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Theory of helium adsorption on noble metals
Solid State Communications, Printed in Great Britain.
Vol. 56, No. 3, pp. 223-225,
1986.
THEORY OF HELIUM ADSORPTION
0038-1098/86 $3.00 + -00 Pergamon Press Ltd.
ON NOBLE METALS
Sun-Mok Paik and S. Das Sarma Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, USA (Received 2 December 1985 by A.A. Maradudin) The physisorption binding energy of a helium adatom on various noble metal surfaces has been calculated using a non-local theory for the metallic dielectric response. We find that non-local corrections to the usual van der Waals interaction modify physisorption binding energy by 30-45% and equilibrium adatom-surface separation by lo-25%. We give results for helium adsorption on Cu, Ag, and Au using a theory that includes nonlocal dielectric response of the metal and quadrupole moments of the atoms and, an effective-medium approximation for the short-range repulsive part.
THERE IS, IN RECENT literature, a growing interest in the improvement of the theoretical calculation of the interaction potential between a solid surface and an external atom [l-7]. It is crucial to be able to calculate this interaction accurately to understand a number of important experiments, such as helium beam diffraction from the surface, physisorption of simple atoms on the solid surface and various phase transitions involving adsorbates on the surface. The interaction potential can be written as a sum of two terms [l, 81, V,(z) and V&r), where V,(z) is the strong, short-range repulsive interaction due to the overlap between the electronic wavefunction of the adatom and the metal wavefunction tails, and, V,(z) is the long-range attractive interaction (van der Waals interaction) arising from the surface plasmon fluctuations in the metal and the multipole fluctuation in the atom. It has been shown that within an effective medium approximation the repulsive part can be written quite accurately by the simple exponential form [l-9] , V,(z) 2 V, exp (- uz) over the region of interest in physisorption. The attractive interaction is usually expressed [ 1, 5, 81 as
c3 bCz)
=
-cz
_zo)3
G -(z_zo)5
’
for large z > l-2 A (but not too large so that we can safely neglect [lo] retardation effects, z Q 200 A). The first term, in equation (1) is the usual van der Waals term arising from the atomic dipole moment. In a local theory [5,8,12], Ca is a constant (independent of the separation z) for the particular metal-atom combination. The leading order correction to the local van der Waals interaction (the second term in equation (1)) comes from the quadrupolar interaction [5,6] between an adatom and the solid surface. However, in non-local approxi-
mations [13, 141, Ca is no longer constant (i.e., it depends on the separation z). There is thus an additional contribution [13] to the O(z’) term arising from the surface response. In equation (l), z. w 0.5 A is the effective dynamical centroid [l ,6-8, 151 of the surface electron density profile from which the adatom separation is usually measured. In two previous papers [13, 141, we pointed out that the non-local theory of van der Waals interaction, associated with wave vector dependence of the dielectric response of the solid (we call this theory non-local approximation throughout this paper), describes the real physical system better than the standard local theory [5,8, 121. We have also shown that the nonlocal higher order, [e.g. O(z-‘)I , correction to the usual van der Waals interaction, is of the same order as the quadrupolar [5] interaction (calculated within local theory). In this paper, we investigate the effects of the non-local correction both on the equilibrium separation and the well-minimum for physisorption binding. For attractive interaction, we use a modified hydrodynamic approximation with an appropriate wavevector cut-off which we have shown earlier [14] to be the best nonlocal approximation which is also computationally tractable. For V,(z) we use an effective medium approximation and follow Nordlander and Harris for the repulsive interaction parameters [I 1. As we discussed in a previous paper [ 141, the model (a sharp, stepdensity surface model) which we use here, is simple and somewhat unphysical. However, it is legitimate to consider the effects of non-locality within this model. We want to emphasize that non-local effects both on the equilibrium separation and on the well-minimum are significant. We expect that these qualitative results will not change significantly in more realistic models.
223
224
THEORY OF HELIUM ADSORPTION
HeKu
ON NOBLE METALS
-
01
i \ \ \ \
./.’ ./ 4 2
tad
1,, , 2
4
1
6
I
I
6
8
IO
z (ad
Fig. 1. The physisorption binding energy of an He adatom on the Cu surface as a function of the adatomsurface separation z for four different approximations on the attractive interaction. (Note, the repulsive interactions are the same for all curves.) Dotted curve: the local van der Waals interaction potential [- Cs/(z - ze)’ ; dashed-dotted curve: the local van der Waals interaction potential plus the quadrupolar contribution of [S] [- Cs/(z - ~e)~ - Cs /(z - ze)’ ] ; dashed curve: the non-local van der Waals potential V,(z - ze) of [ 141 ; solid curve: the non-local van der Waals interaction potential [14] plus the quadrupolar contribution. The chosen parameters for the attractive interaction and the repulsive interaction correspond to [5] and [l] respectively.
j
Vol. 58, No. 3
,
,
,
8
1 IO
2 (ad
Fig. 2. Same as Fig. 1, for He atom on Ag surface. In this paper, we calculate the physisorption binding energy of the adsorbed atom on the three different metals (Cu, Ag and Au). We use the non-local approximation we have developed earlier in [ 141. As we show in that paper [14] this approximation (which uses a suitable wavevector cut-off on the hydrodynamic dielectric function) simulates the results (for the van der Waals interaction) obtained by the more rigorous (and computationally far more complex) Lindhard response theory. In Figs. l-3, we show the physisorption binding energy as a function of the separation z for three dif-
Fig. 3. Same as Figs. 2 and 3 for He atom on Au surface. ferent metal surface, Cu, Ag, and Au, respectively. To make the quantitative contribution of different terms explicit, we have shown four different curves including the local van der Waals potential, the local van der Waals potential plus the quadrupole term [5], the non-local van der Waals potential and the non-local van der Waals potential plus the quadrupole term. The same repulsive potential was used for all four curves. The non-local results are similar in trend to the local results, though quantitatively there are substantial differences. In particular, the well-minimum is 30-45% smaller and the equilibrium separation is lo-25% larger. These trends (shallower wells and larger separations) are expected because the non-local van der Waals potential gives saturation [ 141 on the surface, leading to smaller negative potential V,(z) than the local theory. The quadrupole interaction term lowers the well minimum somewhat, but the final result is still shallower than the result of the local theory. For an Ag metal surface, the local van der Waals potential plus the quadrupole interaction term is so strong that it overwhelms the repulsive V,(z) term around z - 0, and, the strong repulsive back wall does not appear. For Au, the results are even more drastic. The local van der Waals potential is by itself too strong. If we include the quadrupole term to the local van der Waals potential, it does not even give a minimum for the resultant Y(z)! In summary, we have calculated the physisorption binding energy of an adsorbed He atom on three different metal surfaces (Cu, Ag and Au) by considering the non-local dielectric response of the metal. We find that non-local corrections to the physisorption binding energy and to the equilibrium adatom-surface separation are of the order of 30-45% and IO-25%, respectively. Based on this work, we conclude that it is very important to include non-local correction to the physisorption binding energy in order to obtain quantitative (sometimes even qualitative, e.g. He on Au) accuracy.
Vol. 58, No. 3
THEORY OF HELIUM ADSORPTION
Acknowledgements - The work is supported
in part by the Office of Naval Research through contract No. NOO014-84-K0586 and by the National Science Foundation through Grant No. DMR-82-13768. The authors are grateful to the University of Maryland Computing Center for providing the computer time needed for this work.
6. 7. 8. 9.
REFERENCES 1. :: 4. 5.
P. Nordlander & J. Harris, J. Phys C17, 1141 (1984). N.D. Lang, Phys. Rev. Lett. 46,842 (1981). N.D. Lang & J.K. Norskov, Phys. Rev. B27, 4612 (1983). G.G. Kleiman & Uzi Landman, Phys. Rev. B8, 5484 (1973). X.P. Jiang, F. Toigo & M.W. Cole, Surf Sci. 145,
10. 11. 12. 13. 14. 15.
ON NOBLE METALS
225
281(1984) and 148,21(1984). C. Holmberg & P. Appel, Phys. Rev. B30, 5721 (1984). P. Appel8z C. Holmberg, Solid State Commun. 49, 1059j1984). E. Zaremba & W. Kahn, Phys. Rev. B13, 2270 (1976) and B15, 1769 (1977): M. Manninen, J.K. Norskov, M.J. Puska & C. Umrigar, Phys. Rev. B29,2314 (1984). M.J. Mehl & W.L. Schaich, Surf: Sci. 99, 553 (1980). L.W. Bruch, Surj: Sci. 125,194 (1983). E.M. Lifshitz, Sov. Phys. JETP, 2,73 (1956). S. Das Sarma 8c Sun-Mok Paik, Chem. Phys. Lett. 115,525 (1985). S. Das Sarma & Sun-Mok Paik, Chem. Phys. Lett. (to be published). B.N.J. Persson & E. Zaremba, Phys. Rev. B30, 5669 (1984).
Vol. 56, No. 3, pp. 223-225,
1986.
THEORY OF HELIUM ADSORPTION
0038-1098/86 $3.00 + -00 Pergamon Press Ltd.
ON NOBLE METALS
Sun-Mok Paik and S. Das Sarma Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, USA (Received 2 December 1985 by A.A. Maradudin) The physisorption binding energy of a helium adatom on various noble metal surfaces has been calculated using a non-local theory for the metallic dielectric response. We find that non-local corrections to the usual van der Waals interaction modify physisorption binding energy by 30-45% and equilibrium adatom-surface separation by lo-25%. We give results for helium adsorption on Cu, Ag, and Au using a theory that includes nonlocal dielectric response of the metal and quadrupole moments of the atoms and, an effective-medium approximation for the short-range repulsive part.
THERE IS, IN RECENT literature, a growing interest in the improvement of the theoretical calculation of the interaction potential between a solid surface and an external atom [l-7]. It is crucial to be able to calculate this interaction accurately to understand a number of important experiments, such as helium beam diffraction from the surface, physisorption of simple atoms on the solid surface and various phase transitions involving adsorbates on the surface. The interaction potential can be written as a sum of two terms [l, 81, V,(z) and V&r), where V,(z) is the strong, short-range repulsive interaction due to the overlap between the electronic wavefunction of the adatom and the metal wavefunction tails, and, V,(z) is the long-range attractive interaction (van der Waals interaction) arising from the surface plasmon fluctuations in the metal and the multipole fluctuation in the atom. It has been shown that within an effective medium approximation the repulsive part can be written quite accurately by the simple exponential form [l-9] , V,(z) 2 V, exp (- uz) over the region of interest in physisorption. The attractive interaction is usually expressed [ 1, 5, 81 as
c3 bCz)
=
-cz
_zo)3
G -(z_zo)5
’
for large z > l-2 A (but not too large so that we can safely neglect [lo] retardation effects, z Q 200 A). The first term, in equation (1) is the usual van der Waals term arising from the atomic dipole moment. In a local theory [5,8,12], Ca is a constant (independent of the separation z) for the particular metal-atom combination. The leading order correction to the local van der Waals interaction (the second term in equation (1)) comes from the quadrupolar interaction [5,6] between an adatom and the solid surface. However, in non-local approxi-
mations [13, 141, Ca is no longer constant (i.e., it depends on the separation z). There is thus an additional contribution [13] to the O(z’) term arising from the surface response. In equation (l), z. w 0.5 A is the effective dynamical centroid [l ,6-8, 151 of the surface electron density profile from which the adatom separation is usually measured. In two previous papers [13, 141, we pointed out that the non-local theory of van der Waals interaction, associated with wave vector dependence of the dielectric response of the solid (we call this theory non-local approximation throughout this paper), describes the real physical system better than the standard local theory [5,8, 121. We have also shown that the nonlocal higher order, [e.g. O(z-‘)I , correction to the usual van der Waals interaction, is of the same order as the quadrupolar [5] interaction (calculated within local theory). In this paper, we investigate the effects of the non-local correction both on the equilibrium separation and the well-minimum for physisorption binding. For attractive interaction, we use a modified hydrodynamic approximation with an appropriate wavevector cut-off which we have shown earlier [14] to be the best nonlocal approximation which is also computationally tractable. For V,(z) we use an effective medium approximation and follow Nordlander and Harris for the repulsive interaction parameters [I 1. As we discussed in a previous paper [ 141, the model (a sharp, stepdensity surface model) which we use here, is simple and somewhat unphysical. However, it is legitimate to consider the effects of non-locality within this model. We want to emphasize that non-local effects both on the equilibrium separation and on the well-minimum are significant. We expect that these qualitative results will not change significantly in more realistic models.
223
224
THEORY OF HELIUM ADSORPTION
HeKu
ON NOBLE METALS
-
01
i \ \ \ \
./.’ ./ 4 2
tad
1,, , 2
4
1
6
I
I
6
8
IO
z (ad
Fig. 1. The physisorption binding energy of an He adatom on the Cu surface as a function of the adatomsurface separation z for four different approximations on the attractive interaction. (Note, the repulsive interactions are the same for all curves.) Dotted curve: the local van der Waals interaction potential [- Cs/(z - ze)’ ; dashed-dotted curve: the local van der Waals interaction potential plus the quadrupolar contribution of [S] [- Cs/(z - ~e)~ - Cs /(z - ze)’ ] ; dashed curve: the non-local van der Waals potential V,(z - ze) of [ 141 ; solid curve: the non-local van der Waals interaction potential [14] plus the quadrupolar contribution. The chosen parameters for the attractive interaction and the repulsive interaction correspond to [5] and [l] respectively.
j
Vol. 58, No. 3
,
,
,
8
1 IO
2 (ad
Fig. 2. Same as Fig. 1, for He atom on Ag surface. In this paper, we calculate the physisorption binding energy of the adsorbed atom on the three different metals (Cu, Ag and Au). We use the non-local approximation we have developed earlier in [ 141. As we show in that paper [14] this approximation (which uses a suitable wavevector cut-off on the hydrodynamic dielectric function) simulates the results (for the van der Waals interaction) obtained by the more rigorous (and computationally far more complex) Lindhard response theory. In Figs. l-3, we show the physisorption binding energy as a function of the separation z for three dif-
Fig. 3. Same as Figs. 2 and 3 for He atom on Au surface. ferent metal surface, Cu, Ag, and Au, respectively. To make the quantitative contribution of different terms explicit, we have shown four different curves including the local van der Waals potential, the local van der Waals potential plus the quadrupole term [5], the non-local van der Waals potential and the non-local van der Waals potential plus the quadrupole term. The same repulsive potential was used for all four curves. The non-local results are similar in trend to the local results, though quantitatively there are substantial differences. In particular, the well-minimum is 30-45% smaller and the equilibrium separation is lo-25% larger. These trends (shallower wells and larger separations) are expected because the non-local van der Waals potential gives saturation [ 141 on the surface, leading to smaller negative potential V,(z) than the local theory. The quadrupole interaction term lowers the well minimum somewhat, but the final result is still shallower than the result of the local theory. For an Ag metal surface, the local van der Waals potential plus the quadrupole interaction term is so strong that it overwhelms the repulsive V,(z) term around z - 0, and, the strong repulsive back wall does not appear. For Au, the results are even more drastic. The local van der Waals potential is by itself too strong. If we include the quadrupole term to the local van der Waals potential, it does not even give a minimum for the resultant Y(z)! In summary, we have calculated the physisorption binding energy of an adsorbed He atom on three different metal surfaces (Cu, Ag and Au) by considering the non-local dielectric response of the metal. We find that non-local corrections to the physisorption binding energy and to the equilibrium adatom-surface separation are of the order of 30-45% and IO-25%, respectively. Based on this work, we conclude that it is very important to include non-local correction to the physisorption binding energy in order to obtain quantitative (sometimes even qualitative, e.g. He on Au) accuracy.
Vol. 58, No. 3
THEORY OF HELIUM ADSORPTION
Acknowledgements - The work is supported
in part by the Office of Naval Research through contract No. NOO014-84-K0586 and by the National Science Foundation through Grant No. DMR-82-13768. The authors are grateful to the University of Maryland Computing Center for providing the computer time needed for this work.
6. 7. 8. 9.
REFERENCES 1. :: 4. 5.
P. Nordlander & J. Harris, J. Phys C17, 1141 (1984). N.D. Lang, Phys. Rev. Lett. 46,842 (1981). N.D. Lang & J.K. Norskov, Phys. Rev. B27, 4612 (1983). G.G. Kleiman & Uzi Landman, Phys. Rev. B8, 5484 (1973). X.P. Jiang, F. Toigo & M.W. Cole, Surf Sci. 145,
10. 11. 12. 13. 14. 15.
ON NOBLE METALS
225
281(1984) and 148,21(1984). C. Holmberg & P. Appel, Phys. Rev. B30, 5721 (1984). P. Appel8z C. Holmberg, Solid State Commun. 49, 1059j1984). E. Zaremba & W. Kahn, Phys. Rev. B13, 2270 (1976) and B15, 1769 (1977): M. Manninen, J.K. Norskov, M.J. Puska & C. Umrigar, Phys. Rev. B29,2314 (1984). M.J. Mehl & W.L. Schaich, Surf: Sci. 99, 553 (1980). L.W. Bruch, Surj: Sci. 125,194 (1983). E.M. Lifshitz, Sov. Phys. JETP, 2,73 (1956). S. Das Sarma 8c Sun-Mok Paik, Chem. Phys. Lett. 115,525 (1985). S. Das Sarma & Sun-Mok Paik, Chem. Phys. Lett. (to be published). B.N.J. Persson & E. Zaremba, Phys. Rev. B30, 5669 (1984).