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Simple comments on the W(001) surface instability
Volume 164, number 4
SIMPLE COMMENTS
&pan
CHEMICAL PHYSICS LETTERS
I5 December 1989
ON THE W(OO1) SURFACE INSTABILITY
PICK, Mojmir TOMASEK
J. Heyrovsk$ Institute ofPhysical Chemistry and Electrochemisiry. Czechoslovak Academy of Sciences, DolejSkova 3, 182 23 Prague 8, Czechoslovakia
and Marco U. LUCHINI Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CR3 OHE, UK Received 14 July 1989; in final form 18 September I989
A geometrical interpretation of the reconstruction of the ideal W(OO1) surface is suggested, which is based on the use of the generalized Htickel rule. More complicated situations like steps are also mentioned and it is speculated that the increase in the coordination of the surface atom does not mean necessarily an increase in its stability.
The kth moment of the electronic Hamiltonian matrix H is usually defined by the equation mk=TrHk, lz=O, l,.... For nonhomogeneous systems, it is useful to modify this definition, restricting the above sum to the diagonal matrix elements ($1 Hk 1b) with the orbitals 14) localized on the same atom. Consequently, information about local properties is obtained. In this case, another convenient expression for mk is mk =jEkn(E) dE, where IZ(E) is the local density of electronic states (LDS) associated with the atom chosen. The role of the second moment m2 as one of the basic quantities qualitatively characterizing the bonding has been recognized and employed - explicitly or implicitly - by many authors, see e.g. ref. [ 11. The higher-order moments arc incorporated in well known numerical schemes [ 2 1; however, the attempts to use them for simple qualitative predictions are less abundant in the literature (see refs. [ 3-71 and the references therein). Recently, the so-called generalized Hiickel rule (GHR) [5-71 has been applied by the present authors [ 81 to the problem of interatomic interactions on the W (001) surface. According to GHR, it is energetically favourable to increase (decrease) the 0009-2614/89/$
m4k+Z (m4,!) moment of the electronic Hamiltonian. (Note that m2 = m4kf2, li=O.) The reason why the Htickel rule, originally derived for simple models of organic chemistry, admits a wider use is that the underlying picture is the one common to all typical frontier-orbital (HOMO-LUMO, pseudo-JahnTeller) situations [ 5,6]. On the ideal W(O0 1) surface, the frontier orbitals are represented by the partly occupied surface state bands. For the ideal W (00 1) surface atom (as well as for W adatoms in various arrangements on the W (001) surface), the local density of electronic states originating from the surface states contains contributions from both ep (mainly x2-y2) and tlg orbitals. Following refs. [ 81, the tZgorbitals contribute to the interaction between two surface neighbours Si, S2 mainly via the contributions to m4 of the form
(X1=X20rXI#Xd,
(1)
where X,,* stands for substrate atoms. It can be verified that the sum of these contributions is positive and according to GHR, an indirect repulsion (tendency to reduce m4) between S, and SZ is mediated by the subsurface atoms. From steric reasons [ 91,
03.50 0 Elsevier Science Publishers B.V. (North-Holland)
34s
Volume164,number4
CHEMICALPHYSICSLETTERS
the x2-y’ orbitals contribute essentially to the second moment ml by the contributions (S, 1HIS,) * responsible for a direct attraction between surface neighbours. However, even for the x*-y2 orbitals, the repulsive contributions ( 1) to m4, together with the terms [S,lO] (cf. also ref. [5])
(2)
where S, to S, are surface atoms lying on the vertices of a square, are not negligible. Now, let us consider a high-symmetry deformation of the ideal (001) surface, producing a certain number of shortened (S) and the same number of lengthened (L) bonds between the neighbouring surface atoms. For this deformation, the matrix elements V associated with the S-bond (L-bond) are resealed approximately by a factor ( 1+x) ( ( 1-x) ). By performing a L-S pair, the m2 contribution 2 v2 is increased to the value [(l tx)2+(1-x)‘] x Y2> 2 lf2 and the deformation is supported by direct attraction effects. Formally, the interaction between two surface neighbours can be described by the pair potential ~(Sr,)=~Sr,+fb(Sr,)~+..., whcrc 6~6 is the change of the ideal surface interatomic distance. The above m2 effect corresponds to the negative value of b and the indirect repulsion (see also below) points to the negative a value. From @, the surface force constants are derived in the standard way [ 111, resulting in a repulsive stress (a < 0) [ II] and an overall softening (b< 0) of the surface. Together with the L, S-bonds between surface atoms, L, S-bonds between surface and subsurface atoms are created, the scaling factors for the corresponding matrix elements now being (1 ky). As the result of the elementary inequality ( 1+,Y) ( 1-y) < 1, the existence of two neighbouring L- and S-bonds reduces the m4 contribution. Hence! it is desirable to maximize the number of L, S-bonds in neighbouring positions (i.e. minimize the number of L, L- or S, Sneighbours). Note that on the basis of moment analysis, possible stabilization of the W( 001) surface by the L-S bond alternation was suggested in ref. [ 41. There are two high-symmetry deformations obeying the mz criterion. Namely, the zig-zag M, mode of Debe and King [ 9,12 ] is energetically superior to the other candidate, the X3 mode [ 9 ] (denoted also as L2, XZ or the pairing mode in the literature), see 346
I.5DecemberI989
fig. 1. In that figure, the L, S-bond interpretation of the ti, and X3 modes is also shown. (It is more instructive to study the [ 1, 0] polarization of the I& mode rather than the [ 1, l] one observed experimentally. The two polarizations differ in energy only when higher-order anharmonic contributions are taken into account [ 91.) Since the increase of m2, controlled by the number of L-S pairs in our approximation, is the same for the two deformation modes, it is the m4 criterion which determines the higher Ii& reconstruction stability. Namely, the inspection of fig. 1 shows that for the l& mode, the zig-zag geometry leads to a higher number of neighbouring surface-subsurface L-S pairs in the two important situations described below and consequently to a more reduced value of m4. When traced to the product of matrix elements, the following is found: (1) For the indirect interaction contribution ( 1) with S, # SZ, X, =X2, there are more L-S pairs present in the I$ geometry than in the X3 one. In these contributions, both the tZg and eg orbitals are important. (2) The term (2) (S, #SZ#S3#S4) contains a L-S pair in the I$ case, contrary to the L-L or S-S pairs for the X3 mode. (The two deformed bonds connect, say, the atoms S,-S, and S3-S4, respectively, whereas the bonds S,-S, and SZ-S4, unperturbed to the first order in the deformation, are not shown in fig. 1.) In this situation , the x2-y’ orbitals are essential. As a result of our analysis, a natural and concise interrelationship between the
Fig. 1. The geometry ofthe M, and x3 modes, together with their L, S-bond interpretation (left). The circles denote the atoms in the surface and subsurface layers, respectively. The S (L) bonds are shown by heavy (thin) lines connecting the respective atoms.
Volume 164,number 4
CHEMICALPHYSICSLETTERS
surface states behaviour, the m2 and m,, moments, the geometry of the deformation (S, L-bonds), and the phenomenological force constant and pair potential notion is obtained. There are other interesting phenomena for the less coordinated atomic arrangements on the W (001) surface which are not understood completely at present. It is known that the reconstruction is suppressed near the step edge, although the geometry at such a step edge is unknown [ 121. Facilitated vacancy formation is another effect which is suspected to take place [ 12,131 on the (001) surface. Since the competition between the attractive (m2) and repulsive (m,)interactions should be present for any realistic arrangements of W atoms on the W (00 I ) surface, the presence of certain anomalous effects can be expected. A simplified argument based on the m2 reduction would predict the surface state peak narrowing, which, in its turn, might point to an enhanced instability of the less coordinated structures. Actually, the sharp surface state peak in LDS on the ideal surface seems to result from the high translational and point symmetries and it widens or even splits for the less coordinated atoms. As an example, in fig. 2 we compare the LDS for the atom on the [ 1, 0] step edge with that for the atom on the
Fig. 2. The local density n(E) of electronic d states on the [ 1, O] step edge atom (solid line) and on the ideal (001) surfaceatom (dotted line). The surface state feature lies at E- 0.
15 December 1989
(001) ( 1 x 1) surface, both evaluated within the simple LCAO scheme [ 8 1. (The result for the [ 1, 1 ] step orientation is quite similar; for other examples, the reader is referred to refs. [ 13,141.) Hence, one can speculate that many less-coordinated W (00 1) surface structures are less inclined to various kinds of instabilities and reconstructions than the more coordinated ones. This hypothesis is corroborated by preliminary results for the steps in the [ 1, 0] and [ 1, 11 directions obtained in a model analogous to that of ref. [ 81. At present, however, new experimental and theoretical data on the factors governing the interplay between the attractive and the repulsive interactions, and on the magnitude of the underlying effects, are needed. One of the authors (MT) Heine, FRS, for generous help sions. MUL acknowledges the the Fondazione “Angelo Della
thanks Professor V. and valuable discusfinancial support of Riccia”, Florence.
References [ 1] J.Friedel, in: The physics of metals, Vol. 1. Electrons, ed. J. Ziman (Cambridge Univ. Press, Cambridge, 1961) ch. 8. [2] D.W. Bullett, R. Haydock, V. Heine and M.J. Kelly, Solid state physics, Vol. 35 (Academic Press, New York, 1980). [ 3 ] F. Ducastelleand F. Cyrot-Lackmann,J. Phys. Chem. Solids 32 (1971) 285. [4] V. Heine and J.H. Samson, J. Phys. F 13 (1983) 2155. [5] J.K. Burdett, Sttuct. Bonding 65 (1987) 30. [6] S. Pick, CollectionCzech.Chem. Commun. 53 (1988) 1607. [7] Y. Jiang and H. Zhang, Theoret. Chim. Acta 75 (1989) 279. [8] M. TomaSek,S. Pick and M.U. Luchini, Surface Sci. 209 (1989) L99; s. Pick, M. TomaSekandM.U. Luchini, Intern. J. Quantum Chem., in press. 191M. TomPSekand S. Pick, Surface Sci. 140 (1984) L279; Czech. J. Phys.B 35 (1985) 768;PhysicaB 132 (1985) 79. [ lo] S. Pick, Collection Czech. Chem. Commun. 53 ( 1988) 66 1. [ 111S. Lehwald, F. Wolf, H. Jbach, B.M. Hall and D.L. Mills, SurfaceSci. 192 (1987) 131. [ 121D.A. King, Physica Scripta T4 (1983) 34. [ 131D. Singh and H. Krakauer, Surface Sci. 216 (1989) 303. and references therein. [ 141R. Haydockand M.J. Kelly, Surface Sci. 38 (1973) 139.
347
SIMPLE COMMENTS
&pan
CHEMICAL PHYSICS LETTERS
I5 December 1989
ON THE W(OO1) SURFACE INSTABILITY
PICK, Mojmir TOMASEK
J. Heyrovsk$ Institute ofPhysical Chemistry and Electrochemisiry. Czechoslovak Academy of Sciences, DolejSkova 3, 182 23 Prague 8, Czechoslovakia
and Marco U. LUCHINI Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CR3 OHE, UK Received 14 July 1989; in final form 18 September I989
A geometrical interpretation of the reconstruction of the ideal W(OO1) surface is suggested, which is based on the use of the generalized Htickel rule. More complicated situations like steps are also mentioned and it is speculated that the increase in the coordination of the surface atom does not mean necessarily an increase in its stability.
The kth moment of the electronic Hamiltonian matrix H is usually defined by the equation mk=TrHk, lz=O, l,.... For nonhomogeneous systems, it is useful to modify this definition, restricting the above sum to the diagonal matrix elements ($1 Hk 1b) with the orbitals 14) localized on the same atom. Consequently, information about local properties is obtained. In this case, another convenient expression for mk is mk =jEkn(E) dE, where IZ(E) is the local density of electronic states (LDS) associated with the atom chosen. The role of the second moment m2 as one of the basic quantities qualitatively characterizing the bonding has been recognized and employed - explicitly or implicitly - by many authors, see e.g. ref. [ 11. The higher-order moments arc incorporated in well known numerical schemes [ 2 1; however, the attempts to use them for simple qualitative predictions are less abundant in the literature (see refs. [ 3-71 and the references therein). Recently, the so-called generalized Hiickel rule (GHR) [5-71 has been applied by the present authors [ 81 to the problem of interatomic interactions on the W (001) surface. According to GHR, it is energetically favourable to increase (decrease) the 0009-2614/89/$
m4k+Z (m4,!) moment of the electronic Hamiltonian. (Note that m2 = m4kf2, li=O.) The reason why the Htickel rule, originally derived for simple models of organic chemistry, admits a wider use is that the underlying picture is the one common to all typical frontier-orbital (HOMO-LUMO, pseudo-JahnTeller) situations [ 5,6]. On the ideal W(O0 1) surface, the frontier orbitals are represented by the partly occupied surface state bands. For the ideal W (00 1) surface atom (as well as for W adatoms in various arrangements on the W (001) surface), the local density of electronic states originating from the surface states contains contributions from both ep (mainly x2-y2) and tlg orbitals. Following refs. [ 81, the tZgorbitals contribute to the interaction between two surface neighbours Si, S2 mainly via the contributions to m4 of the form
(X1=X20rXI#Xd,
(1)
where X,,* stands for substrate atoms. It can be verified that the sum of these contributions is positive and according to GHR, an indirect repulsion (tendency to reduce m4) between S, and SZ is mediated by the subsurface atoms. From steric reasons [ 91,
03.50 0 Elsevier Science Publishers B.V. (North-Holland)
34s
Volume164,number4
CHEMICALPHYSICSLETTERS
the x2-y’ orbitals contribute essentially to the second moment ml by the contributions (S, 1HIS,) * responsible for a direct attraction between surface neighbours. However, even for the x*-y2 orbitals, the repulsive contributions ( 1) to m4, together with the terms [S,lO] (cf. also ref. [5])
(2)
where S, to S, are surface atoms lying on the vertices of a square, are not negligible. Now, let us consider a high-symmetry deformation of the ideal (001) surface, producing a certain number of shortened (S) and the same number of lengthened (L) bonds between the neighbouring surface atoms. For this deformation, the matrix elements V associated with the S-bond (L-bond) are resealed approximately by a factor ( 1+x) ( ( 1-x) ). By performing a L-S pair, the m2 contribution 2 v2 is increased to the value [(l tx)2+(1-x)‘] x Y2> 2 lf2 and the deformation is supported by direct attraction effects. Formally, the interaction between two surface neighbours can be described by the pair potential ~(Sr,)=~Sr,+fb(Sr,)~+..., whcrc 6~6 is the change of the ideal surface interatomic distance. The above m2 effect corresponds to the negative value of b and the indirect repulsion (see also below) points to the negative a value. From @, the surface force constants are derived in the standard way [ 111, resulting in a repulsive stress (a < 0) [ II] and an overall softening (b< 0) of the surface. Together with the L, S-bonds between surface atoms, L, S-bonds between surface and subsurface atoms are created, the scaling factors for the corresponding matrix elements now being (1 ky). As the result of the elementary inequality ( 1+,Y) ( 1-y) < 1, the existence of two neighbouring L- and S-bonds reduces the m4 contribution. Hence! it is desirable to maximize the number of L, S-bonds in neighbouring positions (i.e. minimize the number of L, L- or S, Sneighbours). Note that on the basis of moment analysis, possible stabilization of the W( 001) surface by the L-S bond alternation was suggested in ref. [ 41. There are two high-symmetry deformations obeying the mz criterion. Namely, the zig-zag M, mode of Debe and King [ 9,12 ] is energetically superior to the other candidate, the X3 mode [ 9 ] (denoted also as L2, XZ or the pairing mode in the literature), see 346
I.5DecemberI989
fig. 1. In that figure, the L, S-bond interpretation of the ti, and X3 modes is also shown. (It is more instructive to study the [ 1, 0] polarization of the I& mode rather than the [ 1, l] one observed experimentally. The two polarizations differ in energy only when higher-order anharmonic contributions are taken into account [ 91.) Since the increase of m2, controlled by the number of L-S pairs in our approximation, is the same for the two deformation modes, it is the m4 criterion which determines the higher Ii& reconstruction stability. Namely, the inspection of fig. 1 shows that for the l& mode, the zig-zag geometry leads to a higher number of neighbouring surface-subsurface L-S pairs in the two important situations described below and consequently to a more reduced value of m4. When traced to the product of matrix elements, the following is found: (1) For the indirect interaction contribution ( 1) with S, # SZ, X, =X2, there are more L-S pairs present in the I$ geometry than in the X3 one. In these contributions, both the tZg and eg orbitals are important. (2) The term (2) (S, #SZ#S3#S4) contains a L-S pair in the I$ case, contrary to the L-L or S-S pairs for the X3 mode. (The two deformed bonds connect, say, the atoms S,-S, and S3-S4, respectively, whereas the bonds S,-S, and SZ-S4, unperturbed to the first order in the deformation, are not shown in fig. 1.) In this situation , the x2-y’ orbitals are essential. As a result of our analysis, a natural and concise interrelationship between the
Fig. 1. The geometry ofthe M, and x3 modes, together with their L, S-bond interpretation (left). The circles denote the atoms in the surface and subsurface layers, respectively. The S (L) bonds are shown by heavy (thin) lines connecting the respective atoms.
Volume 164,number 4
CHEMICALPHYSICSLETTERS
surface states behaviour, the m2 and m,, moments, the geometry of the deformation (S, L-bonds), and the phenomenological force constant and pair potential notion is obtained. There are other interesting phenomena for the less coordinated atomic arrangements on the W (001) surface which are not understood completely at present. It is known that the reconstruction is suppressed near the step edge, although the geometry at such a step edge is unknown [ 121. Facilitated vacancy formation is another effect which is suspected to take place [ 12,131 on the (001) surface. Since the competition between the attractive (m2) and repulsive (m,)interactions should be present for any realistic arrangements of W atoms on the W (00 I ) surface, the presence of certain anomalous effects can be expected. A simplified argument based on the m2 reduction would predict the surface state peak narrowing, which, in its turn, might point to an enhanced instability of the less coordinated structures. Actually, the sharp surface state peak in LDS on the ideal surface seems to result from the high translational and point symmetries and it widens or even splits for the less coordinated atoms. As an example, in fig. 2 we compare the LDS for the atom on the [ 1, 0] step edge with that for the atom on the
Fig. 2. The local density n(E) of electronic d states on the [ 1, O] step edge atom (solid line) and on the ideal (001) surfaceatom (dotted line). The surface state feature lies at E- 0.
15 December 1989
(001) ( 1 x 1) surface, both evaluated within the simple LCAO scheme [ 8 1. (The result for the [ 1, 1 ] step orientation is quite similar; for other examples, the reader is referred to refs. [ 13,141.) Hence, one can speculate that many less-coordinated W (00 1) surface structures are less inclined to various kinds of instabilities and reconstructions than the more coordinated ones. This hypothesis is corroborated by preliminary results for the steps in the [ 1, 0] and [ 1, 11 directions obtained in a model analogous to that of ref. [ 81. At present, however, new experimental and theoretical data on the factors governing the interplay between the attractive and the repulsive interactions, and on the magnitude of the underlying effects, are needed. One of the authors (MT) Heine, FRS, for generous help sions. MUL acknowledges the the Fondazione “Angelo Della
thanks Professor V. and valuable discusfinancial support of Riccia”, Florence.
References [ 1] J.Friedel, in: The physics of metals, Vol. 1. Electrons, ed. J. Ziman (Cambridge Univ. Press, Cambridge, 1961) ch. 8. [2] D.W. Bullett, R. Haydock, V. Heine and M.J. Kelly, Solid state physics, Vol. 35 (Academic Press, New York, 1980). [ 3 ] F. Ducastelleand F. Cyrot-Lackmann,J. Phys. Chem. Solids 32 (1971) 285. [4] V. Heine and J.H. Samson, J. Phys. F 13 (1983) 2155. [5] J.K. Burdett, Sttuct. Bonding 65 (1987) 30. [6] S. Pick, CollectionCzech.Chem. Commun. 53 (1988) 1607. [7] Y. Jiang and H. Zhang, Theoret. Chim. Acta 75 (1989) 279. [8] M. TomaSek,S. Pick and M.U. Luchini, Surface Sci. 209 (1989) L99; s. Pick, M. TomaSekandM.U. Luchini, Intern. J. Quantum Chem., in press. 191M. TomPSekand S. Pick, Surface Sci. 140 (1984) L279; Czech. J. Phys.B 35 (1985) 768;PhysicaB 132 (1985) 79. [ lo] S. Pick, Collection Czech. Chem. Commun. 53 ( 1988) 66 1. [ 111S. Lehwald, F. Wolf, H. Jbach, B.M. Hall and D.L. Mills, SurfaceSci. 192 (1987) 131. [ 121D.A. King, Physica Scripta T4 (1983) 34. [ 131D. Singh and H. Krakauer, Surface Sci. 216 (1989) 303. and references therein. [ 141R. Haydockand M.J. Kelly, Surface Sci. 38 (1973) 139.
347