Introductory Paper Theoretical Aspects of Photoemission, Photodesorption and Photochemistry of Adsorbates BY T. B. GRIMLEY Donnan Laboratories, University of Liverpool, P.O. Box 147, Liverpool L69 3BX Received 10th September, I974 1. INTRODUCTION Photons which interact with solid surfaces, and with adsorbates, can be scattered elastically or inelastically, or they can be absorbed. Scattering is of little interest to us at this Discussion. Elastic scattering is studied experimentally to find the optical constants of the materials (substrate and adsorbed layers) ; the inelastic scattering of optical or U.V. photons is the Raman effect. The absorption of photons by isolated atoms in their ground states leads only to electronic excitation or to photoemission, but photon absorption at surfaces, or in adsorbates, may have many additional consequences ; adsorption, desorption, chemical attack on substrate, chemical changes in the ambient gas, and so on.Just over one half of the papers at this Discussion are concerned with such photochemical phenomena, and just under one half are concerned with photoemission from substrates, and adsorbates, that is to say with UPS and XPS. In spite of its historical importance in the development of quantum theory, only now are microscopic theories of photoemission being developed. The aim of these theories is to relate the experimental data (the energy-resolved angular distribution of the emitted electrons) to the electronic structures of surfaces and adsorbates. I shall discuss some of the problems involved in this task below, but it is important to remind ourselves now, that UPS is only surface sensitive because of the short (- 1 nm) mean free path of 5-100 eV electrons in solids,'* that this short mean free path is a many- body phenomenon, and that consequently a proper many-body theory of photo- emission is the ultimate goal.In the meantime, however, experimental workers are developing photoemission as an empirical diagnostic tool in chemisorption studies. This is amply demonstrated by the papers on photoemission at this Discussion. One of the problems in the theory of photoemission, which is present also when we turn to other photo-effects in adsorbed species, concerns the value of (or the operator for, if we use quantum theory) the vector potential of the radiation field at the surface, or in the adsorbed species; I shall refer to this in section 3.But there are many other problems to be solved, and the fundamental theory of photochemical reactions at surfaces is in no better state than that for homogeneous photochemical reactions. Thus, an ab initio calculation of the rate of a simple process like the photodesorption of CO from tungsten has not yet been attempted because, although the formal theory is straightforward, at least one simplification of doubtful validity is needed to reduce the problem to a simple form involving the familiar potential energy curves of pseudo diatomic molecules (see section 4). 78 INTRODUCTORY PAPER Most of the fundamental theoretical problems in the study of chemical reactions at surfaces remain untouched.As in the theoretical study of gas phase reactions, progress hinges on the calculation of electronically adiabatic potential energy hyper- surfaces for the reaction. This is the task of chemisorption theory. 2. PHOTOEMISSION In view of Lippmann's generalization of Ehrenfest's theorem, the photocurrent can be written in terms of a transition rate. Thus, the transition rate wF to a final state IF) of energy EF is given by WF = h-'xl (Fl TI 1) I 'a(&? - EI) (1) I where T = Y+ VG Y is the T-matrix corresponding to the electron-photon interaction V, and G is the stationary propagator (i.e., the Fourier time transform of the propa- gator) for the electron-photon system ; G(E) = G(E+iO) = (E+iO-H)-I = (E+iO-H, - Y)-', Since Y is small, it is sufficient to replace T by Y in eqn (1).Also, if we treat the radiation field classically or, after evaluating matrix elements of Boson operators otherwise, we can write eqn (1) as a transition rate involving initial and final ezectronic states [ i ) and If) HoIn = &IF), Hot0 = W). Wf = h-'Cl(f[ Vli)l"(&~ +Ao -Ef) (2) i Here hco is the photon energy, and Y = (ieh/mc)V*A, where A is the vector potential of the radiation field. The final state If) in eqn (2) describing photoemission in the presence of condensed matter needs a little thought. This is best illustrated by considering photoemission from a free electron ~ e t a l . ~ - ~ Of course a perfectly free electron cannot absorb a photon because energy and momentum cannot be conserved in the process; the matrix element in eqn (2) vanishes unless momentum is conserved, and the 8-function conserves energy. But the situation is different if the electron is confined to the half- space z < 0 say, by a step potential - V, if z < 0 but zero otherwise, because now the matrix element in eqn (2) acts only to conserve the component of momentum parallel to the surface z = 0.The initial state li) consists of incoming and outgoing plane waves in the metal, and an exponentially decaying state in the vacuum ( z > 0) with the same component of electron momentum parallel to the surface (fig. 1). For a given parallel momentum hfil there are two linearly independent final states with energy cf and orthogonal to li). If we choose them in the forms (3) z > o z t o describing an incoming state in the metal, and both incoming, and outgoing states in the vacuum, and 2 3 0 z < o (4) describing both incoming and outgoing states in the metal, and only an incoming state in the vacuum (fig.l), then since only If) contains the outgoing state exp(i(fz+ fll-p)) describing the photoelectron in the vacuum, this is the final state to be used in9 eqn (2). Had we chosen the two states If) and If’) differently, both would have contained the outgoing state in the vacuum, and the transition rate to both would have had to be calculated to obtain the photocurrent. In eqn (3) and (4) T. B . GRIMLEY if we choose our energy zero to correspond to electrons at rest in vacuum. If > If’> FIG. 1.-Initial and final states for photoemission from a free electron metal.For the step potential, the matrix element in eqn (2) is proportional to the triple product of the values at z = 0 of the initial and final state wavefunctions, and the vector potential. Consequently, for the free electron model, photoemission is a surface effect. This is because it is only at the surface that the potential energy function in Schrodinger’s equation changes its value, and we may note that, because the electron still moves perfectly freely parallel to the surface, there can be no photo- emission if the electric vector in the radiation field is parallel to the surface. If we add a periodic crystal potential to the free electron model, there is, in addition to the surface effect, a “ volume effect ” depending on the energy band structure.This volume effect is unphysical unless allowance is made for the finite mean free path of the electrons in the solid. The simplest way to do this is to use a complex k in eqn (3).5-7 2.1 PHOTOEMISSION FROM ADSORBATES Strictly speaking there is no such thing as photoemission from adsorbates. What we can observe, and derive an expression for (see below), is the difference in the emis- sion with, and without, the adsorbate. It is generally believed that the energy- resolved angular distribution of this difference in photoemission can provide informa- tion on the local geometry of the adsorption site, and on the orbitals used in forming the chemisorption bond. The grounds for this belief are most easily demonstrated by looking at the theory of photoemission from an isolated atom to plane wave final states,* and remembering that in a molecular orbital theory of chemisorption, the atomic orbitals on the adatom will be chosen to belong to irreducible representations of the point group describing the symmetry of the adsorption site.For a plane wave final state the matrix element in eqn (2) is If) = Ik) = exp(ik*r) < f I Vli) = - (ek/mc)Ao(e-k)(k - qli). (6) (7) Here i?A0 exp(i4.r) is the vector potential of the radiation with propagation vector 410 INTRODUCTORY PAPER and polarization specified by the unit vector e. the form For a free atom, the initial state has I9 = f W Ylrn(8, 4) (8) where Ylm is the usual spherical harmonic, so to evaluate the scalar product on the right in eqn (7) we expand the plane wave Ik - q ) in terms of spherical Bessel functions and spherical harmonics ; CQ + L L = O h f = -I, exp{i(k- 4) 1.1 = 4.n 1 iLjL(lk- q1r) x C Y L h f ( f ) Y t M ( & ) .(9) Then Except at low photoelectron energies (less than about eV for the 21.2 eV helium lamp) we can neglect q in comparison with k , and consequently the angular dependence of the emission is described by the terms The first factor simply gives a (cosine)2 distribution about the direction of polarization of the incident radiation, and is the same for all atomic orbitals. The second factor shows how the angular dependence of the emission depends on the I-value of the atomic orbital. For an adsorbed atom, the atomic orbitals will be chosen to belong to the various irreducible representations of the point group describing the symmetry of the adsorp- tion site.Consider, for example, the adsorption of a chalcogen at a site on a crystal where the symmetry is described by the group C41, (fig. 2 is an example). Z t 0 cubic The x FIG. 2.-Adsorption at a site of C4v symmetry. x' adatom orbital p z is now distinguished from the degenerate pair p x and p,,. If 8 is the angle between the direction of the emitted electrons and the surface normal (Lee, the polar angle of i), then since Y,, - cos 8, the angular dependence of the emission from p z is described by (e.i)2 C O S ~ 8. (1 1 ) If the incident radiation is polarized in the plane of incidence, the angular dependence of the emission from p z in the plane of incidence is described by sin2(B+8,) COS* 8 (12) where 8, is the angle of incidence (fig.3). This dependence is shown in fig. 4 forT. R. GRIMLEY z FIG 3.-Photoemission in the plane of incidence by radiation polarized in the plane of incidence. I FIG. 4.-Intensity (arbitrary units) of photoemission from a pz orbital as a function of the final polar angle 6. Angle of incidence of radiation ; (1) 0", (2) 30", (3) 60". 0 30 60 90 flldeg FIG. 5.-Intensity (arbitrary units) of photoemission from the pair of orbitalsp, andp, as a function of the final polar angle 6. Angle of incidence of radiation ; (1) O", (2) 30", (3) 60"12 INTRODUCTORY PAPER 8, = 0 (normal incidence), 30" and 60". The angular dependence of the emission from the degenerate pair of orbitals px and p,, is described by the factors (i%P)2 sin2 e and is shown in fig.5 for the experimental arrangement of fig. 3. If all three orbitals were involved equally in the chemisorption bond, the angular dependence of the photo- emission would reduce to the uninteresting factor (S-r2>2. But, except at tetrahedral, octahedral or icosahedral sites, which we do not in any case expect to encounter in adsorption, equal involvement of p x , p y and pz in the bonding must be " accidental ", and for the adsorption geometry of fig. 2 for example, we expect the emission from pz with the angular distribution of fig. 4 to show at lower electron energies than that from p, and py with the angular distribution of fig. 5. Of course p z is distinguished from p x andp,, at any 4fold site, at the centre of a square for example, and now the emission from px and p,, might well show at lower electron energies than that from pz.Thus, energy resolved angular distribution work is expected to distinguish one local geo- metry from another; it should of course distinguish between C,, and C2, sites because, at the latter, the degeneracy betweenp, and p y is lifted. We note incidentally, that the dependence of the emission on the azimuthal angle 4 (the azimuthal angle of k, the propagation vector of the final state) is not interesting for p-states at a 4-fold site, since it comes entirely from the factor (8&)2. Of course this is not always the case. At an adsorption site with C,, symmetry for example, the angular dependences are and of course all three orbitals are resolved energetically.To summarize the above simple discussion, we can say that, if the measured dzflerence emission is divided by the angular dependent factor ( $ ~ k ) ~ , the remaining energy-resolved angular dependence seems capable of providing information both on the adsorption geometry, and on the adatom orbitals used in forming the chemi- sorption bond. However, there are some important modifications to be made before we can be sure that photoemission is a useful tool in this connection. In particular, both initial and final states are modified by the presence of the substrate, and the final state is also modified by interaction of the ejected electron with the hole it leaves behind. This electron-hole interaction is a feature of atomic photoemission too. The modification of the initial state is of course due to the formation of a chemisorp- tion bond between the adsorbate and the substrate.An electron in the final state contributes little to this bond, and the important modification to make here is to replace the plane wave state by a correct eigenstate of the Hamiltonian for an electron moving in the field of the substrate in the half-space z < 0, as in eqn (3) for example. The scattering of an electron in this state by the adsorbate can be included through an appropriate T-matrix, and in this way an important part of the electron-hole inter- action referred to above can be included. The importance of choosing a final state with the correct Bloch character in the substrate has been demonstrated by Lieb~ch,~ and by Liebsch and Plummer (this Discussion). When this is done, the angular dependence of the emission even from an s-orbital, which would be the trivial ( 2 4 ~ ) ~ factor for a plane wave final state, yields information on the symmetry of the adsorp- tion site, because this information is now contained in the behaviour of the final state wavefunction over the region of space occupied by the adatom.T.B . GRIMLEY 13 The formation of a chemisorption bond is easy to describe in a molecular orbital theory, although other descriptions are possible.1°-12 The molecular orbitals GP of the system adatom+ substrate are written as linear combinations of the adatom orbitals #A and the substrate orbitals 4k A k Of course the substrate orbitals +k are themselves a complete set so the adatom orbitals are not really needed in eqn (14).They are included so that we can get a good approximation to the initial state in the presence of the adatom without including unbound substrate states in the k-summa tion. In practical calculations, the orbitals +k may be linear combinations of a suitable basis set of atomic orbitals on the sub- strate atoms. Using the form (14) for the initial state in eqn (2) we find 1 - = -- Im grS(& + i0) n where @ is the Greenian matrix for use l3 with the non-orthogonal basis set ( A , k ) . The diagonal elements prr(e) of the spectral density matrix are net densities of states. The terms in eqn (15) with r = s = A give the contribution to the transition rate which depends explicitly on the net densities of states on the adatom.This contribution, with PAA approximated either by a single &function at the atomic level or, by a Gaussian distribution, is considered by Liebsch and Plummer (this Discussion). The single &function approximation to PAA is more appropriate to physisorption than to chemisorption, except of course for lone pair orbitals on chemisorbed N, P, As, etc. For orbitals involved in chemisorption, PAA can develop considerable stmcture,14 and for strongly chemisorbed material it will be possible to identify the peaks in PAA below the Fermi level with the orbital energies of electrons in the chemisorption bond. These peaks may however show splitting due to correlation effects. The calculation of pAA is a basic task of chemisorption theory. Grimley and Pisani l6 have obtained results for hydrogen on some cubic tight binding solids using a local Hartree-Fock model, and employing Dyson’s equation to draw the atoms of the semi-infinite substrate into the local self-consistency problem ; more calculations on these lines are in progress.Also at Liverpool, Mr. Newton has (unpublished) results for hydrogen on the nearly free electron metal aluminium, using essentially Anderson’s Hamiltonian, but with overcompleteness of the basis set allowed for. Eqn (1 7) underlines the importance of these calculations in predicting the energy dependence of the emission. Conversely, if the matrix element in eqn (17) can be calculated, the experimental data may provide a method for determining PAA because the angular dependence of the contribution (17) is in the matrix element not in PAA.However, (17) is only one of five contributions to the emission. The other four involve the spectral densities Pkk, PAB, P k l and PAk. From the term involving P k k we can isolate the emission from the clean substrate by using Dyson’s equation14 INTRODUCTORY PAPER 3 = gf + @‘-lrg to express gs, the substrate portion of the Greenian matrix in terms of the interaction matrix V‘ forming the chemisorption bond, and the Greenian matrix @f for the clean substrate. I shall not give the details here (see for example ref. (16)), but in this way we get an expression for the total contribution which the adatom makes to the transition rate to the final state If). This determines the dzflerence photoemission, and as well as the contribution (17) there are contributions involving PAB, Pkl and P& Dr.Bernasconi is working on practical methods of esti- mating the importance of these “ unwanted ” contributions, all of which are energy and angular dependent, because until this can be done, theory cannot tell us whether the “ useful ” contribution (17) can be isolated from the experimental data. Turning now to the final state If) in eqn (15) and (17), since its energy is above the vacuum level, it seems an obvious approximation to take If) to be the same whether there is an adatom on the surface or not, and this is what Liebsch and Plummer (this Discussion) do. But in many cases, and particularly if the initial state hole is well localized on the adsorbate, it may be necessary to correct the final state for scattering by the adsorbate.This is achieved formally by replacing Y in eqn (1 5) by (1 + TG) V where T is the T-matrix corresponding to the adsorbate scattering potential V A ; T = VA+ VAGT where G(Ef+iO) is the stationary propagator. Evidently there is much theoretical work to be done on photoemission. 3. THE PHOTON FIELD In the foregoing discussion, I have assumed that the vector potential A(r, o) is known. For high photon energies (for XPS) the substrate is transparent and A can be assumed to have its vacuum value everywhere. To improve on the approximation, we might consider calculating the (classical) field using the macroscopic electromagnetic constants E , CT and p. But photoemission results from the p A interaction between electrons and photons which also contributes to the electromagnetic constants, so the question arises : How much of the electron-photon interaction can we include in the electromagnetic constants ? By considering the interaction of a radiation field and an electron field we see the problem quite clearly, and actually go some way towards solving it.3.1 THE ELECTRON-PHOTON INTERACTION I N PHOTOEMISSION Plasmon contributions of the substrate to the electromagnetic constants, or even those of an adsorbed overlayer, are allowed for by using an appropriate permittivity E(Z, o) in the usual Helmholtz equation for A . The exact Heisenberg equations of motion of the coupled Boson field A and the Fermion field $ describing the electrons are (1 8) (1 9) 1 V x V x A+,UOEOA = p0.l J = (eh/2im)[$tV$ - (V$)$t] - (e2/m)A$$f and ih$ = - (A2/2m)V2$ + V$ + (ieA/m)AaV$ -k (e2/2m)A2$.In eqn (19) the electron-electron interaction is included in the operator product V@. To make progress with these coupled equations we need to approximate. In the zero order approximation we put J = 0 in eqn (18) so that A is simply the vacuum field, and no account is taken of the presence of condensed matter. We improve on this approximation by calculating J using ground state expectation values of Fermion operators, J -” -(e2/m)n(r)A (20)T . B. GRIMLEY 15 where n(r) is the electron density. time transforms, we find Using this in eqn (18) for A(r, t ) and taking Fourier V x V x A(r, co) - co2pos(r, w)A(r, o) = 0 (21) where E(r, w) = E,[ 1 - m3r)/w2] w:(r) = e2n(r)lmc,. Thus, a position and frequency dependent permittivity depending on the local plasma frequency w, replaces so in the Helmholtz eqn (21) for A .After solving this equation we obtain a vector potential A(r, t ) to use in the perturbing term A.V$ in eqn (19) (the term in A2 can be dropped as usual). In our problem, the substrate occupies the half-space z < 0, and consequently in the first approximation, n(r), and hence E(Y, co), is simply a step function of z at z = 0. We learn from this approximate solution of the coupled eqn (18) and (19) that the inclusion of the plasma contribution to the permittivity in calculating the vector potential to use in the p-A interaction is well- grounded. Further developments are possible but I do not pause to consider them here because even the above simple development, whilst of some fundamental interest, does not seem to have important consequences in our present field of interest. It affects the magnitude, and the spatial variation of A at the surface.The magnitude of A affects the absolute value of the photocurrent in photoemission, and of course the absolute rate of absorption of photons in any photochemical reaction at the surface. The spatial variation of A will be masked in UPS by the short mean free path in the substrate of the ejected electron. 4. PHOTODESORPTION The formal theory of photodesorption is straightforward. Consider the photo- desorption of a single adsorbed molecule. The particles (electrons and nuclei of the substrate and the adsorbate) and the electromagnetic field are in the initial state linit) when the field-particle coupling is switched on.We require the transition rate to final states Ifin) in which there is either an outgoing adsorbate molecule or ion. This is (cf. eqn (1)) w = (finlTlinit)} 2S(Efi, - Einit) (23) where T is the T-matrix corresponding to the field-particle interaction V. Since V is small, T = Vis a good approximation. To make progress we employ the adiabatic approximation for the electronic motion, and write the particle wavefunction Yinit for example as in ternis of the electronic and nuclear coordinates Y and R. If we evaluate the matrix element of the electronic transition i-+fat the equilibrium nuclear configuration in the initial state Ro then (fin1 Ylinit) 21 ( F ( l ) ( f ; R,I Veli; R,) where V , is the p A interaction between electrons and photons, and (FII) is the familiar Franck-Condon factor between initial and final states of the nuclear motion on the potential energy hypersurfaces E,(R) and E,(R) associated with the electronic states t,hi and t,hf.Consequently y i n i t ( r 7 R) = $i(r ; R)XiI(R) w = h-'I(Fll)(f; RolVJi; R,)l26(Ei1+fiW--EfF) (24)16 INTRODUCTORY PAPER where E,, and Efp are the initial and final state energies of theparticles. Further progress seems to me to be quite difficult. If we could introduce a reaction coordinate s which changed only slowly on the time scale of all other nuclear coordinates Q so that x(R) 21 4MQ ; s) we could arrive without difficulty at the concept of potential energy curves Ef(s) and E,(s) along the reaction coordinate.18 It seems quite impossible to justify this concept in our problem, but I employ it simply to draw fig.6 to illustrate photo- desorption. The lowest excited electronic state will often correspond to charge (a) cations (b) neutrals FIG. 6.-Photodesorption of cations and neutrals. transfer between the adsorbate and the substrate, so according to eqn (24), ions are photodesorbed. The threshold frequency for this process on a metal will be ( I - 4 + D)/h for cations, and (6 - A + D)/h for anions. Z and A are the ionization potential and electron affinity of the adsorbate, D is its binding energy on the metal with work function 4. The threshold for photodesorption of neutrals is of course D/h but the process is a second order one requiring T N V+ VGo Y in eqn (23) so that w = 6+ 1 6(EiI + Ao - (25) The charge transfer states giving photodesorbed ions in first order are included in the summation over intermediate states in eqn (25).The second order process with threshold D/h is depicted in fig. 6. Of course neutrals are photodesorbed in first order, but they are in an excited electronic state, and the threshold exceeds D/h by this excitation frequency. For the photodesorption of CO from polycrystalline tungsten there is a threshold near 250 nm (about 4.8 eV) corresponding apparently to CO- ions in the final state.lg It is one task of chemisorption theory to calculate the potential energy hypersur- faces Ei(R) and Ef(R) but I do not think it useful to discuss the problem further because experimental data on photodesorption is scarce, and chemisorption theory is not yet developed to the point where potential energy curves on extended substrates can be calculated to the required accuracy. <FIJ><JlO<f; RolblA ROXA RolVeli; Ro) j J - E j JT.B . GRIMLEY 17 5. PHOTOCHEMICAL REACTIONS ON SURFACES I am not competent to make anything more than some general remarks, and to pose some no doubt obvious questions, on this subject. A review paper will be presented later by Prof. Gerischer. Owing no doubt to our inadequate knowledge of the electronic and geometric structures of surfaces and adsorbates, most of the fundamental theoretical problems in surface photochemistry remain untouched. Consider for example the process CO +2CH4+ Ci +T~co+CH,O+C,H, + Cf where Ci and Cf represent initial and final states of the condensed matter (substrate plus adsorbed species) on which the reaction takes place.The thermal reaction (26) in the homogeneous gas phase is endothermic (AH;98 = 60 kJ/mol CH20) but the photochemical reaction probably takes place on nickel (Bach and Breuer, this Dis- cussion). The fundamental theoretical problem can be posed in connection with any reaction like (26), namely to calculate the rate of reactive transitions out of an initial state /init> to a final state Ifin). Less fundamental but still very challenging problems are involved in the following questions : (a) Where (insofar as this question can be asked in quantum theory) in the condensed matter is the photon absorbed, and what becomes of the hole? (b) Can we begin to calculate electronically adiabatic potential energy hyper- surfaces for the reaction? (c) Can any other coordinates be treated adiabatically? Experiment can help theory here by identifying reaction intermediates, proposing mechanisms, and suggesting a critical elementary step in the overall reaction.As already mentioned, efforts are now being made to develop the theory of chemisorption on extended substates to an accuracy, comparable at least to that obtained in treating small molecules. The possibility of calculating the rate of an elementary hetero- geneous reaction on electronically adiabatic potential energy hypersurfaces depends on the outcome of these efforts. I am grateful to Dr. Bernasconi for some valuable discussions on the theory of photoemission. L. Hedin and S. Lundquist, Solid State Phys., 1969, 21, 1. C. B. Duke, Proc. Int. School of Physics " Enrico Fermi " Course LVIII (1973). To be published in Nuovo Cimento. €3. A. Lippmana, Phys. Rev. Letters, 1965, 15, 11 ; 1966, 16, 135. I. Adawi, Phys. Rev., 1964, 134, A788. G. ID. Mahan, Phys. Rev. By 1970,2,4334. W, L. Schaich and N. W. Ashcroft, Phys. Rev. B, 1971,3,2452. ' D. R. Penn, Phys. Rev. Letters, 1972, 28, 1041. * see for example L. I. S e w , Quantum Mechanics (McGraw-Hill, New York, 1968), p. 420 ; J. W. Gadzuk (J. Vac. Sci. Tech., 1974, 11, 275 and unpublished work) has considered photoemission from adatoms into plane wave final states. A. Liebsch, Phys. Rev. Letters, 1974,32, 1203. K. F. Wojciechowski, Acta Phys. Polonica, 1966,29, 119 ; 1968,33, 363. Physics " Enrico Fermi " Course LVIII (1973). To be published in Nu0z.o Cimenta. LVIII (1973). 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