Vibrations of small cobalt clusters on low-index surfaces of copper : Tight-binding simulations

S. D. Borisova,1 S. V. Eremeev,1 G. G. Rusina,1 V. S. Stepanyuk,2 P. Bruno,2,3 and E. V. Chulkov4,5,* 1Institute of Strength Physics and Materials Science, pr. Academicheskii 2/1, 634021, Tomsk, Russia 2Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany 3European Synchrotron Radiation Facility, BP 220, F38043 Grenoble Cedex, France 4Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal 4, 20018 San Sebastián/Donostia, Basque Country, Spain 5Departamento de Física de Materiales and Centro Mixto CSIC-UPV/EHU, Facultad de Ciencias Químicas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Apartado 1072, 20080 San Sebastián/Donostia, Basque Country, Spain Received 2 April 2008; revised manuscript received 24 June 2008; published 20 August 2008

Despite important role that vibrational properties of small clusters ͑including a single adatom͒ play in understanding of many phenomena such as surface reactions, desorption, adsorbate diffusion, island and film growth, lubrication, and friction 28 significantly less is known about vibrations of metal clusters on metal substrates.This can be accounted for by two reasons.Experimental study of small cluster vibrations requires the use of local technique such as IETS. 13,29owever, until now, the IETS measurements have been done only for some molecules on metals ͑for references see, for instance Ref. 13͒.Theoretical study requires ab initio calculations or evaluations based on accurate interatomic interaction potentials and very big unit cells.The latter factor makes ab initio calculations very time consuming.In the present work we study vibrational properties ͑frequencies, polarizations, and lifetimes͒ of small Co clusters ͑dimer and trimer͒ on low-index Cu surfaces by using many-body interatomic potentials developed by Levanov et al. 30 and Stepanyuk et al. 31 within tight-binding second moment approximation ͑TB-SMA͒.We also present the calculation results for a single Co atom on the Cu surfaces.
The Co/Cu system has been chosen as an important model system since such phenomena as diffusion, growth, and structure were extensively investigated both experimentally and theoretically.Kief and Egelhoff 32 have reported the observation of Co film growth on low-index Cu surfaces, characterized by the formation of compact Co clusters.Using DFT calculations, Pentcheva et al. analyzed possible equilibrium structures for a monolayer coverage of Co on Cu͑001͒. 33They found that a flat Co film on the top of the Cu͑001͒ surface should be a stable structure, whereas the preferred structure was found to be a bilayer Co island covered by a Cu monolayer.However, since island growth is a nonequilibrium process, the surface crystal structure is determined by the kinetic processes.The Co/Cu surface exchange can result in a variety of structures, such as a Co/Cu surface alloy, 34 Cu-covered Co islands, 32 and Co-covered Cu islands. 33,35B-SMA simulations for Co islands and steps on Cu͑111͒ revealed that adsorbate-induced strain in the substrate makes surface diffusion of Co adatom anisotropic: the migration barrier exhibits strong oscillations near the island edges. 36y using accelerated molecular dynamics 37 and TB potentials 30 it was found that upward-transport mechanism at island edges is responsible for the bilayer Co island formation at low temperatures.At high temperatures diffusion of Co into the surface produces qualitative changes in the island structure, leading to the reversal of the low-temperature transport mechanisms and resulting in a monolayer growth.It was found that small Co clusters ͑dimers, trimers, and heptamers͒ significantly affect the low-coverage morphology due to high translational and rotational mobility of the clusters. 38he dynamics of a single Co adatom during lateral manipulation on a copper ͑111͒ surface by a low-temperature scanning tunneling microscope was investigated by Stroscio and Celotta 39 and later on was studied by Liu and Gao. 40It was shown that the frustrated translation ͑FT͒ mode is responsible for the lateral hoping.The calculated frequency shows good agreement with the onset energy for the adatom hopping induced by inelastic tunneling.The detailed study of the FT mode of a single Co adatom on Cu͑111͒ has been carried out in Ref. 41.
Here we give a detailed analysis of horizontal ͑FT͒ and vertical vibrational modes, including lifetimes, for small Co adclusters on Cu͑111͒, Cu͑110͒, and Cu͑001͒.We show that frequencies of these modes, their intensities, and coupling with the substrate atoms strongly depends on the Cu substrate orientation.
The paper is organized as follows.In Sec.II we give a short description of the computational method used.In Sec.III, first, we analyze in details the equilibrium atomic structure of the Cu surfaces with the Co adclusters and the Co-Cu bonding on the Cu substrates since it plays a key role in vibrations of the adsorbed clusters.Then the comparative analysis of vibrations of a single Co adatom, dimer, and trimer on low-index Cu surfaces is carried out.Finally, conclusions are given in Sec.IV.

II. COMPUTATIONAL METHOD
The present calculations of crystal structure and vibrations of small Co clusters on the low-index Cu surfaces are based on interatomic interaction potentials Cu-Cu, Cu-Co, and Co-Co obtained in the second moment tight-binding approximation. 42In this approximation the total energy of a system is given as a sum of band energy ͑electronic cohesive energy or the attractive term͒, E B , and a repulsive energy, E R , of the constituent atoms Here summation is carried out over all positions of atoms of a system.The band energy term represents many-body interatomic interactions the repulsive term is described by a pair modified Born-Mayer repulsive potential

͑4͒
In Eqs.͑2͒-͑4͒ r ij is the distance between atoms i and j, ␣ and ␤ denote different atomic species, r 0 ␣␤ is the first neighbor distance in bulk pure metals ͑␣ = ␤͒ and is an adjustable parameter for the ␣ ␤ case, ␣␤ is an effective hopping integral for the first neighbor atoms.Parameters q ␣␤ and p ␣␤ describe the decay of the interatomic interaction strength.The parameters for the Cu-Cu interaction had been fitted to experimental data for bulk Cu only. 42The Cu-Co and Co-Co potentials were fitted to KKR density functional in the local spin-density approximation calculation results obtained for binding energy of small supported Co clusters on Cu͑001͒ and two Co impurities in bulk Cu. 30,43 Previous studies 37,[44][45][46] have demonstrated that the combination of ab initio and the tight-binding methods allows one to construct many-body potentials which provide reliable description of interatomic interactions, and atomic relaxations in low-dimensional systems.Very recent experimental work has confirmed the scenario of atomic relaxations in Co clusters on Cu͑001͒ predicted by our approach. 47,48o determine the equilibrium atomic structure, we relax the surfaces with adsorbates using standard moleculardynamic technique based on the TB interaction potentials described above.To simulate a semi-infinite Cu metal with a single adatom, dimer, and trimer of Co on the metal surface a thin-film model of a two-dimensional periodic slab consisting of 31 atomic layers of Cu͑111͒ or Cu͑001͒ with supercells ͑3 ϫ 3͒, ͑4 ϫ 4͒, and ͑5 ϫ 5͒ to exclude direct interactions adatom-adatom, dimer-dimer, and trimer-trimer is used.For Cu͑110͒ we use ͑3 ϫ 4͒, ͑3 ϫ 5͒, and ͑3 ϫ 6͒ supercells.For large supercells the calculations have also been carried out for 25 layer film thickness.The chosen thicknesses prevent interactions between two opposite surfaces of the film.The calculations of vibrational spectra are carried out by the dynamical matrix method.Diagonalizing the matrix gives the eigenfrequencies and the polarization vectors of vibrations.The local vibration densities of states were obtained by projecting these eigenmodes onto atoms of interest in a given ͑x + y or z͒ direction.The lifetimes of vibrational modes are calculated by fitting the peaks of the local density of vibrational states by a Lorentzian function.

A. Atomic structure
First, we briefly discuss atomic structure of unsupported ͑free standing͒ small Co clusters ͑dimer and trimer͒ and clean low-index Cu surfaces ͑111͒, ͑001͒, and ͑110͒.The calculated equilibrium bond length ͑distance between Co at-oms͒ of the free dimer is found to be of 2.23 Å.This result is in agreement with the experimental dimer bond length of 2.31 Å ͑Ref.49͒ and slightly differs from the densityfunctional theory ͑DFT͒ calculation values of 1.92-1.96Å ͑Refs.50 and 51͒ as well as from a tight-binding Hubbard Hamiltonian ͑TBHH͒ calculation result of 2.47 Å. 52 The evaluated binding energy of the dimer, 1.43 eV/atom, agrees well with that of 1.45 eV/atom obtained from the DFT calculation. 50Both these values are in between experimental data of 1.73 ͑Ref.49͒ and 1.32 eV. 53The latter energy is the upper limit for the dimer dissociation energy derived from collision-induced dissociation experiments. 53he most stable form of the free Co trimer obtained from our calculation is an equilateral triangle with the bond length of 2.33 Å which is slightly longer than the calculated dimer bond length.The calculated free chain of three Co atoms has bond length of 2.27 Å which is shorter than that in the triangle cluster but slightly longer than the dimer bond length.The calculation shows that the chain lies 0.30 eV/atom higher than the equilibrium triangle.As in the case of the Co dimer our bond length is slightly longer than that obtained from DFT calculations ͓2.06-2.25 Å ͑Refs.50 and 51͔͒ and shorter compared to the TBHH result ͓2.55 and 2.69 Å ͑Ref.
52͔͒.The calculated binding energy of 1.89 eV/atom is in fairly good agreement with the DFT calculation by Datta et al., 50 1.78 eV/atom, and with ab initio calculation by Castro et al., 51 1.53-3.67eV/atom.The equilateral triangle configuration of the Co cluster agrees with both ab initio and TBHH calculation results for Fe 3 and Ni 3 clusters which also form an equilateral triangle. 51,52However, these calculations give an isosceles triangle as ground state for the Co 3 cluster with small difference in the sides length.Castro et al. 51 obtained a triangle whose two equal sides are shorter than the third side; while in the calculation by Datta et al. 50two equal sides are longer than the third one.These distinct results can be accounted for by a very small difference in total energy between different cluster geometries.For instance, our totalenergy calculation gives an isosceles triangle with small difference in the sides length whose energy is only few meV/ atom higher than the equilateral triangle energy.This little inaccuracy of the Co-Co potentials does not affect the Co cluster structure and vibrations on Cu surfaces since the cluster geometry on the substrate is strongly affected by the Cu-Co interactions ͑see Figs.1-3͒.As we show below ͑subsection B, Vibrations͒ the most important consequence for the free standing Co 3 concerning the distinction of the isosceles triangle from the equilateral one is the slight splitting of the lower vibrational mode in two ones.
The evaluated equilibrium atomic structure of the clean Cu͑111͒, Cu͑001͒, and Cu͑110͒ surfaces slightly differs from the ideal one by small alterations of the first and second interlayer spacings.On Cu͑111͒, both interlayer spacings are contracted ͑inward relaxation͒ with the relative values ⌬ 12 = −0.90% and ⌬ 23 = −0.14%which are in excellent agreement with available experimental and theoretical data.The relative contractions of the interlayer distances measured by using medium-energy ion scattering are ⌬ 12 = −1.0%Ϯ 0.4% and ⌬ 23 = −0.2%Ϯ 0.4%. 54The value of the first interlayer contraction obtained from first-principles calculations 55 is −0.9%.On Cu͑001͒, the obtained interlayer spacings are also contracted and, compared to the bulk interlayer spacing, they are: ⌬ 12 = −1.20%and ⌬ 23 = −0.37%.On this surface, even the third interlayer spacing is slightly contracted, ⌬ 34 = −0.05%.These values are well compared to embedded atom model calculation results: ⌬ 12 is −1.21% ͑Ref.56͒ and −1.44%, 57 ⌬ 23 = −0.33%. 57The obtained ⌬ 12 is also in fairly good agreement with experimental results, ⌬ 12 = −1.1% ͑lowenergy electron-diffraction ͑LEED͒, 58 ⌬ 12 = −1.1% ͑MEIS͒, 59 as well as with ab initio value ⌬ 12 = −3.0%͑Ref.8͒ while the experiments give the small expansion of the second interlayer spacing.On Cu͑110͒, the contraction of the topmost interlayer spacing is bigger than that on Cu͑111͒ and Cu͑001͒ due to lower packing of this surface; while the contraction of the second interlayer spacing is smaller than that in the previous cases.Relative contractions found, ⌬ 12 = −2.90% and ⌬ 23 = −0.02%,are smaller than those obtained by using high-energy ion-scattering ͑HEIS͒ ͑Ref.60͒ ͑⌬ 12 = −5.3% and ⌬ 23 = 3.3%͒ and LEED ͑Ref.61͒ ͑⌬ 12 = −7.9% and ⌬ 23 = 2.4%͒.Despite some discrepancies in the layer relaxation with available experimental results, the Cu-Cu potentials give surface phonon modes in good agreement with helium atom scattering and electron energy-loss spectroscopies data ͑see subsection B, Vibrations͒ To simulate a single Co adatom on the Cu͑111͒ surface we used ͑3 ϫ 3͒ and ͑4 ϫ 4͒ unit cells that correspond to coverages ⌰ =1/ 9 ML and ⌰ =1/ 16 ML, respectively ͑see Fig. 1͑a͒ where the ͑3 ϫ 3͒ unit cell is shown͒.A Co adatom was placed at fcc hollow site as it is slightly more preferable than hcp one.The equilibrium distance d Co-Cu between the Co adatom and the nearest-neighbor substrate atom was found to be 2.458 Å. Relaxation results in rearrangement of the substrate atoms, especially in the close vicinity of the adatom.Thus, the Cu atoms which are the first nearest neigh-bors ͑NN͒ to the adatom ͓atoms 1, 2 and 4 in Fig. 1͑a͔͒ have two times larger inward relaxation than the clean Cu͑111͒ surface atoms have.At the same time in the plane they move away from the adatom by 0.012 Å.An atom 5 which is the second-nearest neighbor to the Co adatom shows the outward relaxation and in the plane it moves toward the adatom position by 0.015 Å. Atoms 3 and 7 are the third NNs to Co, they do not show the in-plane displacements and their perpendicular relaxation is almost the same as on the clean surface.
In Fig. 1͑b͒ the simplest adsorbate cluster, dimer, on Cu͑111͒ is shown.The two Co atoms were initially placed at the nearest fcc hollow sites on Cu͑111͒ in the ͑4 ϫ 4͒ unit cell.Such a big unit cell prevents direct interaction between two dimers in any direction.After relaxation the dimer length was found to be 2.295 Å that is 8.5% shorter than the NN distance in bulk Co.The shortening of the dimer length by 0.263 Å moves atoms of the dimer from the ideal threefold hollow positions, however, the relaxation of the NN substrate atoms accommodates them to restore approximately threefold coordination of the adsorbate atoms.In this case the first NN atoms can be classified into two groups.The first one is formed by atom 2 which is NN to both Co adatoms ͑NN-2Co͒ and the second group is formed by atoms 1, 3, 5, and 6 which have only one NN Co adatom ͑NN-1Co͒.The d Co-Cu distance between Co and the NN-2Co atom has been found to be of 2.470 Å and the distance between Co and NN-1Co atoms has been obtained to be of 2.509 Å.Similar to the case of a single adatom, the second NN atoms move toward the adatom positions by 0.021 Å.
To simulate a cluster of three Co atoms the ͑4 ϫ 4͒ unit cell has been used.We have also checked the obtained results for the ͑5 ϫ 5͒ unit cell and found essentially the same results.The Co atoms have initially been placed at three fcc sites on Cu͑111͒ in two different geometries: a triangle and a chain.The molecular-dynamics relaxation of a trimer leads to an equilateral triangle as the most stable configuration ͓Fig.1͑c͔͒ with the bond length of 2.368 Å that is shorter than the initial one but 0.039 Å longer than that in the free cluster.The shortening of bond length in comparison with its unrelaxed value is provided by displacements of Co adatoms from the initial positions in the direction to the center of gravity of the cluster by 0.110 Å.The alteration of the Co positions is accompanied by displacements of the first and second NN atoms of the substrate.Like to the previous case there are two kinds of the first NN atoms: NN-2Co ͑atoms 2, 5, 6͒ and NN-1Co ͑atoms 1, 3, 9͒.The in-plane displacements toward the center of the triangle are 0.004 and 0.002 Å for the first and second group atoms, respectively.The vertical displacements of these atoms are −0.069 and 0.002 Å, respectively, from their relaxed clean surface positions.As a result the d Co-Cu distance for NN-2Co and NN-1Co atoms has been found to be 2.496 and 2.536 Å, respectively.The relaxation of the second NN atoms has been found to be similar to that in the previous cases.
For a single Co adatom on Cu͑001͒ the equilibrium distance d Co-Cu between the Co adatom and nearest-neighbor substrate atoms has been obtained equal to 2.476 Å that is slightly longer than the corresponding d Co-Cu at Cu͑111͒.The first nearest neighbors to the Co adatom ͓atoms 1, 2, 4 and 5 in Fig. 2͑a͔͒ show inward ͑vertical͒ relaxation which is bigger than that of the clean Cu͑001͒ surface by a factor of 1.5.In plane, these atoms move away from the adatom by 0.018 Å. Atoms 3, 6, 7 and 8 ͑the second NNs to the Co adatom͒ undergo inward relaxation which is slightly smaller than that for the surface layer atoms at the clean surface.Contrary to the case of a single Co adatom on Cu͑111͒ where the second NN atoms move toward the adatom, on Cu͑001͒ these atoms do not undergo lateral displacements.The third neighbor atom ͑9͒ undergoes displacements qualitatively similar to those of the clean Cu͑001͒ surface atoms.
On Cu͑001͒ we analyzed two possible dimer geometries.The first one consist of two Co adatoms placed at the nearest-neighbor fourfold hollow sites and the second dimer was placed at the second-nearest-neighbor hollow sites.The first dimer has been found energetically more favorable by 0.791 eV ͓see Fig. 2͑b͔͒.The equilibrium dimer length is of 2.326 Å that is 1.4% longer than the Co dimer length on Cu͑111͒.Similar to the case of a dimer on Cu͑111͒ the first NN atoms can be separated in two groups.The first group is of atoms 2 and 6 which are NNs to both Co adatoms ͑NN-2Co͒ and the second one is of atoms 1, 3, 5, and 7 which have only one NN Co adatom ͑NN-1Co͒.The equilibrium distance d Co-Cu between the Co adatom and the NN-2Co atom is of 2.483 Å while the distance between Co and NN-1Co atoms is of 2.528 Å, i.e., one can see the same trend as for the Co dimer on Cu͑111͒ where the NN-2Co atom is located closer to Co than the NN-1Co one.The NN-2Co atoms undergo significant relaxation displacements.In plane, they move away by 0.044 Å in the direction perpendicular to the dimer and show inward relaxation which is 2.7 larger than that on clean Cu͑001͒.The NN-1Co atoms undergo only minor lateral and vertical relaxation.
To simulate a trimer on Cu͑001͒ we considered two different geometries.The first one was a chain of Co adatoms and the second trimer, shown in Fig. 2͑c͒, was placed as a triangle.After relaxation the triangle configuration of adatoms has been found more stable by 0.987 eV than the chain.Contrary to the case of a trimer on Cu͑111͒, where adatoms form an equilateral triangle, a triangle on Cu͑001͒ is isosceles, i.e., it has two different lengths.The equilibrium lengths d Co 1 −Co 2 ͑Co 2 −Co 3 ͒ and d Co 1 −Co 3 were found to be of 2.328 and 2.794 Å, respectively.As one can see the short length is nearly equal to the dimer length.The long length which is equal to the Cu lattice constant before relaxation becomes 0.823 Å shorter after relaxation.Thus, initially right triangle shape cluster tends to be isosceles triangle one.This trend conflicts with the symmetry of the substrate and result in a complicated relaxation of the nearest-neighbor substrate atoms.Contrary to the case of a trimer on Cu͑111͒, where only two kinds of the NN   3͑a͔͒ show small inward relaxation ͑−0.003Å͒.In plane, these atoms move away from the adatom by 0.018 Å. Atoms 3, 4, 7, and 8 ͑the second-nearest neighbors to the Co adatom͒ undergo small outward relaxation ͑0.007 Å͒.The third ͑9, 10͒ and fourth ͑11, 12͒ neighbor atoms undergo displacements qualitatively similar to those on the clean Cu͑110͒ surface.On Cu͑110͒ we analyzed two possible dimer orientations: the first one along the ͓1 ¯10͔ direction ͓in Fig. 3͑b͒ it corresponds to the x axis͔ and the second one along the ͓001͔ direction ͓in Fig. 3͑b͒ it corresponds to the y axis͔.We have found that the first orientation is lower in energy by 0.827 eV.The equilibrium dimer length is of 2.329 Å that is close to the dimer bond length on Cu͑001͒.The shortest Co-Cu bond length, 2.476 Å, between the adatom and the underlying Cu second layer atoms is slightly longer than that in the single adatom case.As before, the first layer substrate atoms can be separated in two groups.The first group is of atoms 2 and 7 which are NN to both Co adatoms ͑NN-2Co͒ and the second group is of atoms 1, 3, 6, and 8 which have only one NN Co adatom ͑NN-1Co͒.The equilibrium distance d Co-Cu between the Co adatom and the NN-2Co atom is of 2.505 Å while the distance between Co and NN-1Co atoms is of 2.543 Å.The NN-2Co atoms undergo significant lateral and vertical relaxation displacements while the NN-1Co atoms undergo only minor relaxations due to the Co dimer adsorption.
To simulate a trimer on Cu͑110͒ we considered two different geometries.The first one was a chain of Co adatoms along ͓1 ¯10͔ and the second one was a triangle with the legs along the ͓1 ¯10͔ and ͓001͔ axes.After relaxation the chain geometry of adatoms has been found to be more stable by 0.943 eV than the triangle one.This clearly shows that the Co 3 cluster on Cu͑110͒ forms a structure distinct from that of free Co 3 ͓as well as from Co 3 on Cu͑111͒ and on Cu͑001͔͒.This is a consequence of the Co-Cu interactions which can produce the trimer vibrations very distinct from those of free Co 3 .
Similar to the dimer case the shortest distance d Co-Cu has been found between the Co atoms and the second layer Cu atoms lying beneath the adatoms.The equilibrium distances between central Co atom of the chain and its underlying Cu atom is 2.473 Å while the Co-Cu distances in the case of chain end atoms are equal to 2.483 Å, that is the end chain atoms are shifted from the fourfold hollow positions to the center of the chain and lie slightly higher than the central atom.Following the previous cases we will denote the first layer Cu atoms, which are closest to the adatoms, as the NN atoms in spite of that the real NN atoms are the Cu underlying ones.Of the nearest neighbors from the first layer atoms it is possible to pick out those which have only one closest Co adatom ͑NN-1Co͒ and those which have two NN Co adatoms ͑NN-2Co͒.Here the NN-2Co are atoms 2, 3, 8, and 9.They situated at a distance of 2.546 Å from the central Co atom of the chain and at 2.486 Å from the chain end atoms.Atoms 1, 4, 7, and 10 which are the NN-1Co ones have the Cu-Co bond length of 2.556 Å.

Clean Cu low-index surfaces
In Fig. 4 we show the calculated local density of states ͑LDOS͒ for the surface layer of the clean Cu͑111͒, Cu͑001͒, and Cu͑110͒ slabs in comparison with the bulk Cu.The z-direction and xy-plane resolved LDOS are plotted.8][9] The prominent peak at 13.65 meV on Cu͑111͒ corresponds to the Rayleigh mode.The obtained vibrational energies for the Rayleigh mode at the high-symmetry points M ¯͑13.52 meV͒ and K ¯͑14.84 meV͒ agree well both with experimental data 62,63 ͓13.3 ͑Refs.62 and 63͒ and 14.0 ͑Ref.63͒ meV, respectively͔ and ab initio results ͑13.5 and 15.3 meV͒. 8The in-plane polarized vibration peak at 27 meV corresponds to the shear-horizontally polarized surface mode at the M ¯point.This frequency is in a good agreement with ab initio calculation results ͑27.7 and 27.3 meV, for a review see Ref. 8͒.On Cu͑001͒ more features are seen than on Cu͑111͒.The lowest peak on Cu͑001͒ corresponds to the shear horizontally polarized mode S 1 with a frequency of 9.10 meV at the X ¯point.This frequency value agrees well with ab initio calculation results, 9.1-9.9meV. 8he highest frequency corresponds to the in-plane polarized mode S 6 at X ¯.The frequency value of this mode, 26.14 meV is in fairly good agreement with ab initio calculation data, 23.8-26.1 meV. 8The vertically polarized lower frequency mode is the Rayleigh wave at X ¯with a frequency of 12.57 meV which is close to ab initio results, 12.8-14.0meV ͑Ref.8͒ and EELS value 13.4 meV. 64The higher frequency z-polarized mode peak is the contribution from the Rayleigh mode at the M ¯point.For the ͑110͒ surface our results reproduce the dispersion of RW mode, the X ¯gap state and the ⌫ resonance frequencies measured by using HAS. 65The frequencies of all the obtained surface states are also in a good agreement with ab initio 8 and embedded atom method ͑EAM͒ ͑Ref.7͒ calculation results.
Summarizing the obtained results one can note that the semiempiric potentials used in the present work provide a good description of surface vibrations on the low-index Cu surfaces.In Fig. 5 the calculated local density of vibrational states is presented for a single Co adsorbate, Cu atoms ͑1, 2, and 4͒, nearest neighbors ͑Cu s NN͒ to the adatom, and for rest Cu atoms ͑Cu s others͒ of the surface layer.Three peaks at 5.38, 31.34, and 36.98 meV originated from the adsorbate vibrations are clearly seen in the figure.The lower frequency peak, ͑a͒, as well as the peak b, corresponds to vibrations of the Co adatom in the plane parallel to the surface.The highest frequency peak, ͑c͒, is determined by vertical vibrations of the adsorbate.All these adsorbate modes are coupled to the Cu s NN atoms motion causing complicated vibrations of these atoms ͑see Fig. 6 and the middle panel of Fig. 5͒ in the vertical and x , y directions simultaneously.One can note that the vibrations of the Cu s second and third NN atoms are mostly coupled to the clean Cu surface vibrational modes that is illustrated in the lower panel of Fig. 5.This demonstrates a short-range effect of the Co adatom interaction with the Cu substrate.
Vibrations of a single Co adatom on Cu͑111͒ have recently been studied by Liu and Gao 40,41 by using the VASP ab-initio code and semiempiric many-body potentials.The frequencies and polarizations obtained by Liu and Gao are in excellent agreement with our results except the peak b which was missed in Refs.40 and 41.Taking into account that Liu and Gao used in their calculations 41 the similar many-body potentials, 30 we attribute the discrepancy with our results to different details of the calculations.In particular, Liu and Gao used an 11-layer Cu substrate instead of 31-layer film we use.Another possible reason for missing the peak b might be the use not sufficiently big number of k points for the LDOS calculation.The final width of the Co LDOS peaks a, b, and c is caused by coupling of these modes to the substrate vibrations.The respective lifetimes are of 0.63, 0.68, and 1.47 ps for the a, b, and c Co vibrational modes, respectively.The longer lifetime of the c mode is accounted for by location of this mode above the bulk Cu spectrum.This location leads to coupling of the Co adatom motion to vibrations of the Cu atoms from the surface layer only while the modes a and b are coupled to motion of both the surface and bulk Cu atoms.

A single Co adatom on the Cu(001) surface
In Fig. 7 we show the calculated local density of vibrational states for a single Co adatom, for the first NN Cu atoms ͑atoms 1, 2, 4, and 5͒, and the rest Cu atoms ͑Cu s other͒ of the Cu͑001͒ surface layer.As one can see there are three sharp peaks in the Co originated LDOS.The lower one ͑a͒, similar to the case of Co on Cu͑111͒, corresponds to the in-plane motion of Co.Nevertheless, there are two important differences.The frequency of this vibration ͑9.72 meV͒ is significantly higher and the peak width is wider than that on Cu͑111͒.Both these features are naturally explained by stronger interaction of the Co atom with the Cu s NN atoms in comparison with the case of Co/Cu͑111͒.However, the complex structure of the peak reflects non-negligible interactions with the other Cu s atoms of the surface layer.The second peak ͑b͒ also corresponds to the in-plane vibrations of adatom.This vibration with a frequency 31.85 meV is similar to that at 31.34 meV for Co on Cu͑111͒.In contrast to the peak ͑a͒, the peak ͑b͒ is narrower due to interaction of Co mainly with the Cu s NN atoms of the surface layer.The third peak ͑c͒ is the vertically polarized vibration of the Co atom.Due to strong interaction of Co with the Cu s NN atoms, the latter is involved in the complex vibration in the z and x , y directions.The frequency of this mode, 34.33 meV, is approximately 3 meV smaller than that for a single Co adatom on Cu͑111͒.However, despite this decrease the c mode frequency is still higher than the entire bulk Cu spectrum.This position explains the calculated long lifetime, 1.43 ps, of the vertical Co mode which is very close to that for the c mode of Co on Cu͑111͒.The obtained lifetime value of the b mode, 0.68 ps, is equal ͑accidentally͒ to that of the b mode for Co/Cu͑111͒.In contrast, the lifetime of the a mode ͑central peak of the a complex structure͒, 0.31 ps, is shorter by a factor of 2 than that of the a mode for Co on Cu͑111͒.The very short lifetime of the latter mode is accounted for by the interaction of the Co vibration with surface and bulk Cu atoms vibrations ͑qualitatively it is easily seen from the overlap of the a mode frequency with the bulk Cu phonon spec-trum͒.

A single Co adatom on the Cu(110) surface
In Fig. 8 the calculated local density of vibrational states is presented for a single Co adatom, for the substrate second layer atom, ͑Cu 13 ͒, lying beneath the adatom, and for the first NN Cu surface layer atoms ͑atoms 1, 2, 5 and 6͒.As one can see, like for the adatom on Cu͑001͒, there are three sharp peaks in the Co originated LDOS.Peaks ͑a͒ at 6.33 meV and ͑b͒ at 33.63 meV corresponds to the in-plane motion of the Co atom.The polarizations of these vibrations are quite similar to those for Co/Cu͑001͒, however, the frequencies are shifted in comparison with Co/Cu͑001͒ due to anisotropy of the adatom-substrate interaction in the x and y directions on the ͑110͒ surface.The third peak ͑c͒ at 36.11 meV is the vertically polarized vibration of the Co atom.This mode has the higher frequency than the corresponding z-polarized vibration of Co on Cu͑001͒ and is provided by interaction with distinct substrate atoms.Contrary to the Co/Cu͑001͒ case where the z peak is originated from the interaction of adatom with the Cu first layer NN atoms, for Co/Cu͑001͒ it is the result of strong interaction of Co with the second layer Cu atom lying beneath the adatom.
The lifetime of the a mode has been estimated as 0.55 ps that is slightly distinct from the corresponding low-frequency mode a on Cu͑111͒ and Cu͑001͒.The broadening of the b mode is significantly smaller on Cu͑110͒ due to the position FIG.7. Local density of vibrational states for a Co adatom on the Cu͑001͒ surface, for nearest neighbors from the substrate first layer atoms ͑Cu s NN͒, and for second and third nearest-neighbor atoms from the first Cu layer ͑Cu s others͒. of this mode above the entire bulk Cu spectrum.The respective lifetime has been found to be of 1.48 ps.The calculated lifetime of the z-polarized mode c, 1.59 ps, is comparable with that for a single Co adatom on Cu͑111͒ and Cu͑001͒.

Free Co clusters
In this subsection, before discussing vibrations of the Co dimer and trimer on the Cu surfaces, we briefly analyze vibrations of the unsupported ͑free͒ Co clusters.A free dimer has 6 degrees of freedom, of them only one corresponds to vibrational, optical-like, mode with atomic motion along the dimer axis.In equilibrium, the free Co dimer vibrates with a frequency of 36.48 meV.This result is in excellent agreement with photoelectron spectroscopy data 34.74Ϯ2.50 meV and 36 meV obtained by Leopold and Lineberger 66 and by DiLella et al., 67 respectively.Ab initio calculations of Co 2 vibrations 51 give values in the range 52-55 meV which are 50% higher than our computation frequency while a lower vibrational energy of 29.3 meV is obtained from the TBHH evaluation. 52 free equilateral triangle shape trimer has 9 degrees of freedom, of which only three correspond to vibrational modes.The highest frequency mode, 36.36 meV, of Co 3 is symmetric ͑optical-like͒ mode with a fixed center of gravity at the center of equilateral triangle.As one can see the energy of this vibration just slightly lower of that for the stretch mode of the dimer.The lower, degenerate ͑because of C 3v symmetry͒ frequency mode of 25.77 meV is characterized by deformational and antisymmetric vibrations of the cluster.We have also computed vibrations of the isosceles shape Co 3 clusters which are of C 2v symmetry with two long ͑2.36 Å͒ and one short ͑2.26 Å͒ sides as well as with two short ͑2.30Å͒ and one long ͑2.40 Å͒ sides.The difference of 0.1 Å between the length of the long and short sides we used as typically obtained from ab initio calculations.The reduction of symmetry from C 3v to C 2v leads to slight increase to Ϸ38 meV of the higher frequency to slight splitting of the lower degenerate mode, Ϸ22 and 28 meV.
The energies of vibrational modes of Co 3 were calculated by using ab initio51 and TBHH ͑Ref.52͒ theories.The energy of the symmetric stretch mode was found by Castro et al. 51 to be of 49.6 and 45.2 meV in local spin-density approximation ͑LSDA͒ and generalized-gradient approimation ͑GGA͒ approaches, respectively, while it was obtained to be of 28.1 meV in Ref. 52.As in the case of Co 2 these data, respectively, overestimate 51 and underestimate 52 our stretch mode energy.The same trend is for two lower energy vibrations.
A free chain of three atoms has four vibrational modes.Two degenerated deformational modes have frequencies close to zero.The highest frequency mode, 39.98 meV, of the Co chain trimer is antisymmetric one.The lower frequency mode of 25.34 meV is characterized by symmetric vibrations with a fixed center of gravity.Both these energies are quite close to those of the respective vibrations of both equilateral and isosceles triangles of Co.

A Co dimer on Cu(111)
In Fig. 9 we show the calculated local density of states for the Co dimer atoms and for the first Cu layer NN atoms.As was mentioned before, these NN atoms can be classified in two groups.The first group consists of one atom only ͓atom number 2 in Fig. 1͑b͔͒ which has two Co adatoms as the nearest neighbors ͑NN-2Co͒ with d Co-Cu = 2.470 Å.The second group is of atoms 1, 3, 5, and 6.Each of these atoms has only one nearest Co atom ͑NN-1Co͒ with d Co-Cu = 2.509 Å. Contrary to the single adatom case, LDOS for the Co dimer on Cu͑111͒ shows two low-frequency peaks which correspond to vibrations of the cobalt atoms in the plane parallel to the surface.The first peak, ͑a͒, at 4.47 meV is a degenerate vibrational state with two kinds of vibrations ͑a1 and a2 in Fig. 10͒.The a2 vibration is the frustrated rotation mode that corresponds to rotatory vibration of the dimer relative to its center of mass, and the a1 vibration is characterized by FIG. 8. Local density of vibrational states for a Co adatom on the Cu͑110͒ surface, for nearest neighbors from the substrate first layer atoms ͑Cu s NN͒ and for the second layer atom which lies beneath the adatom ͑the lower panel͒.displacements of the entire dimer in the direction perpendicular to the dimer axis ͑frustrated translation mode͒.The second peak, ͑b͒, at 8.56 meV corresponds to the entire Co dimer vibration along the dimer bond.As follows from the figure ͑see the middle and lower panels͒ all these Co vibrations only slightly involve the NN atoms.The third ͑c͒ and fourth ͑d͒ peaks of the Co LDOS at 34.00 and 38.22 meV, respectively, correspond to mixed vibrations of the Co adatoms in the direction perpendicular to the surface and along the dimer bond.The degenerate vibrational mode ͓peak ͑c͔͒ includes two kinds of the dimer motion, c1 and c2.The c1 mode is a rotary vibration of the dimer in the plane perpendicular to the surface.Modes c2 and d result from the stretch vibration of the free dimer, however, due to the interaction with the substrate they have an additional vertical component of motion and distinct frequencies because of different bond-ing of the Co adatoms with the NN-2Co and NN-1Co atoms.In contrast to the a and b modes, the Co atoms motion in the c and d modes strongly couples to the NN substrate atoms motion.In particular, in the c2-mode the Co atoms are mainly coupled to the NN-1Co atoms motion, while in the d mode, the Co atoms are mostly coupled to the NN-2Co substrate atom motion.
The fitting of the Co LDOS peaks by the Lorentzian function gives lifetimes of 0.93, 0.23, 1.32, and 1.49 ps for the a, b, c, and d modes, respectively.The lifetime of the dimer horizontal a mode is the longest one compared to the respective a mode of a single adatom of Co on all the low index surfaces of Cu.This can be explained by an overlap of this state with a smaller number of bulk Cu states that results from the lower energy ͑4.47 meV͒ of the dimer mode.Long lifetimes of the c and d modes are accounted for by the position of these modes above the bulk Cu spectrum.One can note that in the lifetime estimation of the c peak we considered this peak as a sum of the c1 and c2 modes taking into account both the vertical and horizontal polarizations.Both the polarizations have also been included in the lifetime estimation of the d mode.

A Co dimer on Cu(001)
More complicated structure of LDOS arises for the Co dimer on Cu͑001͒ ͑Fig.11͒.First of all the low-frequency peaks become wider that indicates that in-plane vibrations are less localized.Contrary to the dimer on Cu͑111͒ where the lowest vibrational mode is degenerate and corresponds to the rotary vibration of the dimer as well as to the vibration of the entire dimer in the plane parallel to the surface in the direction perpendicular to the dimer axis; on Cu͑001͒ the respective peak ͓at 8.07 meV ͑a͔͒ corresponds to the latter kind of vibration only.It is coupled to the z-polarized vibrations of the Cu surface layer atoms.While a peak at 12.9 meV ͑b͒ corresponds to the dimer frustrated rotation mode FIG. 9. density of vibrational states for a dimer on the Cu͑111͒ surface and for nearest neighbors from the Cu surface layer ͑see the text͒.FIG. 10. ͑Color online͒ The polarization distributions for peaks indicated in Fig. 9.All the notations are as in Fig. 6. strongly coupled to vibrations of the NN-1Co atoms as well as to vibrations of the second-nearest-neighbor atoms.The weak peak at 9.5 meV ͑not denoted in the figure͒ is associated with vibrations of the entire dimer along the dimer bond.Other peaks in the middle frequency region such as that of 24.82 meV correspond to vibrations of the adatoms strongly coupled to the substrate atoms.In the highfrequency region, contrary to the case of the dimer on Cu͑111͒ where only two modes exist, on Cu͑001͒ we have found three sharp peaks.The first peak ͑c in Fig. 11͒ at 32.14 meV corresponds to the z-polarized vibration of the adatoms in opposite directions ͑frustrated rotation vibration in the plane perpendicular to the surface͒ which is coupled to the in-plane vibrations of the NN atoms of the substrate.It is similar to polarization of the c1 vibration of the dimer on Cu͑111͒.The next peak corresponds to two modes with fre-quencies of 33.13 meV ͑d1͒ and 33.38 meV ͑d2͒.The first mode is analogous to the c2 mode at 34.00 meV for the dimer on Cu͑111͒.It corresponds to vibrations of the Co adatoms in the vertical direction and along the dimer bond.The vibrational mode d2 is characterized by strong coupling of the NN-2Co atoms displacements ͑vertical and in the direction perpendicular to the dimer bond͒ to the in-plane vibrations of the dimer atoms across the dimer bond.The highest frequency mode, e, of 38.46 meV is similar to that of 38.22 meV for the Co dimer on Cu͑111͒.Modes d1 and e result from the stretching vibration of the free dimer.However, they are split and show additional vertical component of displacements due to interaction with the substrate atoms.
Since the low-frequency spectrum of the Co dimer on Cu͑001͒ is rather broad we estimate vibrational lifetimes only for narrow high-frequency modes c, d, and e.The fitting of these peaks that includes both horizontal and vertical polarizations gives the lifetimes of 0.91, 0.92, and 1.39 ps, respectively.The highest lifetime value of 1.39 ps is accounted for by the position of the mode out of bulk Cu spectrum.

A Co dimer on Cu(110)
In Fig. 12 we show the LDOS features for the Co dimer atoms, for the Cu͑110͒ surface layer NN atoms, and for the Cu second layer underlying atoms.In the low and middle frequency regions, the Co vibration peaks are broad, they are even broader than those for the Co dimer on Cu͑001͒.It assumes weak localization of these modes at the dimer atoms.Vibrations a and c are characterized by translational motion of the entire dimer along the ͓1 ¯10͔ axis.The Co atoms displacements of the a mode are coupled to the motion of the NN atoms in the same direction whereas vibrations of the c mode are coupled to displacements of the NN atoms in the opposite direction.The b peak at 17.12 meV is originated from the vibration which is absent in the cases of the Co dimer on Cu͑111͒ and Cu͑001͒.This vibration couples the motion of the adatoms and the underlying Cu atoms in the z direction.The peak d at 31.40 meV corresponds to the rotary vibration of the dimer in the plane parallel to the surface, it is coupled mainly to the NN-2Co and underlying Cu atoms.
The peak e at 34.34 meV corresponds to three different kinds of vibrations.The first mode, e1, is similar to the d2 vibration of the dimer on Cu͑001͒.The e2 vibration is the stretch in-plane mode with vertical displacements of the entire dimer.This motion is coupled to the z-polarized motion of the underlying atoms.The e3 mode corresponds to the z-polarized vibration of the adatoms in opposite directions such as the c mode of the dimer on Cu͑001͒.However, on Cu͑110͒ it is about 2 meV higher than that on Cu͑001͒ due to strong coupling with the underlying atoms motion.The highest frequency mode of 36.11meV is similar to that of 38.46 meV for the dimer on Cu͑001͒.So, the e2 and f modes, like to the case of the dimer on Cu͑001͒ and Cu͑111͒, result from the stretch vibration of the free dimer.These modes are split mainly due to strong interaction with underlying Cu atoms ͑and with Cu atoms 2 and 7 in the case of the f mode͒.
As in the case of the Co dimer on Cu͑001͒ we estimated the lifetimes by Lorentzian fitting only for high-frequency peaks of the Co LDOS.The fitting of the e ͑e = e1+e2+e3͒ and f peaks gives 1.47 ps for both features, while the d mode shows sufficiently shorter lifetime, 0.28 ps.

A Co trimer on Cu(111)
In Fig. 13 we show the LDOS features for the Co trimer atoms and for the Cu͑111͒ surface layer NN atoms.As in the previous cases, LDOS's of NN-1Co ͑atoms 1, 3, 9͒ and NN-2Co ͑atoms 2, 5, 6͒ atoms are shown separately.In Fig. 14 we show polarization vectors for the Co-induced modes.In the low-frequency region one can see the two peak structure of in-plane vibrations of the adatoms.The lowest frequency peak at 6.70 meV ͑a͒ corresponds to displacements of the entire cluster and the second one at 8.18 meV ͑b͒ is the rotary vibration of the cluster relative to its center of mass.These in-plane polarized modes ͑a and b͒ are similar to a1 and a2 modes found for the dimer on Cu͑111͒.However, contrary to the dimer case where they are degenerate here, the a and b modes are split by 1.5 eV.As in the previous cases, the low-frequency in-plane vibrations of Co are weakly coupled to vibrations of the substrate atoms while high-frequency Co vibrations which have non-negligible vertical contributions to Co displacements are strongly coupled to the Cu NN atoms vibrations.The peak at 32.14 meV ͑c͒ is formed by two modes.One of them ͑Fig.14, panel c1͒ inherits the antisymmetric vibration from the free Co trimer, while the other mode ͑Fig.14, panel c2͒ inherits the deformational vibration from the free trimer.Both these vibrations, being modified by vertical displacements of Co atoms, are strongly coupled to the substrate atom vibrations.The peak at 34.74 meV ͑d͒ is formed by three modes.The first of these modes ͑Fig.14, panel d1͒ is an optical-like in-plane vibration of the Co trimer modified by vertical displacements of the entire trimer.The second and third modes ͑Fig.14, panels d2 and d3͒ are, respectively, antisymmetric and deformational vibrations also modified by vertical motion of the Co trimer atoms.All these modes are coupled to both the Cu NN-2Co and Cu NN-1Co atoms.The highest frequency mode at 36.98 meV ͑e͒ is an optical-like in-plane vibration of the trimer mixed with vertical displacements of the entire trimer.These Co atoms motions are strongly coupled to the vertical vibration of the entire trimer of the Cu NN-2Co at-oms modified by optical-like in-plane motion of the latter atoms.One can note that the Cu NN-1Co atoms are not involved in this vibration.So d1 and e vibrations inherit their polarization from symmetrical vibration of the free trimer and are split due to different interaction with NN substrate atoms like a stretch mode of the dimer on Cu͑111͒.The same is true for pairs of antisymmetric and deformational types of vibrations.
The lifetime of the broad a and b peaks can be estimated as 0.3-0.4ps.The fitting of the well localized highfrequency peaks gives 0.89, 1.23, and 1.47 ps for the c ͑c = c1+c2͒, d ͑d = d1+d2+d3͒, and e peaks, respectively.One should note that highest stretch mode has the lifetime close to that for the dimer and single adatom on the ͑111͒ surface.

A Co trimer on Cu(001)
As follows from Fig. 2, panel c, Co atoms 1 and 3 occupy symmetrically equivalent positions with respect to a vertical plane that passes through a Co atom 2 and Cu atoms 3, 6, and 9.This leads to identical LDOS features for Co atoms 1 and 3 ͑Fig.15, panel Co-1,3͒ and to distinct LDOS for Co atom 2 ͑Fig.15, panel Co-2͒.In the lower panels we show LDOS for the first layer Cu NN atoms in ascending order with respect to the Cu-Co distance.Similar to the case of the Co trimer on Cu͑111͒ we have found two peaks in the lowfrequency region also for the Co trimer on Cu͑001͒ ͑see Fig. 15, panel Co-1,3͒ which are mostly in-plane vibrations of the Co atoms ͑see Fig. 16, panels a, b1, and b2͒.In spite of this similarity there exist some distinctions in the character of vibrations.The first one is the absence of the rotary vibration of the cluster.The second distinction which arises from incompatibility of symmetry of an isosceles triangle cluster and the ͑001͒ substrate is the polarization of the lowest frequency mode at 4.96 meV ͑Fig.16, panel a͒ which is different from any other found in the previous cases.Another lowfrequency peak at 8.31 meV ͑b͒ is a degenerate mode with the in-plane displacements of the entire cluster ͑Fig.16, panels b1 and b2͒.
In the high-frequency region we have found five sharp peaks.These modes differ in polarization from those found for the Co trimer on Cu͑111͒.This is related to different bonding between atoms of the trimer and between the Co atoms and the Cu substrate atoms.The peak at 33.50 meV ͑c͒ corresponds to the mainly z-polarized vibration of the Co-2 atom that is strongly coupled to horizontal motion of Cu atoms 2, 6, and 7 toward the Co-2 atom.This mode resembles the vibration mode ͑c͒ for the single Co adatom on Cu͑001͒ ͑Fig.7͒.Another difference in the trimer vibrations on Cu͑111͒ and Cu͑001͒ is a splitting of deformational and antisymmetrical trimer vibrations on Cu͑001͒ due to reduction of symmetry of the cluster from C 3v to C 2v .The LDOS peak at 34.74 meV ͑d͒ corresponds to vibration of the trimer with polarization similar to that for the deformational vibration ͓if the Cu͑001͒ symmetry is taken into account͔.The peak at 37.47 meV ͑f͒ can be associated with the antisymmetrical vibration of the trimer coupled to the vertical motion of the Co atoms caused by interaction with the substrate.The LDOS peaks at 35.98 meV ͑e͒ and 41.07 meV ͑g͒ results from the symmetrical vibration of the trimer with additional vertical contribution to polarization caused by mixing with the substrate NN atoms motion.These modes are similar to those found for the Co trimer on Cu͑111͒.
Here besides the high-energy modes the a mode is fairly well localized.The fitting of the a peak gives 0.37 ps.Sharp high-frequency peaks c-g have nearly identical broadening that leads to lifetimes of 1.40, 1.37, 1.41, 1.40, and 1.39 ps for the c, d, e, f, and g modes, respectively.

A Co trimer on Cu(110)
Finally we discuss vibrations of the Co chain trimer on Cu͑110͒.As was reported above the end Co atoms 1 and 3 have distinct positions with respect to the Cu surface plane in comparison with that of the atom 2. They are shifted from fourfold hollow positions to the center of the chain lie slightly higher than the central atom of the trimer.This difference in positions and bonding of the Co atoms with the substrate atoms leads to small splitting of the z-polarized vibrations in two peaks of LDOS ͑a1 at 15.34 meV and a2 at 17.08 meV, Fig. 17͒.These peaks correspond to the coupled adsorbate-underlying atom vibrations analogous to that de-termined by the b peak of the Co LDOS for the dimer on Cu͑110͒.Other weak peaks of the Co LDOS in the lowfrequency region are mainly determined by the in-plane translational vibrations of the Co chain mixed with substrate atoms vibrations.In the middle frequency region, some peaks like that at around 22.7 meV have contribution from the in-plane polarized deformational vibrations of the Co chain.The b peak at 31.4 meV is also formed by deformational and translational vibrations of the trimer.In the highfrequency region we find four sharp peaks.The first of them, c, at 33.34 meV, is originated from the in-plane rotary vibration of the trimer relative to its center of mass.This mode is similar to the d mode found for the dimer on Cu͑110͒.Next peak, d, is formed by three different kinds of vibrations with close frequencies.The d1 vibration with a frequency of 34.30 meV is antisymmetrical vibration of the chain accompanied by vibrations of the first layer NN atoms and mixed with the z-polarized vibrations of the underlying atoms.The d2 vibration at 34.40 meV is deformational vibration of the chain in the plane perpendicular to the surface strongly coupled to the underlying atom vibration.The d3 mode, such as e2 vibration in the dimer on Cu͑110͒, is caused by strong coupling of the NN-2Co atoms vibrations to the in-plane vibrations of the trimer atoms across the chain axis.Other peaks, e at 35.31 meV and f at 36.77 meV are determined by symmetrical ͑e͒ and antisymmetrical ͑f͒ vibrations of the chain coupled to the first layer NN and underlying Cu atoms.Thus, the symmetrical mode energy increases by Ϸ10 meV compared to the free chain due to strong mixing with z-polarized underlying atom vibrations while antisymmetric mode energy decreases and splits due to coupling mainly to laterally polarized vibrations of the NN atoms of the first substrate layer.
Fitting of the sharp high-frequency peaks c-f gives 1.21, 1.11, 1.31, and 2.43 ps, respectively.Contrary, the lifetime of the in-plane polarized b mode is 0.35 ps.

IV. CONCLUSION
We have studied the equilibrium crystal structure and vibrational properties of a single cobalt adatom and simple Co clusters ͑dimers and trimers͒ on the low-index Cu surfaces by using interatomic interaction potentials constructed by using the second moment tight-binding approximation. 30,31We have also done calculations of bond length, binding energy, and vibrations of the free standing clusters Co 2 and Co 3 .
It has been shown that in the case of a single Co adatom on all the Cu surfaces of interest, the lowest and the higher frequency modes are determined by the in-plane motion of the adatom and by mixed motion of the nearest-neighbor Cu surface atoms.The highest frequency mode of Co on Cu͑111͒ and Cu͑001͒ is formed by the z-polarized vibration of the adatom and mixed motion of the Cu NN surface atoms.In contrast, for Co on Cu͑110͒ such mode is characterized by the z-polarized vibrations of the adatom and nearestneighbor Cu second layer atom which is located just beneath the Co adatom.
The calculated Co 2 bond length, binding energy, and the stretch mode frequency are in close agreement with available experimental data.The interaction of the Co dimer with the substrate leads to vibrations which correspond to translational and rotational degrees of freedom of the free dimer.The low-frequency mode corresponds to the frustrated translation vibration of the dimer along the surface plane.The frustrated rotation mode of the Co dimer on Cu͑111͒ and Cu͑001͒ at 32-34 meV is polarized perpendicular to the surface.For the dimer on Cu͑110͒ the mode at 31.4 meV is the in-plane frustrated rotation vibration the dimer mixed with vibrations of the NN-1Co and NN-2Co surface atoms.
For all the low-index surfaces, the highest frequency stretch mode of the dimer is split in two, ones due to different interaction of the dimer atoms with the NN-1Co and NN-2Co substrate atoms.On Cu͑111͒ and Cu͑100͒ this splitting is of 4-5 meV.The lower frequency vibration is determined by the interaction with the NN-1Co atoms while the higher frequency vibration is originated from the interaction of the dimer atoms with the NN-2Co substrate atoms.On Cu͑110͒ such splitting of the stretch mode is smaller ͑ Ϸ2 meV͒ due to stronger interaction of the Co atoms with the Cu NN second layer atoms than with Cu surface atoms.
We have demonstrated that the computed frequencies of the isosceles triangle of Co 3 are close to those of the equilateral triangle cluster with one important difference.The lower frequency degenerate mode in the case equilateral triangle splits in two modes because of reduction in symmetry from C 3v to C 2v .It has been shown that the Co equilateral triangle trimer is stable on the Cu͑111͒ surface.On Cu͑001͒ the equilibrium triangle cluster is distorted ͑isosceles͒ due to distinct symmetry of the trimer and the substrate surface.On the Cu͑110͒ surface the chain of the three Co adatoms is 0.83 eV lower in energy than the triangle cluster while the free chain is 0.90 eV higher.
As in the case of the dimer, the low-frequency modes of the Co trimer on Cu͑111͒ are the frustrated translation and frustrated rotation in-plane polarized modes of the cluster.Deformational, antisymmetrical, and symmetrical highfrequency modes of the trimer on Cu͑111͒ are inherited from the free trimer, however, they are split because of different interaction of Co atoms with NN-1Co and NN-2Co Cu atoms by Ϸ2.5 meV.On the Cu͑001͒ surface, the Co trimer has the reduced symmetry that leads to polarization of Co vibrations slightly different from those of the Co trimer on Cu͑111͒.
The three Co atoms chain on Cu͑110͒ has the in-plane polarized low-frequency vibrations which correspond to translational and rotational degrees of freedom.In contrast to the unsupported chain where deformational vibrations are of low frequency, on Cu͑110͒ they are of high frequency ͑31.4 and 34.4 meV for deformational vibrations with in-plane and perpendicular polarization, respectively͒.Similar to the dimer on Cu͑110͒, the antisymmetrical high-frequency vibration is split by Ϸ2.5 meV due to different interaction of the Co trimer atoms with the NN-1Co and NN-2Co substrate atoms.The highest frequency mode is formed by the Co antisymmetrical vibration coupled to the Cu second layer NN atoms and the Cu surface layer NN-2Co atoms.
The longest lifetimes have been found for high-frequency vibrations of all the Co clusters on the low-index Cu surfaces: these lifetimes are of 1-2.5 ps.The shortest lifetimes, 0.1-0.8ps, have been found for low-frequency horizontal vibrations of the Co clusters.

FIG. 4 .
FIG. 4. Local density of vibrational states for the clean Cu͑111͒, Cu ͑001͒, and Cu͑110͒ surfaces compared to the bulk DOS.

FIG. 5 .
FIG. 5. Local density of vibrational states for a Co adatom on the Cu͑111͒ surface, for the nearest neighbors from the substrate first layer atoms ͑Cu s NN͒ and for the rest atoms of the first Cu layer ͑Cu s others͒.

FIG. 11 .
FIG. 11.Local density of vibrational states for a dimer on the Cu͑001͒ surface and for nearest neighbors from the Cu surface layer ͑see the text͒.

FIG. 12 .
FIG.12.Local density of vibrational states for a Co dimer on the Cu͑110͒ surface, for nearest neighbors from the Cu surface layer ͑see the text͒, and for the Cu second layer atoms which lie beneath the adatoms ͑Cu s−1 underlying͒.

FIG.
FIG. Local density of vibrational states for a Co trimer onCu͑001͒ and for NN substrate atoms.

FIG. 16 .
FIG.16.͑Color online͒ The polarization distributions are shown for peaks indicated in Fig.15.All the notations are as in Fig.6.

FIG. 17 .
FIG. 17. Local density of vibrational states for a Co trimer onCu͑110͒ for nearest neighbors from the Cu surface layer ͑see the text͒, and for Cu second layer atoms which lie beneath the adatoms ͑Cu s−1 underlying͒.
substrate atoms ͑NN-1Co and NN-2Co͒ exist, on Cu͑001͒ more complicated geometry of nearestneighbor Cu atoms arises: five groups of the NN atoms with different relaxations are revealed.The first group is of atoms 1 and 11 which can be characterized by the large ͑0.091Å͒ outward relaxation in the z direction and by the largest distance ͑2.683 Å͒ to the nearest adatom compared to the other NN atoms.The second group consists of one atom ͑6͒ only which undergoes huge ͑−0.242Å͒ inward relaxation due to shortening the long bond length of the cluster.This atom has the shortest distance to the nearest Co atoms ͑2.420 Å͒.Other groups are: Cu 2 and Cu 7 ; Cu 3 ; Cu 5 and Cu 10 .All these atoms have approximately equal inward relaxation, however, they are located at distinct distances from the nearest Co adatoms ͑in the range of 2.488-2.519Å͒due to different lateral relaxations.Moreover, atoms 2 and 7 have the two NN Co atoms and are situated at different distances from the Co 2 and Co 1͑3͒ atoms ͑2.488 and 2.584 Å, respectively͒.Contrary to the case of a single Co adatom on closepacked surfaces, on Cu͑110͒ a nearest to the adatom is the second layer Cu atom lying beneath the adatom.It relaxes inward by 0.065 Å due to the Co adsorption and is situated at a distance of 2.448 Å from the adatom.However, for convenience, as before, we will designate the first layer Cu atoms closest to the adatom as the nearest-neighbor atoms.The equilibrium distance d Co-Cu between the Co adatom and the nearest-neighbor substrate atoms has been obtained equal to 2.493 Å that is slightly longer than the corresponding d Co-Cu at Cu͑111͒ and Cu͑001͒.The first nearest neighbors to the Co adatom ͓atoms 1, 2, 5, and 6 in Fig.