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Based on linear superposition rules and fast discrete Fourier transformation, a semianalytical solution is developed for calculating the elastic fields induced by dislocation loops in an isotropic thin filmsubstrate system. The elastic field problem of thin filmsubstrate system is decomposed into two subproblems: bulk stress due to a dislocation loop in an infinite space, and correction stress induced by free surface and interface of the filmsubstrate system. Correction elastic field is linearly superimposed onto bulk elastic field to produce continuous displacement and traction stress across the interface plane of the perfectlybounded filmsubstrate system. Firstly, calculation examples of dislocation loops in CuNb filmsubstrate system are performed to demonstrate the calculation efficiency of the developed semianalytical approach. Then, elastic fields of dislocation loops within Cu film and Nb substrate of the CuNb filmsubstrate system are analyzed. Finally, effects of film thickness, loop positions are investigated, and it is found that the elastic fields of dislocation loop are influenced remarkably by these two factors.
Keywords 
Elasticity; CuNb; Dislocation loop; Filmsubstrate; Isotropy 
Introduction 
Filmsubstrate structures and systems composed of thin film of finite thickness and substrate of infinite thickness are widely used in microchips, smart electronics, microsensors and manipulators, protective coatings, etc. During the industrial fabrication process and service lifetime of filmsubstrate structures and systems, dislocation clusters will be generated and will result in the microstructure evolution of the filmsubstrate system. The performance, reliability and integrity of thin filmsubstrates are closely related to the material properties of the film and substrate, in particular, the crystal orientation (or texture for polycrystalline material), the dislocation density and distribution. Study of the collective dynamic behaviors of dislocations embedded in thin filmsubstrate system are of great interest to researchers and engineers, which are of critical importance for understanding the microstructure evolution, plastic deformation process of thin filmsubstrate systems. 
Solutions to the elasticity field induced by dislocations within filmsubstrate system are important, because the elasticity solution provides a direct means of determining the PeachKoehler force acting on dislocations, which is of direct relevance in understanding the microstructure evolution and mechanical behaviors of these filmsubstrate systems. In order to simulate the dynamic evolution process of dislocation and dislocation loops, accurate method are needed to calculate the stresses due to dislocations and dislocation loops in a heterogeneous thin filmsubstrate system. Making use of mirror dislocation concept and potential theory, Head [1,2] analyzed the changes in the stress field of a straight screw/edge dislocation caused by differences in the shear modulus on either side of bimetallic medium. Moreover, making use of image dislocation concepts, the equilibrium positions of a group of dislocations within bimaterial and filmsubstrate system are studied. It was found that when dislocations are located on the side with lower modulus and are situated on the slip plane which is perpendicular to the boundary of bimaterial system, dislocations are forced towards the boundary by an applied stress but repelled from the boundary by the change in modulus [3]. Making use of the solution of an edge dislocation in halfplane and that of the reversed traction force prescribed on the interface plane, Weeks et al. [4] presented an exact analysis for the elastic field and the PeachKohler force due to an edge dislocation within the substrate medium of filmsubstrate system. By using Fourier exponential transform, Lee and Dundurs [5] investigated the elastic field of an edge dislocation situated in the surface layer of the filmsubstrate system, and the behavior of edge dislocation is discussed through analyzing the PeachKochler force acting on these dislocations. By extending the classical mirror dislocation solutions for the interface plane between two semiinfinite elastically isotropic media, the image problem for a screw dislocation in the thin films of multilayered filmsubstrate system has been solved analytically [6], and the equilibrium position of a single dislocation in thin films was determined as a function of stress in the films. Kelly et al. [7] studied the stress field induced by an edge dislocation of arbitrary orientation in both the surface layer and substrate medium of the layersubstrate system, and the solution can be employed for deriving the solution of crack problems, together with the PeachKoehler force. Savage [8] studied the solution for the displacement field induced by an edge dislocation in a layered halfspace, and four elementary solutions were considered: the dislocation is either in the halfspace or the layer, and the Burgers vector is either parallel or perpendicular to the interface plane. It was found that the surface displacement field produced by the edge dislocation in the layered halfspace is very similar to that produced by an edge dislocation at a different depth in a uniform halfspace. Moreover, a lowmodulus (highmodulus) layer causes the equivalent dislocation in the halfspace to appear shallower (deeper) than the actual dislocation in the layered halfspace. Wu and Weatherly [9] studied the equilibrium position of misfit dislocations in epitaxial grown systems where the thickness of the epitaxial film is several orders of magnitude smaller than the thickness of the substrate. When the film is elastically stiffer than the substrate, the core of the dislocation is predicted to lie at some distance from the interface in the softer substrate. On the other hand, when the film is softer than the substrate, the core of the dislocation is always predicted to lie close to the interface [9]. The continuous dislocation image dislocation method is used to derive elastic solutions for an edge dislocation in an anisotropic filmsubstrate system, and it was found that elastic anisotropy and material mismatch play an important role in determining the image forces and the stress components due to an edge dislocation located in filmsubstrate systems [10]. The stress and displacement fields due to an edge dislocation in a linearly elastic isotropic filmsubstrate were studied based on image dislocations methods, and it was shown that the film thickness and dislocation position have a significant influence on the image force acting on the dislocation, and the film thickness variation due to an accumulation of dislocations may degrade the performance of optical films [11]. Recently, anisotropic elastic stress fields caused by a dislocation in GexSi1x epitaxial layer on Si substrate are investigated by Wang [12], and effects of layer thickness, the dislocation position and the crystallographic orientation on the stresses of anisotropic filmsubstrate system were investigated extensively. It was revealed that the layer thickness and dislocation position strongly affects the stresses, while the crystallographic orientation play a very weak role in determining the elastic stress fields [12]. However, theses analytical solutions are mainly limited to straight dislocation lines parallel to the interface planes, which is invalid for curved dislocations and dislocation loops within filmsubstrate systems. Making use of potential theory and complete elliptic integrals, the elastic fields of a circular planar dislocation loop in an isotropic twophase material, and the interaction forces between dislocation loop and the second phase were studied analytically, it was found that PeachKoehler forces acting locally on the loop always tend to distort it [13,14]. Besides these analytical solutions, Weinberger et al. [15] and Wu et al. [16] developed semianalytical solutions to calculate the image stress of approaching dislocations and dislocation loops within half space and free standing thin film. It is found that image force has a much stronger effect on 1/2 (111) dislocation loop in anisotropic (111) thin Fe foil, compared to the Voigt equivalent isotropy simulation results. Moreover, for complex dislocation configurations and boundary conditions, the image stress due to free surfaces and interfaces can now, owing to computing power, be computed by finite element method [16]. 
The objective of this paper is to develop an efficient and accurate method to calculate the stress due to dislocation loops in an isotropic heterogeneous thin filmsubstrate system, which is composed of a thin film of finite thickness and a substrate of infinite thickness. Based on linear superposition rules and fast discrete Fourier transformation, the problem is decomposed into two subproblems: the stresses due to a dislocation loop in an infinite space and the correction stresses induced by the upper free surface of film and the filmsubstrate interface. Firstly, calculation examples of dislocation loops in Cu Nb filmsubstrate system are performed to verify the semianalytical approach; Then, elastic fields of dislocation loops within Cu film and Nb substrate of the CuNb filmsubstrate system are analyzed; Finally, effects of film thickness, dislocation loop positions on the elastic field of is investigated, and it is found that film thickness, dislocation loop positions has a remarkable impact on the elastic fields of dislocation loops within filmsubstrate system. 
Stress Fields Induced by Dislocation Loop in Perfect Bonding FilmSubstrate System 
Statement of problem 
As shown in Figure 1, perfect bonding isotropic filmsubstrate system is decomposed into a thin film (A) of finite thickness 2h and a substrate (B) of infinite thickness. The elastic properties of the film and substrate medium are assumed to be (λf, μf, vf) and (λs, μs, vs), respectively. Two sets of local Cartesian coordinates are employed for describing the filmsubstrate system: local Cartesian coordinate (x+,y+,z+) for the upper thin film A, and local Cartesian coordinate (x−,y−,z−) for the lower substrate half space B, where (x+, y+) and (x−,y−) are parallel to the interface plane and simplified as (x,y). The origin of the local Cartesian coordinate for the upper thin film A is located in the middle of the thin film, and the origin of the local Cartesian coordinate for the lower substrate B is located on the interface plane. Accordingly, −h ≤ z+≤ h is valid for the upper thin film A, and z−≤ 0 is valid for the lower substrate B. Dislocation loop L1 is located in the thin film medium A, and dislocation loop L2 is located in the substrate medium B. 
The free surface and interface of the perfect bonding filmsubstrate system should satisfy the following requirements: 
(a) Free traction stress should be satisfied on the top free surface of the filmsubstrate system. 
(1) 
(b) Elastic displacement and traction stress should be continuous across the interface planes of the filmsubstrate system. 
(2) 
and 
(3) 
Where ‘f’ and ‘s’ stand for the film and substrate, and superscripts ‘+’ and ‘−’ distinguish the local Cartesian coordinates of the film and substrate, respectively. 
Solution for Perfect Bonding FilmSubstrate System 
In this subsection, a semianalytical solution is proposed to solve the elastic fields due to dislocation loops within thin film or substrate of perfect bonding filmsubstrate system. Elastic field of perfect bonding filmsubstrate system can be decomposed into two subproblems: bulk stress due to a dislocation loop in an infinite space, and correction stress induced by the free surface and interface of the filmsubstrate system 
As shown in Figure 2a, perfect bonding filmsubstrate system is decomposed into thin film A containing dislocation loops L1 and substrate B containing dislocation loop L2. Bulk traction elastic field and on the top free surface and interface plane can be calculated out. 
As shown in Figure 2b, correction traction stress are linearly superimposed onto the bulk elastic fields. 
As shown in Figure 2c, after linear superposition of bulk stress and correction stress, displacement and traction stress continuity across the interface plane are satisfied. 
Elastic Fields of Dislocation Loops within Infinite Isotropic Medium 
Bulk displacement of dislocation loops can be calculated with Volterra’s formula, and written as surface integration over dislocation area [17,18]: 
(4) 
Where S is the dislocation surface, a cap with its boundary formed by dislocation loop perimeter. C_{ijkl} are the elastic constants, and G_{km,1} is the first order derivative of Green’s function. 
For an isotropic material, following relation exists: 
(5) 
Where R = x−x′ is the vector from dislocation position x′ to the calculated position x, λ, is Lamé’s first parameter of isotropic material, and μ is shear modulus of isotropic material, δ_{ij} is the Kronecker delta operator. 
Bulk stress induced by dislocation loops in a homogenous medium can be calculated with Mura’s formula [18]. Firstly, bulk displacement gradient can be written as: 
(6) 
where the integration is performed along dislocation loop perimeter. For an isotropic material, it appears that: 
(7) 
Where R = XX′ is the distance from the dislocation position x′ to the calculated position x in the space. 
Considering the displacementstrain differential relation, bulk stress of dislocation segment can be generated with isotropic Hooke’s law: 
(8) 
Then, bulk stress induced by a circular dislocation loop can be produced through linear integration along dislocation loop perimeter for a round. 
Correction Stress of Lower Substrate Medium 
In the absence of body forces, the stress equilibrium equation of the lower substrate medium B can be written in terms of the displacement as: 
(9) 
Where λ^{s} = 2 μ^{s}v^{s}/(12v^{s}), μ^{s} and v^{s} are the shear modulus and Poisson’s ratio for the lower substrate medium B. 
Similar to the solutions for the image stress of half space developed by Weinberger et. al [15], an arbitrary correction elastic field written in the form of Fourier series with unknown Fourier coefficients is employed for solving the correction elastic field of the lower substrate medium B. The following correction displacement solution to Eq. (9) is written as sum over different Fourier modes: 
(10) 
where are complex constants. The solution is periodic in the x and y directions and exponential in the z− direction. 
Due to the completeness of the Fourier series, Fourier coefficient components for certain (k_{x}, k_{y}) mode can be written as: 
(11) 
Thus, the correction displacement field is written as: 
(12) 
The correction displacement components on the free surface plane z− = 0 for certain (k_{x}, k_{y}) mode can be written as: 
(13) 
where the details of [N−] are shown in Appendix (A. 1). 
Following the displacement field solution in Eq. (10), it is straightforward to obtain the strain field through differentiation rule and the stress field by using Hooke’s law. The traction stress field can also be written in the form of Fourier series: 
(14) 
The correction traction stress components on the free surface plane z− = 0 for certain ( k_{x} , k_{y} )mode can be written as: 
(15) 
and the details of are shown in Appendix (A. 2). 
The numerical solutions of Eqs. (10)(15) for the substrate medium are considered in the x and y directions with periodic lengths L_{x} and L_{y}.The wave number is set to be k_{x} = 2πn_{x}/L_{x} and k_{y} = 2πn_{y}/L_{y}, where n_{x} = n_{y} = 0, ±1, 2, ±3… 
Correction Stress of upper Thin Film 
Similar to the efficient semianalytical image stress solution derived by Weinberger et al. [15] of isotropic thin foils, an arbitrary correction elastic field written in the form of Fourier series with unknown Fourier coefficients is employed for solving the correction elastic field of the film medium A. 
In the absence of body forces, the stress equilibrium of an isotropic linear medium composed of upper thin film medium A can be written in terms of the displacement as: 
(16) 
where , μ^{f} and v^{f} are the shear modulus and Poisson’s ratio for the upper thin film medium A. The thin film is assumed to have a thickness of 2h in the z+ direction, and −h z^{+} ≤ h. 
The correction elastic field can be written as sum over Fourier series, and the solution is periodic in the x and y directions, and hyperbolic in the z+ direction. 
(17) 
where and the unknown terms (A,B,C) and (E,F,G) are complex constants. 
Due to the mathematical completeness of Fourier series, the Fourier coefficient components for each Fourier (k_{x}, k_{y}) mode is: 
(18) 
Alternatively, the correction displacement field in Eq. (17) can be written as: 
(19) 
and the correction traction stress can be obtained from isotropic Hooke’s law: 
(20) 
Then, Eq. (20) was submitted into the equilibrium Eq. (16), correction displacement fields on the surface planes z+ = ± h can be combined together, and rewritten into two sets of equations on unknown coefficients (A,B,C) and (E,F,G), which correspond to the symmetrical and the asymmetrical parts, respectively. 
The symmetrical correction displacement is: 
(21) 
and the asymmetrical correction displacement part is: 
(22) 
The symmetrical correction traction stress is: 
(23) 
and the asymmetrical correction traction stress is: 
(24) 
The explicit expressions for the correction displacement and traction stress matrices [MS], [MA], [NS] and [NA] are given in Appendix (A. 3)  (A. 6). 
The calculation procedures of Eqs. (17)(24) for isotropic thin film are considered in the x and y directions with periodic lengths L_{x} and L_{y}.The wave number is set to be k_{x} = 2πn_{x}/L_{x} and k_{y} = 2πn_{y}/L_{y}, where nx = n_{y} = 0, ±1, ±2, ±3… 
Elastic Field of Perfect Bonding FilmSubstrate System 
Considering the displacement and traction stress continuity requirements in Eqs. (1)(3) for the perfect bonding filmsubstrate system, following relation stands for each (k_{x}, k_{y}) Fourier mode. 
Free traction stress on the free surface plane of the perfect bonding filmsubstrate system should be satisfied: 
(25) 
Interface traction stress and displacement continuity across the interface plane of the perfect bonding filmsubstrate system should be satisfied: 
(26) 
and 
(27) 
After submitting the bulk elastic field and correction elastic field into Eqs. (25)(27), the following relations stand for each (k_{x}, k_{y}) Fourier mode. 
(a) Free traction boundary condition should be satisfied on the top free surface plane of the thin film A for each (k_{x}, k_{y}) Fourier mode. 
(28) 
(b) Interface traction stress should be continuous for each (k_{x}, k_{y}) Fourier mode. 
(29) 
in which, is a 3*3 diagonal matrix. 
(c) Interface displacement should be continuous for each (k_{x}, k_{y}) Fourier mode. 
(30) 
In summary, Eqs. (28)(30) can be written together as: 
(31) 
Then, unknown coefficient (A,B,C,E,F,G) and of correction displacement can be solved from Eq. (31), and the correction elastic field of the film medium A and substrate medium B can be generated. 
The total elastic field is the sum of the two contribution parts: bulk elastic field ; and correction elastic fields , respectively. 
Total displacement within the upper film A of the filmsubstrate system is: 
(32) 
Total stress within the upper film A of the filmsubstrate system is: 
(33) 
Total displacement within the substrate B of the filmsubstrate system is: 
(34) 
Total stress within the substrate B of the filmsubstrate system is: 
(35) 
Calculation Examples 
In this section, the above semianalytical approach is employed for analyzing the elastic fields induced by dislocation loop within perfect bonding isotropic CuNb filmsubstrate system. The local Cartesian coordinate (x+, y+, z+) is along , and in upper Cu thin foil; and the local Cartesian coordinate (x−, y−, z−) is along , and (110) in lower Nb substrate medium. The origin of the upper Cartesian coordinate is in the middle of the Cu thin foil, and the origin of the lower Nb half space is on the interface plane of the filmsubstrate system. In all the calculation examples below, dislocation loop is segmented into 40 straight dislocation segments along the circular perimeter for a round, and bulk displacement and stress fields are obtained through integrating the dislocation segments around the dislocation loop perimeter, based on Volterra’s [17], Devincre’s [18] and Mura’s integration formulas [19], respectively. The elastic modulus of Cu and Nb is shown in Table 1, and the isotropic equivalent shear modulus and Poisson’s ratio are treated with Voigt isotropic model [20,21]. In the following simulation examples, the periodic length on the interface plane is L_{x} = L_{y} = 200 nm, the meshing density is n_{x} = n_{y} = 200, and the Fourier wave number range is: −30 ≤ k_{x} ≤ 30 and −30 ≤ k_{y} ≤ 30. 
Elastic Field due to a Dislocation Loop in the Substrate Medium 
In this section, elastic displacement and traction stress due to a dislocation loop in the lower Nb substrate medium of the perfect bonding CuNb filmsubstrate system are studied, and the thickness of the upper Cu film is assumed to be 40 nm. The circular 1/2a(111) dislocation loop with radius r = 5 nm is located at a distance d = 10 nm below the interface plane in lower Nb medium, and its habit plane is inclined to the interface plane. Figures 3a, 3b, 3e and 3f are the side view of the elastic field plotted in the σ_{xz} plane (y = 0), and Figures 3c and 3d are the side view of the elastic field plotted in the oyz plane (x = 0) for the perfect bonding CuNb filmsubstrate system. It can be seen that the final interface in plane and out of plane displacement field, traction stress field across the interface plane are identical, and thus continuous displacement and traction stress is generated. 
As shown in Figure 4, correction elastic field are superimposed onto the bulk elastic field, thus generating the final interface elastic field, and the contributions from bulk and correction elastic fields are compared with each other. As shown in Figures 4a4f, interface displacement profile of u and w, and interface traction stress σxz and σzz are plotted along x direction (y = 0) on the interface plane of the perfect bonding CuNb filmsubstrate system; as shown in Figures 4c and 4d, interface displacement profile of v and interface traction stress σyz are plotted along y direction (x = 0) on the interface plane of the perfect bonding CuNb filmsubstrate system. It can be seen from Figures 4b, 4d and 4f that the amplitudes of bulk traction stress are slightly strengthened on the interface plane, as the Voigt shear modulus of upper layer Cu is larger than lower layer Nb, and the traction stress due to a dislocation loop in the lower substrate Nb is slightly strengthened by the upper Cu film at the perfect bonding interface plane. 
Elastic Field due to a Dislocation Loop in the Film Medium 
In this section, elastic displacement and traction stress due to a dislocation loop in the upper Cu film of the perfect bonding CuNb filmsubstrate system are studied, and the thickness of the upper Cu film is assumed to be 40 nm. The circular 1/3a(111)(111) dislocation loop with radius r = 5 nm is located in the middle of upper Cu thin film, and its habit plane is parallel to the interface plane. Figures 5a, 5b, 5e and 5f are the side view of the elastic field plotted in the oxz plane (y = 0), and Figures 5c and 5d are the side view of the elastic field plotted in the oyz plane (x = 0) for the perfect bonding CuNb filmsubstrate system. It can be seen that the final interface in plane and out of plane displacement field, traction stress field across the interface plane are identical, and thus continuous displacement and traction stress is generated. 
As shown in Figures 6a, 6b, 6e and 6f, interface displacement profile of u and w, and interface traction stress σ_{xz} and σ_{zz} are plotted along x direction (y = 0) on the interface plane of the perfect bonding CuNb filmsubstrate system; as shown in Figures 6c and 6d, interface displacement profile of v and interface traction stress σ_{yz} are plotted along y direction (x = 0) on the interface plane of the perfect bonding CuNb filmsubstrate system. It can be seen from Figures 6b, 6d and 6f that the amplitudes of bulk traction stress are slightly weakened on the interface plane, as the Voigt shear modulus of upper layer Cu is larger than lower layer Nb, and the traction stress due to a dislocation loop in the upper Cu film is slightly weakened by the lower Nb substrate at the perfect bonding interface plane. 
Film Thickness Effect on Elastic Field of a Dislocation Loop in the Film Medium 
In this subsection, effects of film thickness on the interface displacement and interface traction stress amplitudes are investigated. Besides Cu film thickness, the physical and material parameters of dislocation loop and filmsubstrate system are identical to the simulated CuNb filmsubstrate system example in subsection 3.2. The thin Cu film thickness is assumed to be: t = 20, 30, 40, 50 and 60 nm, and the dislocation loop is situated in the middle of the upper thin Cu film. The simulation results are shown in Figure 7, and side view of the traction stress σ_{xz} and σ_{zz} are plotted in the oxz plane (y = 0) for the perfect bonding CuNb filmsubstrate system. It can be concluded from Figure 7 that: with the decrease of film thickness, bulk elastic field at interface plane are changed more remarkable by correction stress. 
Loop Depth Effect on Elastic Field of a Dislocation Loop in the Film Medium 
In this subsection, effects of dislocation loop depth within the upper Cu thin film of the CuNb filmsubstrate system on the elastic field are studied. Besides loop depth, physical parameters of the dislocation loop and perfect bonding filmsubstrate system are identical to the simulated CuNb filmsubstrate system example in subsection 3.2. The distance from dislocation loop center to the interface plane is assumed to be: d = 5, 10, 15, 20, 25 and 30 nm, and the film thickness is assumed to be t = 40 nm. The simulation results are shown in Figure 8, and side view of the traction stress σ_{xz} and σ_{zz} are plotted in the oxz plane (y = 0) for the perfect bonding CuNb filmsubstrate system. It can be concluded from Figure 8 that the interface and free surface can influence the bulk elastic field drastically. With the increase of the distance from dislocation loop center to the interface plane, bulk elastic field at interface plane are changed more remarkable by correction stress. 
Conclusion 
A semianalytical calculation approach based on 2D discrete FFT is developed for studying the elastic field due to dislocation loop in perfect bonding thin filmsubstrate system. Final elastic field is calculated as the linear superposition of bulk stress and the correction stress. 
Reliability of the semianalytical solution is verified by studying the elastic field of dislocation loop within perfect bonding CuNb filmsubstrate system. Effects of film thickness and loop depth within thin film on the elastic field are analyzed, demonstrating that these two factors have a significant impact on the elastic fields of dislocation loops in the thin film. 
Acknowledgements 
The research project was Supported by the China Postdoctoral Science Foundation under Grant No. 2015M80091. 
References 

Table 1  Table 2  Table 3  Table 4  Table 5 
Figure 1  Figure 2  Figure 3  Figure 4 
Figure 5  Figure 6  Figure 7  Figure 8 