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Analysis of thermal-treatment-induced dislocation bundles in GaAswafers by means of X-ray transmission topography and complementarymethodsP. Mock

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J. Appl. Cryst. (2001). 34, 65–75 P. Mock � Dislocation bundles

J. Appl. Cryst. (2001). 34, 65±75 P. MoÈck � Dislocation bundles 65

research papers

Journal of

AppliedCrystallography

ISSN 0021-8898

Received 2 March 2000

Accepted 3 November 2000

# 2001 International Union of Crystallography

Printed in Great Britain ± all rights reserved

Analysis of thermal-treatment-induced dislocationbundles in GaAs wafers by means of X-raytransmission topography and complementarymethods

P. MoÈck²

Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, England.

Correspondence e-mail: pmoeck@uic.edu

By means of a heat treatment that was part of a molecular beam epitaxy (MBE)

growth procedure, dislocation bundles have been induced in two-inch-diameter

undoped (001) GaAs substrates. On the basis of contrast variations in

synchrotron-based single-crystal X-ray transmission topograms that were

recorded under conditions of high anomalous transmission, these dislocation

bundles have been classi®ed into three different types. Dislocation bundles of

the majority type start at the sample edges in regions around the four h100iperipheral areas, glide typically up to about 1.5 cm into the bulk of the wafer

following perpendicular h110i line directions, and form a pseudo-symmetric

fourfold set. There are dislocations with two different Burgers vectors in each

majority-type dislocation bundle and the extended segments of all of these

dislocations are of the 60� type. In order to explain complementary experimental

results, it is suggested that dislocation pairs are formed in the majority-type

dislocation bundles. Theoretical support for this hypothesis is derived from a

model of plastic deformation of GaAs wafers during typical MBE growth.

Dislocation bundles of two minority types, on the other hand, are not part of the

fourfold set and originate in peripheral areas at and around h110i.

1. Introduction

It has been demonstrated in a recent X-ray topography survey

that a quite common, radiatively heated, non-indium-bonded

sample-holder design can cause severe plastic deformation in

two-inch-diameter GaAs (001) substrates when the sample is

heated to about 923 K in a molecular beam epitaxy (MBE)

machine and subsequently cooled to room temperature

(MoÈ ck, 2000a). In addition, it is known that heat-treatment-

induced plastic deformation of GaAs substrates is a key factor

that reduces the yield of electronic devices in manufacturing

processes on an industrial scale (Tatsumi et al., 1994; Kiyama et

al., 1997; Sawada et al., 1996).

It is expected that the current market for four-inch-

diameter GaAs wafers will, in the near future, be overtaken by

demand for six-inch-diameter GaAs substrates for industrial

applications (Sawada et al., 1995). Since the thermal-treat-

ment-induced residual strain and plastic deformation is more

dif®cult to avoid in the latter case (Flade et al., 1999) and

because further thermal processing of already dislocated

wafers that possess residual strain leads to more plastic

deformation (Tatsumi et al., 1994; Kawase et al., 1993), it will

be bene®cial to identify the key features of the plastic defor-

mation process.

In our case (MoÈ ck & Smith, 2000), the technical problem of

plastic deformation could be overcome by modi®cations to the

sample holder of a custom-built MBE machine. The science

concerning the nature of the dislocation bundles and the

mechanisms involved in the plastic deformation of two-inch-

diameter substrates, however, remains of interest and it is

likely that the same mechanisms operate more readily in

larger-diameter III±V semiconductor wafers.

The main aim of this paper is to perform a complete Burgers

vector analysis on dislocation bundles of the so-called majority

type for one sample that was part of the earlier X-ray

diffraction topography survey (MoÈ ck, 2000a). For the discus-

sion of the X-ray topography data, results of complementary

analyses that employed scanning infrared polariscopy, visible-

light interferometry, Nomarski microscopy, high-resolution X-

ray diffractometry and transmission electron microscopy will

be drawn upon (MoÈ ck et al., 1999, 2001; MoÈck & Smith, 2000).

In addition to the Burgers vector analysis and in order to

verify its results, the point symmetry group of the crystal-

lographic parameters of the plastic deformation that is

realised by the majority-type dislocation bundles will be

elucidated.² Present address: Department of Physics (MC 273), University of Illinois atChicago, 845 West Taylor Street, Chicago, IL 60607-7059, USA.

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66 P. MoÈck � Dislocation bundles J. Appl. Cryst. (2001). 34, 65±75

The results of the X-ray transmission topography study [and

the results of the complementary studies (MoÈ ck et al., 1999,

2001)] reported herein have been used for the development of

a new model for the plastic deformation of GaAs wafers under

conditions that are typical for MBE growth (MoÈ ck, 2000b).

Since this model accounts for the core of the experimental

observations in a semi-quantitative manner, it lends theore-

tical support to the dislocation-pair formation hypothesis

presented in this paper.

2. Experimental details

The sample that was selected for the X-ray transmission

topography, scanning infrared polariscopy and Makyoh

topography analyses of this study is a double heterostructure

which consists of an undoped vertical-gradient freeze-

Bridgman-grown two-inch-diameter GaAs wafer, with (001)

orientation and 450 mm thickness, a GaAs buffer layer of

thickness 500 nm, an In0.06Ga0.94As epilayer of thickness

60 nm, and a GaAs capping layer of thickness 10 nm. The

epitaxic structure was grown by MBE in a Varian Gen II

machine at a rate of one monolayer per second. While the

GaAs buffer layer was grown at 873 K, both of the hetero-

epitaxial layers where grown at 793 K. Prior to the epitaxial

growth, the surface oxide was desorbed at a temperature of

923 K.

This sample was analysed by means of synchrotron-based

single-crystal X-ray transmission topography under conditions

of high anomalous transmission, employing the experimental

facilities at Daresbury Laboratory (UK). A complete set of

symmetry-related 111, �111, 202, 313 and �313 topograms was

recorded on double-sided Agfa-Gevaert Structurix D4 X-ray

®lm. While topograms of the 111 and �111 re¯ections and two

of the 202 re¯ections were taken at more or less optimized

azimuths at � = 0.13 nm [i.e. just above the Ga K-absorption

edge at 0.1195 nm (International Tables for X-ray Crystal-

lography, 1985)], deliberate azimuthal offsets of about 10�

resulted in the simultaneous diffraction of pairs of 202 and 313

or �313 re¯ections at �1 = 0.073 nm and �2 = 0.053 nm [i.e. just

above the In K-absorption edge at 0.0444 nm (International

Tables for X-ray Crystallography, 1985)]. The optimized

azimuth is de®ned by the condition that the normal of the

diffraction plane, the incident and the diffracted beam are

coplanar. Experimental details on the complementary scan-

ning infrared polariscopy and Makyoh topography studies of

this and other samples of the same sample series (MoÈ ck,

2000a) have been presented by MoÈck et al. (1999).

For comparison purposes only, a 1�11 topogram was

recorded at � = 0.13 nm from a second sample of the same

sample series (MoÈ ck, 2000a) on an Ilford L4 nuclear plate. The

growth conditions of this second sample were identical to

those of the ®rst sample and it differs from the ®rst sample

only in so far as its heteroepitaxial layers are twice as thick as

the heteroepitaxial layers of the ®rst sample. As an 0�22topogram of this second sample shows (MoÈck & Smith, 2000),

its In0.06Ga0.94As layer, of thickness 120 nm, is partly relaxed

by mis®t dislocations at the interface between this layer and

the underlying GaAs buffer layer.

Different kinds of X-ray transmission topography are

distinguished in Appendix A on the basis of the prevalent

absorption conditions. Brief comments on the Burgers vector

determinations by means of X-ray diffraction topography

under conditions of high anomalous transmission are given in

Appendix B.

Epitaxial samples were used for this study because they

were, on the one hand, readily available, and on the other

hand, as shown by Tatsumi et al. (1994) and Kawase et al.

(1993), perfectly adequate for this purpose. This is because the

defects in the GaAs substrates are not caused by the epitaxial

growth process itself, but by the thermal treatment associated

with it (Tatsumi et al., 1994; Kawase et al., 1993).

3. Results of the X-ray transmission topography study

3.1. Distinction between majority- and minority-typedislocation bundles on the basis of their respective spatialarrangements

As Figs. 1, 2 and 3 show, the plastic deformation is mainly

(i.e. typically up to about 98%) realised by bundles of dislo-

cations which start at the sample edges around the four h100iperipheral areas, glide into the bulk of the substrate following

h110i and h1�10i line directions, and form a pseudo-symmetric

fourfold set in undoped GaAs. These dislocation bundles are

called majority-type dislocation bundles and typically extend

up to about �25� around h100i poles (see also MoÈ ck, 2000a).

As the previous studies (MoÈck, 2000a; MoÈ ck et al., 1999)

showed, there can also be dislocation bundles of another type

with a spatial distribution that is centred at and around h110iperipheral regions; these are called minority-type dislocation

bundles (MoÈck, 2000a). This term is used in a generic manner,

being applied to all dislocation bundles that do not belong to

the majority type. While dislocation bundles of the ®rst

minority type tend to be located at or rather close (i.e.��10�)to h110i poles, dislocation bundles of the second minority type

typically deviate by �10� up to about �35� around h110ipoles. Thus, these minority-type dislocation bundles can exist

in areas that are occupied by the pseudo-symmetric set of

majority-type dislocation bundles (see also MoÈ ck, 2000a).

3.2. Assessments of majority-type dislocation bundles

A comparison of Figs. 1(a) and 1(b) shows that the contrast

of the majority-type dislocation bundles is rather similar in

both topograms, despite the fact that opposite surfaces of the

sample have been exposed ®rst to the incoming X-ray beam.

The Borrmann fans (see Appendix B) of the majority-type

dislocation bundles are, thus, rather similar for both re¯ec-

tions. This indicates that the dislocation bundles are distrib-

uted through the whole thickness of the substrate but are

de®nitively not con®ned to the epitaxial structure±substrate

interface.

A comparison of the widths and contrast of individual

quarter subsets of minority-type dislocation bundles in the 0�22

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(Figs. 1a and 2b), 202 (Fig. 2c) and �20�2 (Fig. 1b) topograms

indicates that there must be dislocations of more than one

Burgers vector (bi) in each bundle that belongs to the pseudo-

symmetric fourfold set since there is reduced contrast and

widths (i.e. reduced Borrmann shadow) when gj � bi = 0 is

ful®lled for parts of the dislocations in a bundle. Thus, at least

three re¯ections are required for a determination of the

Burgers vectors in dislocation bundles of the majority type.

In contrast to the intuitive impression of similar `extinction

of shadow contrast' for one quarter subset (markers `A1' and

`A2' in Fig. 1a) of the dislocation bundles in the 1�11, 1�33 and

3�13 re¯ections (Figs. 2a, 2b and 2c), we consider that there is

only partial extinction of the Borrmann shadow in the 1�33 and3�13 topograms, while there is almost complete extinction in

the 1�11 topogram (since the latter plane is the slip plane of all

the dislocations within these bundles). This can be justi®ed by

the cross product [1�33] � [3�13] = �[�334], which does not lead

to an energetically stable Burgers vector in the sphalerite

structure.

The line directions of the quarter subset (markers `A1' and

`A2' in Fig. 1a) of dislocations with `extinction of shadow

contrast' in the 1�11, 1�33 and 3�13 re¯ections (Figs. 2a, 2b and

2c) are�[110]. Since b1 = [1�11]� [1�33] = [1�11]� [0�22] = [1�33]� [0�22] = �1

2[011] and b2 = [1�11] � [3�13] = [1�11] � [202] =

[3�13] � [202] = �12[10

�1], the dislocation type becomes 60�.Table 1 summarizes the parameters of all of the dislocations

that make up the pseudo-symmetric fourfold set of majority-

type dislocation bundles.

Applying Curie's symmetry principle (Shuvalov, 1988;

Pau¯er, 1986; Curie, 1894) to the problem of plastic defor-

mation of GaAs substrates in a typical MBE growth chamber,

one has to conclude that the dislocation parameters as given in

Table 1 have to be related by the symmetry elements of the

point group �42m, which is the result of the intersection of the

two point groups1mm for the temperature ®eld and �43m for

the crystal. Except for the always present symmetry element 1

and including the inverse symmetry elements, the point group�42m consists of a combination of two parallel fourfold inver-

sion axes (also called inverse tetrads) in [001] orientation, with

two mirror planes (also called mirror re¯ections) in [110] and

[�110] orientation, and three mutual perpendicular twofold

axes (also called diads) in h100i orientation (Pau¯er, 1986).

Employing the respective rotation matrices of these

symmetry elements as given in standard text books on physical

crystallography (e.g. Pau¯er, 1986), the crystallographic

parameters of all of the dislocations in Table 1 can be calcu-

lated solely from the experimentally observed dislocation

parameters of the bundle that is denoted as `A1' in Fig. 1(a).

Such calculations were compared with experimental obser-

vations from other dislocation bundles of the pseudo-

symmetric set of majority-type dislocation bundles and

enabled checks on the internal consistency of the crystal-

lographic indices of all re¯ections, and the direct- and reci-

procal-lattice vectors that are given in this paper. In addition,

it becomes obvious from these calculations and from consid-

erations of the multiplicity of general and speci®c poles in the

point group �42m, that the set of majority-type dislocation

bundles is complete, indicating that all other dislocation

bundles in the sample must belong to different types.

From the fact that the widths and contrast of the quarter

subset (markers `A1' and `A2' in Fig. 1a) with the `extinction of

shadow contrast' in the 1�11, 1�33 and 3�13 topograms (Figs. 2a,

2b and 2c) are similar in the 0�22, 202 and �20�2 topograms (Figs.

1a, 2b, 2c and 1b), where dislocations with either one or the

J. Appl. Cryst. (2001). 34, 65±75 P. MoÈck � Dislocation bundles 67

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Figure 1Single-crystal transmission topograms (taken at � = 0.13 nm): (a) 0�22 re¯ection, (b) �20�2 re¯ection. The major diameter of the ellipses is about 4.8 cm.These topograms and those in Figs. 2, 3 and 4(a) represent X-ray analogues of photographic positives (i.e. increased X-ray radiation intensitycorresponds to increased brightness in the images). The Burgers vectors, line directions, glide planes and peripheral areas of origin of the majority-typedislocation bundles, denoted by the markers `A1' to `D2' in (a), are listed in Table 1. All topograms of this paper, except (b) above, are indexed on thebasis of a right-handed axis system with the [001] direction pointing downwards with respect to the observer. For (b), a right-handed axis system has beenused where the [00�1] direction points downwards with respect to the observer. As the indexing of the topograms indicates, different sides of the waferwere initially exposed to the incoming X-ray beam when (a) and (b) were recorded. In (a), the majority-type dislocation bundles follow a clockwisesequence in alphabetical order (`A1', `A2', `B1', `B2', `C1', `C2', `D1', `D2') that starts from the 12 o'clock position. In (b), these dislocation bundlesfollow an anticlockwise sequence in alphabetical order that starts from the 9 o'clock position. There are several artefacts caused by moving and mountingthe sample by means of tweezers. These artefacts are particularly numerous at the region around the major ¯at (10:30 o'clock position in both images)and facilitate comparisons of the contrast of individual dislocation bundles for different re¯ections.

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68 P. MoÈck � Dislocation bundles J. Appl. Cryst. (2001). 34, 65±75

other Burgers vector are extinct, it

is concluded that the number of

dislocations of each of the two

Burgers vectors are similar as well.

There is a clear difference in the

contrast of the quarter subset that

is denoted as `B1' and `B2' in Fig.

1(a) in the 0�22 and 202 topograms

(Figs. 1a, 2b and 2c). While about

half of the dislocations in this

quarter subset seem to be extinct

in the 202 topogram, all of the

dislocations are responsible for the

pronounced Borrmann shadow of

this quarter subset in the 0�22topogram. Based on these and

similar observations from the rest

of the 220-type topograms, it is

concluded that about one quarter

of the dislocations in a pseudo-

symmetric set of majority-type

dislocation bundles is extinct in

any one 220-type re¯ection.

While Figs. 1, 2 and 3 have been

reproduced in order to illustrate

the `extinction of shadow contrast'

effects for the main sample of this

analysis, Figs. 4(a) and 4(b), and

Figs. 4(d) and 4(e) show sections of

several topograms from the same

sample in higher magni®cations.

As can be clearly seen in Figs. 4(a)

and 4(b), the constituent disloca-

tions of a majority-type dislocation

bundle follow more or less straight

lines that are parallel to h110idirections which are perpendicular

to the surface normal. These

dislocations start at the rim of the

wafer and end in segments that

typically form a small hook and

eventually thread to the free wafer

surfaces [i.e. the (001) and (00�1)planes]. Hair-pin-shaped ends of

dislocations were occasionally

observed as well.

Fig. 4(c) shows, for comparison

purposes only, a section of a

majority-type dislocation bundle of

the second sample. Since this

topogram has been recorded on a

nuclear plate rather than a less-

expensive X-ray ®lm, the resolu-

tion is much higher. A majority-

type dislocation bundle with a

signi®cantly higher dislocation

density is shown in Fig. 4(c) to

g

g

g

( )a

( )b

( )c

Figure 2Single-crystal X-ray transmission topograms: (a) 1�11 re¯ection, (b) 0�22 (right) and 1�33 (left) re¯ections,(c) 3�13 (left) and 202 (right) re¯ections. The major diameter of the ellipse of the 1�11 topogram (a) is4.8 cm. The straight line across this topogram is caused by the pasting together of the upper and lowerhalves of the topogram, as the height of the X-ray beam was not suf®cient to record the re¯ected-transmitted intensity from the whole wafer in one image. The 0�22, 1�33, 202 and 3�13 topograms are framedby the edges of a set of slits that reduced the dimensions of the incoming X-ray beam. The wider slitdistance in these topograms is about 4.1 cm for (b) and about 4 cm for (c). The regions of enhanced X-rayintensity (i.e. the bright stripes) at the lower left edge (third quadrant, 7:30 o'clock position) of the 3�13topogram (c), and at the left-hand side (9:00 o'clock position) in the 1�11 topogram (a), are caused bydiffusely scattered white X-ray radiation that passed through the sample mounting at the major ¯at of thesample. The majority-type dislocation bundles in (a) follow a clockwise sequence in alphabetical orderthat starts with `A1' at the 10:30 o'clock position. Since (b) is in the same orientation as Fig. 1(a), the samesequence of majority-type dislocation bundles applies for both images. In (c), the majority-typedislocation bundles follow a clockwise sequence in alphabetic order that starts with `A1' at the 9 o'clockposition. The majority-type dislocation bundles `A1' and `A2' show `extinction of shadow contrast' in the1�11, 1�33 and 3�13 topograms.

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consist of dislocations with the same arrangement as the

dislocations depicted in Figs. 4(a) and 4(b).

3.3. Assessments of minority-type dislocation bundles

Figs. 3(a) and 3(b) show `extinction of shadow contrast' in

the 111 and �111 topograms for a dislocation bundle of the ®rst

minority type (denoted as `MT 1' in Fig. 1a) that exists in the

[110] peripheral area opposite the major ¯at of the wafer. This

dislocation bundle is also extinct in the �133 re¯ection (topo-

gram not shown), leading to a Burgers vector of�12[0

�11]. Since

the line direction of this bundle is [�1�10],the constituent dislocations belong to

the 60� type.Fig. 4(d) shows this dislocation bundle

in �202 re¯ection in a higher magni®ca-

tion. In comparison with all other dislo-

cation bundles in the wafer, this

particular bundle has a very low dislo-

cation density. From a comparison of this

®gure with Figs. 4(a) and 4(b) and the

discussions above, it is obvious that this

particular dislocation bundle belongs to

a type that is different from the majority

type.

The minority-type dislocation bundles

of the second type, e.g. marker `MT 2' in

Fig. 1(a), were more dif®cult to access

since they seem to consist partly of

thermal-treatment-induced dislocations

that have interacted with grown-in dislocations. This led to

ambiguities in the determination of possible `extinction of

shadow contrast' conditions for various re¯ections. There

were, in addition, experimental limitations such as obstruc-

tions caused by the X-ray beam reducing slits in several of the

topograms, which occasionally prevented the imaging of these

dislocation bundles.

Fig. 4(e) shows the area where such a dislocation bundle

comes into close contact with a branch of the majority-type

dislocation bundle `B2', in higher magni®cation for the 1�11re¯ection. From a comparison of Fig. 4(e) with Fig. 4(d) and

the discussions above, it is obvious that this particular dislo-

cation bundle belongs to a type that differs from both the

majority and the ®rst minority type.

4. Discussion

4.1. Nucleation and arrangement of dislocations in majority-type dislocation bundles

Yonenaga & Sumino (1993), Matsui (1987) and Brown et al.

(1983) suggested that dislocation bundles with h110i line

directions and {111} glide planes in plastically deformed GaAs

are nucleated at surface sources. The dislocation arrangements

given in these papers resemble closely the arrangement of the

dislocations in the majority-type dislocation bundles, as given

in x3.2. Therefore, following the suggestion by Yonenaga &

Sumino (1993), Matsui (1987) and Brown et al. (1983), it is

assumed that the nucleation of majority-type dislocation

bundles took place as half-loops at the wafer edges.

After nucleation, the half-loops could have expanded by

glide processes and either the upper or the lower parts of them

could have been lost by gliding out of the specimen, leaving, as

observed experimentally, partial half-loops that consist of

dislocation segments perpendicular to the wafer normal and

segments that are threading either up through the epitaxial

J. Appl. Cryst. (2001). 34, 65±75 P. MoÈck � Dislocation bundles 69

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( )ag

( )bFigure 3Single-crystal X-ray transmission topograms: (a) 111 re¯ection, (b) �111 re¯ection. The majordiameter of these topograms is 4.8 cm. In (a), the majority-type dislocation bundles follow analphabetical sequence (`B1', `B2', `C1', `C2', `D1', `D2', `A1', `A2') clockwise starting with `B1' atthe 10:30 o'clock position. The minority-type dislocation bundle `MT 1' as well as the majority-typedislocation bundles `B1' and `B2' show `extinction of shadow contrast' in (a). In (b), the majority-type dislocation bundles follow an alphabetical sequence (`C1', `C2', `D1', `D2', `A1', `A2', `B1',`B2') clockwise from `C1' at the 10:30 o'clock position. The region of enhanced X-ray intensity (i.e.the bright stripe) at the right-hand side of (b) is caused by diffusely scattered white X-ray radiationthat passed through the sample mounting at the major ¯at of the sample. The minority-typedislocation bundle `MT 1' as well as the majority-type dislocation bundles `C1' and `C2' show`extinction of shadow contrast' in (b).

Table 1Summary of the parameters of all of the dislocations that belong to thepseudo-symmetric fourfold set of majority-type dislocation bundles [asdenoted (A1±D2) in Fig. 1(a)].

(a) Line directions and Burgers vectors.

Linedirection²

Burgers vector,type 1

Burgers vector,type 2

`Effective' Burgersvector³

A1 [110] �12[011] �1

2[10�1] �1

2[110]A2 [�1�10] �1

2[0�1�1] �1

2[�101] �1

2[�1�10]

B1 [�110] �12[0

�11] �12[10

�1] �12[1

�10]B2 [1�10] �1

2[01�1] �1

2[�101] �1

2[�110]

C1 [�1�10] �12[0

�11] �12[

�10�1] �12[

�1�10]C2 [110] �1

2[01�1] �1

2[101] �12[110]

D1 [1�10] �12[011] �1

2[�10�1] �1

2[�110]

D2 [�110] �12[0

�1�1] �12[101] �1

2[1�10]

(b) Glide planes and peripheral area of origin.

Glideplane

Type of plane accordingto Gatos & Lavine (1960)

Peripheral areasof origin

A1 and A2 (1�11) B [0�10], [100]B1 and B2 (111) A [100], [010]C1 and C2 (�111) B [010], [�100]D1 and D2 (�1�11) A [�100], [0�10]

² Sign follows the assumed glide direction from the wafer rim into the bulk of thewafer. ³ After `pairing up' of dislocations, `effective' Burgers vectors are obtained byformally adding the Burgers vector types 1 and 2 of the constituent dislocations of amajority-type dislocation bundle.

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70 P. MoÈck � Dislocation bundles J. Appl. Cryst. (2001). 34, 65±75

structure or downwards to the back side of the substrate.

Taking into account that the radius of the wafer is about 50

times larger than the sample thickness, the result of the gliding

out of one of the 60� segments could be that the partial half-

loops are headed by screw segments that are, as observed

experimentally, trailed by 60� segments.

The sketches of Figs. 5(a), 5(b) and 5(c) illustrate the origin

and ®nal arrangement of a dislocation pair (as discussed

below) in a majority-type dislocation bundle. Note that the

®nal con®guration is in good agreement with the experimen-

tally observed arrangement, as shown in suf®cient detail in

Figs. 4(a), 4(b) and 4(c).

g

1 cm( )a

( )b 0.5 mm

0.5 mm 0.5 mm

0.5 mm

g g

( )c

g g

(d) ( )e

Figure 4Higher magni®cation images ofdifferent types of dislocation bundlesin various re¯ections. (a) Majority-typedislocation bundles `D2' (left) and `A1'(right); 0�22 re¯ection; X-ray analogueof a photographic positive. (b) Sectionof one branch of a majority-typedislocation bundle `D1' where it endsin the bulk of the wafer; �20�2 re¯ection;X-ray analogue of a photographicnegative. (c) Section of one branch ofa majority-type dislocation bundle ofthe second sample; 1�11 re¯ection; X-rayanalogue of a photographic negative.This image is reproduced for compar-ison with (b). As the comparison of (b)and (c) shows, majority-type dislocationbundles have the same microscopicarrangement in thermally processedGaAs wafers. Since a medium with aby-far superior resolution has been usedfor the recording of this topogram, moredetails of the dislocation arrangementcan be seen, although the dislocationdensity is signi®cantly higher. While thedislocations in the majority-type dislo-cation bundle run horizontally from leftto right, one set of 1

2h110i{111}-typemis®t dislocations runs vertically andthe corresponding horizontal set ofmis®t dislocations is extinct for thisparticular re¯ection. The mis®t disloca-tions appear predominantly in theirkinematical image since they arelocated at a depth of only 140 nm belowthe exit surface of the X-ray beam (seealso MoÈ ck & Smith, 2000). The disloca-tions in the majority-type dislocationbundles, however, appear in their dyna-mical images since they are distributedthroughout the whole thickness of theGaAs wafer (see also AppendicesA andB, and MoÈ ck & Smith, 2000). (d)Section of the ®rst minority-type dislo-cation bundle `MT 1' where it ends inthe bulk of the wafer; 1�11 re¯ection; X-ray analogue of a photographic nega-tive. (e) Area in the 1�11 topogramwhere a minority-type dislocationbundle of the second type comes inclose contact with one branch of themajority-type dislocation bundle `B2';X-ray analogue of a photographicnegative. The majority-type dislocationbundle runs vertically and the minority-type dislocation bundle makes a smallangle with the horizontal edge of the®gure.

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4.2. Dislocation pairs in majority-type dislocation bundles

As Fig. 6(a) reveals, majority-type dislocation bundles show

pronounced contrast in infrared polariscopy maps of residual

shear strains. Although the constituent dislocations of

majority-type dislocation bundles are of the 60� type, i.e.

should, if isolated, possess Burgers vector components parallel

to the wafer normal which would cause surface steps, Fig. 6(b)

shows that majority-type dislocation bundles are not revealed

as surface steps in a Makyoh topogram.

Both experimental observations may be explained by the

existence of dislocation pairs with `effective Burgers vectors'

that relieve shear strains effectively, on the one hand, and

cancel the surface steps of the two constituent 60� dislocations,on the other hand (see also MoÈ ck et al., 1999, 2001; MoÈck &

Smith, 2000; MoÈck, 2000b). The `effective Burgers vectors' of

Table 1(a) are obtained by formally adding the Burgers vector

types of the constituent dislocations of a majority-type dislo-

cation bundle. This formal addition can be performed since it

was concluded in x3.2 that there are about equal numbers of

dislocations with either one of two different Burgers vectors in

each majority-type dislocation bundle. As can be easily seen,

the `effective Burgers vectors' of Table 1(a) do not possess

components parallel to the wafer normal, i.e. cannot give rise

to surface steps, but are most effective in relieving shear

strains between the h100i and h010i axes that are perpendi-

cular to the wafer normal.

J. Appl. Cryst. (2001). 34, 65±75 P. MoÈck � Dislocation bundles 71

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Figure 5Sketches that illustrate the origin and arrangement of a dislocation pair ina majority-type dislocation bundle. (a) Nucleation of two half-loops atsurface inhomogeneities. (b) Formation of a dislocation pair and glidingout of one part of a half-loop. (c) Final arrangement after a part of theother half-loop has glided out and the force on the dislocation pair is nolonger suf®cient to sustain its movement.

↑ ↑[ [- -- -1 110] 10]

MT 2

5 × 10–5

( )a ( )b

Figure 6Complementary experimental results (slightly modi®ed after MoÈ ck et al., 1999, 2001); comparison of a scanning infrared polariscopy map with a Makyohtopogram of the wafer from which Figs. 1, 2, 3, 4(a), 4(b), 4(d) and 4(e) were taken. Except for small mounting and imaging artefacts, the whole two-inch-diameter wafer is depicted in both images. (a) Scanning infrared polariscopy map. Residual shear strains between the h100i directions that areperpendicular to the wafer normal are shown. Residual strain maps of complementary strain tensor components are given by MoÈ ck et al. (1999). Thequantifying marker of the strain is linear. The majority-type dislocation bundles follow in alphabetical order clockwise from `A1' at the 1:30 o'clockposition. Majority-type dislocation bundles and two minority-type dislocation bundles are clearly visible, for only they relieved a suf®cient amount ofshear strain during the thermal treatment of the wafer. Careful examination of the original of this ®gure demonstrated that the minority-type dislocationbundle of the second type, denoted as `MT 2' in (b) and Fig. 1(a), did indeed cause a low level of shear strain relief, but the related contrast has been lostin the reproduction of this image. (b) Makyoh topogram. The surface topography is shown with a height sensitivity of about 50 nm and a spatialresolution of about 1 mm (Laczik et al., 1995). Only those minority-type dislocation bundles that consist of a suf®cient number of dislocations with asigni®cant Burgers vector component parallel to the surface normal are visible. Marker `MT 2' stands for the minority-type dislocation bundle of thesecond type, as in Fig. 1(a). The two minority-type dislocation bundles which show pronounced contrast in (a) are also clearly visible. Since majority-typedislocation bundles do not cause discernable surface steps, they are not visible in a Makyoh topogram. The most obvious imaging artefacts in (a) arebright spots, which are caused by dust particles on the free wafer surfaces. The most obvious imaging artefact in (b) is a circular speck in the secondquadrant, which is caused by a surface inhomogeneity in one of the employed optical elements. Both images are in the same orientation and the major¯at of the wafer, i.e. the [�1�10] direction, points up towards the top of the page. It is assumed that the low dislocation density in the minority-typedislocation bundle of the ®rst type, marker `MT 1' in Fig. 1(a), is the reason why there is no discernable contrast for it in both the scanning infraredpolariscopy map (a) and the Makyoh topogram (b).

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72 P. MoÈck � Dislocation bundles J. Appl. Cryst. (2001). 34, 65±75

As the dislocation density in the

majority-type dislocation bundles is too

high for X-ray topography to resolve either

individual dislocations or dislocation pairs,

the proposed formation of dislocation pairs

is so far only a hypothesis. Theoretical

support for the existence of dislocation

pairs has been obtained, however, from the

semi-quantitative modelling of thermal-

treatment-induced plastic deformation in

GaAs wafers (MoÈ ck, 2000b). In short, the

main theoretical arguments are as given in

the following section.

4.3. Theoretical support for the existenceof dislocation pairs in majority-type dislo-cation bundles

For the strain levels that were actually

encountered during the thermal treatment

of the wafer (MoÈ ck, 2000b), the critical

resolved shear stresses on the eight actually

operational slip systems were calculated, as

given in Table 1, employing the formulae

and material data given by MoÈ ck (2000b).

These calculations predict for the outer-

most margin of a GaAs wafer under the

given thermal treatment conditions four

curves for the eight resolved thermal shear

stresses �1 = �2, �3 = �4, �5 = �6 and �7 = �8,over an angular coordinate with respect to

the [100] direction (see Fig. 7a, where the

subscripts are explained).

As Fig. 7(a) shows, there are only eight

small angular regions where the resolved

thermal shear stresses possess values that

are larger than the experimentally derived

threshold for dislocation generation

(Kiyama et al., 1997) in the applicable

deformation geometry (Reid, 1973). In

other words, the model predicts that there

are only eight angular regions where

minority-type dislocation bundles of the

second type may glide after they have been

nucleated by some mechanism. In good

agreement with Figs. 1, 2, 3, 6(a) and 6(b),

and the X-ray topograms that are

presented by MoÈ ck (2000a) and MoÈck &

Smith (2000), the theoretical prediction is

that these areas range from about 10� to

about 35� around both sides of the h110ipoles. It should be noted, and will be brie¯y

discussed in Appendix C, that Figs. 7(a)

and 7(b) are compatible with the experi-

mentally derived point symmetry group�42m (see also MoÈ ck, 2000b).

Figure 7Calculated resolved thermal shear stresses at the rim of the thermally processed wafer as afunction of an angular coordinate with respect to [100]. Both theoretically predicted graphs areentirely compatible with the experimental results that are presented in Table 1. (a) The resolvedthermal shear stresses on isolated dislocations in all symmetry-related minority-type dislocationbundles of the second type are shown. The slip systems of these minority-type dislocationbundles are as follows: (1) �1

2[0�11](111); (2) �1

2[011](�1�11); (3) �1

2[0�11] (�111); (4) �1

2[011](1�11);

(5) �12[10

�1](111); (6) �12[

�10�1](�1�11); (7) �12[

�10�1](�111); (8) �12[10

�1](1�11). Themarker with the annotation `MT 2' at the angular coordinate value 149� indicates the locationof the respective minority-type dislocation bundle with the same denotation as in Figs. 1(a) and6(b). This marker has been annotated in order to indicate that the symmetry of the model isfourfold, as discussed brie¯y in Appendix C, rather than eightfold. (b) The resolved thermalshear stresses on the four types of dislocation pairs that may be formed in the slip planes (111),(�111), (�1�11) and (1�11) are shown. The respective angular positions of the majority-typedislocation bundles that may result from the formation of dislocation pairs are annotated incorrespondence to Fig. 1(a) by the markers `B1', `B2', `C1', `C2', `D1', `D2', `A1' and `A2'.

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Fig. 7(a) indicates that minority-type dislocation bundles

cannot exist in peripheral areas closer than about �10� to a

h110i pole. This prediction shows that dislocation bundles of

the ®rst minority type are not accounted for by the model,

which assumes a perfectly circular wafer. It is believed,

therefore, that minority-type dislocation bundles of the ®rst

type are the result of the in¯uence of the crystal orientation

¯ats on the distribution of the thermal stresses.

The startling conclusion from Fig. 7(a) is, however, if

dislocation pairs could not exist, a few minority-type disloca-

tion bundles of the second type would be all that is possible to

relax the actually encountered thermal strains in the thermally

processed GaAs wafers. If, however, dislocation pairs are

taken into account, the situation changes completely.

The resolved thermal stresses for the four types of dislo-

cation pairs (P�15 = �1 + �5 =

P�26 = �2 + �6,

P�37 = �3 + �7 =P

�48 = �4 + �8) are, for nearly all angular regions, above the

threshold for dislocation glide (Kiyama et al., 1997). The

generation of dislocation pairs is, however, limited by the pre-

existence of dislocation sources, which are most numerous

around the h100i peripheral areas (MoÈck, 2000b). Taking this

limitation into account, the fourfold symmetry of the model

becomes obvious and the angular range of majority-type

dislocation bundles is, as the good agreement of Fig. 7(b) with

Figs. 1, 2, 3 and 6(a), and the X-ray topograms that are

presented by MoÈ ck (2000a) and MoÈck & Smith (2000) indi-

cate, about 22� around both sides of the h100i poles.

4.4. Discussion of minority-type dislocation bundles on thebasis of both the complementary experimental results and thesemi-quantitative model

Since the symmetry of the crystallographic parameters of

the pseudo-symmetric fourfold set of majority-type disloca-

tion bundles is a consequence of the applicable deformation

geometry (Reid, 1973), it is expected that the crystallographic

parameters of the dislocation bundles of the second minority

type will possess the same symmetry. The crystallographic

parameters of the minority-type dislocation bundles of the

®rst type, i.e. those dislocation bundles that are located at and

opposite to the wafer orientation ¯ats, on the other hand, are

expected to possess a lower symmetry since it is believed that

they are caused by an asymmetric disturbance of the applic-

able deformation geometry.

While the minority-type dislocation bundle of the second

type, which is denoted as `MT 2' in Fig. 1(a), is discernable in

the Makyoh topogram (Fig. 6b), it is only hardly visible in the

original of the scanning infrared polariscopy map of residual

shear stains (original of Fig. 6a). According to the model

(MoÈ ck, 2000b), the dislocations in this bundle should have

Burgers vectors of either �12[101] or �1

2[10�1] or possibly a

combination of both.

The two minority-type dislocation bundles in the [110]

peripheral area (adjacent to both the dislocation bundle of the

®rst minority type, which is denoted as `MT 1', and the two

majority-type dislocation bundles, denoted as `B1' and `B2' in

Fig. 1a) show pronounced contrast in both the scanning

infrared polariscopy map (Fig. 6a) and the Makyoh topogram

(Fig. 6b). According to the model (MoÈ ck, 2000b), the Burgers

vectors of the dislocations in one of the bundles should be

either �12[10

�1] or �12[101], or possibly a combination of both.

The other bundle should consist of dislocations with Burgers

vectors of either�12[0

�11] or�12[011], or possibly a combination

of both.

Burgers vectors of minority-type dislocation bundles of the

second type, such as predicted by the model, could explain the

contrast in Figs. 6(a) and 6(b), would be related to each other

by the point symmetry group �42m, and would be compatible

with most of the experimental data that are contained in the

published X-ray topograms.

As far as dislocation bundles of the minority type are

concerned, obviously more work has to be performed to

elucidate the crystallographic parameters of the dislocations in

these bundles. The present X-ray topography study, however,

delivered a ®rst step towards such investigations by identifying

two different types of minority dislocation bundles.

5. Conclusions

Different types of dislocation bundles have been identi®ed in

a plastically deformed two-inch-diameter GaAs wafer that was

subjected to temperatures of up to about 923 K in a

commercial molecular beam epitaxy machine. The dislocation

bundle types differ in the crystallographic parameters of the

constituting dislocations and in their respective spatial distri-

bution. Dislocation bundles of the majority type form a

pseudo-symmetric fourfold set with h100i peripheral-area

orientation and are signi®cantly longer and much more

numerous than dislocation bundles of the minority types that

exist at and around h110i peripheral areas. The crystal-

lographic parameters of the whole set of majority-type dislo-

cation bundles are related by the point symmetry group �42m.

It is assumed that the constituent dislocations of the majority-

type dislocation bundles nucleated as half-loops at surface

irregularities at the rim of the wafer. The formation of dislo-

cation pairs in majority-type dislocation bundles has been

hypothesized in order to explain complementary experimental

results and is theoretically supported by a semi-quantitative

model of plastic deformation of GaAs wafers under conditions

that are typical for molecular beam epitaxy growth.

APPENDIX AKinds of X-ray transmission topography

The standard method for determining the Burgers vectors of

dislocations, as brie¯y outlined in Appendix B, works best for

isolated dislocations in a nearly perfect crystal of a thickness

(t) that is around 0.88 times the extinction distance of the

employed re¯ections (Tanner, 1996). Under such conditions, a

dislocation appears in its kinematical image on a photographic

positive as a brighter region (enhanced X-ray intensity) of

high contrast on a darker background. The magnitude of a

typical �t product (where � is the linear absorption coef®cient

of the diffracted radiation) at which the kinematical image

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74 P. MoÈck � Dislocation bundles J. Appl. Cryst. (2001). 34, 65±75

dominates the contrast is of the order of 1. X-ray transmission

topography is most commonly performed under such �tconditions and may be called `Lang-type topography'

(Authier, 1970).

There is an `awkward' (Lang, 1970) �t range of about 2±5

which should be avoided for straightforward experimental

determinations of Burgers vectors. X-ray transmission topo-

graphy under conditions of �t products of the order of

magnitude 10 is called anomalous transmission topography

(Authier, 1970), Borrmann-effect topography (Klapper &

Smolsky, 1998), or just Borrmann topography (Hildebrandt &

Wagenfeld, 1998), and will be discussed brie¯y in Appendix B.

APPENDIX BBurgers vector determination by means of X-raytransmission topography under conditions of highanomalous transmission

Since GaAs wafers of 0.45 mm thickness were used to record

the X-ray diffraction topograms, the experimental conditions

dictated working with large �t products. These products are

about 11 for � = 0.13 nm (i.e. for the 111, 1�11, �111, �20�2 and oneof the 0�22 re¯ections), 16 for � = 0.073 nm (i.e. for the other

0�22 and the 202 re¯ections) and about 7 for � = 0.053 nm (i.e.

for the 1�33 and 3�13 re¯ections).

This prevalent high anomalous transmission allowed the

usually much weaker dynamical-effect images of the disloca-

tions in a bundle to dominate the contrast in the transmission

topograms, while the usually much stronger corresponding

kinematical images were practically absorbed. Since the

anomalously high transmission depends strongly on the crys-

talline perfection, lattice defects such as dislocation bundles

reduce the effect and lead to a type of dynamical-effect image

that consists of diminished intensity (darker regions of

medium contrast on a brighter background in a photographic

positive), which is often called a Borrmann shadow (Klapper

& Smolsky, 1998). The sharpness of the Borrmann shadow

depends on the distance of the defect to the free surface where

the X-ray beam exits the crystal. As a direct consequence of

the Borrmann fan (Klapper & Smolsky, 1998), the further

away a dislocation is located from the exit surface, the more

blurred its Borrmann shadow will be.

There is another type of dynamical-effect image, also

known as an intermediate image, which arises from the

curvature of the wave®eld rays in the deformed regions

surrounding a dislocation and which leads to enhanced

intensity in some parts of the image (Klapper & Smolsky,

1998). In the cases of �t products of 10 to 20, the intermediate

image arises mainly from the regions near to the free surfaces

of the crystal (Authier, 1970). For thorough discussions of the

contrast in topograms taken under conditions of high anom-

alous transmission, see reviews by Tanner (1996), Petroff

(1983) and Authier (1970). Particular dislocation contrast

features in single-crystal X-ray transmission topograms

resulting from the use of a second-generation synchrotron

radiation source (such as that at Daresbury) have been

discussed by Bowen & Tanner (1998).

In the particular case of dislocation bundles in GaAs, effects

such as As precipitation on the dislocation cores, splitting of

perfect dislocations into partial dislocations, superposition of

strain ®elds of dislocations that are in close proximity to one

another, and stress relaxation in the vicinity of dislocations

that thread towards the free crystal surfaces, may disturb the

extinction effects of both dynamical images for chosen

re¯ections. Some of these disturbances are expected to

increase with increasing �t.Although there is a signi®cant amount of anomalous

transmission present, it is believed that the set of topograms

depicted in Figs. 2(a), 2(b), 2(c), 3(a) and 3(b) allows the

correct determination of the Burgers vectors (bi) and types

[arccos(bi � li)] of the dislocations that are present in a quarter

subset of the pseudo-symmetric fourfold set of majority-type

dislocation bundles (`A1' and `A2' in Fig. 1a) and in a minority-

type dislocation bundle of the ®rst type (`MT 1' in Fig. 1a). The

Burgers vector determinations were performed by applying to

the Borrmann shadows the standard method, i.e. the well

known relation bi = gj � gk for gj � bi = 0 and gk � bi = 0

[isotropic elasticity, gj � (bi � li) = 0, gk � (bi � li) = 0], where giand gk are the reciprocal-lattice vectors of the re¯ections

employed and li are the line directions of the dislocations (see

e.g. Bowen & Tanner, 1998).

APPENDIX CSymmetry properties of the semi-quantitative model byMoÈck (2000b)

As demonstrated by MoÈ ck (2000b), models that are entirely

based on the angular distribution of the resolved thermal

shear stresses (Kiyama et al., 1997; Yamada et al., 1997;

Sawada et al., 1996) are over-simpli®cations since they disobey

Curie's symmetry principle (Shuvalov, 1988; Pau¯er, 1986;

Curie, 1894). The semi-quantitative model reported herein is

an improvement on the above-mentioned models since it

predicts, in agreement with the experimental results (Table 1),

the point symmetry group �42m for the crystallographic para-

meters of the thermal-treatment-induced plastic deformation.

This improvement has been achieved by including in the

model the naturally occurring distribution of dislocation

sources, dislocation pairs as basic constituents of the majority-

type dislocation bundles, and modi®cations of the stress

distribution during plastic ¯ow which is caused by dislocation

interactions in minority-type dislocation bundles.

As an indication of the fourfold symmetry of the model, the

angular coordinate value (149�) of the minority-type disloca-

tion bundle, denoted as `MT 2' in Figs. 1(a) and 6(b), has been

annotated in Fig. 7(a). This value is 8.5� less than the angular

coordinate value (157.5�) at which the local maximum of the

resolved thermal shear stress curves on the glide systems

�12[

�10�1](�111) and �12[10

�1](1�11) occurs. According to the old

models (Kiyama et al., 1997; Yamada et al., 1997; Sawada et al.,

1996), the dislocation bundle that is denoted as `MT 2' in Figs.

1(a) and 6(b) should exist exactly at this local maximum. In

addition, symmetry-related dislocation bundles should exist at

the angular positions of the other seven maxima of the

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resolved thermal shear stress curves in Fig. 7(a), i.e. at 22.5,

67.5, 112.5, 202.5, 247.5, 292.5 and 337.5�, leading to an

eightfold symmetry. As the comparison with the experimental

results (Fig. 1, 2, 3, 6a and 6b) shows, this is obviously not the

case.

The author is indebted to Dr G. W. Smith from the Defence

Evaluation and Research Agency, Malvern, for growing the

samples of this study, and would like to thank Dr D. Laundy

for advice and experimental support at the synchrotron

radiation source (SRS), Daresbury Laboratory. Previous

collaboration with Drs G. R. Booker, Z. J. Laczik and M.

Fukuzawa, as well as Professors M. Yamada and B. K. Tanner,

is kindly acknowledged. The author is grateful to both referees

for their comments. The Engineering and Physical Science

Research Council sponsored the usage of the experimental

facilities at the SRS in the framework of two `Direct Access'

projects (reference numbers 30047 and 31098).

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