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Elastic instability of slip traces in oxidized thin foils

Sami Youssef1

, Mustapha Fnaiech1

, and Roland Bonnet2

1

Unité de Recherche de Physique du Solide, Faculté des Sciences, Boulevard de l’Environnement,

5019 Monastir, Tunisie

2

Science et Ingénierie des Matériaux et Procédés, INPGrenoble-CNRS-UJF, BP 75,

38402 Saint Martin d’Hères, France

Received 22 November 2006, revised 13 December 2006, accepted 21 December 2006

Published online 19 January 2007

PACS 61.72.Nn, 68.55.Ln, 68.60.Bs

Parallel slip traces are often observed in transmission electron microscopy of oxidized and deformed thin

foils, but apparently they have not been reported for foils with thicknesses less than a few tens of nanome-

ters. This situation can be explained by an elastic instability of a dislocation dipole for foil thicknesses

lower than a critical threshold value. This value is derived from the knowledge of the elastic field of a dis-

location in a three-layer laminated medium and the minimization of the elastic potential energy of the thin

foil.

phys. stat. sol. (b) 244, No. 6, 1908–1912 (2007) / DOI 10.1002/pssb.200642587

phys. stat. sol. (b) 244, No. 6, 1908–1912 (2007) / DOI 10.1002/pssb.200642587

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

Elastic instability of slip traces in oxidized thin foils

Sami Youssef 1, Mustapha Fnaiech1, and Roland Bonnet*, 2

1 Unité de Recherche de Physique du Solide, Faculté des Sciences, Boulevard de l’Environnement,

5019 Monastir, Tunisie 2 Science et Ingénierie des Matériaux et Procédés, INPGrenoble-CNRS-UJF, BP 75,

38402 Saint Martin d’Hères, France

Received 22 November 2006, revised 13 December 2006, accepted 21 December 2006

Published online 19 January 2007

PACS 61.72.Nn, 68.55.Ln, 68.60.Bs

Parallel slip traces are often observed in transmission electron microscopy of oxidized and deformed thin

foils, but apparently they have not been reported for foils with thicknesses less than a few tens of nanome-

ters. This situation can be explained by an elastic instability of a dislocation dipole for foil thicknesses

lower than a critical threshold value. This value is derived from the knowledge of the elastic field of a dis-

location in a three-layer laminated medium and the minimization of the elastic potential energy of the thin

foil.

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

1 Introduction

The presence of slip traces or slip bands in the images of deformed thin crystalline specimens observed

in conventional transmission electron microscopy (TEM) has been well known since the first works on

the subject [1]. In particular, their diffraction contrast were investigated positively in [1, 2] from solu-

tions of the elasticity problem of dislocations running parallel to the free surface of a semi-infinite Al

substrate covered by an Al2O3 thin film. Nowadays, surprisingly, the literature dealing with high resolu-

tion TEM images of similar oxidized specimens does not report any evidence of parallel slip traces. Such

traces would be in fact of great interest for workers looking at in plane dislocations to study their atomic

cores, see e. g., [3] for dissociated screw dislocations in silicon. Since this technique requires ultra-thin

foils, the authors suggest that instability of slip traces could take place in such specimens. The suggested

explanation is based on the elastic properties of a dislocation dipole running in the middle crystal of a

three-layer heterogeneous medium. The selected example of an oxidized Al foil clearly illustrates such

an effect for a strong interfacial elastic mismatch.

2 Geometry of the problem

Figure 1 describes the geometry of a thin plate-like specimen of crystal A covered by two nanometric

layers of amorphous oxide denoted B. To simplify the problem, the three-layer system B–A–B is sup-

posed to be deformed by only two opposite dislocations. Crystal A has a thickness h, Young modulus µ h,

Poisson ratio ν h, while the two oxide layers B have the same nanometric thickness e but elastic constants

µe and ν e. The two dislocation lines are placed in crystal A, at the same distance d from an Al/Al2O3

interface and lie in the dotted plane that is at an angle α from Ox1. Each dislocation line is oriented in the

sense of the axis Ox3 of the Cartesian frame Ox1x2x3. Their Burgers vectors are b(1) and b(2). Dislocation

* Corresponding author: e-mail: rbonnet@ltpcm.inpg.fr, Phone: + (0)4 76 82 66 24, Fax: + (0)4 76 82 66 30

phys. stat. sol. (b) 244, No. 6 (2007) 1909

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Original

Paper

b(1) is along Ox3, and the axis Ox2 is directed along the upward normal to the foil. The question is to find

if any d value corresponds indeed to a mechanical equilibrium of the dislocation dipole when the thick-

ness h of the middle crystal A changes.

3 Elastic potential energy of the foil

If the elastic potential energy W(d, h) of the deformed foil can be expressed, the search for its minimum

in function of d should predict what is the equilibrium distance d equ. By definition, the entity W is the

sum of three terms: two of them correspond to the elastic self-energies of dislocation b(1) and b(2), and

another one to the interaction energy W int between the two dislocations. Since the two dislocations are at

the same distance d from an A/B interface, the result is

self int( , ) 2 .W d h W W= + (1)

The term W self is evaluated from the following argument: the elastic field around dislocation b(1)

should be the same as that of a misfit dislocation which is part of a regular dislocation array placed along

the plane x2 = 0, at the limit of a period Λ tending to infinite. This reasoning argument was already

pointed out to treat the elastic field of a translation dislocation in any heterogeneous multilayer [4]. W self

is calculated as the work done to create one misfit dislocation of the array from the application of an

appropriate cut-and-glue operation applied along the plane x2

= 0 over a period. The integral giving W self

is formally the same as that proposed in [5] for a two-layer/substrate system, except that the stresses [σ2k]

(k = 1–3) at x2 = 0 should be calculated for the B–A–B medium described in Fig. 1:

0

0

(1)self 1

2 1 1

1( , 0) d .

2 2

r

k

k

r

b xW x x

Λ

σΛ

-

-Ê ˆ Ê ˆ= -Á ˜ Ë ¯Ë ¯ Ú (2)

This integral is calculated explicitly using the method given in Ref. [5]. As a result, W self is obtained

as a Fourier series, the harmonic coefficients of which depend on all parameters described in Fig. 1,

elasticity constants and on the cut radius r0. Since two or three pages are required to write explicitly its

full expression for the edge components (k = 1 and 2), only the contribution of the screw component is

given:

2(1)

3 0 0 0self

21

( (π ) cos ( ) sin ( )),

2π

h

n

b n r n r n r NW

n D

µ ω ω ω

ω

•

=

Ê ˆ- +È ˘Î ˚= Á ˜Ë ¯

 (3)

Fig. 1 A three-layer thin foil deformed by a dislocation dipole

placed in the middle crystal with Burgers vectors b(1) and b(2) = –b(1)

.

Conventions and symbols: see text. The slip plane is in the dotted

line.

1910 S. Youssef et al.: Elastic instability of slip traces in oxidized thin foils

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

in which

(( 1 ) ( ) ( ) (1 )) ( (1 ( 1 )) (1 )) ,N F B F s B F F G s G s T G s Gs= - + + + - + + + - + + - - - (4)

2 2 2

2 2 2

2 ( 1 ) ( 1 ) 2 ( ( 1 ) (1 ) )

2 ( 1 ) ( 1 ) 2 ( ( 1 ) (1 ) ) ,

D BG FG s FG FG s s

FT FG s BT s FG s

= - + - + - - + - +

+ - + - + - - - + + +

2π/ω Λ= ; /e h

s µ µ= ; (6)

2 ( )e

n d eB

ω +

= ; 2e

n dF

ω

= ; 2 ( )e

n h dG

ω -

= ; 2 ( )e

n h d eT

ω - +

= . (7)

Using data in Refs. [6, 7] and (1)

0 3| |/2r b= , we have verified that the minimization of W self for a large

ratio h/e (i. e. 1000) gives the same distances d equ than those obtained in [6, 7] for a single dislocation

placed in a semi-infinite crystal A, with a Burgers vector b(1)//Ox3 [6] and b(1)//Ox2 [7]. Besides this re-

sult, Fig. 2 exhibits the third and new case for which b(1)//Ox1. The corresponding curve denoted b1 in the

figure can also be derived from formulae given in [5], see Appendix 1.

The second energy term W int is calculated similarly from the integration of the work done to introduce

dislocation b(2) along the plane x2 = h – 2d, in the stress field [σ2k] generated by dislocation b(1). Since the

elastic energy density of dislocation b(1) is assigned to a region nearby the axis Ox3, Wint can be assimi-

lated as the work performed over half a (very large) period during the displacement b(2). If X is the ab-

scissa of dislocation b(2), the integral is

/ 2

int (2)

2 1 1( , 2 ) d

X

k k

X

W x h d b x

Λ

σ

+

= -Ú . (7)

Expression (7) was tested positively from the comparison of the equilibriums of screw and edge disloca-

tion dipoles placed (i), around the middle of a crystal A with a thickness h = 250e and (ii), in an infinite

crystal A. As for W self, the analytical expression of W int is too long to be written explicitly for the edge

components (k = 1 and 2). For the screw dislocation dipole (k = 3):

2int (1) ( 2 ) 2 ( 2d)

3

1

2d2 e [ 1 cos ( π)] cos ( e ) ,

tan ( )

h n h d n h

n

hW b n n

ω ωµ ω ξ ψα

•

- - -

=

-È Ê ˆ ˘= - + ◊ -È ˘Î ˚ Á ˜Í ˙Ë ¯Î ˚ (8)

in which

(( 1 ) ( ) ( ) (1 )) ( ( 1 ) (1 ))/ ,G F B F s B F F G s T s Dξ = - + + + - + - + - + (9)

( ( 1 ) ( ) (1 ) ( )) ( (1 ) (1 ))/ .F B F s F B F G s T s Dψ = - - + + + + - + + + - (10)

Fig. 2 Change of the equilibrium distance d equ of a

single dislocation nearby the interface separating a

layer that covers a semi-infinite crystal A. This dis-

tance changes with the modulus ratio and the disloca-

tion character. Poisson ratios are ν h= ν

e = 1/4.

(5)

phys. stat. sol. (b) 244, No. 6 (2007) 1911

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Original

Paper

To examine if any dipole instability can arise from the minimization of W(d, h) when h decreases step

by step, data relative to the phases Al2O3 and Al were adopted, with Ox2//[001] and Ox3//[110].

The first treated dipole is formed by two mixed dislocations ±1/2[011] lying on a {111} plane with

α = 54.74°. The modulus ratio s is taken as 6, an average value between those found in Refs. [2, 6], and

νe = ν h = 1/4. Besides, e = 4 nm, Λ = 1000e, r0 = |b(1)|/2. The convergence of the Fourier series was veri-

fied for a number of harmonic terms up to 2500. The calculations were conducted with significant digits

as large as 128 to ensure a sufficient precision on the products of exponential terms with large positive

and negative values. As depicted in Fig. 3, an hyperbolic curve is obtained. For large h, d equ/e ≈ 1.02,

while for h/e close to 20 the distance d equ diverges. Below this critical threshold value, the attraction of

the two opposite dislocations becomes larger than the attractions due to free surface relaxation.

The second dipole, formed by two screw dislocations ±1/2[110], presents about the same critical

threshold value. However, for large h, the ratio d equ/e is much larger since it tends to 1.82. This latter

value is that predicted by Head [6] for a semi-infinite substrate within 3%. This small discrepancy arises

probably from the fact that in [6, 7], no term r0 is taken into consideration.

4 Conclusion

This result shows that a dislocation dipole can be unstable for Al foil thicknesses around 70 nm. This

property can explain why parallel slip traces are not easily detected in ultra-thin foils. Beyond this re-

mark, the calculation method proposed in [4, 5] in addition to the present formulae (1, 2, 7) proves to be

a convenient alternative method to face the difficult problem of the energetics of dislocations in multi-

layers [8–10].

Appendix

In Ref. [5], some expressions can be rewritten more simply for a layer on a semi-infinite substrate:

(i) Expression (27) can be simplified to give:

( ) ( )

2 2

1 2

1

0 0 0

2( ( 1 2 ) ) ( 2( 1 ) )

π

[ (π ) cos ( ) sin ( )] ,

h

h h

e

n

E b A C b B Dn

n r n r n r

µν ν

ω ω ω

•

=

-Ê ˆ È ˘= + - + - + - +Á ˜ Î ˚Ë ¯

¥ - +

 (A1)

in which A, B, C and D are defined from the terms R – and S – in Ref. [5]:

Im [ ] ; Re [ ] ; Im [ ] ; Re [ ] .A R B R C S D S- - - -

= = = = (A2)

Fig. 3 When the thickness h of the middle crystal A de-

creases, the equilibrium distance d equ of each mixed dislo-

cation increases slowly. However, when h/e approaches

20, this distance diverges rapidly, giving an elastic instabil-

ity to the mixed dipole.

1912 S. Youssef et al.: Elastic instability of slip traces in oxidized thin foils

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

(ii) Expression (28) is explicit:

( )2

30 0 02

n 1

2 2 2 ( )

2 2 ( )

[ (π ) cos ( ) sin ( )]2π

1 (1 ) (e e ) (1 ) e .

(1 ) e (1 ) e

h

s

n e n d n e d

n d n e d

bE n r n r n r

n

s s s

s s

ω ω ω

ω ω

µω ω ω

ωΛ

•

=

+

+

Ê ˆ-= - +Á ˜Ë ¯

+ + - - - +¥

- + +

 (A3)

In addition, in expression (18), ∆ should read s.

References

[1] A. Howie and M. J. Whelan, Proc. R. Soc. A 267, 206 (1962).

[2] K. Scheerschmidt and U. Richter, Proc. 10th Hungarian Diffraction Conf., Balatonaliga, 1980, p. A33.

[3] K. Suzuki, N. Maeda, and S. Takeuchi, Philos. Mag. A 73, 431 (1996).

[4] R. Bonnet, Phys. Rev. B 53, 10978 (1996).

[5] R. Bonnet, phys. stat. sol. (a) 177, 219 (2000).

[6] A. K. Head, Philos. Mag. 44, 92 (1953).

[7] R. Weeks, J. Dundurs, and M. Stippes, Int. J. Eng. Sci. 6, 365 (1968).

[8] S. W. Kamat, and J. P. Hirth, Scr. Metall. 21, 1587 (1987).

[9] S. T. Choi and Y. Y. Earmme, Int. J. Solids Struct. 39, 1199 (2002).

[10] H. Y. Wang, M. S. Wu, and H. Fan, Int. J. Solids Struct. 39, 1199 (2006).