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Energetics of the 30∘ Shockley partial dislocation in wurtzite GaN
Superlattices and Microstructures 40 (2006) 458–463 www.elsevier.com/locate/superlattices
Energetics of the 30◦ Shockley partial dislocation in wurtzite GaN I. Belabbas a,b,∗ , G. Dimitrakopulos b , J. Kioseoglou b , A. B´er´e a,d , J. Chen c , Ph. Komninou b , P. Ruterana a , G. Nouet a a Laboratoire Structure des Interfaces et Fonctionnalit´e des Couches Minces, UMR CNRS 6176, Ecole Nationale
Sup´erieure d‘Ing´enieurs de Caen, 6 Bld du Mar´echal Juin, 14050 Caen cedex, France b Physics Department, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece c Laboratoire de Recherche sur les Propri´et´es des Mat´eriaux Nouveaux, Institut Universitaire de Technologie
d’Alenc¸on, 61250 Damigny, France d LPCE, Universit´e de Ouagadougou, 03 Boˆıte Postale, 7021 Ouagadougou 03, Burkina Faso
Received 8 September 2006; accepted 8 September 2006 Available online 25 October 2006
Abstract In the present work, we have investigated the relative energy of different core configurations of the 30◦ Shockley partial dislocation in wurtzite GaN. By using a modified Stillinger–Weber potential, we have carried out large scale calculations on models containing many thousands of atoms. Both glide and shuffle configurations have been considered within the two core polarities (Ga, N). Similarly to what was reported for conventional semiconductors, our calculations showed that the reconstructed glide configurations are energetically favoured over the shuffle ones. c 2006 Elsevier Ltd. All rights reserved.
1. Introduction The (0001) c-plane oriented GaN films are the most commonly used for device applications, but the optoelectronic devices grown along this direction suffer from undesirable spontaneous and piezoelectric polarization effects [1]. This leads to a charge separation within quantum ∗ Corresponding author at: Laboratoire Structure des Interfaces et Fonctionnalit´e des Couches Minces, UMR CNRS 6176, Ecole Nationale Sup´erieure d‘Ing´enieurs de Caen, 6 Bld du Mar´echal Juin, 14050 Caen cedex, France. Tel.: +33 (0)2 31 45 26 54; fax: +33 (0)2 31 45 26 60. E-mail address: [email protected] (I. Belabbas).
c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2006.09.013
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
459
wells which decreases the electron–hole recombination efficiency and redshifts the emission wavelengths. These polarization effects, which are undesirable in the operation of shortwavelength visible and ultraviolet emitters, can be eliminated by growing devices on alternative ¯ m-plane or the (1120) ¯ a-plane [2]. Recently, orientations of GaN crystals, such as the (1010) lattice defects in non-polar a-plane GaN layers have been studied by means of High-Resolution Transmission Electron Microscopy (HRTEM) [3]. Basal-plane stacking faults with Shockley partial dislocations were found to be the main defects in these layers. The Shockley partial dislocations were observed to be due to the dissociation of the perfect basal dislocations (pure screw or mixed 60◦ ). The 30◦ Shockley partial dislocation has been extensively investigated in elemental semiconductors like silicon [4] and diamond [5]. These theoretical studies provided consistent results leading to good comprehension of the atomic core structure of this dislocation. For these materials, cores belonging to the glide set were found to be energetically favoured over those in the shuffle set. Moreover, for the glide configurations, an energy lowering could arise from reconstructions occurring along the dislocation line direction [6]. While, this type of reconstruction allows a reduction of the number of dangling bonds, it causes doubling of the translation periodicity along the dislocation line. Similar qualitative results have been obtained for the 30◦ Shockley partial dislocation in compound semiconductors with zincblende structures, like gallium arsenide [7] and silicon carbide [8]. In hexagonal compound semiconductors no report exists on the atomic core structure of the 30◦ partial dislocation. Hence, in the present paper, we present our first results on the investigation of the relative energy of different core configurations of the 30◦ Shockley partial dislocation in wurtzite GaN. 2. Computational details Our approach is based on a modified Stillinger–Weber potential [9]. This potential was adapted to take into account the three different kinds of bonds which may occur in GaN based systems (i.e. Ga–N and Ga–Ga or N–N ‘wrong’ bonds). The modified Stillinger–Weber potential was previously involved in the investigation of the energetic and atomic structures of dislocations [10,11], grain boundaries [12] and planar defects junctions [13] in GaN wurtzite. In hexagonal materials, the mixed 30◦ Shockley partial dislocation has its line direction along ¯ ¯ the [1210], while its Burgers vector is equal to 1/3[1010]. Different core configurations are possible for this dislocation; these are the glide and shuffle configurations. In gallium nitride, each configuration can exist in two different polarities: gallium or nitrogen. ◦ The √ 30 partial dislocation was modelled in a parallelepiped cluster-supercell hybrid, of size ˚ and c = 5.20 A, ˚ are the calculated equilibrium lattice [30 3a × 30c × 4a] (a = 3.19 A parameters), containing about 28 800 atoms. The translational symmetry of the dislocation along its line direction, is maintained by applying periodic boundary conditions along this direction. Initial positions of different core configurations of the dislocation were generated by applying, in the glide or in the shuffle set, the displacement field expected from anisotropic elasticity theory [14]. The equilibrium atomic positions are obtained through a relaxation procedure implemented in a Verlet molecular dynamics scheme [15], where the forces are evaluated by means of the modified Stillinger–Weber potential. The equilibrium is attained when the average thermodynamic temperature of the system is below 10−6 K. In our calculations, the simulation cell is divided into three regions around the dislocation centre. The internal region is delimited by a cylinder of radius 16a and the external region extends beyond a cylinder of radius 20a. Between the two latter, an intermediate region is included.
460
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
Fig. 1. Core configurations of the 30◦ Shockley partial dislocation. (A) Glide configuration with gallium polarity. (B) Shuffle configuration with nitrogen polarity. (C) Side view of an unreconstructed glide configuration. (D) Side view of a reconstructed glide configuration. The black balls represent gallium atoms and the dashed line delimits the stacking fault.
The thickness of both the external and the intermediate regions was chosen to be larger than ˚ for the Ga–Ga interaction). While the atoms in the the maximum range of the potential (3.36 A external region have fixed positions those belonging to the internal and intermediate regions are allowed to relax. The energetic calculations are evaluated taking into account only the energies of the relaxed atoms in the internal region. As the studied dislocation is a Shockley partial, its core is bound by an I2 stacking fault. Then the calculated total strain energy should contain a contribution from as well the dislocation as from its associated stacking fault. In these calculations, the latter contribution vanishes as the used potential is limited to the first neighbours [10]. Indeed, experiencing an energy contribution from the stacking fault, i.e. distinguishing a wurtzite from a zinc-blende structure, requires the inclusion of the second neighbours. As a consequence, all the calculated strain energy is only due to the dislocation. 3. Energetic calculations For the 30◦ Shockley partial dislocation, four configurations have been considered: the reconstructed glide configurations with gallium (G Ga ) or nitrogen (G N ) polarity, and the shuffle configurations with gallium (SGa ) or nitrogen (SN ) polarity (Fig. 1(A), (B)). As the glide configurations are reconstructed their period is doubled along the dislocation line direction (Fig. 1(C), (D)). The energetic calculations have been performed by combining continuum elasticity theory and atomistic calculations based on the modified Stillinger–Weber potential. The total strain energy (E total ) associated with a dislocation when introduced in a medium, could be represented as a sum of an elastic (E elastic ) and a core (E core ) contributions: E total = E elastic + E core .
(1)
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461
Fig. 2. The total strain energy stored in a cylinder of radius R as a function of ln(R) for different core configurations of the 30◦ Shockley partial dislocation.
Within linear elasticity, the elastic strain energy per unit length stored in a cylinder of radius R around the dislocation is given by the relation [14]: E elastic = A ln(R/Rc )
for R > Rc
(2)
where Rc is the dislocation core radius. For a mixed type dislocation, the prelogarithmic factor A is related to both the edge and screw components (be and bs respectively) of the Burgers vector of the dislocation and it is given by the relation [14]: A = (1/4π )(K e be2 + K s bs2 ).
(3)
In the case of a mixed basal dislocation, the energy factors K e and K s associated respectively with the edge and the screw components, are given, under the framework of anisotropic elasticity, by the relations [14]: !1/2 √ p C44 C11 C33 − C13 C11 C33 + C13 Ke = (4a) √ C33 C11 C33 + C13 + 2C44 and Ks =
p C44 C66
(4b)
where Ci j are the elastic constants of the material. In atomistic calculations, one can define the excess energy of a single atom as its difference in energy in the system with presence of the defect and that in bulk material. Hence, the total strain energy contained in a cylinder of radius R around the dislocation is evaluated by summing the excess of energy related to individual atoms belonging to this area. In order to determine the core parameters of the dislocation, i.e. core energy and core radius, we plotted the total strain energy, obtained by atomistic calculations, versus ln(R) (Fig. 2). Usually, in the case of partial
462
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
Table 1 The calculated core energies (E c ), radii (Rc ) and prelogarithmic factors for the 30◦ partial dislocation exp
Configurations
˚ Asw th (eV/A)
˚ Ath (eV/A)
˚ Afit (eV/A)
˚ Rc (A)
˚ E c (eV/A)
˚ ˚ E (eV/A)(R = 4.71 A)
G Ga GN SGa SN
0.203 0.203 0.203 0.203
0.226 0.226 0.226 0.226
0.193 0.192 0.198 0.191
3.31 3.31 4.71 4.71
0.58 0.62 1.15 1.21
0.64 0.68 1.15 1.21
exp
The prelogarithmic factors Asw th , Ath and Afit are given. The latter was obtained by fit and the two first were obtained from anisotropic elasticity by the use of Stillinger–Weber calculated elastic constants and those provided by experiment.
dislocations, the energy contribution of the stacking fault should be subtracted from the total strain energy in order to retain only the energy contribution of the dislocation [5]. However, as the used potential does not experience any energy from the stacking fault, all the calculated strain energy is contributed exclusively from the dislocation. By fitting the data from atomistic calculations to the analytical relation of the elastic strain energy [Eq. (2)], it is possible to identify the core radius of the dislocation as the value from which the curve starts to be linear [5]. The slope of the linear part gives the value of the prelogarithmic factor A. Following the previous methodology, we fitted the data provided by atomistic calculations to the elasticity theory relations. As can be seen from Table 1, for the four configurations, the obtained prelogarithmic factors (Afit ) agree well with those evaluated on the basis of the exp experimental (Ath ) or the calculated (Asw th ) elastic constants. This constitutes an indication on the accuracy of our atomistic calculations. According to the obtained curves, the glide configurations exhibit a core radius (RcG ) of ˚ which corresponds to a core energy of 0.58 eV/A, ˚ for the G Ga configuration, and a 3.31 A, ˚ for the G N configuration. The shuffle configurations were found core energy of 0.62 eV/A, ˚ to have a larger core than the glide ones. Their core radius (RcS ) was evaluated to be 4.71 A, ˚ for the SGa configuration, and a core which corresponds to a core energy of 1.15 eV/A, ˚ for the SN configuration. The ratio of the core radii of the shuffle and energy of 1.21 eV/A, glide configurations is about 1.42. For comparison, the value of this ratio is equal to 1.94 in diamond [5]. Our results show clearly that, for the 30◦ Shockley partial dislocation in GaN, the glide configurations are lower in energy than the shuffle ones. Moreover, in both glide and shuffle sets, the gallium polarity cores have lower energy. An evaluation of the energy of all the configurations ˚ shows that the difference in energy between the glide at a common radius (R = 4.71 A), ˚ and the difference in energy between the shuffle configurations configurations is about 0.04 eV/A ˚ The differences in energy between the glide and shuffle configurations is about 0.06 eV/A. are larger. Indeed, while the difference in energy between the (SGa ) and (G N ) configurations ˚ that between the (SN ) and (G Ga ) configuration is 0.57 eV/A. ˚ These differences in is 0.47 eV/A, energy are maintained asymptotically in the elastic limit (Fig. 2). 4. Conclusion Based on a modified Stillinger–Weber potential, we have investigated the relative energy of different core configurations of the mixed 30◦ Shockley partial dislocation in wurtzite GaN. These calculations show that the reconstructed glide configurations are more energetically favourable than the shuffle ones. These results agree qualitatively with previous reports on the 30◦ Shockley partial dislocation in elemental and compound semiconductors.
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Acknowledgements This work was supported by EU Marie Curie RTN contract MRTN-CT-2004-005583 (PARSEM). The computations were carried out at the CRIHAN (Centre de Ressources Informatiques de HAute Normandie) (http://www.crihan.fr). References [1] P. Lefebvre, A. Morel, M. Gallart, T. Taliercio, J. Allegre, B. Gil, H. Mathieu, B. Damilano, N. Grandjean, J. Massies, Appl. Phys. Lett. 78 (2001) 1252. [2] B.A. Haskel, F. Wu, S. Matsuda, M.D. Craven, P.T. Fini, S.P. DenBaars, J.S. Speck, S. Nakamura, Appl. Phys. Lett. 83 (2003) 1554. [3] D.N. Zakharov, Z. Liliental-Weber, Phys. Rev. B 71 (2005) 235334. [4] J.E. Northrup, M.L. Cohen, J.R. Chelikowsky, J. Spence, A. Olsen, Phys. Rev. B 24 (1981) 4623. [5] A.T. Blumenau, M.I. Heggie, C.J. Fall, R. Jones, Th. Frauenheim, Phys. Rev. B 65 (2002) 205205. [6] S. Marklund, Phys. Status Solidi (b) 92 (1979) 83. [7] R.W. Nunes, L.V.C. Assali, J. Justo, Comput. Mater. Sci. 30 (2004) 67. [8] A.T. Blumenau, R. Jones, S. Oberg, P. Briddon, Th. Frauenheim, Physica B 340–342 (2003) 160. [9] N. Aichoune, V. Potin, P. Ruterana, et al., Comput. Mater. Sci. 17 (2000) 380. [10] A. B´er´e, A. Serra, Phys. Rev. B 65 (2002) 205323. [11] J. Kioseoglou, G.P. Dimitrakopulos, Ph. Komninou, Th. Karakostas, Phys. Rev. B 70 (2004) 035309. [12] J. Chen, P. Ruterana, G. Nouet, Phys. Rev. B 67 (2003) 205210. [13] J. Kioseoglou, G.P. Dimitrakopulos, Ph. Komninou, H.M. Polatoglou, A. Serra, A. B´er´e, G. Nouet, Th. Karakostas, Phys. Rev. B 70 (2004) 115331. [14] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley, New York, 1982. [15] L. Verlet, Phys. Rev. 159 (1967) 98.
Energetics of the 30◦ Shockley partial dislocation in wurtzite GaN I. Belabbas a,b,∗ , G. Dimitrakopulos b , J. Kioseoglou b , A. B´er´e a,d , J. Chen c , Ph. Komninou b , P. Ruterana a , G. Nouet a a Laboratoire Structure des Interfaces et Fonctionnalit´e des Couches Minces, UMR CNRS 6176, Ecole Nationale
Sup´erieure d‘Ing´enieurs de Caen, 6 Bld du Mar´echal Juin, 14050 Caen cedex, France b Physics Department, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece c Laboratoire de Recherche sur les Propri´et´es des Mat´eriaux Nouveaux, Institut Universitaire de Technologie
d’Alenc¸on, 61250 Damigny, France d LPCE, Universit´e de Ouagadougou, 03 Boˆıte Postale, 7021 Ouagadougou 03, Burkina Faso
Received 8 September 2006; accepted 8 September 2006 Available online 25 October 2006
Abstract In the present work, we have investigated the relative energy of different core configurations of the 30◦ Shockley partial dislocation in wurtzite GaN. By using a modified Stillinger–Weber potential, we have carried out large scale calculations on models containing many thousands of atoms. Both glide and shuffle configurations have been considered within the two core polarities (Ga, N). Similarly to what was reported for conventional semiconductors, our calculations showed that the reconstructed glide configurations are energetically favoured over the shuffle ones. c 2006 Elsevier Ltd. All rights reserved.
1. Introduction The (0001) c-plane oriented GaN films are the most commonly used for device applications, but the optoelectronic devices grown along this direction suffer from undesirable spontaneous and piezoelectric polarization effects [1]. This leads to a charge separation within quantum ∗ Corresponding author at: Laboratoire Structure des Interfaces et Fonctionnalit´e des Couches Minces, UMR CNRS 6176, Ecole Nationale Sup´erieure d‘Ing´enieurs de Caen, 6 Bld du Mar´echal Juin, 14050 Caen cedex, France. Tel.: +33 (0)2 31 45 26 54; fax: +33 (0)2 31 45 26 60. E-mail address: [email protected] (I. Belabbas).
c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2006.09.013
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
459
wells which decreases the electron–hole recombination efficiency and redshifts the emission wavelengths. These polarization effects, which are undesirable in the operation of shortwavelength visible and ultraviolet emitters, can be eliminated by growing devices on alternative ¯ m-plane or the (1120) ¯ a-plane [2]. Recently, orientations of GaN crystals, such as the (1010) lattice defects in non-polar a-plane GaN layers have been studied by means of High-Resolution Transmission Electron Microscopy (HRTEM) [3]. Basal-plane stacking faults with Shockley partial dislocations were found to be the main defects in these layers. The Shockley partial dislocations were observed to be due to the dissociation of the perfect basal dislocations (pure screw or mixed 60◦ ). The 30◦ Shockley partial dislocation has been extensively investigated in elemental semiconductors like silicon [4] and diamond [5]. These theoretical studies provided consistent results leading to good comprehension of the atomic core structure of this dislocation. For these materials, cores belonging to the glide set were found to be energetically favoured over those in the shuffle set. Moreover, for the glide configurations, an energy lowering could arise from reconstructions occurring along the dislocation line direction [6]. While, this type of reconstruction allows a reduction of the number of dangling bonds, it causes doubling of the translation periodicity along the dislocation line. Similar qualitative results have been obtained for the 30◦ Shockley partial dislocation in compound semiconductors with zincblende structures, like gallium arsenide [7] and silicon carbide [8]. In hexagonal compound semiconductors no report exists on the atomic core structure of the 30◦ partial dislocation. Hence, in the present paper, we present our first results on the investigation of the relative energy of different core configurations of the 30◦ Shockley partial dislocation in wurtzite GaN. 2. Computational details Our approach is based on a modified Stillinger–Weber potential [9]. This potential was adapted to take into account the three different kinds of bonds which may occur in GaN based systems (i.e. Ga–N and Ga–Ga or N–N ‘wrong’ bonds). The modified Stillinger–Weber potential was previously involved in the investigation of the energetic and atomic structures of dislocations [10,11], grain boundaries [12] and planar defects junctions [13] in GaN wurtzite. In hexagonal materials, the mixed 30◦ Shockley partial dislocation has its line direction along ¯ ¯ the [1210], while its Burgers vector is equal to 1/3[1010]. Different core configurations are possible for this dislocation; these are the glide and shuffle configurations. In gallium nitride, each configuration can exist in two different polarities: gallium or nitrogen. ◦ The √ 30 partial dislocation was modelled in a parallelepiped cluster-supercell hybrid, of size ˚ and c = 5.20 A, ˚ are the calculated equilibrium lattice [30 3a × 30c × 4a] (a = 3.19 A parameters), containing about 28 800 atoms. The translational symmetry of the dislocation along its line direction, is maintained by applying periodic boundary conditions along this direction. Initial positions of different core configurations of the dislocation were generated by applying, in the glide or in the shuffle set, the displacement field expected from anisotropic elasticity theory [14]. The equilibrium atomic positions are obtained through a relaxation procedure implemented in a Verlet molecular dynamics scheme [15], where the forces are evaluated by means of the modified Stillinger–Weber potential. The equilibrium is attained when the average thermodynamic temperature of the system is below 10−6 K. In our calculations, the simulation cell is divided into three regions around the dislocation centre. The internal region is delimited by a cylinder of radius 16a and the external region extends beyond a cylinder of radius 20a. Between the two latter, an intermediate region is included.
460
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
Fig. 1. Core configurations of the 30◦ Shockley partial dislocation. (A) Glide configuration with gallium polarity. (B) Shuffle configuration with nitrogen polarity. (C) Side view of an unreconstructed glide configuration. (D) Side view of a reconstructed glide configuration. The black balls represent gallium atoms and the dashed line delimits the stacking fault.
The thickness of both the external and the intermediate regions was chosen to be larger than ˚ for the Ga–Ga interaction). While the atoms in the the maximum range of the potential (3.36 A external region have fixed positions those belonging to the internal and intermediate regions are allowed to relax. The energetic calculations are evaluated taking into account only the energies of the relaxed atoms in the internal region. As the studied dislocation is a Shockley partial, its core is bound by an I2 stacking fault. Then the calculated total strain energy should contain a contribution from as well the dislocation as from its associated stacking fault. In these calculations, the latter contribution vanishes as the used potential is limited to the first neighbours [10]. Indeed, experiencing an energy contribution from the stacking fault, i.e. distinguishing a wurtzite from a zinc-blende structure, requires the inclusion of the second neighbours. As a consequence, all the calculated strain energy is only due to the dislocation. 3. Energetic calculations For the 30◦ Shockley partial dislocation, four configurations have been considered: the reconstructed glide configurations with gallium (G Ga ) or nitrogen (G N ) polarity, and the shuffle configurations with gallium (SGa ) or nitrogen (SN ) polarity (Fig. 1(A), (B)). As the glide configurations are reconstructed their period is doubled along the dislocation line direction (Fig. 1(C), (D)). The energetic calculations have been performed by combining continuum elasticity theory and atomistic calculations based on the modified Stillinger–Weber potential. The total strain energy (E total ) associated with a dislocation when introduced in a medium, could be represented as a sum of an elastic (E elastic ) and a core (E core ) contributions: E total = E elastic + E core .
(1)
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
461
Fig. 2. The total strain energy stored in a cylinder of radius R as a function of ln(R) for different core configurations of the 30◦ Shockley partial dislocation.
Within linear elasticity, the elastic strain energy per unit length stored in a cylinder of radius R around the dislocation is given by the relation [14]: E elastic = A ln(R/Rc )
for R > Rc
(2)
where Rc is the dislocation core radius. For a mixed type dislocation, the prelogarithmic factor A is related to both the edge and screw components (be and bs respectively) of the Burgers vector of the dislocation and it is given by the relation [14]: A = (1/4π )(K e be2 + K s bs2 ).
(3)
In the case of a mixed basal dislocation, the energy factors K e and K s associated respectively with the edge and the screw components, are given, under the framework of anisotropic elasticity, by the relations [14]: !1/2 √ p C44 C11 C33 − C13 C11 C33 + C13 Ke = (4a) √ C33 C11 C33 + C13 + 2C44 and Ks =
p C44 C66
(4b)
where Ci j are the elastic constants of the material. In atomistic calculations, one can define the excess energy of a single atom as its difference in energy in the system with presence of the defect and that in bulk material. Hence, the total strain energy contained in a cylinder of radius R around the dislocation is evaluated by summing the excess of energy related to individual atoms belonging to this area. In order to determine the core parameters of the dislocation, i.e. core energy and core radius, we plotted the total strain energy, obtained by atomistic calculations, versus ln(R) (Fig. 2). Usually, in the case of partial
462
I. Belabbas et al. / Superlattices and Microstructures 40 (2006) 458–463
Table 1 The calculated core energies (E c ), radii (Rc ) and prelogarithmic factors for the 30◦ partial dislocation exp
Configurations
˚ Asw th (eV/A)
˚ Ath (eV/A)
˚ Afit (eV/A)
˚ Rc (A)
˚ E c (eV/A)
˚ ˚ E (eV/A)(R = 4.71 A)
G Ga GN SGa SN
0.203 0.203 0.203 0.203
0.226 0.226 0.226 0.226
0.193 0.192 0.198 0.191
3.31 3.31 4.71 4.71
0.58 0.62 1.15 1.21
0.64 0.68 1.15 1.21
exp
The prelogarithmic factors Asw th , Ath and Afit are given. The latter was obtained by fit and the two first were obtained from anisotropic elasticity by the use of Stillinger–Weber calculated elastic constants and those provided by experiment.
dislocations, the energy contribution of the stacking fault should be subtracted from the total strain energy in order to retain only the energy contribution of the dislocation [5]. However, as the used potential does not experience any energy from the stacking fault, all the calculated strain energy is contributed exclusively from the dislocation. By fitting the data from atomistic calculations to the analytical relation of the elastic strain energy [Eq. (2)], it is possible to identify the core radius of the dislocation as the value from which the curve starts to be linear [5]. The slope of the linear part gives the value of the prelogarithmic factor A. Following the previous methodology, we fitted the data provided by atomistic calculations to the elasticity theory relations. As can be seen from Table 1, for the four configurations, the obtained prelogarithmic factors (Afit ) agree well with those evaluated on the basis of the exp experimental (Ath ) or the calculated (Asw th ) elastic constants. This constitutes an indication on the accuracy of our atomistic calculations. According to the obtained curves, the glide configurations exhibit a core radius (RcG ) of ˚ which corresponds to a core energy of 0.58 eV/A, ˚ for the G Ga configuration, and a 3.31 A, ˚ for the G N configuration. The shuffle configurations were found core energy of 0.62 eV/A, ˚ to have a larger core than the glide ones. Their core radius (RcS ) was evaluated to be 4.71 A, ˚ for the SGa configuration, and a core which corresponds to a core energy of 1.15 eV/A, ˚ for the SN configuration. The ratio of the core radii of the shuffle and energy of 1.21 eV/A, glide configurations is about 1.42. For comparison, the value of this ratio is equal to 1.94 in diamond [5]. Our results show clearly that, for the 30◦ Shockley partial dislocation in GaN, the glide configurations are lower in energy than the shuffle ones. Moreover, in both glide and shuffle sets, the gallium polarity cores have lower energy. An evaluation of the energy of all the configurations ˚ shows that the difference in energy between the glide at a common radius (R = 4.71 A), ˚ and the difference in energy between the shuffle configurations configurations is about 0.04 eV/A ˚ The differences in energy between the glide and shuffle configurations is about 0.06 eV/A. are larger. Indeed, while the difference in energy between the (SGa ) and (G N ) configurations ˚ that between the (SN ) and (G Ga ) configuration is 0.57 eV/A. ˚ These differences in is 0.47 eV/A, energy are maintained asymptotically in the elastic limit (Fig. 2). 4. Conclusion Based on a modified Stillinger–Weber potential, we have investigated the relative energy of different core configurations of the mixed 30◦ Shockley partial dislocation in wurtzite GaN. These calculations show that the reconstructed glide configurations are more energetically favourable than the shuffle ones. These results agree qualitatively with previous reports on the 30◦ Shockley partial dislocation in elemental and compound semiconductors.
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