Phase Transition and Electronic Structures of All-d-Metal Heusler-Type X2MnTi Compounds (X = Pd, Pt, Ag, Au, Cu, and Ni)

In this work, we investigated the phase transition and electronic structures of some newly designed all-d-metal Heusler compounds, X2MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni), by means of the first principles. The competition between the XA and L21 structures of these materials was studied, and we found that X2MnTi favors to feature the L21-type structure, which is consistent with the well-known site-preference rule (SPR). Under the L21 structure, we have studied the most stable magnetic state of these materials, and we found that the ferromagnetic state is the most stable due to its lower energy. Through tetragonal deformation, we found that the L21 structure is no longer the most stable structure, and a more stable tetragonal L10 structure appeared. That is, under the tetragonal strain, the material enjoys a tetragonal phase transformation (i.e., from cubic L21 to tetragonal L10 structure). This mechanism of L21-L10 structure transition is discussed in detail based on the calculated density of states. Moreover, we found that the energy difference between the most stable phases of L10 and L21, defined as ΔEM (ΔEM = ECubic-ETetragonal), can be adjusted by the uniform strain. Finally, the phonon spectra of all tetragonal X2MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) phases are exhibited, which provides a powerful evidence for the stability of the tetragonal L10 state. We hope that our research can provide a theoretical guidance for future experimental investigations.


INTRODUCTION
Magnetic shape memory compounds (MSMAs) (O'Handley, 1998) are a new type of intelligent materials which integrates magnetic controlled shape memory and magnetic field-induced strain simultaneously. It can be used as key components of sensors and brakes in the future. MSMAs have both thermoplastic martensitic transformation (Oikawa et al., 2001) and magnetic transformation (Oikawa et al., 2002), and their shape memory effect can be controlled by the magnetic field. That is to say, under the effect of magnetic field, their size or volume will be changed, resulting in a great strain, i.e., the magnetostrictive effect (Populoh et al., 2012). In addition, MSMAs enjoy magnetoresistance (Ullakko et al., 1996) and magnetocaloric (GschneidnerJr et al., 2005) effects, so they have been regarded as a research hot spot in recent years.
In the early 1990s, all-d-metal Heusler compounds Zn 2 AuAg and Zn 2 CuAu (Muldawer, 1966) were synthesized experimentally, which proved that all-d-metal Heusler compounds can be successfully prepared. This interesting study opened up a new direction for the research of Heusler compounds. All-d-metal Heusler compounds are different from common Heusler compounds in that they are composed of transition metal elements without the participation of main group elements. Compared with common Heusler compounds, all-d-metal Heusler compounds have the following advantages: (1) all-d-metal Heusler compounds do have not many restrictions on atomic site preference, so they can show more phase space and versatility than traditional materials; (2) they have high strength and toughness; and (3) in addition to magnetic phase transitions, all-d-metal Heusler compounds also have many untouched physical properties, such as spintronics properties.
Recently, an effective method, i.e., adjusting composition, was proposed by Tan et al. (2019) to make regular the phase transition of all-d-metal Heusler compounds. In their work, the atomic ordering, structural stability, tetragonal deformation, magnetism, and electronic structures of the Mn-Ni-V system, including Mn 2−x Ni 1.5+x V 0.5 , Mn 2−x Ni 1+x V, and Mn 2−x Ni 0.5+x V 1.5 compounds, are studied by employing the first principles. They stated that the tetragonal phase is more stable than the cubic phase for these all-d-metal Heusler compounds: MnNi 2 V, Mn 1.25 Ni 1.75 V, MnNi 2.5 V 0.5 , and Mn 0.5 Ni 2 V 1.5 .
Based on the above information, we focus on the electronic structures and phase transition of all-d-metal Heusler compounds X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) with the help of the first principles. To the best of knowledge, the physical properties of the X 2 MnTi system have not been studied yet by other researchers up to now. We will reflect that a L2 1 -L1 0 phase transition can be found in all-d-metal Heusler compounds X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) under the tetragonal distortion. The effect of uniform strain on the cubic-tetragonal transition for this X 2 MnTi system was also discussed in detail in this work. The mechanism of the L2 1 -L1 0 phase transition is discussed according to the calculated density of states in both cubic and tetragonal phases, and finally, we will further prove the stability of the tetragonal phase by calculating the phonon spectrum of the tetragonal L1 0 phase.

COMPUTATIONAL METHODS
Electronic structure calculations were performed using density functional theory (DFT), within the VASP code (Hafner, 2007). The exchange-correlation potential is treated by using the generalized gradient approximation (GGA) (Perdew et al., 1996) with the Perdew-Burke-Ernzerhof (PBE) (Perdew et al., 1998) functional. We also use the projection enhanced wave (PAW) (Kresse and Furthmüller, 1996) method to deal with the interaction between the ion nucleus and valence electron. In the calculations, the cutoff energy was set at 450 eV. A Monkhorst-Pack special 12 × 12 × 12 k-point mesh was used in the Brillouin zone (BZ) integration. The unit cell was optimized until the force and total energy were <0.001 eV/Å and 1 × 10 −5 eV, respectively. The crystal models of cubic L2 1 phase and tetragonal L1 0 phase of Heusler alloys are built via VESTA. To calculate the dynamical stabilities of these alloys, phonon dispersion is obtained by means of the force-constants method using Nanodcal code. For the L2 1type X 2 MnTi alloys, two magnetic states are considered: one is the ferromagnetic state with spin orderings of X-1, X-2, Mn, and Ti that are ↑↑↑↑; the other one is the antiferromagnetic state with spin orderings of X-1, X-2, Mn, and Ti that are ↓↓↑↓, respectively.

RESULTS AND DISCUSSION
The Crystal and Magnetic Structures of the X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) Heusler Compounds Heusler compounds, X 2 YZ, enjoy a highly ordered cubic structure (Han et al., 2019a). There are generally four different positions in a primitive cell, namely, A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D (0.75, 0.75, 0.75), respectively. The transition metal elements X and Y occupy the A, B, and C positions, and the main group element Z is preferred to occupy the D position. Different occupation positions of X and Y atoms will lead to different structures, namely, L2 1 and XA structures (Suzuki and Kyono, 2004) (as shown in Figures 1A,B). In the former, two X atoms occupy A and C positions, and Y and Z atoms enter B and D positions, respectively; in the latter, two X atoms occupy A and B positions, and Y and Z atoms are located at C and D positions, respectively.
Next, we studied the competition between the XA structure and the L2 1 structure of X 2 MnTi. In Figure 2, the total energy FIGURE 1 | Crystal structure of the (A) XA-type X 2 MnTi and (B) L2 1 -type X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni); X is highlighted by gray balls, Ti is marked by blue balls, and Mn is marked by purple balls, respectively.
FIGURE 2 | Total energy difference E(XA) -E(L2 1 ) of X 2 MnTi under the L2 1 and XA structure (in FM state) with the energy of L2 1 set as 0 eV for reference. of X 2 MnTi at the ground state with different structures (L2 1 and XA) is determined, and we set the ground state energy of the L2 1 structure as 0 eV as reference. One can clearly see that, for X 2 MnTi, the total energy of XA is higher than L2 1 , reflecting that the most stable ordered structure is L2 1 for the X 2 MnTi system. In Figure 3, we further give a comparison of the total energy of two magnetic states [ferromagnetic (FM) and antiferromagnetic (AFM)] for the L2 1 -type X 2 MnTi. We set the total energy of AFM to 0 eV as reference. As shown in Figure 3, the energy of X 2 MnTi in the FM structure is lower than 0, which indicates that the X 2 MnTi tends to exhibit the FM magnetic state.
The L2 1 -L1 0 Phase Transition of All-d-Metal Heusler-Type X 2 MnTi In this section, our research goal is to explore the possible competition between the L2 1 (see Figure 4A) and L1 0 (see Figure 4B) of the all-d-metal Heusler compounds X 2 MnTi. We used Bain paths to investigate the reversible transformation between the ordered L2 1 and L1 0 phases during tetragonal distortion. Bain paths (Alippi et al., 1997) have tetragonal states along the geometries which connect the bcc and fcc phases of a material. It is assumed that there is no cell volume change after applying deformation on the cubic phase, then the energy  difference E in respect to c/a ratios is calculated to estimate whether it is a stable or metastable phase. It is a commonly used method to predict reversible transformation, and a number of literature (Barman et al., 2007;Özdemir Kart et al., 2008;Qawasmeh and Bothina, 2012;Zeleny et al., 2014) have used this method to estimate shape memory effect in Heusler FSMAs, such as Ni 2 MnGa and Mn 2 NiGa.
In Figure 5, we can see that all X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) compounds present a L2 1 -L1 0 (possible martensitic transformation) under the effect of tetragonal distortion. In detail, we found that X 2 MnTi (except for X = Au) compounds have two local minimum energies, one locates at c/a < 1 and the other locates at c/a > 1. The value of the local minimum energy with c/a > 1 is lower than that of the local minimum energy with c/a < 1, which means that the local minimum energy with c/a > 1 is the most stable state. Moreover, when c/a < 1, the local minimum energy of X 2 MnTi (except for X = Au) is a metastable phase. For the Au 2 MnTi compound, there is only one local minimum energy, which locates at c/a >1. This local minimum energy is the most stable phase and there is no metastable structure during the cubic-tetragonal transformation. The c/a ratio values of the most stable tetragonal L1 0 structure of these materials are summarized in Table 2.
Next, we studied the influence of uniform strain on the competitiveness of the L2 1 and L1 0 structures. We give Cu 2 MnTi and Ni 2 MnTi as examples in Figure 6 for a detailed discussion.
By adjusting the lattice parameters of X 2 MnTi (X = Cu, Ni), their volume was changed between −2 and +2%, and then tetragonal distortion is applied to the cubic crystal structures. As shown in Figure 6, under the effect of tetragonal distortion, we can see that all substances with a volume of −2 and +2% still have two local energy minimums, which is the same as the optimized volume (V opt ). However, the difference is that during the volume changes from +2 to −2%, the minimum value of local energy gradually becomes lower, that is, the L1 0 state becomes more and more stable as the volume decreases.
In addition, it can be seen that the value of the c/a ratio for the most stable L1 0 phase also changes with the volume changes. During the volume changes from −2% to +2%, the c/a ratio (for the most stable L1 0 phase) gradually decreases. A smaller c/a ratio means a smaller degree of tetragonal distortion.

The Calculated Density of States of All-d-Metal Heusler-Type Compounds X 2 MnTi
We calculated their total density of states (TDOSs) and partial state density of states (PDOSs) (Han et al., 2019b) in cubic and tetragonal states, respectively, and we plotted them in Figures 7, 8.
We found that the TDOS of tetragonal structure around the Fermi level is softer than the cubic TDOS. For example, we can find some strong peaks (see the red lines) near the Fermi level in the spin-down direction and some small peaks in the spin-up direction (see the black line) for the cubic phase (see Figure 7A). Note that one of the contributions to the total energy is the band energy E band = E F E min dEDOS (E)E; a reduction of the DOS near E F in a tetragonal phase, in conjunction with the conservation of the integral for the number of valence electrons , often leads to a lower band energy and, thus, to a lower total energy for the tetragonal phase than for the cubic phase. As shown in Figure 7, we can find that, under the effect of the tetragonal strain, the energy states near E F tend to decrease or move to a high energy level. We give two more detailed explanations as follows: (1) as shown in Figures 7A,C-F, we can clearly observe that the local energy states (blue lines) near the Fermi level of the tetragonal L1 0 phase are significantly lower than those of the cubic L2 1 phase (black lines). Thanks to the TDOSs above, the total energy of the system will be released, resulting in cubic-tetragonal transformation; (2) as shown in Figures 7A,C, we find that the peaks (red line) of the cubic structure near the Fermi level disappeared in the tetragonal structure; however, a very small valley at the same energy (see the pink line) occurred. The peak to valley transition of TDOS near the Fermi level also proves that the tetragonal L1 0 structure is more stable for X 2 MnTi compounds. In Figure 8, we also show the PDOSs of each atom in the cubic and tetragonal phases. As shown in Figure 8A, the PDOS of the L2 1 -type Ag orbitals is almost located in the energy areas of −4.5 to −6 eV. In this region, the spin-up and spin-down PDOSs of Ag-d are almost symmetrical, reflecting that the contribution of Ag atoms to the total magnetism is relatively small. In the energy range from −1.5 to −4 eV, two large energy peaks, which come from the Mn-d orbitals, can be found in the spin-up channel; however, the DOS in the spin-down is nearly flat. Above the Fermi level, the TDOS in the spin-up channel is coming from the Ti-d orbital, and the TDOS in the spin-down channel is arising from the hybridization between the Ti-d and Mn-d orbitals. A similar phenomenon can also be found in L2 1 -type Au 2 MnTi as shown in Figure 8B.
As shown in Figures 8C-F, some large peaks can be obviously found below the Fermi level in the spin-up channel for the L2 1type X 2 MnTi (X = Ni, Cu, Pd, Pt) alloys. The formation of these peaks owes to the hybridization between X-d and Mn-d orbitals. For Ni 2 MnTi, the energy peak of the Ni-d orbital is larger than that of the Mn-d orbital around −1.2 eV; however, for the other three cases, the energy peaks of the X-d orbital are smaller than those of the X-d orbitals. Above the Fermi level, the TDOS is mainly coming from the Mn-d orbital in the spin-up channel and from the Mn-d and X-d orbitals in the spin-down channel.
As shown in Figure 8, near the Fermi level, PDOSs of the Mn atom will produce a strong spin splitting in two spin channels, and then result in strong magnetism. Thus, the total magnetism of X 2 MnTi is mainly coming from the Mn atoms. We also exhibit the total and atomic magnetic moments in Tables 1, 2.
Finally, we calculated the phonon spectrum of the X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) tetragonal L1 0 phase by means of the force-constants method using Nanodcal code and the results are collected in Figure 9. In Figure 9, we can clearly see that there are no virtual frequencies in the phonon spectrum of the X 2 MnTi, and the absence of virtual frequencies further confirms that their tetragonal L1 0 states are theoretically stable. Unfortunately, the possible L2 1 -L1 0 phase transition of X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) has not been studied experimentally, and therefore, a comparison between the theoretical and experimental results cannot be shown in the current work. However, this investigation can help in understanding the physics in all-d-metal alloys. Moreover, we    would like to point out that Bain paths are a sophisticated way to investigate the reversible transformation between the L2 1 and L1 0 phases during the tetragonal distortion. This method has been widely used to design new MSMAs, and some designed MSMAs have been experimentally verified, such as the Mn-Ni-Co-Ti system.

CONCLUSIONS
In this study, we investigated the phase transition and electronic structures of X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) compounds based on first-principle calculations. First, we examined the L2 1 and XA competition of Heusler compounds X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni). The results show that the L2 1 type is the most stable ordered structure for these newly designed materials. Based on the L2 1 structure, we also compared the total energies of the L2 1 X 2 MnTi system with different magnetic states, i.e., FM and AFM. We found that these compounds have lower ground state energy when in the FM state, that is, the most stable state of X 2 MnTi is the FM state under the L2 1 structure. Subsequently, we studied the possible tetragonal transformation of Heusler compounds X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) and found that these materials all feature a stable L1 0 tetragonal phase. The energy difference E M (E T -E C ) in X 2 MnTi can be adjusted by a uniform strain. By analyzing the DOS diagram, it can be found that the magnetic moment of X 2 MnTi mainly comes from Mn atoms, which is due to their strong spin splitting around E F . The lower TDOS of the tetragonal L1 0 state near the Fermi level can be used to explain the stability of the tetragonal L1 0 state of X 2 MnTi. Finally, we gave the phonon spectra of the tetragonal L1 0 phase of X 2 MnTi (X = Pd, Pt, Ag, Au, Cu, and Ni) to further prove their stability.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
MW: conceptualization, methodology, software, and writingoriginal draft preparation. FZ and RK: software and writingoriginal draft preparation. MK and XW: supervision. All authors contributed to the article and approved the submitted version.