Accuracy of ab initio methods in predicting the crystal structures of metals: A review of 80 binary alloys
Introduction
First principles computation, whereby the properties of materials are predicted starting from the principles of quantum mechanics, is becoming well integrated with more traditional materials research. A list of ab initio studies on binary and ternary alloy phase stability up to 1994 can be found in Ref. [1]. Since the earliest, completely ab initio computation of a binary phase diagram [2], the approaches for computing the total energy of a solid have significantly improved, and computing resources have continued to become faster and less expensive. We believe that a point has been reached where, with a reasonable amount of resources, high throughput first principles studies of a large number of alloys can be performed [3], [4], [5], [6], [7], [8]. In this paper we present the results of a first principles study of 14 080 computed total energies on 176 crystal structures in 80 binary alloys. All energies were computed in the Local Density (LDA) or Generalized Gradient Approximation (GGA) to Density Functional Theory, which are standard approaches for first principles studies on solids. To our knowledge this is the largest first principles study of its kind on alloys. As we have compared the results in every system to experimental compilations, this study also offers a statistical test on the accuracy of some current ab initio approaches in correctly predicting the structure of materials.
For 89 compounds we find unambiguous agreement between experiment and the ab initio computation (Table 5), giving some indication of the predictive power of modern ab initio electronic structure methods. For many systems, verification of the ab initio results is difficult, as the systems have been poorly or incompletely characterized, or only high temperature information is available experimentally. For most of these system, we make predictions that are consistent with the limited available information. Even though our library of 176 crystal structures is, to our knowledge, the largest library of ab initio energies ever produced, there are still 27 compounds for which we cannot verify the experimental structures as they are not in our library. We have not included such prototypes because they are extremely rare and complicated (many atoms per unit cell) (Table 8).
Overall, we find remarkably few significant discrepancies between the ab initio predictions and the experimental observations (Table 9). On the basis of the experimental data in Refs. [9], [10], we find only nine compounds for which LDA/GGA seems to predict the ground state incorrectly. For four of these nine systems, the experimental ground state is within less than 10 meV /atom of the ab initio ground state. For the remaining five systems, there are at least two in which further investigation indicates that the experimental structure assignment is poorly justified, leaving three compounds for which a significant disagreement between experiment and ab initio LDA/GGA is likely. Such disagreements are addressed in Section 4.
We believe that the low ratio of unambiguous errors (3) to the number of unambiguous correct predictions (89) is encouraging, and establishes clearly the potential of predicting crystal structure correctly with ab initio methods.
We also predict the stability of five new crystal structures which, to the best of our knowledge, have not yet been observed in any system: an AB3 superstructure of the fcc lattice, stable for CdPt3, PdPt3 and Pd3Pt, an AB bcc superstructure for MoTi, an AB3 bcc superstructure for MoTi3, Mo3Ti, Nb3Tc, RuTi3 and TcTi3, an A2B2 hcp superstructure for RhRu, and an A2B4 hcp superstructure for RhRu2 (Appendix A). In addition, we find two new crystal structures which are not superstructures of fcc, bcc or hcp: Mo5Ti (Mo5Ti and Nb5Ru) (Appendix B).
Section snippets
Binary alloys
Our calculated library contains 80 binary intermetallic alloys. The alloys include the binaries that can be made from row 5 transition metals, as well as some systems with aluminum, gold, magnesium, platinum, scandium, sodium, titanium, and technetium. The alloys are: AgAu, AgCd, AgMg, AgMo∗, AgNa, AgNb∗, AgPd, AgRh∗, AgRu∗, AgTc∗, AgTi, AgY, AgZr, AuCd, AuMo∗, AuNb, AuPd, AuPt∗, AuRh∗, AuRu∗, AuSc, AuTc∗, AuTi, AuY, AuZr, AlSc, CdMo∗, CdNb∗, CdPd, CdPt, CdRh, CdRu∗, CdTc∗, CdTi, CdY, CdZr, CrMg
Ultra Soft Pseudopotential LDA calculations (US-LDA)
Most of the energy calculations of the library were performed using density functional theory in the Local Density Approximation (LDA), with the Ceperley–Alder form for the correlation energy as parameterized by Perdew and Zunger [18] with ultra soft Vanderbilt-type pseudopotentials [19], as implemented in VASP [20]. Calculations are at zero temperature and pressure, and without zero-point motion. The energy cutoff in an alloy was set to 1.5 times the larger of the suggested energy cutoffs of
Discussion and summary of results
In comparing the stable structures predicted by the ab initio computations with available experimental information, we have attempted to classify the results in a few distinct categories. Table 5 gives the compounds where the ab initio result and experiments are in unambiguous agreement. The fact that there are a large number of compounds (89) in Table 5 is a positive statement about the accuracy of LDA/GGA in capturing the close energetic competition between the 176 structures in our library.
Conclusions
Overall, the comparison between experimental data and ab initio results is encouraging. A large number of the ground states are predicted correctly, even though significant competition exists between various structures, indicating that relative energy differences are well reproduced by LDA/GGA. In many cases, the direct comparison between ab initio and experiments is difficult, as systems have often not been studied completely, or have not been studied to low enough temperature to make a
Alloys without ab initio compounds
Table 4 gives the alloys for which we find no compounds with negative formation energy, and the structure with lowest formation energy in the system. All of these agree with experiments except for the ones described below.
Ag–Au (silver–gold)
The system Ag–Au has not been studied in great detail at low temperatures, and no intermetallic compounds have been reported [9], [10], [51], [52], [53], [54], [118]. The solid is reported to be short-range ordered fcc, though it is suggested that long-range order might exist at low temperature. At low temperature we find several stable compounds: Ag4Au, Ag3Au, Ag2Au, AgAu-L10, AgAu2 and AgAu3. In our electronic structure approach, the ground states are degenerate for Ag3Au, Ag2Au, AgAu2 and
Trend for technetium alloys
In our set of calculations, we have noticed that the phase D019 appears in systems MTc3 where M is a transition metal in the columns on the right of Tc (Tc is in column 7B) while D019 is not present if M is in the columns on the left of Tc: D019 is stable in PdTc3, PtTc3, RhTc3, RuTc3, and unstable in NbTc, TcTi, TcY, and TcZr.
Experimental compounds in agreement with ab initio solutions
See Table 5.
Experimentally unknown, non-identified or speculated compounds and ab initio predictions
See Table 6.
Experimental solid solutions, two-phase systems and “not studied” regions and possible ab initio predictions
See Table 7a, Table 7b.
Experimental compounds that could not be checked by our calculations
See Table 8
Experimental compounds in disagreement with other ab initio compounds or two-phase regions
See Table 9.
Acknowledgments
The research was supported by the Department of Energy, Office of Basic Energy Science under Contract No. DE-FG02-96ER45571 and National Science Foundation Information Technology Research (NSF-ITR) Grant No. DMR-0312537.
It has benefited from discussion with John Rodgers, Kristin Persson, Frank Hadley Cocks, Chris Fischer, Aleksey Kolmogorov, Vincent Crespi and Zi-Kui Liu.
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