First Principles Evaluation of Phase Stability in the In-Sn Binary System

J. Phase Equilib. Diffus. https://doi.org/10.1007/s11669-024-01109-8 ORIGINAL RESEARCH ARTICLE First Principles Evaluation of Phase Stability in the In-Sn Binary System Michael Widom1 Submitted: 22 November 2023 / in revised form: 31 March 2024 / Accepted: 3 April 2024 Ó The Author(s) 2024 Abstract The In-Sn binary alloy system exhibits several unusual features that challenge crystallographic and thermodynamic expectations. We combine first principles total energy calculation with simple thermodynamic modeling to address two key points. First, we evaluate energies along the Bain path to interpret the discontinuous transition between the phases a-In (Pearson type tI2) and b-In3Sn (also Pearson type tI2) that are identical in symmetry. Second, we demonstrate that the solid solution phases bIn3Sn and c-InSn4 (Pearson type hP1) exist at high temperatures only, and they exhibit eutectoid decompositions at low temperatures. Keywords Ab Initio methods  enthalpy of formation  phase diagram  thermodynamic stability 1 Introduction In-Sn alloys exhibit lower melting temperatures and improved thermal fatigue as compared with Pb-Sn solders;[1,2] they also exhibit superconductivity,[3,4] and they This invited article is part of a special tribute issue of the Journal of Phase Equilibria and Diffusion dedicated to the memory of Thaddeus B. ‘‘Ted’’ Massalski. The issue was organized by David E. Laughlin, Carnegie Mellon University; John H. Perepezko, University of Wisconsin–Madison; Wei Xiong, University of Pittsburgh; and JPED Editor-in-Chief Ursula Kattner, National Institute of Standards and Technology (NIST). & Michael Widom 1 form the basis for the transparent conductor Indium-TinOxide (ITO). In addition to their practical interest, their experimentally determined alloy phase diagram poses several scientific puzzles. This paper applies first principles total energy and band structure calculations, and simple thermodynamic modeling, to address these questions. The assessed In-Sn binary alloy system[5] exhibits four phases at room temperature and above, all with substantial composition ranges. In order of increasing fraction of Sn, the phases are: a-In, b-In3Sn (both share Pearson type tI2), c -InSn4 (Pearson type hP1), and b-Sn (Pearson type tI4, ‘‘white tin). There exists a different, low temperature, nonmetallic phase a-Sn (Pearson type cF8, ‘‘gray tin’’) that has low In solubility and is stable below 286 K. a-In and bIn3Sn are separated by a discontinuous transition with a narrow coexistence range around Sn fraction x&9-11%, despite sharing the same body-centered tetragonal structure and symmetry space group (I4/mmm). This violates the normal Landau-type model of solid-solid phase transformation that supposes group-subgroup relationships between phases. We resolve the puzzle by evaluating the energies along the Bain path of cubic $ tetragonal deformation. We also explore the differences in interatomic bonding between the a and b structures. The experimentally reported solubility range of c appears nearly temperature-independent and persists towards low temperatures. This suggests a low temperature configurational entropy, in apparent violation of the Third Law of Thermodynamics.[6,7] We propose that c actually decomposes eutectoidly as temperature drops. b-In3Sn also decomposes eutectoidly. The solubility of Sn in a-In vanishes at low temperature, while the solid solution b-Sn transforms to a-Sn with limited In solubility. Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 123 J. Phase Equilib. Diffus. 2 Methods Our calculations follow widely used methods.[8] We utilize the Vienna Ab-Initio Simulation Package VASP[9] to carry out first principles density functional theory (DFT) total energy calculations in the Perdew-Burke-Ernzerhof generalized gradient approximation.[10] We adopt projector augmented wave potentials[11,12] and maintain a fixed energy cutoff of 241.1 eV (the default for Sn). We relax all atomic positions and lattice parameters using the PREC Accurate precision setting, and increase our k-point densities until energies have converged to within 0.1 meV/ atom, then carry out a final static calculation using the tetrahedron integration method. Certain other settings are discussed below as needed. Our structures and phase diagrams are drawn from the ASM phase diagram database[13] and from the Inorganic Crystal Structure Database[14] (ICSD), supplemented with original publications. For solid solution phases we take 16-atom supercells at a variety of compositions and enumerate all possible configurations using enumlib.[15] The supercells of the a and b tI2 structures are 2x2x2. Supercells of c -InSn4.hP1 are based on an orthorhombic supercell of the hexagonal primitive cell. For b-Sn.tI4, we take a H2xH2x2 supercell. All configurations are fully relaxed, and only the lowest energy configuration is employed in the subsequent analysis. Given total energies for a variety of structures, we calculate the enthalpy of formation DHFor, which is the enthalpy of the structure relative to a tie-line connecting the ground state configurations of the pure elements.[16] Formally, for a compound of stoichiometry In1-xSnx with Sn fraction x we define DHFor ¼ H ðIn1x Snx Þð1  xÞH ðInÞxH ðSnÞ where all enthalpies are per atom. Vertices of the convex hull of DHFor constitute the predicted low temperature stable structures. For structures that lie above the convex hull, we calculate the instability energy DE as the enthalpy relative to the convex hull. 2.1 T? 0 K Limit Composition-dependent calculated formation enthalpies are displayed in Fig. 1. Notice that the known stable low temperature phases of pure In and Sn are at DHFor = 0, by definition, while all other formation enthalpies are positive. This implies that there are no thermodynamically stable compounds in the T ? 0 K limit. In particular, it supports the existence of a lower temperature limit for existence of the intermetallic b-In3Sn phase. It also resolves the apparent third-law violation of the c -InSn4 phase by showing that it does not extend to 0 K. 123 Fig. 1 Formation enthalpies DHFor of In-Sn alloys calculated within density functional theory. Solid lines are linear fits to calculated enthalpies Observe that the composition dependent energy of each phase is nearly linear in the composition, x. Small deviations from linearity reflect specific configurations representative of the solid solutions. Straight lines in Fig. 1 are least-squares fits constrained to pass through the endpoint at x = 0 or x = 1. We obtain DHa ¼ 0 þ 0:1x DHb ¼ 0:009 þ 0:1x DHg ¼ 40 þ 0:1ðx  1Þ DHbSn ¼ 41 þ 0:00017ðx  1Þ Although DHa \ DHb at x = 0, the lines cross in the vicinity of x = 0.1 (in the middle of the experimental coexistence range), and b is favored over a for larger amounts of Sn. The enthalpy of c turns up sharply for x \ 3/4 (not included in fit) because In-In neighbors cannot be avoided. (...truncated)