Online available at: http://www.scientific.netCopyright Trans Tech Publications, Switzerland

Dislocation structures in diamond: density-functional based modellingand high resolution electron microscopy

A. T. Blumenau�, T. Frauenheim

�, S. Oberg

�,

B. Willems�

and G. Van Tendeloo�

�Department of Physics, Universitat Paderborn, D-33098 Paderborn, Germany�

Department of Mathematics, University of Lulea, S90187, Lulea, Sweden�Department of Physics-EMAT, University of Antwerp (RUCA), B-2020 Antwerp, Belgium

Keywords: dislocations, diamond, density-functional theory, tight-binding, electron microscopy

Abstract. The core structures of perfect 60�

and edge dislocations in diamond are investigated atom-istically in a density-functional based tight-binding approach, and their dissociation is discussed bothin terms of structure and energy. Furthermore, high resolution electron microscopy is performed ondislocation cores in high-temperature, high-pressure annealed natural brown diamond. HRTEM im-age simulation allows a comparison between theoretically predicted and experimentally observedstructures.

1 Introduction

As in many other crystalline materials, dislocations are common defects in diamond. Chemicallypure type IIa diamond has been reported to contain a dislocation density of up to

����� ��� �and boron

containing type IIb also proves to be dislocation-rich [1, 2, 3]. Natural gem-quality type Ia diamondhowever, where nitrogen is present in an aggregated form, is almost dislocation-free [4]. Also CVD-grown polycrystalline diamond reveals high densities of dislocations — sometimes up to

��� ��� �� � �.

Some originate at the substrate-interface and propagate through the thin film [5], but others lie at ornear grain boundaries [6]. The main slip system in diamond is of the � ����������������� type [7, 8, 9, 10]and in weak-beam electron microscopy, dislocations in type IIa diamond were found to be dissociatedinto glide partials separated by an intrinsic stacking fault ribbon of width 25 – 42 A [10].

These experimental investigations could, however, not resolve the atomistic core structures of thedislocations. An early theoretical study of core structures and their stability was presented by Nand-edkar and Narayan [11] while later ab initio studies were restricted to the 90

�partial glide dislocation

[12, 13, 14]. A more systematic investigation of the core structures and their electronic properties waspresented recently [15, 16] and first core structures could be resolved in high resolution electron mi-croscopy [17]. Related to the latter, the current text focusses on the energetics of dissociation reactionsand on the comparison of theoretical core structures with experimental high resolution images.

The text is organised as follows: In section 2 the basics of the density-functional tight-bindingapproach are briefly introduced. This is followed by results from linear elasticity theory in section 3,which will be of importance later in the text. Section 4 gives an overview over basic dislocationtypes discussed in this work. In Sections 5 and 6 the low energy dislocation structures, partial dislo-cations and the dissociation of the 60

�and the edge dislocation are modelled. High resolution electron

miocroscopy images of dislocation cores are presented in 7 and the experimental images are comparedwith simulated images based on the theoretical models from preceding sections.

2

2 The theoretical method

To model a ceramic / semiconductor crystal on the atomic scale, a wide variety of classical methodsis known — ranging from simple ball and spring models to more sophisticated empirical interatomicpotentials with two- or even three-body contributions [18, 19, 20, 21]. These methods are more or lesssuccessful in describing perfect and strained crystals of tetrahedrally bonded semiconductors. How-ever, when it comes to defects with considerable deviations from the standard bonding configuration(often full four-fold coordination of atoms and tetrahedral bond angles), then quantum mechanical ef-fects, or effects involving charge transfer between atoms, might dominate. Very often this is the casein the very core region of point or line defects. Hence here the use of quantum mechanics is essential.

However, since the modelling of defects involves tens or even hundreds of atoms, the full many-body quantum mechanical wavefunction cannot be solved. The method of choice to reduce the num-ber of parameters is given with density functional theory (DFT), where the electron density is thecentral variable. This implies a reduction from the ��� coordinates of the electron wavefunction toonly three coordinates (if we ignore spin-polarisation), as shown in the next section. The reader in-terested in a more detailed and more exact description may read the original work by Hohenberg andKohn [22] and Kohn and Sham [23] or the rather mathematical treatment of Lieb [24], to give just afew examples.

2.1 A density-functional based tight-binding approach: We will now introduce density-func-tional based tight-binding (DFTB) — the method used in this work to approximately solve the Kohn-Sham equations of DFT. The description will be introductory and lack many details. For furtherinsight the original work of Seifert et al. [25] and Porezag et al. [26, 27] or the review in [28] arerecommended. A compact but nevertheless detailed overview including a sketch of the historicaldevelopment can be found in [29].

To understand the basic idea of the DFTB method, one best starts with the explicit form of thetotal energy in DFT after introducing Kohn-Sham orbitals ������� :�� ������������� � � �������� ��� �"!$# �%'& �% � � ���)(*+ � !,� ( +.- �0/ ( &2143�576 ��8:9;�<�=��8 - �>/ &@?;5 � ���� (1)

Here 143A5�6 is the external potential of the ion cores and ?B5 � the exchange correlation functional. Addi-tionally normalised occupation numbers

� � following Janak et al. [30] have been introduced:� ��8 �C�7���� � � � + �<�����D + �FE � �C�7���� � � � (2)

These occupation numbers� � contain the information how many electrons occupy the respective

Kohn-Sham orbital. In a spin-restricted formalism as used here, the maximum occupation per orbitalis 2.

The idea is now to simplify Eq. (1) by applying a Taylor-expansion around a reference den-sity

��G ���8 and neglecting higher order contributions. Introducing H � ���8 �I� ��8J! �FG ���8 we obtain(Foulkes and Haydock [31]):�� ����K� ������ � � � � � �� ��8�LM N��GO� �P����� - � / ! �% �Q� ��G ��8 ��G ��F(R+ �B!S� ( + - � / ( - � /

&T?;5 � ���GO� ! � 145 � N��G ��8 �U��G ��8 - � /& �% �Q�'� �+ �V!,� ( + &XW � ?;5 � ����W � � 9ZY\[ H � ��8�H � �� ( - � / ( - � / &^] �_H � � (3)

3

Here LM ���� is the Hamiltonian of the Kohn-Sham equations of DFT [23] including the effective poten-tial. The first and second lines give the zero-th order terms in H � and the third line gives the secondand third order. All first order contributions cancel.

Choosing a localised (atom-centred) set of basis functions, the second to fifth terms in Eq. (3)can be expressed as a sum of two-centre integrals4. Subsuming those terms as ? � � 3��=6 and neglectingthree-centre and higher-order contributions yields the total energy in DFTB:�� ������ �7��� � � � � � �� ���� LM ���GO� �<�=���D - � / & ? � � 3��=6 � �������

(4)

DFT calculations show, that the density of an atom surrounded by other atoms in a molecule or asolid is slightly compressed compared to the density of a free atom. Hence in DFTB the startingdensity

��G ��8 is expressed as a superposition of weakly confined neutral atoms, so called pseudo-atoms5. In all DFTB implementations used in this work, the respective pseudo-atomic wavefunctionsare represented by Slater-type orbitals �� Q��� .

Expanding the Kohn-Sham orbitals of Eq. (4) into the atom-centred Slater-type orbitals �� Z���yields:�P�:��8 � � � >���� Z���� (5)

This resembles an LCAO basis (linear combination of atomic orbitals ). Further we approximate thetwo-centre contributions by the sum of short-ranged repulsive pair potentials 1 ���� 3�� :

? � � 3��=6 � �% � ���� � 1 ���� 3�� � + � ��! � � + � ? � 3�� � � � � � (6)

Inserting Eq. (5) and (6) in (4) gives the DFTB energy in terms of the expansion coefficients� >� .

Subsequent variation with respect to the expansion coefficients and subject to normalisation yieldsthe DFTB secular equations:�

� �� � M G� !��4��� � ! � �(7)

If� denotes the atom site the orbital is centred at, then the Hamiltonian and the overlap matrix are

given as:

M � � "##$##%

�'& 6 �)(� * + �-,� � �� ��� LM N��GO� �� ���8 - � / * � /.� � �� * else

(8)

� � � � � �� ����0�� ���8 - � / (9)

Notice, that the diagonal elements ofM � are taken to be atomic orbital energies of a free atom, to en-

sure the right energies for dissociated molecules. Once the pseudo-atom orbitals have been calculated

4For the exchange-correlation terms this only holds approximately.5These pseudo-atom densities result from self-consistent DFT calculations with an additional weak parabolic constric-

tion potential.Just as a note for those familiar with the internal details of the DFTB method: In this work the diamond C–C interactionincludes the superposition of the potentials of the pseudo-atoms. These potentials are compressed by an additional term13254628769�:

with a compression radius of207<;>=@? ACB

A.

4

self-consistently in DFT, the non-diagonal elements of the HamiltonianM � and the overlap matrix� � can be tabulated as a function of distance between the two centres. Thus both matrices can be

calculated in advance. Once these tables have been generated, Eq. (7) can be solved straight away andnon self-consistently for any given coordinates of the nuclei. The DFTB energy then becomes:������� ���7��� � � � �4� &@? � 3�� � � � � � (10)

Apart from the repulsive energy, the total energy is now determined. Assuming approximate transfer-ability of the repulsive potentials 1 � �� 3�� � + � � ! � � + , they are obtained in comparison with self-consistentDFT calculations for selected reference systems (small molecules or infinite crystals): Varying onedistance

+ � �F! � � + the respective repulsive potential is tabulated just like the matrix elementsM �

and � � . In practice, the repulsive pair potentials also contain the respective ion-ion repulsion. Further,the compensating effect of the repulsive potential allows one to use a minimal basis of Slater-type or-bitals, reducing matrix size and thus speeding up all calculations. Further, as has been just shown, allDFT integrals can be calculated in advance. In the calculation of the eigenvalues �Q� this saves a greatdeal of computational effort and speeds up the calculation dramatically.

For some systems with strong ionic bonding character the approximations of standard DFTBmight fail. Especially the second order term in Eq. (3) becomes too large to be transferable and simplytabulated within the repulsive potential. The term has then to be treated self-consistently as explainedin Ref. [28]. In this work however, the self-consistent-charge extension is not used, as test calculationshave shown no considerable improvement for the systems investigated.

2.2 Forces and structural optimisation: With the total energy at hand, one can search for theequilibria of forces in an atomistic model — the local minima of the energy surface. This structuraloptimisation is usually done with the help of algorithms involving atomic forces. The latter can beobtained via the Hellmann-Feynman theorem [32, 33]. Since in this work atom-centred basis sets areused, the so called Pulay corrections have to be applied [34]; and structures are geometrically opti-mised (“relaxed”) using a conjugate gradient algorithm until all forces are well below

������� � �eV/A.

3 Straight dislocations in linear elasticity theory

Linear elasticity theory proves to be a very useful tool to describe the long-range elastic strain ef-fects of dislocations. In the following we will discuss the elastic strain energy of an isolated straightdislocation and the interaction forces between two straight dislocations.

3.1 The elastic strain energy of a straight dislocation: The strain energy of an infinite straightdislocation in an otherwise perfect crystal can be calculated analytically using linear elasticity theory.One writes the energy per unit length of a dislocation, contained in a cylinder of radius � and length�

, as? ��� � ��� � + ��+ ��� ��� � �� ��� & ? ���� ����� � � (11)

where is the angle between�

and the line direction � of the dislocation, � � is the core radiusand ? ��� � the core energy per unit length [9]. The energy factor � �� depends on and the elasticproperties of the material. Assuming an isotropic medium, � � can be evaluated as� � � +�� ! #" � & "�$ � � � ! ,&% (12)

5

ln( / )R

E

c

c

0

E

R

Rc

R

Fig. 1: The elastic strain energy of a dislocation integrated in a surrounding cylinder. The � -axis isscaled logarithmically.

with + being the shear modulus and,

the Poisson’s ratio.The logarithmic term in Eq. (11) diverges for both the limits ��� �

and ��� � . Continuumtheory supposes that the atomic displacements vary slowly over the dimensions of a unit cell and thisbreaks down at the core resulting in the divergence as ��� �

. Also, for � of the order of magnitudeof interatomic distances, continuum theory cannot describe the discrete system correctly. As a result,below a certain radius Eq. (11) does not give a good description of the elastic energy: The core radius� � is defined by the condition that (11) ceases to be applicable for radii ����� � . In other words, thedislocation core is the minimum region which cannot be described by elasticity theory and therefore,discrete (atomistic) models are necessary to evaluate the core energy ? � . If the core energy is known,the elastic strain energy for ����� � can be plotted as shown in Fig. 1.

As already mentioned, Eq. (11) also diverges for ��� � . Consequently, in an infinite crystal wecannot evaluate a finite total elastic energy of a dislocation, but only its core energy, the energy factor� �� and core radius which together describe the variation of the strain energy with � .

3.2 The elastic interaction between two straight dislocations: Let us consider two straight andparallel dislocations A and B of arbitrary Burgers vectors

��and

� in an infinite and isotropic

crystal. Then a somewhat involved calculation yields the elastic interaction energy per unit length:? � � � � � ! +% � ��� � �Z � � � �Z & � ��� � � � � � � �Z� ! , � � � � �� G �! +% � � ! , � � � ��� � � � � � � � � �Z � � � (13)

Here�

is the distance vector between the two dislocations and � its length. � gives the line direc-tion (

+ � + � �). Eq. (13), which was first developed by Nabarro [35], allows the calculation of the

interaction energy except for a constant shift � � � ��� G � ˚� . Similar to the core energy in Eq. (11),this shift cannot be determined in linear elasticity theory. However, this shift does not influence theelastic force between the two dislocations. By differentiation we obtain the radial component of theinteraction force per unit length:� � ��� � � !��� � � ? � � � � % � +% � � ����� � � � � � & � ��� � �Z � � � � � � ! , � (14)

In analogy to this, the angular component can be derived by differentiation ! � � � � � � ��� perpendic-ular to

�. Eq. (14) is very useful in the calculation of equilibrium stacking fault widths.

6

cA

aB

bC

cA

aB

shuffleglide

[112]

[111

]

(111)

a0

a3

a2

a1

Fig. 2: Unit cell of a tetrahedrally bonded cubic semiconductor (zincblende). The structure can beconstructed as a face-centred cubic lattice with a two-atom basis. Left: The cubic unit cell. The (111)plane, the plane with the highest density of lattice sites, is drawn and the minimum lattice translationswithin this plane are shown as dashed lines. Right: The cell embedded in the crystal, looking along ��� ��� �

. The (111) planes lie horizontally in this projection. The stacking sequence of (111) planes isshown and an example of a glide plane and a shuffle plane is indicated.

4 Basic dislocation types in diamond

In the process of plastic deformation one part of the crystal might be sheared macroscopically withrespect to the other part. This so-called crystal slip usually occurs on specific crystallographic planesonly. This can be explained microscopically: Crystal slip involves the formation and propagationof dislocations [9]. Following Eq. (11), those dislocations which have minimum Burgers vectors areeasiest to form. Since a Burgers vector (and also the local line direction) is always given as a linearcombination of lattice translations, dislocations are preferentially formed on planes containing theminimum lattice translations. These planes are the preferred slip planes. A slip plane contains boththe Burgers vector and the line direction of the generated dislocations. Hence the plane of crystal slipis identical with the glide plane of the dislocations involved in the process.

4.1 Perfect dislocations of the � ������� � ����� � slip system in fcc lattices: In the cubic structure theplanes of highest lattice site density are the � 111

�planes. Fig. 2 (left) shows one specific plane of that

family. These planes contain the minimum lattice translations and are thus preferred slip planes. Theslip system is defined by the plane and the direction of slip, conventionally written as � ������� ������� � .For each plane we have three possible directions. Specifically in the (111) plane the minimum latticetranslations (Burgers vectors) are

�� ��� ��� � , �� � ���� � and�� ���� � � . They are drawn as dashed lines in Fig. 2

(left). Basic dislocation types in this slip system are:

� The screw dislocation with line direction and Burgers vector parallel, e.g.��� � ��� ��� �and

��� �� ��� ��� � .

� The 60�

dislocation, where line direction and Burgers vector form an angle of 60�, e.g.��� G;� ��� ��� � and

��G � �� ��� ��� � .

� The edge dislocation with line direction and Burgers vector perpendicular, e.g.� 3 � � � � �4% � and� 3 � �� � ����� � .

As will be shown later, locally this structure is nothing else but a zig-zagged 60�

dislocation.

7

[110

]

[112]

bulk

ISFo o’

Fig. 3: The offset between bulk andfaulted region (ISF) in the glideplane.

4.2 Partial dislocations and dissociation: The preferredslip planes mentioned in the last section are also the planesstacked with the widest separation. As shown in Fig. 2 (right),in an fcc lattice the stacking sequence of these planes is� � ��������� �� + ������ ��� � � �

, where capital and small letters in-dicate planes belonging to one of the two sub-lattices respec-tively. Faults in this stacking sequence are very common planardefects. If the normal sequence is maintained on both sides ofthe fault, then we speak of an intrinsic stacking fault (ISF), oth-erwise of an extrinsic fault. In this work only intrinsic faults areconsidered. In the ISF plane atoms are displaced with respectto the bulk plane in normal stacking. Fig. 3 shows the atom po-sitions of the displaced species in the ISF plane and in the corresponding bulk plane. The offset is � �

� ���% � �

or equivalently�� �����% �

and�� �% ��� �

.Let us now assume a dislocation, which is not surrounded by ideal lattice only, but which borders

an intrinsic stacking fault in its glide plane. This type of dislocation is called Shockley partial disloca-tion [9]. The minimum Burgers vector of such a dislocation is given as the offset vector between thefaulted and unfaulted region of the glide plane. Fig. 4 shows the resulting principal Burgers vectors.The two partials of the 60

�dislocation are:

� The 30�

Shockley partial: e.g. � � G�� � � ��� � and� � G � �

� ���% � �

� The 90�

Shockley partial: e.g. � � G�� � � ��� � and��� G � �

� � � � �4% �

And of the edge dislocation:

� The 60�

Shockley partial: e.g. � 3 � � � � �.% � and��G � �

� � �4% � � �

This means a perfect dislocation can dissociate into its two partials, separated by a stacking fault(Fig. 4 (right)).

ISF

``

b60

b60

beb90

b30

b60 b60

e

b60

bulk ISF

bulkISF

be

60

Fig. 4: The dissociation of perfect dislocations into partials. Left: The Burgers vectors of the perfect60

�dislocation and of the 30

�and 90

�Shockley partials in the (111) glide plane. The vectors of

the partial dislocations are given as the offset vectors between faulted and unfaulted region of theglide plane (compare Fig. 3). The perfect 60

�dislocation can be dissociated into the two Shockley

partials:��G�� � � G & ��� G . Middle: The Burgers vectors of the perfect edge dislocation and of two 60

�Shockley partials in the (111) glide plane. Compared to the figure on the left, here the glide plane isrotated by 90

�to the left. Right: Schematic sketch of the corresponding dissociation reaction of the

edge dislocation in the glide plane.

8

5 Modelling the 60�

dislocation and its dissociation

This section will discuss the atomistic modelling of the 60�

dislocation. The geometry of the modelapplied, the low energy core structures of the dislocation itself and of its partials, and the dissociationof the dislocation are covered. For more details see Ref. [15].

5.1 The atomistic hybrid model: Generally there are two main approaches used to model defectsin semiconductors atomistically: The cluster and the supercell approach. For the case of dislocationsthis means that either an atom cluster containing a single dislocation is considered or a supercellcontaining a dislocation multipole. The long range elastic effects are treated differently in each casebut neither approach treats these effects rigorously as explained in Ref. [15]: In a pure cluster surfaceeffects often are considerable and in particular the regions where the dislocation intersects the surfacepose a problem as here the dislocation core will be heavily distorted. In a pure supercell the interactionbetween the dislocations within the cell and across the cell boundaries with their periodic imagescannot be neglected. Often, if the separation between the dislocations is too small, even the corestructure gets distorted. The effect of the latter on the total energy is impredictable.

Fig. 5: Schematic sketch of a dislo-cation in a supercell-cluster hybridmodel.

To avoid the disadvantages of both the cluster and the super-cell, both can be combined into a hybrid: Here the dislocationis placed in a model which is periodic along the dislocationline, however, it is non-periodic with a hydrogen-terminatedsurface perpendicular to the line direction [36, 37]. This allowsto maintain the ‘natural’ dislocation periodicity as a line defectand at the same time avoid the interactions between dislocationsin different cells. The single dislocation in the hybrid model issurrounded by twice the amount of bulk crystal compared tothe pure supercell containing a dipole — assuming both mod-els are of the same volume and neglecting surface effects. Inother words, in the hybrid model it is by far easier to give a good representation of the surroundingbulk crystal. The latter in combination with the capability of modelling a single and isolated dis-location (avoiding dislocation–dislocation interaction) makes the hybrid model the ideal choice todescribe line defects. The supercell-cluster hybrids used in this work are usually of double-periodlength having an approximate radius of three or more lattice constants.

5.2 Calculating core energies: In the supercell-hybrid approach explained above, as a first guess,the stability of different core structures can be compared by simply comparing the total DFTB en-ergies of the relaxed hybrid models. However, as these models are not embedded in infinite bulk,their surface will respond slightly different to the stress induced. In particular volume expansion orcontraction can vary considerably for different core structures with the same Burgers vector, giving ar-tificial energy differences between the models. Therefore, in this work the differences in core energies? � � � (see Eq. (11)) is calculated directly. As described in detail in Ref. [15], ? ��� � can be obtainedby comparing the elastic energy as given in Eq. (11) with the DFTB formation energy contained inconcentric cylinders around the dislocation core in the hybrid model: From the core radius onwards,in an ? ��� vs. ��� ��� � A plot this formation energy will follow the gradient described by Eq. (11) andthe offset at the core radius yields the core energy (see Fig. 1). To obtain relative energies betweendifferent core structures, one common radius

�� � � � has to be chosen for all cores, and the energiescontained in these cylinders are compared. For the 60

�dislocation and its partials

�� ���A appears

to be sufficient.

9

5.3 Core structures of the 60�

dislocation: One can construct atomistic models of the corre-sponding dislocations by displacing the atoms of an appropriate supercell-cluster hybrid accordingto the Burgers vectors

�and line directions � given in the last sections. For the 60

�dislocation, an

additional half-plane of atoms has to be introduced. Depending on the termination of this half-plane,on a shuffle or a glide � 111

�plane (see Fig. 2 (right)), we speak of a shuffle or a glide dislocation

respectively. Hence�

and � do not uniquely define the dislocation.The so obtained dislocation core structures, are nothing more but a mere first guess. It can be

energetically favourable if the core reconstructs by forming new bonds and possibly breaking existingones. Since core reconstructions can be rather complicated, it is not guaranteed to find the overalllowest energy structures by simply relaxing one starting structure by means of a conjugate gradientalgorithm. The reconstruction found then might just be one of many local minima of the correspondingenergy-surface. Thus, in this work usually several different starting structures for a given combinationof�

and � are structurally optimised. The low energy core structures obtained by means of the DFTBmethod are shown in Fig. 6.

In the following, each structure will be described and discussed briefly:

� The 60�

dislocation (Fig. 6 (a,b)): This dislocation is obtained by inserting (or removing) a� ������� half-plane of atoms. Fig. 6 shows the half-plane for the shuffle and the glide cases framedwith lines.

The relaxed core structure of the glide dislocation shows bond reconstruction of the terminatingatoms of the half-plane with neighbouring atoms. Reconstruction bonds along the dislocationline impose a double-period structure.

The shuffle dislocation does not reconstruct. The terminating atoms possess dangling bondsnormal to the glide plane which give rise to electronic gap states [16]. With an energy differenceof 630 meV/A the shuffle dislocation is found to be less stable than the glide structure.

� The glide set of Shockley partials (Fig. 6 (c–e)): In the glide set of partial dislocations eachdislocation is accompanied by a stacking fault.

The 30�

glide partial reconstructs forming a line of bonded atom pairs. The reconstructionbonds are 18 % stretched compared to bulk diamond and the structure is double-periodic.

For the 90�

glide partial there are two principal structures: A single-period (SP) and a double-period (DP) reconstruction. The SP structure results simply from forming bonds in the glideplane connecting the stacking fault region with the bulk region. These bonds are 13 % stretched.The DP structure can be obtained from the SP structure by introducing alternating kinks. Herethe deviations from the bulk bond length is slightly less. The DP structure was first proposed forsilicon by Bennetto et al. [38]. All atoms in both the SP and the DP structure are fully four-foldcoordinated.

In our approach the DP structure is found 170 meV/A lower in energy than the SP structure.This is in agreement with Nunes et al. [13], who find the DP core to be 235 meV/A lower inenergy in DFT calculations. Similarly, the low-stress quadrupole calculations of Blase et al. [14]yield 169 – 198 meV/A.

� The vacancy structure of the 90�

glide partial (Fig. 6 (f)) was found to be the most stable shufflepartial [15]. It appears to be symmetric in the glide plane and a line of bonded dimers is formed,leaving the dislocation with a double-period. The bonds of the dimers to neighbouring atoms inthe glide plane, however, are weak and 22 % stretched, and one row of dangling bonds remainsunreconstructed.

10

(f) 90° shuffle (V)

(a) 60° glide (b) 60° shuffle (c) 30° glide

(d) 90° glide (SP) (e) 90° glide (DP)

[111

]

[112]

[110

][1

11]

[110

]

[112]

Fig. 6: Dislocation core structures of the 60�

dislocations in diamond. The relaxed core structuresof (a),(b) the undissociated 60

�dislocation and (c) – (e) the glide set of Shockley partials and the

lowest energy shuffle partial are shown. For each structure, the upper figure shows the view along thedislocation line projected into the � � � ��� plane, and the lower figure shows the � ����� glide plane. Theregion of the intrinsic stacking fault accompanying the partials is shaded in the respective structures.In case of the 90

�partial SP and DP denote the single-period and double-period core reconstruction,

and (V) for the 90�

shuffle partial indicates its character as a vacancy structure.

11

(eV

/Å)

E/L

(1)(2)

(3)(4)

ISF width (Å)

680 meV/Å

180 meV/Å(0)0

−0.4

−0.8

0.4

0 10 20 30

Fig. 7: The dissociation energy of the 60�

glide dislocation. The relative atomistic DFTB energiesof the first four dissociation steps and of the undissociated dislocation are labelled (1) to (4) and (0)respectively. Zero energy is set to the undissociated dislocation. The solid line represents the sum ofthe stacking fault energy and the elastic interaction energy as given in continuum theory (Eq. (13)).

5.4 The dissociation energetics of the 60�

glide dislocation: When it comes to the dissociationof perfect dislocations into Shockley partials, then the two competing energy contributions in theelastic limit are the stacking fault energy and the elastic partial–partial interaction energy. The firstgrows proportionally with the partial–partial separation � (energy per unit length: ?���� � � � ��� �with a constant stacking fault energy

�) and the interaction energy lessens logarithmically with �

(Eq. (13)). Unfortunately the offset � ��� �� G � ˚� of the elastic interaction energy is unknown and

consequently a complete energy balance in elasticity theory is impossible. In equilibrium however,the force

����� � � � ! � resulting from the stacking fault and the partial–partial repulsion

� � � � , asgiven in Eq. (14), cancel each other and one easily obtains the equilibrium partial separation:

��G � +� � �

� ! , + � � G +.+ ��� G + (15)

With the DFTB elastic constants + � �#� �GPa and

, � ��� �and the DFTB stacking fault energy� � %�� � mJ/m

�this expression evaluates to

��G � � � A. Consistently, dissociation in the � ����� ��������� �

slip system of diamond has been reported in weak-beam electron microscopy [10]. The dissociationwidths vary from 25 to 42 A.

To obtain the absolute dissociation energy, the offset in the elastic expression can be calculated inatomistic models. The model used here for the dissociation of the 60

�glide dislocation contains about

600 carbon atoms. Without the core regions of the partials getting too close to the hybrid’s surface,this size allows dissociation up to the fourth step — which corresponds to a stacking fault width of8.75 A. Fig. 7 shows the discrete relative energies for the undissociated dislocation and the first fourdissociation distances. With the atomistic results at hand, the unknown energy offset � � � ��� G � ˚

� forthe elastic interaction energy ? � � � in Eq. (13) can be found easily by adjusting ? � � � & ? ��� � � �to the atomistic DFTB energy of the fourth dissociation step, for details see Ref. [17]. The procedureyields � G � � � �

A. In Fig. 7 the resulting continuous energy are drawn as a solid line. One can ob-serve that the atomistic calculation and continuum theory still agree for the third stage of dissociation.For smaller stacking fault widths in the region of overlapping dislocation core radii, however, the de-viation becomes obvious and finally for ��� �

the continuum result diverges. The final dissociationenergy at equilibrium separation between the partials is clearly below the energy of the undissociated

12

[111

]

glide shuffle

Fig. 8: The shuffle and the glide structure of the perfect edge dislocation. The view is approximatelyalong the

� � � �4% �dislocation line direction.

SB

[112

]

[110]

b

Fig. 9: The undissociated edge glide dislocation in the glide plane. Left: The core reconstruction, thesymmetry breaking bond is labelled LB. Right: The Burgers vector and the zig-zagged line directionindicate the local 60

�character.

dislocation ( ! ��� � �� eV/A). Hence dissociation into partial dislocations is strongly favoured6. How-ever, an interesting feature is the presence of an energy barrier of �

� ��meV/A to initiate dissociation

of the 60�

dislocation. This barrier should have a considerable effect, since it is approximately 5 Awide and cannot be overcome in a single step. This might explain the presence of undissociated 60

�dislocations as observed in weak-beam electron microscopy [10] and in HRTEM [17].

6 Modelling the edge dislocation and its dissociation

Just as the 60�

dislocation, the edge dislocation may exist in a shuffle structure, where the insertedhalf plane of atoms terminates between two widely separated � 111

�planes, and the glide structure,

where the half plane terminates between two closely separated � 111�

planes (see Fig. 8). In a 546carbon atom hybrid model we find the shuffle dislocation to be �

� � eV/A higher in line energy.

This is close to the value found for the perfect 60�

dislocation ( ���� �

eV/A) [15]. As can be seen inFig. 9, the reconstruction of the glide edge dislocation resembles, at least locally, that of a zig-zagged

6In the case of the dissociation of the 60�

glide dislocation only the single-period 90�

partial is considered here. Theenergy of the respective systems containing a double-period 90

�instead would be approximately 170 meV/A lower.

13

infinite separationnarrow fault

Fig. 10: The 60�

partial dislocation in the glide plane. The faulted area is shaded. Left: Two closelyseparated partials with a narrow stacking fault. Right: An isolated partial.

(1)(2)

(3)

������������� ������

�����������

�! "#$!%

E/L

?(0)

0 10 20 30

0

0.2

0.4

&('*),+

0.6 Fig. 11: The dissociation en-ergy of the edge glide dislo-cation. The relative atomisticDFTB energies for the firstthree dissociation steps andthe undissociated dislocationare labelled (1) to (3) and (0)respectively. The equilibriumdissociation distance is indi-cated, but no further conclu-sion concerning the energy ����

A can be drawn (see text).

/ staggered perfect 60�

glide dislocation. The symmetry of this structure is broken by a reconstructionbond along

� � � �4% �(labelled SB in Fig. 9), resulting in a double period reconstruction.

As shown in section 4.2, the edge dislocation can dissociate into two 60�

partial dislocationsenclosing a stacking fault. The equilibrium width of the fault evaltuates to 36.5 A (DFTB elasticconstants and stacking fault energy, compare last section). However, atomistic modelling shows asteep 0.5 eV/A increase in line energy for stacking fault widths up to 10 A (see Fig. 11) — the barrierto dissociation appears to be much wider and higher than in the case of the 60

�dislocation in the last

section.

Analysis of the stacking fault for small width reveals the reason for this behaviour: The strain fromthe bordering partials induces a strong, almost sp

�-like distortion into the fault (see Fig. 10 (left)).

This distortion allows one partial to form strong reconstruction bonds and thus lower its energy, whilethe other remains only weakly reconstructed (left and right partial in Fig. 10 (left)) respectively). Ofcourse the energy of the distorted fault is rather high. From about 10 A stacking fault width onwardsthe configuration including one low and one high energy partial plus a high energy fault becomesenergetically less favourable compared to two high energy partials plus a low energy fault. There-fore above 10 A the system undergoes a phase transition and adopts the latter configuration. Due tocurrent computational restrictions in model size, these effects could not be quantified energeticallyyet. However, simulation indicates, that the weakly reconstructed partials are extremely mobile: Oncedissociation proceeds beyond 10 A, the equilibrium widths will be easily and quickly attained.

14

10 Å

30°90°

[112]

[111

]

Fig. 12: HRTEM image of a dissociated 60�

dislocation in natural brown type IIa diamond. The twoBurgers circuits around the 90

�(left) and 30

�(right) partial are drawn projected into the � ��� ��� plane,

yielding a projected Burgers vector of�� � � � �4% �

and���� � � � �.% � respectively. The glide plane is marked by

two small arrows.

7 Electron microscopy

The 60�

and screw dislocations in diamond and their dissociation have been observed early on inweak-beam transmission electron microscopy [10]). These early studies cannot yield any informationconcerning the atomic core structure of dislocations. However, high resolution imaging nowadaysallows to visualise the atomic structure of dislocations in diamond to some extent.

7.1 Experimental setup and sample preparation: For the high resolution transmission electronmicroscopy (HRTEM) investigation of dislocation core structures, a JEOL 4000EX microscope with aScherzer resolution of 0.17 nm was used. The samples of brown natural type IIa diamond were treatedat varying high-pressure, high-temperature (HPHT) annealing conditions. From these samples 2 mm�

2mm�

100 + m diamond slabs were produced and subsequently thinned by ion-beam milling toelectron transparency. All samples were oriented with a

� ���� �axis parallel to the electron beam.

Hence, if the dislocation is well aligned along a� ������

viewing direction the contrast produced inHRTEM enables us to resolve the atomic structure of the dislocation core and compare with thetheoretically predicted dislocation core structures [39].

7.2 The dissociation of the 60�

dislocation: Fig. 12 depicts a HRTEM image of a dissociated 60�

dislocation in diamond. Assuming both dislocations in Fig. 12 are minimum Burgers vector disloca-tions, the projected Burgers vector allows an easy identification of the left one as a 90

�and the right

as a 30�

Shockley partial. However, the contrast pattern does not identify single atoms. Also an exactlocalisation of the two dislocation cores is impossible. The latter is possibly a result of the dislocationnot being straight through the whole thickness of the layer, but kinked back and forth. This then givesa rather diffuse image of the core region. The intrinsic stacking fault in between the two partials canbe identified clearly though. The stacking fault is at least 35 A wide, but might be considerably larger.Given the rather flat energy minimum for the predicted equilibrium width of 35 A in Fig. 7, such vari-ation seems reasonable — especially since theory predicts considerable barriers between two adjacentPeierls valleys [17].

15

Fig. 13: Simulated HRTEM image of the 30�

glide partial with a defocus H�� � ! � � nm and a samplethickness of 4 nm. Atom positions are indicated as circles.

(a) 60° glide (b) 60° shuffle (c) 30° glide

(d) 90° glide (SP) (e) 90° glide (DP) (f) 90° shuffle (V)

Fig. 14: Simulated HRTEM images of low energy dislocation core structures with a defocus H�� �! � � nm and a sample thickness of 4 nm. The input atom coordinates for the simulations are the re-laxed atom positions of the respective hybrid model. Fig. 6 shows the corresponding atomic structuresin the same orientation. Each image is approximately centred on the dislocation core line and the (111)glide plane lies horizontally.

7.3 HRTEM image simulation: As the observed contrast patterns do not directly depict the atompositions, the interpretation of HRTEM images is much easier and safer in comparison with simulatedimages based on atomistic models. Image simulation can be performed using Bloch waves, or in amulti-slice approach as described by Moodie et al. [40]: The atomistic model is divided into slicesperpendicular to the incident beam. For each slice the crystal potential is projected onto a plane. The

16

propagation of the beam through the sample is then described as a succession of scattering eventsof the incident wavefront at the planes representing the slices with intermediate propagation throughvacuum between the slices. The rather complicated scattering theory involved is described in detailin Ref. [41]. The subsequent objective lens is modelled by a Fourier transform. In this procedure, thedefocus caused by the imperfect objective lens can be described by a so called phase contrast transferfunction. This function is specific for each microscope and has to be determined experimentally byimaging the same area of a sample with varying defocus.

Fig. 13 shows a simulated HRTEM contrast pattern for a given defocus and sample thickness gen-erated with the multi-slice method using the commercial Crystal Kit

���and MacTempas

���software

packages. The dislocation is the 30�

glide partial as shown in Fig. 6 (c). The input atom coordinatesfor the simulations are the relaxed atom positions of the corresponding atomistic model. Simulatedimages like these can now be compared with experimental images to identify atomic core structures.

Fig. 14 gives the simulated images for the 60�

dislocation and all low energy Shockley partials 7.The glide and shuffle type of the 60

�dislocation appear to be rather distinct in their HRTEM image.

Also the vacancy shuffle structure of the 90�

partial can be easily distinguished from the two glidestructures. However, the difference between the single and double-period reconstruction of the 90

�glide partial is miniscule — at least they can hardly be distinguished by eye. An effect common to allsimulated images is the bending of the (111) plane. The larger the edge component of the dislocation,the larger the effect. This bending is a result of the limited size of the hybrid models: The insertedmaterial in the upper half (with respect to

����� �) of the model leads to an artificial expansion, which

would not occur if the dislocation was embedded in an infinite crystal. Unfortunately this furthercomplicates the comparison with experimental HRTEM images8.

7.4 Comparing experimental and simulated images: In Fig. 15 the experimental and simulatedcross sectional HRTEM images of a 60

�dislocation are compared. Having the largest edge compo-

nent of all investigated dislocations, the hybrid model of the undissociated 60�

dislocation shows thestrongest artificial bending of the � ����� plane, leading to considerable deviation. However, keepingthis in mind, at least locally the agreement is reasonable. Similarly, in Fig. 16 a 90

�and 30

�partial

are shown. Here the agreement is much better, especially for the 30�

glide partial. Unfortunately thecontrast pattern does not allow a rigorous decision over the periodicity of the core reconstruction ofthe 90

�partial. However comparison with the simulated single-period structure shows slightly less de-

viation. Thus in this specific case the SP reconstruction seems more likely. One should keep in mindhowever, that the diffuse contrast of the 90

�partial might result from the partial not being straight,

but heavily kinked. Then the question of periodicity becomes obsolete anyway, since the DP structurecan be simply seen as a periodically kinked SP structure (Fig. 6).

7Comparing Fig. 14 with Fig. 13 shows us how easily the human eye is deceived: Without the atom positions beingshown, one might easily assume the bright spots to be atoms or atom pairs and the large dark areas to be the empty� =�� =����

channels. However, the real situation for the given defocus and specimen thickness is exactly the opposite. Thisdemonstrates the importance of image simulations when it comes to interpreting experimental HRTEM images.

8To reduce this effect by enlarging the diameter of the hybrid models would result in a large increase in computationalcost. The alternative — to abandon the free relaxation of the hybrid’s surface — is a non-trivial problem: A rigid surface,even with the surface atom positions calculated in elasticity theory, would not allow core structure specific volume expan-sion. With the core that strongly confined, the resulting core energies turn out to be unreasonably large. Hence these twomost simple approaches to improve the geometrical boundary conditions are not really feasible.

17

experiment

simulation

Fig. 15: Comparison of the experimental and the simulated HRTEM image of an undissociated 60�

dislocation. Left: Experimental image. The Burgers circuit shown yields a projected Burgers vectorof

�� � ��� �4% � identifying the dislocation. The overlaying mesh of lines connects the points of highestintensity in the contrast pattern. Right: The mesh of the experimental image superimposed with thecorresponding mesh of the simulated image of a 60

�glide dislocation (Fig. 14 (a)). The latter is mir-

rored to facilitate comparison. The small offset between the two meshes is intentional, to allow bothto be seen clearly. As mentioned in the text, the main deviation between the simulated image and theexperimental one arises from the artificial bending of the � ����� plane in the hybrid models applied inthis work.

90° glide (SP) 90° glide (DP) 30° glide

90°30°

Fig. 16: Comparison of the experimental and the simulated HRTEM image of a dissociated 60�

dis-location. Top: The experimental image. The two regions containing the 90

�and 30

�Shockley partial

(left and right respectively) are framed in white. Apparently the two partials do not lie on the sameglide plane — the two glide planes are indicated by small arrows. Bottom: The mesh derived from thesimulated images of the two types of 90

�glide partials and the 30

�glide partial is shown superimposed

on the respective region of the experimental image.

18

8 Summary and conclusions

The dislocation types investigated in this work are the 60�

dislocation and the edge dislocation andtheir respective partial dislocations.

8.1 The 60�

dislocation: The low energy core structures of the 60�

dislocation and its partialshave been identified theoretically. Comparing the shuffle and the glide structure, the latter is clearlyfound as the more stable structure by 630 meV/A. In the case of the double- and single-period 90

�partial energy differences are much smaller (DP 170 meV/A lower in energy), suggesting that bothstructures might co-exist. The most stable shuffle partial dislocation is the vacancy structure of the90

�partial.At least within the rather crude approach to compare simulated and observed HRTEM images,

the observed core structures are in reasonable agreement with those calculated in the earlier sections.Also, undissociated 60

�dislocations could be identified experimentally, which is in agreement with

the predicted barrier to their dissociation as shown in Fig. 7. Unfortunately it was impossible to dis-tinguish the DP and SP structure of the 90

�partial in HRTEM, since their contrast patterns are very

similar.

8.2 The edge dislocation: Similar to 60�

dislocation, the edge glide dislocation is approximately800 meV/A lower in energy than its shuffle counterpart, and its core structure locally resembles thatof a zig-zagged or staggered 60

�dislocation.

When dissociating the edge dislocation into two 60�

partials, the core structures and the structureof the stacking fault strongly depend on the separation distance: For separations below 10 A the bor-dering partials induce a strong high-energy sp

�-like distortion into the stacking fault while one of the

partials reconstructs to a low energy structure. Above 10 A the system undergoes a phase transitionand adopts a structure with a low-energy stacking fault and two identical weakly reconstructed par-tials, as this lowers the stacking fault energy. Once formed, the weakly reconstructed partials appearto be highly mobile and quickly attain their equilibrium distance. As the sp

�-like distortion of the fault

causes a steep and high barrier to dissociation, in a real diamond crystal most edge dislocations willprobably be undissociated.

Acknowledgments

We thank P. Martineau and D. Fisher at De Beers DTC Research Laboratory, Maidenhead (UK) forsupplying the samples. Sample preparation was carried out by L. Rossou at EMAT. The experimentalpart of this work has been performed within the framework of IUAP V-1 of the Flemish government.

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