Jul 12, 1976 - Top,response function in the representation ofeq. (3); level at Cc and the complete a-level propagator Ga (see remainder, diagrams for the self ...
Volume 57A, number 5
PHYSICS LETTERS
12 July 1976
RELAXATION SHIFTS AND PLASMON EXCITATION IN ABSORPTION SPECTROSCOPY OF ADSORBED ATOMS B. GUMHALTER and D.M. NEWNS Department of Mathematics, Imperial College, London S. W. 7, England Received 18 May 1976 The excitation probability for surface plasmons, and the level shift due to the image force, are investigated when an electron is excited from a core level to an unoccupied virtual level of the adatom. The results differ strongly from the corresponding ones in XPS, in a manner depending on the energy variation of the width of the virtual level.
Spectroscopies in which a bound electron belonging to an adsorbed atom may be excited to a non-localised final state include UPS [1] and XPS [2] together with absorption spectroscopies [3] of the inelastic electron scattering and partial yield types. In the sudden approximation to XPS, where a core electron is assumed to be instantaneously removed from the adatom, the probabiity p(w) of losing energy hw to the surface is governed by the “zero work” sum rule that its first moment be equal to the “relaxation part” v of the shift between the no-loss edge and the gas phase position of the core level [4, 5]. If p(w) is dominated by a peak at the surface plasmon frequency w~,the area under the peak would then be --‘v/w 5 [5 7]. Corrections to the sudden approximation in XPS have been derivedby treating the outgoing electron as a classical particle [7, 8]; there is found to be a smaller plasmon excitation probability which only approaches u/wa for fast electrons [8]. However in UPS and the absorption spectrocopies the dynamics of the final state may in be bulk quantal rather classical. Inreduction X-ray absorption metals the than corresponding in plasmon creation probability during the core —~valence transition has been derived by [.angreth using a quantum framework [9]. In a core valence transition involving an adsorbed atom the final state may be an unoccupied atomic state whose lifetime is Imite due to tunneling into the substrate. An example might be a core -÷ valence 6s transition in adsorbed xenon. We present here a quanturn calculation of the relaxation shifts and surface plasmon creation probabilities when such a transition is observed in absorption spectroscopy [10]. Since we are able to carry the solution of our model quite far, -~
be of wider interest. The Hamiltonian of our model is
it may
0~‘~a + Cc°~1c+
H
Ca
knk + k
+
~ { VakCCk + h.c.} k
+
n
+w b b+X{n +n .~{b +b~+W n {n I a c a Here the Fermion creation operators c, c and c~
represent respectively: the adatom valence orbital Ia) of energy e~,the adatom core orbital Ic) of energy e~°, and the band states 1k) of energy ek belonging to the semi-infinite substrate. n = c~c;Vak describes the hopping between adatom and metal valence states which lifetime broadens the a-level. We assume for simplicity only inelastic losses to surface plasmons, assumed nondispersive and of frequency w~.b+ is the boson creation operator for the appropriate linear combination of surface plasmons to which the adatom is coupled. The coupling is assumed to involve the total charge2 (~a+ n~) onrelaxation the adatomshift with ,where the v iscoefficient approxiX ~vw5)” mately e2/4d for an adatom of radius d [6]. W0 is an intra-atomic Coulomb integral. The optical absorption is proportional to the imaginary part of the Fourier transform of the response function R(t) = —iO I Tx(t)x(0) 10) (2) —
‘
where t = time, 10> = ground state, T = time ordering operator, X = (c~ca+ c;c~);it is assumed that radiation only induces c a transitions. Our method of solution first involves a unitary transformation to H’ = U 1HU, where U = exp X(b+_ b) (~a+ n~)/w~},and —~
~—
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Volume 57A, numberS
PHYSICS LETTFRS
12 July 1976
H’ =Cafla+Ccflc+ ~Ckflk +
~D { VakCCk exp {X(b~ b)/w5} + h.c.}
+
w~b~b + Wnaflc.
(3)
Here Ca = V W°,Cc = U, W = W° 2v. We have so far neglected the purely static 2v, arising from the image of the adatom core [6], between e~ or and the gas phase levels e~or0,e~. Taking and e~= this + into account gives Ca = + v W restoring the correct positive sign of the relaxation shift. We now assume that 6a is sufficiently far above the Fermi level that the W term may be neglected. Then, since X’ X, the response function is just the product of the unperturbed propagator for the core level at Cc and the complete a-level propagator Ga (see fig. 1). Diagrams for Ga involve a and k lines joined at one-body vertices of type yak and Vka; a parity ~ = +1 is assigned to the former and 1 to the latter. A enters through (i) renormalising all Vak to ~k exp( v/2w and similarly Vka (ii) connecting every pair of one- 5), body vertices at times t 1 and t1 by the retarded interaction V ~(t1 t~) exp ( p~p~ exp( ~sItj tj I)} (4) A linked cluster theorem is obtained by intro ducing the Mayer-like f-functions f(r) = 0(r) 1. Now, in terms of frequency w, Dyson’s equation gives = {w c~ ~a(w)} 1, where the self-energy ~a is given by the series whose first few terms are shown in fig. 1. Of this series the first diagram gives ~a(w) = {A(w) ~ exp( v/we), where ~ = x~kIJ’~kI2~(w ~k) and A is the Hilbert transform of & it is dominant if v = 0 or if z~.In the V = 0 case Ga(W) merely represents an a-level shifted to Ca +A and with a lifetime broadening z~.In the latter case one obtains, when v ~‘ a Holstein renormalisation of ~ to ~ exp (— v/w 5) [11]. The one-plasmon approximation, appropriate when v/wa is small, as is the case in the present adsorbed atom problem, is described by the set of diagrams in fig. 1. We give the result only for the case where ~ is symmetric about w = 0, and where the band width D of ~ is large compared with both and c Then the optical absorption at frequency w is prop~tionalto ImR(w) =
+
—
+
±
±
~
~
—
—
—
.
-~-‘~~‘
~
(l—z)~
424 ~
~‘-~ac A)
zz~.
_____
2 ÷~2
(ci.,
Wac
A
w)2 +~2~(5)
Fig. 1. Top, response function in the representation of eq. (3); remainder, diagrams for the self energy ~a in one plasmon approximation. Dot-dash line and full line, unperturbed core level and k-level propagators; heavy line, full a-level propagator Ga. Broken line is f-function.
Here ~ = L~(a v) O(~2/D2), and Wac
~(~‘ = ~‘
0), A A(e~ W0) + W e~.The first term in (5)
describes elastic transitions from c~to the valence level resonance of width ~ the pole in this term occurs at a ferquency w which is shifted by surface screening 2/D2 in A. We can say that the relaxation shift seen only via the small term -vA in this experiment is reduced by .~2/D2relative to that in XPS. The second term in (5) describes core valence transitions accompanied by surface plasmon loss with probability z (v/w 2/~.~D2. This value is re5) relative ~IdA(a)/dCa}~ duced to XPS -~ byv~ the same factor ‘-~2/D2as is the relaxation shift. The reduction factors in relaxation shift and plasmon creation probability however involve the ratio of lifetime broadening z~to bandwidth D, rather than the more intuitively obvious ratio ‘~/~-‘~ In the untransformed representation (1) it is indeed the latter ‘—j
—~
criterion which enters into discussions of the adatom self energy [6]. Our model is more doubtfully valid in the region D, which would correspond to photoemission. In this case the “reduction factors” given by the model are of order unity. We conclude by noting that our principal result, that the reduction factors depend on the w-dependence of A(w) and i~(w)(without which they would be zero), is independent of our special ‘~—
choice of parameters and may have
wider implications.
Volume 57A, number 5
PHYSICS LETTERS
D.M.N. is grateful to S. Andersson for a valuable discussion, and B.G. would like to acknowledge the support of the British Scholarship Trust for Yugoslavs.
References [1] E.W. Plummer, in: Interactions on metal surfaces, ed. R. Gomer (Springer-Verlag, Berlin, New York, 1975). [2] J.T. Yates and N.E. Erickson, Surface Science 44(1974) 489; J.C. Fuggle and D. Menzel, private comm. [3] S. Andersson and U. Jostell, Surface Science 46 (1974) 625;
12 July 1976
D.E. Eastman and J.L. Freouf, Phys. Rev. Lett. 33 (1974) 1601. L4] D.C. Langreth, in: Collective properties of physical systems, eds. S. Lundqvist and B.!. Lundqvist, (Academic Press, New York, 1974). [5] B. Gumhalter and D.M. Newns, Phys. Lett. 53A (1975) 137. 161 A.C. Hewson and D.M. Newns, Japan. J. Appl. Phys., Suppi. 2, part 2 (1974) 121. [7] J. Harris, Solid State Commun. 16 (1975) 671. [8] M. Sunjic and D. Sokcevic, Solid State Commun. 18 (1976) 373. [9] D.C. Langreth, Phys. Rev. Lett. 26 (1971) 1229. [10] B. Gumhalter, Ph.D. thesis, Univ. of London (1976). [11] T. Holstein, Ann. Phys. 8(1959) 325.
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