Theory of Spontaneous Long-Range Order in Semiconductor Alloys

1. BACKGROUND

In the mid eighties, it was known that size mismatch between the atomic constituents leads to positive mixing enthalpy D H^(R) > 0 of random (R) alloys. The question we posed in the Fall of 1984 was: Does the fact that D H^(R) > 0 for all random isovalent semiconductor alloys preclude the formation of long-range order? The time-honored prevailing paradigm in metal alloys was that D H^(R) > 0 reflects the existence of fundamentally repulsive interactions between the alloy constituents, and since ordering requires attractive interactions, D H^(R) > 0 excludes the possibility of ordering. What we found in what became the first published paper on spontaneous long-range order of size-mismatched semiconductor alloys [1] was that this paradigm was incorrect, and that long-range order is, in principle, consistent even with D H^(R) > 0. The basic insight was that D H^(R) > 0 merely reflects the fact that in a random alloy, there is a distribution of many different local environments ("clusters," such as PGa₂In₂, PGa₃In₁, PGa₁In₃), and that the statistical average of their energies is positive, because some of these clusters are strained. But if one were to isolate a single cluster type and repeat it periodically in a strain-minimizing three-dimensional geometric arrangement, long-range order will ensue. This basic observation published in early 1985 [1], started the pursuit by theorists, and experimentalists of long-range order in size-mismatched semiconductor alloys. The first experimental observations of ordering in size-mismatched semiconductor alloys were made by the NEC group of Gomyo and Suzuki, and by the Utah group of G. Stringfellow just a couple of years later.

In the intervening years since this first discovery, we have learned a great deal more about this problem, including the role of surface reconstructions in stabilizing the ordering and the way in which the optical properties of ordered alloys are distinct from those of random alloys. Because ordering doubles the crystallographic unit cell, while altering the point group symmetry from T_d to C_3V, new optical transitions, crystal-field splittings and phonon modes have emerged. This phenomenology created significant experimental and theoretical excitement.

The fact that by inducing ordering one could alter the materials properties of an alloy without altering its chemical composition (thus, retaining its lattice-matching with a given substrate) opened novel technological opportunities. In 1990, a "2% New Initiative" proposal was submitted to OER-BES-DMS on a coordinated growth (Olson, III-V’s and Furdyna, II-VI’s), spectroscopy (Mascarenhas), and theory (Zunger) study of ordering. Because GaInP was, at the time, the alloy of choice for photovoltaic application, most of the experimental work has focused on it.

This section of the write-up summarizes what we have learned from our theoretical studies of ordering. The theoretical work in this project had a few distinct functions:

1. Understanding of the "basic physics" (i) driving ordering; and, (ii) the effects of ordering on material properties.

2. Explaining available experimental observations on ordered materials (mostly GaInP).

3. Predicting hitherto unmeasured "fingerprints" and novel properties of ordered materials.

4. Extending our knowledge and predictions to spontaneous ordering in novel semiconductor/insulator systems, including: (i) vacancy ordering in oxides Li_{x ð 1-x}(CoO₂); (ii) ordering in nitrides; (iii) ordering in chalcopyrites, e.g., Cu_xIn_1-x(Se₂).

The theoretical work has thus focused on the following areas of research:

1. The causes of spontaneous ordering (studying bulk, epitaxial, and surface thermodynamics).

2. The consequences of spontaneous ordering on material properties.

3. The prediction of ordering in new systems.

4. Conceiving new ideas for future work on Ordering.

We will next review our progress in these areas, emphasizing research in the last three years. We will also describe the essential results from earlier years so as to clarify the evolution of the research. Each section title includes references to papers published on that subject.

2. TECHNICAL ACCOMPLISHMENTS

A. THE CAUSES OF ORDERING

(i) Early Work: Bulk and Epitaxial Effects [1-15; 18-20]

Our early work on causes of ordering focused on bulk ordering [1]–[7], [18, 19, 20, 21] and epitaxial ordering [8]–[15], [43]. Concerning bulk ordering, we have used the first-principles pseudopotential method to calculate the total energy of various assumed bulk ordered phases of many III-V and II-VI alloys, contrasting the energies with those of the random phase. The latter was computed as a statistical average of the energies of the various local environments that exist in a random alloy. These studies have shown that even though the random alloy has a positive mixing enthalpy D H^(R)(x)>0, some special 3D ordered atomic arrangements can remarkably minimize strain and maximize charge-transfer, hence become stabler! For AlInP₂ and AlInAs₂ (having strong charge transfer) the local density approximation (LDA) predicted [19] D H^(O) < 0 for the ordered (O) chalcopyrite phase, leading to stability of bulk ordering. For all other III-V compounds, we found that while the ordered phase has positive excess enthalpy D H^(O) > 0, for the chalcopyrite (CH) structure D H^(CH) < D H^(R), so this structure can order metastabily in bulk. We found that other crystal structures (e.g., CuPt) are "topologically frustrated," i.e., do not possess enough geometrical degrees of freedom to permit all chemical bonds to attain their ideal length. Thus, these networks are intrinsically strained and, therefore, bulk unstable. After this work was published, Stringfellow et al., have observed chalcopyrite ordering in III-V alloys. These findings were reviewed in Ref. 29.

Concerning epitaxial ordering [8] – [15], we have discovered that coherence with the substrate can convert the previously predicted metastable bulk ordering into stable epitaxial ordering. The reason is as follows: In the bulk, D H^(CH) > 0, because the relaxed alloy constituents A+B are lower in energy than the ordered AB phase. But on a coherent substrate, phase-separation of AB into A+B is discouraged because these coherently-matched constituents (A-on-substrate and B-on-substrate) are highly strained. Thus, on a substrate, D H^(CH) < 0 not because the ordered chalcopyrite phase is stabilized, but because the alternative to ordering (phase separation) is destabilized! We cause the ordered chalcopyrite phase is stabilized, but because the alternative to ordering (phase separation) is destabilized! We applied these novel concepts to Si-Ge, Si-C [8] and III-V’s [9]-[15] computing, for the first time, temperature-composition phase diagram of epitaxially ordered alloys. The significance of this work was the establishment of novel mechanism for epitaxial stabilization of bulk-metastable structures.

(ii) Recent Work: Reconstruction-induced Ordering

It was clear to us in 1989 that in addition to bulk and epitaxial effects, there must be other driving forces for ordering, since the "CuPt ordered phase" discovered experimentally by Gomyo and Suzuki was shown by us theoretically to be unstable both according to bulk and epitaxial thermodynamics. We then embarked on a new idea: calculation of surface-induced ordering mechanism [22, 25, 26, 27, 48, 49, 55, 69]. What we discovered [22] beginning in 1991 was that the most common feature of semiconductor surfaces – the existing of atomic dimerization at the surface – creates an energetic incentive for the Ga and In atoms in the subsurface layers to adopt ordered positions akin to the CuPt structure. This idea, used earlier by Legous et al., to explain binary ordering in Si-Ge, was applied by us to ternary III-V alloys by performing LDA and valence force field calculations. We considered cation dimerization [22] as well as anion dimerization [25, 26, 27] predicting for the first time a strong surface-thermodynamic driving force for CuPt ordering. (Phillips and Norman et al., in the U.K., repeated a similar suggestion a few years later). The significance of this work was to establish: (i) the first theoretical calculation of a surface-induced ordering mechanism in III-V alloys; (ii) a "dictionary," [48] connecting different reconstruction patterns [(2 x 4); (4 x 2); (4 x 4); (2 x 6)] with different forms of long-range order (CuPt_A, CuPt_B, triple-period); (iii) explaining how surface solubilities can exceed, by many orders of magnitude, bulk solubility [55], and; (iv) relating surface reconstruction with surface segregation [49, 69]. Our work explained the previously observed CuPt_B ordering, and was confirmed by subsequent experiments of Stringfellow et al., and Suzuki et al., who demonstrated the link between reconstruction and ordering: they succeeded in altering the ordering patterns by manipulating the surface reconstruction through changing growth parameters.

B. THE CONSEQUENCES OF ORDERING

This part of our study was aimed at understanding how ordering of the random alloy changes electronic, vibrational, structural, mechanical and magnetic properties of the material.

• A new formula relates the degree of ordering to the band gap reduction [29]. As-grown alloys are not perfectly ordered. The degree of long-range order (LRO) can vary between h =0 (random) to h =1 (fully ordered). While it was known that spectroscopic quantities must depend on h , it was not known what is this dependence. In 1992, using lattice-gas statistical mechanic theories [38], we derived [29] the "h ² rule," leading to simple formulas that relate the valence band splitting D E_VB and the band gap reduction D Eg to h . The significance of this work was that if D Eg(h =1) was given, one could deduce h of a given sample from the measurement of D Eg(h ) or D E_VB(h ). A large number of experimental papers used in subsequent years this theoretical result to determine how different growth conditions result in different h values. Examples include Fluegel et al., PRB 55, 13,647 (1997); Wirth et al., APL 71, 2127 (1997); and Shubert et al., PRB 54, 17,616 (1997). Indeed, the "h ² rule" became a standard tool for spectroscopic interpretation of data.
• Predicting the maximal band gap reduction and valence band splitting upon ordering: Use of the "h ² formula" requires knowing D Eg(h =1) and D E_VB(h =1). These quantities are not known experimentally. We predicted them both at the LDA level and at the LDA-corrected level for many III-V and II-VI alloys [16, 17, 33, 34, 35, 52, 53, 63]. This work allowed experimentalists to determine from their measured D Eg and D E_VB the value of h . We also provided predictions for many yet-unmeasured materials. Those predictions show which materials are expected to have the largest band gap reduction upon ordering, and how different forms of ordering ("CuPt," "chalcopyrite," "CuAu," etc.) leads to different band gaps. The significance of this work was the establishment of magnitudes and trends in spectroscopic quantities pertaining to ordering in a wide range of materials and ordered structures.
• Ordering is predicted to convert GaP_0.5As_0.5 [23a] and Ga_0.5Al_0.5As [23b] into direct-band gap materials. While the respective 50%-50% random alloys, or the (001) monolayer superlattices of GaP/GaAs and GaAs/AlAs have indirect band gaps, we predicted that CuPt ordering will convert these materials to direct band gaps (because the L-folded state repels the G _1c conduction band downwards, so it becomes lower than X_1c). Our prediction was recently confirmed by Yamashita et al. [PRB 53, 15,713 (1996)] for Al_x(GaIn)_1-xP, while for GaPAs it was confirmed by T. Takanohashi and M.Ozeki [JJAP 30, L956 (1991)]. We further showed how deviations from ordering via "interfacial roughness" can reverse the order of conduction-band levels in ordered AlGaAs₂ [30]. This work is significant because it demonstrated a qualitative spectroscopic consequence of ordering.
• Ordering predicted to lower the band gap of InAs-Sb alloys sufficiently to create far IR detectors [24]. While random InAsSb alloys already have a small band gap, we predicted that CuPt ordering can shift significantly the band gap to yet longer wavelengths [24]. Later, this quantitative prediction was seen by the Sandia group, Kurtz et al., APL (1992), who observed ordering-induced-band gap reduction in their IR materials, as described by our theoretical prediction. The significance of this work is that ordering was shown to lead to a technically useful spectroscopic effect.
• Ordering is predicted to lead to spin-polarization [36]. Spin-polarized photoelectrons are needed in a variety of high-energy-physics experiments. They are usually obtained from strained GaAs. It occurred to us that ordering (which leads to a natural crystal-field splitting) can also lead to spin-polarization even without strain. We calculated [Ref. 36] this effect predicting strong spin-polarization in (single-variant) ordered GaInP₂. It was later observed by Kita et al. [PRB 57, R 15,044 (1998)].
• Ordering is predicted to change the X-ray structure factors [63]. In principle, high-precision x-ray diffraction can be used to directly measure t In principle, high-precision x-ray diffraction can be used to directly measure the degree h of LRO. This requires comparison with h =1, fully ordered x-ray structure factors that are not accessible experimentally. To this end, we have computed the Fourier transform of the LDA charge density, thus providing the x-ray fingerprints of full ordering. Recent x-ray measurements of ordered alloys include the data of Forrest et al. The x-ray measured h [Forrest et al., PRB 58, 15355 (1998)] agrees reasonably well with the value deduced optically via fitting the band gap reduction data to our predicted D Eg values.
• Deduction of the degree of ordering from measurement of the polarization anistropy [34, 35]. Ordering leads to an anistropy of the optical transitions [34]. We predicted that the measured anistropy would be fit to our theory of spectral anisotropy, thus deducing via the fit the degree of LRO. The experiment of Kanata et al. [APL 63, 26 (1993)], provided the data that led to a successful determination of h via this method.
• Ordering predicted to alter the pressure dependence of semiconductor band gaps [31]. Since ordering folds L or X states into the Brillouin zone center, and since admixture of these states alters the band gap pressure dependence, we offered that measurement of this effect can serve as a new fingerprint of ordering. We predicted [31] significant changes in pressure coefficient of the first two conduction bands on pressure. These predictions await experimental testing.
• Ordering predicted to increase the electron effective mass, even though k· p models predict the mass to be reduced [47]. The band gap reduction attendant upon ordering will contribute to a reduction of the electron mass. At the same time, the folding and consequent admixture of the heavier L_1c mass into the G _1c state will increase the mass. First-principles calculations [47], which include both effects predict a net increase in mass. Earlier k· p calculations included only the first effect, predicting a reduction. Subsequent experiments by Ernst et al., [JAP, 81, 2814 (1997)] predict an increase in mass, in agreement with first-principles calculations.
• Theory examines whether NMR can be used to determine the degree of ordering [59]. A paper by Mao et al. [PRL 76, 4769 (1996)] offered to combine NMR measurements of the electric field gradient (EFG), with a point-ion model of the effect to deduce the degree of h of LRO. We examined this intriguing possibility by performing all-electron first-principles calculations of the electric field gradients [59] seen by NMR. Unfortunately, we find that the point-ion model used by Mao et al., does not capture correctly the EFG which, according to our calculation, is dominated by covalency effects. Hence, measurement of EFG alone cannot be used to determine h without the intervention of (a complicated first-principles) theoretical model.
• Theory predicts the phonon fingerprints of ordering [61]. Recent experiments in Spain, France, and at NREL showed clear evidence of effects of ordering on the phonon spectra. Using ab-initio linear response theory (LRT), we have calculated phonon frequencies and phonon eigen modes of both ordered and disordered GaInP. While part of our calculation has confirmed previous assignments of phonon modes, we have also offered new assignments of previously observed modes, based on our access to computed polarizations and mode frequencies. Some of our suggested new assignments were contested by experimentalists (Alsina et al.) who offered another sequence of modes ("model 1C"). We have recently shown (unpublished results) that this model is incorrect, and proposed instead a "model 1b". The significance of this work was in providing the first theoretical prediction of phonon signature of ordering, and assisting in experimental assignment.
• Theory predicts Reflectance Difference Spectroscopy (RDS) effects in ordered alloys [41]. The existence of a natural polarization of the interband transitions due to the symmetry reduction attendant upon ordering, suggested to Olson et al., using RDS as a measure of ordering. A theory of RDS for CuPt ordering was developed [41]. It points to the relationships between measured RDS anisotropy and the degree h or LRO. This improves over previous RDS theories, thus allowing a better connection between RDS measurements and the degree of LRO.
• Calculation of fingerprints of ordering on high-energy optical transitions [42]. Although it was first customary to measure the effects of LRO only on the lower energy direct gap E₀, other high-energy transitions (E₁, E₂, E'₀) occurring in the UV could also hold the key to understanding ordering. We have developed a theory that shows how the change in symmetry due to ordering shifts and splits these high energy transitions [42], finding very unusual effects. Recent experiments started probing these high energy transitions, e.g., Alsina et al. [22^nd ICPS Conference Proceedings], and Kita et al. [PRB 54, (1996)], finding good agreement with our calculations.
• Theory contrasts the effects of short-range-order (SRO) vs. long-range-order (LRO) on optical properties [44-46]. While many have focused on detecting the effects of LRO on various optical properties, it is also possible that SRO (e.g., clustering of like atoms, or "anti-clustering" of dissimilar atoms) could have a similar effect on optical properties. In a series of papers [44-46], we constructed models of SRO in semiconductor alloys, establishing the spectroscopic distinguishing features between LRO and SRO. The significance of this research was in theoretically establishing, for the first time, how SRO affects carrier localization and band gap reduction, and how these effects differ from LRO effects.
• Prediction of ordering-induced polarization electric fields and their effects on band offsets [49]. The existence of a unique [111] axis in CuPt ordered alloys suggests the possibility of a piezoelectric field. We have calculated this field for perfect ordering, finding that in the absence of screening of this field by free carriers, it will have a non-negligible magnitude. While experiments by Ernst et al. [ICPS 23 Proceedings, p. 469 (96)] have confirmed the presence of a large field, Perkins et al. [JAP 84, 4502 (98)] place a considerably lower bound on the magnitude of this field (however, their sample had many free-carriers that might have screened part of the field). The question of the magnitude of ordering-induced electric fields is thus still open, awaiting new experiments. In addition to this calculation, we have also predicted the band offset between GaAs and GaInP with varying degrees of LRO. Interestingly, the band offset D E(h ) can change from type-I to type-II by changing h ! Recent experiments by Martinez and Yu have used these results to analyze their data. The significance of discovering that there is a finite band offset between ordered and disordered GaInP₂ is that this suggested the possibility of forming quantum wells and barriers with the same material. This was accomplished experimentally by Stringfellow et al. [APL, 73, 3905 (1998)].

C. THE PREDICTION OF ORDERING IN NEW SYSTEMS

Much of the experimental work on ordering has centered around GaInP, largely because of the availability of good samples, produced in studies of high-efficiency photovoltaic solar cells. A significant part of our theoretical work has, therefore, also centered around GaInP. However, as theorists, we feel that we should attempt to understand a broader set of "generic materials." To this end, we have extended our interest to "ordering in new systems." Examples follow:

• Ordering in conventional III-V’s other than GaInP and in II-VI alloys [6, 16, 17, 18, 19, 37, 52, 63]. The theoretical work described above on optical and structural properties of ordered alloys were extended to conventional III-V alloys other than GaInP₂. These included AlGaAs₂; GaInAs₂; AlInP₂; AlInAs₂; GaInSb₂; Ga₂PAs and Ga₂AsSb; although we have not repeated all calculations for each and every system, we did investigate the main chemical trends in ordering characteristics. While some experimental measurements exist sporadically on such non-GaInP₂ systems (e.g., GaInAs₂), there are no systematic studies in the literature to compare with our calculations. In addition, we extended calculations on ordering-induced optical properties to the II-VI alloys CdZnX₂ (X=Se, Te), HgCdTe₂ and HgZnTe₂. Experiments are unavailable. The significance of these theoretical studies is that they create a "menu" for experimentalists to choose novel systems for studying ordering.

• Ordering in chalcopyrites [28, 32, 57, 68]. We found four types of ordering in CuInSe₂-type chalcopyrite systems: (a) the disorder-order (zincblende to chalcopyrite) transition as a function of temperature [28]; (b) the order-disorder transitions existing in the quaternary system (CuInSe₂)_x(ZnSe)_1-x, predicted in [32] in which the system transforms from chalcopyrite to zincblende as the composition x diminishes; (c) the three-dimensional ordering of Cu vacancies in CuInSe₂-based structures, studied in [57]; and, (d) the ordering of polytypes studied in Ref. [68]. In all cases, we have studied theoretically these ordering transitions using a combination of first-principles total-energy calculations for ordered structures (which yields "effective interactions") with Ising-like description (solved via Monte-Carlo) for the statistical description of order/disorder mediated by the "effective-interactions". The basic computational method is described in Ref. [38]. These studies were partially conducted in collaboration with the PV program. These systems are fascinating, revealing novel types of ordering effects in semiconductor alloys which are very different from those seen in GaInP₂. As part of our future plans (see below), we would like to expand in this direction.

• Ordering in nitride alloys [54, 60]. Nitride alloys such as GaAs-GaN and GaN-InN became very popular because of their importance for blue lasers and photovoltaics. Current experiments have not reached the 50%-50% composition range, where CuPt ordering might be maximal. Nevertheless, we decided to predict theoretically what will ordering do. We find that in random GaAs_1-xN_x, the band gap drops very fast as the nitrogen composition increases, but that it is always positive. However, if we assume CuPt ordering at x = 0.5, we find a negative band gap (metallic system!) This exciting prediction deserves further theoretical and experimental study.

• Vacancy ordering in oxides [64, 65]. LiCoO₂ is a very interesting novel battery material. The electrochemical action is afforded by moving Li ions in-and-out of the lattice. The system is, thus, a "binary alloy" Li_{1-x € x}, where "€ " denotes a vacancy, and where Co and oxygen are fixed "spectator atoms". We speculated that one can find on this lattice ordering of vacancies. We have thus applied to this system the same method [38] we used for GaInP₂ ordering, namely perform total energy calculations on a set of assumed, vacancy-ordered configurations, extract from these calculations "effective interactions," and then solve the ensuing Ising model via Monte-Carlo techniques. We find a fascinating series of vacancy-ordering transitions as a function of temperature [64]. The significance of this work is that a new type of ordering transitions were found in a ABX₂ "semiconducting" system, and that unlike the case of GaInP₂, where the ordering had primarily optical consequences, here ordering has important electrochemical consequences [64, 65]. The results of this theoretical research are pursued experimentally by the battery research group at NREL.

PAST ACCOMPLISHMENTS OF THE THEORY PROGRAM ON ORDERING:

1. Published 23 papers in 1984-1991 (before the start of the BES initiative), and 45 papers 1991– Present (during the initiative). This includes:

   • Sixteen Physical Review Letters

   • 11 Rapid Communication articles

2. Presented two APS March meeting Invited Talks.

3. Delivered sixteen other Invited Talks.

4. Outstanding sustained achievement award in Solid State Physics," given to A. Zunger in 1998 by BES-DMS, in part due to this work on theory of ordering.

5. Cover page of "MRS Bulletin" (Vol. 22, 1997) on our work.

6. Was invited to organize symposia on "ordering in semiconductors," in the Electronics Materials Conference – 1997, 1998, 1999.

7. Published a highly cited book-size review article on the subject [Ref. 37].

ARTICLES PUBLISHED BY NREL’S THEORY GROUP ON ORDERING IN SEMICONDUCTOR ALLOYS (in approximate chronological order):

(Refs. 1-23 are before the start of the new initiative, and were funded by BES "Semiconductor Theory" KC02.0203. Ref. 24-69 are funded fully, or in part, by the new initiative KC02.0103, on ordering.

1. G. P. Srivastava, J. L. Martins, and A. Zunger, "Atomic Structure and Ordering in Semiconductor Alloys," Phys. Rev. B Rapid Communications 31, 2561-2564 (1985).

2. A. Zunger, "Ternary Semiconductors and Ordered Pseudobinary Alloys: Electronic Structure and Prediction of New Materials," Int. J. Quantum. Chem. 19, 629-653 (1986).

3. J. L. Martins and A. Zunger, "Ordering and Decomposition in Semiconductor Alloys," J. Mat. Res. Rapid Communications 1, 523-526 (1986).

4. A.A. Mbaye, L.G. Ferreira and A. Zunger, "First-Principles Calculation of Semiconductor Alloy Phase Diagram," Phys. Review Letters 58, 49-52 (1987).

5. D.M. Wood, S.H. Wei and A. Zunger, "Thermodynamic Instability of Ultrathin Semiconductor Superlattices: The (001) (GaAs)₁ (AlAs)₁ Structure," Phys. Review Letters 58, 1123 (1987).

6. J. E. Bernard, L. G. Ferreira, S.-H. Wei, and A. Zunger, "Ordering of Isovalent Intersemiconductor Alloys," Phys. Rev. B Rapid CommunicationsS.-H. Wei, and A. Zunger, "Ordering of Isovalent Intersemiconductor Alloys," Phys. Rev. B Rapid Communications 38, 6338-6341 (1988).