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First principles calculations of surface phonons on Rh(111)
n l ii iii
surface science ELSEVIER
Surface Science 368 (1996) 222-225
First principles calculations of surface phonons on Rh(111) K . - P . B o h n e n a, A. Eichler b,., j. H a f n e r b a Forschungszentrum Karlsruhe, Institutffir Nukleare Festkrrperphysik, P.O. Box 3640, D-7500 Karlsruhe, Germany b lnstitutffir Theoretische Physik and Center for Computational Materials Science, Technische Universitfit Wien, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria Received 1 May 1996; accepted for publication 1 August 1996
Abstract
Surface phonons on the high symmetry points of the Rh(lll)-surface have been determined using first principles local density functional (LDF) calculations with two different pseudopotentials, once with a planewave and once with a mixed basis-set. Use of the Hellmann-Feynman theorem allowed the calculation of forces and thus the determination of force constants. Due to the lack of experimental or theoretical data for the bulk phonon spectrum, calculations for the bulk properties have also been performed. Both calculations for the bulk phonon spectrum are in excellent agreement with a recently performed experimental study including all observed phonon anomalies. These data do not support the assumption made by Toennies, that the spectrum of a hydrogenated R h ( l l l ) surface would correspond to that of the truncated bulk. The interplanar force constants at the surface have been found to differ rather strongly from the bulk force constants (more than in Pd or Pt, for example), resulting in a very significant influence on the surface phonon frequencies at the zone boundary. Keywords: Ab initio quantum chemical methods and calculations; Density functional calculations; Low index single crystal surfaces; Phonons; Rhodium
Over the past 10 years surface phonons have been measured and/or calculated for many metal surfaces [1,2]. Experimental results from Hescattering and EELS are usually in excellent agreement with the theoretical predictions. Among the systems studied so far, Rh, an important catalyst for nitric oxide reduction, is missing. For Rh the situation is complicated due to the fact that no neutron measurements of the bulk phonon spectrum are available. Knowledge of the bulk phonon modes however is necessary for interpreting changes introduced by the presence of the * Corresponding author. Fax: +43 1 5867760; e-maih eichler@tph2 l.tuwien.ac.at
surface. The only information about bulk phonons are the elastic constants [3] and experimental point contact spectra [4]. Model calculations predict phonon anomalies similar to those observed in Pt and Pd; however, the actual positions of these anomalies depend strongly on the topology of the Fermi surface which is not properly described in these simple models [5]. Due to the presence of these anomalies no short-range force constant model can properly describe the phonon spectrum of Rh. Recently the first measurements of surface modes for R h ( l l l ) have been published [6] but the interpretation of these measurements was hindered by the missing bulk information as well as missing first principles calculations for the surface.
0039-6028/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0039-6028 (96) 01055-2
K.-P. Bohnen et al./ Surface Science 368 (1996) 222-225
In a first step we have carried out first principles total energy density functional calculations for obtaining the bulk phonon spectrum. The method used was based on the determination of the force constants in terms of derivatives of the HellmannFeynman forces. The calculations were performed on the basis of the pseudopotential concept. In the past, structural studies for the bulk as well as for the surface have been carried out with two different pseudopotentials [7,8]. One of these was constructed according to the normconserving concept of Hamann et al. (P1) [91 while the other one (P2) is an ultrasoft one where normconservation has been given up [ 10,11 ]. Due to the very different structure of the potentials, P1 could be treated properly only within the mixed basis formulation [7] while P2 could be treated in a pure plane wave basis [10,11]. Since P2 is of separable type the numerical treatment of potential P2 is much less costly than that of P1. The bulk phonon dispersion branches in the 100, 110 and 111 directions were calculated by using unit cells which contained up to 18 atoms to allow for long-range
,-if,
223
interaction effects. Both potentials gave very similar results, the major difference being in the absolute magnitude of the zone boundary values. Potential P2 gives a 0.5 THz higher maximum frequency which is consistent with a slightly higher bulk modulus. The theoretical predictions have been confirmed by neutron experiments which support the results obtained with potential P1 and differ from those of P2 only by a scaling factor. Based on the calculations and the experimental measurements, atomic force constants could be obtained for the bulk. Interactions up to the twelfth neighbour are necessary to obtain a satisfactory description of the phonon modes. Details of these bulk studies will be published elsewhere [12]. Using these bulk force constants the surface phonon modes could be calculated in the usual way [13,14]. Results obtained with potential P1 are shown in Fig. 1 and frequencies for the surface modes are given in Table 1. Comparing these results with the experimental information [6] it is obvious that substantial changes at the surface must occur. These changes are larger than for
6
~s ¢.) 4
3
0
Fig. 1. Phonon dispersion relation for a Rh(111) slab using the atomic force constant model based on the potential P1 along the path r - M - K - F in the two-dimensional Bz.
16-P. Bohnen et al./ Surface Science 368 (1996) 222-225
224
Table 1 Phonon frequencies (THz) at the R h ( l l l ) surface obtained with bulk force constants [12] excluding surface effects (truncated bulk) and using the calculated force constants for the surface layer (calculation). Experimental data [6] are included. Only true surface modes are shown
bT,l
Truncated bulk
Calculation
Experiment
3.39 4.11 6.63 3.26 4.54 5.38 5.60 6.01
3.95 3.88 6.14 4.30 4.40 4.52 5.24 6.02
3.92 ___0.07 4.21 +0.17 -
other (lll)-surfaces (Cu,Ag,A1) [2,13,14]. We have used total energy density functional calculations to determine these changes. We have determined interplanar surface force-constants at high symmetry points in the two-dimensional Brillouin zone. At these points the distortions induced by the surface phonons are commensurate with the surface unit cell. In such cases, the distorted system still has surface periodicity and we can calculate the change in total energy as well as the atomic forces induced by the distortion. Of course such calculations involve unit cells which are larger than the undistorted cell and it is obvious that such an approach can be used only at special wave vectors, for example the zone center and high symmetry points at the zone boundary. To calculate the surface force constants we start from a relaxed surface geometry [8], so that there are no forces acting on the surface layers. Then the surface layer is slightly distorted by small atomic displacements corresponding to the surface wave vector under consideration. The self-consistent structure for each of the off-equilibrium geometries is calculated and the forces exerted on the atomic layers in the slab evaluated. From this we obtain the interplanar force-constants coupling the surface with the subsurface layers. By moving the atoms in the surface layer in three orthogonal directions, the interlayer force constant matrices coupling the top layer with all the other layers can be determined. These surface force constants are then used together with
the bulk interlayer force constants to construct the full dynamical matrix of the slab and to obtain all vibrational modes of the slab at the given two-dimensional wave vector [ 15 ]. This procedure we have carried out for the F, M and K-point of the two-dimensional Bz. However calculations have been carried out so far only using potential P2 which overestimates the bulk frequencies slightly. Calculations for Rh(100) have shown that the surface force-constants obtained with potential P2 or P1 scale very similarly to the bulk force constants. Using this scaling we have calculated the surface modes given in Table 1. Full calculations with Potential P1 are being carried out at present and will be finished soon. We do not expect substantial changes compared to the results given in Table 1. A number of points are worth mentioning in connection with these results. Since the maximum frequency in the bulk is 7.45 THz, all force constant models for the bulk have to satisfy this constraint. In the models employed in Ref. [6] for analysing their data this constraint is violated. Comparing numbers from Table 1 with results from experiments for Rh(111) with H (1 x 1) Ref. [6] also indicates that the assumption being made that the force constants of the hydrogencovered surface closely resemble those of a bulkterminated surface is not justified. The results presented in Table 1, which include the changes of the force constants at the surface, are in good agreement with the experimental data, clearly indicating that only a strong interplay with first principles calculations of the surface dynamics can lead to an understanding of experimentally determined surface modes.
References [ 1] K.M. Ho. and K.P. Bohnen, J. Electron Spectrosc. Relat. Phenom. 54/55 (1990) 229. [2] K.P. Bohnen and K.M. Ho, Surf. Sci. Rep. 19 (1993) 99. [3] E. Walker, J. Ashkenazi and M. Dacorogna, Phys. Rev. B 24 (1981) 2254. [4] I.K. Yanson, A.V. Khotkevich, S.N. Krainyukov, V.V. Nemoshkalenko, V.N. Antonov, A.V. Zhalko-Titarenko and V.Yu. Milman, in: Physics of Transition Metals, Ed. V.G. Baryakhtar (Naukova Dumka, Kiev, 1988).
K-P. Bohnen et al. /Surface Science 368 (1996) 222-225 [5] V.N. Antonov, V.Yu. Milman, V.V. Nemoshkal¢nko and A.V. Zhalko-Titarenko, Z. Phys. B Condens. matter 79 (1990) 223. [6] G. Witte, J.P. Toennies and Ch. Wodl, Surf. Sci. 323 (1995) 228. [7] C. Elsaesser, N. Takeuchi, K.M. Ho, C.T. Chan, P. Braun and M. Faehnle, J. Phys. Condens. Matter 2 (1990) 4371. [8] A. Eichler, J. Hafner, J. FurthmQller and G. Kresse, Surf. Sci. 346 (1996) 300. [9] G.B. Bachelet, D.R. Hamann and M. Schlueter, Phys. Rev. B 26 (1982) 4199.
225
[10] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [11] G. Kresse and J. Hafner, J. Phys. Condens. Matter 6 (1994) 8245. [12] K.P. Bohnen, G. Nolte, A. Eichler, J. Hafner and W. Reichardt, to be submitted. [13] Y. Chen, S.Y. Tong, K.P. Bohnen, T. Rodach and K.M. Ho, Phys. Rev. Lett. 70 (1993) 603. [14] J. Sch6chlin, K.P. Bohnen and K.M. Ho, Surf. Sci. 324 (1995) 114. [15] R.E. Allen, G.P. Allredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1661.
surface science ELSEVIER
Surface Science 368 (1996) 222-225
First principles calculations of surface phonons on Rh(111) K . - P . B o h n e n a, A. Eichler b,., j. H a f n e r b a Forschungszentrum Karlsruhe, Institutffir Nukleare Festkrrperphysik, P.O. Box 3640, D-7500 Karlsruhe, Germany b lnstitutffir Theoretische Physik and Center for Computational Materials Science, Technische Universitfit Wien, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria Received 1 May 1996; accepted for publication 1 August 1996
Abstract
Surface phonons on the high symmetry points of the Rh(lll)-surface have been determined using first principles local density functional (LDF) calculations with two different pseudopotentials, once with a planewave and once with a mixed basis-set. Use of the Hellmann-Feynman theorem allowed the calculation of forces and thus the determination of force constants. Due to the lack of experimental or theoretical data for the bulk phonon spectrum, calculations for the bulk properties have also been performed. Both calculations for the bulk phonon spectrum are in excellent agreement with a recently performed experimental study including all observed phonon anomalies. These data do not support the assumption made by Toennies, that the spectrum of a hydrogenated R h ( l l l ) surface would correspond to that of the truncated bulk. The interplanar force constants at the surface have been found to differ rather strongly from the bulk force constants (more than in Pd or Pt, for example), resulting in a very significant influence on the surface phonon frequencies at the zone boundary. Keywords: Ab initio quantum chemical methods and calculations; Density functional calculations; Low index single crystal surfaces; Phonons; Rhodium
Over the past 10 years surface phonons have been measured and/or calculated for many metal surfaces [1,2]. Experimental results from Hescattering and EELS are usually in excellent agreement with the theoretical predictions. Among the systems studied so far, Rh, an important catalyst for nitric oxide reduction, is missing. For Rh the situation is complicated due to the fact that no neutron measurements of the bulk phonon spectrum are available. Knowledge of the bulk phonon modes however is necessary for interpreting changes introduced by the presence of the * Corresponding author. Fax: +43 1 5867760; e-maih eichler@tph2 l.tuwien.ac.at
surface. The only information about bulk phonons are the elastic constants [3] and experimental point contact spectra [4]. Model calculations predict phonon anomalies similar to those observed in Pt and Pd; however, the actual positions of these anomalies depend strongly on the topology of the Fermi surface which is not properly described in these simple models [5]. Due to the presence of these anomalies no short-range force constant model can properly describe the phonon spectrum of Rh. Recently the first measurements of surface modes for R h ( l l l ) have been published [6] but the interpretation of these measurements was hindered by the missing bulk information as well as missing first principles calculations for the surface.
0039-6028/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0039-6028 (96) 01055-2
K.-P. Bohnen et al./ Surface Science 368 (1996) 222-225
In a first step we have carried out first principles total energy density functional calculations for obtaining the bulk phonon spectrum. The method used was based on the determination of the force constants in terms of derivatives of the HellmannFeynman forces. The calculations were performed on the basis of the pseudopotential concept. In the past, structural studies for the bulk as well as for the surface have been carried out with two different pseudopotentials [7,8]. One of these was constructed according to the normconserving concept of Hamann et al. (P1) [91 while the other one (P2) is an ultrasoft one where normconservation has been given up [ 10,11 ]. Due to the very different structure of the potentials, P1 could be treated properly only within the mixed basis formulation [7] while P2 could be treated in a pure plane wave basis [10,11]. Since P2 is of separable type the numerical treatment of potential P2 is much less costly than that of P1. The bulk phonon dispersion branches in the 100, 110 and 111 directions were calculated by using unit cells which contained up to 18 atoms to allow for long-range
,-if,
223
interaction effects. Both potentials gave very similar results, the major difference being in the absolute magnitude of the zone boundary values. Potential P2 gives a 0.5 THz higher maximum frequency which is consistent with a slightly higher bulk modulus. The theoretical predictions have been confirmed by neutron experiments which support the results obtained with potential P1 and differ from those of P2 only by a scaling factor. Based on the calculations and the experimental measurements, atomic force constants could be obtained for the bulk. Interactions up to the twelfth neighbour are necessary to obtain a satisfactory description of the phonon modes. Details of these bulk studies will be published elsewhere [12]. Using these bulk force constants the surface phonon modes could be calculated in the usual way [13,14]. Results obtained with potential P1 are shown in Fig. 1 and frequencies for the surface modes are given in Table 1. Comparing these results with the experimental information [6] it is obvious that substantial changes at the surface must occur. These changes are larger than for
6
~s ¢.) 4
3
0
Fig. 1. Phonon dispersion relation for a Rh(111) slab using the atomic force constant model based on the potential P1 along the path r - M - K - F in the two-dimensional Bz.
16-P. Bohnen et al./ Surface Science 368 (1996) 222-225
224
Table 1 Phonon frequencies (THz) at the R h ( l l l ) surface obtained with bulk force constants [12] excluding surface effects (truncated bulk) and using the calculated force constants for the surface layer (calculation). Experimental data [6] are included. Only true surface modes are shown
bT,l
Truncated bulk
Calculation
Experiment
3.39 4.11 6.63 3.26 4.54 5.38 5.60 6.01
3.95 3.88 6.14 4.30 4.40 4.52 5.24 6.02
3.92 ___0.07 4.21 +0.17 -
other (lll)-surfaces (Cu,Ag,A1) [2,13,14]. We have used total energy density functional calculations to determine these changes. We have determined interplanar surface force-constants at high symmetry points in the two-dimensional Brillouin zone. At these points the distortions induced by the surface phonons are commensurate with the surface unit cell. In such cases, the distorted system still has surface periodicity and we can calculate the change in total energy as well as the atomic forces induced by the distortion. Of course such calculations involve unit cells which are larger than the undistorted cell and it is obvious that such an approach can be used only at special wave vectors, for example the zone center and high symmetry points at the zone boundary. To calculate the surface force constants we start from a relaxed surface geometry [8], so that there are no forces acting on the surface layers. Then the surface layer is slightly distorted by small atomic displacements corresponding to the surface wave vector under consideration. The self-consistent structure for each of the off-equilibrium geometries is calculated and the forces exerted on the atomic layers in the slab evaluated. From this we obtain the interplanar force-constants coupling the surface with the subsurface layers. By moving the atoms in the surface layer in three orthogonal directions, the interlayer force constant matrices coupling the top layer with all the other layers can be determined. These surface force constants are then used together with
the bulk interlayer force constants to construct the full dynamical matrix of the slab and to obtain all vibrational modes of the slab at the given two-dimensional wave vector [ 15 ]. This procedure we have carried out for the F, M and K-point of the two-dimensional Bz. However calculations have been carried out so far only using potential P2 which overestimates the bulk frequencies slightly. Calculations for Rh(100) have shown that the surface force-constants obtained with potential P2 or P1 scale very similarly to the bulk force constants. Using this scaling we have calculated the surface modes given in Table 1. Full calculations with Potential P1 are being carried out at present and will be finished soon. We do not expect substantial changes compared to the results given in Table 1. A number of points are worth mentioning in connection with these results. Since the maximum frequency in the bulk is 7.45 THz, all force constant models for the bulk have to satisfy this constraint. In the models employed in Ref. [6] for analysing their data this constraint is violated. Comparing numbers from Table 1 with results from experiments for Rh(111) with H (1 x 1) Ref. [6] also indicates that the assumption being made that the force constants of the hydrogencovered surface closely resemble those of a bulkterminated surface is not justified. The results presented in Table 1, which include the changes of the force constants at the surface, are in good agreement with the experimental data, clearly indicating that only a strong interplay with first principles calculations of the surface dynamics can lead to an understanding of experimentally determined surface modes.
References [ 1] K.M. Ho. and K.P. Bohnen, J. Electron Spectrosc. Relat. Phenom. 54/55 (1990) 229. [2] K.P. Bohnen and K.M. Ho, Surf. Sci. Rep. 19 (1993) 99. [3] E. Walker, J. Ashkenazi and M. Dacorogna, Phys. Rev. B 24 (1981) 2254. [4] I.K. Yanson, A.V. Khotkevich, S.N. Krainyukov, V.V. Nemoshkalenko, V.N. Antonov, A.V. Zhalko-Titarenko and V.Yu. Milman, in: Physics of Transition Metals, Ed. V.G. Baryakhtar (Naukova Dumka, Kiev, 1988).
K-P. Bohnen et al. /Surface Science 368 (1996) 222-225 [5] V.N. Antonov, V.Yu. Milman, V.V. Nemoshkal¢nko and A.V. Zhalko-Titarenko, Z. Phys. B Condens. matter 79 (1990) 223. [6] G. Witte, J.P. Toennies and Ch. Wodl, Surf. Sci. 323 (1995) 228. [7] C. Elsaesser, N. Takeuchi, K.M. Ho, C.T. Chan, P. Braun and M. Faehnle, J. Phys. Condens. Matter 2 (1990) 4371. [8] A. Eichler, J. Hafner, J. FurthmQller and G. Kresse, Surf. Sci. 346 (1996) 300. [9] G.B. Bachelet, D.R. Hamann and M. Schlueter, Phys. Rev. B 26 (1982) 4199.
225
[10] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [11] G. Kresse and J. Hafner, J. Phys. Condens. Matter 6 (1994) 8245. [12] K.P. Bohnen, G. Nolte, A. Eichler, J. Hafner and W. Reichardt, to be submitted. [13] Y. Chen, S.Y. Tong, K.P. Bohnen, T. Rodach and K.M. Ho, Phys. Rev. Lett. 70 (1993) 603. [14] J. Sch6chlin, K.P. Bohnen and K.M. Ho, Surf. Sci. 324 (1995) 114. [15] R.E. Allen, G.P. Allredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1661.