Double rotation mechanism in small Cu clusters concerted diffusion over Cu{1 1 1} surfaces

doi:10.1016/j.susc.2006.11.032

Surface Science

Volume 601, Issue 4, 15 February 2007, Pages 931-935

https://doi.org/10.1016/j.susc.2006.11.032 Get rights and content

Abstract

In this work, we study the role of the double rotation mechanism in the concerted diffusion of two-dimensional small Cu clusters (up to 10 atoms) over Cu{1 1 1} surfaces. Our results show that the necessary energy to diffuse the cluster on any direction over the surface (overall activation energy) increases proportionally to the cluster size. However, the minimum energy necessary to just move the cluster center of mass presents a nonmonotonic increase. The reason for this behavior relies on the double rotation mechanism, which is observed in some clusters with diamond shape configuration. Consequently, clusters as big as hexamers can be expected to be surprisingly mobile with activation energies around 0.15 eV.

Introduction

Thin films have been used in many technological applications including surface coating, microelectronics and optoelectronics. Hence, the importance of studying the growth of thin films has increased over the last decades [1], [2]. The understanding of the growth phenomena requires a complete description of the diffusion mechanisms of small clusters. There exists a wide variety of experimental [3], [4], [5], [6], [7] and theoretical [8], [9], [10], [11], [12], [13], [14], [15], [16] studies of small clusters diffusion on surfaces.

Systematic experimental studies of small clusters diffusion were carried out by Wang and Ehrlich [3], [4], who studied the stability and diffusion of Ir clusters on Ir{1 1 1} surfaces. They observed that activation energies increase proportionally to the cluster size, except for tetramers. Additionally, Kyuno and Ehrlich [5] studied diffusion and dissociation of Pt clusters over Pt{1 1 1} surfaces. In this case a monotonically increase of activation energies was reported for all clusters studied.

Simulation methods were used to show a more detailed description of clusters diffusion on surfaces and to calculate their respective diffusion barriers [2]. Hamilton et al. [9], employing the embedded atom method (EAM) as interatomic potential, described a dislocation mechanism for diffusion of Ni clusters over Ni{1 1 1} surfaces. They suggested that this mechanism may facilitate the diffusion of large clusters. Furthermore, they reported that the most energetically favorable mechanism for tetramers diffusion is a “nearly” simultaneous motion of the four atoms. Chirita et al. [10] described a double-shearing mechanism for the motion of Pt hexamers and octamers on Pt{1 1 1} surfaces. They reported that this mechanism may occur with the same frequency as the gliding motion. Their studies proposed that small clusters migrate by three fundamental mechanisms: gliding, dislocation and double shearing motions. Liu et al. [11] studied the mechanisms of diffusion and clustering over Ti{0 0 0 1} surfaces. They found that trimers and heptamers are more stable than other clusters of comparable size. Additionally, clusters as large as heptamers are highly mobile and cannot be effective nuclei in the three-dimensional growth of thin films.

A nonmonotonic variation of the diffusion barrier with the cluster size was shown by Chang et al. [12] for fcc materials over {1 1 1} surfaces. They proposed that this behavior is due to the zig-zag diffusion mechanism of tetramers, which have similar diffusion barrier than trimers. Recently Marinica et al. [13] studied diverse diffusion mechanisms of Cu clusters over Cu{1 1 1} surfaces. Their results show that the diffusion barrier increases proportionally to the increase of the cluster size up to heptamers. Wang et al. [14] and Coronado and Huang [15] studied the diffusion of adatoms and dimers under diverse substrate configurations. They calculated the mechanisms and the energy barriers for the generalized or three-dimensional Ehrlich–Schwoebel barrier, which is expected to strongly influence the surface morphology.

A recent work of Karim et al. [16] focused on the self-learning kinetic Monte Carlo (SLKMC) method to study the diffusion of small Cu clusters (up to 10 atoms) over Cu{1 1 1} surfaces. The study takes into account both the concerted diffusion, and the single and multiple jumps from one set of fcc sites to another. They found that small clusters diffuse primarily through concerted motion, even if for some cases (nonamers and decamers, principally) single atom mechanisms are more frequent. Additionally, the concerted diffusion leads to an almost linear increase of the diffusion barrier with cluster size.

Even though the energy barriers for small clusters diffusion have been calculated in other studies, using MD simulations to calculate these energies is very difficult and sometimes inaccurate because usually for a given temperature range more than one mechanism can be activated. This drawback did not allow a complete description of the mechanisms of motion [9], [13], [16]. On the other hand, The nudged elastic band (NEB) method, developed by Henkelman and Jónsson [19], can be used to determine precisely the minimum energy path of each atom of the cluster between two previously defined states.

In this work, we use the NEB method to describe the influence of the so-called concerted double rotation mechanism in the diffusion processes of small Cu clusters over Cu{1 1 1} surfaces. Additionally molecular dynamics (MD) simulations are performed to confirm some NEB results. Our calculations suggest that, in opposition to previous insights, small diamond shape clusters can be relatively more mobile, as long as this rotation mechanism operates.

Section snippets

Simulation methods

The interatomic interaction is based on the embedded atom method (EAM) potential [17], [18]. The EAM potential, partly based on DFT concepts, gives a realistic description of fcc metal systems; with the advantage of allowing simulations of systems with up to millions of atoms. The simulation cell used in all calculations is a Cu{1 1 1} slab with 12 layers and with approximately 2300 atoms. The four bottom layers were fixed and the eight upper layers were set up without any constraint. Periodic

Results

We have used the NEB method to calculate the diffusion barriers for adatoms, dimers, and trimers, which are 0.03, 0.10, and 0.12 eV, respectively. These values are in good agreement with experimental and numerical results [1], [7], [11], [12], [13], [14], [15]. A brief review of the diffusion mechanisms observed is as follows. Adatoms can usually move in two possible ways: hopping and exchange process, with the former one being energetically preferred in the case of Cu{1 1 1}. Dimers have two

Conclusions

The minimum energy necessary to move the center of mass of small clusters presents a nonmonotonic dependence with respect to the cluster number of atoms. We found that this low activation energy is basically due to the double rotation mechanism. In the case of tetramers, the double rotation mechanism is the most predominant one and produces a diffusion barrier similar to that observed in trimers. The results obtained using the nudged elastic band (NEB) method also suggest that pentamers and

Acknowledgements

The authors AMC and HH gratefully acknowledge the financial support from Basic Energy Science of Department of Energy (DE-FG02-04ER46167).

References (21)

S.C. Wang et al.
Surf. Sci.
(1990)
K. Kyuno et al.
Surf. Sci.
(1999)
M. Giesen et al.
Surf. Sci.
(2003)
J.J. de Miguel et al.
J. Phys.: Condens. Matter
(2002)
T. Ala-Nissila et al.
Adv. Phys.
(2002)
S.C. Wang et al.
Phys. Rev. Lett.
(1998)
J. Repp et al.
Phys. Rev. Lett.
(2003)
P. Stoltze
J. Phys.: Condens. Matter
(1994)
J.C. Hamilton et al.
Phys. Rev. Lett.
(1995)
V. Chirita et al.
Appl. Phys. Lett.
(1998)

There are more references available in the full text version of this article.

Cited by (9)

Diffusion of Cu adatoms and dimers on Cu(111) and Ag(111) surfaces
2015, Surface Science
Citation Excerpt :
Such systems have been investigated both experimentally [8–10] and theoretically [12–16,20]. In addition to the diffusion of a dimer, there have been also studies concerning adatom clusters of larger size [13–15,20,28]. Hetero-diffusion appears to be easier with lower activation barrier and homo-diffusion is dominated by relatively high energy barriers between neighboring cells, whereas rotation inside each cell can run freely.
Diffusion of Cu adatoms and dimers on Cu(111) and Ag(111) surfaces is analyzed based on ab initio surface potentials. Single adatom diffusion is compared with dimer diffusion on both surfaces. Surface geometry makes the adatoms jump alternately between two states in the same way in both systems, whereas dimers undergo more complex diffusion process that combines translational and rotational motion. Small difference in the surface lattice constant between Cu and Ag crystals results in a completely different energy landscape for dimer jumps. As an effect the character of diffusion process changes. Homogeneous Cu dimer diffusion is more difficult and dimers rather rotate within single surface cell, whereas diffusion over Ag surface is faster and happens more smoothly. The temperature dependence of diffusion coefficient and its parameters: energy barrier and prefactor is calculated and compared for both surfaces.
Structural stabilities and diffusion of small Fe clusters on Fe (1 1 0) surface: A molecular dynamics study
2011, Applied Surface Science
Citation Excerpt :
First-principle calculations should be preferred in principle, but in practice it is severely limited by computing power. Therefore, the calculations including molecular dynamics simulation based on semiempirical potentials are usually performed [5–10]. Although the results obtained from the semiempirical potentials, especially some quantitative results are sometimes not completely in accordance with the experiments owing to the impreciseness of the potentials, the semiempirical potentials method is still very effective to study qualitatively the self-diffusion of adatoms and clusters, and it makes large-scale and long-time molecular dynamics simulation be possible.
Using Embedded-atom-method (EAM) potential of iron, structural stabilities of small Fe clusters on a Fe (1 1 0) surface have been investigated by molecular dynamics studies. It is presented that a tetramer and heptamer clusters are more stable than other sizes. These two clusters have high transition energies. They can be a critical nucleus at low and high temperature, respectively. A dimer diffuses more easily with lower energy barrier than single adatom. The trimer's rotation and dimer shearing mechanisms have been investigated in this paper.
Cu dimer diffusion on strained Cu(0 0 1)
2010, Surface Science
Cu dimer diffusion energy barrier on strained Cu(0 0 1) surfaces has been studied with nudged elastic band method (NEB) and embedded atom method (EAM). Dimer exchange and hopping mechanisms are chosen as the initial diffusion paths in the NEB method. It is shown here that the dimer exchange is dominant on tensile surfaces and the dimer hopping is dominant on compressive surfaces. For most strain conditions Cu dimer diffusion energy barrier is lower than Cu monomer diffusion barrier. The concerted movement of the remaining adatom toward the hopping adatom lowers the dimer hopping barrier. The adsorption induced relaxation makes the dimer exchange barriers lower than the monomer exchange barriers on tensile surfaces. Transition state theory is used to calculate the diffusion frequencies as a function of temperature. No surface crowdion is observed on the shear strained surfaces for the dimer diffusion.
Comparative study of compact hexagonal cluster self-diffusion on Cu(1 1 1) and Pt(1 1 1)
2008, Applied Surface Science
Citation Excerpt :
Over the years, several theoretical methods [3], such as static calculation [8,10–12], MD [13–15], kinetic Monte Carlo (KMC) [16–18] have been used to study the cluster diffusion.
Self-diffusion behaviors of two-dimensional compact hexagonal Cu₇ and Pt₇ clusters on (1 1 1) surface are studied with molecular dynamics simulation. The diffusion barriers ( $E_{m}$ ) and prefactors ( $D_{0}$ ) are determined from the Arrhenius diagram and compared with recent low-temperature field ion microscopy experiment. We find that $D_{0}$ of Pt₇ cluster is about 35 times larger than that of Cu₇ cluster. From the calculated phonon spectral densities of adatoms, we suggest that the different diffusion behaviors of the two clusters may result from strong interactions between the Pt adatoms and surface atoms, which make the nonequivalent self-diffusion processes, which can produce a high prefactor, occur more easily for Pt₇ cluster.
Atomistic simulation of Pt trimer on Pt(1 1 1) surface
2007, Applied Surface Science
Citation Excerpt :
In addition to a providing physical insight into the details of adatom–surface and adatom–adatom interactions on surfaces, information on the mechanism and energetics of cluster motion can contribute to a better understanding for crystal and thin-films growth [1,2]. Over the years, several theoretical methods [1], such as static calculation [3–6], MD [7–9], kinetic Monte Carlo [10] have been used to study the cluster diffusion. The static calculation cannot find all significant diffusion pathways, thus ignoring possible low-barrier processes, and the diffusion coefficient is intrinsically a dynamic quantity, it is the prefactor which contains the true dynamical information.
The diffusion of Platinum trimer on Pt(1 1 1) is studied at different temperatures by molecular dynamics (MD) simulation. The structure stability is studied by cluster binding energy. The interaction between adatoms and surface atoms is discussed based on the calculated phonon density of state of Pt trimer. The diffusion coefficients of Pt trimer are derived from mean square displacement of cluster’s mass-center, which is obtained by long simulation times ( $≃ 0.2$ $μ$ s) and tracing of interstitial atoms on surface. Then the diffusion prefactor and migration energy are deduced from Arrhenius relation. The calculated results are in reasonable agreement with experiment. In addition, using the diffusion prefactor and migration energy, the efficiency of Pt trimer as a critical nucleus for three-dimensional growth of thin films is discussed.
Atom diffusion of small Cu clusters across facet-facet barriers over Cu{1 1 1} surfaces
2007, Modelling and Simulation in Materials Science and Engineering

View all citing articles on Scopus

View full text

Double rotation mechanism in small Cu clusters concerted diffusion over Cu{1 1 1} surfaces

Abstract

Introduction

Section snippets

Simulation methods

Results

Conclusions

Acknowledgements

Surf. Sci.

Surf. Sci.

Surf. Sci.

J. Phys.: Condens. Matter

Adv. Phys.

Phys. Rev. Lett.

Phys. Rev. Lett.

J. Phys.: Condens. Matter

Phys. Rev. Lett.

Appl. Phys. Lett.