phys. stat. sol. (c) 4, No. 8, 2940–2944 (2007) / DOI 10.1002/pssc.200675441

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Investigation of the atomic core structure

of the a+c� �

( ) -mixed dislocation in wurtzite GaN

I. Belabbas*, 1, A. Béré1, 2, J. Chen3, P. Ruterana1, and G. Nouet1

1 Laboratoire Structure des Interfaces et Fonctionnalité des Couches Minces, UMR CNRS 6176,

Ecole Nationale Supérieure d‘Ingénieurs de Caen, 6 Bld. du Maréchal Juin, 14050 Caen cedex, France 2 Laboratoire de Physique et de Chimie de l’Environnement, Université de Ouagadougou,

03 BP: 7021 Ouagadougou 03, Burkina Faso 3 Laboratoire de Recherche sur les Propriétés des Matériaux Nouveaux,

Institut Universitaire de Technologie d’Alençon, 61250 Damigny, France

Received 17 September 2006, revised 22 March 2007, accepted 29 March 2007

Published online 1 June 2007

PACS 61.43.Bn, 61.72.Bb, 61.72.Lk

By using a tight-binding based ab-initio method, we have investigated the atomic and electronic core

structures of the threading ( )a c+� �

-mixed dislocation in wurtzite gallium nitride. For this dislocation, two

core configurations were found to occur: a 5/7-atoms ring and a complex double 5/6-atoms ring configu-

rations. Both core configurations were found to induce in the bandgap as well deep as shallow gap states.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction In gallium nitride, threading dislocations originate from heteroepitaxial growth carried out on highly mismatched substrates, like sapphire (Al2O3) [1]. These threading dislocations have a typi-cal density of 1010 cm–2 and are of three types: a

�

- edge, c�

- screw and ( )a c

� �

+ - mixed [1]. The a

�

- edge dislocations were shown to be predominant, while the c�

- screw dislocations to present the smallest density [2]. However, depending on the growth conditions, the density of the ( )a c

� �

+ - mixed dislocations can vary between those of the two other types. Among the threading dislocations, the a

�

- edge and c�

- screw were the most studied [3]. For these dislocation cores, atomistic models as well as Z-contrast images were provided, resulting in a good comprehension of their behaviour and eventual impact on GaN layers [4]. Arslan et al. succeeded to provide a Z-contrast image of the ( )a c

� �

+ - mixed dislocation core [5]. The observed structure was interpreted as a filled core with 8-atoms ring, just like for the case of the

−a

�

edge dislocation. Recently, based on empirical potential calculations, we reported the first atomistic models for the

( )a c

� �

+ - mixed dislocation core [6]. Two configurations were obtained: a 5/7-atoms ring core and a complex double 5/6-atoms ring core. However, due to the natural limitation of empirical potentials, the electronic structure of the obtained core configurations could not be addressed. In the present article, we have reinvestigated the atomic structure of the ( )a c

� �

+ - mixed dislocation core with a more accurate methodology than that we have adopted in our previous study based on empirical potentials [6]. For this purpose, we have used a tight-binding based ab-initio method (SCC-DFTB method) [7]. Beside the atomic core structure, we will also present and discuss the local electronic structure associated with each core configuration.

* Corresponding author: e-mail: imad.belabbas@ensicaen.fr, Phone: +00 33 231 452 654, Fax: +00 33 231 452 660

phys. stat. sol. (c) 4, No. 8 (2007) 2941

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2 Computational and simulation details The SCC-DFTB (Self-Consistent Charge Density Func-tional Tight-Binding) method can be seen as an approximate density functional scheme [7]. In the total energy expression, beside the two usual tight-binding energetic terms, i.e. the band structure and the short-ranged pair potential, a third one has been introduced where a self-consistent procedure is incorpo-rated at the level of Mulliken charges to account for charge transfer. The electronic wave function is expanded in a linear combination of atomic orbitals involving a minimal basis set of s, p and d confined orbitals. The Hamiltonian and overlap matrix elements are evaluated within the Slater-Koster two-centre approximation. Due to the incorporation of the self-consistent term, the SCC-DFTB method allows to achieve accuracy comparable to ab-initio methods. Moreover, the SCC-DFTB method is numerically efficient as it allows treating large systems of several hundreds of atoms which is attractive in the context of dislocations modelling. Indeed, among all the tight-binding methods, the SCC-DFTB method has proven to be an efficient tool for investigating atomic and electronic structures of defects in wurtzite GaN, like dislocations [8] and grain boundaries [9].

In wurtzite gallium nitride, the ( )a c

� �

+ - mixed threading dislocation lies along the >< 0001 direction with a >< 32113/1 Burgers vector. The presence of both edge and screw components, combined with the large magnitude of the Burgers vector (b = 6.08 Å), leads to a strong strain field which expands far away from the dislocation core. Hence, performing atomistic simulations of such dislocation cores re-quires large sized models. In the present study, the ( )a c

� �

+ - mixed dislocation was modelled in a large supercell-cluster hybrid of about 15 Å of lateral extension containing 750 atoms. Periodic boundary conditions were applied in the three space directions where the physical periodicity is kept along the dislocation line direction, i.e. >< 0001 , in order to reproduce its fundamental extended character. In the directions perpendicular to the dislocation line, a 190 a.u. vacuum thick was included to prevent artificial interactions between the primary supercell-cluster hybrid and those of the repeated cells. The Ga (N) dangling bonds, present at the model lateral surface, were saturated by fractionally charged 1.25e (0.75e) hydrogen atoms [8]. By avoiding surface reconstructions as well as electronic gap states associated with the surface dangling bonds, this procedure allows to mimic the crystalline bulk from where the model was extracted. Doubling the model along the dislocation line direction was found necessary to accurately describe the short ranged potential part contributing to the total energy. Initial positions of the atoms at the dislocation core were generated by applying the displacement field expected from linear isotropic elasticity [10]. The equilibrium positions were obtained through a minimization process based on the conjugate gradient algorithm where the SCC-DFTB was implemented. During this step, the lateral sur-face of the model was allowed to relax freely [11]. For integrals evaluation, the Brillouin zone was only sampled at the Γ point. The equilibrium is reached when the maximum force acting on each atom of the system is well bellow 0.001a.u.

3 Core structure Following Béré et al. [12], different starting core configurations were generated in the framework of linear elasticity, by changing the origin of the displacements field of the edge compo-nent in the two differently spaced { }0110 prismatic planes. The screw component was superimposed to the edge one by conserving the same origin of the displacements field. Viewed along the >< 0001 direc-tion, and similar to the case of pure edge dislocations, we obtained by this procedure two initial configu-rations of the ( )a c

� �

+ - mixed dislocation, that the core structures look like an 8- and 5/7-atoms rings. The relaxation, based on the SCC-DFTB method, of the previous initial models gave the core configura-tions presented in Figs. 1a and 1b. While the configuration with the 5/7-atoms ring was conserved (Fig. 1b), the 8-atom ring was spontaneously transformed to a complex double 5/6-atoms ring core structure (Fig. 1a). These results provide a confirmation of the core structures that we have already obtained using the modified Stillinger-Weber potential [6]. At variance with the 5/7-atoms ring core of the a

�

- edge dislocation, that of the ( )a c

� �

+ - mixed dislocation does not contain any wrong, Ga-Ga and N-N, bonds. This is due to the presence of the screw component. The complex structure of the double 5/6-atoms ring core may be seen as the result of a strong coupling between the edge and screw components inside the core. This configuration presents features that cannot be explained by assuming a simple superimposition

2942 I. Belabbas et al.: Atomic core structure of the ( )a c

� �

+ -mixed dislocation in wurtzite GaN

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

of an edge and screw component. The double 5/6-atoms ring configuration has two major features, the first is the occurrence of a species separation leading to the formation of two sub-columns, 3-Ga and 3-N, made of gallium and nitrogen atoms, respectively. These two sub-columns are only separated by about 0.76 Å. The second feature is the presence of gallium and nitrogen dangling bonds in columns (1) and (2), respectively. Analysis of the atomic structure showed that, in the 5/7-atoms ring core, the most constricted bonds (-5.81%) are established by the atoms of column (1). However, the most stretched bonds (+18.86%) are involved between the atoms at columns (2) and (3). In the double 5/6-atoms ring core, the relaxations that occur around the gallium and nitrogen dangling bonds gives a rise to com-pressed bonds (-8.91%) at the atoms of columns (1) and (2).

By using a microscope with a spatial resolution of 1Å, Arslan et al. have interpreted the core struc-ture of the ( )a c

� �

+ - mixed dislocation as an 8-atoms ring [5]. According to the above results, distin-guishing an 8-atoms ring from a double 5/6-atoms ring core would need a spatial resolution of better than 0.76 Å in order to distinguish the (3-Ga and 3-N) columns (Fig. 1a). Therefore, among the our two calcu-lated configurations, the double 5/6-atoms ring core is likely to correspond to the 8-atoms ring core proposed by the previous authors. Indeed, the experimental validation of the −+ )( ca

��

mixed dislocation core atomic configuration needs a sub-angstrom spatial resolution [13]. 4 Electronic structure Accurate electronic structures would be obtained, in principle, from full ab-initio based calculations. However, in the present case, the size of the models is out of reach of ab-initio techniques, the electronic structure of the ( )a c

� �

+ - mixed dislocation was then explored using the SCC-DFTB methodology. Over several applications, the latter method has proven to provide electronic struc-tures qualitatively in agreement with those obtained by more sophisticated ab-initio methods [14, 15]. Although we end up with deep and shallow levels in the bandgap for the two core configurations, the calculated electronic structure show some different features (Fig. 2a). This is attributed to their specific atomic core structures, (Figs. 1a and 1b).A sub-gap can be defined, for each core configuration, with respect to the position of the two bandgap levels L1 and L2. Level L1 is centred around the peak contrib-uted from the last occupied gap states, while level L2 is centred around the peak from the first unoccu-pied gap states. The two levels L1 and L2 are important, as they are likely to be involved in electronic transitions leading to parasitic luminescence. For the double 5/6-atoms ring core level L1 is located at 0.64 eV, from the top of the valence band maximum, and level L2 is situated at 1.91 eV, leading to a

(a) (b)

Fig. 1 Ball and stick models of the relaxed core configurations of the ( )a c

� �

+ - mixed dislocation. (a) The double 5/6-atoms ring structure, (b) the 7/5-atoms ring structure. The black balls represent gallium atoms while the white ones represent nitrogen atoms.

phys. stat. sol. (c) 4, No. 8 (2007) 2943

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sub-gap of 1.27 eV (Fig. 2b). For the 5/7-atoms ring core, level L1 is located at 0.46 eV, from the top of the valence band maximum, and level L2 is situated at 1.84 eV, leading to a sub-gap of 1.38 eV (Fig. 2c).

The calculation of the projected density of states (PDOS) allowed us to determine the contribution of the atoms located in the dislocation core to the levels L1 and L2. In the double 5/6-atoms ring core, both level L1 and L2 are due to the presence of dangling bonds. Indeed, while the nitrogen dangling bonds at column (2) give rise to the states round level L1, the gallium dangling bonds at column (1) contribute to the states of level L2 (Fig. 2b). Otherwise, in the 5/7-atoms ring core, levels L1 and L2 are due to the

Fig. 2 SCC-DFTB calculated electronic structures of the two core configurations of the ( )a c

� �

+ - mixed disloca-tion. (a) Superimposition of density of states (DOS) of the double 5/6-atoms ring and 7/5-atoms ring cores. (b) Density of sates and projected density of states (PDOS) related to the double 5/6-atoms ring core. (c) Density of sates and projected density of states (PDOS) related to the 5/7-atoms ring core.

(a) (b)

(c)

2944 I. Belabbas et al.: Atomic core structure of the ( )a c

� �

+ -mixed dislocation in wurtzite GaN

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

presence of compressed and stretched bonds. While, level L1 is contributed from the atoms involving compressed bonds at column (1), the level L2 is due to the atoms at columns (2) and (3) with stretched bonds (Fig. 2c).

5 Conclusion In the framework of the SCC-DFTB method, we have investigated the atomic core structure of the threading ( )a c

� �

+ - mixed dislocation. Two core configurations were obtained, with a double 5/6-atoms ring and a 5/7-atoms ring structures, confirming our previous results based on empiri-cal potentials. We have showed that distinguishing our model, with a double 5/6-atoms ring structure, from that, from the 8-atoms ring structure, reported by Arslan et al. would require a sub-angstrom spatial resolution. The investigated electronic structure showed that both core configurations introduce deep and shallow states in the bandgap.

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).

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