Surface Force Constants Research Articles

Scattering of surface waves by inhomogeneities in crystalline structures

In current scientific and technological scenarios, studies of transmittance of surface waves across structured interfaces have gained some wind amidst applications to metasurfaces, electronic edge-waves, crystal grain boundaries, etc. The results presented in the present article shed a light on the influence of material inhomogeneities on propagation of surface waves. Within the framework of classical mechanics, an analogue of the Gurtin–Murdoch model is employed where elastic properties on surface are assumed to be distinct from bulk. Restricting to scalar waves on prototype square lattice half-plane, particles on considered structured surface have piecewise-constant mass and surface force-constants across an interfacial point. Particles in bulk lattice interact with nearest neighbours in a way that involves unequal force-constants parallel to surface versus normal to it. A surface wave band exists for such lattice structure wherein the waveform decays exponentially inside the half-plane. A formula for surface wave transmittance is given based on an exact solution on half-plane, and, thus, previous work (Sharma & Eremeyev 2019 Int. J. Eng. Sci. 143 , 33–38 ( doi:10.1016/j.ijengsci.2019.06.007 )) is extended. An explicit expression for fraction of energy influx leaked via bulk waves is a highlight. Included are graphical results for several illustrative values of surface structure parameters.

Structural, elastic and magnetic studies of the as-synthesized Co1−xSrxFe2O4 nanoparticles

A series of as-synthesized Co1−xSrxFe2O4 nano-ferrites, 0.0 ≤ x ≤ 0.6, were prepared by the chemical co-precipitation method. The techniques used in this investigation were x-ray diffraction, transmission electron microscope, infrared spectrometer and vibrating sample magnetometer. The results proved that the samples have single phase cubic spinel structure. This study revealed that the crystallite size, strain and elastic wave velocities were decreased by increasing strontium content x, whereas the density, theoretical density, specific surface area, force constants, Debye temperature and elastic moduli were increased. The true lattice constant and porosity proved dependence on Sr2+ion content x. TEM images showed agglomeration of the nanoparticles, where the average particle size values ranged 10.56–23.66 nm which agree with R values. Six absorption bands were observed in the IR spectra and assigned to the corresponding site bonds. The saturation magnetization (Ms), coercivity, magnetic moment, anisotropy constant and remnant magnetization were affected by Sr2+ ion content x. The vibrational frequencies; ν1 and ν2 showed decrease with increasing Ms. On the other hand, Ms proved dependence on the crystallite size R, whereas the elastic wave velocities were affected by the sample density.

Interpreting atomic force microscopy measurements of hydrodynamic and surface forces with nonlinear parametric estimation

A nonlinear parameter estimation method has been developed to extract the separation-dependent surface force and cantilever spring constant from atomic force microscope data taken at different speeds for the interaction between a silica colloidal probe and plate in aqueous solution. The distinguishing feature of this approach is that it exploits information from the velocity dependence of the force-displacement data due to hydrodynamic interaction to provide an unbiased estimate of the functional form of the separation-dependent surface force. An assumed function for the surface force with unknown parameters is not required. In addition, the analysis also yields a consistent estimate of the in situ cantilever spring constant. In combination with data from static force measurements, this approach can further be used to quantify the extent of hydrodynamic slip.

Ab initio calculations of the dispersion of surface phonons of ac(2 × 2) CO overlayer on Ag(001)

We examine the phonon dispersion ofc(2 × 2)-CO on Ag(001) by applying density functional perturbation theory withthe generalized-gradient approximation. Our calculations indicate that thec(2 × 2)-CO overlayer on Ag(001) is dynamically stable. We find thatthe bond length of CO is expanded and its stretch mode (ν1) softenedby ∼ 9 meV upon adsorption on Ag(001), in excellent agreement with experiments. We show thatν1 at alone cannot gauge the metal–CO interaction since it is not entirelydetermined by the C–O intramolecular force constant. Further softening ofν1 on Ag(001) is obtained outside , indicative of CO–CO interactions even at a distance of ∼ 4 Å. The frequency of the Ag–CO stretch mode (ν2) is ∼ 30 meV and it is nearly dispersionless, implying that the perturbation correspondingto this mode is short-ranged. The frustrated rotation mode of CO (ν3) overlaps with the bulk band and mixes with substrate modes inside the SBZ,suggesting this as one of the key features for the enhanced diffusivity of CO onAg surfaces over that on Cu surfaces. The frustrated translation mode of CO (ν4) is everywherebelow ∼ 2.8 meV and therefore mixes with substrate modes in the region of the SBZ around . Depending on the q vector, vertical and in-plane surface modes may soften or stiffen with respect to their counterparton clean Ag(001). Although the response of most Ag(001) modes to CO adsorption is similarto that of corresponding Cu(001) modes, there are some contrasting features between thedynamics of the two surfaces concerning the changes in the surface force constants and mixing ofS6 withν3 on Ag(001).

Insights from calculated phonon dispersion curves for an overlayer of H on Pt(1 1 1)

We have calculated the dispersion curves of H vibrational modes on Pt(1 1 1), using first-principles, total energy calculations based on a mixed-basis set and norm-conserving pseudopotentials. Linear response theory and the harmonic approximation are invoked. For 1 ML coverage with H atoms in the fcc hollow sites, we find modes at 73.5 meV and 142.6 meV, polarized, respectively, parallel and perpendicular to the surface, at the Γ point of the surface Brillouin zone. The degeneracy of the parallel mode is lifted at the zone boundaries, yielding modes at 69.6 meV and 86.3 meV, at the M point, and at 79.4 meV and 80.8 meV, at the K point. The changes in Pt surface force constant on H adsorption are found to shift the frequencies of the substrate surface modes from their calculated values for clean Pt(1 1 1). We also consider the case of subsurface adsorption for 2 ML of H and present vibrational frequencies of H atoms adsorbed in several subsurface sites. The appearance of new vertically polarized H modes in the range 98–106 meV (octahedral site occupancy) and 124–162 meV (tetrahedral site occupancy) is discussed in the context of experimental data.

Anomalous surface lattice dynamics of a simple hexagonal close-packed surface

The surface lattice dynamics of Ru(001) has been investigated by a combined first-principles calculation and high-resolution electron-energy-loss spectroscopy. We find very anomalous dispersion curves of low-frequency surface modes and indications of strong softening of longitudinally polarized vibrations. They originate from large modifications of in-plane surface force constants unexpected for such a compact hexagonal surface.

Ab initio lattice dynamics of Ag( [formula omitted

The lattice dynamics of Ag(1 1 0) is investigated by performing ab initio calculations to obtain the bulk and surface interatomic force constants. The results from the ab initio results are used to parametrize a model potential, so as to obtain phonon frequencies and eigenvectors throughout the bulk and surface Brillouin zones. The surface force constants are modified significantly from the bulk values, with alternate interlayer force constants being stiffened and softened. The coupling between next-nearest-neighbor surface atoms is unstable, hinting at a (weak) tendency towards missing-row reconstruction. The calculated surface phonon spectra are in good agreement with experiment.

Surface lattice dynamics of Mg(0001)

A study of the surface lattice dynamics of $s\ensuremath{-}p$ bonded Mg(0001) using momentum-resolved inelastic electron scattering and ab initio calculations is reported. Three phonon modes at $\overline{K}$ and four phonon modes at $\overline{M}$ were observed, all in excellent agreement with calculations. In an attempt to understand the origin of changes in the surface force constants compared to the bulk force constants, the theoretical dispersion of the Rayleigh mode using bulk truncated and real surface force constants was compared for the hexagonal surface of three $s\ensuremath{-}p$ bonded metals, Al, Mg, and Be.

Density-functional calculation of phonons in adsorbate-covered semiconductor surfaces

We present the results from density-functional calculations of the phonon dispersion of adsorbate-covered semiconductor surfaces. The plane-wave method is used together with the slab-supercell description for the surfaces. We focus on vibrational states that characterize the chemisorption site of the adsorbed atoms and phonon modes of the interface. By comparing the vibrational states in the presence of the adsorbates with those of the clean surfaces, adsorption-induced changes of the surface geometry and force constants can be identified. We study the chemisorption of hydrogen, antimony, and group-III elements on the (110) surfaces of III–V compounds, as well as on the (001) and (111) surfaces of silicon.

Deformation of boron networks at the LaB 6(111) surface

The dispersion of the phonons of the LaB 6(111) surface has been measured and reproduced on the basis of lattice dynamical calculations. The surface force constants were found to be different from the bulk force constants. In particular, the force constant of an interoctahedral boron–boron bond at the surface increases by 50% from the corresponding bulk force constant. From the relationship between the force constant and bond length in various hexaborides, we have estimated the contraction of the interoctahedral bond length to be 0.007 nm. The deformation of the boron network on the (111) surface, the behavior of which is similar to that on the (100) surface, is explained by the inherent nature of the boron network.

Surface phonons of Si(111):H-(1 × 1) and Si(111):As-(1 × 1)

The full phonon dispersions of the fully relaxed (Si(111):H-(1 × 1) and Si(111):As-(1 × 1) surfaces are studied within a first-order density-functional perturbation scheme. The results are in excellent agreement with the experiment. It is shown that, despite of significant deviations in the surface force constants, most differences in the two phonon spectra are an effect of the different masses of the surface atoms. Furthermore, the Debye-Waller factors are calculated and discussed. For the atoms in the surface region, they exhibit a pronounced anisotropy. Convergence towards the bulk value is not reached until the eighth atomic layer.

Surface atomic structure and surface force constants: Cu(100)2√2 × √2-R45°-O and LaB 6(100)

Relations between surface force constants and bond lengths are discussed for two surfaces. First, on the oxygen-covered Cu(100) surface, we have obtained the force constants of the O-Cu bonds based on lattice dynamical analysis. The results are consistent with those for the chemisorption site of oxygen atoms in the missing row model proposed by Zeng et al. [Surf. Sci. 208 (1989) L7] using LEED intensity analysis. Concerning the relative position of the oxygen atoms against the topmost Cu atoms, however, there is a small difference between these two results. Second, the surface force constants of the B-B bonds in the LaB 6(100) surface suggest the deformation of the three-dimensional network of boron at the surface, for which no data have been reported so far.

Surface Phonon Dispersion Curves of NiAl(111)

The surface phonon dispersion curves of the NiAl(111) surface have heen measured along \\overlineΓK of the two dimensional Brillouin zone by means of high-resolution energy loss spectroscopy. The observed dispersion curves were compared with lattice dynamical calculations of a slab shaped crystal on the basis of a force constant model including up to the third nearest-neighbor interaction. Good agreement between the experimental and calculated curves was obtained for two different combinations of the surface force constants, which suggests the existence of two different models concerning the surface atomic relaxation.

Comparison of semiempirical and ab initio pseudopotential calculations of the lattice dynamics of aluminum surfaces with He-atom-scattering time-of-flight spectra.

We report a detailed comparison of two theoretical methods for the study of the surface lattice dynamics of sp-bonded aluminum and compare the results with He-atom time-of-flight spectra for the (001), (110), and (111) surfaces of this metal. One of the theoretical methods is a semiempirical Born--von Karman scheme in which the interatomic force constants are related to central pairwise forces and also to ``angle-dependent'' interactions between triplets of atoms. The other method is an ab initio scheme in which the interatomic force constants are calculated from the electron-density response function and second-order pseudopotential perturbation theory. Both schemes require the inclusion of long-range central forces (up to ten nearest neighbors) in order to obtain converged results. In addition, the semiempirical approach must include three-body forces up to second-nearest neighbors to allow for the breakdown of the Cauchy relation, which in the ab initio method arises as a direct consequence of the volume dependence of the effective pair potential. The semiempirical approach reproduces accurately the surface phonon-dispersion curves calculated with the ab initio scheme. The leading surface force constants obtained in both methods have similar values, which provides justification for the physical significance of the semiempirical force constants. Both methods reproduce, with great accuracy, the experimental dispersion relations for the Rayleigh surface phonons, except for the case of the (110) surface. Assuming that the He atom couples only to the ion cores, the reflection coefficients calculated in both methods reproduce well the intensities of the weak resonances observed in the He time-of-flight spectra below the longitudinal edge. These structures can be explained in terms of Fano resonances.

Intrinsic and extrinsic effects in surfaces: Acoustic-phonon softening of capped Si(111) surfaces.

Bond-charge-model calculations of phonon dispersion in the Si(111):As(1\ifmmode\times\else\texttimes\fi{}1) are compared with inelastic helium scattering measurements. The agreement is very good with no surface fitting of model parameters. This permits a novel means of assessing the adlayer-induced changes in the surface: We recalculate the phonon spectrum with the adlayer mass reduced from ${\mathit{m}}_{\mathrm{As}}$ to ${\mathit{m}}_{\mathrm{Si}}$ but with all other parameters unchanged. The resulting dispersion curves agree extremely well with measurements and calculations for a (1\ifmmode\times\else\texttimes\fi{}1) hydrogen-terminated Si(111) surface, demonstrating that the surface force constants are not appreciably altered by these chemisorption processes. Hence the acoustic phonon dispersion in the Si(111):As(1\ifmmode\times\else\texttimes\fi{}1) surface can be fully understood in terms of (1) the intrinsic perturbation of breaking the (111) covalent bond to create the surface and (2) the ``mass defect'' due to As-Si substitution.

Dynamical properties of the hydrogen-covered W(001) surface

Surface phonon dispersion curves have been measured for both high-symmetry directions of the W(001)/H(1 × 1) surface by He atom inelastic scattering. In addition to the Rayleigh mode an intense inelastic peak from a longitudinal surface mode was observed. The measured phonon dispersion curves are shown to be in good agreement with lattice dynamical calculations in which the bulk force constants up to second-nearest neighbors and force constants describing the interaction of the H atoms, both with surface atoms and among each other, are included. Thus on hydrogen adsorption a restoration of surface force constants to bulk values takes place, quite similar to what has been observed for Pt(111) and Si(111).

Vibrational properties of the clean reconstructed W(100) surface.

A force-constant model for the dynamical matrix of the ({radical}2 {times} {radical}2 ){ital R}45{degree} W(100) clean reconstructed surface has been constructed, based on all existing total-energy results for this surface. Such results are available for five distortions around the reconstructed geometry. The dynamical matrix is used to study the vibrational properties of the surface, including the surface modes and their dispersions, the vibrational correlation functions, and the mean squared displacements. The surface is found to exhibit a number of characteristic modes and resonances. These include three strong surface features whose polarizations are mainly perpendicular to one another and which are folded back upon reconstruction, and also a subsurface mode above the bulk bands. The qualitative features of many of these modes and resonances can be understood on the basis of simple, qualitative properties of this surface (such as zone folding), but the values of the frequencies and the dispersions depend on details of the surface force constants. The total-energy results are found to constrain strongly the frequencies and dispersions of the surface modes. The surface mean squared displacements and correlation functions exhibit interesting anisotropies, which are related to the surface modes. The dependence of the surface mode frequencies and eigenvectorsmore » on the force constants is described, and the implications of the present results for electron-energy-loss spectroscopy and He-atom scattering are discussed.« less

Mean-square displacements of surface atoms: c(2 x 2) overlayers on Ni(100).

We present lattice-dynamical calculations of the mean-square displacements of the adsorbate and first-layer substrate atoms for c(2\ifmmode\times\else\texttimes\fi{}2) S and O overlayers on Ni(100). To be consistent with available structural information and to reproduce the observed surface-phonon dispersion data on these systems, surface force constants between Ni atoms are chosen to be different from the value in the bulk. It is found that for sulfur atoms the mean-square displacement in the plane parallel to the surface is larger than that along the surface normal. The opposite is the case for the oxygen atoms. The underlying Ni atoms in the two systems also display dissimilar anisotropies in the three orthogonal directions. We have examined the relative strengths of the mean-square relative displacements between the adsorbate and the first- and second-layer substrate atoms for both systems and explored the effect of the variation of the surface force constants on these quantities. These results have been brought in contact with available surface extended x-ray-absorption fine-structure data on these systems.

Surface phonon spectroscopy of Ni(111) studied by inelastic electron scattering.

We report the experimental determination and theoretical analysis of the surface phonon dispersion relations for clean Ni(111) along the \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{}M\ifmmode\bar\else\textasciimacron\fi{} symmetry direction. The surface phonon spectra were obtained with a high-resolution electron energy loss spectrometer operating in the off-specular impact scattering regime. Kinematic conditions were varied in order to selectively examine the Rayleigh mode and ``gap'' mode, as well as contributions from bulk phonons. Comparison of the experimental surface phonon dispersion relations and inelastic scattering cross sections with lattice dynamical and quantum multiple scattering calculations demonstrate that tensile surface stress is present at the level of 1.6\ifmmode\pm\else\textpm\fi{}0.2 N/m, and that the intraplanar surface force constant is (11\ifmmode\pm\else\textpm\fi{}3)% softer than in bulk nickel.

First principles surface phonon calculations

Advances in computer speeds and computational techniques have enabled us to calculate total energies and forces at crystalline surfaces with a high degree of accuracy from first principles self-consistent calculations. These techniques have been used to study the geometries and surface force constants on the low-indexed faces of Al, Na, Ag, Cu and the (1 × 2) reconstructed (missing-row) Au(110) surface. The results provide detailed information on the frequencies and polarizations of surface phonon modes and resonances which can be used in the interpretation of experimental data.