Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate

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1 Cent. Eur. J. Eng. 3(4) DOI: /s Central European Journal of Engineering Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate Research Article Hossein M. Shodja 1,2, Maryam Tabatabaei 1, Alireza Ostadhossein 3, Ladan Pahlevani 2 1 Department of Civil Engineering, Sharif University of Technology, P.O. Box , Tehran, Iran 2 Institute for Nanoscience and Nanotechnology, Sharif University of Technology, P.O. Box , Tehran, Iran 3 Department of Engineering Science and Mechanics, W 305 Millinum Science Complex, The Pennsylvania State University, University Park, PA 16802, USA Received 21 October 2012; accepted 04 July 2013 Abstract: Certain physical and mechanical phenomena within ultra-thin face-centered cubic (fcc) films containing common types of interacting point defects are addressed. An atomic-scale lattice statics in conjunction with many-body interatomic potentials suitable for binary systems is conducted to analyze the effects of the depth on the: (1) formation energy and layer-by-layer displacements due to the presence of vacancy-octahedral self-interstitial atom (OSIA) ensemble, and (2) elastic fields as well as the free surface shape in the case of vacancy-dopant interaction. Moreover, the effects of the inter-defect spacing for various depths are also examined. To ensure reasonable accuracy and numerical convergence, the atomic interaction up to the second-nearest neighbor is considered. Keywords: Ultra-thin fcc film Interacting point defects Lattice statics method Versita sp. z o.o. 1. Introduction The atomic point defects, within a perfect crystal lattice, separated by only a few interatomic spacings can interact with each other. How the interaction changes with respect to the inter-defect spacing is of particular interest to the community in materials science. This interaction may strongly influence the local electro-mechanical fields, which becomes critical in such devices as sensors and shodja@sharif.edu actuators with high precision [1 3]. Subsistence of even one point defect can adversely affect the properties of the material [4]. The insertion of imperfections and inhomogeneities within a perfect crystal disturbs the host lattice and results in residual stresses [5]. The non-local effects stemming from the residual stresses induced by point defects often play an important role on the long term stability of an alloy. Depending on the crystal lattice structure and the pertinent boundary conditions, the existence of a point defect may lead to different lattice distortions and consequently, various displacement fields will be observed. For instance, the atoms in the vicinity of a vacancy may move towards the vacant site, or move 707

2 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate away due to the attraction of the surrounding atoms. These defect-induced displacement fields can alter the physical and mechanical properties of the material such as electrical resistivity, hardening, and embrittlement. It is interesting to note that some authors have studied lattice distortion due to the presence of a point defect, two interacting point defects, or periodically distorted defects in the bulk only. For instance, using Morse interatomic potential function, [6] for bulk copper, the energies of formation for two configurations of an intrinsic (self) interstitial, body-centered position (octahedral site) and split or dumbbell, were calculated, [7]. The method of lattice statics was used to calculate the strain field due to a pair of interacting point defects in an infinite superlattice of a harmonic discrete face-centered cubic (fcc) model, [8]. On the basis of discrete lattice statics and Hantington s Born-Mayer potential, the interaction between an octahedral self-interstitial atom (OSIA) and the Cu host lattice has been described, [9]. Moreover, Sato et al. [10] calculated the displacement field around a single or multiple octahedral interstitial atoms in a body-centered cubic (bcc) lattice. They presented the numerical results for α iron crystal by utilizing the notion of eigenstrain in lattice theory with the harmonic approximation. However, as mentioned earlier the presence of a single point defect or a pair of interacting point defects in thin films can result in drastic distortions of their lattice structures. The problem of interacting point defects due to their great impact on the physical and mechanical properties of thin films is worthy of examination. The current study focuses on ultra-thin fcc film bonded to a rigid substrate. The induced lattice distortion pertinent to an OSIA near a vacancy located at various sites is studied. Moreover, the interaction between a pair of dopant and vacancy, having the same depth from the free surface, is modeled. The corresponding implications are addressed, and the influence of the distance from the free surface and the film-rigid substrate interface is investigated. 2. Energy considerations Consider an ultra-thin film made of fcc crystal consisting of N atoms. Set the origin of the Cartesian coordinates at the center of the specimen in such a way that periodic boundary condition occurs along the x 1 and x 2 directions. The x 1, x 2, and x 3 -directions are respectively in accordance with the cubic axes of the conventional unit cell, [1 0 0], [0 1 0], and [0 0 1], Figure 1(a). The model may consist of different types of point defects like an interstitial atom, a doped atom, and a vacancy; such defects are marked as at the distance D from the free surface in Figure 1(a). Figure 1(b) involves an OSIA which occupies a site other than the host lattice sites. In Figure 1(c), the substitutional atom is indicated by the dark solid sphere. In both Figure 1(b) and Figure 1(c), a vacant site is illustrated by a cube. The energy considerations associated with the interacting point defects consisting of OSIA-vacancy or dopant-vacancy ensemble are given in Sections 2.1 and 2.2, respectively Calculation of the formation energies of OSIA and OSIA-vacancy interaction For this study, the ultra-thin fcc film with dimensions 11a 11a 6a is simulated in such a way that its free surface coincides with the (0 0 1) plane, where a is the lattice constant of the crystal. The film thickness is 6a and N is nearly 3150 atoms. For a given depth, D, two different OSIA-vacancy configurations are considered. The OSIA is assumed to be at (0, 0, 3 a D; D), whereas the vacant site is placed at any of the two locations, (2.5 a, 0, 3 a D; D) or (2 a, 0.5 a, 3 a D; D); thus, at each specified depth, there are two possible cases which will be referred to as, case I or II, respectively, as shown in Figure 1(b). The formation energies of OSIA, E OSIA and OSIA-vacancy ensemble, E OSIA v are defined as: E OSIA = E T (N + 1, I, 0V ) N + 1 N E T (N, 0, 0), (1) E OSIA v = E T (N, I, 1V ) E T (N, 0, 0), (2) where E T (j, I, iv ) and E T (j, 0, 0) are the total energies of the model consisting of j atoms; the former energy is related to the model with one interstitial atom and i vacancy while the latter energy is pertinent to a perfect specimen without any defects Calculation of the formation energy of a dopant-vacancy interaction In this problem, the dimensions of the simulation box containing N = 2904 atoms are 11 a, 11 a, and 5.5 a in the x 1, x 2, and x 3 directions, respectively. It is assumed that x 3 = 2.5a atomic plane is the free surface and the atomic layer, x 3 = 3a is perfectly bonded to a rigid substrate. To investigate the effect of the distance from the free surface, doping is done at different layers, p = x 3 /a = 0, ±1 and 2. In each configuration, the effect of inter-defect spacing on the interaction is examined by varying the distance between the dopant and vacancy. 708

3 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani Figure 1. (a) The nomenclature of an fcc thin film containing a point defect marked as at the distance D from the free surface, (b) the configuration of OSIA-vacancy for two different positions, indicated as case I and case II, and (c) a dopant-vacancy ensemble. The position of the doped atom and vacant site as shown in Figure 1(c) are symmetrical with respect to the x 1 = 0 plane; and the inter-defect spacing, L is varied from 2a to 6a. For instance, for the case exhibited in Figure 1(c), the vacancy is at ( a, 0, 0; D) and L = 2a. In this type of problem the equilibrium state of distorted lattice and the formation energy of any dopant-vacancy configurations via the atomic approach are of main attempts. The formation energy of dopant-vacancy, E dop v associated with (N 1) atoms is related to the total energy of the defective model with one dopant and one vacancy, E T (N 1, Dop, 1V ) and that of the perfect crystal model, E T (N, 0, 0) as: E dop v = E T (N 1, Dop, 1V ) N 1 N E T (N, 0, 0). (3) Furthermore, the effect of free surface and inter-defect spacing on the stress distribution within the film is studied. 3. The computational technique Throughout this section, the computational details of total energy and stress field, including a suitable potential function which considers the long-range interactions and binary elements are presented. As it was alluded to, the problems are essentially static problems, which can be examined by utilizing a suitable potential function in the framework of lattice statics. In this approach, minimization of the total energy of a system consisting of N atoms leads to Ku = f, (4) where, K is the stiffness matrix, u is the displacement vector, and f is the force vector which accounts for the interaction of each atom with its first and second neighbors. f and K are obtained by consideration of the first and second derivatives of the total potential energy of the system with respect to the atomic position. The displacement vector is determined as u = r r 0 ; where r 0 is the initial atomic configuration and r is the atomic equilibrium position vector after relaxation. The system is assumed to be relaxed when its total potential energy, Φ T is no longer changing. Mathematically, K and f at the initial stage, r = r 0, are given as K = 2 E T r r r=r 0 = 2 Φ T r r r=r 0, f = E T r r=r 0 = F Φ T r r=r0, (5) where F is the applied external force. Let r i and F i denote the position of the ith atom and the external force acting on atom i, respectively. For a system composed of N 709

4 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate Table 1. The parameters associated with the RS potential function for Cu and Al fcc metals. Element a (Å) ε( ev) c m n Cu E Al E atoms, Φ T and E T are linked via the following relation E T (r 1, r 2,..., r N ) = Φ T (r 1, r 2,..., r N ) N F i.r i. (6) The potential energy is calculated using the many-body Rafii-Tabar and Sutton (RS) [11] interatomic potentials, a proper potential function for binary systems. RS potentials have been obtained by modification of Sutton- Chen (SC) potentials [12] which are valid for monoatomic metals only. SC potential is an empirical manybody, long-range potential of a Finnis and Sinclair [13] type which combines the short-range interaction feature of Finnis-Sinclair with a Van der Waals tail, suitable for accounting for the long-range interactions. The many body Finnis and Sinclair potentials describe the cohesion in metals and in contrast to pair potential functions, are capable of modeling the relaxation at the free surfaces of metals. SC potential has proved useful in the studies pertinent to nano-voids, cracks, inclusion within thin films, [14 16]. For demonstration, the values of RS potential function parameters for Cu and Al are given in Table 1. Where, ε is a parameter with the dimension of energy, n and m are positive integers, and c is a positive dimensionless parameter scaling the attractive terms. Associated with two interacting atoms, i and j, f i, f j and K ij, i, j = 1, 2,..., N are calculated and substituted into Eq. (4) from which the atomic positions are obtained. The details of the formulations for the stiffness matrix and the force vector components are given in Appendix A. Once the forces acting on each atom have been calculated, the stress tensor at any atomic site can be evaluated. The concept of atomic level stress field was first developed by Born and Huang, [17] using the method of small homogeneous deformation. Applying a small displacement between a pair of atoms i and j, then σ i αβ = 1 2V a j>i i=1 Φ T (r ij ) rα ij r ij β (7) r ij r ij where, σαβ i is the (αβ)th component of stress at atom i on α-plane and in β-direction, and V a is the local atomic volume. Also, rα ij is the αth component of r ij = r j r i. 4. Results and discussion A three-dimensional simulation technique based on lattice statics in conjunction with RS interatomic potential function for the treatment of an ultra-thin fcc film adhered to a rigid substrate has been developed. Note that, the description of the models including the dimensions of the films has been given in Sections 2.1 and 2.2. The validity of the applied methodology is substantiated throughout Section 4.1. Section 4.2 is devoted to the calculations of the layer-by-layer displacements and surface deformation caused by the OSIA-vacancy; the effect of the depth of the OSIA-vacancy ensemble is addressed. Section 4.3 considers the interaction between a doped atom and a vacancy and gives the induced elastic fields and formation energies for various configurations. Finally, a summary and conclusion are given in Section Lattice distortion and formation energy of an OSIA in Cu bulk: Validity of the current computational methodology In order to examine the validity of the proposed simulation technique, the lattice distortion and the formation energy of an OSIA within Cu bulk are calculated and compared with the previous results reported in the literature. To this end, an OSIA located at the center of a Cu specimen with periodic boundary condition along all three directions is considered. By employing the method of lattice statics, the displacement components of the neighbor atoms next to the interstitial atom are calculated. For verification, the results of certain cases which have been previously obtained by employment of various methods in conjunction with different interatomic potential functions are compared with the results of the present approach. This problem has also been simulated using a three-dimensional molecular dynamics (MD) simulation which was previously developed for studying the inclusion problems associated with thin fcc films, [16]. In this MD simulation RS potential function has been incorporated. During the MD simulation of the present problem the temperature is held near absolute zero using a simple temperature scaling method. After relaxation, the normalized components of displacement at several neighbors of the OSIA are calculated by the proposed method and the threedimensional MD simulation; the results are compared with those of Hoekstra and Behrendt [7] in Table 2. The positions of the neighbors are indicated by (h, k, l)b where b is one-half of the lattice parameter, b = a/2 = Å. For brevity, the normalized component 710

5 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani of displacement in the x i direction is denoted by i which is equal to x i /b, i = 1, 2, 3. The negative value of i associated with an atom denotes its displacement from the normal lattice site in the direction of decreasing x i. In the study of Hoekstra and Behrendt [7] the equilibrium position of the atoms was calculated based on the pair-wise Morse potential function. They have limited their simulation to a small cluster of movable atoms. More specifically, they considered the motion of the 117 nearest neighbors of OSIA enclosed within a sphere of approximate radius 13 b centered at the location of OSIA and bonded to a rigid lattice. It is interesting to note that, the results obtained by the lattice statics, MD simulation, and the simulation proposed by Hoekstra and Behrendt [7] are in good agreement for the first and next neighbors of OSIA. While the results of lattice statics and MD for the atoms beyond second neighbor remain in good agreement, the corresponding results obtained from [7] deteriorate, which is probably due to the additional constraint that has been imposed on the mobility of atoms. Flocken and Hardy [9] have pointed out that the effect of the constraints on the calculated displacement field are quite serious. The displacement components of some atoms, further away from the nearest 117 atoms, calculated from lattice statics are given in Table 3. As it is seen, these displacements are non-zero. In addition, Flocken and Hardy [9] calculated the lattice distortion produced by an OSIA in Cu bulk. The calculations were carried out consistently on the basis of discrete lattice theory using the technique of lattice statics. Furthermore, Seeger and Mann [18], Bennemann and Tewordt [19], and Seeger et al. [20] attempted to calculate the displacement components of several atomic sites due to an OSIA based on a semi-discrete method. In all the above-mentioned references, the interaction is confined to act only between the nearest-neighbor atoms and is assumed to be given by a Huntington s Born-Mayer potential. Their results are summarized in Table 4(a). The corresponding results calculated by the lattice statics and MD employed in the current work are presented in Table 4(b). It is observed that the solutions of lattice statics method, MD simulation, and previous studies show good correspondence, and the observed small discrepancy is due to the fact that the type of employed interatomic potential function has a great influence on the results. Flocken and Hardy [9] have argued that the accuracy of the obtained results is limited by the validity of the interatomic potentials they have assumed, and it is probable that the use of more refined potential will change their numerical results. It should be noted that in the current calculations the utilized many-body interatomic potential considers interaction between atoms up to the second-nearest neighbor. For further verification, the formation energy, E OSIA of an OSIA in Cu bulk is calculated and compared with results of the recently published paper by Jelinek et al. [21]. They have developed a modified embedded atom method (MEAM) potential for Al, Si, Mg, Cu, and Fe alloys and have tested its validity via comparison with ab initio simulations and available experimental results. Jelinek et al. [21] obtained the self-interstitial formation energy for Cu at the octahedral site using both the ab initio calculations and their MEAM potential as 3.5 and 2.72 ev, respectively. Jelinek et al. [21] also incorporated the EAM potential parameters for Cu from the work of Mendelev et al. [22] and obtained the formation energy of OSIA within Cu bulk to equal 2.97 ev. The calculated result for E OSIA via the current lattice statics method is 2.58 ev which is in reasonable agreement with the abovementioned energies Surface deformation and layer-by-layer displacements of the ultra-thin Cu film due to the presence of OSIA-vacancy ensemble as a function of the distance from the free surface One important issue due to the conductivity of metals is volume change introduced in the material under the presence of defects. Thus, in this manuscript, lattice distortion of the specimen due to the interaction between an OSIA and a vacant site is examined. Since the volume change in an ultra-thin film will be affected by the distance of defects from the free surface, the interstitial-vacant site ensemble is placed at different depths of the film; for example, in the vicinity of the free surface and that of the rigid substrate. The other notable feature is to consider the effects of the point defects on the deformation and/or surface configuration. As discussed by Freund and Suresh [23], from the perspective of mechanical behavior, the free energy of thin film changes due to changes in surface shape. Note that, the free energy is calculated by summing the surface energy of the system and the elastic energy associated with the bulk OSIA-vacancy next to the free surface First of all examine the case where an interstitial is at the distance between the second and third atomic layers from the free surface, that is D = 1.5 a. As follows, by employing lattice statics technique in conjunction with RS interatomic potential, the vertical displacements of the atoms normalized with respect to the lattice parameter of the film are computed. The effect of the vacant site with respect to the OSIA on the lattice distortion of the atomic interlayers and the free surface layer 711

6 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate Table 2. The normalized displacement components of the atoms near the OSIA within the Cu host lattice. Present Study Ref. [7] MD Lattice Statics Neighbor Table 3. The normalized displacement components of the atomic sites, beyond the nearest 117 neighbors of an OSIA within the Cu host lattice. Neighbor Neighbor pertinent to cases I and II is demonstrated in Figure 2(a) and Figure 2(b), respectively. These plots show the normalized displacement component along the x 3 -axis, u 3 /a of the atoms located on the interlayers, p = x 3 /a = ±0.5, ±1.5, ±2.5 and on the layers, p = ±3 in (0 1 0) plane with x 2 = 0 in terms of x 1 /a. From Figure 2(a) and Figure 2(b), it is observed that after relaxation the atomic interlayer, p = 1.5, which contains the OSIA undergoes the largest displacement. Though the OSIA moves towards the rigid substrate by a significant amount, all the other atoms move slightly towards the free surface. Note that, the atomic layer, p = 3 is on the rigid substrate and remains fixed. It is evident that the atomic interlayers which are near the OSIA undergo nonuniform displacements. The nonuniformity of the displacements pertinent to atomic interlayer damps out with distance from the OSIA. Figure 2(a) shows the corresponding displacements when the OSIA and vacancy are in the same plane. The results indicate that considerable layer-by-layer displacements take place within the top several surface layers. From Figure 2(a), it is seen that the atoms on the p = 2.5 interlayer move outward by approximately 0.2 a, while all atoms on the p = 2.5 interlayer move outward uniformly by about a. The interatomic spacing which is the distance between the atomic interlayers of p = 2.5 and p = 1.5, expands nearly 2.5% with respect to pure Cu interlayer spacing. Note that, the expansion of deeper interlayers spacing is smaller than that of the upper one. It is seen that, the existence of vacant site disturbs the symmetry of the lattice distortion with respect to the x 3 axis. The maximum expansion of about 7.4% occurs between the interlayers, p = 1.5 and 0.5, at x 1 /a = 1.5, between the OSIA and vacant site. Recall that, in this section D = 1.5a, and so in Figure 2(a) which is pertinent to the case I, the OSIA and the vacant site are located at (0, 0, 1.5a; 1.5a) and (2.5a, 0, 1.5a; 1.5a), respectively. Figure 2(b) corresponds to the case II, and the OSIA-vacancy are located at (0, 0, 1.5a; 1.5a) and (2a, 0.5a, 1.5a; 1.5a), respectively. Comparison of Figure 2(a) and Figure 2(b) indicates that the location of the vacant site has a significant influence on the displacement distributions of the atomic layers, u 3 /a. It is interesting to note that, the magnitude of the interstitial displacement in the case I is 184.5% larger than that of case II. Also, the layerby-layer displacements in the case I are larger than that of case II. But the interlayer expansions for case II are larger as compared to that of case I. From Figure 2(b), it is found that the interatomic spacing associated with p = 1.5 and 0.5 expands approximately 13.5% for the atoms next to the OSIA; the spacing reduces sharply to about 4% for the second neighbor. Another notable effect of the position of the vacancy is that in the case II (Figure 2(b)) the swelling of the free surface accompanies a concave directly above the OSIA; and the observed deformations are nearly symmetric OSIA-vacancy next to the rigid substrate In this section, an OSIA in the ultra-thin Cu film at the depth of D = 4.5 a, between the fifth and sixth atomic layers from the free surface is considered. It is observed that the formation energy of OSIA pertinent to this depth is greater than those of the previous cases. To summon, 712

7 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani Table 4. The normalized displacement components of some selected neighbors of an OSIA within the Cu host lattice. (a) The results of the previous works. Ref. [9] Ref. [18] Ref. [19] Neighbor (b) The results of the present study. MD Lattice Statics Neighbor the formation energies of OSIA (and OSIA-vacancy I) configurations, E OSIA (and E OSIA v ) corresponding to D = 1.5 a, 2.5 a, 3.5 a, and 4.5 a are 2.85 ev (4.41 ev), 4.45 ev (6.15 ev), 4.62 ev (6.32 ev), and 4.75 ev (6.46 ev), respectively. It is seen that, as the distance between the OSIA (OSIA-vacancy I) and the rigid substrate decreases the formation energy increases. Distortion of the lattice within the (0 1 0) plane which contains an OSIA at (0, 0, 1.5a; 4.5a) and a vacant site pertinent to the configuration I is plotted in Figure 3. That is, in this case the OSIA-vacancy system is just above the layer next to the rigid substrate. Comparison of the differences in mode of deformations displayed in Figure 3 and Figure 2(a) is interesting. Recall that the result shown in Figure 2(a) corresponds to the case in which the OSIA-vacancy I system is located at D = 1.5a that is just below the layer next to the free surface. From Figure 3, it is seen that the atoms located at x 1 /a = ±0.5 and on the p = +2.5 atomic interlayer (just below the free surface) undergo an upward movement of about 0.17 a; the upward displacement of the atoms at x 1 /a = ±0.5 and p = 2.5 (just above the rigid substrate) is about a. On the other hand, for the case shown in Figure 2(a), for the atomic sites x 1 /a = ±0.5, u 3 /a 0.19 for p = +2.5, and u 3 /a for p = 2.5. In Figure 3, the maximum value of the interlayer expansion is about 11.5%, which occurs between the interlayers p = 0.5 and p = 1.5. Whereas this maximum value corresponding to the case shown in Figure 2(a) is about 7.4% which occurs between the interlayers p = 1.5 and p = 0.5. As it is observed from Figure 3, the OSIA undergoes a large displacement, u 3 /a 0.324, towards the free surface, while for the case of Figure 2(a), the OSIA moves towards the rigid substrate by a large amount, u 3 /a Also, from Figure 3, it is evident that the interatomic spacings between p = 0.5 and p = 0.5 and those between p = 1.5 and p = 2.5 pertinent to x 1 /a = ±0.5 are subjected to contraction, while the other interatomic spacings expand Interaction between an Al dopant and a vacancy within an ultra-thin Cu film as a function of the distance from the free surface This section considers the elastic fields introduced by interacting Al dopant-vacancy in ultra-thin Cu film. To demonstrate the effect of the inter-defect spacing, L, on the formation energy of these interacting point defects, the distance between the dopant and vacancy is varied from 2a to 6a. In addition, to account for the free surface interaction with these defects, four different depths at which the dopant-vacancy ensemble is located are considered. In all four cases the defects are located on the plane x 2 = 0, but correspond to four different atomic layers, p = x 3 /a = 0, ±1, Lattice distortion This section examines relative displacement of the atoms within the ultra-thin Cu film in the presence of Al dopantvacancy; the relative displacement is normalized with respect to the lattice parameter of the film. Two interdefect spacings L = 2a and L = 4a are considered, and some fundamental results are inferred. In one case Al atom is doped in the (a, 0, 0; 2.5a) lattice site and the vacancy is located at ( a, 0, 0; 2.5a), and in the other case they are placed at (2a, 0, 0; 2.5a) and ( 2a, 0, 0; 2.5a), respectively. Figure 4 shows the relaxed positions of 713

8 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate (a) (b) Figure 2. Lattice distortion within the (0 1 0) plane of an ultra-thin Cu film containing an OSIA at (0, 0, 1.5a; 1.5a), pertinent to cases (a) I, and (b) II. the nearest neighbors of the vacancy and those of the Al dopant, for the case L = 2a. The positions of the nearest neighbors of the vacancy are measured with respect to the initial vacant site, while the distances from the dopant are measured with respect to its equilibrium position. The dopant is depicted with dark solid sphere and the boundaries of the (1 0 0) plane containing the vacant site is shown by dashed lines. For both cases L = 2a and L = 4a it is found that the nearest neighbors of the Al doped atom move away from the dopant. It is also observed that the displacements of the upper neighbors, atoms labeled 2 and 3 are larger than the displacements of the atoms numbered 5 and 4, respectively. Atoms 2 and 3 are within the upper half of the film, whereas atoms 5 and 4 are in the lower half. For the case of L = 2a, the relative displacements of the atoms 2 and 3, 714

9 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani Figure 3. Lattice distortion within the (0 1 0) plane of an ultra-thin Cu film containing an OSIA at (0, 0, 0.5a; 4.5a) and a vacant site at (2.5a, 0, 0.5a; 4.5a). Figure 4. Lattice distortion in the vicinities of a dopant and a vacancy for L = 2a. from their corresponding equilibrium reference state, are measured as and , respectively while they are computed as and for atoms 5 and 4. The different magnitude of the displacements are due to the different boundary conditions existing at the bottom, which is bonded to a rigid substrate, and the top free surface of the specimen. Moreover, it seems that by increasing the inter-defect distance, the outward displacements of the dopant s neighbors, except that of atom 1, decrease. As far as the nearest neighbors of the vacancy are concerned, in both cases of L = 2a and L = 4a, the atoms 6 and 8 which are on the p = 0.5 plane move away from the vacant site. On the other hand, the atoms 7 and 9 which are located on the p = 0.5 plane are attracted towards the vacant site. The in-plane normalized displacements of the atoms located on the p = 0 plane around the vacant site and the substitutional Al atom are shown in Figure 5 for case L = 2a. For both cases L = 2a and L = 4a, it is understood that the doped atom induces a compressive field around itself and so its neighbors are pushed apart. In contrast, the vacancy produces a tensile field, resulting in the nearest neighbors to be attracted. Another attempt is made to examine the effect of the depth of the atomic layer containing the dopant and the vacancy on the distortion of the free surface. To this end, the distance between the free surface and the layer containing 715

10 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate Figure 5. The in-plane displacements of the atoms near the dopant and a vacancy for L = 2a. dopant-vacancy ensemble, D is varied from 1.5a to 3.5a. The free surface shape associated with the cases D = 1.5a, 2.5a and 3.5a are depicted in Figures 6(a) 6(c), respectively. It is seen that in all cases, the maximum swelling of the free surface occurs in the atom located directly above the dopant. The compressive field in the vicinity of the dopant repels the upper layers towards the free surface which subsequently results in local convexity of the free surface. Another mentionable trend is that as the distance between the upper most layer and the layer containing dopant-vacancy decreases the value of the maximum deflection of the free surface increases Stress field and formation energy The normalized stress distribution, σ 33 V a ε 1 Cu along the atomic interlayers, p = x 3 /a = ±0.5, ±1.5, ±2.5 and p = 0 atomic layer in (0 1 0) plane with x 2 = 0, for two cases of L = 2a and 4a, are depicted in Figure 7(a) and Figure 7(b), respectively. Moreover, for clarity of the behavior of the free surface (p = +2.5) the pertinent magnified stress distributions have also been included in Figure 7(a) and Figure 7(b). It should be noted that in the pure ultra-thin Cu film, p = 1.5 atomic interlayer is subjected to uniform tensile stress, σ 33 V a εcu 1 = 45.8; also p = 1.5, 0, ±0.5 are all under uniform tensile stress with σ 33 V a εcu 1 = From Figure 7(a) and Figure 7(b) it is evident that insertion of an Al dopant-vacancy ensemble on the p = 0 atomic layer will significantly disturb the stress distribution along p = 0, ±0.5 in the vicinities of the dopant and vacancy. Note that, perturbations in the stress distributions along p = ±1.5, ±2.5 atomic interlayers are small in comparison with those along p = 0, ±0.5. The dopant located on the p = 0 atomic layer is subjected to the maximum compressive stress; the magnitude of the stress σ 33 in cases L = 2a and 4a is a little above and below 250 Va 1 ε Cu ( ev ), Å 3 respectively. Since the atomic interlayers of p = ±0.5 remain unaffected by the boundary conditions, the free surface and the rigid substrate, the stress distributions associated with these atomic interlayers nearly coincide. The presence of vacancy has induced local tensile stress in addition to the uniform initial tensile stress in the pure film. The effect of the boundary conditions and the distance 716

11 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani (a) (b) (c) Figure 6. The free surface shape induced by a dopant-vacancy system with (a) D = 1.5a, (b) D = 2.5a, and (c) D = 3.5a. 717

12 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate (a) (b) Figure 7. The normalized stress distribution along the atomic layers within the (0 1 0) plane containing a dopant-vacancy system for (a) L = 2a, and (b) L = 4a. between the vacant site and the Al dopant on their formation energy for three cases of L = 2a, 4a, and 6a is given in Table 5. The formation energy of each configuration is calculated with respect to the total energy of the perfect crystal. It is realized that the formation energy of the mentioned pair of defects is larger if they are located on the atomic layers which are closer to the free surface. When the dopant-vacancy ensemble is located 718

13 H. M. Shojda, M. Tabatabaei, A. Ostadhossein, L. Pahlevani Table 5. The formation energy of dopant-vacancy configuration, E dop v (ev). Inter-defect spacing p 2a 4a 6a on the p = 1 or 2 atomic layers, increase of the interdefect spacing does not change the energies of formation; but if located on p = 1 layer, decrease of the inter-defect spacing diminishes the formation energies. To examine the influence of the vacant site, the formation energy of the defect ensemble consisting of a dopant at the atomic site (2a, 0, 2a; 4.5a), in the absence of vacancy, has been compared to that of the ensemble in which a vacancy is also present at the distance of 4a from the dopant. The formation energy of the ensemble without the vacancy equals 0.11 ev and when the vacancy is included it is 1.06 ev as displayed in Table Concluding remarks A new simulation based on the method of lattice statics which incorporates RS interatomic potential function is employed to predict and explain several physical and mechanical phenomena in ultra-thin fcc film. The main effort is focused on assessing the effects of the boundary conditions, consisting of the free surface and the rigid substrate, on the: (1) layer-by-layer displacements in the presence of an OSIA-vacant site ensemble at different depths of the film, and (2) elastic fields caused by a dopant-vacancy ensemble located at various layers. It is shown that the distance of the point defects from the free surface plays an essential role in the results. The effect of inter-defect spacing is also examined. To ensure reasonable precision and numerical convergence, the first and second nearest neighbors have been incorporated in the computations. Corresponding to the case when the free surface of the Cu film is perpendicular to the [0 0 1] direction, the formation energy of an OSIA reduces considerably as the interstitial atom approaches the free surface. It may be partially because of the changes in deformation and/or the free surface shape. It is noteworthy to mention that when the OSIA is near the rigid substrate or the Cu (0 0 1) free surface, it is subjected to a rather large displacement into the material. Present results indicate that in addition to the deformation process of surface layer atoms, considerable layer-by-layer displacements take place within the top several layers. In general by increasing the distance of the OSIA from the free surface, the first interlayer spacing decreases, while the last interlayer spacing increases. In addition, the formation energies due to the interacting dopant and vacancy located at various depths of the film demonstrate the effect of the distance from the free surface. These results for Al dopant in Cu film indicate that the inter-defect (dopant-vacancy) spacing is significant when located on the atomic layer p = 1 (close to the free surface), and insignificant when located on the atomic layers p = 1 or 2 (bottom half of the film). Note that, in contrast with the formation energy of an OSIA-vacancy I which decreases by moving towards the free surface, the formation energy of doped atom-vacancy configuration increases. The lattice distortion induced by interacting Al dopant-vacancy within the film indicates that the nearest neighbors of the Al dopant are affected by its induced compressive field. The atoms in the close vicinity of the dopant are repelled from their initial positions, while the atoms next to the vacant site are sunk towards the vacant site due to its attractive field. Scrutiny of the thin film surface reveals that the swelling of the surface which occurs directly above the dopant and the vacant site increases when the distance between the free surface and the layer containing dopant-vacancy configuration decreases. The study of stresses within the pure Cu film shows that the Cu lattice is subjected to uniform tensile stress, but insertion of Al dopant-vacancy ensemble into the film results in high stress concentrations in the neighborhoods of these point defects; the absolute value of the stress concentration is maximum at the site occupied by Al dopant. For small inter-defect spacing, the absolute value of the local maximum of stress near the vacant site is 31% of the absolute local stress at the dopant site. Note that, by moving away from the mid-layer, p = 0 towards the free surface (p = +2.5) or towards the rigid substrate (p = 3), the perturbations in the stress distributions become small in comparison with those of p = 0, ±0.5 atomic layers. For the pure Cu film, the magnitude of stresses on p = ±2.5 is about two times greater than those exerted on the atoms located over the inner layers. Acknowledgement This work was in part supported by the Sharif University of Technology. 719

14 Elastic fields of interacting point defects within an ultra-thin fcc film bonded to a rigid substrate Appendix A For a specimen consisting of N interacting binary atoms, in the context of the RS interatomic potential function, the αβth component of the stiffness associated with the ith and jth atoms is calculated as: k ij αβ i j = 1 2 rα ij r ij { β ˆp iˆp (r ij ) 4 j n AA ε AA θ(a, n) ( 1 + n AA) + ˆp iˆp j n AA ε AA θ(a, n) ( 1) ˆq iˆq j n BB ε BB θ(b, n) ( 1 + n BB) + ˆq iˆq j n BB ε BB θ(b, n) ( 1) + (ˆp ) iˆq j + ˆp j ˆq i [( 1 4 (naa + n BB ) ( n AA + n BB) ) ε ( 2) AA ε 2 BB θ(a, n) θ(b, n)] d AAˆp i S (ˆp ) ( j, ˆq j ˆp j m AA θ(a, m) ( 1 + m AA) + ˆp j m AA θ(a, m) ( 1) ˆq j [( 1 4 (maa + m BB ) 2 1 ( m AA + m BB) ] θ(a, ) ( 2)) m) θ(b, m) 2 d BB ˆq i S (ˆq ) ( j, ˆp j ˆq j m BB θ(b, m) ( 1 + m BB) + ˆq j m BB θ(b, m) ( 1) + ˆp j [( 1 4 (maa + m BB ) ( m AA + m BB) ) ] θ(a, )} ( 2) m) θ(b, m), (8) 2 k ii αβ = k ij αβ. (9) j i Where (r ij ) 2 = δ αβ, (10) rα ij r ij β 1 S(η, ζ) = ( j i η θ(a, m) + ζ ), (11) θ(a, m) θ(b, m) α, β x, y, or z; i, j = 1, 2,..., N. In Eq. 8 θ(a, n) = (aaa ) naa ; d AA = ε AA c AA, and θ(a, m), θ(b, n), θ(b, m), and (r ij ) naa d BB can be obtained in a similar manner. A and B refer to types A and B atoms, respectively. δ αβ is the Kronecker delta. The site occupancy operator, ˆp i is equal to 1 when the site i is occupied by an A atom and it is 0 when the site i is occupied by a B atom. ˆq i is determined as ˆq i = 1 ˆp i. Moreover, the αth component of the force on the ith atom is obtained from: f i α = 1 2 j i rα ij { ˆp iˆp (r ij ) 2 j n AA ε AA θ(a, n) + ˆq iˆq j n BB ε BB θ(b, n) + 1 ) ( (ˆpiˆq j + ˆp j ˆq i n AA + n BB) ε 2 AA ε BB θ(a, n) θ(b, n) d AAˆp i S (ˆp ) [ˆp j, ˆq j j m AA θ(a, m) ˆq ( j m AA + m BB) θ(a, m) θ(b, m) d BB ˆq i S (ˆq ) [ˆq j, ˆp j j m BB θ(b, m) ˆp ( j m AA + m BB) ] } θ(a, m) θ(b, m). (12) ] 720

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