Abstract
Devising a computational tool that assesses the thermodynamic stability of materials is among the most important steps required to build a “virtual laboratory,” where materials could be designed from first principles without relying on experimental input. Although the formalism that allows the calculation of solid-state phase diagrams from first principles is well established, its practical implementation remains a tedious process. The development of a fully automated algorithm to perform such calculations serves two purposes. First, it will make this powerful tool available to a large number of researchers. Second, it frees the calculation process from arbitrary parameters, guaranteeing that the results obtained are truly derived from the underlying first-principles calculations. The proposed algorithm formalizes the most difficult step of phase diagram calculations, namely the determination of the “cluster expanison,” which is a compact representation of the configurational dependence of the alloy’s energy. This is traditionally achieved by a fit of the unknown interaction parameters of the cluster expansion to a set of structural energies calculated from first principles. We present a formal statistical basis for the selection of both the interaction parameters to include in the cluster expansion and the structures to use to determine them. The proposed method relies on the concepts of cross-validation and variance minimization. An application to the calculation of the phase diagram of the Si-Ge, CaO-MgO, Ti-Al, and Cu-Au systems is presented.
Similar content being viewed by others
References
R. Kikuchi: Phys. Rev., 1951, 81, pp. 988–1003.
J.C. Phillips and L. Kleinman: Phys. Rev., 1959, 116, pp. 287–94.
M. Stone: J. R. Stat. Soc. B Met., 1974, 36, pp. 111–47.
J.W. Connolly and A.R. Williams: Phys. Rev. B, 1983, 27, pp. 5169–172.
J.M. Sanchez, F. Ducastelle, and D. Gratias: Physica, 1984, 128A, pp. 334–50.
K.-C. Li: Ann. Stat., 1987, 15, pp. 958–75.
K. Binder and D.W. Heermann: Monte Carlo Simulation in Statistical Physics. Springer-Verlag, New York, 1988.
V.L. Moruzzi, J.F. Janak, and K. Schwarz: Phys. Rev. B, 1988, 37, pp. 790–99.
A. Qteish and R. Resta: Phys. Rev. B, 1988, 37, pp. 6983–990.
G. Ceder, M. Asta, W.C. Carter, M. Sluiter, M.E. Mann, M. Kraitchman, and D. de Fontaine: Phys. Rev. B, 1990, 41, pp. 8698–9701.
D. Vanderbilt: Phys. Rev. B, 1990, 41, pp. 7892–895.
S. de Gironcoli and P. Giannozzi: Phys. Rev. Lett., 1991, 66, pp. 2116–119.
F. Ducastelle: Order and Phase Stability in Alloys, Elsevier Science, New York, 1991.
L.G. Ferreira, S.-H. Wei, and A. Zunger: Int. J. Supercomput., 1991, 5, pp. 34–55.
A.S. Goldberger: A Course in Econometrics, Harvard University Press, Cambridge, MA, 1991.
D.B. Laks, L.G. Ferreira, S. Froyen, and A. Zunger: Phys. Rev. B, 1992, 46, pp. 12587–2605.
M. Asta, D. de Fontaine, and M. van Schilfgaarde: J. Mater. Res., 1993, 8, pp. 2554–568.
G. Ceder: Comp. Mater. Sci., 1993, 1, pp. 144–49.
G. Ceder, G.D. Garbulsky, D. Avis, and K. Fukuda: Phys. Rev. B, 1994, 49, pp. 1–7.
D. de Fontaine: Solid State Phys., 1994, 47, pp. 33–176.
G.D. Garbulsky and G. Ceder: Phys. Rev. B, 1994, 49, pp. 6327–330.
A. Zunger: First Principles Statistical Mechanics of Semiconductor Alloys and Intermetallic Compounds, in NATO ASI on Statics and Dynamics of Alloy Phase Transformation, Vol. 319, P.E. Turchi and A. Gonis, ed., Plenum Press, New York, 1994, pp. 361–93.
G.D. Garbulksy and G. Ceder: Phys. Rev. B, 1995, 51, pp. 67–72.
C. Wolverton and A. Zunger: Phys. Rev. B, 1995, 52, pp. 8813–828.
G.D. Garbulsky and G. Ceder: Phys. Rev. B, 1996, 53, pp. 8993–9001.
G. Kresse and J. Furthmüller: Comp. Mater. Sci., 1996, 6, pp. 15–50.
G. Kresse and J. Furthmüller: Phys. Rev. B, 1996, 54, pp. 11169–1186.
R. McCormack and D. de Fontaine: Phys. Rev. B, 1996, 54, pp. 9746–755.
P.D. Tepesch, A.F. Kohan, G.D. Garbulsky, and G. Ceder, C. Coley, H.T. Stokes, L.L. Boyer, M.J. Mehl, B.P. Burton, R.J. Cho, and J. Joannopoulos: J. Am. Ceram., 1996, 49, pp. 2033–40.
A.F. Kohan, P.D. Tepesch, G. Ceder, and C. Wolverton: Comp. Mater. Sci., 1998, 9, pp. 389–96.
V. Ozoliņš, C. Wolverton, and A. Zunger: Phys. Rev. B, 1998, 57, pp. 6427–443.
V. Ozoliņš, C. Wolverton, and A. Zunger: Phys. Rev. B, 1998, 58, pp. R5897-R5900.
V. Ozoliņš, C. Wolverton, and Alex Zunger: Phys. Rev. B, 1998, 57, pp. 4816–828.
A. van der Ven, M.K. Aydinol, G. Ceder, G. Kresse, and J. Hafner: Phys. Rev. B, 1998, 58, pp. 2975–987.
A. van de Walle, G. Ceder, and U.V. Waghmare: Phys. Rev. Lett., 1998, 80, pp. 4911–914.
C. Wolverton, V. Ozoliņš, and A. Zunger: Phys. Rev. B, 1998, 57, pp. 4332–348.
A. van de Walle and G. Ceder: Phys. Rev. B, 2000, 61, pp. 5972–978.
A. van de Walle: The MIT Ab initio Phase Stability (MAPS) Code, http://www.mit.edu/navdw/maps.
A. van de Walle and M. Asta: Model. Simul. Mater. Sci. 2002, 10, in press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van de Walle, A., Ceder, G. Automating first-principles phase diagram calculations. JPE 23, 348 (2002). https://doi.org/10.1361/105497102770331596
Received:
DOI: https://doi.org/10.1361/105497102770331596