Elastic models of defects in two-dimensional crystals

ISSN 1063-7834, Physics of the Solid State, 2014, Vol. 56, No. 12, pp. 2573–2579. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A.L. Kolesnikova, T.S. Orlova, I. Hussainova, A.E. Romanov, 2014, published in Fizika Tverdogo Tela, 2014, Vol. 56, No. 12, pp. 2480–2485. GRAPHENES Elastic Models of Defects in Two-Dimensional Crystals A. L. Kolesnikovaa, b, T. S. Orlovab, c, I. Hussainovad, and A. E. Romanovb, c, * a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoi pr. 61, St. Petersburg, 199178 Russia b St. Petersburg National Research University of Information Technologies, Mechanics and Optics, Kronverkskii pr. 49, St. Petersburg, 197101 Russia c Ioffe Physical-Technical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia * e-mail: [email protected] d Tallinn University of Technology, Ehitajate tee 5, Tallinn, 19086 Estonia Received June 17, 2014 Abstract—Elastic models of defects in two-dimensional (2D) crystals are presented in terms of continuum mechanics. The models are based on the classification of defects, which is founded on the dimensionality of the specification region of their self-distortions, i.e., lattice distortions associated with the formation of defects. The elastic field of an infinitesimal dislocation loop in a film is calculated for the first time. The fields of the center of dilatation, dislocation, disclination, and circular inclusion in planar 2D elastic media, namely, nanofilms and graphenes, are considered. Elastic fields of defects in 2D and 3D crystals are compared. DOI: 10.1134/S1063783414120166 1. INTRODUCTION In our investigation, two-dimensional (2D) crystals are films whose thickness can be neglected in a particular stated problem. Among them, there are single-layer crystallites [1], graphenes—crystalline carbon films of single-atomic thickness [1, 2], fullerenes—carbon shells in the form of convex closed polyhedra [3, 4], and biomembranes [5, 6]. From the viewpoint of continuum mechanics, such objects are 2D elastic media or shells. Inelastic and non-film physical objects that form 2D periodic structures, such as liquid crystals [7, 8], ordered ensembles of Abrikosov vortex filaments in type-II superconductors [9], and structured complexes of magnetic moments in ferromagnets [8], are beyond the scope of consideration. Due to the mechanical, thermal, radiation, and chemical effects as well as at the fabrication stage, films, like bulk materials, are able to deform and undergo local transformations. A specific feature of a 2D medium is the property to acquire a nonzero curvature in response to an external action or the appearance of defects in it [10–12]. To be more precise, as defects we do not consider inhomogeneities, i.e., regions with other elastic moduli, and voids, which are not independent sources of elastic fields and only affect their redistribution. Defects are internal sources of the deformation (distortion) of the medium (continuum), which lead to the generation of elastic fields in it. We speak about defects in crystals in the presence of the crystalline order. An important role of defects in 2D crystals can be demonstrated by the example of graphenes: the degree and even the type (electron or hole) of conduction in graphene depend on the defect configuration of grains boundaries [12–14]. The defect configuration determines elastic fields and energies of these boundaries as well as the degree of deviation of their state from equilibrium [15]. We can consider the grain boundary in graphenes as a specific defect of their structure. The goal of this work is a theoretical investigation of defects in a 2D elastic medium. Based on the presented classification of defects, we calculated elastic fields of an infinitesimal dislocation loop, dilatation center, wedge dislocation, and inclusion in films. The fields of individual defects make it possible to calculate the interaction energies between the defects and the energies of ensembles of defects, for example, grain boundaries [15]. We compared fields of defects in 2D and 3D media. It was emphasized that the elastic fields of a biaxial dilatation line, edge dislocation, and other defects in a 2D medium coincide with the fields of similar defects in a 3D plate when the plate thickness tends to zero. 2. CLASSIFICATION OF INTERNAL SOURCES OF ELASTIC DISTORTIONS IN A TWO-DIMENSIONAL CONTINUUM To create mathematical models of actual physical defects of the crystal lattice (point defects, dislocations, and inclusions), the technique of self-distortions is applicable (see, e.g., [16, 17]). Within this 2573 2574 KOLESNIKOVA et al. technique, the source of elastic distortions of the medium can be specified using the tensor of the selfdistortion in the method context. The self-distortion determines the method of specifying the defect and the region where this method is implemented. The method of specifying the defect with the use of self-distortion (or eigenstrain) was first proposed by Eshelby for inclusions [18]. The Eshelby procedure is as follows: a cut over the surface confining a certain volume is made in the material, after which the material is extracted from this volume and is deformed plastically; then forces are applied to this deformed volume in order to insert it back exactly to the cut place; and cut surfaces are glued, and forces are removed. It is noteworthy that the apparatus for finding the elastic fields of the defect by the specified self-distortion is developed well [16, 17]. We can apply the Eshelby procedure to the regions of different dimensionalities and classify the objects in the elastic medium based on the dimensionality for the n-region Ωn of specifying their self-distortion. Let us m n β*ij , where m is the denote the self-distortion as medium dimensionality. It is evident that n ≤ m. According to such a “dimensional principle,” defects in 3D medium are divided into four types [19]; while in the 2D medium, they are divided into three types (see Fig. 1). 2.1. Zero-Dimensional (Point) Defects (n = 0) Let us define an infinitesimal dislocation loop, which is the elementary zero-dimensional defect in 3D and 2D media [19]. Kroupa [20] was the first to calculate infinitesimal dislocation loops in a bulk elastic solid. The self-distortion of an infinitesimal dislocation loop in the 3D medium can be written as follows [19]: 3 0 β*ij = – b j s i δ ( R – R 0 ), i, j = x, y, z, (1) where bj is the Burgers vector of the dislocation placed into point R0, si is the loop area, and δ(R – R0) is the 3D Dirac delta function, which is related to the onedimensional delta functions by equality δ(R – R0) = δ(x – x0)δ(y – y0)δ(z – z0). The self-distortion of the infinitesimal dislocation loop in the film lying in the X0Y coordinate plane will take the form: 2 0 β*ij = – b j l i δ ( r – r 0 ), i, j = x, y. (2) Here, bj is the Burgers vector of the dislocation; r0 is the loop coordinate in plane X0Y; and li is the segment with normal ni, which is the analog of the loop area for the dislocation in the 3D medium (see (1)). The 2D Dirac function is specified by the relationship δ(r – r0) = δ(x – x0)δ(y – y0). By analogy with the specifying dislocation loops in 3D medium, we will consider that if vector b is directed along normal ni, then the infinitesimal loop is prismatic; and if vector b lies along the cut line, the infinitesimal loop is the slip loop. Under the equality condition of the Burgers vectors, two mutually perpendicular prismatic loop form the dilatation center with the following distortion: 2 0 1 β*ii = blδ ( r – r 0 ) = - Δsδ ( r – r 0 ), 2 (3) i = x, y. Such center is an elastic model of the impurity atom, the substitution atom, or vacancy (see Fig. 1a). If l is the initial linear size of a certain area s = l2, then the variation of the linear size by magnitude Δl will lead to varying the area by magnitude s = (l + Δl)2 – l2 ≈ 2lΔl. Assuming that b = Δl, we find bl = -1 Δs and the last 2 equality in formula (3). It is known that the dilatation equiaxial center in the elastic volume possesses the distortion [17] 3 0 β*xx = 3 0 β*yy = 3 0 β*zz = bsδ ( R – R 0 ) (4) = -1 ΔVδ ( R – R 0 ). 3 Here, ΔV is the variation in the volume in the localization point of the defect. It is noteworthy that distortion (3) equals the distortion of the dilatation line in the 3D medium, for 3 1 * is absent [19]. which component βzz For the formal approach, the self-distortion of the point defect in the film can be written in the form 2 0 β*ij = β ij* sδ ( r – r 0 ), i, j = x, y, (5) where β *ij is the plastic distortion of small area s without taking into account its specification place in the medium. An increase in the dimensionality of the specificaΩ1 (Ω1 ≡ L) leads to a tion region of the defect Ω0 one-dimensional defect. 2.2. One-Dimensional Defects (n = 1) In frameworks of the selected classification, the self-distortion of the one-dimensional defect in the 2D medium will take the form 2 1 2 β*ij = β *ij l δ ( L ), (6) where β ij* is the plastic distortion of line L, l is the linear multiplier, 2δ(L) is the delta function in the line in the 2D medium, which is emphasized by the upper- PHYSICS OF THE SOLID STATE Vol. 56 No. 12 2014 ELASTIC MODELS OF DEFECTS IN TWO-DIMENSIONAL CRYSTALS 2575 (a) Selfinterstitial Substitutional impurity atom Vacancy P(x0, y0) 2 0 y β*ij = β *ij sδ ( x – x 0 )δ ( y – y 0 ) Interstitial impurity atom x b y x (b) Dislocation Cutting line y 2|1β∗ x xx= bxδ(x – x0)H(y i n ry r a da G ou n b ≥ y0) 2|1β∗ = β∗ l . 2σ(L) ij ij Disclination −ω y x ω (c) Inclusion 2|2β∗ = 2β∗ . 2δ(S ) = ij ij Inc y β∗ij, r SInc 0, r ∉ SInc y x x Fig. 1. Defects in 2D crystals and their elastic models. (a) Zero-dimensional defect (vacancy, interstitial atom, or impurity substitutional atom), (b) one-dimensional defects (dislocation, disclination, dislocation pile-up, or misorientation boundary), and 2 n (c) 2D inclusion. The main characteristic of the defect model is the self-distortion β*ij , where n = 0, 1, 2 is the dimensionality in the specification region of the distortion and i, j = x, y. 2D crystals with the square and hexagonal lattices are presented. left index 2. For example, for segment [y1, y2], which has coordinate x0, 2δ(L) = δ(x – x0)H(y1 ≤ y ≤ y2), where H(y1 ≤ y ≤ y2) is the Heaviside function. PHYSICS OF THE SOLID STATE Vol. 56 No. 12 One-dimensional defects are the analogs of the Somigliana dislocation [21–23] and the Volterra dislocation and disclination in the elastic 3D medium (see 2014 2576 KOLESNIKOVA et al. Fig. 1b). The segment, at which the distortion is specified, is the analog of the area, at which the self-distortion of the dislocation–disclination loop in the 3D medium is specified. The dislocation-singularity line in the 3D medium is regenerated into the dislocation singularity point in the 2D medium. We refine that the Somigliana dislocations in the classification of defects in the 3D medium, which rests upon the dimensionality of the specification region of their self-distortion, are the surface defects, while the Volterra dislocations and disclinations are the degenerate surface defects having the features of elastic fields on the line rather than on the surface [17, 24, 25]. In the 2D medium, the Volterra dislocations and disclinations are the degenerate linear defects having the singularities of elastic fields in the point. Let us reduce distortion (6) to the form usual for the Volterra dislocations [26]: 2 1 2 β*ij = [ u j ] δ i ( L ) = 2δ (L) i 2 – b j δ i ( L ), i, j = x, y. (7) = ni is the normal to line L, uj is Here, the displacement jump at the cut line upon introducing the dislocation. The specification line of the selfdistortion of defect L in the 2D medium plays the role of area, where the defect distortion is specified, in the 3D medium. The linear defect can be obtained continuously distributing the point defect along certain line with specified linear density ρL. If β *ij in formula (6) is independent of the coordinate of distributed point defects while the linear density is constant, we derive a simple formula 2 β*ij = β ij* sρ L δ ( L ). (8) Here, it is evident that substitution sρL = l transforms formula (8) into (6). With the further increase in the dimensionality of the specification region of distortion Ω1 Ω2 (Ω2 ≡ S), we pass to 2D defects. 2.3. Two-Dimensional Defects (n = 2) The specification region of the self-distortion of 2D defects is a part of the surface. In the film, 2D defects are analogs of an inclusion in the 3D medium (see Fig. 1c). According to (2) and (6), the expression is valid for the distortion of the 2D defect of the general form: 2 2 2 β*ij = β ij* δ ( S Inc ). In the 2D medium, the delta function ⎧ 1, r ∈ S Inc 2 δ ( Ω 2 ) = δ ( S Inc ) = ⎨ ⎩ 0, r ∉ S Inc . 2 ⎧ 1, R ∈ V Inc 3 δ ( Ω 3 ) = δ ( V Inc ) = ⎨ ⎩ 0, R ∉ V Inc . 3 We can also obtain the 2D defect distributing point defects over surface S. We note that in 2D continuum, the defects with dimensionality lower than 2, possess distortions with arbitrary linear multipliers. For point defect, this is s; and for line, this is l. We can accept the sense of the defect size before the plastic deformation to these multipliers (see in detail in [19]). 3. ELASTIC FIELDS OF DEFECTS IN PLANAR 2D MEDIA Based on the self-distortion 2δ(L)n , i 2 1 We note that in the 3D medium, the delta function (9) 2 m β*ij (or eigenstrain 2 m ε*ij ), the Green’s function 2Gij of the elastic medium and its elastic moduli Cilkn is unambiguously determined by the field of complete displacements of the defect as well as its elastic strains and stresses [16, 17]. In the 2D continuum, the field of complete displacements of the defect can be found from the relationship [16] 2 t ui ( r ) ∫ = – C jlkn 2 m 2 ε*kn ( r' ) G ij, l ( r – r' ) dS ', (10) S where the upper-left index 2 indicates the medium dimensionality, derivative 2Gij(|r – r'|) is taken with respect to the undashed variable, S is the film surface, and summation is performed over repeating indices. The Green’s function 2Gij(|r – r'|) and tensor Cjlkn for the isotropic elastic medium have the form [16] 1 Gij ( r – r' ) = ---------------------8π ( 1 – ν )G 2 ⎧ ( x i – x 'i ) ( x j – x 'j ) ⎫ - – ( 3 – 4ν )δ ij ln r ⎬, × ⎨ -----------------------------2 ⎩ ⎭ r 2Gν C jlkn = ------------δ jl δ kn + G ( δ lk δ jn + δ ln δ jk ), 1 – 2ν (11a) (11b) 2 where r = (x – x')2 + (y – y')2, xi and xj are x or y, δkm is the Kronecker symbol, G is the shear modulus, and ν is the Poisson’s ratio. Since film X0Y with free surfaces is the elastic space for us, the stress tensor implies the absence of all components σz j in this case. Thus, we are dealt with a planar stressed state; this means that the Young modulus E in the Green’s function and in the Hooke law should be replaced by relationship E(1 + 2ν)/(1 + ν)2, while the Poisson’s ratio ν should be replaced by the expression ν/(1 + ν) [16]. Since the shear modulus G = PHYSICS OF THE SOLID STATE Vol. 56 No. 12 2014 ELASTIC MODELS OF DEFECTS IN TWO-DIMENSIONAL CRYSTALS E/[2(1 + ν)], the substitution procedure will not affect it. Thus, in order to obtain elastic fields corresponding to the film with free surfaces, we should replace ν by ν/(1 + ν) in the Green’s function (11a), elastic constants (11b), and the Hooke law. The measurement unit of the Young modulus and the shear modulus will change from N/m2 to N/m for the 3D 2D transition. 2 ΔP The field of displacements uj 2 ΔP σij Using relationships (11), let us determine the field of the infinitesimal prismatic dislocation loop and dilatation center. 2 0 IPDL * βxx = b x lδ ( x )δ ( y ). 2 ΔP ux ( 1 + ν )εs x = ------------------ --2 , 2π r (16a) 2 ΔP uy ( 1 + ν )εs y = ------------------ --2 , 2π r (16b) 2 2 IPDL The field of total displacements ui (x, y) of the loop calculated by formulas (10) and (11) allowing for the substitution of the Poisson’s ratio (see explanation to (11)) has the form 2 IPDL ux b x lx 2 2 = --------4 [ – y ( ν – 1 ) + x ( 3 + ν ) ], 4πr (13a) 2 IPDL uy b x ly 2 2 = --------4 [ y ( ν – 1 ) + x ( 1 + 3ν ) ], 4πr (13b) 2 ) ( 1 + ν )εs ( x – y -, = –G --------------------- --------------4 π r 2 2 ΔP σxx (17a) 2 G ( 1 + ν )εs ( x – y ) = – --------------------- ---------------, 4 π r 2 ΔP 2G ( 1 + ν )εs xy σxy = – ------------------------ ---4-, π r 2 ΔP σyy (12) and field of stresses of the dilatation center have the form 2 ΔP σxx 3.1. Infinitesimal Prismatic Dislocation Loop and Dilatation Center in the Film Let us consider the infinitesimal prismatic dislocation loop (IPDL) arranged in the origin of coordinates and having the self-distortion 2577 (17b) (17c) 2 ΔP + σyy = 0. (17d) Field (16), (17) coincides the field of the dilatation line in the 3D medium by its functional part [19]. After the substitution εs = bxl, it is seen from formulas (16), (17) that the dilatation center in the 2D medium is the sum of two mutually perpendicular infinitesimal prismatic loops (13), (14) similarly as the dilatation center in 3D medium is the sum of three mutually perpendicular infinitesimal prismatic dislocation loops [17, 19]. It should be noted that the field of biaxial dilatation L L 3 1 * = 3 1βyy * = εsδ(x)δ(y) line with self-distortion βxx in the 3D medium with the substitution of ν by ν/(1 + ν) in it exactly coincides with field (16), (17). where r2 = x2 + y2. Displacements allow us to calculate the field of 1 elastic strains εij = - (uj, i + ui, j), where uj are elastic 2 displacements, and then, following the Hooke law, in which the corresponding substitutions are also made (see explanation to (11)), stresses are found: 2 IPDL σxx 2 IPDL σyy G ( 1 + ν )b x l 4 2 2 4 = – ---------------------- [ 3x – 6x y – y ], 6 2πr (14a) G ( 1 + ν )b x l 4 2 2 4 = ---------------------- [ x – 6x y + y ], 6 2πr (14b) 2 IPDL σxy 3.2. Dislocation and Wedge Disclination in the Film Let the dislocation distortion is determined by the relationship for definiteness (see Fig. 1b) 2 1 ⎧ 1, H(y ≥ 0) = ⎨ ⎩ 0, = 2 0 ΔP * βyy 2 ⊥ ui = 2 IPDL ( x, i y – y 0 )ρ dy 0 , (19a) 2 IPDL ( x, ij y – y 0 )ρ dy 0 , (19b) ∫u 0 (15) Here, ΔP is the notation of the dilatation center. Vol. 56 y < 0. Such dislocation can be obtained distributing infinitesimal prismatic dislocation loops (12) along axis 0Y: (14c) = εsδ ( x )δ ( y ). PHYSICS OF THE SOLID STATE y≥0 ∞ G ( 1 + ν )b x l 2 2 = – ---------------------- xy [ 3x – y ]. 6 πr ΔP * βxx (18) where the Heaviside function Let us determine the field of the dilatation center with the distortion 2 0 ⊥ * = b x H ( y ≥ 0 )δ ( x ), βxx No. 12 ∞ 2 ⊥ σij = ∫σ 0 where ρ is the linear distribution density of point defects. 2014 2578 KOLESNIKOVA et al. Denoting bxlρ = bx, we derive from (19) 2 ⊥ ux b ( 1 + ν )xy⎞ = ----x- ⎛ 2 arctan ⎛ -y⎞ + ------------------ , 2 ⎝ ⎠ ⎝ ⎠ x 4π r 2 ⊥ uy b ( 1 + ν )x ⎞ = – ----x- ⎛ ( 1 – ν ) ln r + ----------------- , 2 ⎠ 4π ⎝ r (20b) 2 ⊥ σxx 2 G ( 1 + ν )b y x y⎞ - , = – ---------------------x ⎛ --2 + 2 ----4 ⎝r 2π r ⎠ (21a) 2 ⊥ σyy G ( 1 + ν )b y x y⎞ - , = – ---------------------x ⎛ ---2 – 2 ----4 ⎝r 2π r ⎠ (21b) G ( 1 + ν )b x xy ⎞ - . = ---------------------x ⎛ ---2 – 2 ----4 ⎝r 2π r ⎠ (21c) (20a) 2 2 ⊥ σxy 3.3. Dilatation Inclusion in the Film The example of the dilatation inclusion in the 2D medium is shown in Fig. 1c. Knowing the field of the biaxial cylindrical inclusion in the 3D medium [28] having the distortion 3 3 β*xx = ⎧ ε, 3 β*yy = ε δ ( V Inc ) = ⎨ ⎩ 0, R ∈ V Inc R ∉ V Inc , we can write the field of the circular inclusion in the 2D medium with the distortion 2 2 2 2 β*xx = 2 2 ⎧ ε, 3 β*yy = ε δ ( S Inc ) = ⎨ ⎩ 0, R ∈ S Inc R ∉ S Inc , 2 Inc Earlier [22, 27], we presented the solution of the problem on the edge dislocation perpendicular to the surface of a finite-thickness plate. In particular, it was shown in [22] that, when the plate thickness tends to zero, the elastic field generated by the edge dislocation becomes the in-plane stress field and coincides with field (21). The main distinction of dislocation field (20), (21) from the dislocation field in the infinite 3D medium is ⊥ the absence of component σ zz . Other dislocation components in 2D and 3D media coincide accurate to coefficients associated with ν. Thus, the dislocation field in the film can be found by means of substituting ν by ν/(1 + ν) in formulas that describe the dislocation field in the bulk. substituting ν by ν/(1 + ν) and excluding σzz : 2 Inc σxx 2 Inc σyy ⎧ ( x2 – y2 )  – ---------------, r > a 4 2 r = G ( 1 + ν )εa ⎨ 2  -- -2 , r ≤ a, ⎩a (23a) ⎧ ( x2 – y2 )  ----------------, r > a 4 2 r = G ( 1 + ν )εa ⎨ 2  -- -2 , r ≤ a, ⎩a (23b) ⎧ 2xy – ------- , r > a = G ( 1 + ν )εa ⎨ r 4  ⎩ 0, r ≤ a. (23c) 2 ω Let us use this procedure and write expressions σij for the wedge disclination situating in the film based on the stresses of the wedge disclination with the Frank vector ω = ωez in the 3D medium [17]: 2 2 ω σxx ν y G ( 1 + ν )ω = --------------------- ⎛ ln r + ---2 + ---------⎞ , ⎝ 1 – ν⎠ 2π r 2 ω σyy x G ( 1 + ν )ω ν = --------------------- ⎛ ln r + ---2 + ---------⎞ , ⎝ 2π r 1 – ν⎠ (22b) G ( 1 + ν )ω xy = --------------------- ---2-. 2π r (22c) (22a) 2 2 ω σxy 3 3 It is noteworthy that the disclination dipole is a characteristic element of grain boundaries in graphene (see Fig. 1b), see, e.g., [11–13]. It becomes possible to calculate energies of grain boundaries in graphene with the help of disclination series and their elastic fields not applying the computer modeling [15]. It is evident that a similar defect of lower dimensionality occurs for defects inducing in-plane strain in the 3D medium in the film. In this case, fields of defect analogs are equal with the corresponding substitution of elastic moduli. 2 Inc σxy 2 Here, a is the inclusion radius. If we introduce the nucleus radius for biaxial dilatation center a and determine coefficient s in (17) as πa2, then the stresses of the dilatation center will coincide with stresses generated by the circular biaxial inclusion in the surrounding matrix r > a (23). 4. CONCLUSIONS In this paper, we have presented elastic models of defects in planar 2D crystals. The hierarchy of defects of the 2D elastic continuum, which is based on the dimensionality of the specification region of their selfdistortions, was demonstrated. The elastic field of an infinitesimal prismatic dislocation loop and dilatation center in the film was calculated using the self-distortion and the Green’s function. It was found that defects inducing in-plane strain in the 3D medium have analogs in the 2D film. Expressions for defect fields in the isotropic film were derived by means of substituting the coefficients associated with elastic PHYSICS OF THE SOLID STATE Vol. 56 No. 12 2014 ELASTIC MODELS OF DEFECTS IN TWO-DIMENSIONAL CRYSTALS 3 Inc moduli and zeroing the component of stresses σzz in formulas for the fields of defect analogs in the bulk. On this basis, we obtained the fields of the dislocation, wedge disclination, and dilatation inclusion in the film. The determination of the elastic behavior of defects in curved 2D crystals, particularly in spherical shells, remains an important problem. In this case, the effective method for calculating defect fields, most likely, is as follows: first, the boundary problem on the defect in a finite-thickness spherical layer is solved; then, the fields of the defect are found for the layer thickness tending to zero. It was previously shown that this approach is applicable for thin films [22]. 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