Multilayer Relaxation Research Articles

Structural relaxations of Cu vicinals

We have studied multilayer and atomic relaxations of vicinal (regularly stepped or high Miller index) surfaces of Cu with loosely and tightly packed step-edges, using many-body and semi-empirical interaction potential obtained from the embedded atom method. For all investigated surfaces, except for Cu(3 1 1), the calculated multilayer relaxations reveal non-uniform damping magnitudes in interlayer relaxations. Furthermore, we find that the vicinals with wider terrace width display quite different characteristics in the multilayer relaxations compared to the vicinals with narrower terrace width. We also investigate the local atomic relaxations around step and trace the atomic shifts to the changes in the associated local force fields, compared to those in the bulk.

Polarization in a solid body caused by nonuniform deformation

We propose a model of a solid body which takes into account interaction between polarization and deformation fields. We obtain the expression for the free-energy density for the model. Applying Hamilton's variational principle we have derived the corresponding governing equations. Our model describes the phenomenon of multilayer relaxation of crystal surfaces. In addition, we have obtained that the displacement field of a dilatation center has the damped oscillatory character in directions away from the defect.

Molecular‐dynamics simulations of surface and bulk properties of Zn, Cd, and ZnCd systems

AbstractSurface and bulk properties of Zn, Cd, and ZnCd systems have been investigated by performing molecular‐dynamics simulations using a recently developed empirical many‐body potential energy function for these systems, which comprices two‐ and three‐body atomic interactions. Surface reconstruction and multilayer relaxation on clean surfaces, adatom on surface, substitutional atom on surface and bulk materials, and vacancy on surface and bulk materials have been studied extensively. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Enhanced multilayer relaxation at high-index stepped Cu surfaces

Multilayer relaxation at high-index $\mathrm{Cu}(\mathrm{hkl})$ $(hkl=511,$ 320, and 410) stepped surfaces were determined by the first-principles all-electron full-potential linearized augmented plane-wave method within the framework of the local-density approximation and the generalized gradient approximation. The calculated relaxation of the interlayer distances, obtained by a geometry optimization procedure that minimizes the force on each atom, were compared with low-energy electron-diffraction (LEED) analysis of experimental data. In the case of Cu(511), the calculated results are in good agreement with the LEED analyses. On the other hand, for Cu(320) and Cu(410), there are large differences that may be understood from the fact that the LEED analyses of experiments consider up to only three or four layers from the surface, and that whereas even the fifth or sixth layers show large relaxation in our calculations, our results suggest a reanalysis of the LEED data with the inclusion of more layers.

Structure and energetics of clean and hydrogenated Ni surfaces and symmetrical tilt grain boundaries using the embedded-atom method

Multilayer relaxations, surface/interface energies, and hydrogen binding sites and energies were systematically studied using the embedded-atom method, for a series of the clean and hydrogenated Ni surfaces and symmetrical tilt Ni grain boundaries. The hydrogen binding energies were in the range of 2.7--2.9 eV at the surface sites while 2.1--2.6 eV at the grain-boundary sites, both being larger than at the fcc crystal interior site, 2.1 eV. These data are consistent with the conclusions from ab initio calculations that intergranular embrittlement is seen by hydrogen segregation at the Ni grain boundaries. It was found that multilayer relaxation of Ni plays a key role in trapping hydrogen at the grain boundaries.

Multilayer relaxations of (3 1 1), (3 3 1) and (2 1 0) fcc transition metal surfaces studied by pseudopotential DFT calculations

The multilayer relaxations of (3 1 1), (3 3 1) and (2 1 0) surfaces of Ni, Cu, Rh, Pd, Ag, Ir and Pt have been investigated by pseudopotential DFT calculations. Our calculations show a relaxation sequence of − +⋯ for the interlayer spacings of all (3 1 1) surfaces and − − + ⋯ for all (3 3 1) and (2 1 0) surfaces. We also found that the surfaces of Ir and Pt had more significant relaxations than other surfaces. Comparisons with LEED results, where possible, are also made in this paper.

Structural relaxations, vibrational dynamics and thermodynamics of vicinal surfaces

We present here a summary of the calculated structural, dynamical and thermodynamicalproperties of a number of vicinal surfaces of fcc metals with the aim of identifying trends intheir characteristics. In general, multilayer relaxations indicate a contraction in the bondsbetween surface atoms except for that of the least undercoordinated surface atom (‘corner’)whose bond length with the nearest neighbour bulk atom displays an impressive expansion.Electronic structure calculations show a rearrangement of charge densities near thesteps and the layer resolved density of states highlights the characteristics of theundercoordinated atoms. The frequencies of localized vibrational modes on steppedsurfaces point to both softening and stiffening of specific force constants, which lead toenhancement of modes in both the lower and higher ends of the frequency spectrum in thevibrational densities of states. Contributions to the surface vibrational entropy fromundercoordinated atoms are found to depend strongly on their atomic coordination andplay an important role in determining the step excess free energy for certain stepgeometries. Comparisons of results are made with available experimental data.

Multilayer relaxation of Cu(210) studied by layer-doubling LEED analysis and pseudopotential density functional theory calculations

Multilayer relaxation of Cu(210) surface has been studied by layer-doubling low energy electron diffraction (LEED) analysis and pseudopotential density functional theory (DFT) calculations. An excellent agreement between the calculated and measured $I\ensuremath{-}V$ curves has been achieved as judged by direct inspection and a small Pendry R factor of 0.12. We suggest that the layer-doubling method is a suitable choice for quantitative LEED structural studies on high-index metal surfaces with interlayer spacings down to 0.8 \AA{}. Our pseudopotential DFT calculations have reproduced the relaxation sequence determined by the layer-doubling LEED analysis, i.e., $\ensuremath{-}$ $\ensuremath{-}$ $+$ $\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot},$ with the largest quantitative discrepancy of about 0.04 \AA{}. Comparison is made with LEED and DFT studies on other high-index Cu surfaces. Based on this comparison, a general trend for multilayer relaxations of open metal surfaces is proposed.

Relative stability of missing-row reconstructed (1 1 0) surfaces of noble metals

The missing-row reconstruction has been observed experimentally on clean Au, Pt and Ir, while Cu and Ag are dormant cases in that the reconstruction can be provoked only by modest alkali coverage. The experimentally observed deconstruction and roughing transitions of Au(1 1 0) and Pt(1 1 0) are interesting realizations of two-dimensional Ising and XY models. The detailed understanding of the structures and energies of the missing-row configurations is also important for the constructions of many-atom model such as the embedded-atom potentials. We have carried out density-functional molecular-dynamics calculations for the missing-row reconstructed (1×2), (1×3), and (1×∞) surfaces. The calculation results show that the delocalization of d-electrons and the accumulation of sp-electrons near the surface associated with the multilayer relaxation are important factors in stabilizing the experimentally observed (1×2) missing-row structure for Au and Pt. For Ir(1 1 0), a faceted (3 3 1) surface is shown to be the lowest energy configuration, in agreement with experimental observation. These results shed light on the relative stability of the missing-row configurations, and are useful for the construction of improved semi-empirical potentials, especially for the long range interaction.

Local and excess vibrational free energies of stepped metal surfaces

We present a comparative study of the local and excess vibrational thermodynamic functions for the (511), (211), and (331) surfaces of Ag and Cu, in the harmonic approximation, employing interaction potentials based on the embedded atom method. Related thermodynamic functions of the corresponding (100) and (111) surfaces are also examined. For these surfaces, local contributions to the vibrational free energy for atoms in layers down to the fifth are found to deviate from that of the atoms in the bulk, as a result of complex multilayer relaxations. These contributions show a linear dependence on temperature, beyond 200 K, and a correlation with the local coordination but without a simple relationship. The vibrational contribution to the step excess free energy for vicinals of (111) is found to be significant.

Ab initiostudies of stepped Pd surfaces with and without S

Department of Physics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506~Received 10 July 2002; revised manuscript received 18 December 2002; published 28 April 2003!We have performed a comprehensive first-principles study of the electronic and geometric structure of a setof clean and S covered vicinal Pd surfaces of ~111! and ~110!, namely Pd~211!,Pd~331!,Pd~320!, and Pd~551!,each having three-atom wide terraces. The trends in the multilayer relaxation patterns can generally be ex-plained on the basis of charge smoothening which makes the vicinals of Pd~110! prone to larger and deeperrelaxations in comparison to the Pd~111! counterparts. A range of adsorption energies for preferred adsorptionsites for S is found on all four surfaces, with Pd~320! being the most receptive. We show that the adsorbate canhave considerable effect on multilayer relaxations and registry of the substrate atoms. We discuss the nature ofthe S–Pd bonding and how this bonding is affected by the surface geometry and coordination. We relate thepoisoning effect of S to the changes in the electronic structure of the substrate atoms.DOI: 10.1103/PhysRevB.67.165415 PACS number~s!: 68.43.2h, 68.47.De, 73.20.HbI. INTRODUCTION

Chemical termination of the CsCl-structure FeSi/Si(111) film surface and its multilayer relaxation

Metallic FeSi films, epitaxially stabilized on Si(111) in CsCl structure, are investigated experimentally by quantitative low-energy electron diffraction (LEED) and theoretically by total energy calculations using densityfunctional theory (DFT). Both methods show clearly that the surface is Si terminated. Additionally, LEED and DFT agree in retrieving an unusual multilayer relaxation of +6%, - 16%, and +14% from the top layer into the bulk for the first three layer spacings. This relaxation pattern is explained by an enhanced covalent bonding between the subsurface iron and silicon layers.

The phonon spectra and vibrational thermodynamical properties of Cu vicinal surfaces

We present a many-body empirical potential giving the total energy in Cu crystals, the transferability of which is proven by comparing with experiments the phonon band structure for the bulk and the three low index surfaces. After the determination of the multilayer relaxation of a set of vicinal surfaces, we investigate their vibrational spectra by means of their projected phonon band structure and the local spectral densities of states corresponding to a given value of k // which are discussed in relation with recent experiments. Finally, the root mean square displacement of specific atoms for low and high index surfaces are compared with the bulk one and the contribution of the vibrations to the free energy of steps is calculated as a function of temperature and of the terrace width for the p(1 0 0)×(1 1 1) and p(1 1 1)×(1 0 0) surfaces (2⩽ p⩽7). It is found that for these vicinal surfaces the step–step interactions mediated by phonons are attractive and may have the same order of magnitude as the other interactions.

Ab initiocalculations of multilayer relaxations of stepped Cu surfaces

We present trends in the multilayer relaxations of several vicinals of Cu(100) and Cu(111) of varying terrace widths and geometry. The electronic structure calculations are based on density functional theory in the local density approximation with norm-conserving, non-local pseudopotentials in the mixed basis representation. While relaxations continue for several layers, the major effect concentrates near the step and corner atoms. On all surfaces the step atoms contract inwards, in agreement with experimental findings. Additionally, the corner atoms move outwards and the atoms in the adjacent chain undergo large inward relaxation. Correspondingly, the largest contraction (4%) is in the bond length between the step atom and its bulk nearest neighbor (BNN), while that between the corner atom and BNN is somewhat enlarged. The surface atoms also display changes in registry of upto 1.5%. Our results are in general in good agreement with LEED data including the controversial case of Cu(511). Subtle differences are found with results obtained from semi-empirical potentials.

Multilayer relaxation of the Cu(2 1 0) surface

Low-energy electron diffraction (LEED) I– V has been utilized to determine the surface structure of Cu(2 1 0). The surface structure is found to exhibit multilayer relaxation following the trend − − + (− is contraction and + is expansion in the interlayer spacing). The magnitude of interplanar relaxation is found to be damped, where | Δd 12|>| Δd 23|>| Δd 34|, etc. This kind of behavior is quite different from that observed on Al(2 1 0) [Phys. Rev. B 38 (1988) 7913], where the magnitude of interplanar relaxation in the third interlayer is larger than that in the second interlayer (i.e., | Δd 23|<| Δd 34|). The difference of damped multilayer relaxation behavior of Cu(2 1 0) and Al(2 1 0) could be related to a charge density oscillation perpendicular to the surface.

Theoretical and computational study of the interlayer relaxation of surfaces and nanocrystals

The interlayer relaxation of crystal surfaces and nanocrystals is studied for a number of elements in the Periodic Table by using a tight-binding potential based on the second moment approximation. We find that the extent of relaxation strongly depends on the chemical species of elements and is correlated with the ratio of dimer bond-length to nearest-neighbor distances in solids. The oscillatory behaviour of multilayer relaxation is correlated with large relaxation of the interlayer distance between the first and second layers.

Multilayer relaxation of Cu(331)

Both quantitative low-energy electron diffraction (LEED) analysis and theoretical predictions based on the embedded-atom method suggested that Cu(331) has a sequence of interlayer spacing contractions and expansions dissimilar to that expected for a terrace with three atomic layers. To understand the anomalous behavior of Cu(331), we have performed a comparative study of Cu(331) and Cu(211) by using the first-principles full-potential linearized augmented plane-wave method within the generalized gradient approximation. For both surfaces, our calculations reproduce the multilayer relaxation sequences given by LEED measurements. The electronic structure analysis indicates that the seemingly anomalous multilayer relaxation in Cu(331) can also be understood in the charge smoothing picture. However, the trend of charge smoothing in such transition metal surfaces depends not only on how many atomic layers a terrace has, but also on the bond-length--bond-order interplays.

Structure and energetics of Cu(1 0 0) vicinal surfaces

The multilayer surface relaxations of the Cu(1 0 0) vicinals are investigated using a first-principles local density technique. In agreement with the low-energy electron diffraction data for (1 1 5) and (1 1 7) surfaces we find a large inward relaxation of the step atoms and an about −9% reduction of step heights rather independently of the terrace width. For the further layers we predict the interlayer contraction/expansion trends correctly but the calculated values are too high compared with experiments. From the extracted surface energies and step formation energies of (1 1 n) and (1 0 n) surfaces we conclude formation of distorted octahedral Cu islands on Cu(1 0 0) with the longer edges along the 〈1 1 0〉 direction.

Multilayer relaxation and magnetism of a high-index transition metal surface: Fe(310)

Structural, electronic, and magnetic properties of the Fe(310) surface are studied using first-principles full-potential linearized augmented plane wave method within the generalized gradient approximation. Sizable multilayer relaxation is found to extend to the seventh layer from the surface. While low-energy electron diffraction (LEED) and first-principles calculations on multilayer relaxations generally agree for low-index surfaces, there is a disagreement for this high-index surface. We conjecture that this disagreement might be due to the small data set and variational freedom in the LEED analysis. The spin magnetic moment of the Fe(310) surface and subsurface atoms is enhanced to $2.85{\ensuremath{\mu}}_{B}$ and $2.65{\ensuremath{\mu}}_{B},$ from a bulk value of $2.23{\ensuremath{\mu}}_{B}.$ The surface layer enhancement is smaller than that in Fe(100) and larger than that in Fe(111), although all three surfaces have the same coordination number. Subsurface layers are found to play an important role in the magnetization of the surface atoms in the case of an open surface, where the vacuum affects more atomic layers.

Quantitative determination of Cu(117) multilayer surface relaxations by LEED

The relaxation of the first seven atomic layers of Cu(117) was determined by quantitative low-energy electron diffraction (LEED). Intensity versus energy spectra, $I(E),$ were measured for the primary beam at near-normal incidence to the (001) terraces. The data collected cover a cumulative energy range of 3200 eV. Computation of $I(E)$ spectra was performed in the angular momentum representation by considering the surface as a single atomic slab. For the variation of parameters, tensor LEED was applied. The experimental spectra are well reproduced over the entire energy range even for regions of low intensity. The first four layer spacings relax in a way to smooth the surface corrugation, whereby the relative changes $\ensuremath{\Delta}{d/d}_{0}$ amount (from top) to $\ensuremath{-}13%,$ $\ensuremath{-}2.0%,$ $\ensuremath{-}10%,$ and $+7%.$ This leads to a reduction of the vertical distance between step and corner atoms by 0.13 \AA{}. Comparison with experimental results for Cu(115) shows that this modification of the step shape is rather independent of the terrace width. Comparison to theoretical results exhibits, however, some discrepancies with respect to both the expansion/contraction sequence and the amplitudes of the layer relaxations.