Monte Carlo simulations applied to AlxGayIn1xyX quaternary alloys X=As,P,N:A comparative study
M. Marques,1 L. G. Ferreira,2 L. K. Teles,1 and L. M. R. Scolfaro1,*1Instituto de Fsica, Universidade de So Paulo, Caixa Postal 66318, 05315-970 So Paulo, So Paulo, Brazil
2Instituto de Fsica Gleb Wataghin, Universidade Estadual de Campinas, Caixa Postal 6165, 13083-970 Campinas, So Paulo, BrazilsReceived 18 October 2004; published 13 May 2005d
We develop a different Monte Carlo approach applied to the AxByC1xyD quaternary alloys. Combined withfirst-principles total-energy calculations, the thermodynamic properties of the sAl,Ga, IndX sX=As, P, or Ndsystems are obtained and a comparative study is developed in order to understand the roles of As, P, and Natoms as the anion X in the system AlxGayIn1xyX. Also, we study the thermodynamics of specific composi-tions in which AlGaInN, AlGaInP, and AlGaInAs are lattice matched, respectively, to the GaN, GaAs, and InPsubstrates. We verify that the tendency for phase separation is always towards the formation of an In-richphase. For arsenides and phosphides this occurs in general for lower temperatures than for their usual growthtemperatures. This makes these alloys very stable against phase separation. However, for nitrides the In and/orAl concentrations have to be limited in order to avoid the formation of In-rich clusters and, even for lowconcentrations of In and/or Al, we observe a tendency of composition fluctuations towards the clustering of theternary GaInN. We suggest that this latter behavior can explain the formation of the InGaN-like nanoclustersrecently observed in the AlGaInN quaternary alloys.
DOI: 10.1103/PhysRevB.71.205204 PACS numberssd: 61.66.Dk, 64.75.1g, 71.20.Nr, 71.22.1i
In the development of heterostructure-based devices, thedouble requirement of high-quality crystalline layers epitaxi-ally grown on a given substrate with low-misfit-dislocationdensity and an optimized electronic structure are generallyvery difficult to achieve using the common binary and ter-nary compounds. Quaternary semiconducting alloys of theAxByC1xyD kind are very interesting materials, becausetheir use can be considered as an effective approach to re-duce defect density in the heterostructures. Such systems al-low the independent control of both lattice parameter andband-gap energy sthrough x and yd, avoiding the lattice mis-matching and, at the same time, providing an adjustable en-ergy gap for barriers and active layers. In this sense, thelattice-matched systems such as, e.g., AlGaInAs/ InP andAlGaInP/GaAs, have been extensively studied from the ex-perimental point of view. More recently, the AlGaInN/GaNsystem has been studied, showing aspects not observed be-fore. In spite of the great number of experimental works onthe quaternary semiconductor alloys, there are only a fewtheoretical studies,1 mainly due to the complex treatment ofthese systems in a more rigorous way. Particularly, the theo-retical works that involve quaternary alloys make use of verysimplified models, because the more rigorous treatments,such as first principles and Monte Carlo thermodynamics, arevery difficult to apply. Therefore, it is highly desirable tohave a method which gives very rigorous results togetherwith a reasonable computational effort. In the present work,we develop a different Monte Carlo approach, to be usedtogether with first-principles, self-consistent, total-energycalculations for the study of quaternary alloys. We apply thisapproach to study the phase-separation process of the seriesof III-V face-centered-cubic sfccd pseudoternary semicon-ductor alloys AlxGayIn1xyAs, AlxGayIn1xyP, and
AlxGayIn1xyN, in which a microscopic description of thephase separation is performed. We make a comparative studyand an individual study of each alloy for the lattice-matchedsystems of experimental importance.
The motivation for the study of the AlxGayIn1xyAs/InPlattice-matched system arises from its importance for deviceapplications relevant to optical communications such asemitters, waveguides, lasers, and infrared detectors.2,3 This ismainly because the band-gap energy range covered bylattice-matched quaternary alloys overlaps the region ofminimum loss and dispersion current s0.81.2 eVd for opti-cal fibers. The lattice-matched condition with an InP sub-strate is sGa0.47In0.53AsdzsAl0.48In0.52Asd1z, with z from 0 to1, providing a direct energy-gap variation from0.74 to 1.45 eV. However, a shorter lasing wavelength is re-quired for high-density optical information processing sys-tems. Despite the fact that AlP and GaP binary compoundspresent an indirect energy gaps, the phosphide ternaries andquaternaries may have higher direct energy gaps than thearsenides. Particularly, the sAlxGa1xd0.5In0.5P quaternary al-loy is lattice matched to GaAs and, except for the nitrides, ithas the largest direct energy gap among the III-V semicon-ductors, with the emission wavelength being tunable fromred to green by changing the amount of Al. For laser diodes,the AlGaInP forms the barrier with the active layer being theGaInP ternary compound or even the quaternary compoundwith a lower Al concentration. The main commercial interestin devices based on these systems is in the continuing evo-lution of compact-disk technology snow based on theAlGaAs/GaAs system, providing a 780-nm emissiond to-wards the digital-video-disk sDVDd technology. The currentgeneration of DVDs uses an AlGaInP red laser with anemission wavelength of 650 nm. Shorter wavelengthsshigher band gapsd, though desirable, lead to poor AlGaInP-sample quality and an indirect band gap. These two factorsimply a very-low-emission efficiency.
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Bulk AlGaInP, like AlGaInAs, is very stable against clus-tering or phase separation. In the whole compositional rangeof the quaternary lattice matched to GaAs, a good structuralquality and high compositional uniformity is obtained.4However, there are surface effects that lead to differentphases rather than solid solution. Through the combined ef-fects of the surface thermodynamics and kinetics, the com-position modulation and the CuPt-ordered structure can existtogether in the AlGaInP matrix.5 The ordering dramaticallyaffects the electronic properties of the material, and in par-ticular reduces the direct-energy-gap. Therefore, this phase isundesirable for device applications as it leads to longerwavelength emission, and, because the ordering is not uni-form, the resulting large-crystal inhomogeneity likely leadsto inferior device performance. But the generation of orderedstructures can be suppressed by several means, such as theincrease of the growth temperature6 above 700 C, the use ofmisoriented substrates, and the use of p-type doping.7
In the last few years, great progress has been made in theresearch of GaN and related semiconductors, which presentlarger energy gaps than the phosphides and arsenides. As aresult, blue-green light-emission diodes sLEDsd as well asultraviolet sUVd laser diodes sLDsd have been commer-cialized.8 Recently, the AlGaInN quaternary alloys attractedmuch attention due to the fact that lattice-matched materialscan be obtained with a possible energy gap in the deep-UVregion. In addition, the incorporation of Indium in the AlGaNternary alloy, forming the AlGaInN quaternary alloy has nowbeen demonstrated to improve the optical quality of the alloylayer for the UV-emission alloy,9 even when the Al content isincreased. Until now it had not been possible to grow good-quality material in the whole compositional range, and themajority of the samples presents an In concentration lowerthan 4%. But, despite these problems, an increasing numberof experimental works on AlGaInN have been presented. Thesuccessful applications are the recently produced nearlylattice-matched AlGaInN/GaInN UV LDs,10 theAlGaInN/AlGaInN deep-UV LEDs, and the UV LDs.11,12These optical devices can emit a wavelength smaller than400 nm, which is a great improvement in our capacity forinformation storage. Nevertheless, questions concerning thiscomplex system remain still open. For example, the emissionmechanism involved in the UV spectra is frequently associ-ated to the existence of In-rich phases or GaInN-like clusters,and not to the band-to-band transition in the quaternary alloyitself.13,14 Moreover, some samples show a green emissionaround 2.4 eV besides the UV emission.15 These facts lead tothe possibility of a phase-separation process in this alloy.Several works show evidence of alloy inhomogeneities, withthe formation of possible clusters in the matrix of thealloy.13,14 But the questions of how the In nucleation takesplace in the bulk of the AlGaInN quaternary alloys sin whichcomponents the alloy separatesd and what the relation is be-tween the In-separated phases remain under discussion.
Therefore, we not only develop an approach for the studyof AxByC1xyD quaternary systems, but also present a rigor-ous and systematic theoretical study of the thermodynamicproperties of some important quaternary systems, from
which we can obtain new features of their phase-separationprocesses. The paper is organized as follows. In Sec. II wedescribe the details of the calculation methods. In Sec. III wediscuss alloy stabilities and analyze the experimentally rel-evant lattice-matched systems. Finally, a summary is given inSec. IV.
II. CALCULATION METHODS
In this section we describe the main ideas behind the com-putational methods used in this paper.
A. The ternary expansion and the Monte Carlo approach
In a ternary alloy AxByC1xy or pseudoternary squater-naryd alloy AxByC1xyD, the sites of a crystal lattice areoccupied by A, B, and C atoms in different configurations. Toperform Monte Carlo studies one requires, in principle, asampling of the 3N possible configurations of A, B, and Catoms in N lattice sites, where N is about 104. This is aformidable task for first-principles electronic-structure meth-ods, as it involves a huge number of calculations. To circum-vent this problem it is necessary to describe any arrangementof N atoms in terms of a few arrangements of a much smallernumber of atoms. The classical way to accomplish this de-scription is by means of a cluster expansion. In the case ofbinary sAxB1xd or pseudobinary sAxB1xCd alloys, cluster ex-pansions have reached a high degree of sophistication.1618In the case of quaternary alloys, though the theory is welldeveloped,19 cluster expansions are not as useful, becausethey are not as simple.
In this paper, instead of cluster expansions we develop adifferent approach. In particular, we study AxByC1xyD al-loys that crystallize in the zinc-blende structure, where theatoms A, B, and C occupy one fcc sublattice, and the atomsD the other fcc sublattice, although the basic idea of themethod can be applied for other kinds of structures. We as-sume an unstrained alloy, which means that there is no con-straint on the lattice parameter and the alloy is allowedto have its own lattice constant. In our approach, we considerall arrangements of atoms A, B, and C in an enlarged peri-odic fcc lattice with repeating unit vectors s0aad, sa0ad, andsaa0d. These are twice the primitive vectors
FIG. 1. The hexahedron unit cell sstretched cube along a bodydiagonald with the respective eight cation sites. a1W , a2W , and a3W arethe primitive vectors.
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s0,a /2 ,a /2d , sa /2 ,0 ,a /2d , and sa /2 ,a /2 ,0d, of the fcc lat-tice so that the larger unit cell contains eight original fccsites. The unit cell, which is a hexahedron sstretched cubealong a body diagonald, and the eight cation sites are sche-matically shown in Fig. 1. The number of possible arrange-ments of atoms in the enlarged unit cell is 38=6561. Mostarrangements are related by the rotation-inversion transla-tions of the zinc-blende space group. Grouping thesymmetry-related arrangements we obtain 141 classes.20This will be the number of first-principle energy calculationsused to describe all the arrangements sconfigurationsd of thesystem.
The eight-site unit cell can be chosen from any of the fourlisted in Table I and pictured in Fig. 2. The sites of the fourhexahedra are related by the enlarged fcc translations withunit vectors s0aad, sa0ad, and saa0d. The extreme sites of thefour hexahedra are a tetrahedron with vertices at saaad,saaad, saaad, saaad. sIn the case of Table I and Fig. 2 we seta=2.d The many arrangements of atoms in a hexahedron canbe represented by ternary numbers, which unequivocallyidentify the arrangement. In order to have a ternary number,first we call the A, B, and C atoms 0, 1, and 2, respectively.Consider, for example, a possible configuration, where the
eight sites of our unit cell are occupied by the atoms C, A, B,C, C, B, A, and B; in our new notation they correspond to 2,0, 1, 2, 2, 1, 0, and 1, respectively. Each site of the unit cellcorresponds to a number with a basis of 3 and an exponent nfrom 0 to 7 in such a way that the site 1,2,3,,8 correspondsto 37 ,36 ,35 , . . . ,30. Therefore, the example configuration isrepresented by the ternary number 2337+0336+1335+2334+2333+1332+0331+1330=201221013=4843.Thus, there are only 141 ternary numbers corresponding toarrangements not related by symmetry.
Table II lists the configurations of atoms 0, 1, and 2 in acell of N sites, having the largest space groups. The configu-rations are ordered by the size of their space groups ssize ofthe zinc-blende space group/size of the configuration spacegroupd. Many N-site configurations are special arrangementsof atoms in the hexahedral cell. These are indicated in thetable by the corresponding ternary number in the last col-umn. Many configurations are superlattices, as indicated inthe second column of the table. In the third column of thetable we indicate the name of the configuration, when it isknown. For those N-site configurations that are not arrange-ments in the eight-site hexahedral cell, we indicate their lin-ear combinations of ternary number arrangements in the lastcolumn of Table II. In what follows we explain how to findthese linear combinations.
We define the energy of any configuration of atoms in theN sites of a Monte Carlo cell as follows. First, the first-principle total energies for the 141 classes are calculated. Inour case we adopt a first-principles pseudopotential plane-wave code within density functional theory and the localdensity approximation sDFT LDAd, the Vienna Ab InitioSimulation Package sVASPddetails will be given in Sec.II B.21 Since for each ternary T in the range 0T381 weknow to which class it belongs, the first-principle calcula-tions lead to the knowledge of the energy function esTd of aternary. Next, assume that the total energy of the MonteCarlo cell is a sum of the site energies, which in turn is anaverage of the energies of the four...