Phase separation and ordering in group-III nitride alloys

Phase separation and ordering in group-III nitride alloys

L. K. Teles^I; M. Marques^I; L. M. R. Scolfaro^I; J. R. Leite^I; L. G. Ferreira^II

^IInstituto de Física, Universidade de São Paulo, CP 66318, 05315-970 São Paulo, SP, Brazil

^IIInstituto de Física Gleb Wataghin, Universidade Estadual de Campinas, CP 6165, 13083-970 Campinas, SP, Brazil

ABSTRACT

The electronic, structural, and thermodynamic properties of cubic (zinc blende) group-III nitride ternary In_xGa_1–xN and quaternary Al_xIn_yGa_1–x–yN alloys are investigated by combining first-principles total energy calculations and cluster expansion methods. External biaxial strain on the alloys are taken into account in the calculations. While alloy fluctuations and strain effects play a minor role in the physical properties of AlGaN, due to the small lattice-mismatch, in the InGaN alloys we found a remarkable influence of strain on the phase separation. The effect of biaxial strain on the formation of ordered phases has been also investigated, and the results are used to clarify the origin of the light emission process in In_xGa_1–xN based optoelectronic devices. The dependence of the lattice constant and energy bandgap in quaternary Al_xIn_yGa_1–x–yN alloys was obtained as function of the compositions x and y.

PACS: 61.66.Dk, 64.75.+g, 71.20.Nr

1 Introduction

In the last years great progress has been made on the development of optical and electronic devices based on group-III nitride compounds InN, GaN, AlN, and their alloys. Laser diodes (LDs) and light-emitting diodes (LEDs) operating in the short wavelength visible and ultraviolet (UV) spectral regions have been successfully fabricated [1, 2]. An usual feature of these device structures is the utilization of ternary (In_xGa_1–xN, Al_xGa_1–xN, or In_xAl_1–xN) and quaternary (Al_xGa_yIn_1–x–yN) alloys. The alloying between III-nitride compounds allows a wide-range band gap engineering, which covers from infrared, » 0.7-0.9 eV (InN) till deep UV, 6.3 eV (AlN) [3, 4]. Some examples are: i) high-frequency and high-temperature electronic devices, such as field-effect transistors using AlGaN/GaN quantum wells (QWs) [5]; ii) the recently produced UV- LEDs and LDs comprising quaternary AlGaInN/AlGaInN multiple QWs [6, 7, 8, 9]; iii) the light emitters, which usually comprise InGaN as the active layer. Although the devices are currently produced, the mechanism of light emission for example in these InGaN layers is still subject of discussion. It has been argued that self-organized nanometer scale In-rich quantum dots (QDs), originated from In-segregation taking place in the In_xGa_1–xN alloys, are the source of a radiative recombination channel emitting in the blue-green region of the spectrum [10, 11, 12, 13]. It was also shown that the effect of biaxial strain induced by a pseudomorphic growth process in InGaN leads to phase separation suppression effects [14]. In addition to the phase separation process, chemical ordering on the group-III sublattice of InGaN has been reported [15, 16]. Although these ordered phases have been already experimentally observed, there is no systematic theoretical study of the possible existence of bulk ordered structures and the stability of them, as well as, its relation with biaxial strain.

As mentioned, the interest in the quaternary AlGaInN alloys has recently been increased due to their importance to fabricate efficient UV LEDs and LDs. The possibility of alloying using In, besides of Al, makes the Al_xGa_yIn_1–x–yN alloy more flexible, e.g. allowing for the production of lattice-matched AlGaInN to GaN layers. The lack of strain in such structures is responsible for much more efficient luminescence arising from the quaternary alloy layer region [17]. Despite the already existent experimental work on quaternary AlGaInN alloys, very few attempts have been conducted on theoretical predictions of their physical properties [18].

In this work, we present a theoretical study of electronic, structural, and thermodynamic properties of group-III nitride ternary In_xGa_1–xN and quaternary In_xAl_yGa_1–x–yN alloys. The study is performed by carrying out first-principles total energy calculations, in the framework of the density functional theory and the local density approximation, in combination with cluster expansion methods. The energetics and thermodynamic properties of In_xGa_1–xN are analyzed in detail, with emphasis on the role played by the biaxial strain and the formation of ordered phases. In case of quaternary Al_xGa_yIn_1–x–yN alloys, we show the results obtained for the lattice parameter and band gap energy as functions of the alloy compositions x and y.

2 Calculation methods

In order to access the physical properties of the alloys, we used first-principles total energy calculations together with a cluster expansion method (CEM), within two approaches, the generalized quasi-chemical approximation (GQCA) and through Monte Carlo (MC) simulations. The calculated properties shown here for unstrained InGaN alloys were obtained through the GQCA [19, 20]. For the strained InGaN alloy pseudomorphically grown on a rigid zinc blende GaN (001) buffer layer, an Ising expansion for the alloy energy and the MC approach have been adopted [21, 22, 23, 24]. The GQCA method used by us to study the ternary nitrides was generalized to treat the quaternary AlInGaN alloys [19, 20].

Total energy and electronic structure calculations of each basic configuration required by the CEM are carried out by using a pseudopotential method (Vienna Ab-initio Simulation Package - VASP) within the framework of the density functional theory and the local density approximation [25]. Two structures are assumed for InGaN, zinc blende, to describe the fully relaxed (unstrained) alloy and tetragonal, to simulate the biaxially strained alloy. Eight- and sixteen-atom supercells were adopted for the calculations of the ternary and quaternary alloys, respectively. Details of the calculation methods may be found elsewhere [19, 20, 26].

3 Results and Discussion

A. In_xGa_1–xN ternary alloys

Figure 1 shows the calculated alloy mixing free energy, DF, and corresponding phase diagram, T × x, for the unstrained ternary In_xGa_1–xN alloy. The phase diagram (shown in the lower part of Fig. 1) is easily accessed from the curves of DF (shown in the upper part of Fig. 1) [19, 20]. For typical growth temperatures, T » 900 K, there is a large miscibility gap, therefore indicating that fully relaxed InGaN alloys present phase separation effects, or spinodal decomposition, with an In-rich phase of x » 0.8. Very similar features are obtained when MC simulations are used. This result is in very good agreement with x-ray diffraction and Raman spectroscopy measurements performed on cubic InGaN thick layers grown on GaAs(001) substrates by molecular beam epitaxy [27].

Contrary to the case of thick InGaN layer samples, in which the InGaN alloy is unstrained, there are evidences that the presence of a biaxial strain may drive the formation of ordering domains in III-V alloys [28]. The energetics and thermodynamic stability of ordered phases in InGaN were studied by combining the ab initio total energy calculations with the CEM and MC simulations. The generalized Ising Hamiltonian, or the energy expansion in terms of multisite interaction energies allows us to search for the ground state of In_xGa_1–xN, by comparing the energies of different structures at the same alloy composition x. The results can be summarized as follows: i) The biaxial strain induced by the GaN buffer suppresses the tendency of the alloy to phase separate and acts as the driving force to form ordered structures at certain stoichiometric compositions [14]. There are several ordered structures with energies close to the minimum, which leads to the conclusion that a mixture of ordered phases or domains of ordered phases with different In content may arise depending on the growth conditions. ii) The lowest energies for the strained alloys occur at the neighborhood of the [3,3] ordered superlattice (corresponding to alternate planes of In and Ga, in number of three of each kind) implying that the composition x @ 0.5 is the most favorable ordered phase, i.e., In_0.5Ga_0.5N.

Ordering in the alloy takes place only below a critical temperature, the stability limit temperature, T_c. Temperatures higher than T_c drive the alloy to a disordered phase. Fig. 2 shows schematically the ordered/disordered transition for the [3,3] In_0.5Ga_0.5N superlattices, arising from our MC simulations (T_c = 1487 K).

Recently, high-resolution x-ray diffraction (HRXRD) and photoluminescence (PL) experiments were combined to show the presence of strained nanometer scale In-rich phases (QDs) in cubic GaN/In_xGa_1–xN/GaN double heterostructures (DHs) with InGaN layer thickness of 30Å [13]. Fig. 3 shows the distribution of the scattered x-ray intensities in reciprocal space (reciprocal space map) of the asymmetric (3) Bragg reflexes of one of the GaN/InGaN/GaN DHs with alloy layer composition x = 0.33. A strained In-rich phase with x = 0.54 is clearly observed in the map. Since the [3,3] structure is favored in the theoretical calculations, we may conclude that the observed In-rich phase is an ordered structure. Transmission electron microscopy (TEM) experiments on the DHs are underway to confirm this prediction.

B. Al_xGa_yIn_1–x–yN quaternary alloy

We now discuss the results for the quaternary AlGaInN alloy. Fig. 4 shows the calculated lattice constant a(x,y) for Al_xGa_yIn_1–x–yN. We observe, as in the case of other ternary alloys, a linear behavior of a with the composition. If we consider any composition of one of the alloy components as being fixed, the Vegard's law is fulfilled , in other words a(x,y) = xa_AlN + ya_GaN + (1 – x – y)a_InN, with a_XN being the lattice constant of the binary alloy XN, X = Al,Ga,In. Excellent agreement is observed between theory and experimental data obtained from XRD measurements (squares and triangles in the figure) [29, 30].

In Fig. 5 the calculated direct (G – G) energy band gap for Al_xGa_yIn_1–x–yN is depicted as a function of Al and Ga contents, x and y. The available experimental data as obtained from PL measurements [30] are also shown for comparison. Good agreement is seen between the theoretical and experimental values. The energy band gap of the quaternary alloy can be well described by an analytical surface equation, which may be useful in future investigations.

4 Conclusion

By using ab initio total energy calculations and cluster expansion methods we show that: i) fully relaxed In_xGa_1–xN layers undergo phase separation, and ii) biaxial strain induced by thick GaN (001) buffers drives the alloy to separate into ordered phases. Both effects, phase separation and ordering, give rise to self-organized nanometer-scale quantum dots in the In_xGa_1–xN layers. The fact that the ordered structure with lowest energy is a superlattice with a period which comprises 3 planes of Ga and 3 planes of In, leads to the conclusion that the strained alloy ground state has a composition x = 0.5. We suggest that this finding explains recent TEM and XRD experiments which revealed the presence of ordered In_0.5Ga_0.5N phase in the hexagonal In_xGa_1–xN alloy [15, 16]. Besides, it also explains the presence of In-rich phase with composition of about 0.5 detected from recent XRD experiments performed on cubic GaN/In_xGa_1–xN/GaN double heterostructures [13]. Monte Carlo simulations with Metropolis algorithm allowed us to calculate the critical temperature (T_c) for the ordered/disordered transition taking place in the strained alloy. For the ordered structure with lowest energy (In_0.5Ga_0.5N), the value T_c = 1487 K was found. This value is above the usual growth temperatures of the In_xGa_1–xN layers, as expected. The results presented here concerned to the formation of quantum dots in InGaN layers are important to establish the origin of the luminescence in the nitride-based LDs and LEDs, and to extend the operation of these devices to shorter wavelengths.

The understanding of the electronic, structural and thermodynamic properties of quaternary Al_xGa_yIn_1–x–yN alloys is at the beginning. By using the generalized quasi-chemical approximation we calculated the lattice parameter a(x,y) and energy band gap E_g(x,y) of these alloys. While a(x,y) fulfils a Vegard's-like planar law, E_g(x,y) deviates from a planar behavior, displaying a two-dimensional gap bowing in the x,y-plane. The results presented here are very important for tailoring electronic and optoelectronic devices based on Al_xGa_yIn_1–x–yN alloys.

Acknowledgments

We acknowledge K. Lischka, D. J. As, D. Schikora, F. Bechstedt, and J. Furthmüller for the fruitful collaboration. This work was supported by FAPESP, CNPq, and CAPES, Brazilian funding agencies.

Note Added in Proof: ''An extensive Cluster Expansion revealed that the ground state of the InGaN alloy is a [2,2] (not [3,3]) superlattice in the direction (012) (not (001)) and that the critical temperature for ordering is around 1000 K (not 1400 K)".

Received on 9 March, 2003

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Publication Dates

• Publication in this collection
  31 Aug 2004
• Date of issue
  June 2004

History

• Received
  09 Mar 2003