Calculation of displacement fields and ... - Wiley Online Library

Cryst. Res. Technol.

32

1997

1

111- 126

M. LAZAR,G. WAGNER University Leipzig, Faculty of Chemistry and Mineralogy, Institute of Mineralogy, Crystallography and Materials Science, Scharnhorststrasse 20, 04275 Leipzig, F.R.G.

Calculation of displacement fields and simulation of HRTEM images of dislocations in sphalerite type A(III)B(V) compound semiconductors Dedicated to Professor Dr. H e m a n n Neels on the occasion of the 3@ anniversary of his Editorship of C y s t a l Research and Technology

The displacement fields of different kinds of both perfect and dissociated dislocations have been calculated for an isotropic continuum, and by means of linear elasticity. Additionally, the corresponding HRTEM images have been simulated by the well-established EMS program package in order t o predetermine the structural aspects of dislocations, and then to compare it with experimental HRTEM micrographs. The latter ones resulted from plastically deformed GaP single crystals and InAs/(OOl)GaAs single epitaxial layers. It could be established that using the simple approach of linear elasticity and isotropy results can be obtained which correspond well to the experimental images. So, the structure of various Shockley partial dislocations bounding a stacking fault can be detected unambiguously. The splitting behaviour of perfect 30" dislocations (separation into a 0" and 60" partial) and 90" dislocations (separation into two 60" partials) both with line direction along (112), 60" dislocations (separation into 3Oo/9O6 and 90"/30" configuration) and screw dislocations (separation into two 30" partials) along (110) are discussed in the more detail. Moreover, the undissociated sessile Lomer dislocation, glissile 60" dislocation and edge dislocation have been considered too.

Introduction In covalent bonded crystals as Si, Ge, and in groupIII/V compound semiconductors such as GaAs, GaF', InP etc. where the content of ionic bond (fi = ionicity) is fi 5 0.5 the strong PEIERLS potential will cause the dislocations to lie along close packed directions (WESSELand ALEXANDER 1977), i.e. mainly along (110) and (112) (HORNSTRA 1958, HOLT 1962, WAGNER et al. 1993). Moreover, it is well established that dislocations in plastically deformed silicon and/or III/V compounds are dissociated over the most of their length (e.g. COCKAYNE and HONS 1979; GOTTSCHLAK, PATZERand ALEXANDER 1978). Therefore, with respect to plastic deformation (dislocation mobility) the separation of perfect dislocations and core structure of partials and perfect dislocations are of great interest. Using weak-beam technique the separation widths could be detected for various elemental (e.g., GOTTSCHALK 1983, HIRSCH1980, WESSELand ALEXANDER 1977, GOTTSCHALK et al. 1993), III/V compound semiconductors (e.g., GOTTSCHALK, PATZER, ALEXANDER 1978; DE COOMAN et al. 1986; DE COOMAN and CARTER1987; MAKSIMOV et al. 1987; KHODOS et al. 1989; GOMEZand HIRSCH1978; GAIand HOWIE1974; LUYSBERGand GERTHSEN 1994) and n/VI compounds (e.g. LU and COCKAYNE 1986). However, recently high-resolution experiments have been applied more and more to determine as well as the separation width, and, especially, to examine the

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

112

core structure of defects in elemental semiconductors (THIBAULT-DESSAUX and P U TAUX 1989; SATOet al. 1980;HIRSCH1980; HUTCHINSON et al. 1980; BOURRETet al. 1982; ELKAJBAJI and THIBAULT-DESSEAUX 1988; THIBAULT-DESSEAUX et al. 1984; SPENCE1980;OLSENand SPENCE1981;SPENCE and OLSEN1981;SPENCE and OLSEN 1981b, GEIPELet al. 1993) and compound semiconductors (LUYSBERG and GERTHSEN 1994; DE COOMAN et d. 1986; MAKSIMOV et d . 1984;PONCE 1985; MAKSIMOV etal. 1987; PONCE etal. 1986; LUYSBERGetal. 1992; KISELEV1991; HUTCHINSON 1991; PHILLIPP et al. 1994). Moreover, HRTEM was also used to distinguish between the glide or shuffle set coni5guration (OLSENand SPENCE1981, SPENCE and OLSEN 1981) of a separated 60" dislocation in GaAs, and polarity of coherent (111) twin boundaries in GaP (PHILLIPPet al. 1994). However, there is an usual way to identify a particular dislocation on a HRTEM picture by means of the well-known BURGERS-circuit to obtain the edge component of the total BURGERSvector. Thus, because in such HRTEM images the dislocation line lies always end-on its type can be determined (see e.g., PONCEet al. 1986;PONCE1985; KISELEVet al. 1991; SPENCEand OLSEN1981; OLSENand SPENCE1981). It is the aim of this paper to show that by comparing the calculated and experimental HRTEM images it is possible to identify the various kin&- of perfect dislocations and their dissociation into Shockley partials in a simple but straightforward manner. Experimental

Plastically deformed GaP single crystals (.more details see WAGNERet al. 1993, and ROTSCH,PAUFLER and WAGNER1987) and hetereoepitaxial InAs/(OOl)GaAs single layers grown by MOVPE (see GOTTSCHALCH et al. 1995) were thinned by ion milling with Ar+ at 4 kV, 0.5 mA and an incidence angle of 11-13' using a Gatan-Duo-Mill system. (110)cross-sections in the case of layered systems and (110)cuts for deformed single crystals have been prepared for TEM experiments. For examination of 30" and 90" dislocations (112) sections have been prepared from I d s (lattice parameter 0.60584 nm, ADACHI1982) epitaxial layers grown on (001)GaAs substrates. The density of dislocations lying inclined to the interface is extremely high in such heteroepitaxial layers, and therefore, the probability is also increased to find dislocations along (112). In order t o differentiate between the [110]and [liO] directions, and therefore between the (110) and (110) cross-sections the GaP crystals and InAs/(OOl)GaAs layers have been treated by chemical etching as reported by NOLZEet al. 1990. Both types of specimens belong to those phases of the (B3)zinc blende structure type (space group F33m), i.e. there is a polarity in the (111) slip planes. This is necessary with respect to the aand &character of dislocations (cf. WAGNERand PAUFLER 1993). High-resolution experiments (HRTEM) have been performed in a Philips CM200 transmission electron microscope operating at 200 kV. It is equipped with a super-twin objective lens and a Gatan-TV-system. In order to improve the contrast and, therefore the perceptibility of structural details too, the experimental images have been digitally filtered by means of a program package from SIS. At first, an image-was transformed into its power spectrum where the filtering process occurs (filtering in the Fourier space). The choise of filters depends on the characteristic feature to be analysed in the image (cf. NEUMANN 1991). The best results could be achieved if a BRAGG-filter is applied where only the space frequencies of type 220, 111 and 004 (8 beams for beam direction along (110),i.e. with-

Cryst. Res. Technol. 32 (1997)1

113

out the undiffracted beam 000) have been used for image reconstruction. If (112) images should be filtered, then space frequencies of type 111, 131 and 220 (8 beams for beam direction along (112), i.e. without the undiffracted beam 000) were used. As well as the structural pictures and HRTEM images have been exclusively simulated for gallium phosphide (Gap) using the EhIS program package (STADELMANN 1987) on a IBM work station of type RISC 6000.

Calculation and simulation procedure In the following the crystal is treated as an isotropic continuum and linear elasticity is applied. Then, a dislocation within such an infinite body of a crystal is characterized by its displacement field represented by u(r). The position r’ of a certain atom after introducing dislocation is

r’ = r

+ u(r),

(1)

where the vector r characterizes the atom position in the absence (undeformed state) of the dislocation. u(r) is the displacement vector of this atom after the dislocation was introduced into the crystal. The u(r) has the meaning of u = (u,, %, x)in Cartesian coordinates. Now, we set always the dislocation line direction 6 parallel to the positive z direction as shown in Figure 1. The BURGERSvector b (here parallel to x-axis) results from the so-called FS/RH-rule (finish to start/right-hand) with respect to the BURGERS-circuit applied (see HIRTHand LOTHE1982). Then; according to KOEHLER 1941, NABARRO 1952, AUGUST1966, and HIRTHand LOTHE1982 follows for a screw dislocation with Burgers vector b, parallel to z-axis and u,= uy = 0 u, = -arctan 2x bs

- . (1)

In the case of an straight edge dislocation with

5 along z and be parallel x,where uz= 0

Fig. 1. Correlation between the parameters of a dislocation (here an edge dislocation) and the Cartesian coordinates used for calculation. has the meaning of the positive direction of the dislocation line, b is the Burgers vector, and n the normal of glide plane (note, the FS/RH-rule is used here; see HIRTHand LOTHE1982) 8 Cryst. Re. Technol., Vol. 32, No. 1

z

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

114

Table 1 Characteristical parameters of perfect dislocations under discussions. b Burgers vector of perfect dislocation, 6 -positive direction of dislocation line, n - normal of glide plane. be and b, are the edge and screw component of b of the perfect dislocation, where b =be b,

-

+

~~

~

Dislocation type

config.

b

5

be

bs

n

see Fig.

perfect 60"

~ ( s )

1 -[0111 2

1 -[1101

"iizl

1 -[1101

[iii]

3a-c

perfect Lomer

~ ( s )

+[iio]

-[1101 2

-

[Ool]

2a-d

perfect screw

~ ( s )

I[iio]

-

1 pi01 2

perfect 30"

A(S)

-[zi]

1 --[011]

-[ail

[iii]

4a-d

1 -[2li]

1 -[011] 2

-

[lil]

5a-b

A(s)

perfect 90"

2 1 -[110] 2 1 -[011] 2

Jz 1 -[1101 Jz

4

4 1 -

1 --[1101

Jz 1

d5

d3

1

4

4

and lnJm+--

1 2(1 - Y )

52

!?+ y2

}.

(4)

For a mixed dislocations of the Burgers vector b these displacement fields can be superimposed in the framework of linear elasticity, where b, = b sin p and b, = b cos B. /3 is the angle between 6 and b. Then b, and be represent the magnitudes of the screw and edge component of the total Burgers vector b. It is noted, that the expressions given by NABARRO1952, HIRTHand LOTHE1982 and KOVACSand ZSOLDOS 1973 differ in some signs and factors from that given in eqn. (4). However, we use here AUGUST'S expressions (AUGUST 1966, PAUFLER 1978). Table 2 Characteristical parameters of the separated glide dislocations (B(g)) under discussion. b - Burgers vector of perfect dislocation, bp Burgers vector of partial dislocation, 6 - positive direction of dislocation line; b;(l), b;(l), bi(2) and $(2) are the edge and screw components of the partials (1) and (2)

-

perfect disl.

5

separation into partials according to b bP0) +b,(2)

b;(l)

b;(l)

see Fig.

bE(2)

b;(2)

1 -[in] 6

1 -[112] 12

1 -[110] 4

6a-d

1 --11121 12

1 --[110j

7a-b

+

60" 0"

900

1 -[110]

1 1 1 -[011] --[112] +-[121] 6 2 6 60" + 90" +30" 1 --1 -1 -[1101 -pi01 +L[3ii] +s[121~ 2 6 Jz 0" 4 30" + 30" 1 1 1 1 ---[2ii] -[0111+-[1121 +-[1211 2 6 6 d5 90" 60"(120") 60" 1 1 1 ' ' 1 -[[2ii] 2[0ii] +-[121] +-[2ii] 6 6 d5 30" + 60" 0"

Jz

+

30"

+

-[iia]

1 12

L[iio] 4

1 -[0111

1 L(~ii1 -[oil] 12 4

--piij 12

1 q[O1l]

1 -pii] 12

1 [21i]

4-

-

4

1

6

9a-b

'

8a-b

Fig. 2. Arrangement of atoms near the core of Lomer dislocations in shuffle set configuration A(s). b, A and 5 BS in Table 1 and 2. 6 points upwards to the viewer. a - calculated atomic st-mcture; b calculated HRTEM image; c experimental HRTEM image: b = 1/2 [IlO], n = [Ool),5 = l/& [llO]; 5 points into the paper. The position of two Lomer dislocations is marked with arrow tips. Note, each of the other defects on the right-hand side consist of two close-spaced perfect 60' dislocations on non-parallel {lll} glide planes. For one example the additionally inserted lattice planes of type (111) and {OOI}, characteristically for a perfect 60" dislocation, are marked for each single perfect 60" one. Their different glide planes are marked with white lines. d - digitally filtered experimental HRTEM image

-

-

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

116

Using some programs written in C the super-cell-files needed for the EMS program package have been generated for the different dislocation types and separations given in the Tables1 and 2. It is emphasized that the choice of separation width for dissociated dislocations is quite arbitrary and, therefore, it do not correlate to experimental data. The Figures 2a-9a show the arrangements of atoms (in Figure 2a the gallium and phosphorous atoms are marked with Ga and P, respectively) in the neighborhood of the cores of different perfect and partial dislocations, where the core itself is not drawn in these pictures, because it is not possible to calculate the exact atom position in the core using the theory used here. In the case of separated dislocations they are assumed to be in glide (g) set configuration, where perfect dislocations should be in the shuffle (5) set. The nomenclature “glide” and “shuffle” is depending on whether the terminating plane of dislocation ends at the closely spaced (111) crystal planes (glide) or widely spaced (111) planed (shuffle) (cf. HIRTHand LOTHE 1982). However, in all cases only A(s) and

t

C)

Fig. 3. Arrangement of atoms near the core of a perfect (undissociated) 60” dislocation in shuffle set configuration A(s). b, be,4, n and 5‘ as in Table 1 and 2. 5 points upwards to the viewer. a - calculated atomic structure; b - calculated HRTEM image; c - digitally filtered experimental HRTEM image; 5 points into the paper

117

Cryst. Res. Technol. 32 (1997)1

(113)

Fig. 4. Arrangement of atoms near the core of a perfect ( u n d k e ciated) 30" dislocation in shuffle set configuration A(s). b, b,, b,, n and 5 as in Table 1 and 2. 5 points upwards to the viewer. a - calculated atomic structure; b - calculated HRTEM image; c experimental HRTEM image; d digitally filtered experimental HRTEJI image

-

dl

(011)

(13)

118

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

bl Fig. 5 . Arrangement of atoms near the core of a perfect (undissociated) edge dislocation in shuffle set configuration A(s). b. n and as in Table 1 and 2. 5 points upwards to the viewer. a_- calculated atomic structure b calculated HRTEM image

e

Tab. 3. Parameters used for calculation of dislocation structure and simulation of HRTEM images by the EMS program package parameter

values

lattice parameter of GaP space group of GaP accelerating voltage Uo wave lenght I objective aperture diameter spherical aberration of objective lens c. beam semi-convergence ab chromatic &focus spread Debye-Walier factor is assumed to be absorption is assumed to be defocus for simulation (beam direction along (110)) defocus for simulation (beam direction along (112) foil thickness for (110) cros-sections) 512 x 512 beruns foil thickness for (112) cross-sections 512 x 512 beams

0.24512 nm (ADACHI 1982) F43m (space group number 216) 200 kV 2.51 pm

40 nm--'*) 1.2 mm 1.2 mad 2.0 nm 0 0 -70.0 nm

-123.0 t = N x 6 2 = 16 nm; N = 4 0 , 6 2 = 0.4 nm (slice thickness)

t e 3.4 nm; N.= 5, 6 2 = 0.67 nm

-

of the size of the supercell such a large aperture is needed for calculation *) **) Because The Schemer defocs of our CM200 lies at about -67nm. Note, if we have an underdefocus, i.e. negative, then the EMS programm requires the input of a positive value, because the negative sign is already considered within the program

119

Cryst. Re.Technol. 32 (1997) 1

Fig. 6. Arrangement of atom near the core of a dissociated 60" dislocation in glide set 90°/300 configuration B(g). b, be, 4, n and as in Table 1 and 2. 5 points upwards to the viewer. a - calculated atomic structure; b - calculated HRTEM image; c - experimental HRTEM image; d - digitally filtered experimental HRTEM image

90'

120

a)

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

b)

Fig. 7. Arrangement of a t o m near the core of a dissociated screw dislocation in glide set configuration B(g). b, b,, b,, n and as in Table 1 and 2. 6 points upwards to the viewer. a - calculated atomic structure; b calculated HRTEM image

B(g) dislocations have been considered here, i.e. if a perfect A(s) dislocation in the shuffle set (atoms of typeA are in the most distorted core position) dissociates, then the resulting partials should be B(g). In the Tables 1 and 2 the nomenclature of the different types according to the Hunfeld conference (ALEXANDER, HAASEN,LABUSCH,SCHROTER 1979) is considered. The Figures2b-9b show the corresponding simulated HRTEM images for GaP exclusively. In Table 3 all parameters used for simulations are summarized. For calculation of the HRTEM images the multislice method and linear image formation theory have been used. Therfore, within the road map of the EMS program

a)

b)

Fig. 8. Arrangement of atoms near the core of a dissociated 30" dislocation in glide set configuration B(g). b, b,, h,n and 6 as in Table 1 and 2. 6 points upwards to the viewer. a - calculated atomic structure; b - calculated HRTEM image

Cryst. Res. Technol. 32 (1997)1

121

Fig. 9. Arrangement of atoms near the core of a dissociated edge dislocation (60"/60"-configuration)in the shuffle set A(s). b, n and as in Table 1 and 2. C points upwards to the viewer. a - calculated atomic structure; b calciilated HRTEN image

we followed this way: sc3 > msl > lil > tv3 > pr3. Because for the atom situated directly at the inner core (in the super-cell file the position r = (O,O,O)) the theory used here is not valid but for all the atoms outside a inner cut-off radius of approximately b/4. Thus, it has been removed from this position, and therefore, it is not drawn in the calculated structural pictures and simulated HRTEM images too.

Results and discussion From the calculated structure and HRTEM images some well-distinct characteristic can be established for the various perfect and partial dislocations. They are given in Table 4. The calculated HRTEM images correspond well to the digitally filtered experimental micrographs as shown for perfect LOMERdislocations positioned directly at the interface between InAs and GaAs (see Fig. 2), perfect 60" dislocation (Fig. 3), separated 60" dislocation (90"/30" configuration, see Fig. 6 ) , in plastically deformed Gap, and perfect or separated 30" dislocation in InAs with 6 along [112] (see Fig. 4 and 8). As mentioned above, we cannot give exact information about the dislocation core itself. However, the type of linear defect can be clearly determined by comparing with calculated images. Unfortunately we are not lucky to observe experimentally separated screws, glissile perfect and separated edge dislocations with l j along (112), perfect and separated 30" dislocations with l j parallel (112) by HRTEM examinations in plastically deformed Gap. Nevertheless, such types could be observed with considerable quantities in plastically deformed GaP by means of diffraction contrast analyses (PAUFLER et al. 1987, WAGNERet al. 1993). LUYSBERG and GERTHSEN 1994 and LUYSBERG et al. 1992 have presented some HRTEM images of separated edge dipoles and separated edge dislocations in plastically deformed InP single crystals and have measured their separation widths. HRTEM images of separated screws (30°/300configuration) in silicon have been given by SATOet al. 1980, HUTCHINSON et al. 1980, DE COOMAN et al. 1986 and HIRSCH

LAZAR, WAGNER: Simulation of HRTEM Images of Dislocations

122

Tab. 4 Typical charateristic for identification of various perfect and partial dislocations dislocation type

"additional inserted lattice planes" visible by viewing along the different "atomic rows" in the HRTEM image and/or picture of dislocation structure

90" Lomer dislocation perfect 90" dislocation perfect 60" dislocation perfect 30" dislocation 90" partial dislocation 30" partial dislocation 60" partial dislocation

double (110}, 2 non-parellel (111) 2 double (110}, 2 non-parallel double (131) (1101, (1111, (001) (110), 2 non-parallel(131)

{0011 (1111 (110}, 2 non-parallel (131)

1980. However, we have imaged 30" dislocations in epitaxial InAs layers as shown in Figure 4d. Nevertheless, there are some problems in distinguishing perfect and dissociated 30" dislocations with 6 parallel (112). As shown here (see Figure 4 und 8), in the HTREM image it is not possible to decide whether such a dislocation is separated or not, because in both cases (perfect and dissociated 30") only the edge component is visible, and that is equal for both the 30" perfect and 60" partial (see Table 1 and 2). On the other hand, the stacking fault between the both partials is not visible in such (112) sections, and therefore the localization of the 0" partial is impossible. This means, if a 30" dislocation should be separated into a screw and 60" partial, as mentioned, only the latter one can be detected. The former screw partial is not detectable. The same holds for a perfect screw dislocation. So, the separation width of a 30" dislocation lying along (112) is not possible to determine in HRTEM but only by weak-beam imaging in otherwise oriented sections. However, if a perfect screw separates into two 30" partials, then the edge components of both partials are antiparallel, i.e. due to definition of b according to the FS/RH-rule the "additional lattice planes" have to be inserted from above and below (see Figure7) if 5 is kept konstant. As shown there, the resulting intrinsic stacking fault which connects the both 30" SHOCKLEY partials is visible. The most distorted lattice area exists around the core of a perfect 90" slip dislocation with n = (lll),because two additional (110) lattice planes are inserted from the same side partials (see (Figure 5). If such an edge dislocation dissociates into two 60" SHOCKLEY Table 2 and Fig. 9) theire be-components are parallel, but theire b,-components are o p posite. This means, the additional lattice planes are inserted from the same side for both partials where the one 60" partial posses left-handed screw parts and the other one is of right-handed type. In summary, it could be established, that using the simplifications of linear elasticity, isotropy and linear image formation theory the calculated HRTEM images correspond well to the experimental images, and, therefore, it is possible to identify the various types of dislocations in sphalerite type crystals. Due t o the above mentioned assump tions it is not possible to calculate the inner core structure of a dislocation. This can be achieved if the core structure is approximated-fmm three dimensional atomic models as discussed by e.g. OLSENand SPENCE1981. Moreover, the different possibilities of dislocation cores (reconstruction phenomena, interaction with point defects etc.) for various perfect and partial dislocations in elemental and compound semiconductors have been extensively discussed in detail by LOUCHET and THIBAULT-DESSEAUX 1987.

Cryst. Res. Technol. 32 (1997) 1

123

Acknowledgements This work was financially supported by the Deutsche Forschungsgemeinschaft. The authors want to thank Dr. V. Gottschalch and Dr. R. Franzheld for providing epitaxial layers. The Fonds der Chemischen Industrie is greatly appreciated for the generous financial support in improvement of our TEM with hardware and software. References ADACHI, S.: J. Appl. Phys. 53, 877 (1982) ALEXANDER, H., HAASEN,P., LABUSCH,R., SCHROTER,W.: J. De Physique 40, C6 (1979) AUGUST,M. : Theorie statischer Versetzungen, B.G. Teubner Verlagsgesellschaft, Leipzig 1966 BOURRET,A., DESSEAUX, J., RENAULT, A.: Philos. Mag. A 45, 1 (1982) COCKAYNE, D. J . H., HONS. A.: J. Phys., Paris, 40, C6-2 (1979) DE COOMAN,B. C., CARTER,C. B.: Proc. 45th Annual Meet. Electr. Micr. SOC.Amer., San Francisco 1987, p. 308 DE COOMAN, B. C., KUESTERS,K.-H., CARTER,C . B. : Mater. Res. Soc. Symp. 6 2 , 3 7 (1986) ELKAJBAJI, M., THIBAULT-DESSEAUX, J. : Philos. Mag. A 58, 325 (1988) GAI, P. L., HOWIE,A.: Philos. Mag. 30, 939 (1974) GEIPEL,T., XIAO,S. Q., PIROUZ,P . : Philos. Mag. Lett. 67, 245 (1993) GOMEZ,A. M:, HIRSCH,P. B.: Philos. Mag. A 38, 733 (1978) GOTTSCHALCH, V., SCHWABE,R., PIETAK, F., WAGNER,G., FRANZHELD, R., PIETZONKA, I., KRIEGEL,S., HIRSCH,D.: Workshop Booklet of the 6th Europ. Workshop on MOVPE and related growth techniques, Gent, Belgium, June 25-28, 1995, C8 GOTTSCHALK, H. : J. de Physique, Colloque C4, Suppl. 9, Tome 44, Sept. 1983, p. C4-69 GOTTSCHALK, H., HILLER,N., SAUERLAND, S., SPECHT,P., ALEXANDER, H.: phys. stat. sol. (a) 138, 547 (1993) GOTTSCHALK, H., PATZER. G., ALEXANDER, H.: phys. stat. sol. (a) 45, 207 (1978) HIRSCH,P. B.: Micron 11, 243 (1980) HIRTH,J . P., LOTHE, J.: Theory of Dislocations, Second Edition, John Wiley and Sons, New York-Chichester-Brisbane-Toronto-Singapore 1982, p. 22, 60, 78, 378 HOLT, D. B.: J. Phys. Chem. Solids 23, 1353 (1962) HORNSTRA,J.: J. Phys. Chem. Solids 5, 129 (1959) HUTCHINSON, J. : Proc. High-Resol. Electron. Micr.-Fundamentals and Applications, Halle/Saale 1991, Autumn School, Heydenreich and Neumann (eds.), p. 205 J. L., HUMPHREYS, C. J., OURMAZD, A., HIRSCH,P.B.: Electron Microscopy 1, HUTCHINSON, 304 (1980) KHODOS,I. I., SHIKHSAIDOV, h1. SH., SNIGHIREVA, I. I., USHAKOVA, A. P., NIKOLAICHIK, v. I.: phys. stat. sol. (a) 114, 113 (1989) KISELEV,N. A. : Proc. High-Resol. Electron. Micr.-Fundamentals and Applications, Halle/Saale 1991, Autumn School, Heydenreich and Neumann (eds.), p. 116 KOEHLER,J. S.: Phys. Rev. 1, 397 (1941) KOVACS,I., ZSOLDOS,L. : Dislocations and Plastic Deformation, Akademiai Kiado, Budapest 1973, p. 322 LOUCHET,F., THIBAULT-DESSEAUX, J.: Phys. Appl. 22, 207 (1987) Lu, G., COCKAYNE, D. J . H.: Philos. Mag. A 53, 307 (1986) LUYSBERG, M., GERTHSEN, D.: phys. stat. sol. (a) 146, 157 (1994) LUYSBERG, M., GERTHSEN.D., URBAN,K.: Philos. Mag. Lett. 65, 121 (1992) MAKSIMOV, S. K., ZIEGLER,M., KHODOS,I. I., SUIGHIRYOVA, I. I., SHIKHSAIDOV, M. SH.: phys. stat. sol. (a) 84, 79 (1984) MAKSIMOV, S., FILIPPOV,A. P., GAIDUKOV, G . N., HEYDENREICH, J., KHODOS,I. I.: Philos. Mag. A 55, 339 (1987) NABARRO,F . R. N. : The Mathematical Theory of Stationary Dislocations, Advances in Physics 1, 269 (1952) NEUMANN, W., PIPPEL, A., HOFMEISTER,H. : High-Resolution Electron Microscopy-Fundamentals and Applications. Proc. Int. Autumn School 1991, Halle/Saale, p. 95 NOLZE,G., GEIST, V., WAGNER,G., PAUFLER,P., JURKSCHAT, K.: Z. Krist. 193, 111 (1990)

124

LAZAR,WAGNER: Simulation of HRTEM Images of Dislocations

OLSEN,A., SPENCE, J. C. H.: Philos. Mag. A 43,945 (1981) PAUFLER,P. : Physikalische Grundlagen mechanischer Festkorpereigenschaften,Part I, AkademieVerlag Berlin 1978,p. 74 PAUFLER,P., ROTSCH, P., WAGNER,G.: Philos. Mag. A 56,533 (1987) PHILLIPP,F.,H~SCHEN, R., OSAKI,M., MBBUS, G., RUHLE,M.: Ultramicroscopy, 56,1 (1994) PONCE,F. A.: Inst. Phys. Conf. Ser. No 76,Microsc. Semicond. Mater. Conf., Oxford 1985,p. 1 PONCE,F. A., ANDERSON,G. B., HAASEN,P . , BRION,H. G.: Defects in Semiconductors, H. J. von Bardeleben (ed.),Mater. Sci. Forum, 10-12, 775 (1986) SATO, M., HIRAGA,K., SUMINO,K.: Jpn. J. Appl. Phys. 19,L155 (1980) SPENCE,J. C. H.: 38th Ann. Proc. Electr. Microsc. SOC.Amer., SanFrancisco, California 1980, p. 282 SPENCE,J. C.H., OLSEN,A . : in Defects in Semiconductors, Narayan and Tan (eds.), North-Holland Inc. 1981,p. 279 STADELMANN, P . A.: Ultramicroscopy 21,131 (1987) THIBAULT-DESSEAUX, J., PUTAC‘X, J. L.: Inst. Phys. Conf. Ser. No 104,Chpt. 1, Int. Symp. Struc. Propert. Disloc. Semicond., Oxford 1989,p. 1 THIBAULT-DESSEAUX, J., PENISSON,J. M., BOURRET,A. : in Physical Chemistry of the Solid State: Appl. to Metals and their Compounds, Elsevier Sci. Publ. 1984,p. 327 WAGNER,G.,PAUFLER,P . : phys. stat. sol. (a) 138,389 (1993) WAGNER,G.,PAUFLER,P., PONGRATZ,P., SKALICKY, P.: Philos. Mag. A 67,143 (1993) WESSEL,K.,ALEXANDER, H.: Philos. Mag. 35,1523 (1977) (Received August 23,i996;accepted October 5, 1996) Author’s address: Dr. G. WAGNER,Diplom Mineraloge M. LAZAR Universitat Leipzig, Fakultat f i r Chemie und Mineralogie, Institut fur Mineralogie, Kristallographie und Materialwissenschaft, Linnbtr. 3-5 (TA), 04103 Leipzig, F.R.G.