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

Solid State Sciences 4 (2002) 161–166 www.elsevier.com/locate/ssscie

A biperiodic interfacial pattern of misfit dislocation interacting with both free surfaces of a thin bicrystalline sandwich Toufik Outtas a,∗ , Lahbib Adami a , Roland Bonnet b a Département de Génie mécanique, Faculté des Sciences de l’ingénieur, Université de Batna,

Rue Chahid Boukhlouf Med ElHadi, CP 05000 Batna, Algeria b INPG de Grenoble, Laboratoire de thermodynamique et physico-chimie métallurgiques CNRS, Domaine Universitaire,

BP 75, 38402, saint Martin d’Hères, France Received 22 January 2001; received in revised form 10 August 2001; accepted 19 September 2001

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.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Dislocations; Thin bicrystal; Misfit; Elasticity; Free surfaces

1. 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.

* Correspondence and reprints.

E-mail address: [email protected] (T. Outtas).

2. 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 pat−−→ −−→ tern is specified by the base vectors OA and OC and the di−−→ −−→ rected angle θ from OC to OA. The elastic constants of the crystal + (x2 > 0) and of the crystal − (x2 < 0) are denoted

Fig. 1. Hexagonal network of misfit dislocations at the interface of a thin bicrystal.

1293-2558/02/$ – see front matter  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 1 ) 0 1 2 1 2 - 2

162

T. Outtas et al. / Solid State Sciences 4 (2002) 161–166 (G)

(G)

and the coefficients pα and λαk are complex values depending for each medium on the elastic constants Cij kl , a, c, θ , m, n, while the six complex values (G) Pα are unknown depending on m, n, etc. and on the boundary conditions of the problem. For convenience we shall suppress the explicit dependence of new variables on G and introduce some new variables. (G)

Xα = Pα

Fig. 2. A regular hexagonal network of misfit dislocations oriented −→ clockwise H1 H2 H3 H4 H5 H6 around the centre O of a hexagon. OA and −−→ OC are the period vectors. The Cartesian frame work used is Ox1 x2 x3 with −−→ Ox2 normal to the interface towards medium +, and Ox3 OC, the angle θ −→ −− → ◦ between OA and OC is 60 .

Cij+kl

and Cij−kl , respectively, while their thicknesses are by h+ and h− . Two frames of reference will be

by denoted used, both having a common origin O at the centre of a hexagonal cell, see Fig. 2. The orthogonal Cartesian frame Ox1 x2 x3 with Ox2 pointing from crystal − to crystal + and −−→ Ox3 parallel to OC will serve to solve the differential equations of elasticity. The usual oblique frame Ox1 x2 x3 such −−→ that Ox1 OA, Ox2 Ox2 , Ox3 Ox3 will allow the use of “natural” coordinates to express simply the boundary conditions on the displacements in the problem.

3. 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
0 xj + Re uk ( G, x2 ) exp 2πiG R. uk = u0k + vkj (1) G =0

(i)

0 vkj

and 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: (nc − ma cos θ )x1 mx3 G1 x 1 + G3 x = (2) + , ac sin θ c −−→ a and c being the magnitudes of the vectors OA and −−→ OC, while G, m and n are the summation parameters defined in [4b,c,5], (iii) the pre-exponential factor is given in [6] by: uk ( G, x2 ) =

6
α=i

 (G) (G) ( G ) Pα λαk exp 2πipα ,

(4)

for α = 4, 5, 6,

(5)

in to real and imaginary parts as:

= rα ± isα ,

(6)

with sα > 0 for α = 1, 2, 3. It now follows that Eq. (3) becomes: 2πiuk ( G, x2 ) =

3
α=1

 (G) (G)  Xα λαk exp 2πipα x2  (G) ( G)  + Yα λαk exp 2πi p¯ α x2 ,

(7)

which, on introducing this expression into (1), leads to:
  0 xj + Re Cαk exp 2πi rα x2 + G R , (8) uk = u0k + vkj G =0

where (G)

(G)

2πiCαk = Xα λαk exp[−2πsα x2 ] + Yα λαk exp[2πsα x2 ], and it should be remembered that Xα , Yα and sα depend explicitly on G. The stresses are derived from Hooke’s law. Putting (7) into (1) and then differentiating gives: σij = σij0 + Re

3



  Dαij exp 2πi rα x2 + G R ,

(9)

G =0 α=1

with: 0 σij0 = Cij kl vkl ,

Dαij = Xα Lαij exp[−2πsα x2 ] + Yα Lαij exp[2πsα x2 ],

Were: u0k

(G) pα

for α = 1, 2, 3,

2πi,

(G) Yα = Pα 2πi, (G) and resolve pα

(3)

0 where the values vkl , Lαij and Lαij are defined in [4b]. In terms of trigonometric functions, the general expressions for the displacement and stress fields are: 0 xj uk = u0k + vkj

+ Re

3

G =0 α=1

σij = σij0 + Re




  Cαk cos 2π rα x2 + G R   + iCαk sin 2π rα x2 + G R ,

3


G =0 α=1

(10)

  Dαij cos 2πi rα x2 + G R   + iDαij sin 2πi rα x2 + G R . (11)

T. Outtas et al. / Solid State Sciences 4 (2002) 161–166

163

4. 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 Ox1 and Ox3 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 Ox1 x2 x3 , determines the geometry of the pattern. For misfit dislocation, the boundary conditions at the interface are determined by the following sequence of operations. (i) A stress-free crystal slice is generated. (ii) A cut is made along the interface, and, removing or adding matter as necessary, each point of the upper surface of the cut (x2 > 0) is moved with respect to the lower surface (x2 < 0) in such a manner that in each − hexagonal cell the relative displacement [u+ k − uk ]x2 =0 is zero at the centre of the cell and varies linearly inside the cell. Such a variation is indicated in Fig. 3; in particular, the relative displacements at the points A an C are denoted by b(A) /2 and b(C)/2, respectively. Thus, when crossing the segments H1 H2 and H6 H1 along the axes Ox1 and Ox3 , it is clear that the discontinuity vectors for the relative displacements of crystal + relative to crystal − are b(A) and b(C), respectively. Hence, in the frame Ox1 x2 x3 , the relative displacement inside the unit cell centered at the origin O is:     x x1 (A) (C) + − uk − uk = (12) bk + 3 bk . a c This condition has been found in agreement with high resolution electron microscopy observations Ref. [8] of a single family of misfit dislocations along a CdTe/(0 0 1)GaAs heterointerface, details of development in Appendix B of Ref. [5]. (iii) The two surfaces of the cut are now welded together so as to ensure continuity of the stress across the interface. In passing, let us note that the boundary conditions (12) correspond to what is usually meat by the term semicoherent interface. This interface has three types of misfit dislocation segments whose strengths are specified by the perimeter of the cell H1 H2 . . . H6 . It is interesting to note −−→ −−→ that, for very large vectors OA and OC (relative to b(A) and b (C)), the elastic field in the tow media + and − in the vicinity of the perimeter H1 H2 . . . H6 will be close to the field caused by extrinsic volterra dislocations oriented along −−−−→ −−−−→ −−−−→ H1 H2 , H6 H1 , H5 H6 and with Burgers vectors b(C) , b(A) , b(A) − b(C) (RS/RH convention). The boundary conditions for the thin bicrystal are finally: (i) Periodicity of the displacement field in x1 and x3 , so 0 = 0. In the following, the displacement field is that vkj

Fig. 3. Schematic representation of boundary conditions in displacement, which shows the interfacial discontinuity through the pattern, the middles of H1 H2 and H6 H1 are transformed by the Burgers vectors components b(A) and b(C) , respectively.

assumed strictly periodic, as suggested by transmission electron microscopy observations of misfit dislocations using thin foils of bicrystalline material, e.g., Hamelink and Schapink Ref. [9], Tietz et al. Ref. [10] and SagoeCrentsil and Brown Ref. [11], etc. In addition, in O, the displacement uk is assumed zero for both crystal which gives u0k = 0. ± (ii) When x2 = h± , σ2j = 0 for all x1 and x3 . (iii) On the interface x2 = 0 and inside the hexagonal cell + − σ2j = σ2j and the relations (12) hold. These latter relations apply to all hexagonal cells and can be equivalently expressed by a biperiodic Fourier series for an odd function thus,
( G) − u+ (13) Ak sin 2πG R, k − uk = G

where the coefficients A( G ) are derived from a cumbersome calculation, as a result for any hexagonal pattern they can be expressed in terms of θ1 , θ2 , m, n, b(A) , b(C) by: [2π(nθ1 + mθ2 )] π 2 [m + n − 2(nθ1 + mθ2 )]   1 − 2θ1 b(C) (1 − 2θ2)b(A) + . × n − 2(nθ1 + mθ2 ) m − 2(nθ1 + mθ2 )

A( G ) = sin

(14) For a regular hexagon, the cases of zero denominators of (14) are discussed in Appendix B of Ref. [7]. As a result, for each vector G, the following system of 24 real equations determines the 12 complex unknowns Xα+ , Yα+ , Xα− , Yα− for j, k = 1, 2, 3. 3
 + (G) − Re = Ak , i Cαk − Cαk α=1

(15)

164

Re

T. Outtas et al. / Solid State Sciences 4 (2002) 161–166 3


+ − Cαk = 0, − Cαk

(16)

+ −  Dα2j = 0, − Dα2j

(17)

α=1

Re

3
 α=1

Re

3
 + −  = 0, i Dα2j − Dα2j

(18)

α=1

Re

3


+ Dα2j = 0,

(19)

+ iDα2j = 0,

(20)

− Dα2j = 0,

(21)

− iDα2j = 0.

(22)

α=1

Re

3
α=1

Re

3
α=1

Re

3
α=1

+ − + − In Eqs. (15)–(18), the coefficients Cαk , Cαk , Dα2j and Dα2j evaluated at x2 = 0 and in Eqs. (19)–(21), the coefficients + − Dα2j and Dα2j evaluated at x2 = h+ and x2 = h− , respectively. (There are three equations in each of the above expression.) Solving this linear system gives the solution to the problem of finding the elastic field of the misfit dislocation pattern.

5. Application To illustrate the internal elastic field and some thickness effects, a F ORTRAN 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 an ideal system for the study of dislocations by STM. The two crystals have same parallel directions and Burgers vectors of misfit dislocations are from 1/21 1 0 type, the lattice parameters and elastic anisotropic constants are given by Chami [12] as: aInAs = 0.6058 nm and aGaAs = 0.5653 nm, the misfit ε which is given by: ε=2

aInAs − aGaAs = 0.069, aInAs + aGaAs

has a positive value, it means that the crystal + is in bidimensional traction relative to crystal −, Burgers vector √ modulus is given by: |b| = (aInAs + aGaAs)/2 2 and the

components will be as follow:   √ 0 −b 3/2 (C) (A) . and b = b = 0 0 −b/2 −b −−→

−−→

Period vectors OA and OC from hexagonal cell √ have same length as: a = c = aInAsaGaAs/|(aInAs − aGaAs ) 2|. All numerical parameters are resumed in Table 1. For numerical computation, the summation parameters m and n were stopped at 20 because they gives a good convergence of double Fourier series, and for different total thicknesses changing from h = −10aInAs to −2aInAs for x2 < 0, and from h = 2aInAs to 10aInAs for x2 > 0, we represent the aspect of free surface deformations in the Ox2 x3 and Ox1 x2 planes.

6. Results and discussion The validity of the analytical formulae which give the harmonic terms of Fourier series (13) has been verified numerically with a F ORTRAN 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 *U3 and *U1 for points placed, respectively, along the axes Ox3 and Ox1 , we can see that the two results (between expressions (12) and (13)) in each figure differ by less then 2.5% over 3280 harmonics, as expected at a nanometer distance from misfit dislocation core in Fig. 4b and from triple node in Fig. 4a. The harmonics (m, n) are defined by the extended domain D used for the computation of the Fourier series [5]. Fig. 5a,b shows the topology of surfaces deformations for different bicrystals respectively in the plan (x1 , x2 ) and (x2 , x3 ), the bicrystals chosen are (−2aInAs, +2aInAs), (−4aInAs, +4aInAs), (−6aInAs, +6aInAs ), (−8aInAs, +8aInAs), (−10aInAs, +10aInAs), the black points represent the elastically deformed grid with a u2 component exaggerated by a factor 20 for better illustration of the free surface deformation. It is noteworthy that the heterointerface exhibits a hollows directly above the core region indicate by arrows in the plan (x2, x3 ) Fig. 5b, and above the triple node region in the plan (x1 , x2 ) Fig. 5a, we can seen that the different hollows correspond to a full accommodation of the interfacial hexagonal pattern.

7. 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

T. Outtas et al. / Solid State Sciences 4 (2002) 161–166

165

Table 1 Data used for a numerical application for the heterowin interface InAs/(1 1 1)GaAs Crystal InAs GaAs

Lattice parameter (nm) aInAs = 0.6058 aGaAs = 0.5653

Elastic constants (GPa)

Zener factor (anisotropy)

Burgers vectors

2.08 1.82

b(ZU) = (−0.358; 0; 0.2069) b(UV) = (0; 0; −0.414)

C11 = 86.5, C12 = 48.5, C44 = 39.6 C11 = 118, C12 = 53.5, C44 = 59.4

(a)

(b)

Fig. 4. (a) relative displacement along O × 1 towards a triple node of hexagonal cell. (b) relative displacement along O × 3 towards a segment of hexagonal cell.

(a)

(b)

Fig. 5. (a) aspect of free surface deformations in the O × 1 × 2 plan. (b) aspect of free surface deformations in the O × 2 × 3 plan.

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 the interface. The numerical application proposed using the heteroepitaxy InAs/(1 1 1)GaAs, is largely validated in term of relative displacement through the interface with analytical expressions, except near the core or triple node region where the elasticity theory is not applied; the representation of free surfaces deformations is in full accommodation with the hexagonal geometry of the pattern, we can by this mean detect the uncritical thickness which is given by the undeformed sur-

faces at h = ±10aInAs for each crystal, note that the present numerical simulation may be applied to compute the STM (Scanning Tunneling Microscopy) image relative to similar thin bicrystalline sandwich. References [1] J.G. Belk, J.L. Sudijono, X.M. Zhang, J.H. Neave, T.S. Jones, B.A. Joyce, Phys. Rev. Lett. 77 (1997) 475–478. [2] H. Yamaguchi, J.G. Belk, X.M. Zhang, J.L. Sudijono, M.R. Fahy, T.S. Jones, D.W. Pashley, B.A. Joyce, Phys. Rev. B 55 (1997) 1337– 1340. [3] R. Bonnet, Philos. Mag. A 73 (1996) 1193–1209.

166

T. Outtas et al. / Solid State Sciences 4 (2002) 161–166

[4] (a) R. Bonnet, Philos. Mag. 44 (1981) 625; (b) R. Bonnet, Acta Metall. 29 (1981) 437; (c) R. Bonnet, Philos. Mag. A 43 (1981) 1165. [5] A. Bouzaher, R. Bonnet, Philos. Mag. A 66 (1992) 823–837. [6] R. Bonnet, A. Bouzaher, J.M.S.M.94, Fac. Sc. Ain Chock, 1 ENSEM 113–117. [7] A. Bouzaher, R. Bonnet, Acta Metall. Mater. 41 (5) (1993) 1595–1603.

[8] R. Bonnet, M. Loubradou, Phys. Rev. B 49 (1994) 14397–14402. [9] J.J.C. Hamelink, F.W. Schapink, Philos. Mag. 44 (1981) 1219–1228. [10] L.A. Tietz, S.R. Summerfelt, C.B. Carter, Mater. Res. Soc. Symp. Proc. 159 (1990) 209–215. [11] K.K. Sagoe-Crentsil, L.C. Brown, Philos. Mag. A 63 (1991) 447. [12] A.C. Chami, Thèse d’état sur Contraintes et relaxation des couches de semi-conducteurs, Étude par canalisation, Université d’Alger, 1998.