v of the three-bodys dstatesg as intermediate states in Eqs. (25) wel Green’s function, namely:6introduce a set of functionsR a ,b ;t which ap-s dl l

CLAproximately factor the propagator in analogy with G abg,g 9b9a9;v 5B abg,g 9b9a9;vs d s d0

Eqs. (25) according to the ansatz: 2O O O U B abg,jtr ;vmnrt 0s dH F G] ]m,n r,tj

CLA3G mnj,g 9b9a9;vs d† 1 † (28)(2i) a t a ¯O R a ,g ;t2 t9 a t9 as d s ds dK L K La b l l l g bl l l l 2 O U [B abg,njr ;vmnrt 0s dg

] ]m,n,r,t

CLA(26) 3G mjt,g 9b9a9;vs dCLA

1B abg,mtj ;v G jnr,g 9b9a9;v ]s d j0s d] ]andwhere theB functions are:0† 2 †(2i) a t a ¯O R a ,g ;t2 t9 a t9 as d s ds dK L K La b l l g bl l l lgl B abg,g 9b9a9;v 5G abg,g 9b9a9;v 2G bag,g 9b9a9;vs d s d s d0 0 0

B abg,jtr ;v 5G abg,jtr ;v 2G abg,jrt ;v(269) s d s d s d0 0 0] ] ] ] ] ]B abg,njr ;v 5G abg,njr ;v 2G abg,nrj ;vs d s d s d0 0 0 ]] ] ] ] ]where t9 is any time intermediate between 0 andt;B abg,mtj ;v 5G abg,mtj ;v 2G abg,mjt ;vs6d s d s d s d0 0 0] ]the R functions are computed for any t by solving ] ] ] ] (29)

the system fort9→ 0.For an isolated core state Green’s function we

As a shorthand notation, we underline the electronhave that: 6indices that correspond to aR factor; for example:2iE tcg t 5 2 ie ⇒ g ts d s dc c 2 hG abg,jtr ;t2 t 5R g,j ;t2 t S b,t ;ts d ss d0 1 1 0] ]5 ig t2 t9 g t9 ;t9 (27)s d s dc c

h2 t S a,r ;t2 t (30)d s d1 0 1

therefore we call the set of Eqs. (26) core approxi-this is similar to a non-interacting three-body prop-mation (CA).

e 2agator, except that theS has been replaced by aR .The ansatz is also correct in the strong couplingThe second, third and fourth lines of (28) comecase, when localized two-hole resonances develop.

from the (a,b), (c,d) and (e,f) diagrams of Fig. 3,This is appealing, since the strong coupling case isrespectively; while the first two contributions comethe hard one, while at weak coupling practicallyfrom the hole–hole interaction the others come fromevery reasonable approach yields similar results.


Fig. 3. Diagrammatic representation of the contributions of ordern to the CLA (Eq. (28)). The fictitious3 interaction vertex represents a6R function.

sp † †electron–hole interactions and convey information G abg,g 9b9a9;t 5 21 ka a lka a lS a,a9;ts d s d s db g g 9 b 9

on the screening effects due to the electronic cloud (31)which forms as a response to the deep electronionization. where S a,a9;t stands for the time-ordered dresseds d

one-body Green’s function that can be expandedwith:

4 .3. Single-particle contribution ` `

nS a,a9;t 5O 2i E dt . . . Es d s d 1nAt the same level of approximation we must

2` 2`

consider the case when one (or both) the holes †3dt kT a t H t . . . H t a l (32)h js d s d s dn a U 1 U n a 9 cproduced by the Auger transitions has the same spin

as the screening electron. Consider the spin–diagonaland summed with the Dyson’s equation [27] in termscomponentsG abg,g 9b9a9;t , (with s 5s and so of proper self-energy diagrams.s d a 9 a

on); when the holeb has the samez spin component Using the general rules of perturbation theory [27]†as the electron, then, contractinga t with a t and it is possible to show that the self-energy operator iss d s db g

†a with a in (23) one obtains the extra contribu- proportional to a three-body Green’s function; name-b 9 g 9

tion: ly:


ss9 †H 5O n e 1O O T a a (35)0 sis si ij sis s9 jsS a,a9;v 5 S a,a9;v 2 O O U U sis ijss d s d kss9l0 mnrt m 9n 9r 9t 9

m,n,r,tm 9,n 9,r 9,t 9

ss93 [S a,r ;v G mnt,n9t9r9;vh s d s d0 with real T . Next the interactions are described byij2 S (a,t ;v) G(mnr,n9t9r9;v)] S(m9,a9;v)0 an Hubbard term:1 [S (a,t ;v) G(mnr,m9t9r9;v)0

2 S (a,r ;v) G(mnt,m9t9r9;v)] S n9,a9;vs dj H 5U O n n 1O n n (36)0 Hub si↑ sj↓ s1s s2sF Gss(33) sij

subtracted of a Hartree–Fock mean field average ofso we get a conserving approximation for the proper

the interaction so that the total hamiltonian reads:self-energy, using the CLA for the three-bodyGreen’s functions of Eq. (33). H5H 1H 2V (37)0 Hub H–F

By solving Dyson’s Eq. (33) one can model XPSspectra from valence bands with low band filling; and in this way, the Hartree–Fock contribution to thethis is another field of application of our approach. self-energy would be automatically embodied in the

The deep hole attracts a screening electron that bare propagators.can be directly involved in the Auger decay; this is By considering various possible electron popula-the physical origin of the contribution (31) to the tions N# 20 in the cluster, the mean occupation ofthree-body Green’s function. Locally, such processes the Auger site in the non-interacting ground state:leave the system with one hole in the final state. The

1presence of an one-body contribution in the Auger]knl5 Okn 1 n l (38)51s 52s4spectra from transition metals like Ti or Sc has been s

pointed out already [16,17]. Besides the three-bodywas varied. The test becomes more severe whenknl(28) and one-body (31) diagrams, there are mixedis reduced towards half filling andU /W is increased.contributions as well. Physically they represent inter-Many different densities of states are obtained fromference terms in which the system evolves fromthe matrix elements ofG(v); in the context of ourone-hole states to two-holes-one-electron states andtheory, the density:back. However such terms can be neglected [18]; the

final result is: † †D (v);kc ua a a d v 1Hs1h 51↑ 52↓ 51↑CLAG abg,g 9b9a9;v 5G abg,g 9b9a9;vs d s d †

2E a a a ucl (39)d 51↑ 52↓ 51↑sp1G abg,g 9b9a9;v . (34)s d

is of special interest because in the diagrammaticseries the annihilation of the spin-up electron by the4 .4. A model cluster calculationshole of the same spin is particularly strong so wemay expect that the one-body term is important. By

CLA results have been tested against those of acontrast, in the density:

model system that can be diagonalised exactly [18].The model was a five atom cluster, with two levels † †D (v);kc ua a a d v 1Hs2h1e 51↑ 51↓ 52↑for each atom; the one-body basis elements areusisl

†with s51, . . . ,5 the site index, i5 1,2 the level 2E a a a ucl (40)d 52↑ 51↓ 51↑index ands 5 ↑ ,↓ for the spin direction; the one-body energies are denoted bye . The atoms1 . . . 4 the same contribution should be smaller and possiblysj

occupy the vertices of a square, and the Auger atom absent. Although the analytic development can beis at site 5 above the center. The system’s dynamics somewhat boring, the maximum size of the matricesis given by an Hubbard-like Hamiltonian composed involved is just 8.of a one-body HamiltonianH , of the same level of Fig. 4 showsD for U /W51 with a population0 1h

sophistication as a tight-binding model of a solid, knl5 0.86 and knl5 0.72 on the Auger site. Thewith a nearest neighbor hopping term: density is dominated by a single peak at binding


Fig. 4. Binding energy dependence ofD for U /W51 andknl5 0.86 (left frame) andknl5 0.72 (right frame). Heavy line: exact result;1h

light: CLA; dashed: 1-body contribution to the CLA result; dotted: 3-body contribution to the CLA result. The line shapes have beenconvolved with a Lorentzian (FWHM50.75 eV).

energy ¯ 4 eV, but also shows a pair of wings. The stable. The CLA also explains the increase of thevalue of U is well outside the scope of weak relative weight of the 3-body contribution withcoupling approaches but forknl50.72 the CLA reducing band filling.reproduces the exact results rather well. A shift (¯ 5 As one could expect,D (v) has much more2h1e

eV) and an increase of the structure at high binding weight at high binding energies thanD (v), as one1h

energy is understandable because the filling is high can see in Fig. 5; in both framesknl5 0.72 but theand the screening ineffective. From these first results interaction increase fromU /W50.25 (left frame) towe see that the performance of the CLA does not U /W51 (right frame).break down quickly with increasingU as weak- ForU /W50.25 the agreement is fairly good. Forcoupling approaches tend to do, but remains fairly U /W51 the exact line shape shows three partially

Fig. 5. Binding energy dependence ofD for knl5 0.72 andU /W50.25 (left frame) andU /W51 (right frame). Heavy line: exact result;2h1e

light: CLA; dotted: 3-body contribution to the CLA result. The line shapes have been convolved with a Lorentzian (FWHM50.75 eV). Note†that for knl5 0.72 no single-particle contribution exists becausekc ua a ucl50.51↑ 52↑


resolved broad peaks covering a range of¯ 10 eV in treat the screening electron and the Auger holes onbinding energy. The CLA misses the main peak equal footing.position by ¯ 1 eV and underevaluates the low Further applications of CLA are possible: first thebinding energy shoulder; however, we can still claim Coster–KronigCC9V decay followed by Augerat least a qualitative agreement with the results of the C9VV transition that leaves three holes in the valenceexact calculation in such severe conditions. bands; CLA provides a method to calculate the final

In all cases we verify that the Herglotz property is state dynamics. Moreover the sum of the perturba-fully preserved; this is a most valuable feature which tion series of the three-body Green’s function can beis not easily obtained for approximate three-body used to construct a self-energy operator that canpropagators. For instance, the approach of Ref. [17] works outside the low-density regime of the usualfails in this respect at strong coupling. T-matrix approach.

Comparing the two frames of Fig. 5 we observethe apparent negative-U behavior: increasing theinteraction U, the main peak shifts towardslower

A cknowledgementsbinding energies; this is a consequence of theinteraction of the screening electron with the two

This work has been supported by the IstitutoAuger holes. We remark that a high enoughn ish Nazionale di Fisica della Materia. One of us (AM)necessary to build up a localized screening cloud.has been supported by the INFM scholarship ‘AnalisiThis is why the negative-U behavior is observed inteorica della Tecnica A.P.E.C.S.’, prot. 1350.the early transition metals, but not in the late ones.

R eferences5 . Conclusions

¨[1] O. Gunnarsson, K. Schonhammer, Phys. Rev. B 22 (1980)3710.We have proposed a model where the ‘U , 0’

[2] P. Weightman, Rep. Prog. Phys. 45 (1982) 753;behavior of the Auger and APECS line shapes ofJ.C. Riviere, Atomic Energy Research Report AERE-early transition metals is described by a three-bodyR10384, 1982.

Green’s function dynamics. The solution of the two-[3] M. Cini, Solid State Commun. 20 (1976) 605.

holes-one-electron perturbation series is a formidable [4] M. Cini, Solid State Commun. 24 (1977) 681.problem that we solve proposing the core-ladder- [5] G.A. Sawatzky, Phys. Rev. Lett. 39 (1977) 504.

[6] C. Verdozzi, M. Cini, J.F. McGilp, G. Mondio, D. Norman,approximation: a new approach based on physical`J.A. Evans, A.D. Laine, P.S. Fowles, L. Duo, P. Weightman,approximations and results valid for the two-holes

Phys. Rev. B 43 (1991) 9550.case. The CLA is the first step of a procedure which[7] R.J. Cole, C.Verdozzi, M. Cini, P. Weigthmam, Phys. Rev. B

eventually leads to the exact summation of the 3- 49 (1994) 13329.body ladder series. Like perturbation theory and [8] C. Verdozzi, M. Cini, J.A. Evans, R.J. Cole, A.D. Laine, P.S.

`Fowles, L. Duo, P. Weightman, Europhys. Lett. 16 (8) (1991)other approaches to the many-body problem, it743;allows systematic improvements at the cost of moreM. Cini, C. Verdozzi, Physica Scripta T 41 (1992) 67;computation.C. Verdozzi, M. Cini, Phys. Rev. B 51 (1995) 7412.Physically CLA is well motivated and well bal-

[9] M. Cini, Surf. Sci. 87 (1979) 483.anced, and cluster calculations provide evidence that [10] V. Galitzkii, Soviet Phys. JEPT 7 (1958) 104.it correctly describes the effect of the screening [11] M. Cini, C. Verdozzi, J. Phys. C 1 (1989) 7457.

[12] M. Cini, M. De Crescenzi, F. Patella, N. Motta, M. Sastry, F.electron over the two final-state holes. The propertyRochet, R. Pasquali, A. Balzarotti, C. Verdozzi, Phys. Rev. Bof the CLA of remaining qualitatively correct even at41 (1990) 5685.rather strong coupling, conserving the Herglotz

[13] J.E. Houston, J.W. Rogers, R.R. Rye, F.L. Hutson, D.property, is an important feature. The success of Ramaker, Phys. Rev. B 34 (1986) 1215;CLA depends on the fact that the core approximation M. Cini, A. D’Andrea, in: K. Wandelt, G. Mondio, G.becomes accurate at strong coupling, and permits to Cubiotti (Eds.), Auger Spectroscopy and Electronic Struc-


ture, Springer Series in Surface Science, Vol. 18, Springer, [21] G.A. Sawatzky, Auger Photoelectron Coincidence Spectros-Heidelberg, 1989. copy, Treatise On Material Science and Technology, Vol. 30,

˚[14] P. Hedegard, F.U. Hillebrecht, Phys. Rev. B 34 (1986) 3045. Academic Press, New York, 1990.[15] D.K.G. de Boer, C. Haas, G.A. Sawatzky, J. Phys. F 14 [22] E. Jensen, R.A. Bartynski, S.L. Hulbert, E.D. Johnson, Rev.

(1984) 2769. Sci. Instrum. 63 (1992) 3013.[16] D.D. Sarma, S.R. Barman, S. Nimkar, H.R. Krishnamurthy, [23] G.A. Sawatzky, A. Lenselink, Phys. Rev. B 21 (1980) 1390.

Phys. Scripta 41 (1992) 184. ¨[24] O. Gunnarsson, K. Schonhammer, Phys. Rev. B 26 (1982)[17] M. Cini, V. Drchal, J. Phys. C 6 (1994) 8549; 2765.

M. Cini, V. Drchal, J. Electron Spectrosc. Relat. Phenom. 72 [25] M. Cini, C. Verdozzi, Solid State Commun. 57 (1986) 657.(1995) 151. [26] P.W. Anderson, Phys. Rev. Lett. 18 (1967) 1049.

[18] A. Marini, M. Cini, Phys. Rev. B 60 (1999) 11391. [27] R.D. Mattuck, A Guide To Feynman Diagrams in the Many-[19] H.W. Haak, G.A. Sawatzky, T.D. Thomas, Phys. Rev. Lett. Body Problem, McGraw-Hill, New York, 1976, Chapter 10.

41 (1978) 1825.[20] H.W. Haak, G.A. Sawatzky, L. Ungier, J.K. Gimziewsky,

T.D. Thomas, Rev. Sci. Instrum. 55 (1984) 696.

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