Sami Youssef

Roland Bonnet
email: rbonnet@ltpcm.inpg.fr

A dissociated dislocation in an ultra thin silicon plate

Keywords: thin-film silicon, dislocation theory, nanomechanics thin plate, silicon, free surfaces, dislocation, elasticity .manuscriptcentral

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A dissociated dislocation in an ultra thin silicon plate

Introduction

The use of thinner and thinner plates to investigate the fine structures of materials by high resolution transmission electron microscopy (HRTEM) puts new questions relative to free surface effects due to the confinement of dislocations in the plate. Here, the authors report on a description of the elastic displacement field of a dislocation of any Burgers vector b with the aid of a twodimensional theory leading to fully explicit expressions. In the past, the stress field of a dislocation of direction U running parallel to the boundary was found by three different authors : Leibfried and Dietze [1] for the screw component, then Kroupa [2] for the edge component parallel to the plate surfaces, and finally Lee and Dundurs [3] for the edge component parallel to the normal N to the surfaces (see Appendix 1). One of the present authors also solved this problem from two other different approaches, the interest of which is to provide the full elastic field (stresses and displacement u). One is based on a Fourier series solution [4] and applies to a periodic family of misfit dislocations at the interface of a heterophase bicrystal and the other is expressed from the solution of a system of two integral equations [5]. However, the numerical convergence of all these solutions is not as rapid as one would wish for applications. The Fourier series is slowly converging close to the dislocation line (Gibbs phenomenon), while the solutions in [2,3,5] involve integration quadratures and, for the dummy variable of integration, inaccessible limits at zero and infinite.

In the first part of the present work, it is shown that, for an elastically isotropic plate, the expressions of u derived in [4] can be greatly simplified so as to obtain short and explicit formulae and therefore a much better numerical efficiency. In addition to providing these simplified expressions, the present study investigates the change of the separation distance S between two partials of a dissociated dislocation in silicon (parameter a= 0.54282 nm [START_REF] Cullity | Elements of X-ray diffraction[END_REF]). Since the first observations of the separation of perfect dislocations [START_REF] Cockayne | [END_REF]8], dislocations in silicon have been investigated in numerous works by weak-beam microscopy and HRTEM, e. g., [9][10][11][12][13][14][15][16][17][18]. In particular, the intrinsic mobility and the possible atomic structures of the (near) screw dislocations with Burgers vector 1/2<110> are topics of intense activity, see, e. g., [19][20][21][22]. When some thermomechanical conditions are fulfilled, the screw dislocation appear to dissociate into two 30° Shockley partials moving on the so-called "glide" or "shuffle" set [START_REF] Hirth | Theory of dislocations[END_REF]. In particular, Cai et al. [20] Corresponding author. Email : rbonnet@ltpcm.inpg.fr have shown that its mobility depends strongly on the spacing S separating the partials, which stresses the importance to measure precisely this parameter. Most measurements of S were performed from weak-beam TEM observations on screw dislocations running in a direction <110> roughly parallel to the average plane of the thin plate. They often lie either on a {111} gliding plane [9][10][11][12] or a {001} low-angle twist boundary [17,18,[START_REF] Belov | [END_REF]25]. Plan view images taken under HRTEM conditions were also used [16] in the hope to visualize possible kinks along the partials, the plate thickness being extremely small, about 4 nm. On the other hand, previous measurements of the separation distance S of the dissociated screw dislocation reveals a large spread in the values since it lies in the range 2.3 nm [START_REF] Belov | [END_REF] to 5.8 nm [11]. High internal stresses can doubtless contribute to the cause of this large spread of S, among which free surface effects very difficult to handle on a rigorous basis, despite some qualitative or quantitative observations of elastic relaxation due to these surfaces, e. g., [26][27][START_REF] Williams | Transmission electron microscopy. Imaging III[END_REF][START_REF] Korner | [END_REF][30]. It will be shown in §3 that S can depend drastically, in an ultra thin plate, on the position and orientation of the fault plane. §2. DISLOCATIONS IN AN ISOTROPIC THIN PLATE As outlined above, the theoretical approach starts from the knowledge of the displacement field u of a family of undissociated, infinitely long, misfit dislocations [4], periodically distributed along the interface of a thin bicrystal with the period Λ, figure 1. The boundary conditions in stresses and displacements along the interface are those taken in [31], while at the two free surfaces the external applied forces are zero.

For the sake of clarity, symbols and conventions are now presented. Figure 1 represents the Cartesian frame Ox 1 x 2 x 3 attached to one of these dislocations for which Ox 3 // U. Its Burgers vector is b with components (b 1 ,b 2 ,b 3 ). The normal to the plate is accordingly N // Ox 2 . The interface separates two crystals denoted + and -according to their positions with respect to x 2 > 0 or x 2 < 0. Their thicknesses are respectively h + and h -, while that of the bicrystal is consequently h

= h + + h - .
If Λ tends to infinity the misfit between the two crystals tends to zero and the elastic field around the dislocation lying along Ox 3 tends asymptotically towards that of a translation dislocation [START_REF] Hirth | Theory of dislocations[END_REF][START_REF] Volterra | Sur l'équilibre des corps élastiques multiplement connexes[END_REF]. Since the shear modulus µ and the Poisson ratio ν of the two crystals are identical, closed form formulae can be obtained from [4]. Its u field can be written, for each crystal, as

1 x ω n i 0 n (n) k k e U u ∑ ∞ ≠ ∞ - = (1)
expression in which ω= 2π/Λ and

U 1 (n ) = ( P + Q n ω x 2 )e -n ω x 2 + ( R + S n ω x 2 )e n ω x 2 (2) U 2 (n ) = i ( P + Q( 3 -4ν ) + Q n ω x 2 ) [ ] e -n ω x 2 -i ( R -S( 3 -4ν ) + S n ω x 2 ) [ ] e n ω x 2
(3)

U 3 (n ) = 2 2 x x V T ω ω - + n n e e
(4) The simplified expressions of the complex constants P, Q, R, S, T, V written in [4] have been derived after corrections of some minor typographical errors (see Appendix 1). For the general term of order n they are the following Corresponding author. Email : rbonnet@ltpcm.inpg.fr

P + = 2 i b 1 (-1+ν ) + b 2 (1 -2ν ) 8 π n (-1+ν ) + P -, (6)
Q + = i b 1 -b 2 8 π n (-1+ν ) + Q -, (7)
R + = 2 i b 1 (1 -ν )+ b 2 (1 -2ν ) 8 π n (1 -ν ) + ( P -( 3 -4ν + 2 n ω h -) + 2 Q -( 2 -6ν + 4 ν 2 -n 2 ω 2 (h -) 2 ) ) e 2 h -n ω (8) S + = i b 1 + b 2 8 π n (1 -ν ) + ( 2P -+ Q -( 3 -4ν -2 n ω h -) ) e 2 h -n ω , (9)
R -= R + - 2 i b 1 (1 -ν )+ b 2 (1 -2ν ) 8 π n (1 -ν ) , (10)
S -= S + - i b 1 + b 2 8 π n (1-ν ) , ( 11
) ∆               ω ω ω ω ω ω ω ω = ω ω ω ω ω ω - / 2 e e 2 e 2 e 2 e 2 e Q 2 2 2 2 2 )))) n h + (-1 n h 2 + ) n h 2 + (-1 + (1 + ) n h + (1 + (-1 b i - ))) n h + (1 n h 2 + (1 + ) n h - (1 + ) n h + (1 - (-1 b + - n h n h + n h 1 + n h + n h - )n h + 2(h 2 + - + + + , 12) ∆                       ω + - ω ω + ω ω ν + - ω + ω ω ν + ω ν ω ω ν ω + - ω ω - ω ω ν + + ω ν ω ω ω ν ν ν = ω ω ω ω ω ω - / ) 2 3 ( 2 )) 2 ( 2 ( 2 1 ( e ) ( 4 ( 1 e 2 e ) 2 3 ( )) 2 ( 2 ( 1 ( e )) e 2 e P 2 2 2 2 2 2 ))) n h h h h n + (-1 n h n h + 1 n h + 1 + ) n h 1 n h 2 + ) n h - 2 (- + )) n h 2 + (1 2 - ) n h + (-1 n h - (1 + 2 - (1 b + ))) n h h h -h n + (2 n h n h - 1 n h + -1 n h 2 + (1 - ) n h + ((1 + ) ) n h - (1 - n h - ( + + (-1 b i 2 - + + - + n h + + + n h - - - n ) h + (h 2 - + + - + n h + 2 + n h 2 - - n ) h + (h 1 + + + + (13) ∆ = 8 π n (-1+ ν ) (-1 -e 4 h n ω + e 2 h n ω (2 + 4 h 2 n 2 ω 2 )), ( 14
) ) ))/(8 n Coth(h + (-1 ) + (-1 b n h n h 3 - + n e e i T 2 2 π ω = ω ω + , ( 15
) ) ))/(8 n Coth(h + (-1 ) + (-1 b n h 3 - n e i V 2 π ω = ω + , ( 16
) ) ))/(8 n Coth(h + (-1 ) + (-1 b n h 3 + n e i T 2 π ω - = ω - , ( 17
) ) ))/(8 n Coth(h + (-1 ) + (-1 b n h n h 3 + - n e e i V 2 2 π ω - = ω ω - .
(18) Application of the Hooke law and extraction of real part then permits derivation of the stress tensor σ ij from Eq. (1). To improve the slow numerical convergence of the Fourier series nearby the dislocation lines, due to the Gibbs phenomenon of discontinuous series, the Fourier harmonics of large order can be reduced to their principal terms, i. e. those for which n > N, N being an integer sufficiently large. The analytic forms of these principal terms are sufficiently simple for them to be summed exactly when n goes to infinity, thanks to analytical well-known functions [START_REF] Wolfram | Mathematica, software[END_REF] Corresponding author. Email : rbonnet@ltpcm.inpg.fr S is therefore changing with α because of the elastic relaxation effect. A small virtual displacement dS of dislocation 2 along the fault plane will work in the stress field σ ij (1) generated by dislocation 1 and produce the following small energy change per unit length (with the usual summation convention)

)dS b n σ (γ dW (2) i j (1) ij - = . ( 19
)
The equilibrium is reached when the second member of this equation is zero.

Since the question here is to treat the equilibrium of an isolated screw dislocation, the stress field σ ij (1) must be that of misfit dislocations extremely distant from each other to approach asymptotically that of a translation dislocation when the period Λ tends to infinity.

3.2 Application to the dissociated screw dislocation in silicon. Data. In this section, Eq. ( 19) is applied to determine the equilibrium of the dissociated screw dislocation in silicon. The fault plane is assumed to be of intrinsic type. Partials 1 and 2 have the Burgers b

(1) = 1/ 6[1 1 2] and b

(2) = 1/6[1 2 1].
For our application to ultra thin plates, the geometries are the following : Λ = 100 h with h = 4 to 20 nm. For the summation of the Fourier series, the value N= 350 was retained because larger N values do not produce any sensitive effect on the graphs of the stress field around a dislocation line. With these values, it has been verified that graphs representing various stress fields around Ox 3 are not discernible (by eye) from other graphs derived from application of expressions given in ref. [1][2][3] (see Appendix 1).

The isotropic elastic constants µ and ν taken for silicon were evaluated from the anisotropic constants C ij given in [START_REF] Hirth | Theory of dislocations[END_REF], en GPa, C 11 =165.7, C 12 = 63.9, C 44 = 79.6, and a procedure proposed in [START_REF] Fribourg-Blanc | [END_REF] to average the elastic behaviour of an anisotropic crystal submitted to virtual tensile tests in all spatial directions. The results are µ = 66.11 GPa and ν = 0.229. In passing, the elastic anisotropy of silicon is small since its anisotropy factor is 2 C 44 /(C 11 -C 12 ) = 1.56.

The adopted value γ of the intrinsic fault energy needs a brief comment. For the infinite silicon crystal, a mean separation of 4.0 nm was measured in [12,13], values very close to those given in [8,20], i. e., respectively 4.1 nm and 10 a/ 2 = 3.84 nm. Using elastic anisotropy and a separation distance of 4 nm, the solution of a relation similar to (19) gives γ= 51.39 mJ/m2, the value adopted in the present work. With this fault energy, the application of relation (19) to an infinite isotropic medium requires a separation distance S ∞ = 4.28 nm, slightly different from the mean separation of 4.0 nm. Note in passing that, from the isotropic elastic constants given in [START_REF] Hirth | Theory of dislocations[END_REF], a larger value 4.46 nm is obtained.

On the other hand, to avoid numerical difficulties due to products of exponential functions with very large and opposite exponents, the calculation programme is built to include digits of precision equal to 32 in the numbers [START_REF] Wolfram | Mathematica, software[END_REF]. With our desk computer, and for the smaller thicknesses h investigated, a consuming time of a few minutes is enough to draw a curve exhibiting the locus of partial 2 for all orientations of the fault plane.

To obtain a better insight on the independent effect of each surface on the locus of partial 2, calculations have been performed with h + much larger than h -. In this case, the elastic field tends to that of a straight dislocation located parallel and close to the bottom free surface of a semi-infinite substrate, see [START_REF] Hirth | Theory of dislocations[END_REF] for the screw component and [35,36] for the two edge components (see also Appendix 1). The quasi-identity of the graphs corresponding to a dislocation located in a bottomlimited semi-infinite crystal and in a thin plate with h + = 25 nm and h -= 5 nm were verified numerically. The converse situation of a top-limited semi-infinite substrate (h -tends to infinity) and a plate with h + = 5 nm and h -= 25 nm was also verified. Corresponding author. Email : rbonnet@ltpcm.inpg.fr §4. RESULTS AND DISCUSSION Free surface relaxation effects of the dissociated [0 1 1] screw dislocation are investigated for five isotropic silicon plates (thicknesses h=4 to 20 nm) and illustrated in figures 2(a-e). The plain curve of each graph describes the change of the separation distance S when partial 1 is fixed in the median plane of the plate and partial 2 is turning around partial 1. In figure 2(d), n is the fault normal and the distance 1-2 in dotted lines represents the equilibrium distance S between partials 1 and 2 for a given polar angle α. The plain curves are consequently the locus of partial 2 when α is changing. The case of an infinite silicon crystal is quite different since this locus is simply the circle in dotted lines (radius S ∞ = 4.28 nm).

Since isotropy is assumed, the locus of partial 2 exhibits symmetry with respect to the axes Ox 1 and Ox 2 . The equilibrium calculation can therefore be limited inside the interval 0 < α < 90°. Three types of configurations are possible:

-If h < 15 nm (approximately), the locus of partial 2 in the plate describes a vertical barrelshape profile. Therefore, partial 2 cannot find any equilibrium in the plate if the fault plane is too largely inclined with respect to Ox 1 . In figures 2(a-c), h= 4, 8 and 12 nm, stability is only possible for | α | < 45°, 55° and 63° respectively. In figure 2(a), h = 4 nm and α = 0, the separation distance falls down to 3.08 nm. The contraction of S is 27.9% with respect to S ∞ . For h= 3 nm and α = 0, i. e., a situation very close to that taken in the experimental work by HRTEM of Suzuki et al. [16], S= 2.6 nm (39% contraction).

-If 15 nm < h < 17.8 nm, e. g. figure 2(d), the locus of partial 2 looks like a symmetric egg-cup with two symmetrical necks nearby the free surfaces. For h approaching 17.8 nm, the necks become narrower and narrower until they transform in points.

-If h > 17.8 nm, e. g. figure 2(e), the locus of partial 2 is complex since three disconnected curves are possible according to the value of α. In this figure, three possible dissociations 1-2 are represented. A closed curve surrounds the central circle in dotted line (infinite case), while two circular segments are placed near the two free surfaces. Therefore, a fault plane oriented with an angle | α | close to 90° give rise to two possible equilibriums.

-For larger and larger h, the central curve tends gradually to the circle in dotted line while the circular-shape segments nearby the free surfaces have lengths tending to 0.

For an easier comparison of the independent effect of the two free surfaces on the separation S, the case of a bottom-limited semi-infinite crystal is also considered in figures 3(a-e). The distance h -between partial 1 and the bottom free surface takes the same values as for the plate in figures 2(ae). The geometrical forms of the loci of partial 2 for x 2 < 0, in figures 2(a-e) and 3(a-e), look at first sight similar. A unique plain curve describes the locus of partial 2 for h -< 11.45 nm, against h -< 8.8 nm for a plate. These critical values of h -give an idea of the intensity of the relaxation phenomena, much stronger for the plate.

In view of the variety of the equilibrium curves depicted in figures 2(a-e) and 3(a-e), a further investigation was undertaken to determine if any unstable mechanical balance can be present. To simplify, calculations were only performed for a semi-infinite crystal and a thickness range including a necking phenomenon in the locus of partial 2. Besides, despite the small anisotropy ratio of silicon, anisotropic elasticity was also applied to calculate the locus of partial 2. The corresponding elastic field is that already tested in [START_REF] Bonnet | Fundamentals of Diffusion Bonding[END_REF], see also Appendix 1. Of course, for these latter calculations, the elastic constants relative to Ox 1 x 2 x 3 change with α. From Eq. ( 19), numerical maps of the second derivatives d 2 W / dS 2 were calculated for each case, using:

-isotropy and h In these figures, the thicker bottom curves are obtained for d 2 W / dS 2 = 0. They separate regions where stability ( d 2 W / dS 2 > 0) or instability ( d 2 W / dS 2 < 0, shaded regions) of the mechanical equilibrium occurs for partial 2.

In isotropy, partial 2 is stable in figure 4(a) for any n, except those orientations for which the polar radius 1-2 cuts the bottom free surface. An extreme separation distance equal to about 2.5 S ∞ for α= -70° is therefore predicted for a partial 2 very close to the free surface, a case for which the elastic field of partial 2 would vanish. For figures 4(b,c), the equilibriums corresponding to partial 2 along the plain curves and inside the shaded regions are unstable. In figure 4(c), two possible dissociation planes 1-2 are depicted in dotted lines, one is stable (shorter distance S) and the other is unstable (larger distance S).

For the infinite anisotropic silicon, the circles in dotted lines in figures 4(d-f) correspond to S ∞ = 4.0 nm, the measured values in [12,13].

For the semi-infinite anisotropic silicon, the graphs in figures 4(d-f) are roughly similar to figures 4(a-c). For each graph, a slight asymmetric behaviour of the dissociation is visible with respect to the vertical plane x 1 = 0, also visible in the limiting curves of the shaded regions associated to instability. In figures 4(d,e), on the left hand side, very small branches of the plain curves are outside the shaded region and could give rise to a stable partial 2 very close to the free surface. In figure 4(f), stability is always obtained with a partial 2 on the closed loop of plain curve, while instability is predicted for a partial 2 located on the plain curve inside the shaded region. Anisotropy influences slightly the zero necking phenomenon for the semi-infinite silicon (h -= 10.6 nm in anisotropy against 11.45 nm in isotropy). §5. CONCLUSION The displacement field u of a straight translation dislocation lying parallel to the two free surfaces of an ultra thin isotropic plate can be found as the limit of a solution valid for a periodic family of misfit dislocations. Its interest is to provide an explicit and simple solution for u, contrary to the way followed in previous works where integration quadratures with a dummy variable tending to zero or infinity are required. It is potentially able to treat a number of partial dislocations greater than two, coplanar or not, with any Burgers vector.

An illustration of this solution to a dissociated [011] screw dislocation in silicon shows that the distance S separating the two 30° Shockley partials depends largely on the geometry of the dissociation in the plate. For the calculation, one partial (partial 1) is fixed in the middle of the plate and the other (partial 2) is positioned around partial 1 by the orientation of the fault plane and mechanical equilibrium.

For h < 5 nm, which is of use in high resolution TEM for silicon [16], and a fault plane almost parallel to the median plane of the plate, the distance S is much lower than in an infinite medium, by about 20 or 30%. Such a situation can be met in high resolution TEM, see e. g. [16]. If the fault plane is inclined by angles greater than about 45° with respect to the median plane, partial 2 can escape from the plate by glide because of strong image forces associated to free surfaces.

For 5 nm < h < 17.6 nm, the locus of partial 2 resembles successively a barrel-shape profile (h < 15 nm) and an egg-cup (15 nm < h < 18 nm).

For h > 17.6 nm, the locus is described by three disconnected curves, the central one being very close to a circle describing the locus of partials 2 for an infinite crystal.

It has been shown that instability of the mechanical equilibrium may occur for h > 17.6 nm and inclined fault planes with a Shockley partial located nearby one free surface.

The anisotropic elasticity of silicon, only taken into account for a dissociated dislocation nearby a free surface of silicon, is proved to be of small impact on the mechanical instability of the dissociation, although some asymmetry properties arise in the separation distance between the two partials. It is noteworthy that such thickness scales and inclined {111} planes are commonly encountered in TEM investigations in silicon, the interface normal being close to <001> or <111>, see [8][9][10][11][12][13][14][15][16][17][18]. It is also expected that the relaxation phenomena described in the present work should be amplified for materials with much lower γ fault energies than that of silicon. Dissociated dislocations with edge character should also interact more strongly with the free surfaces of the plate.

Finally, it has been shown in the present work that the influence of foil thickness and dislocation depth in a thin foil can contributes severely to a systematic uncertainty in the measurement by TEM of the separation distance between two partials. This fact, already stressed years ago in [START_REF] Cockayne | [END_REF], remains to clarify for highly anisotropic plates, a work in progress.

  . §3. DEPENDENCE OF THE DISLOCATION SPACING ON THE FAULT PLANE ORIENTATION 3.1 The mechanical equilibrium Let us suppose that two partial dislocations 1 and 2, running parallel to the two free surfaces of the plate, are linked by a fault plane with the fault energy per unit surface γ. Their separation distance S can be determined from mechanical equilibrium. Dislocation 1 has the Burgers vector b (1), is supposed fixed along Ox 3 in figure1, and is oriented by this axis. When the crystallographic direction of the normal N to the plate changes, the normal n to the fault plane turns around Ox 3 with the polar angle α= (N,n), and drags dislocation 2 with Burgers vector b

- = 10 ,

 10 11 and 11.5 nm (very close to zero necking), figures 4(a-c), -anisotropy and h -= 10, 10.5 (very close to zero necking), and 11.5 nm, figures (d-f).

6 -

 6 Corresponding author. Email : rbonnet@ltpcm.inpg.fr

  author. Email : rbonnet@ltpcm.inpg.fr

  author. Email : rbonnet@ltpcm.inpg.fr

Fig. 1

 1 Fig. 1

  author. Email : rbonnet@ltpcm.inpg.fr

Fig. 2

 2 Fig. 2 Dissociation of a [011] screw dislocation located in an isotropic silicon plate. Different possible loci (plain curves) of partial 2 when the ) 1 1 1 ( fault plane (normal n) takes different orientations around the fixed partial 1. The normal is N and the thickness is (h + + h -) = 4 to 20 nm from (a) to (e)). The locus of partial 2 for an unlimited silicon crystal is a circle (in dotted line).

1 -

 1 Corresponding author. Email : rbonnet@ltpcm.inpg.fr

Fig. 3 2 -

 32 Fig. 3Possible dissociation schemes of a [011] screw dislocation located in a bottom-limited semi-infinite silicon crystal (isotropic elasticity). The separation distance S between the two Shockley partials 1 and 2 depends on the fault plane orientation n and on the position of partial 1 (h -= 2 to 10 nm). The successive distances h -are the same in graphs (a-e) of figures 2 and 3.

Fig. 4

 4 Fig. 4

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments

The authors would like to thank the Centre National de la Recherche Scientifique (Paris) and the Direction Générale à la Recherche Scientifique et Technique (Tunis) for constant financial support.

Appendix 1

Some cited papers contain misprints. They are listen as follows : Réf. [3] : in expression (10a) the term (κ 2 -1) should be 2(κ 2 -1). In expression (12c) replace the term

) by 2(q 2 -t 2 ). Réf. [4] : in expressions ( 5) and ( 7) replace (1+k) by (-1+k). In the second line of expression (8) replace the factor +k by -k. In expression ( 14) replace +2h nω by -2h nω. In expression (16) replace the factor k by -k. In the fourth line of expression (20) replace +(1+2hnω) by x(1+2 h + nω) . Réf. [31] : in expressions (15) the signs before the two monomials in B 2 shoud be minus. In expression (16) the second monomial in B 3 should be the same monomial but in B 2 . In expression (20a), the sign before the second monomial in B 2 should be negative. In the system of linear equations ( 26), line 12, the second monomial in B 3 should be the same monomial but in B 1 . In expression (32c) and function F 6 , the first term of the denominator should be written with a positive exponent (+2π g |x 2 |). Réf. [35]: in p. 797, the numerator of the third fraction giving τ xx should be written without any curly brackets. Réf. [START_REF] Bonnet | Fundamentals of Diffusion Bonding[END_REF]: in p. 333, equation ( 9), the denominator of the last fraction should include a bar above the variable + β Z (to form the complex conjugate of + β Z ).


Karsten ; Tillmann
email: k.tillmann@fz-juelich.de

Lothar Houben

Andreas Thust

† Ernst

ATOMIC-RESOLUTION IMAGING OF LATTICE IM-PERFECTIONS IN SEMICONDUCTORS BY CON-JOINED ABERRATION-CORRECTED HRTEM AND EXIT-PLANE WAVEFUNCTION RETRIEVAL

Keywords: GaAs, GaN, HRTEM, dislocation structures, high-resolution imaging, lattice defects, semiconductors, transmission electron microscopy user supplied): spherical aberration correction, exit-plane wavefunction reconstruction, dislocation core structure High-resolution transmission electron microscopy, spherical aberration correction, exit-plane wavefunction reconstruction, lattice defects, disloca-

Atomic-resolution imaging of lattice imperfections in

semiconductors by conjoined aberration-corrected HRTEM and exit-plane wavefunction retrieval

INTRODUCTION

Over the past fifty years, transmission electron microscopy has become the technique of choice whenever structural properties associated with lattice imperfections in solid state materials have been investigated experimentally. Even though enormous progress has been made in the characterization of dislocations, stacking faults and internal boundaries in these common semiconductor materials, there is still much to learn about the precise atomic configuration in the vicinity of these structural imperfections. Above and beyond purely scientific curiosity this task is also of technological interest as further improvements in applied semiconductor research stand in need of an amended understanding of the interaction between structural and electronic properties with advancing miniaturization [i, ii]. To put it another way, the detailed evaluation of electronic features induced by lattice imperfections and the elucidation of the atomic structure of these imperfections are interdependent [iii, iv].

Recently of particular interest with high-resolution transmission electron microscopy is (i) the pursuit of sub-Ångström resolution at medium acceleration voltages [v, vi, vii] in close communion with (ii) the straightforward interpretability of micrographs [viii, ix, x] characterized by (iii) a rather low image delocalization in the vicinity of lattice imperfections [xi, xii]. Two sophisticated techniques to accomplish all three of these objectives simultaneously, not within reach during operation of ''traditional'' medium voltage instruments equipped with field emission gun emitters, have attracted much interest in recent years. On the one hand, double hexapole corrector elements enabling a compensation of detrimental lens aberrations are practically usable these days [xiii, xiv]. As a result, the aberration coefficient C S emerges as an additional parameter for extending the point resolution at phase contrast imaging conditions to the information limit together with the simultaneous maximization of phase contrast and the minimization of contrast delocalization [xv]. Moreover, when employing a negative C S value combined with a certain overfocus setting Z opt > 0, not only a substantial contrast improvement is achieved but low-nuclear charge elements can also be imaged at bright atom contrast features in the vicinity of highnuclear charge atoms [viii, ix].

On the other hand, numerical techniques enable the restoration of the phase Φ(r) and the amplitude Α(r) of the exit-plane wavefunction from a focal series [xvi, xvii, xviii] or an illumination tilt direction series [xix, xx] of experimental micrographs. By this means, all spatial information up to (focal series) or even beyond (tilt direction series) the axial information limit of the instrument can be retrieved, also allowing for a subsequent elimination of residual lens aberrations still present in aberration corrected microscopy [xxi, xxii, xxiii, xxiv].

The objective of the present contribution is to demonstrate the benefits accruing from the deliberated combination of both of these state-of-the-art techniques for the atomic scale imaging of structural imperfections in semiconductor materials. Special emphasize will be put on conceptual vantages for the characterization of lattice defects at atomic resolution and for

METHODICAL BACKGROUND: NEGATIVE SPHERICAL-ABERRATION CORRECTED IMAGING IN TANDEM WITH THE NUMERICAL RETRIEVAL OF THE EXIT-PLANE WAVEFUNCTION

In aberration-corrected microscopy a high amount of negative phase contrast may be obtained simultaneously with a widely minimized image delocalization, when setting the constant of spherical aberration to an optimized value [ix, xv] C S = -64 / 27 λ -3

g max -4 (1)
and choosing an overfocus setting of the objective lens

Z opt = 16 / 9 λ -1 g max -2 (2)
with λ and 1/ g max denoting the electron wavelength and information limit of the instrument, respectively. A corresponding set-up of the instrument yields direct interpretable high-resolution micrographs accompanied by a residual image delocalization of

R = 16 / 27 g max -1 . ( 3
)
Using an aberration corrected CM-200 FEG ST instrument, the aforementioned parameters amount to Z opt = 11.6 nm, C S = --40.6 µm and R = 80 pm when putting to use an electron wavelength of λ = 2.51 pm and an information limit of 1/g max = 125 pm with the latter value measured from a Young's fringe analysis of images taken with the microscope utilized in this study [xv]. The associated partially coherent phase contrast transfer function, displayed in Figure 1, extends up to 1/g max and is characterized by a broad pass-band with a phase deviation close to •/2. Simplifying the imaging process to linear theory, a weak phase object is then imaged under bright atom contrast conditions [xxv] and a substantial contrast improvement is achieved since the linear phase contrast and the nonlinear dark field signal add rather than subtract as is usual when applying ''traditional'' high-resolution imaging modes [xxvi]. In the strict sense Eqns. ( 1) to ( 3) base upon a first order expansion of the aberration function, thus neglecting any detrimental impact of higher order lens aberrations. As the hexapole corrector of the CM-200 FEG ST allows for a sufficient correction of aberrations up to the third order [xxvii] and higher order aberrations will become relevant not until 1/g max becomes smaller than 100 pm [xxviii, xxix], these aberrations do not need to be considered for the instrument under consideration.

The numerical retrieval of the exit-plane wavefunction Ψ(r) from a focal series of micrographs provides auxiliary improvements as it is basically free from nonlinear imaging artifacts and the impact of the low frequency gap in information transfer is considerably reduced by the combination of many images taken at different foci. As improvements of the instrumental resolution limit still gather pace, this aspect is of special importance when imaging common semiconductors at atomic resolution since the structural information is predominantly carried by the low-frequency {111} reflections. Furthermore, by extracting information from about N = 20 ... 30 images, the signal-to-noise ratio at high spatial frequencies can be substantially increased. Even the application of small C S values, which is a prerequisite to obtain phase contrast, induces a parasitic delocalization whereas Ψ(r) is ideally free from any delocalization induced artifacts.

Moreover, the availability of Ψ(r) allows for the numerical a posterior measurement of residual lens aberrations. This aspect is of special practical importance as experience shows that not all aberrations of the microscope are sufficiently constant over the period of operation or cannot be determined before the experiment with sufficient accuracy. In detail, the measurement of even aberrations, i. e. the defocus Z and the twofold astigmatism A 1 , may be carried out by processing the weak signal originating from amorphous overlayers [xxi] while odd aberrations, i. e. the axial coma B 2 and the threefold astigmatism A 2 , may be determined by sampling a variety of discrete aberration azimuths and magnitudes up to twice their arguable limits according to [xxvii] followed by handpicking those aberrations at which the symmetry properties of the corresponding Φ(r)

and Α(r) images correlate best with the supposed lattice structure [xxii]. Irrespective of this procedure even more sophisticated approaches are available to tackle the problem of measurement of odd aberrations [xxx].

Finally, since Ψ(r) is complex-valued, we may evaluate ''local'' diffraction patterns from specimen areas as small as desired. When evaluated during operation of the microscope, the judgement of the symmetry properties of these ''local'' diffraction patterns is a most Moreover, image delocalization in a non-corrected instrument will generally not be less than [xi]

R L = C S / 4 λ 3 g max 3 (4)
and only to be within reach at Lichte's defocus of least confusion

Z L = -3 / 4 C S λ 2 g max 2 (5

EXPERIMENTAL SETUP

This study utilizes a Philips CM-200 FEG instrument equipped with a computer-controlled electromagnetic hexapole system for the correction of the spherical aberration of the objective lens [xxvii]. An optical coarse adjustment of the hexapole corrector elements was performed by analysing Zemlin tableaus taken from amorphous specimen areas with an incident electron beam tilted up to 20 mrad from the optical axis [xxxii] thus yielding an initial estimate on the quality of higherorder aberration corrections. Setting down an optimized negative C S = --40 µm the magnitudes of residual aberration coefficients were limited to A 1 ≤ 2.0 nm for the two-fold astigmatism, to A 2 ≤ 170 nm for the three-fold astigmatism and to B 2 ≤ 55 nm for the axial coma [xxvii].

Focal series of about N = 20 ... 30 images were recorded using a 1k x 1k CCD camera system at a sampling rate of about 20 pm per picture element. This corresponds to an image discretization well below the Nyquist frequency with respect to half of the instrumen- applying a doughnut-shaped restoration filter.

ATOMIC RESOLUTION IMAGING AT

/GAAS HETEROINTERFACES

As a first materials science example of use, we report on the core structure of Lomer type pure edge dislocations [xxxiii] at the heterointerfaces between latticemismatched sphalerite semiconductors. In view of the multiplicity of conceivable lattice positions of the pulled out half-plane's terminal point, numerous core models have been proposed for this type dislocations [xxxiv, xxxv] but owing to the hitherto insufficient instrumental resolution previous high-resolution analyses required elaborate support by numerical image simulations merely to draw indirect conclusions on the corresponding dislocation core structures [xxxvi, xxxvii, xxxviii].

In the present analysis we focus on an In 0.3 Ga 0.7 As epilayer with a thickness of 35 nm that was grown by molecular beam epitaxy on a GaAs(001) substrate and subsequently overgrown with a GaAs capping layer of 55 nm in thickness whereby growth conditions were chosen as specified in a previous analysis [xxxix]. reconstruction process. By contrast, the phase image Φ(r) is free of any distortions after the above specified aberrations A i and B i have been eliminated [xxxi].

Above and beyond this, the phase image is also richer in contrast compared to the Z opt micrograph since there are practically no limitations with respect to the temporal and spatial coherence, thus providing a superior signal-to-noise ratio at high frequencies up to the information limit. Additionally, the retrieved phase benefits from the averaging of information from a complete series of micrographs. When quantifying the signal-to-noise-ratios σ Φ and σ Ι of the Φ(r) and Ι(r) im- ages, respectively, by applying analysis procedures outlined in [xl] an improvement through the restoration of the exit-plane wavefunction by a factor of (σ Φ /σ Ι ) exp = 4.1 is obtained. This value is in excellent agreement with the ideal factor of (σ Φ /σ Ι ) th = •N/2 = •30/2 = 3.9

to be expected for information retrieval from a focal series of N = 30 micrographs.

When also determining the polarity of the sample by the evaluation of a set of relevant reciprocal-space amplitudes from the ''local'' diffraction pattern and the subsequent trimming of simulated data, individual bright contrast dots in the Φ(r) image may be directly associated with gallium and arsenic columns [xxxi]. From the exit-plane wavefunction a core structure model according to Figure 3(a) can be deduced as contrast features are intuitionally interpretable. The dislocation core shows some similarity to the glide set type model of a Lomer dislocation originally proposed by Hornstra [xxxiv] which is displayed in Figure 3(b) and characterized by an eight-atomic inner ring. The present analysis, however, reveals a less compact core, which is build up of a ten-sided polygon also traversing the detached atomic column with a coordination different from the surrounding matrix. This structure also deviates from rather complex mirror symmetric configurations observed at GaAs/Si interfaces [xxxvi, xli] and in Ge/Si heterosystems [xxxviii]. As surface effects in the vicinity of the highly strained dislocation core are likely to occur in thin electron microscopy specimens, the retrieved core structure may certainly deviate from that of a dislocation core embedded in bulk material. This problem, however, exists in any highresolution analysis concerned with lattice defects raising rather strong lattice distortions. Referring to a detailed discussion of the matter [xxxi] we briefly summarize the causal connections impeding the resolvability of the dislocation core in the Z opt image but evidently not in the retrieved exit-plane wavefunction. Numerical image calculations give clear evidence that this non-resolvability in the single Ι(r) image will neither be due to (i) the residual image delocalization or (ii) a decreased sample thickness in the vicinity of the dislocation core. Moreover, a potential impact of (iii) a locally inhomogeneous strain relaxation of the thin specimen may be ruled out since, otherwise, the Α(r) and Φ(r) images would similarly be affected. Instead, the practically improved resolution of the amplitude and phase images will be rather due to (iv) an increased resistibility against time-dependant electron beam induced degeneration of the sample. While numerical retrieval techniques purge information from intrinsically unstable amorphous overlayers by averaging, a single Z opt micrograph simply represents merely a snap-shot of a sample under illumination with a certain probability of an adverse atomic overlayer assembly during the time of exposure [xxxi].

GEOMETRIC LINEARITY AT HIGH-LEVEL PRECESSION: LATTICE DISTORTIONS ASSOCIATED WITH DISSOCIATED 60° DISLOCATIONS IN GAAS (110)

In sphalerite materials 60° dislocations with a Burgers vectors of type b = a/2 < 011> may dissociate into two partials bounding either an intrinsic or an extrinsic stacking fault [xlii, xliii, xliv, xlv] [xlvi, xlvii, xlviii], it is only recently that efforts have been put forth to investigate the core structure of 30° and 90° partials, both theoretically applying ab initio calculations [xlix, l, li] and experimentally employing advanced electron microscopy techniques [lii, liii].

In the following we shall focus on a dissociated 60° dislocation bounding a double stacking fault ribbon. overall dislocation core, which is in perfect agreement with a 90° partial dislocation composed of two 30° partials separated by two atomic layers [xlv].

To put it a step further, we may also quantify local distortions of the atomic dumbbells across the faulted ribbon at some distance of the dislocation core. For these purposes contrast dot positions have been measured from the 5.5 nm x 4.2 nm phase image Φ(r) area displayed in Figure 5(a), whose centre is 5.2 nm left of the dislocation core. For theses purposes a digital image processing algorithm [lx] has been applied which basically employs (i) dedicated image intensity thresholding operations, (ii) a centre-of-mass analysis inside individual bright contrast dots supported by (iii) adequate refining procedures as specified in [lxi, lxii]. From the measured contrast dot positions, the der investigation [liv]. Hence, a more likely explanation could be that the terminating partial dislocation biases the rearrangement of atomic columns in the vicinity of the faulted ribbons and that next but one neighbour interactions between different atomic species will indeed play an important role during this process.

ATOMIC STRUCTURE OF CHROMIUM IMPLANTATION INDUCED LAT-TICE DEFECTS IN GAN (1120)

As a finale example, we report on dislocations introduced by chromium implantation into GaN. Implantation of transition metals into semiconductors is a candidate process for the fabrication of diluted magnetic semiconductors for spintronic applications [lxiv]. Based on these observations a tentative 5/7 ring configuration for the core is presented in Figure 7(c). The Z opt image level may therefore contain complementary but not necessarily identical information when compared to Φ(r). Indeed for the present case the Z opt image with its inferior noise level compared to Φ(r) also complies an alternative structure which contains two wrong Ga-Ga bonds connecting threefold coordinated Ga atoms at positions 1-5 and 2-3, respectively.

CONCLUSIONS

In summary, a combination of negative spherical aberra-

INTRODUCTION

Over the past fifty years, transmission electron microscopy has become the technique of choice whenever structural properties associated with lattice imperfections in solid state materials have been investigated experimentally. Even though enormous progress has been made in the characterization of dislocations, stacking faults and internal boundaries in these common semiconductor materials, there is still much to learn about the precise atomic configuration in the vicinity of these structural imperfections. Above and beyond purely scientific curiosity this task is also of technological interest as further improvements in applied semiconductor research stand in need of an amended understanding of the interaction between structural and electronic properties with advancing miniaturization [START_REF] Alexander | Teichler in Handbook of Semiconductor Technology[END_REF][START_REF] Jones | [END_REF]. To put it another way, the detailed evaluation of electronic features induced by lattice imperfections and the elucidation of the atomic structure of these imperfections are interdependent [3,4].

Recently of particular interest with high-resolution transmission electron microscopy is (i) the pursuit of sub-Ångström resolution at medium acceleration voltages [5,6,7] in close communion with (ii) the straightforward interpretability of micrographs [8,9,10] characterized by (iii) a rather low image delocalization in the vicinity of lattice imperfections [11,12]. Two sophisticated techniques to accomplish all three of these objectives simultaneously, not within reach during operation of "traditional" medium voltage instruments equipped with field emission gun emitters, have attracted much interest in recent years.

On the one hand, double hexapole corrector elements enabling a compensation of detrimental lens aberrations are practically usable these days [13,14]. As a result, the aberration coefficient C S emerges as an additional parameter for extending the point resolution at phase contrast imaging conditions to the information limit together with the simultaneous maximization of phase contrast and the minimization of contrast delocalization [15]. Moreover, when employing a negative C S value combined with a certain overfocus setting Z opt > 0, not only a substantial contrast improvement is achieved but low-nuclear charge elements can also be imaged at bright atom contrast features in the vicinity of high-nuclear charge atoms [8,9]. On the other hand, numerical techniques enable the restoration of the phase Φ(r)

and the amplitude Α(r) of the exit-plane wavefunction from a focal series [16,17,18] or an illumination tilt direction series [19,20] of experimental micrographs. By this means, all spatial information up to (focal series) or even beyond (tilt direction series) the axial information limit of the instrument can be retrieved, also allowing for a subsequent elimination of residual lens aberrations still present in aberration corrected microscopy [21,[START_REF] Thust | Proceedings ICEM-15[END_REF][START_REF] Meyer | [END_REF]24].

The objective of the present contribution is to demonstrate the benefits accruing from the deliberated combination of both of these state-of-the-art techniques for the atomic scale imaging of structural imperfections in semiconductor materials. Special emphasize will be put on conceptual vantages for the characterization of lattice defects at atomic resolution and for the quantification of atomic column positions in the vicinity of lattice imperfections in semiconductor materials.

METHODICAL BACKGROUND: NEGATIVE SPHERICAL-ABERRATION COR-RECTED IMAGING IN TANDEM WITH THE NUMERICAL RETRIEVAL OF THE EXIT-PLANE WAVEFUNCTION

In aberration-corrected microscopy a high amount of negative phase contrast may be obtained simultaneously with a widely minimized image delocalization, when setting the constant of spherical aberration to an optimized value [9,15] !

C S = " 64 / 27 # "3 g max "4 (1)
and choosing an overfocus setting of the objective lens

! Z opt = 16 / 9 " #1 g max #2
(2)

with λ and 1/ g max denoting the electron wavelength and information limit of the instrument, respectively. A corresponding set-up of the instrument yields direct in- Using an aberration corrected CM-200 FEG ST instrument, the aforementioned parameters amount to Z opt = 11.6 nm, C S = -40.6 µm and R = 80 pm when putting to use an electron wavelength of λ = 2.51 pm and an information limit of 1/g max = 125 pm with the latter value measured from a Young's fringe analysis of images taken with the microscope utilized in this study [15].

The associated partially coherent phase contrast transfer function, displayed in Figure 1, extends up to 1/g max and is characterized by a broad pass-band with a phase deviation close to π/2. Simplifying the imaging process to linear theory, a weak phase object is then imaged under bright atom contrast conditions [START_REF] Williams | Transmission Electron Microscopy[END_REF] and a substantial contrast improvement is achieved since the linear phase contrast and the nonlinear dark field signal add rather than subtract as is usual when applying "traditional" high-resolution imaging modes [START_REF] Lentzen | [END_REF]. In the strict sense Eqns. ( 1) to ( 3) base upon a first order expansion of the aberration function, thus neglecting any detrimental impact of higher order lens aberrations. As the hexapole corrector of the CM-200 FEG ST allows for a sufficient correction of aberrations up to the third order [27] and higher order aberrations will become relevant not until 1/g max becomes smaller than 100 pm [28,29], these aberrations do not need to be considered for the instrument under consideration.

The numerical retrieval of the exit-plane wavefunction Ψ(r) from a focal series of micrographs provides auxiliary improvements as it is basically free from nonlinear imaging artifacts and the impact of the low frequency gap in information transfer is considerably reduced by the combination of many images taken at different foci. As improvements of the instrumental resolution limit still gather pace, this aspect is of special importance when imaging common semiconductors at atomic resolution since the structural information is predominantly carried by the low-frequency {111} reflections. Furthermore, by extracting information from about N = 20 ... 30 images, the signal-to-noise ratio at high spatial frequencies can be substantially increased.

Even the application of small C S values, which is a prerequisite to obtain phase contrast, induces a parasitic delocalization whereas Ψ(r) is ideally free from any delocalization induced artifacts. Moreover, the availability of Ψ(r) allows for the numerical a posterior measurement of residual lens aberrations. This aspect is of special practical importance as experience

shows that not all aberrations of the microscope are sufficiently constant over the period of operation or cannot be determined before the experiment with sufficient accuracy. In detail, the measurement of even aberrations, i. e. the defocus Z and the twofold astigmatism A 1 , may be carried out by processing the weak signal originating from amorphous overlayers [21] while odd aberrations, i. e. the axial coma B 2 and the threefold astigmatism A 2 , may be determined by sampling a variety of discrete aberration azimuths and magnitudes up to twice their arguable limits according to [27] followed by handpicking those aberrations at which the symmetry properties of the corresponding Φ(r) and Α(r) images correlate best with the supposed lattice structure [START_REF] Thust | Proceedings ICEM-15[END_REF]. Irrespective of this procedure even more sophisticated approaches are available to tackle the problem of measurement of odd aberrations [30].

Finally, since Ψ(r) is complex-valued, we may evaluate "local" diffraction patterns from specimen areas as small as desired. When evaluated during operation of the microscope, the judgement of the symmetry properties of these "local" diffraction patterns is a most convenient tool for the proper orientation of specimen areas under investigation. By this means, a proper zone axis alignment of the samples with an accuracy below 3 mrad is ensured while the vast majority of samples aligned by "traditional" procedures demonstrates off-zone-axis orientations above 10 mrads [31].

To set aright the common fallacy that the aforementioned benefits coming along with the restoration of the exit-plane wavefunction do not necessitate the use of spherical aberration corrected instrumentation but rather suggest an increase of the instrumental resolution solely, e. g. by monochromatisation of the electron source, it is emphasized that the a posterior measurement of residual lens aberrations requires a sufficient pre-reduction of the very same aberrations. Moreover, image delocalization in a non-corrected instrument will generally not be less than [11] !

R L = C S / 4 " 3 g max 3 (4)
and only to be within reach at Lichte's defocus of least confusion which, alas, comes along with the great disadvantage that the phase contrast transfer function exhibits a large number of rapid contrast oscillations at medium spatial frequencies, and the corresponding contrast reversals make the resulting image quite difficult to interpret. Moreover, image delocalization according to Eqn. ( 4) does not only have a detrimental effect on the spread out of structural information in the vicinity of lattice imperfections but also to data acquisition at marginal areas of the recording medium. In other words, the rather large image delocalization of uncorrected instruments represents a serious limitation for the usable field of view of the charge-coupled device (CCD) camera detector.

! Z L = " 3 / 4 C S # 2 g max 2 (5)

EXPERIMENTAL SETUP

This study utilizes a Philips CM-200 FEG instrument equipped with a computercontrolled electromagnetic hexapole system for the correction of the spherical aberration of the objective lens [27]. An optical coarse adjustment of the hexapole corrector elements was performed by analysing Zemlin tableaus taken from amorphous specimen areas with an incident electron beam tilted up to 20 mrad from the optical axis [32] thus yielding an initial estimate on the quality of higher-order aberration corrections. Setting down an optimized negative C S = -40 µm the magnitudes of residual aberration coefficients were limited to A 1 ≤ 2.0 nm for the two-fold astigmatism, to A 2 ≤ 170 nm for the three-fold astigmatism and to B 2 ≤ 55 nm for the axial coma [27].

Focal series of about N = 20 ... 30 images were recorded using a 1k x 1k CCD camera system at a sampling rate of about 20 pm per picture element. This corresponds to an image discretization well below the Nyquist frequency with respect to half of the instrumental information limit 1/(2 g max ) = 65 pm and represents an adequate sampling to tap the full potential of the instrument facing that the modulation transfer function of many CCD cameras is known to exhibit a rather poor transfer at higher spatial frequencies. The focal range of each series included the focus setting with Z opt = 11.6 nm for optimized phase contrast. From these series the exit-plane wavefunction was retrieved for the frequency band between 1 nm -1 and 8 nm -1 applying a doughnut-shaped restoration filter.

EROINTERFACES

As a first materials science example of use, we report on the core structure of Lomer type pure edge dislocations [33] at the heterointerfaces between lattice-mismatched sphalerite semiconductors. In view of the multiplicity of conceivable lattice positions of the pulled out half-plane's terminal point, numerous core models have been proposed for this type dislocations [34,35] but owing to the hitherto insufficient instrumental resolution previous high-resolution analyses required elaborate support by numerical image simulations merely to draw indirect conclusions on the corresponding dislocation core structures [36,37,38].

In the present analysis we focus on an In 0.3 Ga 0.7 As epilayer with a thickness of 35 nm that was grown by molecular beam epitaxy on a GaAs(001) substrate and subsequently overgrown with a GaAs capping layer of 55 nm in thickness whereby growth conditions were chosen as specified in a previous analysis [39]. However, two conspicuous contrast artifacts become evident from close inspection of the Z opt micrograph of the GaAs matrix displayed in Figure 2(e). Firstly, nonstructure related spurious intensity peaks are observed in between the atomic column positions which are due to an expended transfer of low spatial frequencies at severe phase offsets, especially of the strongly excited {111} beam amplitudes [START_REF] Lentzen | [END_REF].

Secondly, a kidney-shaped distortion of the dumbbells is observed. As can be shown from quantitative measurements on the associated "local" diffraction pattern dis- rentheses indicating the respective azimuth angles inclined with the [001] direction [31]. For that reason, the phase Φ unc (r) of exit-plane wavefunction retrieved from a focal series of N = 30 micrographs and displayed in Figure 2(e) still shows distortions of the contrast dumbbells as long as these residual aberrations are not considered during the initial step of the reconstruction process. By contrast, the phase image Φ(r) is free of any distortions after the above specified aberrations A i and B i have been eliminated [31].

Above and beyond this, the phase image is also richer in contrast compared to the Z opt micrograph since there are practically no limitations with respect to the temporal and spatial coherence, thus providing a superior signal-to-noise ratio at high frequencies up to the information limit. Additionally, the retrieved phase benefits from the averaging of information from a complete series of micrographs. When quantifying the signal-to-noise-ratios σ Φ and σ Ι of the Φ(r) and Ι(r) images, respectively, by applying analysis procedures outlined in [40] an improvement through the restoration of the exit-plane wavefunction by a factor of (σ Φ /σ Ι ) exp = 4.1 is obtained. This value is in excellent agreement with the ideal factor of (σ Φ /σ Ι ) th = √N/2 = √30/2 = 3.9 to be expected for information retrieval from a focal series of N = 30 micrographs.

When also determining the polarity of the sample by the evaluation of a set of relevant reciprocal-space amplitudes from the "local" diffraction pattern and the subsequent trimming of simulated data, individual bright contrast dots in the Φ(r) image may be directly associated with gallium and arsenic columns [31]. Corresponding atomic species are indicated in Figure 2(e).

The essential benefit of the combined use of the numerical retrieval of the exit-plane wavefunction with aberration-corrected imaging, however, becomes evident when having a closer look at the atomic structure of the dislocation core, which is basically not resolved in the single optimum focus image Ι(r) but only in the amplitude and phase images, respectively. Both, Α(r) and Φ(r), show the atomic arrangement down to the dislocation core and reveal a detached atomic column at the point of intersec- From the exit-plane wavefunction a core structure model according to Figure 3(a) can be deduced as contrast features are intuitionally interpretable. The dislocation core shows some similarity to the glide set type model of a Lomer dislocation originally proposed by Hornstra [34] which is displayed in Figure 3(b) and characterized by an eight-atomic inner ring. The present analysis, however, reveals a less compact core, which is build up of a ten-sided polygon also traversing the detached atomic column with a coordination different from the surrounding matrix. This structure also deviates from rather complex mirror symmetric configurations observed at GaAs/Si interfaces [36,41] and in Ge/Si heterosystems [38]. As surface effects in the vicinity of the highly strained dislocation core are likely to occur in thin electron microscopy specimens, the retrieved core structure may certainly deviate from that of a dislocation core embedded in bulk material. This problem, however, exists in any high-resolution analysis concerned with lattice defects raising rather strong lattice distortions.

Referring to a detailed discussion of the matter [31] we briefly summarize the causal connections impeding the resolvability of the dislocation core in the Z opt image but evidently not in the retrieved exit-plane wavefunction. Numerical image calculations give clear evidence that this non-resolvability in the single Ι(r) image will neither be due to (i) the residual image delocalization or (ii) a decreased sample thickness in the vicinity of the dislocation core. Moreover, a potential impact of (iii) a locally inhomogeneous strain relaxation of the thin specimen may be ruled out since, otherwise, the Α(r) and Φ(r) images would similarly be affected. Instead, the practically improved resolution of the amplitude and phase images will be rather due to (iv) an increased resistibility against time-dependant electron beam induced degeneration of the sample. While numerical retrieval techniques purge information from intrinsically unstable amorphous overlayers by averaging, a single Z opt micrograph simply represents merely a snap-shot of a sample under illumination with a certain probability of an adverse atomic overlayer assembly during the time of exposure [31]. may dissociate into two partials bounding either an intrinsic or an extrinsic stacking fault [42,43,44,[START_REF] Amelinckx | Dislocations in Solids[END_REF]. The dissociation reaction is:

a/2 ! [011] -> a/6 ! [121] + a/6 ! [112] (6)
with the Burgers vectors of both partials inclining angles of 30° and 90° with the dislocation line along the ! [11 0] direction , respectively. While most analyses at the out- set of quantitative high-resolution microscopy were primarily concerned with distinguishing glide and shuffle set dislocations, i. e. discriminating whether dislocations cores lie on the narrowly or widely spaced {111} planes, by interpreting characteristic contrast features in the vicinity of the dislocations cores [START_REF] Olsen | [END_REF]47,48], it is only recently that efforts have been put forth to investigate the core structure of 30° and 90° partials, both theoretically applying ab initio calculations [49,50,51] and experimentally employing advanced electron microscopy techniques [52,53].

In the following we shall focus on a dissociated 60° dislocation bounding a double stacking fault ribbon. Both of these limiting partials are in fact again superpositions of 30° and 90° partials separated by two atomic layers [START_REF] Amelinckx | Dislocations in Solids[END_REF] with the extrinsicality expressing in two adjacent ribbons of {111} lattice planes with differently aligned atomic [START_REF] Hirth | Theory of Dislocations[END_REF].

To check the validity of this model hypothesis, the elastic displacement component u ! 112 along the ! [112] direction has been evaluated from the Φ(r) image using geomet- rical phase analysis algorithms [START_REF] Hÿtch | [END_REF]. Calculations have been performed applying Gaussian shaped masks with decay parameters σ = |g 111 |/3 centred around the g 111 beams thus averaging information over regions of about r dir ≈ 3 / (2 |g 111 |) ≈ 490 pm in radius at most [56]. The correspondingly measured u ! 1 1 1 ) extracted from this distribution along the ! [111 ] direction, cf. Figure 4(e). For the latter, data values have been aver- aged over a width of 1 nm, i. e. over the width of the drawn-in dashed blue frame.

The line profile shows two conspicuous plateaus separated by an abrupt discontinuity of ∆u ! 11 2 = 234 pm ± 60 pm. This quantity is in excellent agreement with the closing vector associated with the 90° partial of an extrinsic stacking fault for which a total shift of lattice planes of Δu ! 11 2 ,30° = 2 b proj,30°-A/B = 231 pm is expected, cf. above. We emphasize that the specified margin of error in the ∆u ! 11 2 value rather reflects inhomogeneous lattice distortions of the sample but not the fundamental accuracy to quantify atomic column positions which may be measured to a precision of about a few picometres [57,58].

In Figure 4 111) planes. As these planes are narrowly spaced, the partial dislocations are of glide set type. Both dislocation cores are made up of gallium columns leading to a so-called β dislocation [59]. Moreover, the figure also reveals a symmetrical six-fold structure of the overall dislocation core, which is in perfect agreement with a 90° partial dislocation composed of two 30° partials separated by two atomic layers [START_REF] Amelinckx | Dislocations in Solids[END_REF].

To put it a step further, we may also quantify local distortions of the atomic dumbbells across the faulted ribbon at some distance of the dislocation core. For these purposes contrast dot positions have been measured from the 5.5 nm x 4.2 nm phase image Φ(r) area displayed in Figure 5(a), whose centre is 5.2 nm left of the dislocation core. For theses purposes a digital image processing algorithm [60] has been applied which basically employs (i) dedicated image intensity thresholding operations, (ii) a centre-of-mass analysis inside individual bright contrast dots supported by (iii) adequate refining procedures as specified in [61,62]. From the measured contrast dot positions, the length and orientation of individual dumbbells have been In order to gauge whether the observed antisymmetric distortions represent a genuine structural property or not, potential implications of scattering and imaging artefacts need to be ruled out. Since a hypothetic global misalignment of the sample cannot give rise to any local torsion and dilatation of projected atomic dumbbells we may disenfranchise from this explanation. An argument of the same kind holds true of the potential impact of not fully compensated aberrations as they would take effect on the entire image and, hence, would distort all atomic dumbbells of the same orientation equally. Strictly speaking any potential impact of both, on-axial and offaxial aberrations would need be evaluated separately. In practise this conceptional differentiation can, however, be neglected because of the rather limited field of view of the CCD detector and the semiamplanatic imaging properties of the double hexapole corrector [13]. Beyond electron-optical reasoning, a supposed strictly antisymmetric lattice distortion caused by a dedicated combination of higher order aberrations with the faulted ribbon incidentally acting as the symmetry plane, may be ruled out from a probability point of view because of an almost immense number of possible combinations which will not give raise to the observed distortion behaviour.

The observed alteration of atomic dumbbell lengths in the vicinity of stacking faults ribbons of about 10 pm is in fair agreement with recent measurements of lattice distortions along intrinsic stacking faults ribbons in heavily beryllium doped GaAs. This analysis revealed an average expansion of dumbbell lengths up to 158 pm in the faulted lattice plane which was explained by the segregation of beryllium dopant atoms and the subsequent formation of antisite defects inside of the plane of the faulted ribbon [53]. Additionally, lattice displacements around the central stacking fault of Z-shaped dipoles connecting two stair rod dislocations in indium doped GaAs revealed the very same antisymmetric distortion of dumbbell related contrast dots as were observed in the present analysis. Lateral displacements of gallium and arsenic atom pair related contrast features along the ! [112] direction were found to be as much as 20 % to 50 % of the dumbbell length [63].

As the specimen investigated in the present study was undoped we may, in the first instance, rule out the aforementioned explanations but only speculate about an indium diffusion from the underlying In 0.3 Ga 0.7 As layer to the GaAs layer along the faulted ribbon at most. As long as an inhomogeneous incorporation of indium at- oms is not considered this approach, thus, cannot explain the observed lattice distortions. Admittedly, the antisymmetric distortions become perspicuous when considering the elastic distortions associated with the 90° partial dislocations which basically follow the measured characteristics of the dumbbell's measured expansion and torsion but, alas, yield only alterations of dumbbells lengths smaller than ± 3 pm and misorientation angles smaller than ± 0.4° for the image area under investigation [START_REF] Hirth | Theory of Dislocations[END_REF].

Hence, a more likely explanation could be that the terminating partial dislocation biases the rearrangement of atomic columns in the vicinity of the faulted ribbons and that next but one neighbour interactions between different atomic species will indeed play an important role during this process.

ATOMIC STRUCTURE OF CHROMIUM IMPLANTATION INDUCED LATTICE DEFECTS IN GAN (1120)

As a finale example, we report on dislocations introduced by chromium implantation into GaN. Implantation of transition metals into semiconductors is a candidate process for the fabrication of diluted magnetic semiconductors for spintronic applications [64]. A by-product of the implantation is the creation of ion-induced lattice defects: Extrinsic and intrinsic basal plane stacking faults, bounded by Frank partial or Shockley-Frank partial dislocations, are formed as the result of the precipitation of excess interstitials [START_REF] Hirth | Theory of Dislocations[END_REF]65].

The GaN films presented in this study were grown by metal organic chemical vapour deposition on (0001) oriented sapphire substrates under Si doping to a concentration of 2 × 10 17 cm -3 . Cr+ ions were implanted with an energy of 200 kV and a dose of 5 × 10 16 cm -2 at a temperature of 350 °C to avoid amorphization in the asimplanted samples. A high density of basal plane stacking faults is observed besides spherical Cr-rich precipitates in surface near regions after chromium implantation and rapid thermal annealing at 700 °C for 5 min in N 2 atmosphere [66]. Based on these observations a tentative 5/7 ring configuration for the core is presented in Figure 7(c). The nitrogen atoms shared by the 5-and 7-membered rings are threefold coordinated with Ga and N. Due to the wrong bond and the presence of unpaired electrons, this structure is not expected to be the most stable configuration.

Since Φ(r) is retrieved from a focal series taken over a period of a few ten seconds, the image conceivably reflects a transient state of the core that gathers further interstitials during observation. The Z opt image level may therefore contain complementary but not necessarily identical information when compared to Φ(r). Indeed for the present case the Z opt image with its inferior noise level compared to Φ(r) also complies an alternative structure which contains two wrong Ga-Ga bonds connecting threefold coordinated Ga atoms at positions 1-5 and 2-3, respectively. It has been demonstrated that recent improvements in the resolution power of transmission electron microscopes enable the imaging of finest structure details at directly interpretable contrast features also coming along with a widely minimized image delocalization when micrographs are taken under optimized imaging conditions.

Additionally, the numerical retrieval of the exit-plane wavefunction allows for the elimination of artificial contrast features still visible in micrographs taken under optimized focusing conditions. Beyond it's genuine purpose, the retrieval of the exitplane wavefunction was demonstrated to be a most suitable tool for the measurement and numerical correction of residual lens aberrations as well as for proper orientation of specimens during operation of the electron microscope.

Above and beyond the individual vantages coming along with either technique their combined use allows for the accurate identification of lattice imperfections in semiconductor materials at hitherto unequalled imaging. Experimental analyses conspicuously benefit from putting to use aberration corrected input data, i.e. from an increase of the instrumental information limit together with a decrease of image delocalization and a pre-reduction of lens aberrations, as well as from the complete elimination of residual aberrations by the numerical retrieval of the exit-plane wavefunction, which is also free from non-linear imaging artefacts and yields images characterized by a decent signal-to-noise ratio.

  the proper orientation of specimen areas under investigation. By this means, a proper zone axis alignment of the samples with an accuracy below 3 mrad is ensured while the vast majority of samples aligned by ''traditional'' procedures demonstrates offzone-axis orientations above 10 mrads[xxxi].To set aright the common fallacy that the aforementioned benefits coming along with the restoration of the exit-plane wavefunction do not necessitate the use of spherical aberration corrected instrumentation but rather suggest an increase of the instrumental resolution solely, e. g. by monochromatisation of the electron source, it is emphasized that the a posterior measurement of residual lens aberrations requires a sufficient pre-reduction of the very same aberrations.

  ) which, alas, comes along with the great disadvantage that the phase contrast transfer function exhibits a large number of rapid contrast oscillations at medium spatial frequencies, and the corresponding contrast reversals make the resulting image quite difficult to interpret. Moreover, image delocalization according to Eqn. (4) does not only have a detrimental effect on the spread out of structural information in the vicinity of lattice imperfections but also to data acquisition at the recording medium. In other words, the rather large image delocalization of uncorrected instruments represents a serious limitation for the usable field of view of the charge-coupled device (CCD) camera detector.

tal information limit 1 /

 1 (2 g max ) = 65 pm and represents an adequate sampling to tap the full potential of the instrument facing that the modulation transfer function of many CCD cameras is known to exhibit a rather poor spatial frequencies. The focal range of each series included the focus setting with Z opt = 11.6 nm for optimized phase contrast. From these series the exit-plane wavefunction was retrieved for the fre-

Figure 2

 2 Figure 2(a) displays an optimum focus micrograph I(r) of a Lomer dislocation with a Burgers circuit yielding a projected closing vector b = a/2 [11 0] in the proximity of the interface between the In 0.3 Ga 0.7 As epilayer and the GaAs(001) substrate viewed with a [11 0] zone axis orientation of the sample. Far-off the dislocation core, the image contrast is dominated by clearly separated pairs of contrast dots constituting atomic dumbbells at distances of a/4 • 141 pm with a = 565 pm denoting the lattice parameter of the substrate material.

  Figure 2(e).

Figure 4 (Figure 4 Figure 4

 444 Figure 4(a) displays an optimum micrograph Ι(r) of the terminating zone of such an extended dislocation in GaAs viewed along the crystallographic [11 0] direction. The image has been taken from the same specimen investigated in the previous section focusing on the GaAs capping layer in this case. The associated phase Φ(r) and amplitude A(r) images displayed in Figure 4(b) and (c) allow for an accurate identification of the atomic arrangement of the stacking fault ribbons and, especially, the core structure of the dislocation terminating the defect.

  length and orientation of individual dumbbells have been calculated and normalized to mean values associated with the lower (planes no. 0 -3) and upper (planes no. 11-13) areas of the Φ(r) image. For the improvement of statistics 15 data points have been averaged along the [112] direction and median values are plotted in Figure 5(b) and (c) in dependence on specific (111) lattice plane positions.As can be seen from both plots, the dumbbells on either side of the double ribbon rearrange roughly antisymmetric in the vicinity of the faulted (111) planes. In this context, the most conspicuous feature is that the dumbbells of the bottommost lattice plane of the upper crystal domain (plane no. 8) are compressed to a length of ‹d› = 133 pm ± 4 pm and turn towards the double ribbon with a misorientation angle ‹δ› • 2.6° ± 1.3° com-reference lattice planes. In contrast, the uppermost dumbbells of the lower crystal area (plane no. 5) are stretched to a length of ‹d› = 150 pm ± 4 pm and bend away from the double ribbon at a misorientation angle of ‹δ› • --4.9° ± 1.5°.In order to gauge whether the observed antisymmetric distortions represent a genuine structural property or not, potential implications of scattering and imaging artefacts need to be ruled out. Since a hypothetic global misalignment of the sample cannot give rise to any local torsion and dilatation of projected atomic dumbbells we may disenfranchise from this explanation.An argument of the same kind holds true of the potential impact of not fully compensated aberrations as they would take effect on the entire image and, hence, would distort all atomic dumbbells of the same orientation equally. Strictly speaking any potential impact of both, on-axial and off-axial aberrations would need be evaluated separately. In practise this conceptional differentiation can, however, be neglected because of the rather limited field of view of the CCD detector and the semiamplanatic imaging properties of the double hexapole corrector[xiii]. Beyond electron-optical reasoning, a supposed strictly antisymmetric lattice distortion caused by a dedicated combination of higher order aberrations with the faulted ribbon incidentally acting as the symmetry plane, may be ruled out from a probability point of view because of an almost immense number of possible combinations which will not give raise to the observed distortion behaviour.The observed alteration of atomic dumbbell lengths in the vicinity of stacking faults ribbons of about 10 pm agreement with recent measurements of lattice distortions along intrinsic stacking faults ribbons in heavily beryllium doped GaAs. This analysis revealed an average expansion of dumbbell lengths up to 158 pm in the faulted lattice plane which was explained by the segregation of beryllium dopant atoms and the subsequent formation of antisite defects inside of the plane of the faulted ribbon[liii]. Additionally, lattice displacements around the central stacking fault of Z-shaped dipoles connecting two stair rod dislocations in indium doped GaAs revealed the very same antisymmetric distortion of dumbbell related contrast dots as were observed in the present analysis. Lateral displacements of gallium and arsenic atom pair related contrast features along the [112] direction were found to be as much as 20 % to 50 % of the dumbbell length[lxiii].As the specimen investigated in the present study was undoped we may, in the first instance, rule out the aforementioned explanations but only speculate about an indium diffusion from the underlying In 0.3 Ga 0.7 As layer to the GaAs layer along the faulted ribbon at most. As long as an inhomogeneous incorporation of indium atoms is not considered this approach, thus, cannot explain the observed lattice distortions. Admittedly, the antisymmetric distortions become perspicuous when considering the elastic distortions associated with the 90° partial dislocations which basically follow the measured characteristics of the dumbbell's measured expansion and torsion but, alas, yield only alterations of dumbbells lengths smaller than ± 3 pm and misorientation angles smaller than ± 0.4° for the image area un-

3 .

 3 A byproduct of the implantation is the creation of ioninduced lattice defects: Extrinsic and intrinsic basal plane stacking faults, bounded by Frank partial or Shockley-Frank partial dislocations, are formed as the result of the precipitation of excess interstitials [liv, lxv]. The GaN films presented in this study were grown by metal organic chemical vapour deposition on (0001) oriented sapphire substrates under Si doping to a concentration of 2 × 10 17 cm -Cr+ ions were implanted with an energy of 200 kV and a dose of 5 × 10 16 cm -2 at a temperature of 350 °C to avoid amorphization in the asimplanted samples. A high density of basal plane stacking faults is observed besides spherical Cr-rich precipitates in surface near regions after chromium implantation and rapid thermal annealing at 700 °C for 5 min in N 2 atmosphere [lxvi].

  the basal plane stacking faults is extrinsic bound by Frank partial dislocations with Burgers vector b = 1/2 [0001]. Figure 6 displays the amplitude image A(r) and phase image Φ(r) of a perfect nanometre sized interstitial loop in the end of range depth of implantation. Gallium and nitrogen columns at a dumbbell distance of 114 pm, considerably smaller than the information limit of the instrument, are not fully resolved. Nonetheless the N-polarity is unambiguously visible from the directly interpretable bright contrast in the phase image. The tetrahedral coordination across the faulted layer stacking is confirmed, asserting that no foreign chromium-gallium alloy phase is connected with the planar defects. A magnified view of the core of a Frank partial dislocation at the terminating zone of a several 10 nm large dislocation loop is shown in the Z opt image and the Φ(r) image retrieved from a corresponding focal series displayed in Figure 7(a) and (b). Although the Ι(r) and Φ(r) appear very similar at a first glance, the improved signal-to-noise ratio in the phase image enhances the visibility of the nitrogen positions and the dumbbell orientation up to the core of the dislocation. The phase image Φ(r) fortifies the presence of further nitrogen atoms within the core surrounded by the cage of the five marked gallium columns. The faint phase shift in Φ(r), in which non-linear image components and image delocalization are eliminated, indicates a nitrogen filled core in favour of a chromium or gallium rich core.

  shared by the 5-and 7-membered rings are threefold coordinated with Ga and N. Due to the wrong bond and the presence of unpaired electrons, this structure is not expected to be the most stable configuration. Since Φ(r) is retrieved from a focal series taken over a period of a few ten seconds, the image conceivably reflects a transient state of the core that gathers further interstitials during observation.

Figure 1 :

 1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: Analysis of a Lomer type dislocation with b = a/2 [11 0] at an In 0.3 Ga 0.7 As/GaAs heterointerface. a: Optimum focus micrograph Ι Ι Ι Ι(r) taken along the [11 0] zone axis orientation of the sample. b: ''Local'' diffraction pattern evaluated from the exit-plane wavefunction of the image area. c: Restored phase image Φ(r) and d: amplitude image Α(r). e: Magnified clipping of the GaAs related image area together with the associated phase images Φ unc (r) and Φ(r) obtained when neglecting and considering measured residual lens aberrations, respectively.

Figure 3 :

 3 Figure 3: Dislocation core models for symmetrical Lomer dislocations. a: Model retrieved from the exit-plane wavefunction displayed in Figure 2 together with b: the symmetrical glide set type core model according toHornstra[xxxiv].

Figure 4 :

 4 Figure 4: Analysis of an extrinsic stacking fault in GaAs bound by two adjacent 30° partial dislocations with b proj = [112] in total. a: High-resolution micrograph Ι(r) taken under Z opt conditions. b: Phase image Φ(r) and

Figure 5 :

 5 Figure 5: Locally inhomogeneous distortions of the atomic dumbbells in the vicinity of the faulted double ribbon. a: Clipping from the phase image displayed in Figure 4(b) with atomic column positions superimposed and dumbbell distortions indicated exemplarily in dependence on specific positions along the [111] direction. b: Average projected bond length ‹d› and c: misorientation angle ‹δ› of the dumbbells along the (111) direction. The lattice planes belonging to the double stacking fault ribbon are indicated in lighter grey colour.

Figure 6 :

 6 Figure 6: Interstitial loop in Cr-implanted GaN viewed along the [112 0] direction. a: Amplitude image Α(r) and b: phase image Φ(r) calculated from a focal series of micrographs. Simulated images of the periodic Npolarity GaN structure assuming a sample thickness of 3.2 nm at a resolution of 7.5 nm -1 are superimposed in the lower left part of the images. The stacking sequence is ABABCABAB, where the underlined part highlights sphalerite stacking.

Figure 7 : 24 ABSTRACT

 724 Figure 7: Frank partial dislocation with the Burgers vector b = 1/2 [0001] projected along the [112 0] direc-

Figure 2 ( 2 !

 22 Figure 2(a) displays an optimum focus micrograph I(r) of a Lomer dislocation with a Burgers circuit yielding a projected closing vector b = a/2

  2(b), these distortions are not due to any potential misorientation of the sample but instead result from parasitic lens aberrations amounting to A 1 = 2.2 nm (83°), B 2 = 110 nm (83°), and A 2 = 150 nm (43°) with the values in pa-

  line positions indicated by the horizontal and vertical arrows.Relevant contrast features show mirror symmetry with respect to the medial ( by the upright arrows. This global symmetry behaviour ranges down to the dislocation core.

Figure 4 ( 6 ! 6 ! 6 !

 4666 a) displays an optimum micrograph Ι(r) of the terminating zone of such an extended dislocation in GaAs viewed along the crystallographic ! [11 0] direction. The image has been taken from the same specimen investi- gated in the previous section focusing on the GaAs capping layer in this case. The associated phase Φ(r) and amplitude A(r) images displayed in Figure 4(b) and (c) allow for an accurate identification of the atomic arrangement of the stacking fault ribbons and, especially, the core structure of the dislocation terminating the defect. As the overall lattice defect is composed of two adjacent (111) lattice planes showing similar misorientations of the atomic dumbbells, it appears inceptively reasonable to describe the entire defect as a sequence of two adjoining likewise congenerous faulted fault ribbons. Both ribbons need then to be characterized by a projected component of the terminating Burgers vector amounting to b proj /2 with b proj denoting the closing vector of the entire defect when considering a projection to the ( a configuration build-up by 30° partials, characterized by b 30°-A = a/[121] as well as b 30°-B = a/[ 211] and yielding b proj,30°-A/B = a/12 ! [112], the as- sociated total lattice displacement along the ! [112] direction between the upper and the lower areas of the crystal far from the dislocation core should be Δu ! 11 2 ,30° = 2 b proj,30° = 231 pm. This description is fully compatible with the definition of an extrinsic stacking fault bound by a 90° partial with b 90° = b proj,90° = a/[112] = 231 pm and a 30° partial with b proj,30° = a/12 ! [112] = 115 pm at the opposite core position.

  (d) single atomic columns, as allocated from the determination of the polarity of the sample, are superimposed to a magnified clipping of the phase image.The global displacement of the upper and lower crystal areas against each other is found to be mediated by the local re-organization of the atomic columns within those two (111) lattice planes forming the faulted double ribbon. Hence, the associ-of both partials, indicated by red circles in the figure, also lie on these very same (

  calculated and normalized to mean values associated with the lower (planes no. 0 -3) and upper (planes no. 11-13) areas of the Φ(r) image. For the improvement of statistics 15 data points have been averaged along the ! [112] direction and median values are plotted in Figure 5(b) and (c) in dependence on specific (111) lattice plane positions.As can be seen from both plots, the dumbbells on either side of the double ribbon rearrange roughly antisymmetric in the vicinity of the faulted (111) planes. In this context, the most conspicuous feature is that the dumbbells of the bottommost lattice plane of the upper crystal domain (plane no. 8) are compressed to a length of ‹d› = 133 pm ± 4 pm and turn towards the double ribbon with a misorientation angle ‹δ› ≈ 2.6° ± 1.3° compared to the reference lattice planes. In contrast, the uppermost dumbbells of the lower crystal area (plane no. 5) are stretched to a length of ‹d› = 150 pm ± 4 pm and bend away from the double ribbon at a misorientation angle of ‹δ› ≈ -4.9° ± 1.5°.

  The majority of the basal plane stacking faults is extrinsic bound by Frank partial dislocations with Burgers vector b = 1/2[0001]. Figure6displays the amplitude image A(r) and phase image Φ(r) of a perfect nanometre sized interstitial loop in the depth of implantation. Gallium and nitrogen columns at a dumbbell distance of 114 pm, considerably smaller than the information limit of the instrument, are not fully resolved. Nonetheless the N-polarity is unambiguously visible from the directly interpretable bright contrast in the phase image. The tetrahedral coordination across the faulted layer stacking is confirmed, asserting that no foreign chromium-gallium alloy phase is connected with the planar defects.A magnified view of the core of a Frank partial dislocation at the terminating zone of a several 10 nm large dislocation loop is shown in the Z opt image and the Φ(r) image retrieved from a corresponding focal series displayed in Figure7(a) and (b). Although the Ι(r) and Φ(r) appear very similar at a first glance, the improved signal-to-noise ratio in the phase image enhances the visibility of the nitrogen positions and the dumbbell orientation up to the core of the dislocation. The phase image Φ(r) fortifies the presence of further nitrogen atoms within the core surrounded by the cage of the five marked gallium columns. The faint phase shift in Φ(r), in which non-linear image components and image delocalization are eliminated, indicates a nitrogen filled core in favour of a chromium or gallium rich core.

  In summary, a combination of negative spherical aberration imaging and the restoration of the exit-plane wavefunction from a focal series of micrographs has been applied for the investigation of lattice imperfections in semiconductor materials at atomic resolution and, in particular, for the extraction of quantitative information on local lattice distortions.

  By means of the deliberated combination of both techniques, Lomer dislocation cores at In 0.3 Ga 0.7 As/GaAs heterointerfaces have been imaged at atomic resolution and directly identified as being composed of a mirror symmetric ten-atomic ring, thus rendering extensive image simulations superfluous. The quantification of local lattice distortions in the vicinity of a double stacking fault ribbon in GaAs by means of the analysis of experimental phase images Φ(r) brought clear evidence of an anti-expansion of atomic dumbbells with respect to the faulted ribbon. A similar quantitative analysis focusing on lattice imperfections in GaN at atomic resolution based on the synergetic combination of negative spherical aberration corrected imaging and the numerical retrieval of the exit-plane wavefunction is presently merely hampered by the instrumental resolution but will come along with forthcoming aberration corrected sub-Ångström resolution microscopes.

Figure 1 :

 1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: Analysis of a Lomer type dislocation with b = a/2

Figure 3 :

 3 Figure 3: Dislocation core models for symmetrical Lomer dislocations. a: Model retrieved from the exit-plane wavefunction displayed in Figure 2 together with b: the symmetrical glide set type core model according to Hornstra [34].

Figure 4 :

 4 Figure 4: Analysis of an extrinsic stacking fault in GaAs bound by two adjacent 30° partial dislocations with b proj =

Figure 5 :

 5 Figure 5: Locally inhomogeneous distortions of the atomic dumbbells in the vicinity of the faulted double ribbon. a: Clipping from the phase image displayed in Figure 4(b) with atomic column positions superimposed and dumbbell distortions indicated exemplarily in dependence on specific positions along the [111] direction.

  Average projected bond length ‹d› and c: misorientation angle ‹δ› of the dumbbells along the (111) direction. The lattice planes belonging to the double stacking fault ribbon are indicated in lighter grey colour.

Figure 6 :

 6 Figure 6: Interstitial loop in Cr-implanted GaN viewed along the

Figure 7 :

 7 Figure 7: Frank partial dislocation with the Burgers vector b = 1/2 [0001] projected along the

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(r) distribution is displayed in Figure4(e) together with a line profile u

ACKNOWLEDGEMENTS

The authors are grateful to Arno Förster and Vitaly Guzenko for making available the samples investigated in this study as well as Doris Meertens for indefatigable specimen preparation work.

ACKNOWLEDGEMENTS

The authors are grateful to Arno Förster and Vitaly Guzenko for making available the samples investigated in this study as well as Doris Meertens for indefatigable specimen preparation work.