Stress field of a near-surface basal screw dislocation in elastically anisotropic hexagonal crystals

In this study, we derive and analyze the analytical expressions for stress components of the dislocation elastic field induced by a near-surface basal screw dislocation in a semi-infinite elastically anisotropic material with hexagonal crystal lattice. The variation of above stress components depending on “free surface–dislocation” distance (i.e., free surface effect) is studied by means of plotting the stress distribution maps for elastically anisotropic crystals of GaN and TiB2 that exhibit different degrees of elastic anisotropy. The dependence both of the image force on a screw dislocation and the force of interaction between two neighboring basal screw dislocations on the “free surface–dislocation” distance is analyzed as well. The influence of elastic anisotropy on the latter force is numerically analyzed for GaN and TiB2 and also for crystals of such highly elastically-anisotropic materials as Ti, Zn, Cd, and graphite. The comparatively stronger effect of the elastic anisotropy on dislocation-induced stress distribution quantified for TiB2 is attributed to the higher degree of elastic anisotropy of this compound in comparison to that of the GaN. For GaN and TiB2, the dislocation stress distribution maps are highly influenced by the free surface effect at “free surface–dislocation” distances roughly smaller than ≈15 and ≈50 nm, respectively. It is found that, for above indicated materials, the relative decrease of the force of interaction between near-surface screw dislocations due to free surface effect is in the order Ti > GaN > TiB2 > Zn > Cd > Graphite that results from increase of the specific shear anisotropy parameter in the reverse order Ti < GaN < TiB2 < Zn < Cd < Graphite. The results obtained in this study are also applicable to the case when a screw dislocation is situated in the “thin film–substrate” system at a (0001) basal interface between the film and substrate provided that the elastic constants of the film and substrate are the same or sufficiently close to each other.


Introduction
The dislocation strain/stress field substantially affects mechanical, optical, electrical, and other physical properties of bulk materials and their thin films [1,2,3]. Therefore, the theoretical investigations of the dislocation stress field are of practical importance and enable to quantify the influence of dislocations on structure-property relationships. The character of the stress distribution produced by an individual dislocation in a crystalline material strongly depends on the type of dislocation (screw, edge or mixed), elastic properties of the material (elastic constants), and relative location of the dislocation with respect to free surface(s) [1]. Regardless of the type of dislocation and elastic properties of the material, the dislocation elastic field drastically changes with approach of the dislocation position to a free surface [1,4]. This necessitates, in the calculations of the stress field of a near-surface dislocation, a proper account of the interaction of dislocation with the free surface.
The influence of a free surface on the dislocation elastic field and the Peierls stress have been analyzed within the frameworks of the theory of isotropic elasticity in a large number of publications: particularly in studies [1,[4][5][6][7][8] and other investigations.
However, the review of the results reported in above cited references and other similar studies in literature show that, for hexagonal crystals with considerable elasticity anisotropy, the theoretical investigations of the stress field produced by near-surface individual basal dislocations have received a little attention. A broad variety of elastically anisotropic compounds with hexagonal crystal lattice (BN, AlN, GaN, InN, SiC, TiB 2 , graphite, etc.) has extensive applications in production of optical, semiconductor, microelectronic, and other type devices [2,22,23]. Production of these devices is mostly accompanied by formation of unavoidable dislocations that are located at hetero-interfaces (misfit dislocations) and/or in the vicinity of a free surface (near-surface dislocations). It is of particular interest to study the peculiarities of the stress field of near-surface basal dislocations in the case when the free surface by itself is a (0001) basal plane. In the technologies of heteroepitaxial deposition of semiconductor thin films with hexagonal crystal lattice, a frequently used deposition surface both for epitaxial films and substrates is the (0001) basal surface [2,22,24]. Especially for materials with pronounced elastic anisotropy, the above specified investigations will be helpful for precise quantification of the dislocation energy and interaction of the dislocation with neighboring near-surface defects (other individual dislocations, dislocation arrays, and point defects) and the (0001) free surface. It is also of interest to quantify and compare the effect of the elastic anisotropy on dislocation stress field for materials with hexagonal crystal lattice exhibiting different degrees of elastic anisotropy. In above comparative analysis, it may be helpful to use the so-called elasticity anisotropy parameters that are in detail analyzed in studies [25,26,27] and quantified for group III-nitrides in our earlier work [28].
In this study, on the basis of the Chou's theoretical results [11,12], we derive and analyze the analytical expressions for stress components of the dislocation elastic field induced by a near-surface basal screw dislocation in a semi-infinite elastically anisotropic material with hexagonal crystal lattice. The variation of above stress components depending on "free surface-dislocation" distance (i.e., free surface effect) is studied by means of plotting the stress distribution maps for elastically anisotropic crystals of GaN (gallium nitride) and TiB 2 (titanium diboride) that exhibit different degrees of elastic anisotropy. The dependence both of the image force on a screw dislocation and the force of interaction between two neighboring basal screw dislocations on the "free surface-dislocation" distance is analyzed as well. The influence of elastic anisotropy on the latter force is numerically analyzed for GaN and TiB 2 and also for crystals of such highly elastically-anisotropic materials as Ti, Zn, Cd, and graphite.

Statement of the Problem
Our theoretical study is restricted to the following conditions: (i) a basal-plane perfect screw dislocation is situated near a free surface of an elastically anisotropic bulk single crystal with hexagonal crystal lattice, (ii) the dislocation line that is directed along the   0 2 11 crystallographic direction is parallel to the free surface, which coincides with a (0001) basal plane, (iii) in terms of dislocation near-surface position (or extension of the dislocation elastic field), the crystal is considered as a semi-infinite medium.
The main aim of this study is presented at the end of Section 1. It should be clarified that, among the materials of interest in this study, the GaN may crystallize both in the zinc blende and wurtzite crystal structure with cubic and hexagonal crystal lattice, respectively [29], so for GaN the present study relates to the latter type of the crystal lattice.

Stress Components of a Near-surface Screw Dislocation
According to Section 2, Figure 1 schematically shows a basal-plane perfect screw dislocation (solid circle) in a semi-infinite single crystal, with dislocation line situated at a distance h from the free (0001) surface. The dislocation line that coincides with   0 2 11 direction is perpendicular to the plane of figure. We introduce a right-handed Cartesian coordinate system xyz with the z axis directed along the dislocation line and y axis oriented normal to the free surface (this means that the z and y axes are parallel to   0 2 11 and [0001] crystallographic directions, respectively). The Burgers vector of this dislocation, , is directed along the z axis (a is the lattice constant in the basal plane). The magnitude of the Burgers vector s b is equal to above specified lattice constant, a b s  [1]. It is assumed that the dislocation is also right-handed. In elastically anisotropic infinite medium with hexagonal crystal lattice, the non-zero stress components of the elastic field induced by above specified basal screw dislocation are given as [11,12]: where 2 / 1 44 12 11 ] In Eqs. (1)-(4), s b is the magnitude of the Burgers vector, s K is the dislocation energy factor, and ij c are the elastic stiffness constants of the crystal. Note that both stress components given by Eqs. (1) and (2) are of shear type. According to a known approach [1], in order to take into account the influence of the free surface on above stress components, it is necessary to introduce an image screw dislocation, with the opposite Burgers vector, at the same distance h from the free surface (in Figure 1, the image dislocation is shown by an empty circle). In the defined coordinate system xyz (see Figure 1), the dislocation lines of the real and image dislocations are situated at positions ) in Eqs. (1) and (2) and changing the signs of both stress components into opposite ones, we obtain the stress components of the image dislocation in infinite medium: Finally, the shear stress components of a real screw dislocation at a distance h from the free surface, zx  and zy  , are determined as a superposition of the corresponding stress components given by Eqs. (1), (2), (5), and (6):  [11,12], Eqs. (7) and (8) reduce to known expressions for dislocation stress components in the case of an elastically isotropic medium [1]: where G is the isotropic shear modulus. Eqs. (1), (2), and (5)−(10) are applicable in the spatial region out of the dislocation core. In the first approximation, the radius of the dislocation core is estimated to be

Quantification of the Effect of Elastic Anisotropy on Dislocation Stress Field
In order to quantify solely the effect of elastic anisotropy on dislocation stress field, we consider the dislocation stress distribution in an infinite medium, since in this case the stress distribution is not influenced by a free surface. We numerically compare for zx  component the spatial stress distribution with account of elastic anisotropy [Eq. (1) in combination with Eqs. (3) and (4)] with that in the approximation of elastic isotropy. In the approximation of elastic isotropy, the zx  stress component is achieved from Eq. (1) owing to above specified replacements G K s  and 1   : In Eq. (11), as in Eqs. (9) and (10), G is the isotropic shear modulus. The above comparative analysis is conducted for two materials, GaN and TiB 2 , that exhibit mutually different degrees of elastic anisotropy (see Section 8). For a basal screw dislocation in an infinitely large GaN crystal, Figure 2a shows the plots for dependences ) (x zx  according to Eqs. (1) and (11)  application of anisotropic and isotropic elasticity theory, respectively. The above ) (x zx  dependences are presented for (x,y) points that lie on the lines , Analogously with the case of GaN, Figure 2b shows for an infinitely large TiB 2 crystal the plots of (1) and (11) along the radial directions , , 3 . 0 x y x y   and x y 5  . In the plots presented in Figure 2, for the sake of convenience the sign minus in Eqs. (1) and (11) was omitted and the following parametric data have been used: for GaN and TiB 2 , 19 .
, respectively, and the data for stiffness constants ij c [31,32] and shear modulus G from Table 1 and Table 2 where R G and V G are the isotropic lower and upper limits of the shear modulus resultant from the Reuss and Voigt averaging schemes, respectively. Along with the data for modulus G , Table 2 also lists the calculated data for moduli R G and V G that are achieved with the use of ij c stiffness constants presented in Table 1. For the sake of brevity, we omit here the details of these calculations that may be found particularly in study [34]. The data for the dislocation energy factor s K and parameter  , calculated according to Eqs. (3) and (4) with the use of ij c values from Table 1, are presented in Table 2 as well.   (3)), isotropic shear moduli R G , V G , and G (Eq. (12)), shear anisotropy parameter A (Eq. (17)), and parameter  (Eq. (4)).

Effect on zx  Stress Component
The plots presented in Figure 2 show that, in the whole, the account of the elastic anisotropy both for GaN and TiB 2 crystals results in a non-negligible contribution into the dislocation stress distribution quantified in the approximation of isotropic elasticity. The above contribution is appreciable along the lines for GaN ( Figure 2a) and is considerable along the lines x y 3 . 0  and x y  for TiB 2 (Figure 2b). Therefore, it is more realistic to analyze/quantify the effect of the "free surface-dislocation" distance on dislocation stress field within the scope of the anisotropic elasticity theory.
For the case of a basal screw dislocation situated at a distance h from the free surface of a GaN crystal (Figure 1 . In all stress maps presented in Figure 3, the solid circle centered at position ) 0 , 0 (   y x schematically shows the region occupied by the dislocation core. In the plotting of these stress maps, we used the values of the Burgers vector specified in Section 4 and the data for ij c stiffness constants listed in Table 1.

Effect on zy  Stress Component
For the case of a dislocation situated at a distance h from the free surface of a GaN crystal, In all stress maps presented in Figure 4, the solid circle centered at position ) 0 , 0 (   y x schematically shows the region occupied by the dislocation core. In the plotting of these stress maps, we used the values of the Burgers vector specified in Section 4 and the data for ij c stiffness constants listed in Table 1.

Interaction of Dislocation with Free Surface
In the case under consideration (Figure 1), the interaction force of a real dislocation with the free surface is quantified via the interaction force exerted by the image dislocation on the real dislocation [1]. In our case, the absolute value of this force (per unit length of the dislocation line) is determined with use of Eq. (5) as follows: is the stress produced by image dislocation at position occupied by the real dislocation, ) 0 , 0 (   y x (see Figure 1). Not that, according to Eq. (6), at position of the real stress component of the image dislocation is zero and, hence, no force component is produced by this stress component on real dislocation. As the force between unlike screw dislocations (i.e., between the real and image dislocations) is attractive, this means that the force F (Eq. (13)) is directed along the positive direction of the y axis (see Figure 1). From the physical standpoint, this is equivalent to a statement that the free surface attracts the near-surface dislocation. Figure 5 shows the plots for dependence ) (h F according to Eq. (13) for the cases when a near-surface screw dislocation is situated in semi-infinite crystals of GaN and TiB 2 .

Interaction of Two Neighboring Basal Screw Dislocations
In terms of dislocation slip in the (0001) basal plane, it is of particular interest interaction of two near-surface screw dislocations with parallel dislocations lines situated in the same basal plane. Figure 6a schematically shows these two dislocations and it is assumed that they have the same Burgers vector. In this case, the interaction force F exerted parallel to basal plane on each of these dislocations is repelling. This force may be determined with the use of Eq.
where d is the distance between dislocation lines. Figures 6b and 6c show the plots of the dependence F(d) given by Eq. (14) at different values of parameter h for location of dislocations in TiB 2 and GaN crystals, respectively. It is helpful to represent Eq. (14) in the following equivalent form: The parameter f S expressed through Eq. (16)  (no force suppression in infinitely large crystal) and 0  F (force is gradually suppressed with approach of dislocations to free surface), respectively. The above discussed trends associated with the suppression of the dislocation interaction force by a free surface are well reflected by the plots presented in Figures 6b and 6c. The degree of influence of parameter  on parameters f S and F [Eqs. (15) and (16)] is discussed in the next Section 8.

Discussion
For analysis of the influence of elastic anisotropy on distribution of the dislocation stress field, it is helpful for materials considered in this study to quantify the so-called elastic anisotropy parameter. There should be distinguished two types of elastic anisotropy for a material: compressive and shear anisotropy. The compressive anisotropy is associated with compressive deformation and quantifies the difference in the compressibility of the material in different crystallographic directions. The shear anisotropy is defined in analogous way but in terms of the shear deformation. Since this study deals with the screw type dislocation (both non-zero stress components of the dislocation stress field are of shear type), there is a sense to quantify for materials of interest, GaN and TiB 2 , only the shear anisotropy parameter. The shear anisotropy parameter (expressed in per cent units) quantifies the degree of elastic anisotropy possessed by a crystal and is defined by the expression [25] % 100 where R G and V G are the isotropic shear moduli defined in Eq. (12). For GaN and TiB 2 , the values of parameter A calculated according to Eq. (17) are presented in Table 2 and show that the degree of elastic anisotropy of a TiB 2 single crystal, % 53 , is about twice higher than that of a GaN single crystal, . It is worth noting that in the case of an elastically isotropic material , all contours expand in size; the lower stress contours split one after one into two separate parts with termination at the free surface [particularly, Figure 3d shows that at h = 40 Å the lowest (−1 GPa) and intermediate (−1.5 GPa) stress contours are split with termination at the free surface], 2) at the same time, in the spatial range of 0  y , all stress contours shrink towards the dislocation line/core position with no splitting. Figures 4b and 4c (for the case of GaN) and Figures 4e and 4f (for the case of TiB 2 ) show that with approach of the dislocation position to a free surface (i.e., with decrease of the distance h) all zy  stress contours shrink towards the dislocation core position with no splitting. In comparison to the case with a GaN crystal, in the case of a TiB 2 crystal both zx  and zy  stress contours are extended over wider spatial regions (see the stress maps in Figures 3 and 4) as a result of a comparatively larger value in the energy factor s K in Eqs. (7) and (8) (see the data for s K in Table 2). The plots according to Eq. (13) ( Figure 5) show that with approach of the dislocation to the free surface the image force (i.e., the force of interaction of dislocation with the free surface) monotonically increases. As it follows from Eq. (13), this force is larger in the case of a TiB 2 crystal compared to the case of a GaN crystal owing to a comparatively larger energy factor s K in the former case.
The plots according to Eq. (14) (Figures 6b and 6c) show that with approach of a pair of dislocations to the free surface ( 0  h ) the interaction force of dislocations, F, strongly decreases at all distances between dislocations, d. According to Eqs. (8) and (14) where, In Eq. (18) (or equivalently in Eq. (4)), p G and b G are understood as the prism plane and basal plane shear moduli, respectively. From Eq. (18), it follows that parameter  is also valid for quantification of the shear anisotropy. However, it should be clarified that parameter Z A (or parameter  ) is sufficient only for characterization of the degree of shear anisotropy in cubic crystal lattice. For characterization of the overall shear anisotropy in hexagonal crystal lattice, it is recommended [25] to use the parameter A (Eq. (17)), which takes into account the difference/anisotropy in the shear modulus on all possible crystallographic planes. Meanwhile, for hexagonal crystal lattice, the parameter  characterizes the shear anisotropy only in terms of the shear moduli p G and b G defined in Eq. (18). In terms of the shear anisotropy parameter  , (i)  Table 3. Parameter  increases for these materials in the order Ti < GaN < TiB 2 < Zn < Cd < Graphite (see Tables 2 and 3) and this results in decrease of the force-suppressing parameter f S in the reverse order Ti > GaN > TiB 2 > Zn > Cd > Graphite (Figure 7). The plot in Figure 7 shows that parameter f S varies in a broad range, from 003 . 0  (for graphite) up to 0.25 (for Ti), and thereby demonstrates the importance of the account of elasticity anisotropy in the quantitative analysis of the interdislocation interaction forces.  (16)). However, at large interdislocation distances d, the force F by itself drastically decreases (see Eq. (15)). From the physical standpoint, in graphite a very small effect of the (0001) free surface on the force of interaction between near-surface basal screw dislocations results from a weak bonding between crystallographic (0001) basal planes. Some important peculiarities of the dislocation-induced stress distribution in above considered highly anisotropic materials, Ti, Zn, Cd, and graphite, will be analyzed in our next study.
It is important to mention that, in the plots of stress maps of a near-surface screw dislocation, a very good agreement was achieved in the isotropic approximation between the results obtained particularly from Eq. (10) and corresponding atomistic simulation [18]. It would be also important the validation of analytical stress expressions for a near-surface screw dislocation in elastically anisotropic materials (Eqs. (7) and (8)) by atomistic simulation models.
The results obtained in this study are also applicable to the case when a screw dislocation is situated in the "thin film-substrate" system at a (0001) basal interface between the film and substrate provided that the elastic constants of the film and substrate are the same or sufficiently close to each other.   Table 3.

Conclusion
Both the zx  and zy  dislocation stress components (Eqs. (7) and (8)) strongly depend on the position of dislocation with respect to a free surface. For a dislocation positioned in an infinitely large crystal, both the zx  and zy  stress contours (Figures 3a and 3e and Figures 4a and 4d) are symmetric in shape with respect to both x and y coordinate axes. With approach of the dislocation position to the free surface, the above stress contours drastically change in shape (Figures 3b-d, 3f-h and Figures 4b, 4c, 4e, 4f) preserving the spatial symmetry only with respect to the y axis. In comparison to the case of a crystal with a smaller value of the dislocation energy factor (GaN), in a crystal with a larger value of this parameter (TiB 2 ) the stress contours are extended over wider spatial regions (Figures 3 and 4) and the interaction of a near-surface dislocation with the free surface (resultant from the zx  stress component) is stronger ( Figure 5). For GaN and TiB 2 , the dislocation stress distribution maps are highly influenced by the free surface effect at "free surface-dislocation" distances roughly smaller than ≈15 and ≈50 nm, respectively. In terms of the shear anisotropy parameter A (Eq. (17)), the comparatively stronger effect of the elastic anisotropy on dislocation-induced stress distribution in TiB 2 is attributed to the higher degree of elastic anisotropy of this compound ( % 53 . , the force-suppressing factor, at comparable distances d and h (Figure 6a), is negligibly small, 003 . 0  f S (Figure 7).