crystals

Article Theoretical Study of the Electronic and Magnetic Properties and Phase Stability of the Full Heusler Compound Pd2CoAl

1, 1, 2 3 1 Liyu Hao †, Jiaxue You †, Rabah Khenata , Yanfeng Wang , Xiaotian Wang and Tie Yang 1,* 1 School of Physical Science and Technology, Southwest University, Chongqing 400715, China 2 Laboratoire de Physique Quantique de la Matière et de Modélisation Mathématique, Université de Mascara, Mascara 29000, Algeria 3 College of Sciences, Hebei North University, Zhangjiakou 075000, China * Correspondence: [email protected] Authors contributed equally. †  Received: 30 July 2019; Accepted: 13 August 2019; Published: 14 August 2019 

Abstract: Based on first principles calculation, a systematical investigation has been performed to study the electronic, magnetic, dynamic, and mechanical properties of the full Heusler compound Pd2CoAl. It is found that the L21-type structure is energetically more stable than the XA-type due to the lower total energy. The obtained lattice constant in cubic ground state is 6.057 Å, which matches well with previous study. The calculated electronic band structure reveals the metallic nature of Pd2CoAl and its total magnetic moment of 1.78 µB is mainly contributed by Co atom from strong spin splitting effect, as indicated with the distinctive distributions of the density of states in two spin directions. Under uniform strains from 5% to +5%, the variation of total magnetic moment − has been obtained and it is still caused by the much larger change from Co atom, compared with Pd and Al atoms. The tetragonal structure has further been analyzed and we found that there is possible martensitic phase transformation because the total energy can be further reduced when the cubic structure is varied into the tetragonal one. The large energy difference of 0.165 eV between the tetragonal and cubic phases is found at the c/a ratio of 1.30. The total density of states has been compared between the cubic and tetragonal phases for Pd2CoAl and results show tetragonal phase transformation could reduce the states at the Fermi energy level in both directions. In addition, the dynamic and mechanical stabilities have also been evaluated for Pd2CoAl in both cubic and tetragonal structures and results confirm that the tetragonal phase shows good stability against the cubic phase, which further verifies that the tetragonal phase transformation is highly expected. In the end, the strong elastic anisotropy in the tetragonal structure has been clearly shown with the calculated directional dependence of the Young’s modulus and .

Keywords: first principles calculation; Heusler compounds; electronic band structure; phase transformation

1. Introduction The family of Heusler alloys has attracted much scientific attention and research interest during recent years, and their various and special functionalities have been extensively studied and explored in many different areas. The was firstly observed in Heusler compounds with no ferromagnetic elements and, afterwards, many more new properties have been found, such as half metallicity [1–10], spin gapless semiconductivity [11–17], thermoelectricity [18–23], [24–26], and topological insulativity [27,28]. Also, these properties can be easily

Crystals 2019, 9, 422; doi:10.3390/cryst9080422 www.mdpi.com/journal/crystals Crystals 2019, 9, 422 2 of 14 tuned by simple element substitution within the periodic table. Therefore, ongoing investigations to enhance their performance or even search for new property are still very intense in the scientific community from both theoretical calculation and experimental synthesis. Most of the current applications for Heusler compound are related with the magnetic effect just in different forms, including mainly and magnetoelectronics. Normally, Heusler compounds have highly ordered cubic crystal structures with different atomic ordering configurations. However, several recent studies have found many Heusler materials have tetragonal structure as the ground state, e.g., Felser et al. [29] found that Mn3-xFexGa and Mn3-xCoxGa have tetragonal structure over the whole range of compositions from experiment results, and then Faleev et al. [30,31] performed an extensive investigation on the origin of the tetragonal ground state for 286 Heusler compounds by theoretical calculation, which shows 62% have tetragonal structure. More recently, Han et al. [32,33] and Wu et al. [34] studied the phase competition between cubic and tetragonal structures in a series of conventional Heusler compounds Pd2YZ (Y = Co, Fe, Mn; Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb) and all-d-metal Heusler compounds X2-xMn1+xV (X = Pd, Ni, Pt, Ag, Au, Ir, Co; x = 1, 0) and Zn2MMn (M = Ru, Rh, Pd, Os, Ir) and found that many of them also exhibit tetragonal phases, especially the most of Pd2Co-based Heusler alloys. Compared with the cubic structures, the tetragonal ones have several special properties, like large perpendicular magnetic anisotropy [35–37] and ferromagnetic shape memory behavior [38–43], which are very important for the development of spin-transfer torque magnetic random-access memory, ferromagnetic shape memory alloys and other spintronic applications [29,44–46]. In this work, a detailed study on the full Heusler compound Pd2CoAl has been carried out with first principles calculation. The electronic and magnetic properties have been calculated based on both cubic and tetragonal structures. Except the general total energy evaluation, the phase stability was further examined by means of both dynamic and mechanical manners and results show that the tetragonal structure has good stability against the cubic counterpart, which further confirms the possible tetragonal phase transformation could be highly expected in this material. Moreover, the elastic anisotropy in the tetragonal structure is revealed with the directional dependence of Young’s modulus and shear modulus. This study investigates the phase stability from several different perspectives and thus can give valuable information for the tetragonal state in the palladium-based full Heusler materials or even inspire other similar investigations.

2. Computational Methods and Details

The electronic, magnetic, and mechanical properties of the full Heusler compound Pd2CoAl have been investigated with the first principles calculations by applying the pseudopotential plane wave methods based on density functional theory [47], as implemented with the CASTEP codes [48]. The Perdew–Burke–Ernzerhof (PBE) 96 in the frame of the generalized gradient approximation (GGA) [49] was selected to describe the electronic exchange correlation energy. The ultrasoft pseudopotential [50] was used to treat the interactions between the valence electrons and the atomic core. The configurations of the valence electrons for Pd, Co, and Al are 4d10, 3d74s2, and 3s23p1, respectively. After an initial convergence test, a place wave cutoff energy of 500 eV has been used together with a k mesh of 12 12 12 Monkhorst–Pack grid for the Brillouin zone sampling. For the × × self-consistent field iteration convergence, the difference of the total energy was set to smaller than 1 10 6 eV/atom. For the phonon property, the finite displacement method within the density × − functional perturbation theory [51] has been used.

3. Results and Discussions

3.1. Crystal Structure and Equilibrium Lattice The Heusler alloys generally have two variants of stoichiometric compositions: the half-Heusler compound with generic formula of XYZ [52–57] and the full Heusler compound X2YZ [5,58,59], Crystals 2019, 9, 422 3 of 14 where X and Y are from elements and Z from main group elements. In both half and fullCrystals Heusler 2019, 9,alloys, x FOR PEER they REVIEW normally crystalize in a cubic structure yet with different configurations3 of [60 13]: non-centrosymmetric cubic structure C1b for half-Heusler; two different types for full Heusler: space ground 𝐹43𝑚, also known as “inverse structure” or XA-type. For the currently studied full the Cu2MnAl structure with space group Fm3m, also known as L21-type, and the Hg2CuTi structure Heusler compound Pd2CoAl, the two structural types have been taken into consideration and they with space ground F43m, also known as “inverse structure” or XA-type. For the currently studied full are shown in Figure 1. It can been seen that the full Heusler Pd2CoAl has the general FCC-like Heusler compound Pd2CoAl, the two structural types have been taken into consideration and they are symmetry with four interpenetrating sublattices defined by Wyckoff coordinates: A(0,0,0), shown in Figure1. It can been seen that the full Heusler Pd 2CoAl has the general FCC-like symmetry B(0.25,0.25,0.25), C(0.5,0.5,0.5), and D(0.75,0.75,0.75). In L21-type structure, the two Pd atoms occupy with four interpenetrating sublattices defined by Wyckoff coordinates: A(0,0,0), B(0.25,0.25,0.25), the A and C sites, which are symmetrically identical, and the Co and Al atoms enter into the B and D C(0.5,0.5,0.5), and D(0.75,0.75,0.75). In L21-type structure, the two Pd atoms occupy the A and C sites, sites, respectively; see Figure 1a. In XA-type structure, the two Pd atoms occupy the A and B sites which are symmetrically identical, and the Co and Al atoms enter into the B and D sites, respectively; with different surrounding environments, and the Co and Al atoms enter into the C and D sites; see see Figure1a. In XA-type structure, the two Pd atoms occupy the A and B sites with different Figure 1b. surrounding environments, and the Co and Al atoms enter into the C and D sites; see Figure1b.

Figure 1. Schematic illustrations of the atomic configurations for the full Heusler compound Pd2CoAl Figure 1. Schematic illustrations of the atomic configurations for the full Heusler compound Pd2CoAl in (a) L21-type and (b) XA-type structures. in (a) L21-type and (b) XA-type structures. In order to determine the ground state structural configuration and also obtain the equilibrium In order to determine the ground state structural configuration and also obtain the equilibrium lattice constant, we have calculated the total energy of Pd2CoAl with different structures under different lattice constant, we have calculated the total energy of Pd2CoAl with different structures under lattice constants. It should be noted that only the magnetic state is considered, as in the literature [34], different lattice constants. It should be noted that only the magnetic state is considered, as in the since there is Co atom present. The obtained result is reported in Figure2. It is found that the literature [34], since there is Co atom present. The obtained result is reported in Figure 2. It is found calculated total energy of Pd2CoAl in L21-type structure is always smaller than in XA-type structure that the calculated total energy of Pd2CoAl in L21-type structure is always smaller than in XA-type throughout the whole lattice variation range, which means the L21-type structure is more energetically structure throughout the whole lattice variation range, which means the L21-type structure is more preferable and should be the ground state. This observation is in a good agreement with the general energetically preferable and should be the ground state. This observation is in a good agreement with site preference rule found in other palladium-based Heusler compounds [32,34], i.e., since the Pd the general site preference rule found in other palladium-based Heusler compounds [32,34], i.e., since atom has more valence electrons than the Co atom, it is more electronegative and prefers the A and C the Pd atom has more valence electrons than the Co atom, it is more electronegative and prefers the sties. Thus, the L21-type structure is formed for Pd2CoAl. After the crystal structural configuration of A and C sties. Thus, the L21-type structure is formed for Pd2CoAl. After the crystal structural Pd2CoAl has been determined, we further calculated its equilibrium lattice by polynomial fitting and configuration of Pd2CoAl has been determined, we further calculated its equilibrium lattice by minimization searching of the total energy and the derived equilibrium lattice is 6.057 Å, see Table1, polynomial fitting and minimization searching of the total energy and the derived equilibrium lattice which coincides very well with previous study [30,34]. is 6.057 Å, see Table 1, which coincides very well with previous study [30,34]. Table 1. The calculated equilibrium lattice constant, total and atom-resolved magnetic moment.

Magnetic Moment (µB) Compound Lattice (Å) MTotal (µB) MPd(A) MPd(B) MCo MAl Current 6.057 1.78 0.06 0.06 1.95 0.05 Pd2CoAl − − − Ref [34] 6.06 1.79 0.03 0.03 1.75 0.03 −

Crystals 2019, 9, x FOR PEER REVIEW 3 of 13 space ground 𝐹43𝑚, also known as “inverse structure” or XA-type. For the currently studied full Heusler compound Pd2CoAl, the two structural types have been taken into consideration and they are shown in Figure 1. It can been seen that the full Heusler Pd2CoAl has the general FCC-like symmetry with four interpenetrating sublattices defined by Wyckoff coordinates: 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). In L21-type structure, the two Pd atoms occupy the A and C sites, which are symmetrically identical, and the Co and Al atoms enter into the B and D sites, respectively; see Figure 1a. In XA-type structure, the two Pd atoms occupy the A and B sites with different surrounding environments, and the Co and Al atoms enter into the C and D sites; see Figure 1b.

Figure 1. Schematic illustrations of the atomic configurations for the full Heusler compound Pd2CoAl

in (a) L21-type and (b) XA-type structures.

In order to determine the ground state structural configuration and also obtain the equilibrium lattice constant, we have calculated the total energy of Pd2CoAl with different structures under different lattice constants. It should be noted that only the magnetic state is considered, as in the literature [34], since there is Co atom present. The obtained result is reported in Figure 2. It is found that the calculated total energy of Pd2CoAl in L21-type structure is always smaller than in XA-type structure throughout the whole lattice variation range, which means the L21-type structure is more energetically preferable and should be the ground state. This observation is in a good agreement with the general site preference rule found in other palladium-based Heusler compounds [32,34], i.e., since the Pd atom has more valence electrons than the Co atom, it is more electronegative and prefers the A and C sties. Thus, the L21-type structure is formed for Pd2CoAl. After the crystal structural configuration of Pd2CoAl has been determined, we further calculated its equilibrium lattice by Crystals 2019, 9, 422 4 of 14 polynomial fitting and minimization searching of the total energy and the derived equilibrium lattice is 6.057 Å, see Table 1, which coincides very well with previous study [30,34].

Crystals 2019, 9, x FOR PEER REVIEW 4 of 13

Figure 2. The calculated total energies of the full Heusler compound Pd2CoAl with different crystal structures under different lattice constants.

Table 1. The calculated equilibrium lattice constant, total and atom-resolved magnetic moment.

Magnetic Moment (μB) Compound Lattice (Å ) MTotal (μB ) MPd(A) MPd(B) MCo MAl Figure 2. The calculatedCurrent total energies 6.057 of the full Heusler1.78 compound−0.06 Pd2−CoAl0.06 with1.95 di fferent−0.05 crystal Pd2CoAl structures under differentRef lattice[34] constants.6.06 1.79 0.03 0.03 1.75 −0.03 3.2. Electronic and Magnetic Properties 3.2. Electronic and Magnetic Properties With the obtained equilibrium lattice constant, we can then investigate the corresponding electronic With the obtained equilibrium lattice constant, we can then investigate the corresponding and magnetic properties of the full Heusler compound Pd2CoAl. Firstly, the spin-polarized electronic electronicband structure and magnetic has been properties calculated andof the it full is shown Heusler in Figurecompound3. The Pd Fermi2CoAl. energy Firstly, is the shifted spin-polarized to the zero electronicenergy level. band It structure can be immediately has been calculated observed and that it there is shown are energy in Figure bands 3. The crossing Fermi theenergy Fermi is shifted energy tolevel the in zero both energy spin directions, level. It can meaning be immediately that this Heusler observed compound that there behaves are energy like metallicbands crossing material the at Fermiground energy state. level in both spin directions, meaning that this Heusler compound behaves like metallic material at ground state.

Figure 3. The calculated spin-polarized electronic band structures for the full Heusler compound Figure 3. The calculated spin-polarized electronic band structures for the full Heusler compound Pd2CoAl in L21-type cubic structure at the equilibrium lattice constant. Pd2CoAl in L21-type cubic structure at the equilibrium lattice constant. In order to further elucidate the origin of the and also access the correlation states In order to further elucidate the origin of the magnetism and also access the correlation states between different atoms in Pd2CoAl, the total and partial densities of states have been also computed betweenand the resultdifferent is shownatoms in in Pd Figure2CoAl,4. the We total can and find partial that the densities main groupof states element have been Al hasalso negligiblecomputed andcontribution the result and is theshown total in density Figure of4. statesWe can is mainlyfind that contributed the main fromgroup the elementd states Al of has the negligible transition contributionmetal elements and Pd the and total Co: density a roughly of symmetricstates is mainly distribution contributed of states from between the d states the two ofspin the transition directions metal elements Pd and Co: a roughly symmetric distribution of states between the two spin directions around −5 to −2 eV from Pd and a strongly asymmetric distribution between the two spin directions around −2 to 1 eV from Co. In particular for the states near the Fermi energy level, they play a very important role in the determination of the electronic and magnetic properties for Heusler compounds. From the total density of states, there is a strong spin splitting effect in the two spin directions with a high peak below the Fermi level in the spin-up direction and a high peak just at the Fermi level in the spin-down direction. A closer look at the partial density of states reveals that this strong spin splitting is caused by the d-d hybridization between the Pd and Co atoms and the strong exchange splitting of the Co atom. The calculated magnetic moments are shown in Table 1, and it is found the total magnetic moment of Pd2CoAl is about 1.78 μB per formula unit and it is mainly from

Crystals 2019, 9, 422 5 of 14 around 5 to 2 eV from Pd and a strongly asymmetric distribution between the two spin directions − − around 2 to 1 eV from Co. In particular for the states near the Fermi energy level, they play a very − important role in the determination of the electronic and magnetic properties for Heusler compounds. From the total density of states, there is a strong spin splitting effect in the two spin directions with a high peak below the Fermi level in the spin-up direction and a high peak just at the Fermi level in the spin-down direction. A closer look at the partial density of states reveals that this strong spin splitting is caused by the d-d hybridization between the Pd and Co atoms and the strong exchange splitting of Crystalsthe Co 2019 atom., 9, x The FOR calculated PEER REVIEW magnetic moments are shown in Table1, and it is found the total magnetic5 of 13 moment of Pd2CoAl is about 1.78 µB per formula unit and it is mainly from the Co atom. These results thefor theCo magneticatom. These moments results are for consistent the magnetic with moment the aboves are analysis consistent from electronicwith the above densities. analysis from electronic densities.

Figure 4. The calculated total and partial densities of states for the full Heusler compound Pd CoAl in Figure 4. The calculated total and partial densities of states for the full Heusler compound Pd2 2CoAl L2 -type cubic structure at the equilibrium lattice constant. in L21 1-type cubic structure at the equilibrium lattice constant.

Furthermore, the electronic and magnetic properties of Pd2CoAl compound under uniform strains Furthermore, the electronic and magnetic properties of Pd2CoAl compound under uniform have been evaluated with lattice variation from 5% to +5% with respect to the equilibrium condition. strains have been evaluated with lattice variation− from −5% to +5% with respect to the equilibrium The cubic L21-type structure for Pd2CoAl under different uniform strains is always maintained. condition. The cubic L21-type structure for Pd2CoAl under different uniform strains is always The calculated band structures always exhibit overlaps with the Fermi energy level in both spin maintained. The calculated band structures always exhibit overlaps with the Fermi energy level in directions, indicating the metallic nature of Pd2CoAl is preserved under uniform strain at the studied both spin directions, indicating the metallic nature of Pd2CoAl is preserved under uniform strain at range. The total and atom-resolved magnetic moments under uniform strains are displayed in Figure5. the studied range. The total and atom-resolved magnetic moments under uniform strains are It is found that the magnetic moment of Co atom has quite large variation while the moments of the displayed in Figure 5. It is found that the magnetic moment of Co atom has quite large variation while Pd and Al atoms stay almost constant with strain applied. Note the different scale in different parts the moments of the Pd and Al atoms stay almost constant with strain applied. Note the different scale of the vertical axis. The total magnetic moment follows the trend of Co atom, which indicates that in different parts of the vertical axis. The total magnetic moment follows the trend of Co atom, which the strong exchange splitting remains under the studied whole uniform strain. With strain variation indicates that the strong exchange splitting remains under the studied whole uniform strain. With from 0% to /+5% side, the lattice decreases/increases and the distance between every atom also strain variation− from 0% to −/+5% side, the lattice decreases/increases and the distance between every decreases/increases. Thus, the interaction of the valence electrons from different atoms is decreased atom also decreases/increases. Thus, the interaction of the valence electrons from different atoms is with lattice increase and they become more localized, which results in the partial regain of the spin decreased with lattice increase and they become more localized, which results in the partial regain of moments from Co atom. It is worth mentioning that the magnetic moments of two Pd atoms are the spin moments from Co atom. It is worth mentioning that the magnetic moments of two Pd atoms identical and overlap with each other through the whole uniform strain, which is because they have are identical and overlap with each other through the whole uniform strain, which is because they the same surrounding environments in L21-type structure have the same surrounding environments in L21-type structure

Crystals 2019, 9, x FOR PEER REVIEW 5 of 13 the Co atom. These results for the magnetic moments are consistent with the above analysis from electronic densities.

Figure 4. The calculated total and partial densities of states for the full Heusler compound Pd2CoAl

in L21-type cubic structure at the equilibrium lattice constant.

Furthermore, the electronic and magnetic properties of Pd2CoAl compound under uniform strains have been evaluated with lattice variation from −5% to +5% with respect to the equilibrium condition. The cubic L21-type structure for Pd2CoAl under different uniform strains is always maintained. The calculated band structures always exhibit overlaps with the Fermi energy level in both spin directions, indicating the metallic nature of Pd2CoAl is preserved under uniform strain at the studied range. The total and atom-resolved magnetic moments under uniform strains are displayed in Figure 5. It is found that the magnetic moment of Co atom has quite large variation while the moments of the Pd and Al atoms stay almost constant with strain applied. Note the different scale in different parts of the vertical axis. The total magnetic moment follows the trend of Co atom, which indicates that the strong exchange splitting remains under the studied whole uniform strain. With strain variation from 0% to −/+5% side, the lattice decreases/increases and the distance between every atom also decreases/increases. Thus, the interaction of the valence electrons from different atoms is decreased with lattice increase and they become more localized, which results in the partial regain of the spin moments from Co atom. It is worth mentioning that the magnetic moments of two Pd atoms areCrystals identical2019, 9, 422and overlap with each other through the whole uniform strain, which is because 6they of 14 have the same surrounding environments in L21-type structure

Figure 5. The calculated total and atom-resolved magnetic moments for the full Heusler compound

Pd2CoAl in L21-type under different uniform strains. Note the different scale of the two parts in the vertical axis.

3.3. Tetragonal Structure and Phase Stability In full Heusler compounds with possible phase transformation, the Bain paths have been widely utilized to study the reversible structural transformation between the austenitic and martensitic phases. A recent comprehensive theoretical study [30] of 286 full Heusler alloys from Faleev et al. has found that 62% have tetragonal ground states at zero temperature, including 15 Pd2-based full Heusler compounds. It is also well known that the physical properties of solid crystals are strongly related with the corresponding structures. To investigate the phase stability and also its effect in the full Heusler Pd2CoAl, we further calculate its electronic and magnetic properties under tetragonal structures, which is simply obtained by varying the lattice unproportionally; see Figure6. Di fferent tetragonal structures are formed by the different c/a ratios, yet the unit cell volume is kept constant, which is the same as the cubic structure. The total energies of the tetragonal phases for Pd2CoAl under different structure types have been calculated with different c/a ratios and the result is shown in Figure6. It is seen that there is energy decrease when c/a is deviated from 1 for both structure types, that is, the tetragonal structure could further reduce the system energy and then lead to possible martensitic phase transformation. In comparison, the minimal total energy in the tetragonal XA-type structure is even higher than that in the cubic L21-type structure about 0.103 eV, meaning that the L21-type structure is energetically stable even under tetragonal structures. The energy minimum in tetragonal L21-type structure is found at c/a of 1.30 with energy difference of 0.165 eV relative to the cubic structure; see Figure6. The energy di fference between the cubic austenitic phase and the tetragonal martensitic phase and the c/a ratio at which the tetragonal structure reaches the minimum energy are commonly used to evaluate whether the phase transformation will occur or not in these Heusler compounds. The larger the energy difference is, the more probable the occurrence of the phase transformation is and the more stable the martensitic phase is. Some examples for the energy difference required for stable phase transformation are 0.14eV and 0.12 eV for Mn3Ga [61] and Mn2FeGa [29,31], respectively. The obtained value of 0.165 eV for Pd2CoAl is even bigger, which implies the stable tetragonal phase could be highly expected. As for the degree of shape memory, the c/a ratio can be used to access this effect and a larger value is preferred. Crystals 2019, 9, x FOR PEER REVIEW 6 of 13

Figure 5. The calculated total and atom-resolved magnetic moments for the full Heusler compound Pd2CoAl in L21-type under different uniform strains. Note the different scale of the two parts in the vertical axis.

3.3. Tetragonal Structure and Phase Stability In full Heusler compounds with possible phase transformation, the Bain paths have been widely utilized to study the reversible structural transformation between the austenitic and martensitic phases. A recent comprehensive theoretical study [30] of 286 full Heusler alloys from Faleev et al. has found that 62% have tetragonal ground states at zero temperature, including 15 Pd2-based full Heusler compounds. It is also well known that the physical properties of solid crystals are strongly related with the corresponding structures. To investigate the phase stability and also its effect in the full Heusler Pd2CoAl, we further calculate its electronic and magnetic properties under tetragonal structures, which is simply obtained by varying the lattice unproportionally; see Figure 6. Different tetragonal structures are formed by the different c/a ratios, yet the unit cell volume is kept constant, which is the same as the cubic structure. The total energies of the tetragonal phases for Pd2CoAl under different structure types have been calculated with different c/a ratios and the result is shown in Figure 6. It is seen that there is energy decrease when c/a is deviated from 1 for both structure types, that is, the tetragonal structure could further reduce the system energy and then lead to possible martensitic phase transformation. In comparison, the minimal total energy in the tetragonal XA-type structure is even higher than that in the cubic L21-type structure about 0.103 eV, meaning that the L21- type structure is energetically stable even under tetragonal structures. The energy minimum in tetragonal L21-type structure is found at c/a of 1.30 with energy difference of 0.165 eV relative to the cubic structure; see Figure 6. The energy difference between the cubic austenitic phase and the tetragonal martensitic phase and the c/a ratio at which the tetragonal structure reaches the minimum energy are commonly used to evaluate whether the phase transformation will occur or not in these Heusler compounds. The larger the energy difference is, the more probable the occurrence of the phase transformation is and the more stable the martensitic phase is. Some examples for the energy difference required for stable phase transformation are 0.14eV and 0.12 eV for Mn3Ga [61] and Mn2FeGa [29,31], respectively. The obtained value of 0.165 eV for Pd2CoAl is even bigger, which Crystalsimplies2019 the, 9 stable, 422 tetragonal phase could be highly expected. As for the degree of shape memory,7 of the 14 c/a ratio can be used to access this effect and a larger value is preferred.

Figure 6. The tetragonal L21-type crystal structure for the full Heusler compound Pd2CoAl and the calculatedFigure 6. The total tetragonal energies withL21-type two crystalcrystal structuresstructure for under the difullfferent Heusler tetragonal compound structures. Pd2CoAl and the calculated total energies with two crystal structures under different tetragonal structures. To examine the effect of phase transformation on the electronic properties, we calculate the electronicTo examine band structure the effect of the of tetragonalphase transformationL21-type Pd2 CoAlon the at celectronic/a ratio of 1.30properties, and the we result calculate is reported the inelectronic Figure7 .band Note structure that the diofff theerent tetragonal high symmetry L21-type point Pd2CoAl path isat adopted c/a ratio in of the 1.30 tetragonal and the structureresult is Crystals 2019, 9, x FOR PEER REVIEW 7 of 13 fromreported the cubicin Figure one. 7. It Note is seen that that the the different metallic high feature symmetry maintains point for path the tetragonalis adopted structure in the tetragonal because therestructure are stillfrom bands the cubic crossing one. theIt is Fermiseen that energy the metallic level in bothfeature spin maintains directions. for Thus,the tetragonal the ferromagnetic structure ferromagnetic metallic property of the full Heusler Pd2CoAl preserves even with the tetragonal metallicbecause propertythere are of still the bands full Heusler crossing Pd CoAlthe Fermi preserves energy even level with in the both tetragonal spin directions. phase. Thus, the phase. 2

Figure 7. The calculated spin polarized electronic band structures for the full Heusler compound Figure 7. The calculated spin polarized electronic band structures for the full Heusler compound Pd2CoAl in L21-type tetragonal structure under c/a of 1.30. Pd2CoAl in L21-type tetragonal structure under c/a of 1.30. To understand the possible reasons for this tetragonal phase stability, we explore the total densities of statesTo understand in both cubic the austenitic possible phase reasons and tetragonalfor this tetragonal martensitic phase phase; stability, see Figure we8 .explore We can seethe theretotal densitiesis strong of spin states splitting in both of cubic the states austenitic in the phase two spin and directions,tetragonal especiallymartensitic around phase; thesee FermiFigure energy 8. We canlevel, see indicating there is strong the ferromagnetic spin splitting behavior of the states for bothin the phases. two spin However, directions, the especially two strong around spin split the Fermidistribution energy areas level, in indicating the austenitic the ferromagnetic phase have been behavior compressed for both into phases. much widerHowever, and shallowerthe two strong ones spinin the split martensitic distribution phase, areas which in meansthe austenitic the strong phased-d hybridization have been compressed and exchange into splitting much inwider the cubic and shallowerphase have ones been in weakened the martensitic in the tetragonalphase, which phase. means As suggested the strong by d- previousd hybridization study [30 and], the exchange density splittingof states atin thethe Fermicubic energyphase have plays been an important weakened role in inthe regulating tetragonal the phase. phase As stability. suggested Thus, by we previous further studyprovide [30], the the values density of states of states in the at twothe Fermi spin directions energy plays for both an important phases at therole Fermiin regulating level, as the shown phase in stability.the right Thus, vertical we axis further in Figure provide8. It the is found values that of states the values in the in two the spin-upspin directions and spin-down for both phases channels at thehave Fermi been level, reduced as shown from 1.02 in the states right/eV vertical and 5.66 axis states in Figure/eV in the8. It austenitic is found that phase the to values 0.38 states in the/eV spin- and up and spin-down channels have been reduced from 1.02 states/eV and 5.66 states/eV in the austenitic phase to 0.38 states/eV and 2.85 states/eV in the martensitic phase. According to the Jahn–Teller effect, this decreasing behavior of the density of states proves that the martensitic phase is more stable than the austenitic phase.

Crystals 2019, 9, x FOR PEER REVIEW 7 of 13 ferromagnetic metallic property of the full Heusler Pd2CoAl preserves even with the tetragonal phase.

Figure 7. The calculated spin polarized electronic band structures for the full Heusler compound

Pd2CoAl in L21-type tetragonal structure under c/a of 1.30.

To understand the possible reasons for this tetragonal phase stability, we explore the total densities of states in both cubic austenitic phase and tetragonal martensitic phase; see Figure 8. We can see there is strong spin splitting of the states in the two spin directions, especially around the Fermi energy level, indicating the ferromagnetic behavior for both phases. However, the two strong spin split distribution areas in the austenitic phase have been compressed into much wider and shallower ones in the martensitic phase, which means the strong d-d hybridization and exchange splitting in the cubic phase have been weakened in the tetragonal phase. As suggested by previous study [30], the density of states at the Fermi energy plays an important role in regulating the phase stability. Thus, we further provide the values of states in the two spin directions for both phases at theCrystals Fermi2019 level,, 9, 422 as shown in the right vertical axis in Figure 8. It is found that the values in the 8spin- of 14 up and spin-down channels have been reduced from 1.02 states/eV and 5.66 states/eV in the austenitic phase to 0.38 states/eV and 2.85 states/eV in the martensitic phase. According to the Jahn–Teller effect, 2.85 states/eV in the martensitic phase. According to the Jahn–Teller effect, this decreasing behavior of this decreasing behavior of the density of states proves that the martensitic phase is more stable than the density of states proves that the martensitic phase is more stable than the austenitic phase. the austenitic phase.

Crystals 2019, 9, x FOR PEER REVIEW 8 of 13

Figure 8. The calculated total densities of states for the full Heusler compound Pd2CoAl in L21-type Figure 8. The calculated total densities of states for the full Heusler compound Pd2CoAl in L21-type cubic and tetragonal structures. cubic and tetragonal structures.

With different tetragonal structures, the total and partial magnetic moments have also been With different tetragonal structures, the total and partial magnetic moments have also been calculated and are displayed in Figure9. Note the di fferent scale in the two parts of the vertical axis. calculated and are displayed in Figure 9. Note the different scale in the two parts of the vertical axis. We can see that the magnetic moment of Co atom always decreases at both sides when c/a ratio is We can see that the magnetic moment of Co atom always decreases at both sides when c/a ratio is changed from 1, while the moments of Pd atoms decrease at first and then increase, leading to the changed from 1, while the moments of Pd atoms decrease at first and then increase, leading to the switch from very weakly antiparallel to weakly parallel. In particular at c/a ratio of 1.30, the magnetic switch from very weakly antiparallel to weakly parallel. In particular at c/a ratio of 1.30, the magnetic moments of Co and Pd atoms are 1.85 µB and 0.07 µB, respectively. The total magnetic moment is also moments of Co and Pd atoms are 1.85 μB and 0.07 μB, respectively. The total magnetic moment is also increased from 1.78 µB in the cubic austenitic phase to 1.91 µB in the tetragonal martensitic phase. increased from 1.78 μB in the cubic austenitic phase to 1.91 μB in the tetragonal martensitic phase.

Figure 9. The calculated total and atom-resolved magnetic moments for the full Heusler compound Figure 9. The calculated total and atom-resolved magnetic moments for the full Heusler compound Pd2CoAl in L21-type structure under different tetragonal strains. Note the different scale of the two Pd2CoAl in L21-type structure under different tetragonal strains. Note the different scale of the two parts in the vertical axis. parts in the vertical axis. For the full Heusler compound, the possible martensitic phase transformation can be examined not onlyFor the with full energetic Heusler perspectivecompound, butthe alsopossible by the mart dynamicensitic phase and mechanical transformation stability. can be The examined phonon not only with energetic perspective but also by the dynamic and mechanical stability. The phonon dispersion spectra along the high symmetry points in the Brillouin zone for L21-type Pd2CoAl in both dispersion spectra along the high symmetry points in the Brillouin zone for L21-type Pd2CoAl in both cubic austenitic and tetragonal martensitic phases have been calculated with the finite displacement method and the results are plotted in Figure 10. It is clearly seen that there is a strong softening behavior present in the cubic structure with a lot of imaginary frequencies, suggesting the structural instability of the austenitic phase. Whereas, the phonon spectrum in the tetragonal structure exhibit no imaginary frequencies and, thus, the system is dynamically stable. This structural instability in the austenitic phase could lead to the martensitic phase transformation.

Crystals 2019, 9, x FOR PEER REVIEW 8 of 13

Figure 8. The calculated total densities of states for the full Heusler compound Pd2CoAl in L21-type cubic and tetragonal structures.

With different tetragonal structures, the total and partial magnetic moments have also been calculated and are displayed in Figure 9. Note the different scale in the two parts of the vertical axis. We can see that the magnetic moment of Co atom always decreases at both sides when c/a ratio is changed from 1, while the moments of Pd atoms decrease at first and then increase, leading to the switch from very weakly antiparallel to weakly parallel. In particular at c/a ratio of 1.30, the magnetic moments of Co and Pd atoms are 1.85 μB and 0.07 μB, respectively. The total magnetic moment is also increased from 1.78 μB in the cubic austenitic phase to 1.91 μB in the tetragonal martensitic phase.

Figure 9. The calculated total and atom-resolved magnetic moments for the full Heusler compound

Pd2CoAl in L21-type structure under different tetragonal strains. Note the different scale of the two parts in the vertical axis.

CrystalsFor2019 the, 9 ,full 422 Heusler compound, the possible martensitic phase transformation can be examined9 of 14 not only with energetic perspective but also by the dynamic and mechanical stability. The phonon dispersioncubic austenitic spectra and along tetragonal the high martensitic symmetry phases points havein the been Brillouin calculated zone for with L2 the1-type finite Pd2 displacementCoAl in both cubicmethod austenitic and the and results tetragonal are plotted martensitic in Figure phases 10. ha Itve is clearlybeen calculated seen that with there the is finite a strong displacement softening methodbehavior and present the results in the cubicare plotted structure in withFigure a lot10. of It imaginary is clearly frequencies,seen that there suggesting is a strong the structuralsoftening behaviorinstability present of the austenitic in the cubic phase. structure Whereas, with the a lot phonon of imaginary spectrum frequencies, in the tetragonal suggesting structure the structural exhibit no instabilityimaginary of frequencies the austenitic and, phase. thus, theWhereas, system the is dynamicallyphonon spectrum stable. in Thisthe tetragonal structural structure instability exhibit in the noaustenitic imaginary phase frequencies could lead and, to the thus, martensitic the system phase is dynamically transformation. stable. This structural instability in the austenitic phase could lead to the martensitic phase transformation.

Figure 10. The calculated phonon dispersion spectrum along high symmetry points for the full Heusler

compound Pd2CoAl in L21-type cubic and tetragonal structures.

By employing the stress strain method [62,63], we also computed the various elastic constants for the full Heusler compound Pd2CoAl in both cubic and tetragonal structures. For a simple cubic structure, there are only three independent elastic constants, namely, C11,C12, and C14. For the tetragonal structure, there are six: C11,C12,C13,C33,C44, and C66. The derived values for all constants are summarized in Table2. Based on the generalized Born–Huang elastic stability criteria [ 64], the mechanical stability of Pd2CoAl in cubic and tetragonal phases should fulfill the following two corresponding conditions: C + 2C > 0, C C > 0, C > 0 (1) 11 12 11 − 12 44 2 C11 > C12 , 2C13 < C33(C11+C12),C44 > 0, C66 > 0 (2) From these, we can derive that the cubic phase is not mechanically stable but the tetragonal phase is stable. In combination, the cubic phase Pd2CoAl is not stable from both the mechanical and dynamic point of view while the tetragonal phase shows good stability, which further elevates the martensitic phase transformation possibility.

Table 2. The calculated various elastic constants (Cij) for the full Heusler compound Pd2CoAl in both cubic and tetragonal structures.

Elastic Constants (GPa) Compound Structure C11 C12 C13 C14 C33 C44 C66 Cubic 143.2 167.4 - 100.2 - - - Pd CoAl 2 Tetragonal 133.7 38.1 60.3 - 231.2 93.0 86.4

Moreover, with the obtained elastic constants we can further evaluate the elastic anisotropy in the tetragonal L21-type Pd2CoAl compound. The calculated 3D directional dependence of the Young’s modulus and shear modulus with ELATE program [65] are plotted in Figures 11 and 12, respectively. Besides this, the corresponding 2D projections in different planes are also provided together. We can Crystals 2019, 9, x FOR PEER REVIEW 9 of 13 Crystals 2019, 9, x FOR PEER REVIEW 9 of 13 Figure 10. The calculated phonon dispersion spectrum along high symmetry points for the full

FigureHeusler 10. compound The calculated Pd2CoAl phonon in L21-type dispersion cubic and spectrum tetragonal along structures. high symmetry points for the full Heusler compound Pd2CoAl in L21-type cubic and tetragonal structures. By employing the stress strain method [62,63], we also computed the various elastic constants for theBy full employing Heusler thecompound stress strain Pd2CoAl method in both [62,63], cubic we and also tetragonal computed structures. the various For elastic a simple constants cubic forstructure, the full there Heusler are compoundonly three Pdindependent2CoAl in both elastic cubic constants, and tetragonal namely, structures. C11, C12 ,For and a simpleC14. For cubic the tetragonalstructure, therestructure, are onlythere threeare six: independent C11, C12, C13 , elasticC33, C 44constants,, and C66. Thenamely, derived C11 ,values C12, and for allC14 constants. For the tetragonalare summarized structure, in Table there 2. are Based six: Con11, theC12 , generalizedC13, C33, C44 ,Born–Huang and C66. The elasticderived stability values forcriteria all constants [64], the mechanicalare summarized stability in Table of Pd 2.2CoAl Based in on cubic the generalizedand tetragonal Born–Huang phases should elastic fulfillstability the criteria following [64], twothe correspondingmechanical stability conditions: of Pd 2CoAl in cubic and tetragonal phases should fulfill the following two corresponding conditions: C11 + 2C12 > 0, C11 − C12 > 0, C44 > 0 (1) C11 + 2C12 > 0, C11 − C12 > 0, C44 > 0 (1) 2 C11 > |C12|, 2C13 < C33(C11+ C12), C44> 0, C66 > 0 (2) 2 C11 > |C12|, 2C < C33(C11+ C12), C44> 0, C66 > 0 (2) From these, we can derive that the cubic13 phase is not mechanically stable but the tetragonal phaseFrom is stable. these, In we combination, can derive thethat cubic the cubicphase phasePd2CoAl is notis not mechanically stable from stableboth the but mechanical the tetragonal and phasedynamic is stable. point ofIn viewcombination, while the the tetragonal cubic phase phase Pd 2showsCoAl is good not stable stability, from which both furtherthe mechanical elevates andthe dynamicmartensitic point phase of transformationview while the possibility.tetragonal phase shows good stability, which further elevates the martensitic phase transformation possibility. Table 2. The calculated various elastic constants (Cij) for the full Heusler compound Pd2CoAl in both Tablecubic and 2. The tetragonal calculated structures. various elastic constants (Cij) for the full Heusler compound Pd2CoAl in both cubic and tetragonal structures. Elastic Constants (GPa) Compound Structure C11 C12Elastic C13 ConstantsC14 (GPa)C33 C44 C66 Compound Structure Cubic 143.2C11 167.4C12 C-13 100.2C14 C-33 C -44 C -66 Pd2CoAl TetragonalCubic 143.2133.7 167.4 38.1 60.3- 100.2 - 231.2- 93.0 - 86.4 - Pd2CoAl Moreover, with the obtainedTetragonal elastic 133.7 constants 38.1 we can 60.3 further - evaluate 231.2 the93.0 elastic 86.4 anisotropy in the tetragonalMoreover, L 2with1-type the Pd obtained2CoAl compound. elastic constants The calculated we can 3Dfurther directional evaluate dependence the elastic of anisotropy the Young’s in Crystalsthemodulus tetragonal2019 and, 9, 422 shear L21-type modulus Pd2CoAl with compound. ELATE program The calculated [65] are 3Dplotted directional in Figures dependence 11 and 12, of respectively. the Young’s10 of 14 Besidesmodulus this, and the shear corresponding modulus with 2D ELATE projections program in di fferent[65] are planes plotted are in alsoFigures provided 11 and together. 12, respectively. We can clearlyBesides see this, that the the corresponding tetragonal Pd 2D2CoAl projections compound in di hasfferent strong planes elasti arec anisotropy. also provided The together.maximum We of canthe clearly see that the tetragonal Pd2CoAl compound has strong elastic anisotropy. The maximum of the clearly see that the tetragonal Pd2CoAl compound has strong elastic anisotropy. The maximum of the Young’sYoung’s modulusmodulus andand shearshear modulusmodulus areare foundfound inin thethe [111][111] andand [001][001]directions, directions, respectively. respectively. Young’s modulus and shear modulus are found in the [111] and [001] directions, respectively.

Figure 11. The calculated directional dependent Young’s modulus for the full Heusler compound Figure 11. The calculated directional dependent Young’s modulus for the full Heusler compound PdFigure2CoAl 11. in The L21 -typecalculated tetragonal directional structure. dependent Young’s modulus for the full Heusler compound Pd22CoAl in L2 1-type-type tetragonal tetragonal structure. structure.

Figure 12. The calculated directional dependent shear modulus for the full Heusler compound Pd2CoAl in L21-type tetragonal structure.

4. Conclusions With first principles calculation, we have systematically investigated the electronic, magnetic, dynamic, and mechanical properties of the full Heusler compound Pd2CoAl and also examined its phase stability from different perspectives. Results show that the L21-type structure is energetically more preferable than the XA-type because of the lower total energy. The derived equilibrium lattice constant is 6.057 Å and it is in very good agreement with previous study. At ground state in cubic phase, Pd2CoAl behaves like ferromagnetic metal with total magnetic moment of 1.78 µB, which is mainly contributed by the Co atom from strong spin splitting effect of the density of states in two spin directions. Under uniform strains from 5% to +5%, the variation of total magnetic moments has been − obtained and it is still caused by the much larger change from the Co atom, compared with Pd and Al atoms. The tetragonal structure has further been considered and it is found the total energy can be reduced when the cubic structure is varied into the tetragonal one, leading to the possible tetragonal martensitic phase transformation. The large energy difference of 0.165 eV between the tetragonal and cubic phases is found at the c/a ratio of 1.30. To evaluate the phase stability, the total density of states has been compared between the cubic and tetragonal phases for Pd2CoAl, and results show that the tetragonal phase transformation could reduce the states at the Fermi energy level in both directions and thus enhance the phase stability. In addition, the dynamic and mechanical stabilities have also been accessed for Pd2CoAl in both cubic and tetragonal structures and it is found that the tetragonal phase shows good stability against the cubic phase, which further confirms that the tetragonal phase transformation is highly expected. Lastly, the strong elastic anisotropy has been clearly shown with the 3D representation of the directional dependence of the Young’s modulus and shear modulus.

Author Contributions: T.Y. and X.W. conceived the work; L.H. and J.Y. performed the calculations and result analysis and wrote the paper; Y.W. and R.K. provided valuable comments on this work and the manuscript. Funding: This research was supported by “Fundamental Research Funds for the Central Universities” grant number [XDJK2018C078] and “Doctoral Fund Project of Southwest University” grant number [SWU117037]. Crystals 2019, 9, 422 11 of 14

Acknowledgments: The authors thank the anonymous reviewers for their constructive suggestions. Conflicts of Interest: The authors declare no conflict of interest.

References

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