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A soluble relaxation model for core level spectroscopy on adsorbed atoms
Surface Science 17 (1978) 421-448 0 North-Ho~and ~bIish~g Company
ASOLUBLERELAXATIONMODELFORCORELEVELSPECTROSCOPYON ADSORBEDATOMS J.W. GADZUK NationalBureau of Standards, Washington,DC 20234, USA
S. DONIACH * Department of Applied Physics, Stanford University,Stanford, California94305, USA Received 10 March 1978; manuscript received in final form 1 June 1978
The photoelectron spectrum of core levels in adsorbed atoms is calculated taking into account the screening of the core hole due to both displaced surface plasmons (image charges) and actual transfer of screening charge into the lowest unfiied valence levels of the adsorbate. The theory is detailed on a simplified but exactly soluble model, permitting parametric numerical studies of the spectrum while maintaining close contact with physical understanding. Recent theory on adsorbate relaxation effects due to Gumhalter and Newns, Gunnarsson and Schonhammer, Lang and Williams, and others are considered in the light of this model.
1. Introduction Core level X-ray photoelectron spectroscopy (XPS) in or on metals cannot be satisfactorily understood in terms of single electron theories, due to so-called final state relaxation effects. Within the domain of extra-atomic relaxation, the localized core hole is screened by the free electrons of the conduction band. The Coulomb interaction between the screening charge, often written in simplified form in terms of displaced plasmon density ~uctuations, and the hole charge gives rise to the extra-atomic relaxation or screening energy shift of the ejected electron. Depending upon the dynamics of the hole creation and screening, it is also possible to leave the electron gas (metal) in an excited state, in which case the ejected electron must emerge with a correspondin~y reduced energy. Features in a spectrum which are due to this effect include shakeup satellites. In the limit in which the hole potential is instantaneously switched on (the sudden limit), there is a one to one corre* Research supported by Army Research Office, Durham, NC, and NSF through Stanford Synchrotron Radiation Laboratory. 427
428
J. W. Gadzuk, S. DonL2ch /Soluble
relaxation (node1
spondence between the photoelectron spectrum and the spectral function (imaginary part of the Green’s function) for a transient core hole propagating in the reactive electron gas [2-61. A sudden limit sum rule has been derived which relates the intensity in the relaxation shifted main peak with the intensity and position of the satellites [ 1,2,7]. At least for XPS within the bulk, the short time response to the core hole involving real and virtual plasmons results in the screening of the long range bare Coulomb potential and the creation of plasmon satellites. The long time response to the screened core potential shows up as the excitation of electron-hole pairs, causing a skewing, to the higher binding energy side, of the main XPS line [6]. For the most part, the electron gas based theories are given in terms of response functions which do not depend upon atomic-like properties of the metal. Exceptions to this are the excitonic [8] or lowest-unfilled-atomic orbital [9] models in which the screening charge is built up from wave functions which are orthogonal to the core state. Another important class of relaxation effects occurs in XPS from atoms which are adsorbed on metal surfaces. Considerable theoretical work has depended upon the proposition that the extra-atomic screening can be represented in terms of image charges which are in turn expressed as a coherent displacement of surface plasmons [ 11,121. Although this is obviously valid for large adatom surface separations, some interesting new features must be considered for separations which are small enough to be chemically significant. Numerical calculations by Lang and Williams [13] have shown that under some circumstances, the screening charge associated with an adsorbed atom more closely resembles the lowest-unfilled-atomic orbital [9] on the adatom than a surface or image screening charge within the substrate [ 11,121 which would appear to vitiate the reliability of the surface plasmon models. However, since these calculations pertain to the purely static aspects of the problem, some dynamical effects are missed which are quite important in understanding the surface screening process, the resulting photoelectron spectrum, and the relation between the surface plasmon and Lang-Williamns relaxation response theories [ 10-141. The following tempera1 sequence of events can be envisioned occurring within the life of an adatom core hole. Upon switching on the core hole, surface plasmons respond on a time scale of the order of w,-‘, the inverse surface plasmon frequency, establishing an image potential shift and surface plasmon sateliites. Since the adatom is in chemical contact with the substrate, valence electrons may transfer between the two at a rate which is roughly proportional to the bonding-antibonding level separation. Usually this rate is much slower than the surface plasmon screening rate. Once an electron hop from the substrate to the lowest-unfilled adatom orbital occurs, charge neutrality on the adatom is restored, coupling to the surface plasmons ceases, and the surface version of Herbst screening [9] is set up, as pointed out by Lang and Williams [13]. However, since the reverse hop can still occur, the dynamics of the competing surface plasmon screening and valence electron hopping must be considered. This has already been done to a certain
J. W. Gadzuk, S. DonL;rch/Soluble relaxation model
429
extent by Gumhalter and Newns [12] who considered modification to X-ray absorption on the adatom, due to the possibility of electron hopping and surface plasmon screening. In this work, we consider a highly simplified, but exactly soluble model for the core hole spectral function in which all the dynamical features are treated in a democratic way. As the spectral functions can be given in closed algebraic form and the relaxation process is completely characterized by three parameters, systematic studies into the observable consequences of various parameter sets are trivial. Thus this communication should serve as a useful complement to that of Schonhammer and Gunnarsson on a similar topic and allows an extension of their ideas to include shakeup effects. The structure of the paper is as follows. In section 2, the general model is established and simplifying approximations are introduced which make possible closed form calculations of the transient. hole spectral function introduced in section 3. Possible electron-hole pair shakeup of the Mahan-Nozieres-De Dominicis [3,4] (MND) type which skews XPS lines in the bulk is discussed in terms of GumhalterNewns formulae in section 4. The resulting spectral functions are discussed in detail for several limiting sets of parameters in section 5 and are interpreted in the light of already existing knowledge. General conclusions are given in section 6. 2. Theoretical model
The basic relaxation processes associated with sudden core hole creation on an adatom can be uderstood in terms of the model Hamiltonian advocated by Gumhalter and Newns [1,11,12]: &,t
= e,& + [e, - &(l
- $)I&
+T
eknk + T
(v&&
+ h.c.)
(1) which has the following meaning. The first term describes the core state with eigenvalue E, (quite negative with respect to a zero at the Fermi level) whose occupation is either n, = 1 in the initial state or n, = 0 in the final state. The second term (f fYd in eq. (1) is mainly responsible for the interesting possibilities suggested in the recent theories [ 12-141. The eigenvalue E, is that of the lowest unoccupied adatom orbital in the initial state, hence n, = 0. The intra-atomic Coulomb integral U,, (positive in sign) is a measure of the attraction between the positive hole charge and the negative charge that could occupy the e, orbital. Since n, = 1 in the initial state I$,” = esn, whereas n, = 0 in the final photoionized state, implies Hafin = (E, U&r,. The occupation number of the adatom orbital in the final state depends sensitively on the sign of E”,E E, - U,, with lim;O,_,
n, = 1
and
lim~~~+,n,
=0 .
(2)
430
J. W. Gadzuk, S. Doniach /Soluble relaxation model
The third term in eq. (1) represents the conduction band states of the substrate. Hopping between the adatom orbital and the substrate is allowed for by the fourth term where the strength of the coupling depends upon V;J, the Anderson-Newns matrix elements. The next two terms describe the surface plasmons and their coupling to the ion formed in the photoionization event. The strength of this interaction which under certain limits, reduces to the quantum mechanical version of the image potential shift [ 1 l] is characterized by the coupling function b, to be explicitly detailed in section 5. Note that in the initial state n, = 1, n, z 0 and 1 n, -n, = 0 whereas in the final state n, = 0 so the strength of the interaction is proportional to X(1 - n,). Since n, is an operator with a non-trivial time dependence due to the hopping term, the interplay between the surface plasmon and hole hopping dynamics becomes all important in understanding the relaxation process. Finally we include a phenomenological term IY~__~,to be discussed in section 4, which allows for hole decay and MND-like effects. Assuming [17, He-r,] = 0, the total spectral function is then the convolution of independent spectral functions for Hand&-h [5,6]. To illustrate the various physical features which are implicit in eq. (1) we will adopt a well defined set of model assumptions which hopefully permit sufficient new insight to justify the simplicity of the model. Specifically we consider: (a) A single level substrate which implies &e& -+ e&z,; this jUSt says that the electron which hops between the adatom and substrate via Vak in eq. (l), is always in state E, when in the substrate; as discussed later, this is a narrow band limit. (b) Completely unoccupied adatom orbital E, in the initial state, so n, = 0; equivalent to assuming a discrete state rather than a broadened resonance [13-151; f, sufficiently high that V,k mixing in initial state can be neglected. (c) Discrete final state, whose occupation number depends largely on the sign of f”, and is equivalent to assuming no broadened resonance. (d) Dispersionless surface plasmons with a Q * qll cutoff if necessary (411being parallel to the plane of the surface). (e) Only one surface plasmon satellite; not bad since satellites are rather weak anyway. With this set of specifications, the initial no hole and final one hole state Hamiltonian, from eq. (l), are respectively Hfin = emnm +
V(C~C,
+
h.c.) + w,btb + A(1 - n,) (bt + b) + &n, .
(3)
The Hilbert space of possible states, within the present model defined by conditions a-e is taken to be completely spanned by the four eigenfunctions of the number operators, In,, n,, n,) with nP = 0, 1 corresponding to zero or one intrinsic plasmon and with the constraint n, + n, = 1. Adopting this notation, the initial state, before photoionization, is I Gin)
=
I Oa9lm, 0,) 7
(4)
J. W. Gadzuk, S. Doniach /Soluble
relaxation model
431
where ($inlHinlJ/in) = E,,, and the energy zero has been displaced by e,. The final states can be written as linear combinations
+ Y(ej)lO,,1,~
1,) + h(ej)l la9 Orn9 lp)
(5)
9
where the four coefficients and four eigenvalues ej (j = l-4) can be given in closed algebraic form. At this point, the eigenstates of the final one hole Hamiltonian, eq. (3), are determined by the 4 X 4 matrix representation of the Schrodinger equation: V
V
Em -
0 \o
a
0
0
E,--E
0
EX
x
E,+W,
0
V
--E
fl
1‘
v
CO
’
(6)
and the normalization condition lcrl2 + IpI t ly12 t 16 I* = 1. The eigenvalues of eq. (6) are given as the solutions of the quartic equation obtained by setting the secular determinant equal to zero: e4 + cres + C2EZ+ C3f
+ c4 = 0 )
(W
where Cl = -2&
+ E, + w,) )
0)
+2Cd~)+EmC~“,A2-2V29(7c)
C2-(E,+W,)(E”,+W,)+(Em+E”,)(Em+~~
C3 E -iFaEm(&
+ Em +
20,)
- (Em + W,) (E”,+ a,) (E”,+ Wm)
+ P(2E”, + w,) + 2 VZ(E”,+ Em + w,) , c4 =zaem(E"a+
-
P(g&,
W,)(E,
+ a,)-
(74
PE",
+ (& + c.d,)(E, + w,)-
V2).
(7e)
The four eigenvalues
ei which are roots of eq. (7a) can be expressed explicitly as The algebraically cumbersome nature of these expressions prohibits any additional physical insights however. The coefficients in the eigenfunction expansion, eq. (5), are eji= Ej(E",,Em9 a,, X, v) [161.
P(Ej) = [l + V2/U:,j + A2(V2 + a:,j)/(a3,]ff4,j a(Ej) = vD(Ej)lal,j
2
- V2)2]-1’2 9
(84 @b)
432
J. W. Gadzuk, S. Doniach /Soluble relaxation model
YCej) = -A a4,jP(~ji)l(a3,~4,j - v2)
9
s(fj) =
VAP(S)l("3,fa4,j - v2)
(84
7
with al,j=
fl3,j
E", -
=
Ej
,
Em + Cd, -
a*,j=
Ej
,
Em
cl4,j
-
C?j ,
=
E”,+ Cd,
-
Ej
.
Eqs. (5), (7) and (8) comprise a working solution to the problem represented by the eq. (3) Hamiltonian. The ensuing sections will relate these solutions to photoelectron spectra, both for well known limiting choices of parameters as well as for general cases.
3. Spectral functions The photoelectron spectrum is readily calculated only in the two extreme limits of adiabatic and sudden switching on of the hole potential [ 1,6,7,17,18]. Creation of the core hole changes the many-body Hamiltonian by an amount AH * VhOrex r(Sf/at) where r is the time duration of the hole switch-on and U-I’) a mean time rate of change of the Hamiltonian due to removal of the photoelectron [ 171. These considerations relate to eq. (l), since U&l - nJ n, and h(1 - n3 (b t b+) are in effect time dependent potentials due to the implicit time dependence of n, = n,(t). The adiabatic limit leaves the system always in the ground state of the potential determined by the instantaneous charge distribution of the ion cores plus relaxing electrons. This condition is satisfied when T S h/e*, where E* is a “typical” excitation energy of the relaxing system. If the adiabatic limit is achieved, all the photoelectrons emerge with the same “totally relaxed energy” and there are no interesting dynamical problems to consider. This limit is not realized under actual experimental conditions which are better thought of in the sudden limit where the hole switch on time r + 0, corresponding to instantaneous removal of the photoelectron. Thus although r(FN/&) can be represented as a discontinuous stepfunction in time, the many-body wave function is continuous, which facilitates spectral decomposition of the initial state within the manifold of eigenstates (eq. (5)) of Grin (eq. (3)). In general, the sudden limit hole spectral function is simply [5] : N+(E)=C
~(e-Ej)I(J/inl~fin,j)l*
*
(9)
i Specifically for the model characterized and (5), eq. (9) becomes N+(E) =
C &(Ei
ej)
IP(e
3
by the initial and final states of eqs. (4)
(10)
J. W. Gadzuk, S, Doniach /Soluble relaxation model
433
which is a distribution of delta functions whose positions and strengths are given by eqs. (7) and (8). As mentioned earlier, both hole decay and MND edge effects are allowed for through the last term, &_h, in eq. (1). By ansatz, [He-r,, Hi”] = [Hea, Hfin] = 0, sothe total spectral function can be written as a convolution - e’) N+(E’) de’ ,
NT&E) =,-D&e
which with eq. (10) is NT&E)
= 7
Ii+j)?
De-
h(E -
Ej) .
(11)
In eq. (1 l), De_his the spectral function associated with &_h, to be discussed in more detail in the next section. As an example, finite core hole lifetimes are usually considered simply by taking De_hto be a Lorentzian of width y [6,14]. 4. Pair shakeup and hole decay The still somewhat controversial role of electron-hole pair edge effects in X-ray emission and absorption spectroscopy [3,4,19] is much more firmly established in XPS of metals [6,20]. The sudden creation of the core hole excites a large number of low energy electron hole pairs, which result in the characteristic MND infrared divergent spectral function
D(e)- IE~-~+~~(--E),
(12)
where QI is determined by the properties of the screened core hole potential and equals (6r=e/n)* for s-wave scattering only with 6 the phase shift. Although the MND derivations of this formula are strictly speaking valid only as E -+ 0, Minnhagen has numerically verified eq. (12) for all energies less than the plasmon energy of the simple metal in question. Doniach and Sunjic [6] realizing that core hole lifetime effects [22] would soften the divergence, convolved eq. (12) with a Lorentzian to obtain a now well confirmed [20] expression for the asymmetric XPS lineshape, skewed to the higher binding energy (lower kinetic energy) side of the spectrum. In an exactly analogous manner, core level XPS on adsorbed atoms is accompanied by pair production, which in turn influences the shape of the photoelectron line. Obviously the coupling of the core hole outside the metal with the pair excitations within takes on a much different functional form, but still the principles remain the same. In fact Gumhalter and Newns [23] have worked out a first order theory for the adsorbate analog of the MND expression, which is, in the ThomasFermi approximation to the RPA [24a], &N(E)
= [WE)
exp(f/w,)
I E I-’
‘fl I(W9 w$,>,
(13)
434
J. W. Cadzuk, S. Doninch f Soluble relaxation model
where w, = r&/2s with bF the substrate Fermi velocity and s the ion core-image plane separation, f= log&l.59 KF~ s)/@k$ with KFT and k~ the Fermi-Thomas and Fermi wavevectors, and I’ the gamma function. Again, finite core hole lifetimes can be accounted for by convolving BGN(e) with a Lorentzian. Such an intergral has been previously given [6b] as
1 exPW%)
_
rrwf,
J-V -n (e* + y2)f l -flP
cos @_ &L)
>
(14)
where
e = (f-
1) tan-i(e/r)
- 7rf/2 - $6~~ ,
and 6D is a small correction that can be neglected for 1~15 a few y. In the limit of wm+w, eq. (14) has an algebraic structure identical with the Doniach-Sunjid expression. Parenthetica~y note that the asymmetry of the adsorbate line is a function of
0.8
0.6
Fig. 1. Asymmetric XPS lineshape for an adsorbate core level (eq. (14)) as a function of energy, in units of the level width y. A “typical” value for the asymmetry parameter f(= 0.1) is used is treated parametrically. The inset shows the Gumhalterand the s-dependent cutoff q/w, Newns estimates offversus s, taking r, as a parameter.
J. W. Gadzuk, S. Doniach /Soluble relaxation model
435
distance from the surface, due to the fact that f, y, and w, are functions of s. To get a feel for the qualitative variations to be expected in the lineshapes as a function of atom-metal separations, eq. (14) has been evaluated as a function of e/y, treating y/w, parameterically. Some resulting lineshapes are displayed in fig. 1, together with values of fversus s shown in the inset. Such experimental adsorbate lineshapes have been observed [24b]. Finally we note that an alternative (and sometimes even more dominant) asymmetry mechanism has been proposed also by Gumhalter [24b] in which the suddenly created core hole causes electron-hole pair creation on the adatom, as well as within the substrate. For this to be significant, the position of the final state screening orbital energy ‘E”,must be near the Fermi level as the asymmetry index is given by [24c] f= (U,, p,,J2 where pa0 is a Lorentzian-like adatom local density of states centered at E”, and broadened by A - I V12. Since Y is a function of s, this mechanism will also produce an XPS lineshape which becomes more symmetric as s decreases.
5. Results The numerical consequences of the preceding four sections shall now be explored for two limiting ranges of parameters and then for some arbitrary (but physically significant) sets. 5.1. Surface plasmon response
A popular model of the surface relaxation response [25] is the so-called surface plasmon or image charge theory [ 111. The dielectric reponse of the substrate to the appearance of a fxed charge outside results in a screening charge localized in the surface region. If this induced charge is constrained to remain within the substrate, then it can be decomposed into a superposition of surface density fluctuations of the metal-adsorbate interface, which in the long wavelength limit are just surface plasmons [ 11,241. The observable consequences, in a photoelectron spectrum, of the surface plasmon response are an upward extra-atomic relaxation or screening energy shift, which for large atom-substrate separations is the classical image potential shift, and also shakeup satellites corresponding to the intrinsic excitation of surface plasmons. The important criterion which determines whether the screening charge remains in the substrate is the value of the surface plasmon energy compared to the level width A = p(ef) I VI2 of the unoccupied orbital, where p is the Fermi level density of states. When A 4 AU,, the surface plasmons respond as if the adparticle retains one unit of positive charge throughout the entire screening process. Only when the hopping time (--A/A) for an electron to transfer from the substrate to the unfilled orbital & becomes comparable with the plasmon response time does the Herbst-Lang-Williams screening dominate. This limit will be discussed in section 5.2.
436
J. W. Gadzuk, S, Doniach /Soluble relaxation model
Within the context of the present model, the surface plasmon screening limit occurs when V* + 0 in eqs. (l)-(3). Realizing that n, = 0 for all times of importance, the final state Hamiltonian is just + w,btb + X(bt + b) ,
Hrrn = e,n,
which is easily diagonal&d, since CY= 6 = 0 in the eigenstate expansion, eigenvalues of the ionized system, from eq. (6) are
eq. (5). The
t 4hz/w,Z)r/Z )
E = E, + &JS L!Z &(I
(154
where for x/w, Q 1 ) E_~:E,
- X2/% ,
(15b)
E+=:f,+W,+P/o,.
(15c) The E_ state is the eigenstate of the screening electron, having a reduced energy due to the additional attraction between the induced charge and the ion core. The resulting photoelectron energy is thus “relaxation upshifted.” From eqs. (8), (lo), and (15a), the spectral function is [lt (1 t b2)‘j212
N+(f) = 6(E - E_)
with b2 G 4h2/w,?. The interpretation written in the form: N+(E)“8
A2
(
E-f,
[l - (1 t b2)‘j212
[1 t (1 t b2)‘j212 + b2 + ‘(e -et)
t-
as
[1 _ (1 + b2)‘/2]2 +
of N+(E) is particularly
I[
l--$tO($J]
b2
transparent
3 (16)
when
+#-$)
4 [1+6(9] ,
(17)
which is valid when X/w, < 1. First consider the schematic functions shown in fig. 2. The first and second panels show the Koopmans theorem unrelaxed and the adiabatically relaxed limits. Due to the equivalence X2/o, = Aq the relaxation energy, the hole-plasmon coupling function can be specified in terms of an acceptable image-like interaction; that is X = (oS Vi/image(s))‘12 where for example, Ae, = vimage (s) could be taken equal to e2/4s , e2/4(s + Jje2
(18a) K &)
)
%ds -
WI
e?ll, 0)
%ds+ E(4ll,O)
exp(-%s)
3
(18~)
J. W. Gadzuk, S. Doniach /Soluble relaxation model
(a)
Unrelaxed
(b)
Adiabatic
1
relaxed
i
431
’ N’(c’
l,..
~~~~~
Fig. 2. Schematic unbroadened hole spectral functions and thus photoelectron energy distributions in various degrees of approximations, all of which ignore electron-hole pair excitations. (a) Unrelaxed spectrum which applies when Koopmans theorem is satisfied. (b) Adiabatically relaxed spectrum which occurs if the hole potential is slowly switched on. (c) Sudden limit spectrum in which all satellites are forced into one “mean” surface plasmon shakeup peak. (d) Infinite order independent boson model spectrum which is a Poisson distribution.
depending upon the level of accuracy required [ 11 ,12,23,24]. In eq. (18) e(q ,,, 0) is the transverse wave mimber dependent metal-vacuum surface dielectric function and e,& is the dielectric constant of the adlayer material [11,24,26]. Detailed numerical consequences have been worked out elsewhere [24]. The complete spectrum, as given by eq. (17), is shown in fig. 2c. The relaxation shifted main peak whose intensity is reduced by a factor of l-a, with a E X2/~ = AE,/w,, is compensated for by an effective surface plasmon satellite of intensity a, thus preserving the normalization IN+(e) de = 1. Since the two peaked spectrum given by eq. (17) is an approximation to the Poisson distributed spectrum of the infinite order independent boson model (IBM) [ 1,151,
N_(E) = ema
n$o (an/n!)
ti(e - em + au, - nw$ ,
(19)
shown in fig. 2d, the low energy satellite is displaced by an amount “X*/o, = aw, below the position of the first satellite in the complete IBM spectrum. In effect, the total spectral weight for all the satellites (n > 1 in eq. (19)) is lumped into one
438
J. W. Gadzuk, S. Doniach /Soluble relaxation model
satellite whose position [2,5,7,18,27,28] s
and intensity
is consistent
with the sudden limit sum rule
e N+(e) de = era ,
(20)
where ef,, is the frozen orbital or unrelaxed energy shown in fig. 2a. As the holesurface plasmon coupling increases, the upward relaxation shift of the leading peak also increases. However this is accompanied by a redistribution of intensity into the satellite system, which when approximated by a single structure, must shift downards in energy below o,, as shown in fig. 2c in order to maintain eq. (20), the model independent sum rule. A similar situation arose in the lowest order hole self energy calculations (in a homogeneous electron gas) of B. Lundqvist [2] compared to the infinite order IBM presented by Langreth [5]. In the theory of Lundqvist all the satellites were coalesced into one “effective plasmon satellite” whose separation from the main peak increased with increasing coupling constants in an exactly analogous manner to the case considered here. The numerical information implicit in eqs. (15) and (16) is shown in fig. 3 for the case in which w, = 0. The two-peaked spectrum can be ascertained in the following way. For a specific value of X/w,, the energies of the main and satellite peaks are given by the dashed curves, plotted as a function of h/w,, on the right hand axis. The intensities of the peaks are determined from the full curves and left hand axis at the applicable values of e = e+(h) and e__(X). As an example, with (X/o,)2 = 2, E-/O, = 1, E+/ C+ = -2, I/3_12 = 2/3, and I/3+1’= l/3. In this figure, the energy scale has been shifted so that the zero coincides with the Koopmans theorem unrelaxed energy. Thus the main peak is precisely at the surface plasmon relaxation energy. The energy scale increases from left to right when considered in terms of the kinetic energy of the photoelectron. I
I
w,=O \ Satellite l.O- \ (c+) L \
I
I
I
/-
\
Main peak
5
/
Fig. 3. Numerical consequences of eqs. (15) and (16) which show the energies and relative intensities of the main and satellite peaks, for the “surface plasmon screening only” limit. The horizontal axis gives the kinetic energy of the photoelectron with respect to a zero at the unrelaxed energy. To use this curve, fist ascertain the coupling constant. The right-hand vertical axis and dashed curves give the energies for the main and satellite peaks. The left-hand axis and full curves show the intensities at the specified energies.
J. W. Gudzuk, S. Donriich /Soluble relaxation model
439
One feature of these “data” is a direct result (artifact?) of the two-state-only model; namely that the intensity in the surface plasmon satellite is always
+
F (?‘,,gcftc, +h.c.1.
(22)
440
J. W. Gadzuk, S. Don&h /Soluble relaxatioti model
With approximations a-c introduced in section 2, the initial no hole and final one hole state Hamiltonians, from eq. (22) are Hi, = E,n, + e,n, Hfin
= ~,n,
+
3
t
V(C,C,
(234
+
h.c.) + e,n,.
G=)
The initial and final eigenstates are given by eq. (4) and eq. (5) (with y = 6 = 0) respectively. The fundamental process treated by both the SC theory and model implicit in eq. (23) is the following. Initially one unit of electron charge which is to do the screening resides in the metal. Upon photoionization, the extra attraction on the adatom due to the core hole pulls the unfilled level E, downards, maybe to the Fermi level or lower, and the renormalized energy is E”a= ea - U,,. In the final state, the Valakterm couples the metal state E, with the renormalized excited adatom state E”,to form what amounts to bonding and anti-bonding metal-adatom states whose eigenvalues are given by the secular determinant of the upper left-hand 2 X 2 matrix in eq. (6); that is - Em)]2 + V2}1’2 .
e* = 1,& + em) i: {[&
(24)
The “screening electron” must now occupy one or the other of these states. If the screening charge occupies the lowest energy bonding state e_, then the photoelectron must be ejected with its adiabatic relaxed energy. On the other hand, if the photoionized system is left in an excited state, then the photoelectron emerges with a kinetic energy that is precisely the excitation energy below the adiabatic peak. In the model described by eq. (23), the only excited state is one in which the screening electron occupies the “anti-bonding” state E+. From eq. (lo), the two-peaked spectral function is N+(e) = 8(E - E+) IP(
•t S(E - E_) IP(
5
(25)
which can be interpreted as an adiabatic relaxed main peak at E = E- plus a shakeup satellite at e = E+. The intensities in the two peaks, obtained by inserting eq. (24) into eq. (8) (with X = 0) are given by
IB(E*)12 = [I + (E*/y121-l
(264
with e*/V=p p=ZJ2v.
f (1 t p2)“2
,
Wb) (26~)
In eqs. (26a-c), E, has been set to zero since the absolute value of the ground state of the system is not required. All the information pertaining to photoelectron spectra which is contained in the two-state limit of the GS model, as given by eqs. (2&-c), is compactly displayed in fig. 4. For a given value of p on the right hand axis (whose sign depends upon the particular system), the positions of the energy
J. W. Gadzuk, S. Doniach /Soluble relaxation model
-
I .oc)-
441
2
0.7: i0.5c )-
I
0.25 Y ‘i z
0
-3
-2
/I /
/ / / e_ /
0
/I ,
/
2
:
/
0
3
k/V)
-
> N ’ 2 Y” VI a
-I
Is+ I
-. -2
Fig. 4. Numerical consequences of eqs. (26a-c) which show the energies and relative intensities of the main and satellite peaks, for the “unfiied atomic orbital only” limit. The same procedure as described in fig. 3 is to be invoked for using this figure, substituting the parameter p for the coupling constant. Note that if p > 0, the largest intensity is in the main peak whereas for p < 0, the largest intensity is seen in the satellite. In this figure, the photoelectron kinetic energy is the negative of E.
main and satellite peaks are given by the so-labelled dashed curves and the intensities by the value of the full curve (left hand axis) at the energies just determined. Note that when p > 0, the intensity in the leading peak is less than that in the satellite whereas for p < 0, the adiabatic peak carries the majority intensity. To further illustrate the points to be made here, some two-peaked spectra are shown in fig. 5 for a range of parameters. These figures can be regarded as an attempt to
‘.OOl
1
’
1
I
’
’ I
(-c/V) Fig. 5. Some explicit “unfiied orbital only” spectra, from eqs. (26a-c) or fig. 4, as a function of photoelectron kinetic energy, treating p parametrically. The related main and satellite peaks are connected by light horizontal lines.
442
J. W. Gadzuk, S. Don&h
/Soluble relaxatioi2 model
reproduce the results of GS within our model. To the extent that this has been successful, the two-peaked structure of the SG spectra can be viewed as a relaxed peak and a shakeup satellite (rather than an “unrelaxed” peak). The satellite corresponds to the excited state in which the electron occupies a basically antibonding orbital formed by the linear combination of the renormalized unfilled atomic orbital with E”, and the metal “orbital” with E,. If this is a correct interpretation, then in this state the screening charge density should display nodal tendencies between the adatom and the substrate. As mentioned earlier, the crucial feature which determines the relative intensities of the satellite to adiabatic peak is the sign of p. This is easily illustrated within the physical picture depicted in fig. 6. When p < 0, the bonding adatom-substrate screening orbital ~-O,~O)=(YIla,O,~+pIOa,
1,)
is more strongly localized on the adatom (i.e., Ia_ I* > Ifl_ I*) and thus the intensity in the leading peak, proportional to the modules squared overlap of Gin with I/_, is less than that in the satellite (whose intensity = ]<$in]$+)1* > 0.5). As E”, exceeds E,, either due to a larger value of E, or smaller U,, or effective metal energy e,, then the majority intensity
is found in the adiabatic peak, as can be understood from the right hand column in fig. 6. Thus there should be, no mystery in understanding the manner in which a spectrum can undergo a smooth transition from mostly “main” to mostly satellite intensity (or relaxed to unrelaxed) as the model parameters are varied. Finally, we note the dependence of the spectrum on the magnitude of V. As V greatly exceeds & (and thus by inference the screening electron bandwidth in a real solid, since E, was taken to be zero), the magnitude of V and not the properties of the indivudual entities (as represented by E, and Q determines the final state spectral function. In this limit, the substrate and adatom must contribute in a completely symmetric (gerade) or antisymmetric (ungerade) way to the possible system final eigenstates, in which case the projection of a unit charge onto the adatom state (or “symmetrically,” the substrate) requires equal contributions from the gerade (E_) and ungerade (E+) states. The consequences of this as manifested in our model photoelectron spectrum are that lim V-large orp-tcllS@*)12= 0.5 2
as is apparent in fig. 4. Cederbaum and Domcke have given a very nice discussion of a similar point with regard to core level photoionization in simple diatomic molecules [35]. The present (X = 0 limit) model can be considered as a narrow-band or strongcoupling limit of the SG theory. This becomes clear upon contemplation of the V dependence of the SG spectrum. In the weak-coupling limit, V& merely converts
443
J. W. Gadzuk, S. Doniach /Soluble relaxation model
,-.E+ ** Em -.
-.
. . E
_‘-0”
.
-, .-
*in
;a
JI_
&y m
a
+in
*+ nn
&+ m
*I-
a
Jlin
a
m
E
Fig. 6. Molecular orbital analog of the unfiied orbital relaxation model, illustrating the differences which arise depending upon the sign of ra - em, negative for the left-hand column and positive for the right. The fist row shows the final state eigenvalues E+and E_. The second and third rows depict the wavefunctions of the initial state, $i,,, and the possible final states, either bonding ($ _) or antibonding (IL+), from which it is possible to see the origins of the relative weights in the satellite versus main peaks in the spectra, shown in the bottom row. For esthetic reasons, the delta finctions have also been smoothed out.
the excited state Ea into a single broadened virtual level. As the coupling strength becomes comparable to or exceeds the bandwidth of the screening electrons, new virtual resonances and/or localized states which are split off below or above the band may form [ 151. These split-off states are in fact our screening states, e+, which suggests that the two-state model achieves closer union with quantitative reality when applied to systems in which I VI is the largest energy parameter. Nonetheless, the physical insights gained from the model remain valid over the entire parameter space. 5.3. General case The general solution to the “4-state relaxation model” is contained in eqs. (7), (8), and (10). Complete numerical consequences, in the limits of either pure surface
444
J. W. Gadzuk,
S. Don&h
/Soluble
relaxation
model
plasmon or unoccupied atomic orbital screening have been given in the preceding sections. As a change of pace, in fig. 7 we display some calculated spectra in the form given by eq. (1 l), where each discrete line is broadened into a Doniach-SunjiC lineshape (w, -+ 00 limit of eq. (14)). DS parameter values are y = 0.3 and f= 0.3. The 4-state model is characterized by the atomic orbital parameter F E V2/(em - g& = 0.1 and surface plasmon parameter A E X*/(e, - &) = 0.6,2.0, and 3.0, as labeled.
-75
-7.5
-5.0
- 5.0
-2.5
0
-2.5
0
2.5
2.5
E
Fig. 7. Some general spectra, obtained from eqs. (7), (8) and (10) which have been broadened with a DS lineshape. The atomic orbital coupling parameter r (defined in the text) is held constant and the surface plasmon coupling has been varied, as shown in the figure.
J. W. Gadzuk, S. Doniach /Soluble relaxation model
445
The object of this exercise is to follow the change in the spectrum as the plasmon coupling is increased relative to the atomic orbital hopping. In tig. 7a, for the smallest value of the surface plasmon relaxation shown here, the two peaks at E a 0.25 and 2.0 can be regarded as LW or GS structures that have been weakly modified by surface plasmon hybridization. The structure at e - -5.5 is the observably-intense-one of two surface plasmon satellites. As A is increased, the twopeaked unfilled orbital spectrum is altered first into peaks of equal intensity, as seen in the progression from figs. 7a to 7c and then into an apparent single line (actually a doublet which is unresolvable on the scale of plasmon relaxation energies). As this goes on, both the intensity of the surface plasmon satellite increases and its energy decreases, due to the stronger repulsion between the zero and one plasmon portions of the state vectors with larger A and thus h. Being now in a position of understanding the basic mechanisms which contribute to the hole spectral function and thus photoelectron energy distribution, one should be able to proceed with some degree of confidence in simulating such spectra as shown in fig. 7 using eqs. (7) (8), (1 l), and (14) and a hand calculator.
6. Summary In the preceding chapters we have explored the physical basis of the relaxation response to adatom core hole creation in the limits in which the total response can be modeled in terms of either displaced surface plasmons or occupation of unfilled atomic orbitals on the adatom or some combination of the two. The degree to which one or the other relaxation or screening process dominates in real situations is a question of considerable current interest. From our studies some semiquantitative guidelines have emerged. The strength of the surface plasmon response depends upon both the couplihg constant h and the surface plasmori energy o, in two observable ways; namely through a relaxation shift Ae, 7 h*/o, in the main line and the generation of a satellite line with an intensity - AZ/o:. Thus even though a large surface plasmon shift might be experienced, due to a large coupling constant A, the intensity in the satellite could be small due to a large value of w,. However the position of the satellites, in the infinite order IBM, depends only upon w, and not A. In contrast, the degree to which the occupation of the unfilled orbital contributes to the screening of the hole is determined by the ease with which screening charge may flow from the substrate to the adatom. The parameter p = E”,/2V is a measure of this quality, with small values of p implying best charge flow. Strongest coupling occurs for transfer between states E, and ;a which are at the same energy. Hence with E, = 0 (the convention which has been adopted) optimal mixing occurs as 7, and thus p approaches zero. Secondly, since the hopping increases as V - l/p does, the small p limit again indicates efficient charge flow. For the complete system in which both processes are possible, the numerics of eqs. (7) (8) and (10)
446
J. W. Gadzuk, S. Doniach /Soluble relaxation model
show that mixing between the two processes is most significant in the parameter range in which l/p = h’/o$, in agreement with our enlightened expectations. The question will still be asked whether it is definitively possible to ascertain which screening mechanism dominates in a true experimental situation or even if such a separation is meaningful. We made a (dogmatic?) statement in the introduction that this question could not be resolved on the basis of a comparison of calculated relaxation energy shifts with experimentally determined ones (assuming that relaxation and chemical shifts could be separated, which is rarely done legitimately), since the numbers are reasonalby similar [36]. What Is not similar and thus may be useful as a diagnostic tool is the energy position of the satellites with respect to the main line. Whilst the surface plasmon satellite is at lest w, down, the doublet separation, from eq. (26), is AE = 2V(l + (E”a/2V)2)1/2 , which will probably be much smaller than the surface plasmon energy. Observation of such a doublet will provide encouragement (although not unique proof) for the operation of atomic-like relaxation mechanisms [37]. However, life is not all that simple since the non-observation of the surface plasmon satellite does not permit the conclusion that surface plasmon or image charge screening is not occurring, only that the intensity of the shakeup satellite is too low to be observed. We are thus left in the happy position in which more work, from either point of view, will certainly be welcomed and needed in our quest for a better understanding of the relaxation response in XPS of adsorbed atoms [38,39].
References [ l] Review and survey articles:
[2] [3] [4] [S] [a] [7]
L. Hedin and S. Lundqvist, Solid State Phys. 23 (1969) 1; L. Hedin, in: X-ray Spectroscopy, Ed. L.V. Azaroff (McGraw-Hill, New York, 1974); S. Lundqvist and G. Wendin, J. Electron Spectr. 5 (1974) 513; G.D. Mahan, Solid State Phys. 29 (1974) 75 ; D.C. Langreth, in: Collective Properties of Physical Systems, Eds. B.I. Lundqvist and S. Lundqvist (Academic Press, New York, 1974), and X-ray Emission and Absorption, NORDITA-76/21 (Copenhagen); M. Sunjic, 2. Crljen and D. SokEevid, Surface Sci. 68 (1977) 479; J.W. Gadzuk, in: Photoemission and the Electronic Properties of Surfaces Eds. B. Feuerbacher, B. Fitton and R. Willis (Wiley, New York, 1978). B.I. Lundqvist, Phys. Kondens. Mater. 6 (1967) 193; 6 (1967) 206; 7 (1968) 117; 9 (1969) 236. G.D. Mahan, Phys. Rev. 163 (1967) 612. P. Nozieres and C.T. DeDominicis, Phys. Rev. 178 (1969) 1097. DC. Langreth, Phy? Rev. Bl (1970) 471. S. Doniach and M. Sunjii, J. Phys. C3 (1970) 285; J.W. Gadzuk and M. Sunjic, Phys. Rev. B12 (1975) 524. J.W. Gadzuk, J. Electron Spectr. 11 (1977) 355.
J. W. Gadzuk, S. Doniach /Soluble relaxation model
441
[8] D.A. Shirley, Chem. Phys. Letters 16 (1972) 220; D.A. Shirley, R.L. Martin, S.P. Kowalczyk, F.R. McFeely and L. Ley, Phys Rev. B15 (1977) 544. [9] R.E. Watson, J.F. Herbst and J.W. Wilkins, Phys. Rev. B14 (1976) 18. [lo] See also A.R. Williams and N.D. Lang, Phys. Rev. Letters 40 (1978) 954, for a discussion of the philosophy espoused in refs. [S] and [ 91. [ 111 G.E. Laramore and W.J. Camp, Phys. Rev B9 (1974) 3270; A.C. Hewson and D.M. Newns, Japan. J. Appl. Phys., Suppl. 2, Part 2 (1974) 121; J. Harris, Solid State Commun. 16 (1975) 671; J.W. Gadzuk, Phys. Rev B14 (1976) 2267; A. Datta and D.M. Newns, Phys. Letters 59A (1976) 326. [ 121 B. Gumhalter and D.M. Newns, Phys. Letters 57A (1976) 423; B. Gumhalter, J. Phys. (Paris) 38 (1977) 1117. [ 131 N.D. Lang and A.R. Williams, Phys. Rev B16 (1977) 2408. [ 141 K. Schonhammer and 0. Gunnarsson, Solid State Commun. 23 (1977) 691. [15] P.W. Anderson, Phys. Rev. 124 (1961) 41; D.M. Newns, Phys. Rev. 178 (1969) 1123. [ 161 R.A. Beaumont and R.S. Pierce, The Algebraic Foundations of Mathematics, (AddisonWesley, Reading, Mass., 1963) p. 361. [ 171 H.W. Meldner and J.D. Perez, Phys. Rev. A4 (1971) 1388. [18] S.A. Flodstrom, R.Z. Bachrach, R.S. Bauer, J.C. McMenanin and S.B.M. Hagstrom, J. Vacuum Sci. Technol. 14 (1977) 303; D.S. Rajoria, L. Kovnat, E.W. Plummer and W.R. Salaneck, Chem. Phys. Letters 49 (1977) [19] ?D. Mahan Phys Rev. Bll (1975) 4814.B14 (1976) 3702; J.D. Dow and D.R. Franceschetti, Phys. Rev. Letters 34 (1975) 1320; P. Longe, Phys. Rev. B14 (1976) 3699; J.D. Dow, D.L. Smith, D.R. Franceschetti, J.E. Robinson and T.R. Carver, Phys. Rev. B16 (1977) 4707. [20] P.H. Citrin,Phys. Rev. B8 (1973) 5545; S. Htifner and G.K’ Wertheim, Phys. Rev. Bll (1975) 678; P.H. Citrin, G.K. Wertheim, and Y. Baer, Phys. Rev. B16 (1977) 4256. [21] P. Minnhagen, Phys. Letters 56A (1976) 327; J. Phys. F7 (1977) 2441. (221 A controversial issue as to the interplay between phonon relaxation and hole lifetime effects has been much discussed 1ateJy. See: M. Sunjic and A. Lucas, Chem. Phys. Letters 42 (1976) 462 and subsequent work on questions raised here. P. Minnhagen, J. Phys. F6 (1976) 1789; J.W. Gadzuk, Phys Rev. B14 (1976) 5458; P.H. Citrin and D.R. Hamann, Phys. Rev. B15 (1977) 2923; G.D. Mahan, Phys. Rev. B15 (1977) 4587; C.O. Almbladh and P. Minnhagen, Phys. Rev. B, in press. [23] B. Gumhalter and D.M. Newns, Phys. Letters 53A (1975) 137. [24] (a) The Fermi-Thomas dielectric function is quite adequate for the surface calculations considered here, as detailed in J.W. Gadzuk, Surface Sci. 67 (1977) 77. (b) J.C. Fuggle, E. Umbach, D. Menzel, K. Wandelt and C.R. Brundle, Solid State Commun., to be published; (c) B. Gumhalter, J. Phys. Cl0 (1977) L219. [25] For another perspective see: H. Benson, The Relaxation Response (Morrow, New York, 1975).
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J. W. Gadzuk, S. Doniach / Soluble relaxation model
[26] R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234; J. Heinrichs, Phys. Rev. B8 (1973) 1346; J. Harris and R.O. Jones, J. Phys. C6 (1973) 3585. [ 271 R. Manne and T. Aberg, Chem. Phys. Letters 7 (1970) 282; C.S. Fadley, Chem. Phys. Letters 25 (1974) 225; S. Manson, J. Electron Spect. 9 (1976) 21. [281 Note that eq. (17), as written, fulftis eq. (20) only to order AZ/w: since E+ and E_ given in eq. (15a) have been approximated by eq. (15b and c). The sum rule is satisfied to all orders in h*/w$ if the exact eigenvalue expressions are used. 1291 Although this would be accompanied by changes in w m,f, and 7 in eq. (14) in addition to the desired variation in KFT or e(ql1) in eq. (18), all of which affect h. [301 H. Morawitz and M.R. Philpott, Phys. Rev. BlO (1974) 4863; R. Silbey, Ann. Rev. Phys. Chem. 27 (1976) 203. [311 A.M. Bradshaw, W. Domcke and L.S. Cederbaum, Phys. Rev. B16 (1977) 1480. 1321 P.H. Citrin and D.R. Hamann, Phys. Rev. BlO (1974) 4948. [33] (a) Some progress on composite response functions has been presented by one of us in ref. [24] and in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977. (b) One can conceive of some pathological situations where this assertion may break down; for instance in the case of a transition metal adsorbate where the screening electron occupies an unfilled d orbital. Since the radius of the d shell is less than that of the still occupied s valence electron and since the mean distance of the alternative induced surface charge from the core hole must be greater than the s radius, the resulting screening energies for the two situations could differ by a significant (-factor of two) amount. Undoubtedly other exceptions of this type could be found, and in fact might be very useful as candidates for investigative studies aimed at the elucidation of the basic physics of the relaxation process. [34] Because, as pointed out by LW amongst others, Aer is the Coulomb energy between the core hole and a unit of electron charge roughly an atomic distance away. This volume integrated quantity is not too sensitive to the shape of the charge distribution. [35] L.S. Cederbaum and W. Domcke, J. Chem. Phys. 66 (1977) 5084. [36] It should be reemphasized that the atomic screening has been proposed mainly as a mechanism for determining the “true” leading edge position which is not to say that this feature dominates. On the other hand, the point of the present study has been emphasis on the rich structure of the total spectrum, which if correctly analyzed, provides additional checks and confirmations on whatever relaxation model and characteristic parameters are chosen. For a more detailed exposition of this philosophy, see J.W. Gadzuk, in: Proc. Intern. Conf. on Solid Films and Surfaces, Tokyo, 1978 (to be published in Surface Sci.). [37] Fuggle et al. (ref. [24b]) have observed an -6 eV doublet structure in XPS on the 1s core state of N2 adsorbed on both W(110) and Ni(lOO) and also on the C 1s core state of CO adsorbed on Cu(100) and W(110). This excitation energy is similar to the lowest gas phase molecular ion excitation energy. Since the satellite structure is substrate invariant, this suggests a localized molecular-like excitation mechanism, possibly a SG satellite peak occurring in the unfilled orbital screening process. Another alternative that cannot be ruled out on the basis of the reported data is pure intra-molecular shakeup which has little to do with the extra-molecular relaxation. Additional experiments will be required to resolve this question. Parenthetically we also note that the leading peak for the N2 was further split into an -1.4 eV doublet, possibly due to differential relaxation shifts experienced by the inequivalent N atoms for N2 adsorbed in the standing up position, as discussed in ref. [ 24aJ. [38] Since completion of this manuscript, we have become aware that the unfilled atomic orbital model was fist introduced, in another context, by A. Kotani and Y. Toyozawa, J. Phys. Sot. Japan 37 (1974) 912. [39] S.S. Hussain and D.M. Newns (Solid State Commun., 25 (1978) 1049) have independently worked on a relaxation model similar to ours.
ASOLUBLERELAXATIONMODELFORCORELEVELSPECTROSCOPYON ADSORBEDATOMS J.W. GADZUK NationalBureau of Standards, Washington,DC 20234, USA
S. DONIACH * Department of Applied Physics, Stanford University,Stanford, California94305, USA Received 10 March 1978; manuscript received in final form 1 June 1978
The photoelectron spectrum of core levels in adsorbed atoms is calculated taking into account the screening of the core hole due to both displaced surface plasmons (image charges) and actual transfer of screening charge into the lowest unfiied valence levels of the adsorbate. The theory is detailed on a simplified but exactly soluble model, permitting parametric numerical studies of the spectrum while maintaining close contact with physical understanding. Recent theory on adsorbate relaxation effects due to Gumhalter and Newns, Gunnarsson and Schonhammer, Lang and Williams, and others are considered in the light of this model.
1. Introduction Core level X-ray photoelectron spectroscopy (XPS) in or on metals cannot be satisfactorily understood in terms of single electron theories, due to so-called final state relaxation effects. Within the domain of extra-atomic relaxation, the localized core hole is screened by the free electrons of the conduction band. The Coulomb interaction between the screening charge, often written in simplified form in terms of displaced plasmon density ~uctuations, and the hole charge gives rise to the extra-atomic relaxation or screening energy shift of the ejected electron. Depending upon the dynamics of the hole creation and screening, it is also possible to leave the electron gas (metal) in an excited state, in which case the ejected electron must emerge with a correspondin~y reduced energy. Features in a spectrum which are due to this effect include shakeup satellites. In the limit in which the hole potential is instantaneously switched on (the sudden limit), there is a one to one corre* Research supported by Army Research Office, Durham, NC, and NSF through Stanford Synchrotron Radiation Laboratory. 427
428
J. W. Gadzuk, S. DonL2ch /Soluble
relaxation (node1
spondence between the photoelectron spectrum and the spectral function (imaginary part of the Green’s function) for a transient core hole propagating in the reactive electron gas [2-61. A sudden limit sum rule has been derived which relates the intensity in the relaxation shifted main peak with the intensity and position of the satellites [ 1,2,7]. At least for XPS within the bulk, the short time response to the core hole involving real and virtual plasmons results in the screening of the long range bare Coulomb potential and the creation of plasmon satellites. The long time response to the screened core potential shows up as the excitation of electron-hole pairs, causing a skewing, to the higher binding energy side, of the main XPS line [6]. For the most part, the electron gas based theories are given in terms of response functions which do not depend upon atomic-like properties of the metal. Exceptions to this are the excitonic [8] or lowest-unfilled-atomic orbital [9] models in which the screening charge is built up from wave functions which are orthogonal to the core state. Another important class of relaxation effects occurs in XPS from atoms which are adsorbed on metal surfaces. Considerable theoretical work has depended upon the proposition that the extra-atomic screening can be represented in terms of image charges which are in turn expressed as a coherent displacement of surface plasmons [ 11,121. Although this is obviously valid for large adatom surface separations, some interesting new features must be considered for separations which are small enough to be chemically significant. Numerical calculations by Lang and Williams [13] have shown that under some circumstances, the screening charge associated with an adsorbed atom more closely resembles the lowest-unfilled-atomic orbital [9] on the adatom than a surface or image screening charge within the substrate [ 11,121 which would appear to vitiate the reliability of the surface plasmon models. However, since these calculations pertain to the purely static aspects of the problem, some dynamical effects are missed which are quite important in understanding the surface screening process, the resulting photoelectron spectrum, and the relation between the surface plasmon and Lang-Williamns relaxation response theories [ 10-141. The following tempera1 sequence of events can be envisioned occurring within the life of an adatom core hole. Upon switching on the core hole, surface plasmons respond on a time scale of the order of w,-‘, the inverse surface plasmon frequency, establishing an image potential shift and surface plasmon sateliites. Since the adatom is in chemical contact with the substrate, valence electrons may transfer between the two at a rate which is roughly proportional to the bonding-antibonding level separation. Usually this rate is much slower than the surface plasmon screening rate. Once an electron hop from the substrate to the lowest-unfilled adatom orbital occurs, charge neutrality on the adatom is restored, coupling to the surface plasmons ceases, and the surface version of Herbst screening [9] is set up, as pointed out by Lang and Williams [13]. However, since the reverse hop can still occur, the dynamics of the competing surface plasmon screening and valence electron hopping must be considered. This has already been done to a certain
J. W. Gadzuk, S. DonL;rch/Soluble relaxation model
429
extent by Gumhalter and Newns [12] who considered modification to X-ray absorption on the adatom, due to the possibility of electron hopping and surface plasmon screening. In this work, we consider a highly simplified, but exactly soluble model for the core hole spectral function in which all the dynamical features are treated in a democratic way. As the spectral functions can be given in closed algebraic form and the relaxation process is completely characterized by three parameters, systematic studies into the observable consequences of various parameter sets are trivial. Thus this communication should serve as a useful complement to that of Schonhammer and Gunnarsson on a similar topic and allows an extension of their ideas to include shakeup effects. The structure of the paper is as follows. In section 2, the general model is established and simplifying approximations are introduced which make possible closed form calculations of the transient. hole spectral function introduced in section 3. Possible electron-hole pair shakeup of the Mahan-Nozieres-De Dominicis [3,4] (MND) type which skews XPS lines in the bulk is discussed in terms of GumhalterNewns formulae in section 4. The resulting spectral functions are discussed in detail for several limiting sets of parameters in section 5 and are interpreted in the light of already existing knowledge. General conclusions are given in section 6. 2. Theoretical model
The basic relaxation processes associated with sudden core hole creation on an adatom can be uderstood in terms of the model Hamiltonian advocated by Gumhalter and Newns [1,11,12]: &,t
= e,& + [e, - &(l
- $)I&
+T
eknk + T
(v&&
+ h.c.)
(1) which has the following meaning. The first term describes the core state with eigenvalue E, (quite negative with respect to a zero at the Fermi level) whose occupation is either n, = 1 in the initial state or n, = 0 in the final state. The second term (f fYd in eq. (1) is mainly responsible for the interesting possibilities suggested in the recent theories [ 12-141. The eigenvalue E, is that of the lowest unoccupied adatom orbital in the initial state, hence n, = 0. The intra-atomic Coulomb integral U,, (positive in sign) is a measure of the attraction between the positive hole charge and the negative charge that could occupy the e, orbital. Since n, = 1 in the initial state I$,” = esn, whereas n, = 0 in the final photoionized state, implies Hafin = (E, U&r,. The occupation number of the adatom orbital in the final state depends sensitively on the sign of E”,E E, - U,, with lim;O,_,
n, = 1
and
lim~~~+,n,
=0 .
(2)
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J. W. Gadzuk, S. Doniach /Soluble relaxation model
The third term in eq. (1) represents the conduction band states of the substrate. Hopping between the adatom orbital and the substrate is allowed for by the fourth term where the strength of the coupling depends upon V;J, the Anderson-Newns matrix elements. The next two terms describe the surface plasmons and their coupling to the ion formed in the photoionization event. The strength of this interaction which under certain limits, reduces to the quantum mechanical version of the image potential shift [ 1 l] is characterized by the coupling function b, to be explicitly detailed in section 5. Note that in the initial state n, = 1, n, z 0 and 1 n, -n, = 0 whereas in the final state n, = 0 so the strength of the interaction is proportional to X(1 - n,). Since n, is an operator with a non-trivial time dependence due to the hopping term, the interplay between the surface plasmon and hole hopping dynamics becomes all important in understanding the relaxation process. Finally we include a phenomenological term IY~__~,to be discussed in section 4, which allows for hole decay and MND-like effects. Assuming [17, He-r,] = 0, the total spectral function is then the convolution of independent spectral functions for Hand&-h [5,6]. To illustrate the various physical features which are implicit in eq. (1) we will adopt a well defined set of model assumptions which hopefully permit sufficient new insight to justify the simplicity of the model. Specifically we consider: (a) A single level substrate which implies &e& -+ e&z,; this jUSt says that the electron which hops between the adatom and substrate via Vak in eq. (l), is always in state E, when in the substrate; as discussed later, this is a narrow band limit. (b) Completely unoccupied adatom orbital E, in the initial state, so n, = 0; equivalent to assuming a discrete state rather than a broadened resonance [13-151; f, sufficiently high that V,k mixing in initial state can be neglected. (c) Discrete final state, whose occupation number depends largely on the sign of f”, and is equivalent to assuming no broadened resonance. (d) Dispersionless surface plasmons with a Q * qll cutoff if necessary (411being parallel to the plane of the surface). (e) Only one surface plasmon satellite; not bad since satellites are rather weak anyway. With this set of specifications, the initial no hole and final one hole state Hamiltonian, from eq. (l), are respectively Hfin = emnm +
V(C~C,
+
h.c.) + w,btb + A(1 - n,) (bt + b) + &n, .
(3)
The Hilbert space of possible states, within the present model defined by conditions a-e is taken to be completely spanned by the four eigenfunctions of the number operators, In,, n,, n,) with nP = 0, 1 corresponding to zero or one intrinsic plasmon and with the constraint n, + n, = 1. Adopting this notation, the initial state, before photoionization, is I Gin)
=
I Oa9lm, 0,) 7
(4)
J. W. Gadzuk, S. Doniach /Soluble
relaxation model
431
where ($inlHinlJ/in) = E,,, and the energy zero has been displaced by e,. The final states can be written as linear combinations
+ Y(ej)lO,,1,~
1,) + h(ej)l la9 Orn9 lp)
(5)
9
where the four coefficients and four eigenvalues ej (j = l-4) can be given in closed algebraic form. At this point, the eigenstates of the final one hole Hamiltonian, eq. (3), are determined by the 4 X 4 matrix representation of the Schrodinger equation: V
V
Em -
0 \o
a
0
0
E,--E
0
EX
x
E,+W,
0
V
--E
fl
1‘
v
CO
’
(6)
and the normalization condition lcrl2 + IpI t ly12 t 16 I* = 1. The eigenvalues of eq. (6) are given as the solutions of the quartic equation obtained by setting the secular determinant equal to zero: e4 + cres + C2EZ+ C3f
+ c4 = 0 )
(W
where Cl = -2&
+ E, + w,) )
0)
+2Cd~)+EmC~“,A2-2V29(7c)
C2-(E,+W,)(E”,+W,)+(Em+E”,)(Em+~~
C3 E -iFaEm(&
+ Em +
20,)
- (Em + W,) (E”,+ a,) (E”,+ Wm)
+ P(2E”, + w,) + 2 VZ(E”,+ Em + w,) , c4 =zaem(E"a+
-
P(g&,
W,)(E,
+ a,)-
(74
PE",
+ (& + c.d,)(E, + w,)-
V2).
(7e)
The four eigenvalues
ei which are roots of eq. (7a) can be expressed explicitly as The algebraically cumbersome nature of these expressions prohibits any additional physical insights however. The coefficients in the eigenfunction expansion, eq. (5), are eji= Ej(E",,Em9 a,, X, v) [161.
P(Ej) = [l + V2/U:,j + A2(V2 + a:,j)/(a3,]ff4,j a(Ej) = vD(Ej)lal,j
2
- V2)2]-1’2 9
(84 @b)
432
J. W. Gadzuk, S. Doniach /Soluble relaxation model
YCej) = -A a4,jP(~ji)l(a3,~4,j - v2)
9
s(fj) =
VAP(S)l("3,fa4,j - v2)
(84
7
with al,j=
fl3,j
E", -
=
Ej
,
Em + Cd, -
a*,j=
Ej
,
Em
cl4,j
-
C?j ,
=
E”,+ Cd,
-
Ej
.
Eqs. (5), (7) and (8) comprise a working solution to the problem represented by the eq. (3) Hamiltonian. The ensuing sections will relate these solutions to photoelectron spectra, both for well known limiting choices of parameters as well as for general cases.
3. Spectral functions The photoelectron spectrum is readily calculated only in the two extreme limits of adiabatic and sudden switching on of the hole potential [ 1,6,7,17,18]. Creation of the core hole changes the many-body Hamiltonian by an amount AH * VhOrex r(Sf/at) where r is the time duration of the hole switch-on and U-I’) a mean time rate of change of the Hamiltonian due to removal of the photoelectron [ 171. These considerations relate to eq. (l), since U&l - nJ n, and h(1 - n3 (b t b+) are in effect time dependent potentials due to the implicit time dependence of n, = n,(t). The adiabatic limit leaves the system always in the ground state of the potential determined by the instantaneous charge distribution of the ion cores plus relaxing electrons. This condition is satisfied when T S h/e*, where E* is a “typical” excitation energy of the relaxing system. If the adiabatic limit is achieved, all the photoelectrons emerge with the same “totally relaxed energy” and there are no interesting dynamical problems to consider. This limit is not realized under actual experimental conditions which are better thought of in the sudden limit where the hole switch on time r + 0, corresponding to instantaneous removal of the photoelectron. Thus although r(FN/&) can be represented as a discontinuous stepfunction in time, the many-body wave function is continuous, which facilitates spectral decomposition of the initial state within the manifold of eigenstates (eq. (5)) of Grin (eq. (3)). In general, the sudden limit hole spectral function is simply [5] : N+(E)=C
~(e-Ej)I(J/inl~fin,j)l*
*
(9)
i Specifically for the model characterized and (5), eq. (9) becomes N+(E) =
C &(Ei
ej)
IP(e
3
by the initial and final states of eqs. (4)
(10)
J. W. Gadzuk, S, Doniach /Soluble relaxation model
433
which is a distribution of delta functions whose positions and strengths are given by eqs. (7) and (8). As mentioned earlier, both hole decay and MND edge effects are allowed for through the last term, &_h, in eq. (1). By ansatz, [He-r,, Hi”] = [Hea, Hfin] = 0, sothe total spectral function can be written as a convolution - e’) N+(E’) de’ ,
NT&E) =,-D&e
which with eq. (10) is NT&E)
= 7
Ii+j)?
De-
h(E -
Ej) .
(11)
In eq. (1 l), De_his the spectral function associated with &_h, to be discussed in more detail in the next section. As an example, finite core hole lifetimes are usually considered simply by taking De_hto be a Lorentzian of width y [6,14]. 4. Pair shakeup and hole decay The still somewhat controversial role of electron-hole pair edge effects in X-ray emission and absorption spectroscopy [3,4,19] is much more firmly established in XPS of metals [6,20]. The sudden creation of the core hole excites a large number of low energy electron hole pairs, which result in the characteristic MND infrared divergent spectral function
D(e)- IE~-~+~~(--E),
(12)
where QI is determined by the properties of the screened core hole potential and equals (6r=e/n)* for s-wave scattering only with 6 the phase shift. Although the MND derivations of this formula are strictly speaking valid only as E -+ 0, Minnhagen has numerically verified eq. (12) for all energies less than the plasmon energy of the simple metal in question. Doniach and Sunjic [6] realizing that core hole lifetime effects [22] would soften the divergence, convolved eq. (12) with a Lorentzian to obtain a now well confirmed [20] expression for the asymmetric XPS lineshape, skewed to the higher binding energy (lower kinetic energy) side of the spectrum. In an exactly analogous manner, core level XPS on adsorbed atoms is accompanied by pair production, which in turn influences the shape of the photoelectron line. Obviously the coupling of the core hole outside the metal with the pair excitations within takes on a much different functional form, but still the principles remain the same. In fact Gumhalter and Newns [23] have worked out a first order theory for the adsorbate analog of the MND expression, which is, in the ThomasFermi approximation to the RPA [24a], &N(E)
= [WE)
exp(f/w,)
I E I-’
‘fl I(W9 w$,>,
(13)
434
J. W. Cadzuk, S. Doninch f Soluble relaxation model
where w, = r&/2s with bF the substrate Fermi velocity and s the ion core-image plane separation, f= log&l.59 KF~ s)/@k$ with KFT and k~ the Fermi-Thomas and Fermi wavevectors, and I’ the gamma function. Again, finite core hole lifetimes can be accounted for by convolving BGN(e) with a Lorentzian. Such an intergral has been previously given [6b] as
1 exPW%)
_
rrwf,
J-V -n (e* + y2)f l -flP
cos @_ &L)
>
(14)
where
e = (f-
1) tan-i(e/r)
- 7rf/2 - $6~~ ,
and 6D is a small correction that can be neglected for 1~15 a few y. In the limit of wm+w, eq. (14) has an algebraic structure identical with the Doniach-Sunjid expression. Parenthetica~y note that the asymmetry of the adsorbate line is a function of
0.8
0.6
Fig. 1. Asymmetric XPS lineshape for an adsorbate core level (eq. (14)) as a function of energy, in units of the level width y. A “typical” value for the asymmetry parameter f(= 0.1) is used is treated parametrically. The inset shows the Gumhalterand the s-dependent cutoff q/w, Newns estimates offversus s, taking r, as a parameter.
J. W. Gadzuk, S. Doniach /Soluble relaxation model
435
distance from the surface, due to the fact that f, y, and w, are functions of s. To get a feel for the qualitative variations to be expected in the lineshapes as a function of atom-metal separations, eq. (14) has been evaluated as a function of e/y, treating y/w, parameterically. Some resulting lineshapes are displayed in fig. 1, together with values of fversus s shown in the inset. Such experimental adsorbate lineshapes have been observed [24b]. Finally we note that an alternative (and sometimes even more dominant) asymmetry mechanism has been proposed also by Gumhalter [24b] in which the suddenly created core hole causes electron-hole pair creation on the adatom, as well as within the substrate. For this to be significant, the position of the final state screening orbital energy ‘E”,must be near the Fermi level as the asymmetry index is given by [24c] f= (U,, p,,J2 where pa0 is a Lorentzian-like adatom local density of states centered at E”, and broadened by A - I V12. Since Y is a function of s, this mechanism will also produce an XPS lineshape which becomes more symmetric as s decreases.
5. Results The numerical consequences of the preceding four sections shall now be explored for two limiting ranges of parameters and then for some arbitrary (but physically significant) sets. 5.1. Surface plasmon response
A popular model of the surface relaxation response [25] is the so-called surface plasmon or image charge theory [ 111. The dielectric reponse of the substrate to the appearance of a fxed charge outside results in a screening charge localized in the surface region. If this induced charge is constrained to remain within the substrate, then it can be decomposed into a superposition of surface density fluctuations of the metal-adsorbate interface, which in the long wavelength limit are just surface plasmons [ 11,241. The observable consequences, in a photoelectron spectrum, of the surface plasmon response are an upward extra-atomic relaxation or screening energy shift, which for large atom-substrate separations is the classical image potential shift, and also shakeup satellites corresponding to the intrinsic excitation of surface plasmons. The important criterion which determines whether the screening charge remains in the substrate is the value of the surface plasmon energy compared to the level width A = p(ef) I VI2 of the unoccupied orbital, where p is the Fermi level density of states. When A 4 AU,, the surface plasmons respond as if the adparticle retains one unit of positive charge throughout the entire screening process. Only when the hopping time (--A/A) for an electron to transfer from the substrate to the unfilled orbital & becomes comparable with the plasmon response time does the Herbst-Lang-Williams screening dominate. This limit will be discussed in section 5.2.
436
J. W. Gadzuk, S, Doniach /Soluble relaxation model
Within the context of the present model, the surface plasmon screening limit occurs when V* + 0 in eqs. (l)-(3). Realizing that n, = 0 for all times of importance, the final state Hamiltonian is just + w,btb + X(bt + b) ,
Hrrn = e,n,
which is easily diagonal&d, since CY= 6 = 0 in the eigenstate expansion, eigenvalues of the ionized system, from eq. (6) are
eq. (5). The
t 4hz/w,Z)r/Z )
E = E, + &JS L!Z &(I
(154
where for x/w, Q 1 ) E_~:E,
- X2/% ,
(15b)
E+=:f,+W,+P/o,.
(15c) The E_ state is the eigenstate of the screening electron, having a reduced energy due to the additional attraction between the induced charge and the ion core. The resulting photoelectron energy is thus “relaxation upshifted.” From eqs. (8), (lo), and (15a), the spectral function is [lt (1 t b2)‘j212
N+(f) = 6(E - E_)
with b2 G 4h2/w,?. The interpretation written in the form: N+(E)“8
A2
(
E-f,
[l - (1 t b2)‘j212
[1 t (1 t b2)‘j212 + b2 + ‘(e -et)
t-
as
[1 _ (1 + b2)‘/2]2 +
of N+(E) is particularly
I[
l--$tO($J]
b2
transparent
3 (16)
when
+#-$)
4 [1+6(9] ,
(17)
which is valid when X/w, < 1. First consider the schematic functions shown in fig. 2. The first and second panels show the Koopmans theorem unrelaxed and the adiabatically relaxed limits. Due to the equivalence X2/o, = Aq the relaxation energy, the hole-plasmon coupling function can be specified in terms of an acceptable image-like interaction; that is X = (oS Vi/image(s))‘12 where for example, Ae, = vimage (s) could be taken equal to e2/4s , e2/4(s + Jje2
(18a) K &)
)
%ds -
WI
e?ll, 0)
%ds+ E(4ll,O)
exp(-%s)
3
(18~)
J. W. Gadzuk, S. Doniach /Soluble relaxation model
(a)
Unrelaxed
(b)
Adiabatic
1
relaxed
i
431
’ N’(c’
l,..
~~~~~
Fig. 2. Schematic unbroadened hole spectral functions and thus photoelectron energy distributions in various degrees of approximations, all of which ignore electron-hole pair excitations. (a) Unrelaxed spectrum which applies when Koopmans theorem is satisfied. (b) Adiabatically relaxed spectrum which occurs if the hole potential is slowly switched on. (c) Sudden limit spectrum in which all satellites are forced into one “mean” surface plasmon shakeup peak. (d) Infinite order independent boson model spectrum which is a Poisson distribution.
depending upon the level of accuracy required [ 11 ,12,23,24]. In eq. (18) e(q ,,, 0) is the transverse wave mimber dependent metal-vacuum surface dielectric function and e,& is the dielectric constant of the adlayer material [11,24,26]. Detailed numerical consequences have been worked out elsewhere [24]. The complete spectrum, as given by eq. (17), is shown in fig. 2c. The relaxation shifted main peak whose intensity is reduced by a factor of l-a, with a E X2/~ = AE,/w,, is compensated for by an effective surface plasmon satellite of intensity a, thus preserving the normalization IN+(e) de = 1. Since the two peaked spectrum given by eq. (17) is an approximation to the Poisson distributed spectrum of the infinite order independent boson model (IBM) [ 1,151,
N_(E) = ema
n$o (an/n!)
ti(e - em + au, - nw$ ,
(19)
shown in fig. 2d, the low energy satellite is displaced by an amount “X*/o, = aw, below the position of the first satellite in the complete IBM spectrum. In effect, the total spectral weight for all the satellites (n > 1 in eq. (19)) is lumped into one
438
J. W. Gadzuk, S. Doniach /Soluble relaxation model
satellite whose position [2,5,7,18,27,28] s
and intensity
is consistent
with the sudden limit sum rule
e N+(e) de = era ,
(20)
where ef,, is the frozen orbital or unrelaxed energy shown in fig. 2a. As the holesurface plasmon coupling increases, the upward relaxation shift of the leading peak also increases. However this is accompanied by a redistribution of intensity into the satellite system, which when approximated by a single structure, must shift downards in energy below o,, as shown in fig. 2c in order to maintain eq. (20), the model independent sum rule. A similar situation arose in the lowest order hole self energy calculations (in a homogeneous electron gas) of B. Lundqvist [2] compared to the infinite order IBM presented by Langreth [5]. In the theory of Lundqvist all the satellites were coalesced into one “effective plasmon satellite” whose separation from the main peak increased with increasing coupling constants in an exactly analogous manner to the case considered here. The numerical information implicit in eqs. (15) and (16) is shown in fig. 3 for the case in which w, = 0. The two-peaked spectrum can be ascertained in the following way. For a specific value of X/w,, the energies of the main and satellite peaks are given by the dashed curves, plotted as a function of h/w,, on the right hand axis. The intensities of the peaks are determined from the full curves and left hand axis at the applicable values of e = e+(h) and e__(X). As an example, with (X/o,)2 = 2, E-/O, = 1, E+/ C+ = -2, I/3_12 = 2/3, and I/3+1’= l/3. In this figure, the energy scale has been shifted so that the zero coincides with the Koopmans theorem unrelaxed energy. Thus the main peak is precisely at the surface plasmon relaxation energy. The energy scale increases from left to right when considered in terms of the kinetic energy of the photoelectron. I
I
w,=O \ Satellite l.O- \ (c+) L \
I
I
I
/-
\
Main peak
5
/
Fig. 3. Numerical consequences of eqs. (15) and (16) which show the energies and relative intensities of the main and satellite peaks, for the “surface plasmon screening only” limit. The horizontal axis gives the kinetic energy of the photoelectron with respect to a zero at the unrelaxed energy. To use this curve, fist ascertain the coupling constant. The right-hand vertical axis and dashed curves give the energies for the main and satellite peaks. The left-hand axis and full curves show the intensities at the specified energies.
J. W. Gudzuk, S. Donriich /Soluble relaxation model
439
One feature of these “data” is a direct result (artifact?) of the two-state-only model; namely that the intensity in the surface plasmon satellite is always
+
F (?‘,,gcftc, +h.c.1.
(22)
440
J. W. Gadzuk, S. Don&h /Soluble relaxatioti model
With approximations a-c introduced in section 2, the initial no hole and final one hole state Hamiltonians, from eq. (22) are Hi, = E,n, + e,n, Hfin
= ~,n,
+
3
t
V(C,C,
(234
+
h.c.) + e,n,.
G=)
The initial and final eigenstates are given by eq. (4) and eq. (5) (with y = 6 = 0) respectively. The fundamental process treated by both the SC theory and model implicit in eq. (23) is the following. Initially one unit of electron charge which is to do the screening resides in the metal. Upon photoionization, the extra attraction on the adatom due to the core hole pulls the unfilled level E, downards, maybe to the Fermi level or lower, and the renormalized energy is E”a= ea - U,,. In the final state, the Valakterm couples the metal state E, with the renormalized excited adatom state E”,to form what amounts to bonding and anti-bonding metal-adatom states whose eigenvalues are given by the secular determinant of the upper left-hand 2 X 2 matrix in eq. (6); that is - Em)]2 + V2}1’2 .
e* = 1,& + em) i: {[&
(24)
The “screening electron” must now occupy one or the other of these states. If the screening charge occupies the lowest energy bonding state e_, then the photoelectron must be ejected with its adiabatic relaxed energy. On the other hand, if the photoionized system is left in an excited state, then the photoelectron emerges with a kinetic energy that is precisely the excitation energy below the adiabatic peak. In the model described by eq. (23), the only excited state is one in which the screening electron occupies the “anti-bonding” state E+. From eq. (lo), the two-peaked spectral function is N+(e) = 8(E - E+) IP(
•t S(E - E_) IP(
5
(25)
which can be interpreted as an adiabatic relaxed main peak at E = E- plus a shakeup satellite at e = E+. The intensities in the two peaks, obtained by inserting eq. (24) into eq. (8) (with X = 0) are given by
IB(E*)12 = [I + (E*/y121-l
(264
with e*/V=p p=ZJ2v.
f (1 t p2)“2
,
Wb) (26~)
In eqs. (26a-c), E, has been set to zero since the absolute value of the ground state of the system is not required. All the information pertaining to photoelectron spectra which is contained in the two-state limit of the GS model, as given by eqs. (2&-c), is compactly displayed in fig. 4. For a given value of p on the right hand axis (whose sign depends upon the particular system), the positions of the energy
J. W. Gadzuk, S. Doniach /Soluble relaxation model
-
I .oc)-
441
2
0.7: i0.5c )-
I
0.25 Y ‘i z
0
-3
-2
/I /
/ / / e_ /
0
/I ,
/
2
:
/
0
3
k/V)
-
> N ’ 2 Y” VI a
-I
Is+ I
-. -2
Fig. 4. Numerical consequences of eqs. (26a-c) which show the energies and relative intensities of the main and satellite peaks, for the “unfiied atomic orbital only” limit. The same procedure as described in fig. 3 is to be invoked for using this figure, substituting the parameter p for the coupling constant. Note that if p > 0, the largest intensity is in the main peak whereas for p < 0, the largest intensity is seen in the satellite. In this figure, the photoelectron kinetic energy is the negative of E.
main and satellite peaks are given by the so-labelled dashed curves and the intensities by the value of the full curve (left hand axis) at the energies just determined. Note that when p > 0, the intensity in the leading peak is less than that in the satellite whereas for p < 0, the adiabatic peak carries the majority intensity. To further illustrate the points to be made here, some two-peaked spectra are shown in fig. 5 for a range of parameters. These figures can be regarded as an attempt to
‘.OOl
1
’
1
I
’
’ I
(-c/V) Fig. 5. Some explicit “unfiied orbital only” spectra, from eqs. (26a-c) or fig. 4, as a function of photoelectron kinetic energy, treating p parametrically. The related main and satellite peaks are connected by light horizontal lines.
442
J. W. Gadzuk, S. Don&h
/Soluble relaxatioi2 model
reproduce the results of GS within our model. To the extent that this has been successful, the two-peaked structure of the SG spectra can be viewed as a relaxed peak and a shakeup satellite (rather than an “unrelaxed” peak). The satellite corresponds to the excited state in which the electron occupies a basically antibonding orbital formed by the linear combination of the renormalized unfilled atomic orbital with E”, and the metal “orbital” with E,. If this is a correct interpretation, then in this state the screening charge density should display nodal tendencies between the adatom and the substrate. As mentioned earlier, the crucial feature which determines the relative intensities of the satellite to adiabatic peak is the sign of p. This is easily illustrated within the physical picture depicted in fig. 6. When p < 0, the bonding adatom-substrate screening orbital ~-O,~O)=(YIla,O,~+pIOa,
1,)
is more strongly localized on the adatom (i.e., Ia_ I* > Ifl_ I*) and thus the intensity in the leading peak, proportional to the modules squared overlap of Gin with I/_, is less than that in the satellite (whose intensity = ]<$in]$+)1* > 0.5). As E”, exceeds E,, either due to a larger value of E, or smaller U,, or effective metal energy e,, then the majority intensity
is found in the adiabatic peak, as can be understood from the right hand column in fig. 6. Thus there should be, no mystery in understanding the manner in which a spectrum can undergo a smooth transition from mostly “main” to mostly satellite intensity (or relaxed to unrelaxed) as the model parameters are varied. Finally, we note the dependence of the spectrum on the magnitude of V. As V greatly exceeds & (and thus by inference the screening electron bandwidth in a real solid, since E, was taken to be zero), the magnitude of V and not the properties of the indivudual entities (as represented by E, and Q determines the final state spectral function. In this limit, the substrate and adatom must contribute in a completely symmetric (gerade) or antisymmetric (ungerade) way to the possible system final eigenstates, in which case the projection of a unit charge onto the adatom state (or “symmetrically,” the substrate) requires equal contributions from the gerade (E_) and ungerade (E+) states. The consequences of this as manifested in our model photoelectron spectrum are that lim V-large orp-tcllS@*)12= 0.5 2
as is apparent in fig. 4. Cederbaum and Domcke have given a very nice discussion of a similar point with regard to core level photoionization in simple diatomic molecules [35]. The present (X = 0 limit) model can be considered as a narrow-band or strongcoupling limit of the SG theory. This becomes clear upon contemplation of the V dependence of the SG spectrum. In the weak-coupling limit, V& merely converts
443
J. W. Gadzuk, S. Doniach /Soluble relaxation model
,-.E+ ** Em -.
-.
. . E
_‘-0”
.
-, .-
*in
;a
JI_
&y m
a
+in
*+ nn
&+ m
*I-
a
Jlin
a
m
E
Fig. 6. Molecular orbital analog of the unfiied orbital relaxation model, illustrating the differences which arise depending upon the sign of ra - em, negative for the left-hand column and positive for the right. The fist row shows the final state eigenvalues E+and E_. The second and third rows depict the wavefunctions of the initial state, $i,,, and the possible final states, either bonding ($ _) or antibonding (IL+), from which it is possible to see the origins of the relative weights in the satellite versus main peaks in the spectra, shown in the bottom row. For esthetic reasons, the delta finctions have also been smoothed out.
the excited state Ea into a single broadened virtual level. As the coupling strength becomes comparable to or exceeds the bandwidth of the screening electrons, new virtual resonances and/or localized states which are split off below or above the band may form [ 151. These split-off states are in fact our screening states, e+, which suggests that the two-state model achieves closer union with quantitative reality when applied to systems in which I VI is the largest energy parameter. Nonetheless, the physical insights gained from the model remain valid over the entire parameter space. 5.3. General case The general solution to the “4-state relaxation model” is contained in eqs. (7), (8), and (10). Complete numerical consequences, in the limits of either pure surface
444
J. W. Gadzuk,
S. Don&h
/Soluble
relaxation
model
plasmon or unoccupied atomic orbital screening have been given in the preceding sections. As a change of pace, in fig. 7 we display some calculated spectra in the form given by eq. (1 l), where each discrete line is broadened into a Doniach-SunjiC lineshape (w, -+ 00 limit of eq. (14)). DS parameter values are y = 0.3 and f= 0.3. The 4-state model is characterized by the atomic orbital parameter F E V2/(em - g& = 0.1 and surface plasmon parameter A E X*/(e, - &) = 0.6,2.0, and 3.0, as labeled.
-75
-7.5
-5.0
- 5.0
-2.5
0
-2.5
0
2.5
2.5
E
Fig. 7. Some general spectra, obtained from eqs. (7), (8) and (10) which have been broadened with a DS lineshape. The atomic orbital coupling parameter r (defined in the text) is held constant and the surface plasmon coupling has been varied, as shown in the figure.
J. W. Gadzuk, S. Doniach /Soluble relaxation model
445
The object of this exercise is to follow the change in the spectrum as the plasmon coupling is increased relative to the atomic orbital hopping. In tig. 7a, for the smallest value of the surface plasmon relaxation shown here, the two peaks at E a 0.25 and 2.0 can be regarded as LW or GS structures that have been weakly modified by surface plasmon hybridization. The structure at e - -5.5 is the observably-intense-one of two surface plasmon satellites. As A is increased, the twopeaked unfilled orbital spectrum is altered first into peaks of equal intensity, as seen in the progression from figs. 7a to 7c and then into an apparent single line (actually a doublet which is unresolvable on the scale of plasmon relaxation energies). As this goes on, both the intensity of the surface plasmon satellite increases and its energy decreases, due to the stronger repulsion between the zero and one plasmon portions of the state vectors with larger A and thus h. Being now in a position of understanding the basic mechanisms which contribute to the hole spectral function and thus photoelectron energy distribution, one should be able to proceed with some degree of confidence in simulating such spectra as shown in fig. 7 using eqs. (7) (8), (1 l), and (14) and a hand calculator.
6. Summary In the preceding chapters we have explored the physical basis of the relaxation response to adatom core hole creation in the limits in which the total response can be modeled in terms of either displaced surface plasmons or occupation of unfilled atomic orbitals on the adatom or some combination of the two. The degree to which one or the other relaxation or screening process dominates in real situations is a question of considerable current interest. From our studies some semiquantitative guidelines have emerged. The strength of the surface plasmon response depends upon both the couplihg constant h and the surface plasmori energy o, in two observable ways; namely through a relaxation shift Ae, 7 h*/o, in the main line and the generation of a satellite line with an intensity - AZ/o:. Thus even though a large surface plasmon shift might be experienced, due to a large coupling constant A, the intensity in the satellite could be small due to a large value of w,. However the position of the satellites, in the infinite order IBM, depends only upon w, and not A. In contrast, the degree to which the occupation of the unfilled orbital contributes to the screening of the hole is determined by the ease with which screening charge may flow from the substrate to the adatom. The parameter p = E”,/2V is a measure of this quality, with small values of p implying best charge flow. Strongest coupling occurs for transfer between states E, and ;a which are at the same energy. Hence with E, = 0 (the convention which has been adopted) optimal mixing occurs as 7, and thus p approaches zero. Secondly, since the hopping increases as V - l/p does, the small p limit again indicates efficient charge flow. For the complete system in which both processes are possible, the numerics of eqs. (7) (8) and (10)
446
J. W. Gadzuk, S. Doniach /Soluble relaxation model
show that mixing between the two processes is most significant in the parameter range in which l/p = h’/o$, in agreement with our enlightened expectations. The question will still be asked whether it is definitively possible to ascertain which screening mechanism dominates in a true experimental situation or even if such a separation is meaningful. We made a (dogmatic?) statement in the introduction that this question could not be resolved on the basis of a comparison of calculated relaxation energy shifts with experimentally determined ones (assuming that relaxation and chemical shifts could be separated, which is rarely done legitimately), since the numbers are reasonalby similar [36]. What Is not similar and thus may be useful as a diagnostic tool is the energy position of the satellites with respect to the main line. Whilst the surface plasmon satellite is at lest w, down, the doublet separation, from eq. (26), is AE = 2V(l + (E”a/2V)2)1/2 , which will probably be much smaller than the surface plasmon energy. Observation of such a doublet will provide encouragement (although not unique proof) for the operation of atomic-like relaxation mechanisms [37]. However, life is not all that simple since the non-observation of the surface plasmon satellite does not permit the conclusion that surface plasmon or image charge screening is not occurring, only that the intensity of the shakeup satellite is too low to be observed. We are thus left in the happy position in which more work, from either point of view, will certainly be welcomed and needed in our quest for a better understanding of the relaxation response in XPS of adsorbed atoms [38,39].
References [ l] Review and survey articles:
[2] [3] [4] [S] [a] [7]
L. Hedin and S. Lundqvist, Solid State Phys. 23 (1969) 1; L. Hedin, in: X-ray Spectroscopy, Ed. L.V. Azaroff (McGraw-Hill, New York, 1974); S. Lundqvist and G. Wendin, J. Electron Spectr. 5 (1974) 513; G.D. Mahan, Solid State Phys. 29 (1974) 75 ; D.C. Langreth, in: Collective Properties of Physical Systems, Eds. B.I. Lundqvist and S. Lundqvist (Academic Press, New York, 1974), and X-ray Emission and Absorption, NORDITA-76/21 (Copenhagen); M. Sunjic, 2. Crljen and D. SokEevid, Surface Sci. 68 (1977) 479; J.W. Gadzuk, in: Photoemission and the Electronic Properties of Surfaces Eds. B. Feuerbacher, B. Fitton and R. Willis (Wiley, New York, 1978). B.I. Lundqvist, Phys. Kondens. Mater. 6 (1967) 193; 6 (1967) 206; 7 (1968) 117; 9 (1969) 236. G.D. Mahan, Phys. Rev. 163 (1967) 612. P. Nozieres and C.T. DeDominicis, Phys. Rev. 178 (1969) 1097. DC. Langreth, Phy? Rev. Bl (1970) 471. S. Doniach and M. Sunjii, J. Phys. C3 (1970) 285; J.W. Gadzuk and M. Sunjic, Phys. Rev. B12 (1975) 524. J.W. Gadzuk, J. Electron Spectr. 11 (1977) 355.
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[8] D.A. Shirley, Chem. Phys. Letters 16 (1972) 220; D.A. Shirley, R.L. Martin, S.P. Kowalczyk, F.R. McFeely and L. Ley, Phys Rev. B15 (1977) 544. [9] R.E. Watson, J.F. Herbst and J.W. Wilkins, Phys. Rev. B14 (1976) 18. [lo] See also A.R. Williams and N.D. Lang, Phys. Rev. Letters 40 (1978) 954, for a discussion of the philosophy espoused in refs. [S] and [ 91. [ 111 G.E. Laramore and W.J. Camp, Phys. Rev B9 (1974) 3270; A.C. Hewson and D.M. Newns, Japan. J. Appl. Phys., Suppl. 2, Part 2 (1974) 121; J. Harris, Solid State Commun. 16 (1975) 671; J.W. Gadzuk, Phys. Rev B14 (1976) 2267; A. Datta and D.M. Newns, Phys. Letters 59A (1976) 326. [ 121 B. Gumhalter and D.M. Newns, Phys. Letters 57A (1976) 423; B. Gumhalter, J. Phys. (Paris) 38 (1977) 1117. [ 131 N.D. Lang and A.R. Williams, Phys. Rev B16 (1977) 2408. [ 141 K. Schonhammer and 0. Gunnarsson, Solid State Commun. 23 (1977) 691. [15] P.W. Anderson, Phys. Rev. 124 (1961) 41; D.M. Newns, Phys. Rev. 178 (1969) 1123. [ 161 R.A. Beaumont and R.S. Pierce, The Algebraic Foundations of Mathematics, (AddisonWesley, Reading, Mass., 1963) p. 361. [ 171 H.W. Meldner and J.D. Perez, Phys. Rev. A4 (1971) 1388. [18] S.A. Flodstrom, R.Z. Bachrach, R.S. Bauer, J.C. McMenanin and S.B.M. Hagstrom, J. Vacuum Sci. Technol. 14 (1977) 303; D.S. Rajoria, L. Kovnat, E.W. Plummer and W.R. Salaneck, Chem. Phys. Letters 49 (1977) [19] ?D. Mahan Phys Rev. Bll (1975) 4814.B14 (1976) 3702; J.D. Dow and D.R. Franceschetti, Phys. Rev. Letters 34 (1975) 1320; P. Longe, Phys. Rev. B14 (1976) 3699; J.D. Dow, D.L. Smith, D.R. Franceschetti, J.E. Robinson and T.R. Carver, Phys. Rev. B16 (1977) 4707. [20] P.H. Citrin,Phys. Rev. B8 (1973) 5545; S. Htifner and G.K’ Wertheim, Phys. Rev. Bll (1975) 678; P.H. Citrin, G.K. Wertheim, and Y. Baer, Phys. Rev. B16 (1977) 4256. [21] P. Minnhagen, Phys. Letters 56A (1976) 327; J. Phys. F7 (1977) 2441. (221 A controversial issue as to the interplay between phonon relaxation and hole lifetime effects has been much discussed 1ateJy. See: M. Sunjic and A. Lucas, Chem. Phys. Letters 42 (1976) 462 and subsequent work on questions raised here. P. Minnhagen, J. Phys. F6 (1976) 1789; J.W. Gadzuk, Phys Rev. B14 (1976) 5458; P.H. Citrin and D.R. Hamann, Phys. Rev. B15 (1977) 2923; G.D. Mahan, Phys. Rev. B15 (1977) 4587; C.O. Almbladh and P. Minnhagen, Phys. Rev. B, in press. [23] B. Gumhalter and D.M. Newns, Phys. Letters 53A (1975) 137. [24] (a) The Fermi-Thomas dielectric function is quite adequate for the surface calculations considered here, as detailed in J.W. Gadzuk, Surface Sci. 67 (1977) 77. (b) J.C. Fuggle, E. Umbach, D. Menzel, K. Wandelt and C.R. Brundle, Solid State Commun., to be published; (c) B. Gumhalter, J. Phys. Cl0 (1977) L219. [25] For another perspective see: H. Benson, The Relaxation Response (Morrow, New York, 1975).
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[26] R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234; J. Heinrichs, Phys. Rev. B8 (1973) 1346; J. Harris and R.O. Jones, J. Phys. C6 (1973) 3585. [ 271 R. Manne and T. Aberg, Chem. Phys. Letters 7 (1970) 282; C.S. Fadley, Chem. Phys. Letters 25 (1974) 225; S. Manson, J. Electron Spect. 9 (1976) 21. [281 Note that eq. (17), as written, fulftis eq. (20) only to order AZ/w: since E+ and E_ given in eq. (15a) have been approximated by eq. (15b and c). The sum rule is satisfied to all orders in h*/w$ if the exact eigenvalue expressions are used. 1291 Although this would be accompanied by changes in w m,f, and 7 in eq. (14) in addition to the desired variation in KFT or e(ql1) in eq. (18), all of which affect h. [301 H. Morawitz and M.R. Philpott, Phys. Rev. BlO (1974) 4863; R. Silbey, Ann. Rev. Phys. Chem. 27 (1976) 203. [311 A.M. Bradshaw, W. Domcke and L.S. Cederbaum, Phys. Rev. B16 (1977) 1480. 1321 P.H. Citrin and D.R. Hamann, Phys. Rev. BlO (1974) 4948. [33] (a) Some progress on composite response functions has been presented by one of us in ref. [24] and in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977. (b) One can conceive of some pathological situations where this assertion may break down; for instance in the case of a transition metal adsorbate where the screening electron occupies an unfilled d orbital. Since the radius of the d shell is less than that of the still occupied s valence electron and since the mean distance of the alternative induced surface charge from the core hole must be greater than the s radius, the resulting screening energies for the two situations could differ by a significant (-factor of two) amount. Undoubtedly other exceptions of this type could be found, and in fact might be very useful as candidates for investigative studies aimed at the elucidation of the basic physics of the relaxation process. [34] Because, as pointed out by LW amongst others, Aer is the Coulomb energy between the core hole and a unit of electron charge roughly an atomic distance away. This volume integrated quantity is not too sensitive to the shape of the charge distribution. [35] L.S. Cederbaum and W. Domcke, J. Chem. Phys. 66 (1977) 5084. [36] It should be reemphasized that the atomic screening has been proposed mainly as a mechanism for determining the “true” leading edge position which is not to say that this feature dominates. On the other hand, the point of the present study has been emphasis on the rich structure of the total spectrum, which if correctly analyzed, provides additional checks and confirmations on whatever relaxation model and characteristic parameters are chosen. For a more detailed exposition of this philosophy, see J.W. Gadzuk, in: Proc. Intern. Conf. on Solid Films and Surfaces, Tokyo, 1978 (to be published in Surface Sci.). [37] Fuggle et al. (ref. [24b]) have observed an -6 eV doublet structure in XPS on the 1s core state of N2 adsorbed on both W(110) and Ni(lOO) and also on the C 1s core state of CO adsorbed on Cu(100) and W(110). This excitation energy is similar to the lowest gas phase molecular ion excitation energy. Since the satellite structure is substrate invariant, this suggests a localized molecular-like excitation mechanism, possibly a SG satellite peak occurring in the unfilled orbital screening process. Another alternative that cannot be ruled out on the basis of the reported data is pure intra-molecular shakeup which has little to do with the extra-molecular relaxation. Additional experiments will be required to resolve this question. Parenthetically we also note that the leading peak for the N2 was further split into an -1.4 eV doublet, possibly due to differential relaxation shifts experienced by the inequivalent N atoms for N2 adsorbed in the standing up position, as discussed in ref. [ 24aJ. [38] Since completion of this manuscript, we have become aware that the unfilled atomic orbital model was fist introduced, in another context, by A. Kotani and Y. Toyozawa, J. Phys. Sot. Japan 37 (1974) 912. [39] S.S. Hussain and D.M. Newns (Solid State Commun., 25 (1978) 1049) have independently worked on a relaxation model similar to ours.