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An Ab Initio Study of the 90° Partial Dislocation Core in Diamond
P. Sitch, R. Jones, S. Öberg, M. Heggie
To cite this version:
P. Sitch, R. Jones, S. Öberg, M. Heggie. An Ab Initio Study of the 90° Partial Dislocation Core in Diamond. Journal de Physique III, EDP Sciences, 1997, 7 (7), pp.1381-1387. �10.1051/jp3:1997193�.
�jpa-00249651�
An Ab Initio Study of the 90° Partial Dislocation Core in Diamond
P-K- Sitch (~>*), R. Jones (~), S. Oberg (~) and M-I- Heggie (~)
(~) Institut ffir Physik, Technische Umversitit, Chemnitz, 09127, Germany (~) Department of Physics, University of Exeter, Exeter, EX4 4QL, UK (~) Department of Mathematics, University of Lulel 595187, Sweden
(~) School of Chemistry and Molecular Sciences, University of Sussex, Brighton BNI 9QJ, UK
(Received 3 October 1996, accepted 2 April 1997)
PACS.61.72.Lk Linear defects: dislocations, dischnations PACS.61.82.Fk Semiconductors
PACS.62.20.Fe Deformation and plasticity (including yield, ductility, and superplasticity)
Abstract. The electronic and structural properties of the 90° glide partial dislocation in diamond are investigated using an ab imho local density functional cluster method. The core C-C bond is found to be reconstructed with a bond length 5% longer than that in bulk diamond.
The formation and migration energy of the kink on the dislocation are calculated to be 0.32 and 2.97 eV respectively. Further, the shift of the gap levels during kink motion suggests that p-type doping will lead to an increase in the mobility of the partial
1. Introduction
We describe here the first large scale self-consistent ab initio calculations of the structure, electronic and dynamic properties of the 90° glide partial dislocation in diamond. The moti- vation for this research is the high degree of current interest in diamond, due in part to its
extreme hardness, high thermal conductivity and chemical inertness. There is a great need to
understand the microscopic structure of dislocations in diamond and their dynamics 11,2].
The 60° dislocation lying on (III) planes is a common line defect in groups IV and III-V semiconductors and also exists in diamond. It is known to be dissociated into 30° and 90°
partials separated by a stacking fault [3] according to the reaction:
~[l10] ~
~ [iii] + ~[2@.
It is likely that the partials are of the glide type [4].
There are, as yet, no definitive theoretical studies of dislocations on diamond. Early work by Lodge et al. [5] using a valence force potential showed the the reconstructed geometry to be favoured by 1.33 eV per core atom over the unreconstructed geometry for a 90° partial dislo- cation. No structural relaxation was included in this study. This conclusion was subsequently
(* Author for correspondence (e-mail: p.sitch©physik tu-chemnitz de)
© Les #ditions de Physique 1997
1382 JOURNAL DE PHYSIQUE III N°7
supported by Bonapasta et al. [6] using a molecular Hartree-Fock method on CioHis clusters.
Again no structural relaxation was undertaken. Using a Keating potential, Jones [7] showed the formation energy of a single kink on a reconstructed 90° dislocation was about 0.5 eV and its migration energy was 2.7 eV. There is a need to repeat these calculations using a modern
technique which describes quantum mechanically the bonding between core atoms and does not rely on empirical information.
Here we shall present the results of a large scale investigation into the structure and motion of the 90° partial dislocation using an Ab Initio Modelling PRogram (AIMPRO) [8], where full relaxation of the atoms within the core is included.
2. Method
The technique used is based on local density functional theory and incorporates norm-conserving pseudopotentials [9] in a cluster of atoms whose surface is terminated by H. The electronic
wavefunctions are expanded in a basis of Gaussian functions. The total energy of the cluster is evaluated and forces on each atom calculated simultaneously from an analytic expression.
All atoms, including the terminating hydrogens, are then allowed to move to their minimum energy configurations using a conjugate gradient technique. Thus we ignore the effect of the lattice beyond the termination of the cluster on the dislocation core. However, we have found
previously for Si and GaAs [10] that changing the boundary conditions in such large clusters,
for example by fixing the >hydrogen terminating atoms in positions determined by linear elastic-
ity theory has little effect on core structure: core bond lengths and angles are local properties.
This method has previously been successful in describing dislocations in Si ill], GaAs [12] and SiC [13], as well as the interaction of dislocation cores with impurity atoms and the generation
of kinks at dislocation cores. Earlier calculations have shown the method accurately describes the structure of bulk diamond, giving a bulk C-C bond of length within I% of the experimental
value [14].
3. The Straight Dislocation
The dislocation cluster used in this work was constructed by displacing the atoms in a very
large (m 2000) atom cluster, containing a 90° partial dislocation, according to linear elasticity theory. A small cylinder centred on the dislocation axis was then extracted and its dangling
bodds terminited with hydrogen atoms. The resultant cluster is shown in Figure I. It has
196 atoms with composition CiioH86 and contains three C-C core bonds. The outer two
ark 8ubject to surface forces and thu8 only one internal d1810cation bond is present. After relaxation, this has length 1.61 I: 3% longer than the bulk bond found by our method. The overall- dislocation core structure is, therefore, a strongly reconstructed one. The back bonds of the[two C atoms have lengths 1.47, 1.60, 1.61 and 1.48 I along the glide plane. These
nearest neighbour bonds are therefore compressed/dilated by up to 6% from their values in
the bulk material. The nearest neighbour bond lengths perpendicular to the glide plane are
1.54 and 1.571. The
process of reconstruction can be viewed as a competition between two
effects. Firstly, the electronic energy is lowered by the formation of the reconstructed core C-C bond from two dangling C bonds. Secondly, the reconstruction is opposed by the strain field arising from the distortion of the surrounding lattice structure. Our results show then that the electronic effect is sufficient for a strong C-C bond to be formed, but that the repulsive strain field prevents the ~ideal' C-C bond length from being realized, the core bond being 3% longer
than the bulk value.
Fig. 1. The 196 atom cluster, CiioH86, containing a 90° partial dislocation with a reconstructed
Core. ~
~ - - ~
~ ~ ~ ~
fl~ ~
~
~
_
~
-
~ ~ ~ ~
~~j ~ ## ~
~ - ~ ~
@ ~ ~ ~
/~ + ~ @
~ ~ ~
fi fi fi
- ~
(a) (b) (C)
Fig. 2. Cluster Energy levels for (al straight (b) kinked (c) saddle point dislocation, structures in diamond. The broken lines show the three lowest unoccupied levels.
The energy levels for this cluster are shown in Figure 2. They show a clear gap of 5.07 eV between the highest occupied and lowest unoccupied levels. This compares favourably with the experimental result for diamond of 5.50 eV. However, this agreement is fortuitous. The local density approximation used in this work is found to underestimate band gaps by m 50%.
On the other hand, the cluster method results in a confinement of the cluster wavefunctions by the terminating hydrogen atoms that tends to enlarge the band gap. This results in a
cancellation of errors. We can therefore not make any definitive predictions about the exact nature of shallow levels arising from the dislocation. However, it is clear that the dislocation does not introduce deep levels. This is the same as was found for dislocations in Si, GaAs and
SiC, and is a particularly important result as it shows that the ideal dislocation core cannot per se act as a killer centre for electron /hole pair8.
1384 JOURNAL DE PHYSIQUE III N°7
Ii
$
xi
'
x2 j3
Kink separation -
Fig. 3. Schematic representation of the energy involved in the formation of a kink pair of width nb, and the migration energy, Wm of a kink. Ei and E2 represent the activation barrier and the formation
energy of a double kink of width n
= 1
4. Dislocation Motion
We next d18cuss the mobility of the dislocation. Dislocation8 move through the crystal matrix
by the nucleation and sub8equent propagation of kink8 along the dislocation line. According to the Hirth-Lothe model [4], the velocity for a long dislocation 8egment in a high stress regime
is determined by the formation and migration energy of 8ingle kinks given by the equation.
~ flvdTb~~-(Q-TS)/kT
~'~ kT
where b is the primitive lattice vector and fl is a geometrical factor of the order of unity. T is the applied stress, vd an attempt frequency and S is an entropy term. The activation energy for ~bis, Q, 18 the sum of the single-kink formation energy, Fk, and the kink migration energy Wm. Fk and Wm are represented schematically in Figure 3.
We estimate Fk and Wm using a calculational scheme previously successfully applied to Si, GaAs and SiC [12,13]. The method is only briefly sketched here a more detailed account is given by 0berg et al [13]. The energies
we calculate, El and E2 are shown in Figure 3. E2 is the formation energy of the shortest possible kink pair and is therefore twice the single pair
formation plus an (attractive) interaction term:
E2 " 2Fk E,nt(n
= 1).
Eint is approximately given by elasticity theory [4] by:
where G is the 8hear modulus, bp the partial Burgers vector, v is Po1880n's ratio and h and b
are the kink height and al12 respectively. For a 90° partial, h
= bp = al16 and taking v to be 0.068, the interaction energy is:
Eint(n)
#
° °~~~~
a) b)
Fig. 4. The atoms on the glide (111) plane, i.e. parallel to the horizontal plane in Figure 1, for
(a) the relaxed straight 196 atom diamond cluster and (b) the relaxed kinked 196 atom cluster. The full line in (a) is the dislocation core along [10i] and is perpendicular to the partial Burgers vector of the 90° partial along [111]. In (b), the double kink has moved part of the dislocation in the (111) slip plane and in the direction of the Burgers vector.
With b and Gb~ taken to be 2.51 I and 53.3 eV respectively, we get E,nt(n) to be 0.27/ n eV.
We are assuming that the expression for E,nt(n = I) based on elasticity theory is valid for kink-pairs with width of about 2.5 I. The kink migration energy Wm refers to the barrier
associated with the motion of one end of a kink-pair with critical separation n* [4]. This separation depends on the applied stress but is usually of the order of 3-10 b. This separation is too large to be modelled by a calculation carried out on a cluster of the size considered
here. However, as the atomic processes in this critical migratory step are very similar to those involved in the formation of the kink-pair from a straight dislocation, we expect that Wm is
approximately the average of Ei and Ei E2.
5. Double Kink Formation
Having calculated the structure of the straight dislocation, the next step is to form a kink pair along the dislocation line. This was achieved by rotating the core C atom and one of its nearest
neighbours in the glide plane (numbered 26 and 8 in Fig. 4 by 90° about an axis normal to the slip plane. This causes a small part of the dislocation to advance into the stacking fault,
and creates the smallest possible kink-pair. Similar movements of adjacent atoms lying in the
slip plane, would lead to the expansion of the kink-pair and the progressive movement of the dislocation in the slip plane.
The dislocation containing the kink-pair was then relaxed, giving a reconstructed kink bond of length 1.66 I: about 6% longer that of the bulk C-C bond. Nevertheless, this still represents
a strong reconstruction. The back bonds of the kink core atoms are 1.51, 1.61, 1.55 and 1.45 I
in the glide plane and 1.57 and 1.54 I out of the glide plane, representing a compression /dilation
of up to 7% compared to that of bulk bonds. It can therefore be seen that the strain at the kink pair is about the same as that at the dislocation core, although the reconstructed core bond is slightly longer. The kink pair structure is higher in energy than the straight dislocation
1386 JOURNAL DE PHYSIQUE III N°7
C_2
C_1
Fig. 5. Contour plot of the energy barrier, Ei (eV), to kink pair activation for the 90° partial in diamond. The lower left hand corner is close to the configuration of the straight dislocation, whereas the upper right hand corner is close to that of the double kink.
by 0.36 eV. This is the energy E2 as described earlier and hence from our discussion of the
relationship between E2 and Fk, the single kink formation energy is 0.32 eV.
The barrier to the formation of this kink can be calculated [12,13j by finding the energies of clusters with structures intermediate between those of the kink pair and the straight dislocation.
These intermediate structures are generated by defining two constraints cl and c2 that confine the atoms involved in the kink pair formation process to be a fixed distance from each other.
Cl " (R26 R27(~ (R26 R28(~
C2 " (R8 R28(~ (R8 R27(~.
In the 8traight dislocation these constraints are both negative a8 the atoms 26 and 8 are bonded to 27 and 28 respectively. However, they become positive for the kink-pair structure where 26 and 8 are now bonded to 28 and 27 respectively as shown in Figure 4. For fixed values of cl, c2, these equations are solved for one of the coordinates of atoms 8 and 26, and then all the other degrees of freedom are relaxed using the conjugate gradient method. The resulting
energy contours are shown in Figure 5. A saddle point is found near the origin of (cl,c2) and leads to the activation barrier El of 3.15 eV. Thus we find Wm to be 2.97 eV and this yields
an activation energy, Q, for the dislocation mobility of 3.3 eV.
The occupied and three lowest unoccupied energy levels for the straight, kinked and saddle point configurations for the dislocation are 8hown in Figure 2. The straight and kinked dis-
locations have band gaps devoid of deep level states, consistent with a strong reconstruction which displaces apart the bonding and antibonding states. However, the saddle point structure has a highest occupied level displaced to about mid-gap. Thus it is predicted that in p-type
diamond, when this level is only partially or wholly unoccupied, the energy barrier will be reduced. This is similar to the Fermi-level effects found for dislocations in Si and GaAs [13].
6. Summary
We have presented the first ab iniho calculations of the structure of the 90° partial dislocation in diamond. We find a core structure that is strongly reconstructed with a C-C core bond
length comparable with that in bulk diamond. The band structure shows that there are no deep levels associated with the dislocation core, but there are possibly shallow level electron
(hole) traps just below (above) the conduction (valence) band edge. In addition, we find an activation energy for dislocation motion of 3.3 eV. This estimate is close to the previous one found using a Keating potential [7]. The shift in the energy levels during the kink formation procedure also indicates that p-type doping will reduce the energy barrier and hence lead to
an increased dislocation mobility. This suggests that boron doped diamonds, I.e. type IIa, should be softer than type Ia or IIb assuming that there is no effect due to pinning by the boron impurities.
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