A biperiodic interfacial pattern of misfit dislocation interacting with both free surfaces of a thin bicrystalline sandwich

doi:10.1016/S1293-2558(01)01212-2

Solid State Sciences

Volume 4, Issue 2, February 2002, Pages 161-166

https://doi.org/10.1016/S1293-2558(01)01212-2 Get rights and content

Abstract

The elastic field, of a thin bicrystalline sandwich with parallel faces when the interface between the two anisotropic crystal components is parallel to the faces and is composed of a biperiodic pattern of defects is numerically computed, using a double Fourier series method, an explicit formulation of each harmonic term of series is derived that depends explicitly on the anisotropic elasticity coefficients and thickness of each crystal. Application is made to the particular case of intrinsic dislocation meshes using a InAs/( $1 1 1$ )GaAs related system by a regular array of MDS at the interface, cf. J.G. Belk et al., Phys. Rev. Lett. 77 (1997) 475 and H. Yamaguchi et al., Phys. Rev. B 55 (1997) 1337, to determine free surface deformations through different thickness of the bicrystal.

Graphic

Introduction

Advances in the understanding of the structures of grain or phase boundaries are mainly due to observations made on thin bicrystalline foils by means of an electron microscope. For particular relative orientations of both crystals, the interfaces exhibit the presence of periodic arrays of defects. For a number of purposes and, in particular, in order to obtain a convincing identification of these defects a precise analysis of the elastic field of such interacting defects is required, especially if this field is influenced by one or more free surfaces as is certainly the case where high resolution electron microscopy is used. A first contribution to this problem has been proposed recently for the epitaxy case [3].

This paper deals with a similar problem but with a general hexagonal array of defects lying in the interface; a case which is more frequently encountered in practice.

Section snippets

Specification of the problem

In the following calculations most of the notations and conventions used in [4] are again adopted. The interfacial hexagonal pattern of linear defects, which could be dislocations, is illustrated in Fig. 1. The geometry of this pattern is specified by the base vectors $OA$ and $OC$ and the directed angle θ from $OC$ to $OA$ . The elastic constants of the crystal $+ (x_{2} >0)$ and of the crystal $− (x_{2} <0)$ are denoted by C_ijkl⁺ and C_ijkl⁻, respectively, while their thicknesses are denoted by h⁺ and h⁻. Two frames

General expressions for the elastic field of a periodic pattern of defects

The following “a priori” expressions for the displacements in each crystal were tried in [4] and found to solve the problem. Successfully for $j, k=1,2,3$ $u_{k} =u_{k}^{0} +v_{kj}^{0} x_{j} + Re ∑ G≠0 u_{k} (G,x_{2}) exp 2πi G R .$ Were:

(i)
u_k⁰ and v_kj⁰ are constants to be determined, the symbol Re means “real part of”,
(ii)
$G$ is a vector determined by two integers m, n (see Ref. [4b]) and such that: $G_{1} x_{1} +G_{3} x= (nc−ma cos θ)x_{1} ac sin θ + mx_{3} c,$ a and c being the magnitudes of the vectors $OA$ and $OC$ , while $G$ , m and n are the summation parameters defined in

A hexagonal pattern of misfit dislocations

The geometry of a typical general hexagonal cell of dislocation segments is given in Fig. 2. The origin O is at the centre of the cell while the axes Ox′₁ and Ox′₃ cut the cell, respectively, at points A₁ and C₁ on adjacent sides of the hexagon. H is the triple node closest to the origin in the planar area defined by the angle θ. The choice of θ and of the coordinates (θ₁,θ₂) of H in the frame Ox′₁x′₂x′₃, determines the geometry of the pattern. For misfit dislocation, the boundary conditions at

Application

To illustrate the internal elastic field and some thickness effects, a Fortran program is established to resolve the precedent linear system, for data, recent works as InAs/( $0 0 1$ )GaAs Belk et al. [1], InAs/( $1 1 1$ )GaAs Yamaguchi et al. [2], had study STM (Scanning Tunnelling Microscopy) images which exhibit the presence of an hexagonal mesh of edge dislocations parallel to the free surfaces with Burgers vector parallel to the interface, so it appears that the heteroepitaxiy of InAs on ( $1 1 1$ )GaAs is

Results and discussion

The validity of the analytical formulae which give the harmonic terms of Fourier series (13) has been verified numerically with a Fortran program which computes the relative displacement at a given point (x,y) of the interface with the two equivalent expressions: the analytical expression (12) and the Fourier series one (13).

Fig. 4a,b depicts, respectively, the representation of the relative displacement ΔU₃ and ΔU₁ for points placed, respectively, along the axes Ox₃ and Ox₁, we can see that

Conclusion

The formal solution in isotropic elasticity theory for the elastic displacement field caused by a planar hexagonal pattern of misfit dislocations found previously [4a], is developed and adapted to the anisotropic case and resolved numerically for the first time. The main result is to express explicitly the displacement and stress field as an analytical formulation under a double Fourier series for the important case where the hexagonal network is regular, and the Burgers vectors are parallel to

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Cited by (4)

Initial stages of misfit stress relaxation in composite nanostructures through generation of rectangular prismatic dislocation loops
2015, Acta Materialia
Citation Excerpt :
In the framework of our current investigation, we consider only the initial stage of the relaxation process when the rectangular shape of PDLs still exists. The critical conditions for the formation of the final relaxed configurations that are circular misfit dislocation loops around the cores in core–shell nanoparticles and in the cross sections of core–shell nanowires were calculated in Refs. [54,55,57,58] and [31,33,35–37,41,50], respectively, while those for straight edge misfit dislocations in the longitudinal sections of core–shell nanowires were calculated in Refs. [28,33,35,51,52], in bi-nanolayers in Refs. [22,59–69] and in tri-nanolayers in Ref. [22]. It is worth noting that in the literature, there are many experimental observations of the role of PDLs in misfit stress relaxation in planar heterostructures (see, for example, Ref. [70] and references therein).
Critical (necessary) conditions of misfit stress relaxation in bulk and hollow core–shell nanoparticles, bulk and hollow core–shell nanowires, and flat bi- and tri-nanolayers (composing of a substrate and one or two thin films, respectively) through generation of rectangular prismatic dislocation loops (PDLs) at the inner and outer interfaces of the composite nanostructures with subsequent extension of the PDLs into cores (substrates) or shells (films) are analyzed theoretically and discussed in detail. The critical conditions are considered for different shapes and places of generation of the PDLs. It is shown that the misfit parameter is the principal factor, which determines the energetic preference of PDL generation: the latter is energetically favorable when the misfit parameter reaches its critical value, which depends on the geometry parameters of a composite nanostructure and the PDL type. Regardless of the place of PDL generation, the most energetically favorable case is when the PDL is elongated along the interface. The generation of PDLs from the free surface is the most preferable for small thicknesses of the shells (films). Growth of the shell (film) leads to a change of places of PDL generation while maintains their preferential shape – elongated along the interface. Among the six composite nanostructures under consideration, the tri-nanolayers are the least stable to the PDL generation. The hollow nanoparticles and nanowires are more stable to the PDL generation than the bulk ones. The bi-nanolayers are the most stable to the PDL generation.
Stress and displacement fields around misfit dislocation in anisotropic dissimilar materials with interface stress and interface elasticity
2015, Journal of Applied Mechanics, Transactions ASME
Free surface effect on displacement and relative interfacial displacement fields of misfit dislocations at the bicrystal interface
2007, Transactions of the Indian Institute of Metals
Anisotropic elasticity of multilayered crystals deformed by a biperiodic network of misfit dislocations
2007, Physical Review B - Condensed Matter and Materials Physics

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A biperiodic interfacial pattern of misfit dislocation interacting with both free surfaces of a thin bicrystalline sandwich

Abstract

Introduction

Section snippets

Specification of the problem

General expressions for the elastic field of a periodic pattern of defects

A hexagonal pattern of misfit dislocations

Application

Results and discussion

Conclusion

Philos. Mag. A

Philos. Mag.

Phys. Rev. Lett.

Phys. Rev. B

Philos. Mag. A

Philos. Mag.

Acta Metall.

Philos. Mag. A