Surface states of Sc(0001) and Ti(0001)

17. Solid State Communications, Vol.31,inpp.413—4 Pergan~nPress Ltd. 1979. Printed Great Britain.

SURFACE STATES OF Sc(0001) AND TI(0001) Peter J. Feibelman Sandia Laboratories~ Albuquerque, New Mexico 87185 and D. R. Hamann Bell Laboratories Murray Hill, New Jersey 07974 (Received 16 April 1979 by A. G. Chynoweth) ABSTRACT Calculated electronic properties are compared for 11-layer Sc(0001) and 11(0001) films. Prominent surface states are found whose locations conform to expectations based on the respective bulk band structures establishing a roughly rigid band picture of the surface bands. Surface core-level shift and work function results are qualitatively explained.

We report here a comparison of the electronic structures of Sc(0001) and Ti(0001) films. This work represents a first step in establishing the nature of trends in surface dcctronic structure across the periodic table,1 a key element in developing an understanding of the role of surface states in the chemistry of transition metal surfaces. The picture which emerges from our self-consistent linear combination of Gaussian orbitals calculations2’3 is of remarkable simplicity in its qualita. tive features. Strongly surface-peaked surface states are predicted to lie in all the lowest gaps of the projected bulk band structures of Sc and Ti, which are in fact quite similar. In the former case the surface states are unoccupied while in the latter they are partially occupied, because Sc has three valence dcctrons per atom while Ti has four. However, occupation of the surface states is evidently not important to their existence nor to their strong surface localization. The occupation of the surface states does appear to be important in determining the values of work function, 3.9 eV for Ti as against 2.1 eV for Sc, which emerge from the calculations.4 The work function of the Ti is larger at least in part because of an increased surface dipole contribution associated with the occupation of the Ti surface states, which places valence charge relatively farther outside the first layer of Ti nuclei. The predictability of the relation between Ti and Sc surface electronic properties including

crystal structure, one expects a “rigid band” picture of their bulk electronic structures to be roughly correct. This expectstion is borne out by previous non-self-consistent band structure calculations6 as well as by our own self-consistent ones.3 The calculated band structure of bulk Sc is shown in Fig. 1. Apart from the width of its s-bands, the corresponding plot for Ti differs mainly in small details, i.e., band crossings and ordering at and near symmetry points. Since surface states necessarily lie at energies where no bulk bands of the same symmetry are found, we focus on the gaps in the Sc and Ti and structures. In order to understand the relation between the bulk and surface band structures consider the Sc bands (Fig. 1) along the lines F—A—A, H—P—K, and M—U—L, which upon projection yield the locations of the bulk states of the (0001) film at the points F, K, and M of the surface Brillouin zone7 (SBZ). Along the line F—A—A, there is a gap of — .1 au (-‘- 3 eV) between the s-bands below and the d-bands above the Fermi level. (For Ti this gap is also — .1 au, but the Fermi level is almost up to the lowest d-band.) This gap is a potential location for a surface state. If such a state is to be d-like in character it will appear in the upper pottion of the gap. Along the line H—P—K in the SBZ there are three d-like bands of P 8 just below the Sc Fermi level. The next1,higher is of P P2 andband P3 symmetry 3 symmetry. States of the symmetry P1 and P2 do not appear until the next band, leaving a “symmetry gap” of —. .1 au for the P1, P2 states. (Again, a similar situation obtains for Ti.) This close-to 3 eV symmetry gap is a potential location for a surface state of P1—P2 symmetry, and indeed it is in this gap at point K of the SBZ that the most strongly-surface peaked state is found for both Sc and Ti. Consider now the line M—U—L in Fig. 1. Again we find a substantial (—‘ .05 au) gap in the projected bulk band structure, a possible location for d-like surface states. Finally, note the gap about the Fermi level in the bulk Sc band structure along the line M—T’—K. This gap makes the region of the tial location SBZ near for surface its M—T’—K states. line However, appear toward to be another K this potengap is filled in in the projected band structure by the states near the top of the bulk BZ, as is seen from the dispersion curves along L—S’—H. Consequently, whatever surface states are seen for

surface state location and work function magnitude is a phenomenon whose generality is important to determine, In addition to valence electron spectra, we have calculated core-level positions for the atoms of the Sc and Ti films. According to a simple model, discussed further below, one anticipates that the core electrons of the surface layer Sc and Ti atoms will be more tightly bound than the electrons in the corresponding levels of the interior atoms, with the shift being larger for Sc than for Ti. Our results bear out this prediction with the core-level shifts being — .5 CV for Sc and .2 eV for Ti. These results are of particular interest because, as is predicted, 5 and the predicted shifts are forofCu.2 the opposite sign to those observed for Au Both Sc and Ti are hcp metals at room temperature and atmospheric pressure, with the Sc lattice parameters being —. 12% larger than those of Ti. Because Sc and Ti have the same

A US Department of Energy facility. Work at Sandia was supported by the US Department of Energy (DOE) under Contract DE-ACO4-DP00789. *

413

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2a and b, we show the even parity states of the Sc and Ti films, The corresponding band-structure plots for the odd states are very similar, which indeed 9is the reason we wished to study a The lowest-lying, most surface-peaked states for the Sc film of as many as 11 layers. and Ti films are indicated in Figs. 2 by heavy lines. (Surface localization or peaking is judged by comparing the relative wave function square averaged over the outer film layer and over the central layers.) In both cases the following statements describe these states:

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the Sc film along M—T’---K of the SBZ will be broadened into surface resonances by coupling to bulk bands, in the region nearer K. Having examined the Sc bulk band structure and identified several regions of the SBZ where surface states might be expected, we now turn to our calculated band structures for 11-layer Sc and Ti films. Since the calculations represent the film wave functions using a2’3 basis Gaussians, has been ourofresults are notas reliable at described in detail energies close to orelsewhere, above the vacuum level, where the wave functions have appreciable amplitude outside the films. Consequently we limit our discussion to the lower-lying surface states. An (0001) hcp film of an odd number of layers is symmetric under reflection in its central plane. Therefore the states of the 11-layer Sc(0001) and Ti(0001) films can be

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The state which is most strongly localized at the surface is the state of P1 —P2 symmetry at the point K, which at —middle .139 auoffora Ti andof—bulk .063 states au for of Sc.P3 symlies in 8the band Proceeding along the line K—T’—M, this state mixes metry, with the states of the bulk band, becoming a surface resonance, until one arrives at the point where the bulk states turn up rather sharply in energy. At this point the surface resonance splits off the bulk band and exists as a rather well-defined surface state, with only a mild

dispersion, in the large gap which exists about the point M. (c) Well-defined surface states exist at the point r and split off the lowest d-band. However, as one proceeds along either F—Z—M or I’—T—K, these states mix rapidly with the bulk bands and lose their surface character. Thus these states can be expected to contribute little to physical phenomena which are described terms of SBZ integrals over the surface states, in incomparison with the states which exist in the outer regions of the zone, near M—T’—K. Notice that all the low-lying gaps of Sc and Ti contain surface states. The main difference between the Sc and Ti results appears to be that the surface states in Ti are somewhat higher in energy relative to the lowest d-bands than in Sc. For example, at K the P 1 —P2 symmetry surface state in Ti lies in the middle of the gap between the lowest and next higher bulk band of P1 —P2 states, while the corresponding state in the Sc film lies at only one-third the gap up from the lower P1 — P2 split the difference bulk bandsindicates by a somewhat surfaceinperturband.offThis that thestronger surface states Ti are bation which may be related to the higher surface barrier p0tential. This surface state band is identifiable along a portion of the K—T—1’ line in Ti, but because of its weaker splitting, mixes too strongly with the continuum to be seen as a surface state or distinct resonance in Sc. The extra surface state in Ti at K, at -.166 au is of P 3 symmetry and lies in a gap of -~ I eV between the lowest bulk P3 bands. In Sc the corresponding gap is only —. .5 eV. The absence of a corresponding state in Sc Since state toexists over only a small portion ofsmaller the SBZgap. in may bethis related the weaker surface barrier or the Ti, its absence in Sc should have little effect on zone-integrated physical properties. A population analysist°of the wave functions for the Sc and Ti films permits the evaluation of their layerwise local densities of states (LDOS), the results of which are shown in Fig. 3. In both cases we note the strong surface state peak in the

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first (surface) layer, which falls at an energy corresponding to a valley in the interior layer (i.e., “bulk’) factfall simply corresponds to the observation thatLDOS. surfaceThis ‘states in gaps of the bulk band structure and consequently occur when the bulk DOS is lowest. The fact that the surface states lie higher in the valley between the first major d-band peaks for Ti than for Sc is plain in Fig. 3, as is the fact that the strength of the surface peak is not greatly affected by its occupation. Regarding this latter point, note that the Sc surface feature lies well above the Sc Fermi level while that for Ti coincides with

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Vol. 31, No. 6

analysis explains, moreover, why in the angular2—r2 momentum characdecomposition of the LDOS (notsmall shown”) the 3z ter of the surface peaks is very -- only the surface states near r are appreciably of this symmetry. We turn now to a discussion of the surface atom corelevel shifts which emerge from our calculations. A simple model which predicts the sign and relative magnitude of such shifts is as follows: In simple tight-binding models for isolated narrow bands, bandwidths are proportional to coordination number. Consequently one expects the total d-band LDOS for a surface atom to be narrower than that for a bulk atom. If one assumes that this narrowing occurs roughly without a change in the shape of the LDOS, and if one imposes the condition, for a metal surface, of layerwise charge neutrality, then it follows that a redistribution of charge will occur that shifts the surface LDOS up if the Fermi level was above the midpoint of the band and down if it was below. This screening effect will necessarily also shift the core levels of the surface atoms relative to those of the bulk, in the same way. A selfconsistent calculation similar to those reported here for a

Cu(111) film predicted a 0.5 eV core level shift toward lower binding energy, consistent with this simple argument.2 A shift to lower binding energy of the core levels of surface Au atoms

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has recently been verified experimentally.5 For Sc and Ti the model implies that the surface atom core levels should be shifted to a higher binding energy than

the corresponding bulk atom levels, with the effect being larger for Sc. This prediction is satisfied by our results, which are that the Ti surface atom core states are .22 eV deeper than the

bulk ones, while the shift is of the same sign and equal to .48 eV for the case of Sc. Free atom core level binding energies are several eV lower than those of bulk atoms for Cu, and several eV higher for Ti and Sc. This is consistent with the simple idea that the surface layer changes in the direction of

the free atom, the surface core shift being 10-20% of the bulk free atom shift in all three cases)2

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predictedThe work lastfunctions set of of results Sc and weTi,wish respectively to discuss 2.1 are eV and the 3.8 eV. Both of these predicted values are low compared to the corresponding measured results, 3.1—3.5 eV for Sc’3 and 4.04.5 for Ti.14 Comparison between theory and experiment for these numbers is probably not a fair test of the theory since Sc

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and Ti are extremely reactive materials, making it unlikely that any of the data in print correpsond to clean surfaces. We do wish to remark on the fact that the work function of Ti is

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Layerwise LDOS’s for the Sc and Ti films. In both cases note the strong peak in the outer (first) layer LDOS aligned in energy with the minimum between the lowest large (d-band) peaks of the inner layer LDOS. In these figures, thedashed lines represent the Fermi level and 0.0 eV is the vacuum level,

its Fermi energy. Finally, an analysis of the contributions to the LDOS from different regions of the SBZ confirms the judgment based on Figs. 2 that the main contributions to the surface peak come from the neighborhood of M—T’—-K. The

in higher the Sc fact they than thatare that in completely Tithe ofdo Sc,the surface however. unoccupied. states This are result partially Since the occupied surface inthe part while states from have considerably more amplitude outside thestems outer atom layer of the films than bulk states, occupation of surface states enhances the dipole layer in the direction which increases the work function. In summary, virtually all of the qualitative features of the surface electronic structures of Ti(0001) and Sc(0001) films

are related in a straightforward way to the bulk band structures of Ti and Sc. The ‘rigid band” relation of the bulk electronic structures of Ti and Sc therefore accounts for the similarity of the surface spectra of Sc(000l) and Ti(0001) seen in Fig. 3. The strong influence of the Sc and Ti bulk properties on those of the surface is related to the thinness of their surface regions, which ensures strong communication between their surface and

bulk states. For systems in which strong surface reconstruction occurs, matters should be expected to be quite different. We wish to acknowledge stimulating discussions of surface core shifts with P. H. Citrin.

REFERENCES 1.

To date there has been no systematic attempt to correlate surface electronic properties across the

periodic table, although self-consistent surface band structures have been obtained for Cu, by GAY, J.G., SMITH, J.R. and ARLINGHAUS, F.J., Phys. Rev. Lett. 38, 561 (1977) and APPELBAUM, J.A. and HAMANN, D.R., Solid State Comm. 27, 881 (1978), for Nb, LOUIE,

Vol. 31, No. 6

2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13.

14.

SURFACE STATES OF Sc(0001) AND Ti(000l) S.G., HO, K.M., CHELIKOWSKY, J.R. and COHEN, M.L., Phys. Rev. Lett. 40, 1593 (1978) and for Pd, by LOUIE, S.G., Phys. Rev. Lett. 40, 1525 (1978). APPELBAUM, iA. and HAMANN, D.R., op. cit. FEIBELMAN, P.J., APPELBAUM, J.A. and HAMANN, D.R. (to be published). See below for a discussion of the comparison of these numbers to experiment. CITRIN, PH., WERTHEIM, G.K. and BAER, Y., Phys. Rev. Lett. 41, 1425 (1978). For Sc, see, e.g., DAS, S.G., Phys. Rev. B13, 3978 (1976); for Ti, see JEPSEN, 0., Phys. Rev. B12, 2988 (1975). The labeling of the symmetry points in the BZ is standard; see KOSTER, G.F., Space Groups and Their Representations (Academic Press, NY, 1957). The labeling of the symmetry points in the SBZ bears the obvious relation to that of the bulk BZ. The symmetry notation is that of SLATER, J.C., Quantum Theory ofMolecules and Solids, Vol. 2 (McGraw-Hill, NY, 1965). For thinner films the interaction between the surface states of the two surfaces may be expected to split the energies of their odd and even parity linear combinations. See Ref. 3, Footnote 22 and also DEMPSEY, D.G. and KLEINMAN, L., Phys. Rev. B16, 5356 (1977). See Ref. 3 for partial for the Ti film.z is The surface states arethe found to benormal. predominantly of 2—y2) and (xz, LDOS yz) character. Here the coordinate along surface (xy, x

Experimental bulk, surface, and atom core binding energies need not follow this trend unless correction is made for interatomic relaxation energies, which are present and presumably nearly

equal for bulk and surface atoms, and absent for free atoms. SAVITSKIY, EM., TEREKHOVA, V.F. and MASLOVA, E.V., Rad. Eng. and Electr. Phys. 7, 1233 (1967); EASTMAN, D.E., Phys. Rev. B2, 1 (1970). ANDERSON, J.R. and THOMPSON, N., Surf. Sci. 26, 397 (1971); D’AROY, R.J. and SURPLICE, N.A., Surf. Sci. 36, 783 (1973); EASTMAN, D.E., op. cit.

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