Structure and dynamics at the Pd(100) surface

surface science i

ELSEVIER

Surface Science 346 (1996) 127-135

Structure and dynamics at the Pd(100) surface A. W a c h t e r a, K.P. B o h n e n a,,, K . M . H o b a Forschungszentrum Karlsruhe, lnstitutffir Nukleare FestkOrperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany b Ames Laboratory, US Department of Energy and Department of Physics, Iowa State University, Ames, IA 50011, USA Received l I May 1995; accepted for publication 14 September 1995

Abstract

Using first-principles total-energy calculations the lattice relaxation, surface energy, work function and surface phonons have been determined for the Pd(100) surface. Calculation of forces allows for a very efficient determination of equilibrium geometries and interplanar force constants. Results will be presented and compared with available experimental information as well as with other theoretical treatments.

Keywords: Density functional calculations; Low index single crystal surfaces; Palladium; Phonons; Surface electronic phenomena; Surface energy; Surface relaxation and reconstruction; Work function

1. Introduction

A basic question in surface science is the determination of structural and dynamical properties of surfaces. The location of atoms at the surface and the knowledge of surface vibrations belong to the most important surface information necessary for the understanding of a wide class of surface phenomena. Because of the missing neighbor atoms in the topmost layer, the electronic density at the surface differs from the one in the bulk. The total energy of the system is no longer minimal and so the atoms reorder in such a way that the forces occurring disappear. In most cases this leads to relaxations or reconstructions. These changes in geometry lead to significant effects on the physical properties of surfaces: for example the interplanar force constants change. This has important consequences for the surface phonon spectrum. Surface * Corresponding author. 0039-6028/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0039-6028 (95)04937-0

vibrations are involved in many processes on surfaces. A detailed knowledge of the surface phonon spectrum is essential in studies of surface diffusion, phase transitions and desorption processes. Low energy electron diffraction (LEED) [ 1] and ion scattering [ 2] experiments are common experimental methods to examine geometric structures of surfaces while electron e~lergy loss spectroscopy (EELS) [3] and inelastic helium beam scattering experiments [4] provide information about the surface phonon dispersion curves. On the theoretical side over the past few years substantial progress has also been made in determining lattice relaxation, reconstruction and surface phonons for metals from first principles. It is well known that first-principles total-energy calculations in the framework of local-density approximation (LDA) [5] are a useful method for calculating structural and dynamical properties of a large variety of bulk and surface materials [6]. Our studies of the Pd(100) surface as well as

A. Wachter et aL /Surface Science 346 (1996) 127-135

128

energy calculations. The use of the pseudopotential concept in the framework of density functional theory has been applied to a large variety of bulk systems and is also suitable for surfaces• Although most pseudopotential calculations have been done using a plane wave basis, this is not necessary• The transition metals are systems with strongly localized orbitals. Therefore it is helpful to choose a mixed basis consisting of plane waves and localized orbitals [ 11 ]. We chose an energy cut-off for the plane waves of 16.5 Ry. The decays of the functions to describe the localized orbitals can be seen in Fig. 1. To simulate the surface, a periodic slab geometry as shown in Fig. 2 consisting of seven bulk layers and a vacuum region of five bulk layers was used. The top layers in this periodic slab do not interact and the inner layers behave like those in the bulk. For a given trial geometry the total energy of the system is calculated within the local-density formalism [5] using the Hedin-Lundqvist interpolation formula [ 12] for the electronic exchange and correlation energy• The norm conserving pseudopotential [ 13] in the present calculation has been used in previous calculations of the bulk structural properties [ 14]. In this work we used the scalar

other theoretical treatments [7] show, as expected, a small inward relaxation of the first layer. The effects in the deeper layers are nearly zero. These results differ from LEED data of Quinn et al. [8] who predicted an expansion of the top interlayer spacing• This could probably be a consequence of adsorbed elements such as hydrogen, for example, or a consequence of ferromagnetism of the surface layer. In contrast to the Cu(100) and Ag(100) surface structures [6] so far no agreement between experiment and theory has been obtained• The calculation of interplanar force constants allowed the determination of surface phonon modes at the high symmetry points F, X and 1VI of the surface Brillouin zone (SBZ). As a result of the modified force constants the frequencies of surface modes deviate from those obtained with bulk force constants• These surface phonons also disagree with experimental data [9]. This is also in contrast to the cases of Cu(100) and Ag(100) [10].

2. Method of calculation The determination of the properties for the Pd(100) surface are based on first-principles total0.9

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A. Wachter et aL ISurfate Science 346 (1996) 127-135

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until the total energy differences are stable to within 5x 10 - 6 Ry. In order to accelerate the convergence to the self-consistent solution a modified Broyden scheme [ 17-1 is used. In the past the determination of phonon modes in the bulk has mostly been carried out using the 'frozen phonon' method [18]. A straightforward application to surfaces is not possible due to the fact that even at a high symmetry point in the SBZ the vibration pattern of the surface modes is not completely determined by symmetry. The decay of modes into the interior is not known. To avoid any ambiguity we calculated the intra- and interplanar force constant matrices for our siab at the high symmetry points F, X and M of the SBZ. This could be achieved by distorting the equilibrium geometry in a way corresponding to the respective high symmetry point. The differences in forces between the equilibrium geometry and the distorted geometry allow calculation of the force constant matrices, which enable us to determine the surface phonon spectrum at the respective high symmetry point by diagonalizing the dynamical matrix [ 19]. n

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3. Bulk properties Fig. 2. Periodic slab geometry of Pd(100) consisting of seven layers of atoms. The vacuum has a width of five Pd layers.

relativistic pseudopotential discussed in Ref. [ 14]. The sampling grid for the surface calculations of Pd(100) was a uniform mesh shifted away from the r point. To take into account that the states near the Fermi energy are only partially occupied and to avoid the use of a great number of k points to describe the Fermi surface, a Gaussian smearing scheme has been applied. This method is very useful as previous calculations have shown. In this work we used a smearing constant of 0.2 eV. For the influence of this k point smearing see Ref. [ 15]. Beside the total energy it is very helpful to calculate the forces on each atomic layer. While this is an easy task for a plane wave basis, it is by no means simple within a mixed basis description. Details can be found in Ref. [ 16]. In our calculation, self-consistency is reached

In order to check the quality of the pseudopotential, the equilibrium lattice constant, bulk modulus, elastic constants, cohesive energy and the phonon dispersion curves in the (100) direction have been determined and compared with experimental [20-23] and other theoretical data [ 14]. We found 3.88 A ,"or the equilibrium lattice constant, 2.08 Mbar tor the bulk modulus (via elastic constants) and 4.63 eV for the cohesive energy, in agreement with the mentioned references. For more detailed comparison see Table 1. In addition, the electronic band structure was calculated and found to be in good agreement with results obtained with other treatments [24,25]. To determine the phonon dispersion in the (100) direction it was necessary to calculate the interplanar force constants for the longitudinal and transversal branches. This was achieved by calculating the forces for the equilibrium geometry and several distorted situations. Having calculated the

130

A. Wachter et al./Surface Science 346 (1996) 127-135

Table 1 Calculated bulk properties in comparison with experimental data Parameter

Own calculation

Experimental

ao (,A) E¢oh(eV) B (Mbar)

3.88 4.63 2.08

3.89 [23] 3.89 [23] 1.955 [21]

1.93 [22]

10t2 cm2] 2.32

2.3412 [21]

2.26 [22]

dyn'~ 1.96 ct2 lOt ~m2]

1.7614 [21]

1.76 [22]

(10t2 dyn'~ cm2] 0.60

0.7117 [21]

0.717[22]

ctt

C44

( \

interplanar force constants, the longitudinal and transversal branches could be found by a Born-yon K~irm/m analysis. The result can be seen in Fig. 3. We obtained these results taking into consideration the interaction with the 3. neighboring plane. Beside the possibility of determining interplanar force constants by microscopic calculations, they also can be calculated via atomic force constants derived from fitting neutron scattering data [20]. The dispersion curves determined in the two different ways are shown in Fig. 3 and are compared with neutron scattering measurements at 120 K [20]. The calculated curves are in good agreement with the available experimental data and show that the bulk can be described well within the pseudopotential [ 14].

4. Equilibrium geometry The first step when calculating surface properties is to determine the equilibrium situation. The results of our calculations for relaxation of Pd(100) are given in Table 2. We obtained these results starting from ideal unrelaxed geometry and deducTable 2 The calculated data for the relaxation of the Pd(100) surface Adt2 (%)

Ad23 (%)

Ad34 (%)

-

~ 0.0

~

1.20

0.0

ing the equilibrium geometry via a force constant matrix. The relaxed geometry is given by Ad12= - 1 . 2 % of the interlayer spacing and Ad23= Ad34 ~0%, where Ado is the change in interlayer separation between layers i and j. We only find relaxation effects in the topmost layer, the effects in inner layers are nearly zero. While these results are in agreement with other theoretical studies [7] they are in serious disagreement with experimental LEED data [8]. Quinn et al. [81 measured an expansion of the top interlayer spacing from about Adt2 = +3% and a contraction of Ad23= -- 1% of the second layer. This differs seriously from measured or calculated properties of most fcc (100) metal surfaces. Having found the equilibrium geometry we calculated the work function and the surface energy which is the energy associated with the creation of the surface. The result obtained for the work function in comparison with another theoretical treatment and experimental data is given in Table 3. The determined 5.80 eV is in acceptable agreement with the measured [26] and the calculated [7] value. For the surface energy we find 1.006 eV/atom in comparison to another theoretical value of 0.89 eV/atom [7].

5. Surface phonons Beside the structural examination, the studies of the lattice dynamics also play an important role in surface science. It is well known that the discrete phonon dispersion curves in the bulk become broad bands when projected onto the surface reciprocal lattice. In this work we considered a slab consisting of 50 atomic layers. Solving the matrix equation the vibrational modes in dependence of Table 3 Comparison of calculated data for work function (in eV) of the Pd(100) surface with another theoretical result [7] and photoemission data [26] Pd (100)

Reference

5.30 5.55_+0.1 5.80

[72 [26] This work

131

A. Wachter et aL /Surface Science 346 (1996) 127-135

their wave vector in the surface Brillouin zone could be found. The calculation was carried out using bulk force constants which were determined by fitting inelastic neutron scattering data [20] by an axially symmetric Born-von Kfirmfin model [27]. Figure4 shows the modes obtained along

the directions FX, XM and 1VIF. At the X point, three discrete modes are remarkable: for the Rayleigh mode, which is the lowest mode and transversally polarized, we obtained 2.1 THz. The next mode with 2.7 THz is longitudinally polarized and for the gap mode we found a value of 5.7 THz

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132

A. Wachter et al./Surface Science 346 (1996) 127-135

Pd(100)-surface p h o n o n s p e c t r u m

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133

A. Wachter et al./Surface Science 346 (1996) 127-135

(see also Table 5). At the 1VI point we obtain 2.8 THz for the lowest mode which is now polarized perpendicularly to the surface. In a second step we determined the intra- and interplanar force constants coupling the surface layers by microscopic calculations. The microscopic force constants are determined via the calculation of force differences. The two geometries used for these calculations are always the equilibrium geometry and a situation where the whole layer of atoms has been moved. The nonequilibrium geometries and the unit cells considered are shown in Fig. 5. In order to eliminate the anharmonic part, the magnitude of the distortions was chosen as 1-2% of the bulk interlayer spacing. In addition, the distortions perpendicular

to the surface are done inward and outward to eliminate the anharmonic part. As shown in Fig. 5, the unit cells for the high symmetry points and lVl have to be doubled in order to realize the respective vibration pattern. The calculated interplanar force matrices in comparison with the bulk parameters are given in Table 4. At the point the changes of force constants parallel to the surface are very small while those perpendicular to the surface are < 10%. At the X and M points, force constant elements which involve the surface normal are also stiffer than in the bulk, however, for the other elements such a general statement cannot be made. For the X point it seems that the asymmetry between the x and y directions is being reduced, while for the M point

Table 4 Force constant matrices {in N/m) at the high symmetry points F, X and lVl describing the coupling between the topmost layers; these interplanar force constants are linear combinations of the interatomic force constants [27]; ~t and [3 refer to Cartesian coordinates; force constants from microscopic calculation are compared with those of the bulk parameter model; axes are chosen as

x = (011), y= (011), and z= (100) Bulk parameters

Microscopic calculation

r point

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0 36.81 0

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0 208.77 0

0 1 0 80.81

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0 143.74 0

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0 0 0

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I 193.76 0 0 42.15 0

0 0

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° 1

134

A. Wachter et al./Surface Science 346 (1996) 127-135

Table 5 Phonon frequencies and polarization at the Pd(100) surface from microscopic calculation; for comparison, results from the bulk parameter model are included as well as experimental data I-9]

1~1

Experimental frequency (THz)

Microscopic calculation

Bulk parameters

Frequency (THz)

Polarizaton vector first layer

Frequency (THz)

Polarization vector first layer

2.3 2.9 3.1 5.3 3.3

(0.00, 0.99, 0.00) (0.00, 0.00, 0.84) (0.58, 0.00, 0.00) (0.71, 0.00, 0.00) (0.00, 0.00, 0.97)

2.1 2.7

(0.00, 0.99, 0.00) {0.00, 0.00, 0.92)

2.7

5.7 2.8

(0.77, 0.00, 0.00) (0.00, 0.00, 0~99)

2.8

the in-plane elements, ~ 1~, ~ = x , y are substantially reduced. Details can be seen in Table 4. Using the microscopically calculated surface force constant matrix elements together with matrices determined with the bulk atomic force constants for the inner layers, we obtained the surface modes given in Table 5. For comparison also the results using bulk atomic force constants up to the surface are indicated. The polarization vector in the outermost layer is also given. At the X point we find a frequency of 2.3 THz for the transversally polarized Rayleigh mode. This mode has not been determined until now. Compared with the value of 2.1 THz obtained using bulk parameters this is an upward shift of 0.2 THz. Similarly the longitudinally polarized mode is shifted by 0.2 THz to a value of 2.9 THz when calculated with microscopic surface force constants. The experimental value (EELS) is 2.7 THz [9]. At the lVl point we find the Rayleigh mode lying at 3.3 THz while the calculated value using bulk parameters was 2.8 THz. For comparison with EELS measurements see Table 5. A detailed comparison of calculated and measured EELS spectra will be presented in a future publication [28]. The frequencies obtained by using the surface force constants are higher than those of the experiment while the results obtained using bulk atomic force constants are in good agreement with the experimentally observed modes.

6. Summary We have used first-principles self-consistent total energy calculations to investigate the surface struc-

ture and surface phonon spectrum of the Pd(100) surface. Agreement between experiment and theory was not obtained. This was the case for the structural LEED data [8] as well as for the phonon results [28 ]. Generally, microscopically calculated lattice relaxations as well as surface phonons for fcc (100) surfaces agree well with experimental results [10,16]. This supports the hypothesis that the observed discrepancy for the Pd(100) surface is due to surface contamination (probably mostly hydrogen), for example, or may be a consequence of ferromagnetism of the surface layer, which has not been considered in this work.

References [1"1 G. Ertl and J. Kiippers, Low Energy Electrons and Surface Chemistry, 2nd ed. (VCH, Weinheim, 1985) p. 201. [2] K.H. Rieder, in: Topics in Current Physics, Vol. 41, W. Schommers and P. yon Blankenhagen, Eds. (Springer, New York, 1986) p. 17; C. Varlas, in: Topics in Current Physics, Vol. 41, W. Schommers and P. yon Blankenhagen, Eds. (Springer, New York, 1986) p. 111; J. Als-Nielsen, in: Topics in Current Physics, Vol. 43, W. Schommers and P. yon Blankenhagen, Eds. (Springer, New York, 1987) p. 181. [31 H. lbach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982); H. lbach and T.S. Rahman, in: Chemistry and Physics of Solid Surfaces, R. Vanselow and R. Howe, Eds. (Springer, New York, 1984). [-4] J.P. Toennies, J. Vac. Sci. Technol. A5 (1987) 440; in: Proc. Solvay Conf. on Surface Science, F. de Wette, Ed. ( Springer, Heidelberg, 1988). 1"5.1 P. Hohenberg and W. Kohn, Phys. Rev. B 136 {1964) 864; W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133.

A. Wachter et al./Surface Science 346 (1996) 127-135

135

[6] K.P. Bohnen and K.M. Ho, Surf. Sci. Rep. 19 (1993) 99. [7] M. Methfessel, D. Henning and M. Scheflter, Phys. Rev.

[18] K.M. Ho, C.L. Fu and B.N. Harmon, Phys. Rev. B 29

B 46 (1992) 4816. J. Quinn, Y.S. Li, D. Tian, H. Li and F. Jona, Phys. Rev. B 42 (1990) 11348. L. Chen and L.L. Kesmodel, Surf. Sci. 320 (1995) 105; private communication. S.Y. Tong, Y. Chen, K.P. Bohnen, T. Rodach and K.M. Ho, Surf. Rev. Lett. 1 (1994) 97. S.G. Louie, K.M. Ho and M.L. Cohen, Phys. Rev. B 19

[19] F.W. de Wette and G.P. Alldredge, in: Methods in Comp.

(1979) 1774.

[23]

[8] [9] [10] [11]

(1984) 1575, and references therein.

[20] [21]

[22]

[12] L. Hedin, B.I. Lundqvist, J. Phys. C: Solid State Phys. 4 (1971) 2064.

[24]

[13] D.R. Hamann, M. Schliiter and C. Chiang, Phys. Rev. Lett. 43 (1979) 1494.

[25]

[14] C. Els~isser, N. Takeuchi, K.M. Ho, C.T. Chan, P. Braun and M. Ffihnle, J. Phys.: Condensed Matter 2 (1990) 4371.

[15] K.M. Ho and K.P. Bohnen, Phys. Rev. B 32 (1985) 3446. [16] K.M. Ho, C. Els~isser, C.T. Chan and M. F/ihnle, J. Phys.: Condensed Matter 4 (1992) 5189.

[26] [272

[17] C.G. Broyden, Math. Comput.

19 (1965) 577; D. Vanderbilt and S.G. Louie, Phys. Rev. B 30 (1984) 6118.

[28]

Phys., Vol. 15, Lattice Dynamics of Surfaces of Solids (New York, 1976) p. 163. Landolt-B6rnstein: Zahlenwerte aus Natur und Technik, Band 13a (Heidelberg, 1981). T. Rayne, Phys. Rev. 118 (1960) 1545. S. Allard, Int. Tables of Selected Constants, Vol. 16 (Pergamon, Oxford, 1969). C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1975). D.A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum, New York, 1986). V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, Oxford, 1978). J. H61zl and F.K. Schulte, Work Function of Metals, in: Solid Surface Physics 85 (Springer, Heidelberg, 1979). B.N. Brockhouse, T. Arase, G. Gaglioti, K.R. Rao and A.D.B. Woods, Phys. Rev. 128 (1962) 1099. L. Chen, L.L. Kesmodel, K.P. Bohnen and A. Wachter, to be published.