Jun 2, 1975 - One of us (B.G.) would like to acknowledge the support of the British Scholarship Trust for Yugoslavs. [1] D.E. Eastman and J.E. Dernuth, Proc.
Volume 53A, number 2
PHYSICS LETTERS
2 June 1975
DYNAMIC SCREENING IN PHOTOEMISSION FROM ADSORBED SPECIES B. GUMHALTER and D.M. NEWNS Department of Mathematics, Imperial College, London, S. W. 7., England Received 12 April 1975 The level shift, plasmon satellites and electron-hole structure in photoemission from an adsorbate energy level are estimated within the framework of the surface response function.
The current importance of photoemission spectra in identifying the energy levels of chemisorbed species has brought to the fore the problem of relaxation shifts [1] and inelastic effects in such spectra. This problem is considered here for a long lifetime adsorbate level located outside the surface, such as the carbon a levels discussed by Eastman, or an adsorbate core level. We first introduce the retarded surface response function R~(w) q~(w)/qQ((J.)for a metal. Here the metal, taken to be jeffium, is assumed to lie entirely in the region z < 0 (z = O’Ts the sffrface), and qQ(w) is the Fourier transformof a charge distribution in a plane z = z0 (z0 > 0, w = frequency and Q = parallel c~mponentof wave18(~)ImRQ(w), where e = step function. The pervector); q’ is the image of q in z = 0. We define SQ(w) = —ir fect screening sum rule [2] fsQ(~)w1dw=j.;
Q-~0,
(1)
follows because on multiplying (1) by 2qQ(w) the left hand side is minus the static total image charge hmQ+ 0 q~(0)which must equal the scource charge for a conductor. RQ(w) can be expressed in terms of density operators, when the usual commutator arguments [2,31yield a useful foim of the f-sum rule o~
~,
jr S~(w)w dw = 2ireQ m
o
0
jp exp (2Qz) p(z) dz.
(2)
-~
Here p(z) is the substrate electron density at z. On substituting the Ansatz SQ(w) = A~i(w w5) into (1) and (2), we obtain in the limit Q -÷0the well known results c~= w~/-s/~A = where is the bulk plasma frequen. cy, in a strikingly simple way. We assume that the spectral density seen in photoemission is approximately N~(w),the spectral density of occupied states in the adsorbate orbital [4]. For simplicity the orbital is taken to be spherical with centre at = d, and its energy level is defined to be zero at d = Following Langreth [5] and neglecting (i) internal excitations of the adsorbate, and (ii) vibrational degrees of freedom, N~(w)is given by —
~,
00•
N~(~) =
~
Re
f exp[i(w
—
is)t
+
C(t)] dt;
,
s
=
0~,
(3)
where 2 dco, (4) C(t) = v~0 SQ(w)[l exp (iwt) + i~t] w and V~= 2ire2Q1 exp (—2Qd). We note that dC(t)/dt = 0 when t = 0, so that the first moment of N~(w)vanishes. The third term in brackets in (4) gives rise to the upward relaxation shift [6] in the adatom level from zero to v, where according to(l) u —~e2/4din the small Q, or large d, approximation. The remainder of(4) gives rise to satellites, caused by excitations left in the metal after ionization, whose first moment about w = v must be —u, to —
~
f
—
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Volume 53A, number 2
PHYSICS LETTERS
2 June 1975
satisfy the moment strength of the first
theorem. Use of the Ansatz SQ(w) =-~w~b(w ~c~)in(4) yields plasmon satellites, but the of these is of order v/w~which is usually small. The electron-hole pair contribution is considered to be more important because it forms a continuum in the w v region and needs to be estimated in interpreting experimental data. The contribution of the first two terms in brackets in (4) to the small ~ electron-hole structure in iV+(w) is dominated by the small w part of SQ(w) [5]. The Ansatz [5] SQ(w) = S~(0)wexp (_w/wQ) enables the w-integral to be done; here w0 Qu1~,where vv is substrate Fermi velocity. In this letter we perform (3) exactly [7] by putting WQ wm = vFJ2d since Q l/2d is the dominant region of Q space. Assuming the RPA and an infinite square potential barrier at the surface of the metal, it has been shown [8] that RQ(w) = [1 Q(W)I /[l + where CQ(w) is related to a matrix D + A. In a Fermi-Thomas approximation we (a) neglect the off diagonal elementsA, when e~(w) becomes an integral involving the Lindhard dielectric function, and (b) use the FermiThomas approximation to the real part of the latter. Evaluating the Q-sum to lowest order in d~, we obtain ‘—
—
N~(w)= exp[(w
(v w)~~’YO(v w)/F(y)w~. (5) 2k~,with kTF = Fermi-Thomas screening wave-vector and kF = Fermi wavevector Here y =substrate; log~(l.S9 for the FkTrd)/8d = gamma function. The first moment ofN~(w)in (5) is —k~’y/2d in atomic units. Our somewhat crude result (5), which is not valid for too small d, give an asymmetric spreading out towards negative w of the shifted adsorbate level, over an energy range given by the first moment and estimated as some fraction of a volt. This is small compared with v, which can be of the order of two volts. Comparison is also needed with the envelope of the Franck-Condon structure, arising from the neglected atomic vibrations, which may dominate the line shape in many cases and tends to obscure the electron-hole structure. The neglected adsorbate internal electronic structure should resemble plasmons in giving a level shift and resolvable satellites; non-cancellation of the level shift and chemical shift between adsorbate and gas phases may confuse the interpretation of —
U)/Wm]
-~
—
V.
In conclusion, the effect we calculate is very distance dependant and is probably most easily detectable for very small d, such as in the case of an adatom-surface bonding orbital, which would however be quantitatively outside the range of the present model. For example the very weak photoemission structure seen for hydrogen on nickel and some other transition metals [9] coincides with the predicted position [10] of such a level. Brenig [7] has suggested that the exceptionally weak intensity is attributable to coupling to spin fluctuations. The electron hole effect should lead to a very significant line broadening in the manner of(S) with Wrn several volts and ‘y ~ 1, there will be only a modified effect for levels lying in the metal d band. We shall not report here on results for finite lifetime levels, except to note that arguments [6] for the image screening of the intra-adatom Coulomb integral U in the adiabatic regime are verified by an extension of the methods of ref. [6] to include spin degeneracy; for example the unitary transformation in eq. (7) and (8) of ref. [6] now replaces U by U 2v. —
One of us (B.G.) would like to acknowledge the support of the British Scholarship Trust for Yugoslavs. [1]
D.E. Eastman and J.E. Dernuth, Proc. 2nd ICSS, Kyoto 1974, in press [21 D. Pines and P. Noziires, Theory of quantum liquids I (\V.A. Benjamin Inc., New York, 1966). [31 J. Harris and A. Griffin, Can. J. Phys. 48 (1970) 2592; J.E. Inglesfield and E. Wikborg, Solid State Commun. 15(1974)1727. 141 5. Doniach, Phys. Rev. B 2 (1970) 3898. [5] D.C. Langreth, Phys. Rev. Bi (1970) 471; P. Nozi~resand C.J. de Dominicis, Phys. Rev. 178 (1969) 1072. [6] A.C. Hewson and D.M. Newns, Proc. 2nd ICSS, Kyoto 1974, in press. [7] W. Brenig, private communication. [8] D.M. Newns, Phys. Rev. B 1(1970)3304. [9] G. Ertl, private communication. [10] D.M. Newns, Phys. Rev. 178 (1969) 1123.
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