A. VALENCE-ELECTRON LEVELS OF ADSORBED MOLECULES Theory of the Angular Dependence of the Photoemission Line Shape from an Adsorbate BY A. LIBBSCH AND E. W. PLUMMER" Department of Physics, University of Pennsylvania, Philadelphia, Pa. 191 74 Received 17 June 1974 A theory of the angular-resolved photoemission from localized adsorbate orbitals is presented in which the effects of the final state are discussed in detail. This theory is applied to the problem of line shapes and peak positions in a photoemission energy distribution from an orbital localized at the surface. The interference of the wave scattered elastically from the periodic potential of the solid with the wave emitted directly from the adsorbate can cause apparent splitting of a single adsorbate level and l e v shifts in peak positions.In a previous paper a " one-step '' model for photoemission was developed and applied to photoemission from ads0rbates.l The essential ingredient of this model was an appropriate description of the Bloch character of the final state of the excited electron by using a multiple-scattering theory such as has been applied in EEED calculations.2 The purpose of the original paper was to demonstrate that the angular distribution of the photoemitted electrons from a localized level is dominated by the symmetry of the adsorption site as well as the symmetry of the bonding orbital^.^ In this paper the theory, which will be described below, will be applied to a more practical problem: the line shape and position of the peak in the photoemissioii spectrum from an adsorbate as a function of the angle of detection and photon energy. Recent experimental and theoretical l * work has stimulated considerable interest in angular-resolved photoemission from surface adsorbates.The angular dependence of the emission from a localized surface orbital may well yield information about the symmetry of the bonding site as well as the symmetry of each 0rbita1.~ On the other hand, if an experiment is conducted with a fixed angle of collection these effects may become a liability when comparing other experimental data or theoretical calculations of the ground-state orbitals. What one measures will depend upon the angle of collection as well as the incident photon energy. Calculations for a simple model of a hydrogenic adsorbate on a cubic crystal predict that the measured photo- emission spectra from this localized surface orbital may have its peak shifted by 1 eV or in some cases appear as a double peak, depending upon the final-state energy, adatoni position and angle of collection.These types of effects seem to have been observed in two cases : (1) CO adsorption on Ni( 11 1) where the CO peak splits as the crystal is rotated, and (2) Egelhoff seems to see a splitting in the hydrogen level at 5 eV below the Fermi energy on (1OO)W when he collects 14" off normal, whereas Pluinmer and Waclawski do not see a splitting when collecting over all azimuthal angles from zero to 45" in polar angle.4 THEORY The photoemission process is characterized in our microscopic " one-step '* model by the description of the initial and the final state of the system.It is con- venient to use a linear combination of atomic orbitals for the initial state as has been 1920 ANGULAR DEPENDENCE OF THE PHOTOEMISSION LINE done previously by several other This approach permits a simple incorporation of the adsorption geometry and can easily be extended to hybridized adsorbate-substrate orbitals as well as LCAO's appropriate to the substrate itself The new feature of the model lies in the description of the final state. As it is to be expected on the basis of low-energy electron diffraction experiments, the coherent elastic scattering of the excited electron by the lattice potential is strong in the energy range of interest in photoemission. This fact indicates that the final state must be described by its full Bloch wave-function, since the plane-wave final state approxima- tion does not incorporate the multiple scattering.Furthermore, the electronic mean free path, Ze, at these energies is of the order of a few lattice spacings as a consequence of strong inelastic electron-electron interactions. We therefore employ a multiple scattering formalism similar to those that are currently being used in analysis of LEED intensities.2 The matrix elements between the initial and the complete final state are then evaluated numerically without any further assumptions. Several points regarding the adequacy of the above model and its relation to existing photoemission calculations are noteworthy. First, no attempt has been made at this point to include the presence of the hole that is left behind by the excited electon.Thus, possible relaxation effects as well as exciton-like interactons of the outgoing electron with the bole are so far not taken into consideration. Secondly, for infinite systems (i.e., extended initial states) and in the limit of weak electronic damping, the present model becomes similar to band structure calculations of photoemission intensities.' Since most materials, however, show rather short or medium mean free paths and since the band-structure approach is by its very nature limited to bulk photoemission, the above described model is, in principle, more appropriate. VACUUM SOU D 0 0 0 e 0 1" FIG. 1 direct .-Illustration of two processes contributing to photoemission from adsorbate pz orbital : (1) emission into plane-wave final state, (2) indirect emission via backscattering from substrate.Only single scattering from the first layer is indicated. In order to illustrate the influence of the coherent scattering, we consider the excitation of an electron from a localized s or p orbital adsorbed on the substrate. The two processes that contribute to the scattering amplitude for the adsorbate signal are the direct emission from the orbital into a plane-wave final state, lkf>, and the indirect emission from the orbital via backscattering from the substrate lattice potential. Both are indicated schematically in fig. 1. For a given polarizationA . LIEBSCH AND E. W. PLUMMER 21 vector A inside the solid and a photon energy ha, the cross-section is a function of final electron energy Ef and detector angles 0, and Qf.It is given by the expression Here, 4R is the wave function of the orbital adsorbed at the site R and Ei its energy level. In the case of surface molecule^,^ (bR is to be replaced by the linear combination of orbitals appropriate to the adsorbed atom or molecule and one or several substrate atoms. T = VL+ VLGT is the T-matrix corresponding to the lattice potential VL and G is a free-electron propagator with an appropriate self-energy inserted to include inelastic effects. The first term of the matrix element in eqn (l), i.e., the direct emission in the absence of scattering, is simply proportional to the Fourier transform of the initial state : (2) where Fn,, and Ylm are the radial and the angular part of 4(kf), respectively. The second term of the matrix element, i.e., the indirect emission from the adsorbate via backscattering from the substrate, can be evaluated for the general case of multiple elastic scattering and arbitrary atomic potentials. For simplicity, however, we present here only the result for single scattering from s-wave scatterers : 4 E f , @,, Qf) I(kfl(1 + T a p = A I 4 R ) l 2 ~ ( E f - - E i - ~ ~ ) .(1) YIP A I ~ R ) = exp(-ikf R )hkf A F A ~ ~ I ) Ylrn(k,>, ( ~ ~ I T G P AIW + ~ X P (-ikf * ~)~,l(IkfI)t(IkfI>(l/a~) C Y,rn(kfll +O, - k f l ( g ) > 0 where The g are the reciprocal lattice vectors of the Bravais net parallel to the surface and a is the lattice constant. The normal and parallel position of the adatom relative to the substrate are denoted by dl and dll, respectively.The quantity t( lkfl) represents the s-wave component of a single-site scattering vertex in the substrate, Vo is the inner potential and T(E) is the imaginary part of the optical potential. The above result exhibits the following physical features. (1) The contribution to the matrix element due to backscattering is no longer proportional to the angular part of the Fourier transform Yzrn(kf) of the initial state. Instead, the expression involves the Fourier components corresponding to directions in k-space that differ from kf by a reciprocal lattice vector parallel to the surface and that point into the crystal rather than to the detector. (2) Similarly, the matrix element is no longer pro- portional to kf * A as in the absence of scattering but rather involves the factors (k,ll +g, -kfL(g)) .A.(3) For a given detector angle, the structure in the intensity as function of final energy Ef is determined by the band structure of the substrate, In the specific case considered in eqn (3), resonance energies occur for Re rc(g)a = 2nn, y1 integer, i.e., where En = (li2~nfa)~/2m, Eg = (lig)'/2m, and Eli = E sin2 Of. These energies coincide with band crossings in the corresponding free electron band structure. (4) The geometry of the adsorption site enters the matrix element only via two phase- factors, one for the normal position dL and one for the parallel displacement d , ~ of the Ef + Yo = Ell + (En + Ee + kf 11 gh2 Jm)2 J4En, ( 5 )22 ANGULAR DEPENDENCE OF THE PHOTOEMISSION LINE adatom relative to the underlying substrate.This is a particularly attractive feature since it permits the separation of adsorbate and substrate geometry. The positions at which extrema in the intensity occur are entirely determined by the symmetry of the substrate whereas the relative intensities of these extrema are determined by the adsorption site. RESULTS In order to illustrate some of the consequences of the theory outlined above, we show in fig. 2 the intensity as function of final energy for emission along the surface normal from an s orbital bound in two configurations.8 The curves represent the intensity for the case of single scattering (solid curves), multiple scattering (dotted curves), and in the absence of scattering (dashed curves), The three-dimensional FIG.^.-Intensity (arbitrary units) as function of final electron energy for emission from an s orbital adsorbed in position (0, 0, a), (a) ; and ( 4 2 , a/2, a/2), (6) relative to the substiate : Single scattering (solid curves), multiple scattering (dotted curves), and no scattering (dashed curves). The arrows indicate the resonance energies specified by the reciprocal lattice vectors, and the corresponding band crossings in the free-electron band structure at the tap of the figure.I I I 1 + S ORBITAL Ef =6eV A =(0,0,1) (b) CENTRES - 0.5 - 0 30 60 0 30 60 90 final polar angle 6'' FIG. 3.-Photoemission intcnsity (arbitrary units) as function of final polar angle for an s orbital ad- sorbed in top (a) and centre (b) positions ; single scattering (solid curves), multiple scattering (dotted curves), and no scattering (dashed curves).2.5 2 .o * h 1.5 5 .- v) * .- E: .C( *i 1 .o 0.5 '3.4 4.4 5.4 6.4 ' final energy/eV I FIG. 4.-Photoemission intensity (arbitrary units) as function of final electron energy for an s orbital adsorbed in top position. The solid curves represent the intensity along four different detector angles : 6''. The dashed curve shows the intensity in the absence of scattering. The arrows indicate the effective peak position caused by final-state effects.24 ANGULAR DEPENDENCE OF THE PHOTOEMISSION LINE reciprocal lattice vectors specify the resonance energies, eqn (9, and the corresponding band crossings as indicated in the free-electron band structure at the top of the figure.The effect of backscattering from the substrate is seen to be of the order of 20-50 % relative to the intensities in the absence of ~cattering.~ It is, however, rather remarkable that the multiple scattering intensities agree so closely with the single scattering intensities. We believe this to be a consequence of the fact that the single scattering resonances coincide not only with the reflection point of the first band, i.e., the Bragg energies as is the case in LEED, but also with the intersections of the first band with all higher bands.1° This is due to the circumstance that the adatom acts as a spherical source in contrast to the incoming plane wave in LEED. While this point requires further detaiIed study, it might prove to be of considerable practical interest in that single- or double-scattering approximations to the final state in photoemission from adsorbates are much more adequate than in LEED.Comparing panels (a) and (b), we notice that the change of the adsorption site from (0, 0, a) to (a/2, 4 2 , a/2) inverts some of the maxima to minima due to the presence of the phase factors as described above. It is apparent from eqn (5) that in normal direction all resonances are degenerate with regard to various vectors g whose components have opposite signs. At finite angles 0, and @,, however, these resonances split very rapidly indicating that the photoemission intensity exhibits a strong angular dependence. As an example we show in fig. 3 a figure from ref. (1) in which the intensity is plotted as function of polar angle 0, for two adsorption sites at a fixed energy E,.In the absence of S OR81TAL W = 2 e V 4= 40" + f = 0 3.4 4.4 5.4 6.4 7.4 - - . _ _ kina1 energy/eV angle is 4Q" ; otherwise as in fig. 4. FIG. 5.-Intensity as function of final energy for two different electronic mean free path. The detectorA . LIEBSCH AND E. W. PLUMMER 25 scattering, the intensity is a smooth function proportional to k, A (dashed curves). In the limit of single scattering (solid curves), the maxima that are seen in panel (a) for the top position are inverted into minima in panel (b) for the centre position because of the phase factor associated with the parallel displacement. In both cases, the effect of multiple scattering (dotted curves) tends to smooth out the single-scattering intensities.The intensity also varies with azimuthal angle and the type of orbital chosen for the ads0rbate.l In the example above and in ref. (I), we have assumed that the initial state has a sharp energy level, Ef. For a given photon energy, hco, we accordingly observe only electrons with a kinetic energy given by Ef = Ei+ttcv (no relaxation). We now consider the case where the initial level can be described by a Gaussian distribution centred around Ei with a full-width at half-maximum given by W. For a particular photon energy, we then observe emitted electrons with kinetic energies approximately in the range E,+ko- W < Ef 5 Ei+ ha+ W. Because of the coherent scattering in the final state, the photoemission intensities exhibit a considerable amount of structure over this energy range, thus causing the observed peak to deviate in shape and position from the Gaussian distribution centred around Ei + hco that one would obtain for plane-wave final states.This effect is shown in fig. 4, in which intensities are plotted for emission from an s orbital for four different detector directions 0, (solid curves). The dashed curve gives the corresponding distributioii in the absence of scattering, i.e., for plane-wave final states. Over the indicated range of 60°, the peak is seen to greatly vary its shape and to change its position by about 2 eV. At I .5 CI A I .o * 8 -g .- c .I E .I UI 0 . 5 n ,---1- v 3.4 4.4 5.4 6.4 7.4 final energy/eV detector angle is 40" ; otherwise as in fig. 4.FIG. 6.-Intensity as function of final electron energy for two different initial state widths. The26 ANGULAR DEPENDENCE OF THE PHOTOEMISSION LINE 0, = 60°, a weak splitting of the peak takes place. Fig. 5 illustrates the dependence of the observed peak on the electron damping length. Since strong damping (i.e., small Ze) diminishes the importance of scattering, the peak distortion and shift are less pronounced for Ze = 12 A than for Ze = 20 A. 2.5 2 .o x -3 1.5 8 c.. .* c c 0 .I-# v1 3 1.0 0.5 n - S ORBITAL U - 12 - 10 -8 initial energy/eV detector angle is 40"; otherwise as in fig. 4. FIG. 7.--lntensity as function of initial electron energy for three different photon energies. The We show in fig. 6 the influence of the width of the initial level on the observed peak.It is obvious that the final state effects will be more significant for broad levels than for narrow ones. (For infinitely sharp levels, the peak position is unaffected and only the relative intensities are modified.) Finally, in fig. 7 we show the effect of changing the photon energy. The scattering effects depend upon the final energy so they will change as the photon energy is varied, as is seen in fig. 7. This points out an obvious check for initial or final state effects in line shape. In summary, we have shown that : (a) the correct final state can cause considerable structure in the measured intensities, as a function of the angle of detection ; (b) this structure is closely related to the substrate bonding symmetry; and (c) for a reasonably wide adsorbate level (1-2 eV) the peak position and peak shape in the photoemission spectra depends on the angle of detection, the electron escape depth and the final-state energy.A. Liebsch, Phys. Rev. Letters, 1974, 32, 1203. For a review of LEED theories see, for example, G. E. Laramore, J. Yuc. Sci. Tech., 1972,9,625.A . LIEBSCH AND E . W. PLUMMER 37 For photoemission from surface molecules into plane-wave final states, Gadzuk (J. Vac. Sci. Tech., 1974, 11, 275, and Solid State Comm. (in press)) has demonstrated that the angular re- solved intensity is highly anisotropic reflecting both the adsorption geometry and the orientation of the chemisorption orbitals. While we believe the adequate description of the initial state to be crucial for a quantitative analysis of photoemission spectra, we restrict our discussion in the present paper to localized adatom levels in order to isolate the angular anisotropy due to the Bloch character of the final state.U. Gerhardt and E. Dietz, Phys. Rev. Letters, 1971, 26,1477 ; T. Gustafsson, P. 0. Nilson, anti L. Wallden,Phys. Letters, 1971,27A, 121 ; N. V.Smith and M. M.Traum,Phys. Rec. Letters, 1973, 21, 1247 ; L. Wallden and T. Gustafsson, Physica Scripta, 1972, 6, 73 ; R. Y. Koyama and I-. R. Hughey, Phys. Rev. Lett, 1972,29,1518 ; R. H. Williams, J. M. Thomas, M. Barber and N. Alford, Chem. Phys. Lett., 1972, 17, 142. Also, private com~nunications with B. J. Waclawski (0, on (100) W) and W. F. Egelhoff (HZ on (100) W). D. E. Eastman and J. E. Demuth, to be published in J. Vuc. Sci. Tech. W. L. Schaich and N. W. Ashcroft, Phys. Rev. B, 1971,3,2452 ; and D. R. Penn, Piiys. Rev. Let- ters, 1972, 28, 1041. ' See, for example, N. V. Smith, Phys. Rev. Letters, 1969,23, 1222 ; N. E. Cristensen and B, 0. Seraphin, Php. Rev. B, 1971, 4, 3321 ; and A. R. Williams, J. F. Janak, and V. L. Morruzzi, Phys. Rev. Letters, 1972, 28, 671. * A11 calculations are for the (001) surface of a simple cubic crystal and, unless otherwise specified, the following parameters are used : Yo = 10 eV (inner potential), 1, = 6A (mean free path), 6, = 77/2 (s-wave phase shift), a = 4&lattice constant), and A = (O,O,l). All curves are divided by [kflF&(lkfl) so that the intensity in the absence of scattering is energy independent. The initial energy can then be taken as arbitrary. For realistic potentials, thz magnitude of these effects is likely to be even higher. ances. ' O The remaining crossings between higher-lying bands correspond to multiple-scattering reson -