Abstract

1
First-principles calculations of Pt surface
phonon dispersion Sampyo Hong,
Talat S. Rahman (Department of Physics, Kansas
State University, Manhattan, KS 66506) Rolf
Heid, Klaus Peter Bohnen (Forschungszentrum
Karlsruhe, IFF, Karlsruhe, Germany) Email
likedew_at_phys.ksu.edu
Abstract We have calculated dispersion curves
for surface phonons of Pt(100), Pt(110) and
Pt(111), using first-principles, total energy
calculations based on a mixed-basis set and
norm-conserving pseudopotentials. Linear response
theory and the harmonic approximation of lattice
dynamics are also invoked. For bulk Pt our
calculations show the experimentally observed
anomaly along (110) direction, and the calculated
relaxations on the three surfaces are also in
agreement with previous results. The dispersion
of the Rayleigh wave and the resonance modes on
Pt(111) are in good agreement with data from He
scattering measurements. Our results of phonon
dispersion for Pt(100) and Pt(110) are for their
unreconstructed surfaces. The richness in the
phonon dispersion curves for the three surfaces
are compared and conclusions presented about the
changes in the surface force constants from the
bulk values. The propensity of two of the
surfaces to reconstruct is discussed. Results are
compared with those for Pd surfaces.
Abstract We have calculated dispersion curves
for surface phonons of Pt(100), Pt(110) and
Pt(111), using first-principles, total energy
calculations based on a mixed-basis set and
norm-conserving pseudopotentials. Linear response
theory and the harmonic approximation of lattice
dynamics are also invoked. For bulk Pt our
calculations show the experimentally observed
anomaly along (110) direction, and the calculated
Relaxations on the three surfaces are also in
agreement with previous results. The dispersion
of the Rayleigh wave and the Resonance modes on
Pt(111) are in good agreement with data from He
scattering measurements. Our results Of phonon
dispersion for Pt(100) and Pt(110) are for their
unreconstructed surfaces. The richness in the
phonon dispersion curves for the three surfaces
will be compared and conclusions presented about
the changes in the surface force constants from
the bulk values. The propensity of two of the
surfaces to reconstruct will also be discussed.
2

• Introduction
• The low Miller index surfaces of Pt have been
  studied extensively because of the widespread use
  of Pt as a catalyst and a need to understand the
  microscopic details of the bonds between surface
  atoms.
• Surface phonon dispersion curves reflect the
  dynamical behavior of atoms at the surface of the
  crystal. Particularly, they provide direct
  information on the forces acting between atoms in
  the surface layers. Such details are essential
  for understanding processes like adsorption,
  dissociation and eventually chemical reactions at
  surfaces. Additionally, at the fundamental level,
  Pt surfaces have been the subject of controversy
  about the possible existence of phonon anomalies
  1-4. There have been theoretical studies
  based on semi-empirical methods 2 force
  constant parameterization. As these methods have
  limited predictive power, their validity has to
  be checked by more accurate methods like ones
  based on density functional theory 5.
  Therefore, we have carried out first-principles
  calculations of the phonons of Pt surfaces using
  on density functional perturbation theory6 in
  the mixed-basis representation. A summary of our
  procedures and results is presented below.
• Some Interesting Observations
• Kohn anomaly in bulk Pt phonon dispersion
  curves 1, 2.
• Kohn-like anomaly in Pt(111) surface phonon
  dispersion 1-4.
• Existence of Longitudinal Resonance modes in
  Pt(111) 2, 8.
• Prediction of strong softening of lateral force
  constant in Pt(111)2.

3

• Reference
• 1 U. Harten, J. P. Toennies, C. Woell, and G.
  Zhang, Phys. Rev. Lett. 55, 2308(1985).
• 2 V. Bortolani, A. Franchini, G. Santoro, J. P.
  Toennies, Ch. Well, and G. Zhang, Phys. Rev. B
  40, 3524 (1989).
• 3 W. Di, Kevin E. Smith, and S. D. Kevan, Phys.
  Rev. B 45, 3652(1992).
• 4 K. Kern, R. David, R. L. Palmer, G. Comsa and
  T. S. Rahman, Phys. Rev. B 33, 4334(1986).
• 5 P. Hohenberg and W. Kohn, Phys. Rev. B 136,
  864(1964).
• 6 S. Baroni, S. de Gironcoli, A. Dal. Corso,
  and P. Giannozzi, Rev. Mod. Phys. 73, 515(2001).
• 7 J. P. Toennies, in Surface Phonons, Vol. 21
  of Springer Series in Surface Sciences, edited by
  W. Kresse and F. W. de Wette ( Springer-Verlag,
  Berlin,New York,1991), Chap. 5.
• 8 R. Heid, K. -P. Bohnen, to be published.
• 9 B. Meyer, C. Elsaesser, and M. Faehnle
  (unpublished).

4
Details of Theoretical Calculations Electronic
Structure Calculations were performed within a
pseudopotential approach to density-functional
theory(DFT) in the local-density
approximation(LDA)5. To calculate the total
energy and equilibrium structure of the ground
state of the system, a program developed by Meyer
et al9, based on a mixed-basis representation
of wave functions, was used. To represent
ion-electron interaction a norm-conserving
pseudopotential for Pt was used while for
electron-electron interaction in LDA, a
Hedin-Lundqvist form of the exchange-correlation
functional was employed. For the valence states
of Pt, d-type local functions at each Pt site,
smoothly cut off at a radius of 2.1 a.u., and
plane waves with a kinetic energy of 16.5 Ryd
were applied. Integration over an irreducible
Brillouin-zone were carried out using a number of
special kpoints (1242 points were found to be
sufficient depending on the systems). A Fermi
level smearing of 0.2 eV was also employed. For
simulating surfaces, supercells containing 911
layers(920 atoms) with inversion symmetry were
used. Structure relaxation was carried out by
calculating forces using Hellmann-Feynman theorem
until forces on all atoms were less than 0.001
Ryd/a.u. Experimental observations have shown
that Pt(100) and Pt(110) are unstable and
reconstruct into (5x1) superstructure and (1x2)
missing-row structure, respectively, while
Pt(111) does not reconstruct below 1300K. Our
calculations of structural relaxation which are
appropriate for very low temperatures, were
carried out for unreconstructed Pt(100) and
Pt(111), both unreconstructed and reconstructed
Pt(110).
5
Phonon Calculations were performed within a
density functional perturbation theory(DFPT)
approach 6 and by invoking the harmonic
approximation of lattice dynamics. The phonon
calculations were carried out with the fully
relaxed Pt(100), Pt(110) and Pt(111) slabs, as
described above. Dispersion curves were obtained
by interpolating of (4x4),(4x6) and (6x6) q-point
meshes of Pt(100), Pt(110) and Pt(111),
respectively. Surface force constants were
combined with bulk force constant by adding an
asymmetric bulk slab of 50 layers to obtain
projected bulk phonon modes. For bulk phonon
calculation (8x8x8) q-point mesh was used.
Surface modes were identified by a weight larger
than 20 in the first two layers. Experimentally
surface phonons are measured by inelastic
scattering techniques. The most widely used
methods are inelastic Helium Atom
Scattering(HAS)7 and Electron Energy loss
spectroscopy(EELS)7.
6
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7
Surface Relaxations
10 Lee at al, PRB 59, 1673(1999) 11 N.
Materer et al, Surf. Sci. 325, 207(1995)
8
Bulk phonon dispersion A weak Kohn anomaly is
observed in the calculated TA phonon branch along
?- ? direction. This anomaly comes from couplings
beyond second nearest neighbors.
Kohn Anomaly
9
Surface phonons
12 S. Hong, R. Heid, K.-P. Bohnen, and T. S.
Rahman, unpublished. 13 A. Wachter, K.-P.
Bohnen, and K. M. Ho, Surf. Sci. 346,
127(1996). 14 A. M. Lahee, J. P. Toennies, and
C. Woell, Surf. Sci. 191, 529 (1987).
10
For Ideal Pt(100) and Pt(110) Phonon dispersion
curves for ideal Pt(100) and Pt(110) show
anomalous softening of the projected bulk phonon
modes, which cause the surface modes to exist as
resonances. Large separation of the surface modes
from the bulk bands comes from strong softening
of surface force constants, up to -40 for
Pt(100) and -36 Pt(110). However, the large
softening may not be responsible for the
reconstruction of these surfaces. Both Pd(100)
and Pd(110) also show very similar softening of
surface phonons but they do not reconstruct8.
The richness of surface modes of Pt(110) is
interesting. Rayleigh waves for these surfaces
become unstable at long wavelengths, in our
calculations.

11
Pt(111) In our calculation the projected bulk
phonon modes along most of ?- ? and ?- M
directions are softened significantly to cause
the Rayleigh mode(S1) to become a resonance for
long wavelengths. Experiments show the existence
of two sets of modes which are assigned as
Rayleigh wave and longitudinal resonance. Our
calculations are at odds with these conclusions.
The figure below shows the theoretical surface
modes(black dots)8 and the HAS data(open
circles)2.
.
For comparison we present below the calculated
and measured phonon dispersion curve for Pd(111).

12
The figure below shows the theoretical surface
modes(black dots)8 and the HAS data(open
rectangles)14.
13

• Summary
• The weak Kohn anomaly in the transverse
  acoustic mode of bulk phonon dispersion has
  strong effect on the characteristics of the
  surface modes. For example, the softening of the
  bulk phonon spectrum causes the Rayleigh wave to
  immerse into the bulk spectrum and appear only as
  a surface resonance.
  
• The experimentally measured second surface mode
  (classified as longitudinal resonance) above the
  Rayleigh wave in the Pt(111) was not found in our
  calculation, raising questions about its origin.
• There are subtle differences between the
  calculated surface phonon dispersion curve of
  Pt(111) and Pd(111)
• Although Pt(100) and Pt(110) reconstruct, their
  surface phonon dispersion curves do not display
  any dynamical instability, implying that the
  reconstruction may not be phonon driven.