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Copyright Trans Tech Publications, Switzerland
Dislocation structures in diamond: density-functional based modelling
and high resolution electron microscopy
,
A. T. Blumenau , T. Frauenheim , S. Öberg
B. Willems and G. Van Tendeloo
Department
of Physics, Universität Paderborn, D-33098 Paderborn, Germany
Department
of Mathematics, University of Luleå, S90187, Luleå, 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 atomistically in a density-functional based tight-binding approach, and their dissociation is discussed both
in terms of structure and energy. Furthermore, high resolution electron microscopy is performed on
dislocation cores in high-temperature, high-pressure annealed natural brown diamond. HRTEM image simulation allows a comparison between theoretically predicted and experimentally observed
structures.
1
Introduction
As in many other crystalline materials, dislocations are common defects in diamond. Chemically
pure 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 diamond
however, 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 or
near 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 dissociated
into glide partials separated by an intrinsic stacking fault ribbon of width 25 – 42 Å [10].
These experimental investigations could, however, not resolve the atomistic core structures of the
dislocations. 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 was
presented recently [15, 16] and first core structures could be resolved in high resolution electron microscopy [17]. Related to the latter, the current text focusses on the energetics of dissociation reactions
and 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-binding
approach 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 dislocation
types 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 compared
with 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 methods
is known — ranging from simple ball and spring models to more sophisticated empirical interatomic
potentials with two- or even three-body contributions [18, 19, 20, 21]. These methods are more or less
successful in describing perfect and strained crystals of tetrahedrally bonded semiconductors. However, 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 effects, or effects involving charge transfer between atoms, might dominate. Very often this is the case
in 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 manybody quantum mechanical wavefunction cannot be solved. The method of choice to reduce the number of parameters is given with density functional theory (DFT), where the electron density is the
central variable. This implies a reduction from the coordinates of the electron wavefunction to
only three coordinates (if we ignore spin-polarisation), as shown in the next section. The reader interested in a more detailed and more exact description may read the original work by Hohenberg and
Kohn [22] and Kohn and Sham [23] or the rather mathematical treatment of Lieb [24], to give just a
few examples.
2.1 A density-functional based tight-binding approach: We will now introduce density-functional based tight-binding (DFTB) — the method used in this work to approximately solve the KohnSham equations of DFT. The description will be introductory and lack many details. For further
insight the original work of Seifert et al. [25] and Porezag et al. [26, 27] or the review in [28] are
recommended. A compact but nevertheless detailed overview including a sketch of the historical
development can be found in [29].
To understand the basic idea of the DFTB method, one best starts with the explicit form of the
total energy in DFT after introducing Kohn-Sham orbitals :
0/
>/
)(*
+.- ( &2143576 8
:9;<=8
(1)
"!$# %'& % +
&@?;5
!, (
Here 143A56 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:
+ FE
C 7 +
C 7 8
<D
(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 orbital
is 2.
The idea is now to simplify Eq. (1) by applying a Taylor-expansion around a reference denG
I
FG
sity 8
and neglecting higher order contributions. Introducing H 8
8
J!
8
we obtain
(Foulkes and Haydock [31]):
G
G
/
K
M NGO
8
F(R
/ /
+
+ - ( 8
L
P
! % Q
B!S (
/
GO
NG
UG
! 145
8
8
&T?;5
/
/
?;5
+ &XW 9ZY\[ H 8
H ( - ( (3)
& % Q' +
&^] _H
V!, (
W
3
M Here L
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 second
and 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 neglecting
three-centre and higher-order contributions yields the total energy in DFTB:
7 /
M GO
L
<=D
-
? 3 =6
&
(4)
DFT calculations show, that the density of an atom surrounded by other atoms in a molecule or a
solid is slightly compressed compared to the density of a free atom. Hence in DFTB the starting
G
density 8
is expressed as a superposition of weakly confined neutral atoms, so called pseudoatoms5 . In all DFTB implementations used in this work, the respective pseudo-atomic wavefunctions
are represented by Slater-type orbitals Q
.
Expanding the Kohn-Sham orbitals of Eq. (4) into the atom-centred Slater-type orbitals Z
yields:
(5)
P:8
>
Z
This resembles an LCAO basis (linear combination of atomic orbitals ). Further we approximate the
two-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 yields
the 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:
-,
6
*
+
(
'& )
$
/
"##
M
M NGO
*
8
/.
(8)
L
##%
08
-
*
/
else
(9)
M
Notice, that the diagonal elements of are taken to be atomic orbital energies of a free atom, to ensure the right energies for dissociated molecules. Once the pseudo-atom orbitals have been calculated
4
For the exchange-correlation terms this only holds approximately.
These pseudo-atom densities result from self-consistent DFT calculations with an additional weak parabolic constriction potential.
Just as a note for those familiar with the internal details of the DFTB method: In this work the diamond C–C interaction
includes
the superposition of the potentials
of the pseudo-atoms. These potentials are compressed by an additional term
13254628769:
207<;>=@? ACB
Å.
with a compression radius of
5
4
M
self-consistently in DFT, the non-diagonal elements of the Hamiltonian 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 and
non self-consistently for any given coordinates of the nuclei. The DFTB energy then becomes:
7 (10)
4 &@? 3 Apart from the repulsive energy, the total
energy
+ is now determined. Assuming approximate transfer + , they are obtained in comparison with self-consistent
ability of the repulsive potentials 1 3 !
DFT calculations
for
+
+ selected reference systems (small molecules or infinite crystals): Varying M one
distance F!
the respective repulsive potential is tabulated just like the matrix elements 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 orbitals, reducing matrix size and thus speeding up all calculations. Further, as has been just shown, all
DFT integrals can be calculated in advance. In the calculation of the eigenvalues Q this saves a great
deal of computational effort and speeds up the calculation dramatically.
For some systems with strong ionic bonding character the approximations of standard DFTB
might fail. Especially the second order term in Eq. (3) becomes too large to be transferable and simply
tabulated within the repulsive potential. The term has then to be treated self-consistently as explained
in Ref. [28]. In this work however, the self-consistent-charge extension is not used, as test calculations
have shown no considerable improvement for the systems investigated.
2.2 Forces and structural optimisation: With the total energy at hand, one can search for the
equilibria of forces in an atomistic model — the local minima of the energy surface. This structural
optimisation is usually done with the help of algorithms involving atomic forces. The latter can be
obtained via the Hellmann-Feynman theorem [32, 33]. Since in this work atom-centred basis sets are
used, the so called Pulay corrections have to be applied [34]; and structures are geometrically
opti eV/
Å.
mised (“relaxed”) using a conjugate gradient algorithm until all forces are well below
3
Straight dislocations in linear elasticity theory
Linear elasticity theory proves to be a very useful tool to describe the long-range elastic strain effects of dislocations. In the following we will discuss the elastic strain energy of an isolated straight
dislocation and the interaction forces between two straight dislocations.
3.1 The elastic strain energy of a straight dislocation: The strain energy of an infinite straight
dislocation 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 radius
the core energy per unit length [9]. The energy factor depends on
and ?
and the elastic
properties of the material. Assuming an isotropic medium, can be evaluated as
!#" +
&
"$ ,&%
!
(12)
5
E
R
Ec
Rc
0
ln( R/Rc)
Fig. 1: The elastic strain energy of a dislocation integrated in a surrounding cylinder. The -axis is
scaled logarithmically.
,
with + being the shear modulus and the Poisson’s ratio.
The logarithmic term in Eq. (11) diverges for both the limits and . Continuum
theory supposes that the atomic displacements vary slowly over the dimensions of a unit cell and this
breaks down at the core resulting in the divergence as . Also, for of the order of magnitude
of 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, the
dislocation 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 we
cannot 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 and
parallel 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 the
elastic force between the two dislocations. By differentiation we obtain the radial component of the
interaction 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
B
a
[111]
a0
a3
A
c
C
b
shuffle
glide
(1
11
B
a
)
a2
A
c
[112]
a1
Fig. 2: Unit cell of a tetrahedrally bonded cubic semiconductor (zincblende). The structure can be
constructed 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 translations
within
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 is
shown 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 with
respect to the other part. This so-called crystal slip usually occurs on specific crystallographic planes
only. This can be explained microscopically: Crystal slip involves the formation and propagation
of dislocations [9]. Following Eq. (11), those dislocations which have minimum Burgers vectors are
easiest to form. Since a Burgers vector (and also the local line direction) is always given as a linear
combination of lattice translations, dislocations are preferentially formed on planes containing the
minimum lattice translations. These planes are the preferred slip planes. A slip plane contains both
the Burgers vector and the line direction of the generated dislocations. Hence the plane of crystal slip
is identical with the glide plane of the dislocations involved in the process.
slip system in fcc lattices: In the cubic structure the
4.1 Perfect dislocations of the planes 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. The
slip 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 lattice
,
and . They are drawn as dashed lines in Fig. 2
translations (Burgers vectors) are (left). Basic dislocation types in this slip system are:
The screw dislocation with
and .
line
direction
and
Burgers
vector
parallel,
e.g.
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
4.2 Partial dislocations and dissociation: The preferred
slip planes mentioned in the last section are also the planes
stacked with the widest separation. As shown in Fig. 2 (right),
bulk
in an
fcc lattice the stacking sequence of these planes is
+ ISF
, where capital and small letters ino’
o
dicate planes belonging to one of the two sub-lattices respectively. Faults in this stacking sequence are very common planar
[112]
defects. If the normal sequence is maintained on both sides of
the fault, then we speak of an intrinsic stacking fault (ISF), oth- Fig. 3: The offset between bulk and
erwise of an extrinsic fault. In this work only intrinsic faults are faulted region (ISF) in the glide
considered. In the ISF plane atoms are displaced with respect plane.
to the bulk plane in normal stacking. Fig. 3 shows the atom positions 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 dislocation [9]. The minimum Burgers vector of such a dislocation is given as the offset vector between the
faulted 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 4% The 90 Shockley partial: e.g. G and G [110]
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)).
` 60
bulk
`e
b60
ISF
b30
be
b90
bulk ISF
b60
b60
ISF
be
b60
b60
Fig. 4: The dissociation of perfect dislocations into partials. Left: The Burgers vectors of the perfect
60 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 the
glide plane
3). The perfect 60 dislocation can be dissociated into the two Shockley
G(compare
Fig.
G
G
partials: . 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 is
rotated 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 model
applied, the low energy core structures of the dislocation itself and of its partials, and the dissociation
of 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 defects
in semiconductors atomistically: The cluster and the supercell approach. For the case of dislocations
this means that either an atom cluster containing a single dislocation is considered or a supercell
containing a dislocation multipole. The long range elastic effects are treated differently in each case
but neither approach treats these effects rigorously as explained in Ref. [15]: In a pure cluster surface
effects often are considerable and in particular the regions where the dislocation intersects the surface
pose a problem as here the dislocation core will be heavily distorted. In a pure supercell the interaction
between the dislocations within the cell and across the cell boundaries with their periodic images
cannot be neglected. Often, if the separation between the dislocations is too small, even the core
structure gets distorted. The effect of the latter on the total energy is impredictable.
To avoid the disadvantages of both the cluster and the supercell, both can be combined into a hybrid: Here the dislocation
is placed in a model which is periodic along the dislocation
line, however, it is non-periodic with a hydrogen-terminated
surface perpendicular to the line direction [36, 37]. This allows
to maintain the ‘natural’ dislocation periodicity as a line defect
and at the same time avoid the interactions between dislocations
in different cells. The single dislocation in the hybrid model is
surrounded by twice the amount of bulk crystal compared to Fig. 5: Schematic sketch of a dislothe pure supercell containing a dipole — assuming both mod- cation in a supercell-cluster hybrid
els are of the same volume and neglecting surface effects. In model.
other words, in the hybrid model it is by far easier to give a good representation of the surrounding
bulk crystal. The latter in combination with the capability of modelling a single and isolated dislocation (avoiding dislocation–dislocation interaction) makes the hybrid model the ideal choice to
describe line defects. The supercell-cluster hybrids used in this work are usually of double-period
length 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 energies 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 or
contraction can vary considerably for different core structures with the same Burgers vector, giving artificial 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 obtained
?
by comparing the elastic energy as given in Eq. (11) with the DFTB formation energy contained in
concentric cylinders around the dislocation core in the hybrid model: From the core radius onwards,
in an ? vs. Å plot this formation energy will follow the gradient described by Eq. (11) and
the offset at the core radius yields the core energy (see Fig. 1). To obtain relative energies between
different core structures, one common radius to be chosen for all cores, and the energies
has
contained in these cylinders are compared. For the 60 dislocation and its partials Å appears
to be sufficient.
9
5.3 Core structures of the 60 dislocation: One can construct atomistic models of the corresponding dislocations by
according
displacing the atoms of an appropriate supercell-cluster hybrid
to 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 existing
ones. Since core reconstructions can be rather complicated, it is not guaranteed to find the overall
lowest energy structures by simply relaxing one starting structure by means of a conjugate gradient
algorithm. The reconstruction found then might just be one of many local minima of the corresponding
energy-surface.
Thus, in this work usually several different starting structures for a given combination
of and are structurally optimised. The low energy core structures obtained by means of the DFTB
method 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 framed
with lines.
The relaxed core structure of the glide dislocation shows bond reconstruction of the terminating
atoms of the half-plane with neighbouring atoms. Reconstruction bonds along the dislocation
line impose a double-period structure.
The shuffle dislocation does not reconstruct. The terminating atoms possess dangling bonds
normal to the glide plane which give rise to electronic gap states [16]. With an energy difference
of 630 meV/Å 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 each
dislocation is accompanied by a stacking fault.
The 30 glide partial reconstructs forming a line of bonded atom pairs. The reconstruction
bonds 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 doubleperiod (DP) reconstruction. The SP structure results simply from forming bonds in the glide
plane 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. Here
the deviations from the bulk bond length is slightly less. The DP structure was first proposed for
silicon by Bennetto et al. [38]. All atoms in both the SP and the DP structure are fully four-fold
coordinated.
In our approach the DP structure is found 170 meV/Å 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/ Å lower in
energy in DFT calculations. Similarly, the low-stress quadrupole calculations of Blase et al. [14]
yield 169 – 198 meV/Å.
The vacancy structure of the 90 glide partial (Fig. 6 (f)) was found to be the most stable shuffle
partial [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 in
the glide plane, however, are weak and 22 % stretched, and one row of dangling bonds remains
unreconstructed.
[111]
10
[110]
[112]
(b) 60° shuffle
(c) 30° glide
(e) 90° glide (DP)
(f) 90° shuffle (V)
[111]
(a) 60° glide
[110]
[112]
(d) 90° glide (SP)
Fig. 6: Dislocation core structures of the 60 dislocations in diamond. The relaxed core structures
of (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 the
dislocation line projected into the plane, and the lower figure shows the glide plane. The
region 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.
E/L (eV/Å)
11
0.4
(1)
(2)
(0)
0
180 meV/Å
(3)
(4)
680 meV/Å
−0.4
−0.8
0
10
20
30
ISF width (Å)
Fig. 7: The dissociation energy of the 60 glide dislocation. The relative atomistic DFTB energies
of 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 of
the 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 dissociation
of perfect dislocations into Shockley partials, then the two competing energy contributions in the
elastic limit are the stacking fault energy and the elastic partial–partial interaction energy. The first
grows proportionally with the partial–partial separation (energy per unit length: ?
with a constant stacking fault energy ) and the interaction energy lessens logarithmically with G
(Eq. (13)). Unfortunately the offset
˚ of the elastic interaction energy is unknown and
consequently a complete
energy balance in elasticity theory is impossible. In equilibrium however,
, as
!
the force resulting from the stacking fault and the partial–partial repulsion
given 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 Å. Consistently,
dissociation in the slip system of diamond has been reported in weak-beam electron microscopy [10]. The dissociation
widths vary from 25 to 42 Å.
To obtain the absolute dissociation energy, the offset in the elastic expression can be calculated in
atomistic 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 of
8.75 Å. Fig. 7 shows the discrete relative energies for the undissociated dislocation and the first
four
G
˚
dissociation distances. With the atomistic
results
at
hand,
the
unknown
energy
offset
for
the 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 procedure
G yields Å. In Fig. 7 the resulting continuous energy are drawn as a solid line. One can observe 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 dissociation
energy at equilibrium separation between the partials is clearly below the energy of the undissociated
[111]
12
glide
shuffle
[112]
Fig. 8: The shuffle and the glide structure of the perfect edge dislocation. The view is approximately
4% along the
dislocation line direction.
SB
b
[110]
Fig. 9: The undissociated edge glide dislocation in the glide plane. Left: The core reconstruction, the
symmetry breaking bond is labelled LB. Right: The Burgers vector and the zig-zagged line direction
indicate the local 60 character.
dislocation ( !
eV/Å). Hence dissociation into partial dislocations is strongly favoured 6 . How ever, an interesting feature is the presence of an energy barrier of
meV/ Å to initiate dissociation
of the 60 dislocation. This barrier should have a considerable effect, since it is approximately 5 Å
wide 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 inserted
half 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 546
carbon atom hybrid model we find the shuffle dislocation to be
eV/ Å higher in line energy.
This is close to the value found for the perfect 60 dislocation (
eV/Å) [15]. As can be seen in
Fig. 9, the reconstruction of the glide edge dislocation resembles, at least locally, that of a zig-zagged
6
In the case of the dissociation of the 60 glide dislocation only the single-period 90 partial is considered here. The
energy of the respective systems containing a double-period 90 instead would be approximately 170 meV/Å lower.
13
narrow fault
infinite separation
Fig. 10: The 60 partial dislocation in the glide plane. The faulted area is shaded. Left: Two closely
separated partials with a narrow stacking fault. Right: An isolated partial.
0.6
E/L
$!%
"#
!
(3)
0.4
(2)
(1)
0.2
0
(0)
?
&('*),+
0
10
20
30
Fig. 11: The dissociation energy of the edge glide dislocation. The relative atomistic
DFTB energies for the first
three dissociation steps and
the undissociated dislocation
are labelled (1) to (3) and (0)
respectively. The equilibrium
dissociation distance is indicated, but no further conclusion concerning the energy
Å can be drawn (see text).
/ staggered perfect 60 glide dislocation. The symmetry of this structure is broken by a reconstruction
4% bond along
(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 dislocations
enclosing a stacking fault. The equilibrium width of the fault evaltuates to 36.5 Å (DFTB elastic
constants and stacking fault energy, compare last section). However, atomistic modelling shows a
steep 0.5 eV/Å increase in line energy for stacking fault widths up to 10 Å (see Fig. 11) — the barrier
to 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 from
the 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, while
the other remains only weakly reconstructed (left and right partial in Fig. 10 (left)) respectively). Of
course the energy of the distorted fault is rather high. From about 10 Å stacking fault width onwards
the configuration including one low and one high energy partial plus a high energy fault becomes
energetically less favourable compared to two high energy partials plus a low energy fault. Therefore above 10 Å the system undergoes a phase transition and adopts the latter configuration. Due to
current computational restrictions in model size, these effects could not be quantified energetically
yet. However, simulation indicates, that the weakly reconstructed partials are extremely mobile: Once
dissociation proceeds beyond 10 Å, the equilibrium widths will be easily and quickly attained.
14
90°
10 Å
[111]
30°
[112]
Fig. 12: HRTEM image of a dissociated 60 dislocation in natural brown type IIa diamond. The two
plane,
Burgers circuits around the 90 (left) and 30 (right) partial are drawn projected into the 4% .% yielding a projected Burgers vector of 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 in
weak-beam transmission electron microscopy [10]). These early studies cannot yield any information
concerning the atomic core structure of dislocations. However, high resolution imaging nowadays
allows to visualise the atomic structure of dislocations in diamond to some extent.
7.1 Experimental setup and sample preparation: For the high resolution transmission electron
microscopy (HRTEM) investigation of dislocation core structures, a JEOL 4000EX microscope with a
Scherzer resolution of 0.17 nm was used. The samples of brown natural type IIa diamond were treated
at 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 to
electron 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 in
HRTEM enables us to resolve the atomic structure of the dislocation core and compare with the
theoretically 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 exact
localisation of the two dislocation cores is impossible. The latter is possibly a result of the dislocation
not being straight through the whole thickness of the layer, but kinked back and forth. This then gives
a rather diffuse image of the core region. The intrinsic stacking fault in between the two partials can
be identified clearly though. The stacking fault is at least 35 Å wide, but might be considerably larger.
Given the rather flat energy minimum for the predicted equilibrium width of 35 Å in Fig. 7, such variation seems reasonable — especially since theory predicts considerable barriers between two adjacent
Peierls valleys [17].
15
Fig. 13: Simulated HRTEM image of the 30 glide partial with a defocus H
thickness 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)
nm and a sample
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 structures
in 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 atom
positions, the interpretation of HRTEM images is much easier and safer in comparison with simulated
images based on atomistic models. Image simulation can be performed using Bloch waves, or in a
multi-slice approach as described by Moodie et al. [40]: The atomistic model is divided into slices
perpendicular 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 events
of the incident wavefront at the planes representing the slices with intermediate propagation through
vacuum between the slices. The rather complicated scattering theory involved is described in detail
in Ref. [41]. The subsequent objective lens is modelled by a Fourier transform. In this procedure, the
defocus caused by the imperfect objective lens can be described by a so called phase contrast transfer
function. This function is specific for each microscope and has to be determined experimentally by
imaging the same area of a sample with varying defocus.
Fig. 13 shows a simulated HRTEM contrast pattern for a given defocus
and sample thickness
generated 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 coordinates
for the simulations are the relaxed atom positions of the corresponding atomistic model. Simulated
images 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 glide
structures. 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 all
simulated 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 inserted
) of the model leads to an artificial expansion, which
material in the upper half (with respect to
would not occur if the dislocation was embedded in an infinite crystal. Unfortunately this further
complicates the comparison with experimental HRTEM images8 .
7.4 Comparing experimental and simulated images: In Fig. 15 the experimental and simulated
cross 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 the
plane, leading to considerable deviation. However, keeping
strongest artificial bending of the this 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 the
contrast pattern does not allow a rigorous decision over the periodicity of the core reconstruction of
the 90 partial. However comparison with the simulated single-period structure shows slightly less deviation. Thus in this specific case the SP reconstruction seems more likely. One should keep in mind
however, 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 structure
can be simply seen as a periodically kinked SP structure (Fig. 6).
7
Comparing Fig. 14 with Fig. 13 shows us how easily the human eye is deceived: Without the atom positions being
shown,
== 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. This
demonstrates the importance of image simulations when it comes to interpreting experimental HRTEM images.
8
To reduce this effect by enlarging the diameter of the hybrid models would result in a large increase in computational
cost. 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 expansion. With the core that strongly confined, the resulting core energies turn out to be unreasonably large. Hence these two
most 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 vector
4% of identifying the dislocation. The overlaying mesh of lines connects the points of highest
intensity in the contrast pattern. Right: The mesh of the experimental image superimposed with the
corresponding mesh of the simulated image of a 60 glide dislocation (Fig. 14 (a)). The latter is mirrored to facilitate comparison. The small offset between the two meshes is intentional, to allow both
to be seen clearly. As mentioned in the text, the main deviation between the simulated image and the
experimental one arises from the artificial bending of the plane in the hybrid models applied in
this work.
90°
90° glide (SP)
30°
90° glide (DP)
30° glide
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 same
glide plane — the two glide planes are indicated by small arrows. Bottom: The mesh derived from the
simulated 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 and
their respective partial dislocations.
8.1 The 60 dislocation: The low energy core structures of the 60 dislocation and its partials
have been identified theoretically. Comparing the shuffle and the glide structure, the latter is clearly
found as the more stable structure by 630 meV/Å. In the case of the double- and single-period 90
partial energy differences are much smaller (DP 170 meV/Å lower in energy), suggesting that both
structures might co-exist. The most stable shuffle partial dislocation is the vacancy structure of the
90 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 approximately
800 meV/Å lower in energy than its shuffle counterpart, and its core structure locally resembles that
of a zig-zagged or staggered 60 dislocation.
When dissociating the edge dislocation into two 60 partials, the core structures and the structure
of the stacking fault strongly depend on the separation
distance: For separations below 10 Å 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 Å the system undergoes a phase transition
and adopts a structure with a low-energy stacking fault and two identical weakly reconstructed partials, as this lowers the stacking fault energy. Once formed, the weakly reconstructed
partials appear
to 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 will
probably be undissociated.
Acknowledgments
We thank P. Martineau and D. Fisher at De Beers DTC Research Laboratory, Maidenhead (UK) for
supplying the samples. Sample preparation was carried out by L. Rossou at EMAT. The experimental
part of this work has been performed within the framework of IUAP V-1 of the Flemish government.
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