phys. stat. sol. (a) 204, No. 9, 3126–3131 (2007) / DOI 10.1002/pssa.200622169

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Numerical simulation of anisotropic elastic fields

of a GaAs/GaAs twist boundary

Salah Madani*, Toufik Outtas, and Lahbib Adami

Département de Mécanique, Faculté des Sciences de l’Ingénieur, University of Batna,

Avenue Chahid Boukhlouf, 05000 Batna, Algeria

Received 6 April 2006, revised 9 April 2007, accepted 16 April 2007

Published online 20 June 2007

PACS 61.72.Mm, 68.35.Fx, 68.35.Gy, 68.55.Jk, 68.65.Hb

Self-assembled nanostructures are particularly interesting for optoelectronic and photonic applications,

especially on silicon and GaAs substrates. Nevertheless, their long-range spatial distribution is random,

their density is difficult to control, their size distribution can be large and their shapes can be different. By

overcoming these drawbacks, it should be possible to improve the performances of existing devices or to

fabricate new ones. This work studies the possibility to order on a long range self-assembled nanostruc-

tures on a GaAs substrate, by means of the elastic fields induced at the surface by shallowly buried peri-

odic dislocation networks. The needed strain and stress fields, generated by a square network of screw

dislocations located between a finite layer of GaAs bonded onto a semi-infinite GaAs substrate, are calcu-

lated using anisotropic elasticity. The results obtained are compared to those obtained using isotropic elas-

ticity.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The technology of self-assembled quantum dots (SAQDs), with their nanoscale and discrete energy spec-

trum, has become very attractive to provide a variety of advances for electronics, such as increased effi-

ciency, reduced power requirements, increased speeds of operation and many novel electronic character-

istics [1, 2].

Some potential applications have already been expected to be explored with quantum dot (QD) tech-

nology, such as nanosize memory devices, intersubband photodetectors, quantum dot based lasers and

cavity quantum electrodynamics (QED). However, further developments of various applications depend

on overcoming the major challenge of the problem; that is, how to make quantum dots assemble them-

selves in a very precise manner, and how to control their size distributions during the fabrication process.

One way to control the location of quantum-dot nucleation and their size is the use of a functional

substrate inducing a lateral self-organization by a buried dislocation network (DN) [3–5]. The network

periodicity is controlled by the orientation angle between the substrate and the bonded film. Let us note

that the basic relationship between the dislocation network and SAQD nucleation is not yet fully investi-

gated.

In this work, anisotropic elasticity theory, based on the analysis of the elasticity of dislocations in

subsurface layers [6, 7], is used to determine the strain field and energy density, generated by a square

network of screw dislocations, at the free surface of a finite layer of GaAs bonded onto a semi-infinite

GaAs substrate. The effects of the thickness of the layer and the periodicity of the dislocations are also

* Corresponding author: e-mail: madani_s@hotmail.com, Phone: +213 33 81 21 43, Fax: +213 33 81 21 43

phys. stat. sol. (a) 204, No. 9 (2007) 3127

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Original

Paper

Crystal 1 Crystal 2

Edge dislocations

ϕ

Crystal 1

Crystal 2

Screw dislocations

(a) (b)

Fig. 1 Different kinds of dislocation networks. (a) A parallel network of edge dislocations. (b) A square

network of screw dislocations.

shown. Finally, computational results and discussions are given and then compared to those obtained

using isotropic elasticity.

2 Geometry of the elastic model

The molecular bonding of two GaAs(001) crystals creates two kinds of dislocation networks [8–12],

depending on the out-of-plane and in-plane disorientations. The first network, made up of parallel mixed

dislocation lines, is induced by the flexion between the two crystals (Fig. 1a). The second network is

a square network of screw dislocations (Fig. 1b), whose periodicity λ is related to the in-plane rotation ϕ

by Frank’s formula /(2 2 sin ( / 2)),aλ ϕ= where a is the silicon lattice parameter. Screw dislocations

have Burgers vectors of the type /2 110a · Ò . In this paper we have considered only the second case. Figure 2 shows the geometry of our problem

where a square network of screw dislocations is located at the interface separating the thin layer and the

substrate. In this analysis most of the notation and conventions used in previous papers [13, 14] are again

Substrate

Layer

Dislocationslines

X1

X2

X3

h

Fig. 2 Geometry of the problem of a square

network of screw dislocations.

3128 S. Madani et al.: Numerical simulation of anisotropic elastic fields

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-a.com

adopted. To resolve this problem let us consider first one family of screw dislocations parallel to OX3.

The displacement field that results from strain is periodic and can be described quite generally by the

function

( ) ( )( )

2 1

0

exp 2i ,n

k k

n

u U x gnxπ

= p (1)

where U( )n

k(x2) is a complex function depending on x2.

In order to fulfil Hooke’s law and the local equilibrium requirements, the displacement in each

medium must be such that

,

0 .ijkl k jiC u◊ = (2)

To simplify the resolution of the problem, and also to perform the numerical calculation, the final

expression of the displacement field can be written as

3

1 2

0 1

( ) ( )

2 2

1{cos [ ( )]

Re [( i ) exp ( ) + ( i ) exp ( )]}

k

n

n n

kk

u n x r xn

X n s x Y n s x

α

α

αα α α α α

ω

λ ω λ ω

− − −

�> =

Ê ˆ= +Ë ¯p

¥

 Â

( ) ( )

1 2 2 2+{sin [ ( )] Re [( ) exp ( ) + ( ) exp ( )]} ,n n

kkn x r x X n s x Y n s xα

α α α α α αω λ ω λ ω − �+ (3)

where 2ω λ= p .

The strain εij and stress σij fields are derived from Eq. (3) using Hooke’s law:

1, ,2

( ) .ij i j j iu uε = + (4)

For the stress field, we obtain

3

( ) ( )

1 2 2 2

0 1

2 {cos [ ( )] Re [ exp ( ) exp ( )]}n nijij ij

n

g n x r x X L n s x Y L n s xαα α α α α α

α

σ ω ω ω �> =

= + - +Â Â

( ) ( )

1 2 2 2{sin [ ( + )] Re [i exp ( ) i exp ( )]}n nijijn x r x X L n s x Y L n s xα

α α α α α αω ω ω �+ - + (5)

with

[ ]1 2= , , = 1, 2, 3 , = 1, 2 ,kl j klj kljL C p C i j l

α α αλ +

where , X Yα α

+ +

and Yα

-

(α = 1, 2, 3) are nine complex unknowns to be found from the following boundary

conditions:

(i) The Heaviside step function for the discontinuity of each component of the interfacial relative

displacement.

( ) ( )2 2

1 10 0

1

, 1/ sin . 2

k k k

k k k kx x

n

b b bu u x u u n n xω

Λ

•

+ - + -

= =

=

Ê ˆ- = - - = -È ˘ È ˘Î ˚ Î ˚Ë ¯ p  (6)

(ii) The continuity of the stresses σ2k along the interface (x2 = 0).

2 2

2 20 0.

k kx xσ σ

+ -

= =

= (7)

(iii) The nullity of the stresses σ2k at the free surface (x2 = h).

2

20 .

k x hσ

+

=

=È ˘Î ˚ (8)

phys. stat. sol. (a) 204, No. 9 (2007) 3129

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Original

Paper

X1

X3

X2

M

Family I

Family II

Finally, a simple way to evaluate the elastic energy E stored in the layer and the substrate when the

interfacial shift uk

+ − uk

− is produced is given by

1 12 2

,

.ij ij

i j

E σε σ ε= = Â (9)

Due to the principle of superposition [13], the elastic field of two networks of dislocations on the same

interface can be resolved simply. The second family II is deducted from the previous one (family I) by a

rotation of +π/2 around the axis OX2, see Fig. 3. So, the point M ′ (−x3, x2, x1) corresponds to the point M

(x1, x2, x3) and the total displacement field becomes

( ) ( )

( ) ( )

( ) ( )

1 1 3

2 2 2

3 3 1

u u M u M

u u M u M

u u M u M

+ ¢È ˘È ˘Í ˙Í ˙ = + ¢Í ˙Í ˙

Í ˙ Í ˙- ¢Î ˚ Î ˚

. (10)

In a similar way, the total stress field is calculated by

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

I I I I I I

11 33 12 23 13 13

(I+II) I I I I I I

12 23 22 22 23 12

I I I I I I

13 13 23 12 33 11

.ij

M M M M M M

M M M M M M M

M M M M M M

σ σ σ σ σ σ

σ σ σ σ σ σ σ

σ σ σ σ σ σ

+ + -¢ ¢ ¢È ˘Í ˙

= + + -¢ ¢ ¢È ˘Î ˚ Í ˙Í ˙- - +¢ ¢ ¢Î ˚

(11)

3 Application to GaAs/GaAs(100) and interpretation

The GaAs/GaAs(100) system has been largely studied experimentally [11, 12]. The elastic constants of

gallium arsenic alloy and other parameters used in the present simulation are given in Table 1. The

thickness of the thin layer is chosen equal to 20 nm and the orientation angle between the substrate and

the layer is taken as 0.87°.

Figure 4 present the strain ε13 contours in the plane (x1, x3) for x2 = 20 nm (i.e. at the free surface). The

dislocation lines are indicated by arrows. We can observe in this figure that the strain is equal to zero on

Table 1 GaAs alloy experimental elastic constants [8, 10].

elastic constants

[GPa]

lattice parameter a

[nm]

Burgers vector b

[nm]

period λ

[nm]

C11 = 118, C12 = 53.5, C44 = 59.4 0.5653 0.3838 25

Fig. 3 Two families of screw dislocations.

3130 S. Madani et al.: Numerical simulation of anisotropic elastic fields

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-a.com

0 5 10 15 20 25

X1(nm)

0

5

10

15

20

25

X3(

nm

)

-0.00065

-0.00050

-0.00035

-0.00020

-0.00005

0.00010

0.00025

0.00040

0.00055

the diagonals of the square network and that extremes are located above the medium of the dislocation

lines. The maximum strain calculated at this free surface is 0.065%. This value is greater than the value

obtained by Coelho [11], which is arround 0.035% in the case of isotropic elasticity.

Figure 5 shows the energy density on a surface of 2 × 2 periods (3D surface). Extremes are, like strain,

located above the medium of the dislocation lines. The maximal energy value calculated at the free sur-

face is in the order of 1.43 × 10−5 J/m2.

In the previous study, the periodicity of the network of dislocations and the thickness of the thin layer

were fixed. What happens when the period of the network varies?

Figure 6 shows the evolution of maximum energy density in the surface (x2 = h) for various periods of

the network (λ varies from 5 to 300 nm). For a fixed thickness, maximal energy density increases with

the periodicity of the network, until it reaches a limiting value. For example, the domain of periodicity

between 50 and 100 nm is very interesting to obtain sufficient elastic energy in the surface.

Fig. 5 3D surface plot of energy density on a surface of 2 × 2 periods.

Fig. 4 ε13 strain contours for GaAs/GaAs(100).

phys. stat. sol. (a) 204, No. 9 (2007) 3131

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0 50 100 150 200 250 300Period

-0.0010

0.0000

0.0010

0.0020

0.0030

0.0040E

nerg

y

λ (nm)

E (J/m )

h = 20 nm

2

Fig. 6 Energy as a function of period λ for fixed thickness of the thin layer.

4 Conclusion

In this paper, for the first time, anisotropic elasticity is used to determine the strain contour and energy

density, induced by a square network of screw dislocations, at the free surface of a finite layer of GaAs

bonded onto a semi-infinite GaAs substrate. The result shows that the anisotropic effect on strain field

and energy density could not be neglected because the extreme values are smaller than in the isotropic

case. The effect of periodicity of the dislocation network on the strain field was also discussed. This

result shows that by using a strain-selective etching, we can manage to define a nanopatterned surface in

a well-controlled way.

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