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Interatomic Force Constants and the Reconstruction of Transition Metal Surfaces
191
Journal of Electron Spectroscopy and Related Phenomena, 39 (1986) 191-193 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
INTERATOMIC FORCE CONSTANTS AND THE RECONSTRUCTION OF TRANSITION METAL SURFACES Dani~l
P. Joubert
Cavendish Laboratory, Mading1ey Road, Cambridge CB3 OHE, England
EXTENDED ABSTRACT During the past few years it has been demonstrated that interatomic force constants
and phonon
spectra
of
transition
metals
can
be adequately
investigated on the basis of a tight-binding (TB) description of the electronic structure (ref.1-5). The present method, similar to that used in (ref.3,5), works entirely in real space, making use of the recursion method to evaluate the electronic Green function.
It could therefore be applied to arbitrary
configurations of atoms, and is excellently suited to the investigation of surface effects.
A minimal basis of nine s, p and d orbitals fitted to a first
principles band structure calculation for a five layer slab of W(ref.?) within the non-orthogonal two-centre TB approximation (ref.B), was used in the description of the electronic structure.
In Fig. I the d-band contribution to
the second neighbour force constants, i.e. neighbouring atoms on the (001) surface, are ill ustrated as a function of band fi 11 ing for the fi rst seven (001) layers of an un-relaxed 5-d bcc transition metal.
We find that only the
surface interactions are significantly different from those in the bulk, the most pronounced changes occurring at the centre of the d-band, i.e. for a band filling of six electrons per atom.
For first neighbour interactions this is
also the case, but the changes are much smaller.
At the surface is xz non-zero and of comparable magnitude to the tangential force constants at the cI>
centre of the d-band, evidence that non-central interactions are important at the surface.
In the bulk these interactions are prevented by the higher
symmetry and they rapidly vanish as a function of distance from the surface. The non-central nature of the interaction at the surface is also evident from the difference between yy and zz cI>
cI>
192
The behaviour of the surface force constants as a function of band filling is particularly relevant to the well known reconstruction of the bcc group IV
Surface
zx
xx
20 -20
-40
20 -20
20 -20
20 -20
20 -20
20 -20
20 -20
-40 4
10
4
10
4
10
4
10
Electrons/atom Fig. 1. The d-band contribution to the second neighbour force constants for an ideal 5-d bcc transition metal as a function of band filling for the first seven (001) layers.
193
transition metal surfaces (W,Mo,Cr) (ref. 15-19).
It is only at the centre of
the d-band where the surface force constants differ s i gnifi cantly from thei r bulk values, suggestive of a localised instability in the ideal
surface
structure for which the calculation was performed (ref.11,12). In the present approach it is possible to express the interactions in terms of one-, two-, three- and four-body terms and to extract the indivi dua 1 contributions, giving information on a microscopic scale of the interactions involved.
An analysis of the individual contributions is given in ref. 10.
REFERENCES 1 C.M. Varma and W. Weber, Phys. Rev. B19 (1979) 6142-6154 2 D~H. Lee and J.D. Joannopoulos, Phys. Rev. Lett. 48 (1982) 1846-1849 3 M.W. Finnis, K.L. Kear and D.G. Pettifor, Phys. Rev. Lett. 52 (1984) 291-294 4 Douglas C. Allan and E.J. Mele, Phys. Rev. Lett. 53 (1984) 826-829 5 M.W. Finnis and D.G. Pettifor in D.G. Pettifor and D.L. Weaire (Eds.), The Recursion Method and its Applications, Springer Verlag, Berlin, 1985, pp. 120-131 6 R. Haydock, V. Heine and M.J. Kelly, J. Phys. C8 (1975) 2591 7 L.F. Mattheiss and D.R. Hamann, Phys. Rev. B29 (1984) 5372-5381 8 L.F. Mattheiss and W. Weber, Phys. Rev. B25 (1982) 2248-2269 9 A.R. Mackintosh and O.K. Andersen in M. Springford (Ed.), Electrons at the Fermi Surface, Cambridge University Press, Cambridge, 1980, pp. 147-222 10 D.P. Joubert, to be published 11 A. Fasolino, G. Santoro and E. Tosatti, Phys. Rev. Lett. 25 (1980) 1684-1687 12 A. Fasolino, G. Santoro and E. Tosatti, Surface Sci. 125 (1983) 317-325 13 Volker Heine and J.H. Sampson, J. Phys. F13 (1983) 2155-2168 14 T.E. Felter, R.A. Barker and P.J. Estrup, Phys. Rev. Lett. 38 (1977) 1138-1141 15 R.A. Barker and P.J. Estrup, J. Chern. Phys. 74(1981) 1442-1452 16 L.D. Roelofs and P.J. Estrup, Surface Sci. 125 (1983) 51-73 17 M.K. Debe and D.A. King, Surface Sci. 81 (1979) 193 18 D.A. King and G. Thomas, Surface Sci. 92 (1980) 201-236 19 G. Gewinner, J.C. Peruchetti, A. Jaegle and R. Riedinger, Phys. Rev. Lett. 43 (1979) 935-938
Journal of Electron Spectroscopy and Related Phenomena, 39 (1986) 191-193 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
INTERATOMIC FORCE CONSTANTS AND THE RECONSTRUCTION OF TRANSITION METAL SURFACES Dani~l
P. Joubert
Cavendish Laboratory, Mading1ey Road, Cambridge CB3 OHE, England
EXTENDED ABSTRACT During the past few years it has been demonstrated that interatomic force constants
and phonon
spectra
of
transition
metals
can
be adequately
investigated on the basis of a tight-binding (TB) description of the electronic structure (ref.1-5). The present method, similar to that used in (ref.3,5), works entirely in real space, making use of the recursion method to evaluate the electronic Green function.
It could therefore be applied to arbitrary
configurations of atoms, and is excellently suited to the investigation of surface effects.
A minimal basis of nine s, p and d orbitals fitted to a first
principles band structure calculation for a five layer slab of W(ref.?) within the non-orthogonal two-centre TB approximation (ref.B), was used in the description of the electronic structure.
In Fig. I the d-band contribution to
the second neighbour force constants, i.e. neighbouring atoms on the (001) surface, are ill ustrated as a function of band fi 11 ing for the fi rst seven (001) layers of an un-relaxed 5-d bcc transition metal.
We find that only the
surface interactions are significantly different from those in the bulk, the most pronounced changes occurring at the centre of the d-band, i.e. for a band filling of six electrons per atom.
For first neighbour interactions this is
also the case, but the changes are much smaller.
At the surface is xz non-zero and of comparable magnitude to the tangential force constants at the cI>
centre of the d-band, evidence that non-central interactions are important at the surface.
In the bulk these interactions are prevented by the higher
symmetry and they rapidly vanish as a function of distance from the surface. The non-central nature of the interaction at the surface is also evident from the difference between yy and zz cI>
cI>
192
The behaviour of the surface force constants as a function of band filling is particularly relevant to the well known reconstruction of the bcc group IV
Surface
20 -20
-40
20 -20
20 -20
20 -20
20 -20
20 -20
20 -20
-40 4
10
4
10
4
10
4
10
Electrons/atom Fig. 1. The d-band contribution to the second neighbour force constants for an ideal 5-d bcc transition metal as a function of band filling for the first seven (001) layers.
193
transition metal surfaces (W,Mo,Cr) (ref. 15-19).
It is only at the centre of
the d-band where the surface force constants differ s i gnifi cantly from thei r bulk values, suggestive of a localised instability in the ideal
surface
structure for which the calculation was performed (ref.11,12). In the present approach it is possible to express the interactions in terms of one-, two-, three- and four-body terms and to extract the indivi dua 1 contributions, giving information on a microscopic scale of the interactions involved.
An analysis of the individual contributions is given in ref. 10.
REFERENCES 1 C.M. Varma and W. Weber, Phys. Rev. B19 (1979) 6142-6154 2 D~H. Lee and J.D. Joannopoulos, Phys. Rev. Lett. 48 (1982) 1846-1849 3 M.W. Finnis, K.L. Kear and D.G. Pettifor, Phys. Rev. Lett. 52 (1984) 291-294 4 Douglas C. Allan and E.J. Mele, Phys. Rev. Lett. 53 (1984) 826-829 5 M.W. Finnis and D.G. Pettifor in D.G. Pettifor and D.L. Weaire (Eds.), The Recursion Method and its Applications, Springer Verlag, Berlin, 1985, pp. 120-131 6 R. Haydock, V. Heine and M.J. Kelly, J. Phys. C8 (1975) 2591 7 L.F. Mattheiss and D.R. Hamann, Phys. Rev. B29 (1984) 5372-5381 8 L.F. Mattheiss and W. Weber, Phys. Rev. B25 (1982) 2248-2269 9 A.R. Mackintosh and O.K. Andersen in M. Springford (Ed.), Electrons at the Fermi Surface, Cambridge University Press, Cambridge, 1980, pp. 147-222 10 D.P. Joubert, to be published 11 A. Fasolino, G. Santoro and E. Tosatti, Phys. Rev. Lett. 25 (1980) 1684-1687 12 A. Fasolino, G. Santoro and E. Tosatti, Surface Sci. 125 (1983) 317-325 13 Volker Heine and J.H. Sampson, J. Phys. F13 (1983) 2155-2168 14 T.E. Felter, R.A. Barker and P.J. Estrup, Phys. Rev. Lett. 38 (1977) 1138-1141 15 R.A. Barker and P.J. Estrup, J. Chern. Phys. 74(1981) 1442-1452 16 L.D. Roelofs and P.J. Estrup, Surface Sci. 125 (1983) 51-73 17 M.K. Debe and D.A. King, Surface Sci. 81 (1979) 193 18 D.A. King and G. Thomas, Surface Sci. 92 (1980) 201-236 19 G. Gewinner, J.C. Peruchetti, A. Jaegle and R. Riedinger, Phys. Rev. Lett. 43 (1979) 935-938