Ordered structures in III-Nitride ternary alloys

https://doi.org/10.1016/j.commatsci.2016.02.036Get rights and content

Highlights

  • The ordering in III-Nitride ternary alloys is investigated with evolutionary prediction methods.

  • Energetically preferable configurations are determined for the full range of compositions.

  • Highly ordered structures at x = 0.25, 0.50 and 0.75 induce minima on the formation enthalpy.

  • Results are verified by large scale molecular dynamics calculations.

  • The bandgaps of the ternary alloys are calculated and new composition-dependent bowing parameters are established.

Abstract

An efficient evolutionary structure prediction algorithm in combination with ab initio calculations is implemented in order to reveal energetically favorable superstructures of the III-Nitride ternary alloys. Several 2 × 2 × 2 32-atom supercells are used to explore the full range of concentrations, from x = 0 to 1. The formation enthalpies, bandgaps, clustering and/or ordering of the atoms are investigated and the results are discussed.

The formation enthalpy plots show local minima at specific concentrations, namely for x = 0.25, 0.50 and 0.75, that correspond to ordered structures. The valance band maxima, conduction band minima, bandgaps and the composition-independent bowing parameters for 2nd order Vegard’s equation are calculated. The bandgap deviations from 1st order Vegard’s law show total maxima at specific concentrations. The formation enthalpies and the bandgaps cannot be accurately described by single composition-independent bowing parameters, but the bandgaps are sufficiently described by composition-dependent bowing parameters, that are established.

In order to verify the rationality of the results against the size of the ab initio supercells, molecular dynamics calculations of 14 × 14 × 14 supercells of ∼104 atoms using bond-order interatomic potentials are performed. The obtained local minima of the formation enthalpy for the specific alloy compositions concur with those predicted by ab initio calculations proving the results are not influenced by the supercell size.

Introduction

Nitride alloys have been recognized as materials of significant importance for optoelectronic devices [1]. Although various studies on wurtzite nitride alloys have been reported [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], the effect of the metallic mixing (ordered-disordered structures) on the electronic and optical characteristics is still under investigation. There have been many theoretical studies for the bandgap dependence on the alloy concentrations [12], [13], [14], [15], [16], but the results are based on a large set of different structural configurations, disordered and ordered. The disordered structures can be described as randomly distributed metallic atoms in a supercell, while on the ordered structures the atoms occupy specific crystallographically defined sites in the lattice. In the experimentally observed cases disordered as well as ordered structural configurations may coexist in these alloys. However, in all the cases the arrangement of the metallic atoms in the structures is driven by the several growth parameters. In order to investigate the distribution of the metallic atoms in both the disordered and ordered structures, various computational approaches have been implemented [17], [18], [19], [20] taking into account the required formation energy. Initially Northrup et al. [21], proposed a model in order to explain simple 1 × 1 ordering with alternating Ga and Al/In rich layers. Recently Xu et al. [5], concluded by the use of first principles calculations in seven low-energy order structural configurations for 2 × 2 × 1 16-atom GaxAl1xN supercells. The concept of Special Quasirandom Structures (SQS) has also been employed [17]. The method generates structures by occupying all possible lattice sites in a given supercell and successfully provides optimal disordered states within the frame of a periodic supercell approximation. The exhaustive algorithms scale poorly with the number of atoms, limiting the method to small supercells. Gan et al. [15] used this method to calculate the thermodynamic properties of wurtzite and zinc-blende InxGa1xN alloys producing optimal, but still disordered, 32-atom structures for x = 1/4, 1/2 and 3/4. An alternative approach by de Carvalho et al. [16] uses the cluster expansion method [18], [19], in which InxGa1xN and InxAl1xN alloy structures are divided into a number of clusters, which are grouped into classes according to lattice symmetry. The authors used 16-atom 2 × 2 × 1 supercells and found more or less ordering along the main crystallographic directions. Atomic ordering has been identified also experimentally in GaxAl1xN and InxGa1xN alloys [6], [7], [8], [9], [10], [11], [22], [23], [24], [25]. Ordering in GaxAl1xN films grown by plasma assisted molecular beam epitaxy has been investigated by Moustakas et al. [9], [10], [11]. The researchers have found that GaxAl1xN exhibits a 1 × 1 monolayer cation ordering along the [0 0 0 1] direction. They have identified three types of spontaneously formed superlattice structures with periodicities of 2, 7, and 12 monolayers.

In an effort to clarify the issues of alloy ordering, we combine powerful evolutionary algorithms, density functional theory (DFT) calculations and molecular dynamics (MD) simulations with interatomic potentials (IP) in the investigation of the three III-Nitride ternary alloys. DFT models consisting of 32 atoms and MD models consisting of 104 atoms are used. The symmetry of the 2 × 2 × 2 supercell model used in the present study concluded on specific ordered structures exhibiting local formation energy minima by both IP and DFT based calculations at x = 0.25, 0.50 and 0.75 of the metallic atoms for the GaxAl1xN, InxGa1xN and InxAl1xN ternary alloys.

Section snippets

Computational methods

The USPEX structure prediction code [20] in combination with the VASP ab initio simulation package [26] is implemented in order to reveal energetically favorable structures of the GaxAl1xN, InxGa1xN and InxAl1xN alloys for the full range of compositions. USPEX employs a powerful evolutionary algorithm to predict the stable structures of a given composition of atoms. In this case bulk crystal enthalpy optimizations with fixed compositions are performed for the ternary alloys. In order to

Results

In Table 1, the lattice constants and bandgap values for AlN, GaN and InN calculated under LDA, GGA and HSE approaches are presented. In agreement with the literature [40] the obtained values show that LDA underestimates and GGA overestimates the equilibrium lattice parameters, while the parameters obtained from HSE are in good agreement with the experimental values. As shown in Table 1, the bandgaps are underestimated in LDA and GGA, i.e. InN has a negative bandgap in both methods. Again, the

Conclusions

In this work we investigate primarily the preferable structures of the nitride ternary alloys and secondly their energetics and structural and electronic properties. A 2 × 2 × 2 32-atom supercell is used and the energetically preferable structures are determined using the USPEX code in conjunction with the VASP package. The obtained structures are isomorphic for the three alloys and are described in detail. The formation energies of the structures show local minima when plotted against the cation

Acknowledgements

The authors are grateful to Prof. Th. Kehagias and Prof. G.P. Dimitrakopulos for valuable discussions. This research was co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: THALES, Project: NITPHOTO. Computations were supported by the LinkSCEEM-2 project, funded by the European Commission under the 7th Framework

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