Theory of the angular-resolved photoemission from TaS2

Solid State Communications, Vol. 19, pp. 1193—1196, 1976.

Pergamon Press.

Printed in Great Britain

THEORY OF THE ANGULAR-RESOLVED PHOTOEMISSION FROM TaS2 * A. Liebscht Department of Physics, University of Pennsylvania, Philadelphia, PA 19174, U.S.A. (Received 19 April 1976 by J. Tauc) The energy and angular resolved photoemission spectra of TaS2 are discussed using a model involving delocalized initial states. A multiplescattering scheme is used to determine both the ground state and the excited state wave function in the presence of the surface. Theoretical calculations of azimuthal spectra for TaS2 compare favorably with recent experimental results and demonstrate the non-plane wave character of the final state as well as the importance of matrix elements. 4 for the band structure of thin proposed by Kar eta!. films. The excited state of the system is derived from a similar multiple scattering techniqueusing the same muffin tin potential which however now includes a uniform complex optical potential to describe inelastic

A PREVIOUSLY DEVELOPED one-electron theory is of photoemission from atomic-like adsorbate levels1 extended to delocalized initial states such as bulk-like states, surface states as well as those adsorbate levels that hybridize with substrate states to form two dimensional bands. In this paper, we apply the theory to the case of TaS 2 and compare our predictions with recently 2 The results performed experiments onstate this system. show that initial and final features in the observed spectra can clearly be identified. They furthermore demonstrate the non-plane wave character of the final state and the importance of matrix element effects, The realization that angular resolved photoemission spectra information not only the momentumcontain dispersion of electronic statesabout but also about their spatial character3 has been the cause of recent interest in this technique as a tool to analyze surface phenomena. The information, however, is available only in a rather convoluted form as the transition matrix element involves the ground state and an excited state of the system. In order to be able to determine the

electron—electron interactions and the vacuum side, goes smoothly into thewhich, imageonpotential. For For the purpose of this paper, we limit the description of the formalism to systems having a slab geometry so that the bound states are characterized by twodimensional Bloch vectors k 1. This situation applies to the metallic d-bands of the layer-type5compound TaS2 show essentially for which band structure calculations no dispersion in the direction perpendicular to the surface. In the one-step model’ the wave function of the outgoing photoelectron at the position R of the detector may be written as ‘1’O~) = G p A I ‘1’k (l a)

1)

transition strength as function of the detector direction and of the energy and polarization of the incident radiation, it is necessary to have a scheme that allows one to calculate both the initial and final wave function of the system in the presence of the surface. We assume the one-electron potential to beand of wave the muffin tin form. The bound state energy bands functions are evaluated using a multiple scattering scheme recently

eulhi ~ ~ (‘I’rIP’AI’I’~), R

Research supported by the Laboratory for Research on the Structure of Matter and the Advanced Research Projects Agency of the Department of Defense as monitored by the Air Force Office of Scientific Research under Contract No. F44620-75C-0069 and the NSF under Grants DM R 2-03025 and 73-07682. t Present address: Institut für Festkorperforschung, Kernforschungsanlage, 517 Julich, W. Germany. *

4

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(lb)

where “ks represents the initial state wave function, G the (retarded) one-electron propagator and 2 =excited Ef the state kinetic energy of the photoemitted electron. K The function ~ appearing in the matrix element which is usually referred to as final state wave function of the photoexcitation process, Consists of two terms: an outgoing plane wave with momentum k~ ,c.k and incoming sphencal waves. Thus, ‘I’~~ represents the time-reversed LEED wave function. The main task now lies in finding a suitable representation for the initial and final state wave functions such that the matrix element can be evaluated. Only the essential steps are indicated here; a more detailed analysis wifi be given elsewhere.6 Let us first discuss the

1193

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ANGULAR-RESOLVED PHOTOEMISSION FROM TaS2 K

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K M’ K Fig. I. Theoretical azimuthal distribution (arbitrary units) in the plane wave final state approximation (solid curve) for Ef = 8 eV, ~ = 48°,and A in the surface plane, unpolarized. The dot—dashed circle indicates the position of the parallel momentum component within the two dimensional surface Brilouin zone. The dashed lines indicate the occupied regions.

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Vol. 19, No. 12

Bravais net: denotevalue the expansion by 4~g (lc~). By we matching and slope coefficients of these various representations along all boundaries which define the three potential regions, one obtains a large hermitian matrix whose determinant is zero at the bound state energy E 1(k1) for a fixed k,. Through successive elimination of most expansion coefficients (either ~ or 4~)the matrix can actually be reduced to a reasonably small dimension. In practice, it is convenient to solve for the eigenvalue that vanishes at the bound state energy since its corresponding eigenvector contains the desired expansion coefficients and, furthermore, the slope with which it goes through zero determines the overall normalization of the initial state wave function. Whereas the ground state in the external region consists solely of evanescent waves, two types of solutions exist, however, at the excited energy above the vacuum level. Due to the additional coefficients, the matching leads to a linear system of equations rather than to a determin~.

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antai prooiem. me somution ot tnese equations proviues the various expansion coefficients that completely characterize the final state.

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Fig. 2. Comparison of experimental azimuthal spectra (dashed lines) with corresponding theoretical results (solid curves) using actual initial and final state (arbitrary units). (a) A in plane, polarized, A I kfl1, Ef = 10eV, ~f = 42°; (b) A in plane, polarized, ~ (A, k111) 36°,Ef = 8 eV, O~= 57°(theoretical curve: 48°);(c) A in plane, unpolarized, Ef=l5eV,Of=60 bound state problem. The wave function within the atomic spheres consists of a linear combination of solutions to the spherically symmetric potential. We

denote the expansion coefficients by A~(k1)where a is a layer index and L = (1, m) the angular momentum quantum numbers. In the external region, the wave function is a superposition of solutions to the onedimensional potential V(z). The wave function in the interstitial region can be obtained by applying Green’s 4 or by expanding, along planes between the theorem atomic layers, in terms of a set of plane waves individual characterized by reciprocal lattice vectors of the surface

It is convenient to utilize an identity which replaces p A in the matrix element, equation (lb), by (E, Ef)’AVV(r). This form has the advantage that the matrix element vanishes in the interstitial region. The contribution to the matrix element from the volume within the muffin tin spheres has the form: 1

~ rn-i

Yi”,rn(A) LL ~ A~1~(kfil)Raif(Ef, E1) x A~j,’(k 1)J(lm;L, L’)

(2)

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ANGULAR-RESOLVED PHOTOEMISSION FROM TaS2

wnere Rd,’ denotes the radial matrix element for the ath layer between angular momenta I and 1’, I the Gaunt integrals and k, is subject to the condition k1 + g = k~where g is a reciprocal lattice vector. An analogous expression holds for the contribution from the external region. The above expression emphasizes the spatial character of the initial and final state in that it describes the allowed transitions between the various partial wave components in each sphere. Since the A~,however, are related to the clag it is evident that the above matrix element can also be written in terms of the planar expansion coefficients. This form focuses more on the Bloch nature of the two states in that their corresponding expansions in terms of reciprocal lattice vectors parallel to the surface appear explicitly, We have applied the above outlined formalism to calculate photoemission angular distributions from the layer-type material 1T-TaS2. The metallic d-bands which extend about 1 eV below the Fermi energy exhibit little dispersion in the direction normal to the basic 3-layer unit consisting of a hexagonal plane of Ta atoms sandwiched between two (antiprismatic 5 Thus, thesulfur wave planes functions do not overconfiguration). lap between individual sandwiches but are Bloch-like in the direction parallel to the film. A band calculation for a single sandwich gave results essentially identical to those obtained for bulk systems.5 The potential was derived by overlapping atomic charge densities within the slab geometry.4 The exchange parameter was taken to be unity, the same as that in reference 5. The ground state wave function is predominantly of d character on the Ta spheres and ofp character on the S spheres. The remaining partial wave components, however, are generally not negligible, In order to separate final state effects on the experimentally observed spectra from those caused by the initial state, we have carried out a calculation of the matrix element (using the p A form) for a planewave final state and unpolarized, normally incident light. In this limit, one probes the Fourier transform of the ground state wave function.3 The results for a photon energy of 14 eV and a polar exit angle of 48°are shown in Fig. 1, superimposed on the two dimensional surface Brillouin Zone. The azimuthal pattern exhibits three lobes along the ~—M’ direction. Similar distributions, with only small changes in amplitude and width of these lobes, are obtained throughout the entire range of final energies and polar angles for which the experiments were performed. These results suggest that the experimentally observed splittings of the lobes as well as the considerable emission along the f—M direction2 is caused by the final state or, more precisely, by the combined effects of

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ground state wave function, excited wave function, and polarization of the incident radiation as indicated by the matrix element equation (2). Indeed, theoretical calculations using the full final state reproduce most of the structure in the observed spectra. In Fig. 2, we compare azimuthal patterns for three different polarizations, final energies and polar angles. Since the vector potential A in all three cases lies in the plane of the surface, the contribution to the matrix element due to the gradient of A near the surface vanishes. The discrepancies between experimental and theoretical spectra can arise from the following four sources. (a) Quasi-elastic scattering. For kfii in the first Brillouin Zone, there are no occupied states along the f—K direction (see Fig. 1). Furthermore, on the basis of symmetry arguments, the intensity along the directions f—M and 1’—M’ for the polarization in Fig. 2(a) should vanish. The experimentally observed intensities at these angles therefore gives a direct measure of quasi-elastic electron scattering which itself can lead to angular anisotropies. Also, no experimental broadening was included in the calculations. The very narrow double peak itincarries Fig. 2(c) would be reduced to some extent since little weight. (The splitting decreases rapidly at higher polar angles.) (b) The use of a non-self-consistent muffin tin potential. It is known that electrons are most sensitive to the detailed distribution of the valence charge density in the low energy region above the vacuum. (c) Furthermore, there is some uncertainty in the choice of the mean free path at these low energies. The calculations were performed for an imaginary part of the self-energy equal to 3 eV which corresponds to a mean free path of approximately 12 A. (d) We have included in the calculations only the emission from the first sandwich. In principle, the emission from the subsequent layers should be added incoherently and although the signal is attenuated, it could lead to slightly different structure in the azimuthal profiles. The overall agreement between experimental and theoretical results suggests that in those cases in which relaxation effects and the localization of the hole can be ignored a quantitative description of angle and energy-resolved photoemission is indeed possible. The main problem appears to be the availability of accurate, if possible self-consistent one-electron potentials. We feel that this technique should prove to be a valuable tool to gain a better understanding of the ground state wave function of clean as well as adsorbate covered surfaces. Acknowledgement — The author is grateful to Professor P. Soven for many helpful discussions and to N. Kar for his program of the structure constants.

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Vol. 19, No. 12

REFERENCES 1.

LIEBSCHA.,Phys. Rev. Lett. 32,1203 (1974);Phys. Rev. B13, 544 (1976).

2.

SMITH N.y. & TRAUM M.M.,Phys. Rev. Bil, 2087 (1975); SMITH N.V., TRAUM M.M. & DISALVO F.J., Solid State Commun. 15, 211(1974); SMITH N.V., TRAUM M.M., KNAPP J.A., ANDERSON J. & LAPEYRE G.J. Phys. Rev. B13, 4462 (1976).

3.

GADZUK J.W., Solid State Commun. 15, 1011 (1974);Phys. Rev. BlO, 5030 (1974).

4. 5.

KAR N. & SOVEN P.,Phys. Rev. Dli, 3761 (1975). MATTHEISS L.F., Phys. Rev. B8, 3719(1973); MYRON H.W. & FREEMAN A.J., Phys. Rev. B! 1,2735 (1975). SMITH N.y. & LIEBSCH A. (to be published).

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