First-principles high-throughput screening of shape-memory alloys based on energetic, dynamical, and structural properties

First-principles-based materials screening is systematically performed to discover new combinations of chemical elements possibly making shape-memory alloys (SMAs). The B2, D03, and L21 crystal structures are considered as the parent phases, and the 2H and 6M structures are considered as the martensitic phases. 3,384 binary and 3,243 ternary alloys (6,627 in total) with stoichiometric composition ratios are investigated by the materials screening in terms of energetic and dynamical stabilities of the martensitic phases as well as structural compatibility between the parent and the martensitic phases. 187 alloys are found to survive after the screening. Some of the surviving alloys are constituted by the chemical elements already widely used in SMAs, but other various metallic elements are also found in the surviving alloys. The energetic stability of the surviving alloys is further analyzed by comparison with the data in Materials Project Database (MPD) to examine the alloys which may occur phase separation or transition.


I. INTRODUCTION
Shape-memory alloys (SMAs) constitute an important class of materials in industrial use because of their shape-memory effects and pseudoelasticity [1]. Already various kinds of SMAs have been well known.
Ni-Ti (Nitinol) alloys are now widely used because of its working range around room temperature, good pseudoelastic property, phase stability, and so on [2]. The Ni-Ti alloys, however, suffer from large thermal hysteresis unless additional elements are included [3]. Toxicity of Ni in these alloys is also problematic for bioapplications [4]. Cu-based SMAs such as Cu-Zn, Cu-Al, and their multicomponent alloys are commercially available because of their low price, but they also have disadvantages such as instability of martensitic phase, brittleness [5], and poor thermomechanical performance [6]. Discovery of new SMAs are therefore still needed for better functional stability, design of working temperature [7,8], and other special purposes such as nontoxic biomaterials [4] and ferromagnetic SMAs [9].
To design SMAs, their working temperature and functional stability may be the most important engineering properties. Working temperature is critical especially to design high temperature SMAs (HTSMAs). Functional stability is also important to improve the reliability of SMAs. We can expect the working temperature is related to the energetic stability of the martensitic phase relative to that of the parent phase (see Sec. III.A). Meanwhile, some reports have found that better structural compatibility between the parent and the martensitic phases results in smaller thermal hysteresis, which gives better functional stability [10,11].
We can adjust working temperature and functional stability of SMAs by replacing their constituent elements with others. The working temperature of Ni-Ti alloys can be much increased to the range of 400-1200 K by the total or partial replacement of Ni and Ti with the same group elements, namely Pd or Pt [12,13] and Zr or Hf [14], respectively. These reports imply that the martensitic phases become energetically more stable than the parent phases at the low temperature by replacing constituent elements. Meanwhile, better functional stability, which is associated with smaller thermal hysteresis and functional fatigue, can be achieved by adding Cu [11] or Pd [10] in Ni-Ti alloys and Cr in Ti-Pd alloys [15].
To discover new materials by modifying their constituent elements, computational research is useful because of its efficiency compared with experimental search. Especially, materials screening based on systematic first-principles calculations prior to experimental synthesis is useful to limit the searching space.
Thanks to the recently improved computational machine power, now we can perform high-throughput firstprinciples calculations for various combinations of chemical elements with various crystal structures for searching new materials. Actually many different materials systems have been investigated in this manner [16][17][18][19]. Hautier et al. [16] and Hinuma et al. [17] have identified new ternary-oxide and zinc-nitride semiconductors, respectively, using the "prototype" crystal structures of Inorganic Crystal Structure Database (ICSD) [20]. Carrete et al. [18] have reported the semiconductors with the half-Heusler structure that show low thermal conductivity. Greeley et al. [19] have reported the binary surface alloys to show a good electrocatalytic property.
In this study, we perform the first-principles high-throughput screening to discover new combinations of chemical elements that are possibly available as SMAs. As the screening conditions, we consider the energetic and dynamical stabilities of the martensitic phase, as well as the structural compatibility between the parent and the martensitic phases, which is probably related to the functional stability. These screening conditions require relatively low computational costs and are therefore suitable for the high-throughput materials screening. 6,627 alloys are initially considered, and 187 survive as the candidates of new SMAs after the screening. We investigate which elements mainly participate in the surviving alloys. Finally, the energetic stability of surviving candidates are further analyzed by comparison with the first-principles data in Materials Project Database (MPD) [21] to examine the alloys which may occur phase separation or transition.

A. Elements and crystal structures of investigated alloys
We consider binary and ternary alloys with stoichiometric composition ratios composed of 48 metallic elements between Li and Bi. Specifically, we consider 1,128 (= 48 C 2 ) XY, 2,256 (= 48 P 2 ) X 3 Y, and 3,243 (= 3 × 47 C 2 ) X 2 YZ (X = Ti, Cu, Zn) alloys with the B2, D0 3 , and L2 1 parent-phase structures, respectively. These parent-phase structures are derived from the body-centered cubic (bcc) structures [22] as described in Fig. 1. For the martensitic phases, we consider the orthorhombic 2H (or B19 for the binary alloys with the B2 parent phase) and monoclinic 6M [in the Otsuka notation [23,24], which will be used hereafter to correctively refer to 9R (for B2) and 18R (for D0 3 and L2 1 ) in the Ramsdell notation [25]] structures. Both the 2H and 6M martensitic-phase structures have the close-packed basal plane, but they have different stacking orders: "AB" for the 2H structure and "ABCBCACAB" for the 6M structure. Figure 2(a) shows the crystal structures of the 2H and 6M for ternary X 2 YZ alloys with the L2 1 parent phase. The structure of the 2H is explicitly calculated in this study, while the structure of the 6M is estimated from that of the 2H as described later. The space-group type of the 2H structure is Pmma for the B2-parent XY alloys and Pnma for the D0 3 -parent X 3 Y and for the L2 1 -parent X 2 YZ alloys.
The lattice basis for the 2H martensitic structure is given as where a 2H , b 2H , and c 2H are the lattice constants of the 2H structure. The unit cell of the parent phase that changes to L 2H after the martensitic transformation may be given as and where a p , b p , and c p are the lattice basis of the conventional unit cell for the parent phase structures, and a p is their lattice constant. The deformation gradient [26] F 2H is then obtained as Figure 2(b) describes the martensitic transformation between the L2 1 parent and its 2H martensitic phases. It should be emphasized that the martensitic transformation path is similar to the Burgers path [27,28] for pure metals, which describes the transformation between the bcc and the hexagonal close-packed (hcp) structures.
In order to reduce computational costs, the crystal structure of the 6M is estimated from that of the 2H as follows. We first assume that their basal-plane structure and layer distance along the stacking direction are the same for both the 2H and 6M. We further assume that the stacking position of each layer is different exactly by a 2H /3 along a 2H . Then, the lattice basis for the 6M structure is given as The unit cell of the parent phase that changes to the L 6M after the martensitic transformation may be given as and The deformation gradient F 6M is then obtained as Figure 3 shows the flowchart of the materials screening.

B. Materials-screening conditions
Firstly, we check whether the space-group type of the optimized structure of the 2H martensitic phase (G m opt ) is actually the same as the expected one (G p init ). For many investigated alloys, the 2H structure is optimized to the parent-phase structure or to some other structure. Such alloys are excluded from the screening because they cannot have the assumed martensitic structure. The space-group types of the optimized structures are checked using the SPGLIB library inside the PHONOPY code [29,30].
Secondly, we investigate the energetic stability. Here we check whether the energy of the martensitic phase (E m ) is smaller than that of the parent phase (E p ). Only the alloys satisfying ΔE m-p ≡ E m − E p < 0 survive. We also guarantee that the 2H martensitic phase is energetically more stable than pure metals as references. Here, the formation energy of the martensitic phase relative to those of pure metals in their most stable crystal structures, E f m , must be smaller than zero.
Thirdly, the structural compatibility between the parent and the martensitic phases is considered. For this purpose, we use the transformation stretch tensors [26] U 2H and U 6M , which are positive-definite and symmetric matrices. These matrices are obtained from F 2H and F 6M using the polar decomposition as and where R 2H and R 6M are rotation matrices. James et al. have shown using their model that when the second largest eigenvalue λ 2 of U (hereafter the U 2H or U 6M are collectively referred to as U) is equal to one, the two phases can make a distortionless interface [26], which is intuitively expected as an advantage for showing better functional stability. Actually, several SMAs with small thermal hysteresis and functional fatigue are found by modifying the composition ratios to realize λ 2 close to one [10,15,31,32]. Based on these reports, we adopt |λ 2 − 1| < 0.01 as a screening condition. We also consider the volume difference between the parent and the martensitic phases, because the large volume difference is expected to cause huge stress between the two phases and to result in low functional stability [32]. Actually, several materials systems such as lithiumion batteries, whose applications are related to their phase transitions, are known to have good advantage of reliability when they show small volume differences [33]. The relative difference between the volume of the parent (V p ) and the martensitic (V m ) phases are obtained as det(U), and hence we adopt |det(U) − 1| < 0.01 as another screening condition.
Lastly, the dynamical stability of the martensitic phase is investigated. For this purpose, we analyze the phonon frequencies of the martensitic phases ω ph m . The alloys with imaginary phonon frequencies, i.e., {ω ph m } 2 < 0 for some phonon modes, are screened out, because the existence of the imaginary phonon frequencies indicates that the crystal structure is dynamically unstable. The phonon calculations are performed under the harmonic approximation on the lattice Hamiltonian. Force constants of the alloys are calculated from their supercell models based on density functional perturbation theory [34] at the Γ point, and then phonon frequencies are calculated from the force constants. Phonon dispersion curves and density of states (PhDOSs) are used to confirm the dynamical stability of martensitic phase by the existence of the imaginary phonon frequencies. Note that the dynamical stability of the 6M martensitic structure is assumed to be the same as that of the 2H martensitic structure. The phonon calculations are performed using the PHONOPY code [29,30].

C. First-principles calculations
The first-principles calculations are performed by the project augmented wave (PAW) method [35,36] implemented in the Vienna Ab-initio Simulation Package (VASP) [37,38] within the framework of the generalized gradient approximation of Perdew-Burke-Ernzerhof form [39]. The cutoff energy is set to 400 eV. The volume and shape of the cells and internal atomic coordinates are fully relaxed until residual forces acting on atoms reach below 0.005 eV/Å. The structure optimization is performed for a primitive-cell model to reduce computational costs. Table I shows detailed computational conditions for the primitive cell, supercell for the phonon calculations, and k-space sampling. Both the nonmagnetic (NM) and the ferromagnetic (FM) states are calculated for each system, and the lower-energy states are investigated in the subsequent materials screening.

A. SMAs reported in experiments
Prior to the materials screening, we first investigate ΔE m-p for 13 alloys that were reported to show the shape-memory effects near the stoichiometric composition ratios in experiments.  [46,47], which is significant particularly at high temperature, is probably essential to make the parent phases dynamically stable.
According to these results, we expect that the screening conditions for the energetic and dynamical stabilities described in Sec. II are suitable for the materials selection of SMAs. Later we will also discuss the structural compatibility of the 13 investigated alloys.

B. Candidates for new SMAs
In order to identify candidates for new SMAs, we apply the screening conditions described in Sec.
II.B and Fig. 3 to 6,627 alloys. Figure 7 shows  (Table II) show that the thermal hysteresis of these alloys, except for Zn 2 AuCu, is larger than 20-30 K, which is not small [3,49].
The larger thermal hysteresis should result in the worse functional stability. Inclusion of point defects in offstoichiometric composition ratios [31] or solute elements may be essential for these alloys for better functional stability or smaller thermal hysteresis. [10,11,15].
Frequency of chemical elements in the surviving 111 binary alloys is summarized in Fig. 8. The ternary X 2 YZ alloys are excluded for this analysis because the X component for the ternary in this study is restricted to three elements, i.e., Ti, Cu, and Zn. Among the 48 chemical elements, Cu, Zn, Ag, Au, and Pd are the most frequently included elements in the descending order. The result is natural since many Cu-based alloys are known to exhibit the shape-memory effects. Cu-Zn-based SMAs are found in the form of either binary alloys of 60-64 at.% Cu [50] or ternary alloys incorporating Al, Si, Ga, and Mn [5]. Cu-Al-based SMAs are found in the form of ternary alloys incorporating Ni [51], Mn [52], Be [53], and Zn [5]. Cu-Snbased binary alloys also show the shape-memory effects for 74-91 at.% Cu [5]. These alloys are not included in the 13 SMAs analyzed in Sec III.A. This may be because their shape-memory effects were reported experimentally only for the off-stoichiometric composition ratios.
Besides these popular alloy systems, many other elements and their combinations are found in the surviving alloys. Among such elements, Li and Sc have been rarely used as the constituent elements of the SMAs reported in experiments. Only recently, the Mg 80 Sc 20 alloy was discovered as a SMA having technological advantages with its light weight [54]. The In-Tl nanowire was also reported to exhibit the shape-memory effects [55]. Much opportunity to discover new SMA from the less popular systems can be expected.
According to several review papers [7,8], the SMAs with the T c above 370-400 K are categorized in

IV. CONCLUSION
We perform the first-principles-based materials screening to discover so far unknown combinations We also examine the correlation between the martensitic transformation temperature T c in experiments and the ΔE m-p obtained from the first-principles calculations for the 13 SMAs with nearly stoichiometric composition ratios. Strong correlation is found between the experimental T c and the computed ΔE m-p . This implies that ΔE m-p can be used to roughly estimate the working temperature range as SMAs.
The findings in this study may help the new discovery of SMAs that overcome the problems of costs, toxicity, and poor functional stability. Furthermore, this study shows the strategy for the design of SMAs based on first-principles calculations. Although in this study we focus on only the alloys with stoichiometric composition ratios, the strategy in this study may be able to be applied also to the alloys with nonstoichiometric composition ratios. Actually, the off-stoichiometric Zn 45 Au 30 Cu 25 alloy shows smaller