Long jumps contribution to the adatom diffusion process near the step edge: The case of Ag/Cu(110)

Long jumps contribution to theadatom diffusion process near the step edge:The case of Ag/Cu(110)

Khalid Sbiaai*,1, Yahia Boughaleb*,2,3, Abdelkader Kara4, Samira Touhtouh5, and Bouchta Sahraoui6

1 Faculté Polydisciplinaire de Khouribga, Université Hassan 1, 26000 Settat, Morocco2 Ecole Normale Supérieure, Université Hassan II, Casablanca, Morocco3Hassan II Academy of Science and Technology, Rabat, Morocco4Department of Physics, University of Central Florida, Orlando, Florida USA5 Ecole Nationale des Sciences Appliquées, Université Chouaïb Doukkali, El Jadida, Morocco6 LUNAM Université, Université d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 blvd Lavoisier,49045 Angers Cedex, France

Received 10 September 2013, revised 12 November 2013, accepted 13 November 2013Published online 27 December 2013

Keywords adatom, embedded-atom method, molecular dynamic simulation, silver surface diffusion

* Corresponding author: e-mail [email protected], Phone: þ212 6 69 84 44 79, Fax: þ212 522 98 53 26** e-mail [email protected], Phone: þ212 6 62 25 54 44

In this work, the diffusion of a single Ag adatom on a low-indexCu surface (110) in the presence of a step edge is studied usingthe embedded-atom method (EAM). Molecular static simula-tion is carried out in order to calculate the activation energy ofdifferent diffusion processes. Our findings are in a goodagreement with results existing in the literature indicating thatadatom diffusion via jump process is more favored than theother mechanisms. The activation energy corresponding todiffusing via hopping is found to be 0.25 eV (at 0K). On theother hand, the activation barrier calculated by molecular

dynamics (MD) simulation for a large range of temperature(310–500K) is found to be around 0.25 eV for both upper andlower position leading to a good agreement between static anddynamic calculations. The prefactor for Ag adatom self-diffusion via hopping on Cu(110) surface near the step edge isexamined. The results show that the prefactors are 2.7 and3.6� 104 cm2 s�1 for the upper and lower position, respective-ly. This is in line with the value of 10�3 cm2 s�1 that is generallyadopted. We also found that long jumps occur frequently in thissystem and their contribution cannot be neglected.

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Diffusion of adatoms on metal surfa-ces has been the subject of many studies for many years dueto their practical applications in thin-film growth [1–3].Therefore, obtaining better device performance depends onunderstanding thin-film growth at an atomic level.

For this purpose, many tools have been developed,such as scanning tunneling microscopy (STM) [4, 5] andfield ion microscopy (FIM) [6, 7]. On the other hand, theheteroepitaxial of metals on the metallic substrates is ofgreat interest from the point of view of applications.While, the bimetallic compound (heterosystem) is still thesubject of active research [8–11]. In such systems, we haveto take into account the lattice mismatch between thegrowing layer and substrate (the bond-length misfit in Cu

and Ag is 12%) and the disparity of binding energybetween the two metals.

The (110) surface of a fcc metal is very interesting due tothe presence of channel geometry and also for its anisotropy.For instance, the existence of the step on the surface plays acrucial role for the determination of the growth mode, whichis directly related to the Ehrlish–Schwoebel (ES) barrier [12]usually defined as the extra energy needed for an adatom todiffuse from the upper terrace to the lower terrace of the step.The later is a characteristic of diffusion near the step edge.The presence of the ES barrier leads to a three-dimensional(3D) growth mode. On the other hand, its absence enablesa layer-by-layer growth mode [13]. For Cu on Cu(100),the growth mode is more complicated. Experimental

Phys. Status Solidi B 251, No. 4, 838–844 (2014) / DOI 10.1002/pssb.201350324 p s sbasic solid state physics

b

statu

s

soli

di

www.pss-b.comph

ysi

ca

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

investigations report a 3D growth mode with the presence ofthe mounds [14], while theoretical studies led to theconclusion that the growth could be via a layer-by-layermode where the ES barrier via a hopping process is found tobe in the range 125–320meV [15–20], and lies between 30and 130meV [16, 19, 20] through an exchange process. ForAg on Ag(100), the ES barrier via exchange deduced fromthe first-principles calculations is 0meV, whereas the ESbarrier via the jump process is of significant importance(with a value of 100meV). For Ag/Ag(100), and based onthis result, one could expect that the growth could proceedvia the layer-by-layer mode.

As mentioned above, adatom diffusion is very importantin growth of thin films. One of crucial ingredient is thereforethe diffusion coefficient. For this reason, several experimen-tal studies have been done using field ion (FIM) [21] andSTM [22, 23] in order to determine this physical quantity.From the theoretical point of view, elements controllingdiffusion have been deduced from transition-state theory(TS) [24]. Using molecular dynamic simulations for Ag onCu(110) [25], it was found that the prefactor of the diffusioncoefficient on terrace is D0¼ 6.68� 10�3 with thecorresponding activation energy ED¼ 0.23 eV. In general,if adatoms jump only between nearest-neighbor sites and ifwe neglect entropy contribution, the prefactor of diffusioncoefficient D0 should be of the order of 10�3 cm2 s�1.

The aim of this work is to study different Ag adatomsdisplacements on a Cu surface near the step. We will presentthe contribution of long jumps (double, triple, etc.) on theadatom diffusion. The choice of Ag/Cu(110) based on thefact that this system is still the subject of recent studies [26–28] and for their large applications in industrial areas (such assolar cells, interconnectors, etc.). Therefore, we calculate theactivation energy of each diffusion process in order to predictits predominance. The diffusion coefficient has also beenexamined in this work for the jump process along straightstep both for the upper and the lower position.

This paper is organized as follows. After a briefpresentation of the model of energy used in this work(Section 2), we present an analytic expression of diffusioncoefficient along straight steps deduced from moleculardynamic simulations (Section 3). Section 4 contains ourresults with discussions. Conclusions are summarized inSection 5.

2 Theoretical models In this investigation, ourmolecular dynamics (MD) simulation is based on semiem-pirical many-body potentials derived from the embedded-atom method (EAM) [29].

In this method, the total energy is given by the followingequation:

Etot ¼ 12

Xij

V ijðRijÞ þXi

FiðriÞ; ð1Þ

Rij is the scalar distance between atom i and j, and ri is thesum of individual electron density provided by the otheratoms of the metal [29]:

ri ¼Xiði 6¼jÞ

raj ðRijÞ: ð2Þ

3 Difusion coefficient along straight stepsdeduced from molecular dynamics (MD)simulations The adatom diffusion is studied usingmolecular dynamic simulation. The simulation is carriedout from 310 to 600K. The trajectory of an adatom isanalyzed after the resolution of the Newton equation ofadatom displacement. The adatom movement is assumed tooccur via uncorrelated and discrete jumps. In the statisticalinvestigation, a simple jump is considered when the adatomtravels between two adjacent adsorption sites separated by a(A), while a long jump correspond to na(A).

The diffusion coefficient is expressed by the Einsteinequation as follows:

DðTÞ ¼ limt!1h½rðtÞ�2i

2t; ð3Þ

where

rðtÞ ¼XNs

i¼1

niri; ð4Þ

Ns is the total number of jumps during the interval time t. niriis the vector displacement corresponding to the ith jump(ni¼ 1: simple jump, ni¼ 2: double jump, ni¼ 3: triplejump, etc.). The new expression of diffusion coefficientD(T)is:

DðTÞ ¼ a2

2GMDðTÞ; ð5Þ

where Nn is the number of jumps corresponding to the lengthna(A) observed during the time of simulation trun with:

GMDðTÞ ¼Xn

Nn

trunn2: ð6Þ

Finally, the diffusion coefficient could be written as:

D ¼ 12a2si

Xl

l2pl; ð7Þ

where a is the distance between adjacent in channel minima,and pl is the probability to have an l-site jump. si is thejump rate. The usual method to analyze the temperaturedependence of si is to verify the Arrhenius law based onthe following equation:

si ¼ s0j exp

�Ea

kBT

� �: ð8Þ

During diffusion, the adatom is kept in an equilibriumposition for tth, the so-called thermal time which is relatedto the temperature by [30]

Phys. Status Solidi B 251, No. 4 (2014) 839

www.pss-b.com � 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

tth ¼ am

kBT

� �1=2

: ð9Þ

For example, tth¼ 1 ps at T¼ 500K [31]. If the diffusingadatom spends more than 1 ps in the same site, then we canconsider that the adatom has visited this adsorption position.If not, the adatom flies over this position until reaching a newadsorption position (this mechanism yields to a long jumpprocess).

4 Results and discussion4.1 Systemand geometry Our simulation boxes are

formed by 480 atoms, each layer contains 64 except the toplayer (32 atoms). We suppose also that the periodic boundaryconditions are applied in the parallel direction to the surface(x and y directions).

In Fig. 1, we present our system geometry where thediffusion channels are clearly observed. During adatomdeposition, two adsorption sites are available for adatomreception. The first position is on the lower surface (low)while the second one is on the upper terrace (up) of the step.Such channeled geometry gives preference to the in-channeldiffusion. If we add to this criterion the atomic sizedifference between silver and copper (r� ¼ rCu/rAg¼ 0.886)and the lattice mismatch between Ag and Cu (12%), it isclear that the insertion of an Ag adatom on the Cu surface isnot easy. This analysis is in a good agreement with ourfindings in Refs. [32, 33].

4.2 Adatom diffusion along straight step Asmentioned above, the jump process is more favored inAg/Cu(110), which means that the statistical events areeasy to collect for this mechanism. Thus, in this section wewill calculate the diffusion coefficient via jumps (eitherfor simple or long jumps) and the activation energy usingEq. (7). This equation enables us to introduce thecontribution of long jumps in the diffusion coefficient andcalculate the prefactor of the diffusion coefficient.

Figure 2a and b represents the adatoms positions andthe all diffusion processes that can occur in our system. The

adatom diffusion processes on the upper position near thestep are noted by (a), (b), and (c). It is worth noting that theprocess (a) is the adatom diffusion along the step edge andalong the diffusion channel. On the other hand, the exchangeprocess is illustrated by the process (b). After this, theadatom can integrate the step. In order to reach the lowerposition, the adatom should follow the direction (c). Whenthe adatom is deposited in the lower position near the stepedge, two processes could be observed: process (d), which isequivalent to the process (a) (on the upper surface), and theexchange process is given by the process (e).

The diffusion processes on the terrace (perfect surface)are illustrated by the processes (f) for diffusion via jumpmechanism, (g) for exchange phenomena, and (h) fordiffusion across the channel.

The occurrence of long jumps (so-called correlatedjumps) has been investigated using molecular dynamicsimulations [25, 32, 33] and observed experimentally indifferent system [34, 35]. The contribution of long jumps, inthe total jump diffusion, is very significant in the case of Pd(about 20%) and is rare in Ir and Rh (�3%). In the case ofAu/Au(110), at 450K, long jumps are practically about 3%and reach 6% in Ag/Ag(110) at the same temperature. ForCu/Cu(110), long jumps are very frequent and reach 15% at600K. The diffusion of Pt on a Pt(110)(1� 2) missing rowreconstructed surface shows that long jumps follow anArrhenius with an activation energy 0.89 and 0.81 eV for

Figure 1 The studied system formed by atoms. The adatom ispresented by the red sphere. Yellow spheres present the uppersurface of the edge. The substrate is formed by the gray spheres.

Figure 2 (a) Representation of the adatoms position on terrace(dark circle) and near the step edge (red circles). The narrow circlesrepresent atoms forming the step edge. The gray circles are theatoms of the second layer. (b) Different diffusion processes on the(110) surface near the step edge for an adatom in the upper andlower positions. Yellow circles illustrate atoms of the upper surfaceof edge. The lower surface is represented by the gray circles.Adatoms are illustrated by dark and red circles.

840 K. Sbiaai et al.: Adatom diffusion process near the step edge of Ag/Cu(110)

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

a ssp stat

us

solid

i b

simple jumps. In the case of heterogeneous systems, longjumps are demonstrated using MD simulations. For Ag/Cu(110), (on (110) terrace), the contribution of correlatedjumps is about 14% [36].

Near the step edge (Fig. 2), the diffusion of Ag onCu(110) is dominated by jump process (simple and long).The exchange is less probable according to the descriptionpresented in Section 4.1. This is clearly seen if we comparethe activation energy of each process. For diffusion via asimple jump (Fig. 3(a)–(b)), the activation energy is 0.25 eV.However, the diffusion via the exchange process, where theadatom is incorporated to the edge, needs more energy(0.44 eV) (Fig. 3(b)–(c)).

Figure 4 represents the ratio of the double-jumpcontribution to the total jump of the adatom in both upperand lower position near the step edge. It is clearly seen thatthe simple jump is predominant. For the upper position(Fig. 2, adatom (1)), the simple-jump percentage can reach

95.4%. We remark also that the ratio of simple jumpsincreases when the temperature increases especially between310 and 500K. The same observation is noted for an adatomon the lower position where the simple jump is alwayspredominant and can present 93.5%. Concerning doublejumps, their presence is observed for the two adatompositions and their contribution increases where thetemperature increases (from 310 to 500K). However, itcan exceed 5% and reach 12%. This important contributionof long jumps is due to the lowest activation energy (seeTable 1) and also to weak energy dissipation between the Agadatom and the Cu surface [25].

In this way, our EAM results are compared to otherresults (see Table 1), obtained by the Rosato–Guillope–Legrand model (RGL) [15, 37], effective medium (EM) [38],corrective effective medium (CEM) [39], embedded atom inthe Adams–Foiles–Wolfer parameterization (AFW) [40], theVoter–Chem parameterization (EA(VC)) [41], the aniso-tropic broken band model (ABBM) [37], and molecularstatic (MS) calculations [42]. In this later reference, theauthors have used the nudged elastic band (NEB) method inorder to calculate the activation energy of each process [42].

Table 2 reports some results concerning the activationenergy for some diffusion processes illustrated in Fig. 1 but

Figure 3 Schematic representation of adatom diffusion (graysphere) by simple jump (a and b), and via an exchange mechanism(b and c) near the step edge (for upper position). The black circleindicates the initial position of the adatom before diffusion.

Figure 4 Jump process distribution for adatom near the edge. (a)For an adatom in the upper position whereas (b) is for an adatom inlower position. The gray bar corresponds to the simple-jumpcontribution and the dark one represents the double-jumpcontribution.



Original

Paper

Table 1 Activation energy for Cu on Cu(110) calculated by the embedded-atom method (EAM) and compared to other results.

process EAM RGL EM CEM, MD/MC-CEM EA (AFW) EA (VC) ABBM MS

a 0.28 0.26a) – – 0.28e) – 0.265a) –

b 0.22 0.40a) – – – – 0.325a) –

c 1.20 – – – – – – 0.66g)

d 0.28 – – – – – – 0.29g)

e 0.36 – – – – – – –

f 0.23 0.23b) 0.18c) 0.08, 0.26d) 0.24e) 0.53f) 0.23a) –

g 0.29 0.29b) 0.26c) 0.09, 0.49d) 0.30e) 0.31f) 0.29a) –

h 1.18 – – – – – – –

a)From Mottet et al. [37]; b)From Montalenti and Ferrando [15]; c)From Hansen et al. [38]; d)From Perkins and DePristo [39];e)Liu et al. [40]; f)From Karimi et al. [41]; g)From Stepanyuk et al. [42].

Table 2 Activation energy for Ag on Cu(100) calculated by EAMand compared with Perdew–Wang (PW91) and Predew–Burke–Ernzerhof (PBE) results.

process EAM PW91, PBE [43]

a 0.195 0.20b 0.71 0.69c 0.63 0.54d 0.20 –

e 0.82 –

f 0.48 0.37

Table 3 Activation energy for Ag on Cu(100) calculated by thedrag method (eV).

process activation barrier

a 0.25b 0.44c 1.15d 0.25e 0.42f 0.24g 0.32h 0.98

Table 4 Statistical events collected at T¼ 400K during 10 ns(simulation time).

position simple double triple tetra

upper 404 35 4 2lower 394 41 5 1

Figure 5 Arrhenius plot of coefficient diffusion of adatom situatedin the upper surface near the step edge (10�7 cm2 s�1).

Figure 6 Arrhenius plot of coefficient diffusion of adatom situatedin the lower surface near the step edge (10�7 cm2 s�1).

Table 5 Diffusion energy and prefactors for jump process on Cu(110) near the step. Estatic is the zero-temperature value of the activationenergy.

s0 (ps�1) D0 (cm

2 s�1) Esa (eV) ED

a (eV) Estatic (eV)

upper 4.22 2.7� 10�4 0.236� 0.005 0.25� 0.01 0.25lower 3.63 3.6� 10�4 0.24� 0.01 0.25� 0.01



ph

ysic

a ssp stat

us

solid

i b

on the (100) surface of heterogeneous system. All values inthis table are compared with Predew and Wang 91 (PW91)and Predew–Burke–Ernzerhof (PBE) methods [43] in orderto enhance the reliability of our method. We remark that ourfindings are in reasonable agreement with those presented inRef. [43].

In Table 3, we summarize for all processes the activationenergy for adatom diffusion near the step edge. All theseactivation energies are calculated using the drag method at0K where the thermal activation is neglected. The energybarrier calculated in Refs. [35, 36] are in good agreementwith our findings for diffusion via processes (f) and (g). Ourcalculation shows that activation energy for the process (f),where the adatom diffuses along a straight step (lowerposition), is found to be 0.28 eV for Cu/Cu(110) and 0.25 eVfor Ag/Cu(110). For diffusion via the exchange mechanism(process (g)), we found that the activation energy (0.42 eV)is higher than via the jump process, which is in goodagreement with the analysis in Section 4.1. We note also agood agreement between the activation energy of process (d)calculated by our method and the nudged elastic band (NEB)method [42].

During the diffusion process, in addition to doublejumps, other long jumps could be observed by reaching thethird and the fourth positions. By observing long jumps,one can see that the adatom flies along the in-channeldiffusion along a straight step. This nature of jumps is alsodemonstrated when Ag adatom diffuses on Cu(110)terrace [25] where an adatom diffuses along 4a (a is thedistance between the nearest site along diffusion channel in a(110) surface). Statistical events summarized in Table 4 areinjected into Eq. (7) in order to determine accurately theactivation energy and the prefactor of diffusion coefficient.

An Arrhenius plot of the diffusion coefficient is given inFigs. 5 and 6; the corresponding parameters are listedin Table 5. The prefactor of the jump process is found to be4.22 and 3.63 ps�1 for position, respectively. We can easilysee that the activation energy calculated from Arrheniusplot of the diffusion coefficient is exactly the same as thatcalculated by the drag method. One can conclude that thecontribution of long jumps in the diffusion coefficient usingEq. (7), act as a correction and brings the dynamic activationbarrier close to the static one.

The prefactors obtained from the Arrhenius plots inFigs. 5 and 6 are 2.7 and 3.6 (�10�4 cm2 s�1) for the upperand lower positions, respectively. These findings are in agood agreement with experimental results carried out for Cu/Cu(100) where the prefactor diffusion is 1.4� 10�4 cm2

s�1 [44]. For hopping on the (001), (110), and (111) of Cu,Ag, and Ni, the prefactors are found to be in the range 10�1

–

10�4 cm2 s�1 [45], in line with the value of 10�3 cm2 s�1 thatis generally used. Yildirim et al. demonstrated that for Cuadatom on Cu(110) the prefactor of diffusion is found to be6.39� 10�4 cm2 s�1 [46].

5 Conclusions In this paper, we summarized adetailed study of the diffusion of the adatom on (110)

surface in the case of heterogeneous systems. The numericaltool used in this investigation is MD simulations based onthe EAM. Our results are discussed in the context of recentstudies that envisage understanding of heteroepitaxy of ametal on a metal. The values we obtain of activation barriersfor diffusion indicate that diffusion via hopping along astraight edge is more favored than the others processes. ForAg on a Cu(110) surface, the activation energy of the jumpprocess near the step edge (0.25 eV) is slightly larger thanthis on a terrace (0.23 eV). On the other hand, a goodagreement between statistical and static calculations foractivation energy is found. While the presence of a largeactivation energy for diffusing over the step both via anexchange process and via hopping, indicating lower masstransport from the upper to the lower surface. This result canenable us to predict a (3D) growth for such a system. Wealso note that a calculation of the prefactor of the diffusioncoefficient has been made. The prefactor is found to be2.7 and 3.6� 10�4 cm2 s�1 for the upper and the lowerpositions, respectively. One cannot forget the contributionof long jumps in the total jump process, which can reach12% for double jumps as mentioned before. In effect, wefound that the adatom can move further than nearest-neighbor sites and reach third and even sometimes fourthpositions. This is due to the small activation energy neededfor the jump and to the weak energy dissipation between theAg adatom and Cu substrate in agreement with our previousfindings [23].

Acknowledgement We acknowledge the financial supportfrom CNRST-Morocco to the University of Chouaïb Doukkali(contract: 104/2009) under the project: Sciences of Materials andthe Framework of Averroes II Program.

References

[1] A. Yokotani, Y. Okazaki, K. Nurulhusna, M. Tode, andY. Takigawa, Opt. Mater. 32, 759 (2010).

[2] R. Ferrando, J. Jellinek, and R. L. Johnston, Chem. Rev. 108,845 (2008).

[3] A. Hader, I. Achik, A. Hajjaji, and Y. Boughaleb, Opt. Mater.34, 1713 (2012).

[4] B. Geisler, P. Kratzer, T. Suzuki, T. Lutz, G. Costantini, andK. Kern, Phys. Rev. B 86, 115428 (2012).

[5] X. Liu, M. Hupalo, C.-Z. Wang, W.-C. Lu, P. A. Thiel, K.-M.Ho, and M. C. Tringides, Phys. Rev. B 86, 081414 (2012).

[6] G. Ehrlich, Surf. Sci. 246, 1 (1991).[7] G. Antczak and G. Ehrlich, Surf. Sci. Rep. 62, 39–61

(2007).[8] Y. Jin, C. Jia, S.-W. Huang, M. O’Donnell, and X. Gao,

Nature Commun. 1, 41 (2010).[9] M. Liu and P. Guyot-Sionnest, J. Phys. Chem. B 108, 5882

(2004).[10] Y. Boughaleb, R. O. Rosenberg, M. A. Ratner, and A. Nitzan,

Solid State Ion. 18–19, 160–168 (1986).A. Asaklil, Y. Boughaleb, M. Mazroui, M. Chhib, and L. ElArroum, Solid State Ion. 159, 331–343 (2003).

[11] O. O. Brovko, N. N. Negulyaev, and V. S. Stepanyuk, Phys.Rev. B 82, 155452 (2010).



Original

Paper

[12] G. Ehrlich and F. G. Hudda, J. Chem. Phys. 44, 1039 (1966);R. L. Schwoebel and E. J. Shipsey, J. Appl. Phys. 37, 3682(1966).

[13] J. Villain, J. Phys. I (France) 1, 19 (1991).[14] R. Gerlach, T. Maroutian, L. Douillard, D. Martinotti, and

H.-J. Ernst, Surf. Sci. 480, 97 (2001);F. Rabbering, H. Wormeester, F. Everts, and B. Poelsema,Phys. Rev. B 79, 075402 (2009).

[15] F. Montalenti and R. Ferrando, Phys. Rev. B 59, 5881 (1999).[16] M. Karimi, T. Tomtowski, G. Vidali, and O. Biham, Phys.

Rev. B 52, 5364 (1995).[17] H. Yildirim, A. Kara, S. Durukanoglu, and T. S. Rahman,

Surf. Sci. 557, 190 (2004).[18] J. Wang, H. Huang, and T. S. Cale, Modell. Simul. Mater. Sci.

Eng. 12, 1209 (2004).[19] S. Youb Kim, I.-H. Lee, and S. Jun, Phys. Rev. B 76, 245408

(2007).[20] F.-M. Wu, H.-J. Lu, and Z.-Q. Wu, Chin. Phys. 15, 807

(2006).[21] G. Antczak and G. Ehrlich, Phys. Rev. Lett. 92, 166105

(2004).[22] J. Repp, G. Meyer, K. H. Rieder, and P. Hyldgaard, Phys.

Rev. Lett. 91, 206102 (2003).[23] M. Giesen, Prog. Surf. Sci. 68, 1 (2001).[24] M. C. Marinica, C. Barreteau, M. C. Desjonquères, and

D. Spanjaard, Phys. Rev. B 70, 075415 (2004).[25] K. Sbiaai, Y. Boughaleb, J.-Y. Raty, A. Hajjaji, M. Mazroui,

and A. Kara, J. Opt. Adv. Mater. 14, 1059 (2012).[26] C. Langlois, Z. L. Li, B. Jun Yuan, D. Alloyeau, J. Nelayah,

D. Bochicchio, R. Ferrando, and C. Ricolleau, Nanoscale 4,3381 (2012).

[27] G. Garzel, J. Janczak-Rusch, and L. Zabdyr, Calphad 369,52 (2012).

[28] Suk. Jun. Kim, E. A. Stach, and C. A. Handwerker, Appl.Phys. Lett. 96, 144101 (2010).

[29] M. S. Daw and M. I. Baskes, Phy. Rev. B 29, 6443 (1984).[30] R. Ferrando and G. Tréglia, Phys. Rev. B 50, 12104 (1994).[31] M. C. Marinica, C. Barreteau, D. Spanjaard, and M. C.

Desjonquères, Phys. Rev. B 72, 115402 (2005).[32] K. Sbiaai, Y. Boughaleb, M.Mazroui, A. Hajjaji, and A. Kara,

Thin Solid Films 548, 331 (2013).[33] K. Sbiaai, Y. Boughaleb, M. Mazroui, and A. Kara, Surf.

Interface Anal. 45, 1702 (2013).[34] J. Ellis and J. P. Toennies, Phys. Rev. Lett. 70, 2118 (1993).[35] D. Cowell Senft and G. Ehrlich, Phys. Rev. Lett. 74, 294

(1995).[36] K. Sbiaai, A. Eddiai, Y. Boughaleb, A. Hajjaji, M. Mazroui, and

A. Kara, Opt. Mater. (2013), DOI: 10.1016/j.optmat.2013.06.016.[37] C. Mottet, R. Ferrando, F. Hontinfinde, and A. C. Levi, Surf.

Sci. 417, 220 (1998).[38] L. Hansen, P. Stoltze, K. W. Jacobsen, and J. K. Norskov,

Phys. Rev. B 44, 6523 (1991).[39] L. S. Perkins and A. E. DePristo, Surf. Sci. 317, L1152

(1994).[40] C. J. Liu, J. M. Cohen, J. B. Adams, and A. F. Voter, Surf. Sci.

253, 334 (1991).[41] M. Karimi, T. Tomkowski, G. Vidali, and O. Biham, Phys.

Rev. B 52, 5364 (1995).[42] O. V. Stepanyuk, N. N. Negulyaev, A. M. Saletsky, and

W. Hergert, Phys. Rev. B 78, 113406 (2008).[43] H. Yildirim and T. S. Rahman, Phys. Rev. B 80, 235413

(2009).[44] G. Boisvert and L. Lewis, Phys. Rev. B 56, 7643 (1997).[45] L. T. Kong and L. J. Lewis, Phys. Rev. B 74, 073412 (2006).[46] H. Yildirim, A. Kara, S. Durukanoglu, and T. S. Rahmann,

Surf. Sci. 600, 484 (2006).



ph

ysic

a ssp stat

us

solid

i b

Documents

Long jumps contribution to the adatom diffusion process near the step edge: The case of Ag/Cu(110)