research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

IUCrJ
Volume 6| Part 2| March 2019| Pages 218-225
ISSN: 2052-2525

Site preference and tetragonal distortion in palladium-rich Heusler alloys

aSchool of Physical Science and Technology, Southwest University, Chongqing 400715, People's Republic of China, bLaboratory for Developing New Materials and Their Characterization, University Ferhat Abbas Setif 1, Setif 19000, Algeria, cInstitute for Superconducting and Electronic Materials (ISEM), University of Wollongong, Wollongong 2500, Australia, dLaboratoire de Physique Quantique de la Matière et de Modélisation Mathématique (LPQ3M), Université de Mascara, Mascara 29000, Algeria, and eApplied Physics, Division of Materials Science, Department of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå SE-971 87, Sweden
*Correspondence e-mail: wangxt45@126.com

Edited by A. Fitch, ESRF, France (Received 25 October 2018; accepted 12 December 2018; online 24 January 2019)

In this work, two kinds of competition between different Heusler structure types are considered, one is the competition between XA and L21 structures based on the cubic system of full-Heusler alloys, Pd2YZ (Y = Co, Fe, Mn; Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb). Most alloys prefer the L21 structure; that is, Pd atoms tend to occupy the a (0, 0, 0) and c (0.5, 0.5, 0.5) Wyckoff sites, the Y atom is generally located at site b (0.25, 0.25, 0.25), and the main group element Z has a preference for site d (0.75, 0.75, 0.75), meeting the well known site-preference rule. The difference between these two cubic structures in terms of their magnetic and electronic properties is illustrated further by their phonon dispersion and density-of-states curves. The second type of competition that was subjected to systematic study was the competitive mechanism between the L21 cubic system and its L10 tetragonal system. A series of potential tetragonal distortions in cubic full-Heusler alloys (Pd2YZ) have been predicted in this work. The valley-and-peak structure at, or in the vicinity of, the Fermi level in both spin channels is mainly attributed to the tetragonal ground states according to the density-of-states analysis. ΔEM is defined as the difference between the most stable energy values of the cubic and tetragonal states; the larger the value, the easier the occurrence of tetragonal distortion, and the corresponding tetragonal structure is stable. Compared with the ΔEM values of classic Mn2-based tetragonal Heusler alloys, the ΔEM values of most Pd2CoZ alloys in this study indicate that they can overcome the energy barriers between cubic and tetragonal states, and possess possible tetragonal transformations. The uniform strain has also been taken into consideration to further investigate the tetragonal distortion of these alloys in detail. This work aims to provide guidance for researchers to further explore and study new magnetic functional tetragonal materials among the full-Heusler alloys.

1. Introduction

Since the first series of Heusler compounds, of the general formula Cu2MnX (X = Al, In, Sn, Sb, Bi), was proposed by Heusler in 1903, the passion for research in Heusler alloys has continued to rise because of their numerous excellent properties and potential for many applications in numerous technical fields. They act as promising candidates for spin-gapless semiconductors (Wang, Chang, Liu et al., 2017[Wang, X., Cheng, Z., Liu, G., Dai, X., Khenata, R., Wang, L. & Bouhemadou, A. (2017). IUCrJ, 4, 758-768.]; Bainsla et al., 2015[Bainsla, L., Mallick, A. I., Raja, M. M., Nigam, A. K., Varaprasad, B. C. S., Takahashi, Y. K., Alam, A., Suresh, K. G. & Hono, K. (2015). Phys. Rev. B, 91, 104408.]; Wang et al., 2016[Wang, X., Cheng, Z., Wang, J., Wang, X. L. & Liu, G. (2016). J. Mater. Chem. C. 4, 7176-7192.]; Gao & Yao, 2013[Gao, G. Y. & Yao, K. L. (2013). Appl. Phys. Lett. 103, 232409.]; Skaftouros et al., 2013a[Skaftouros, S., Özdoğan, K., Şaşıoğlu, E. & Galanakis, I. (2013a). Appl. Phys. Lett. 102, 022402.]), thermoelectric materials (Wehmeyer et al., 2017[Wehmeyer, G., Yabuki, T., Monachon, C., Wu, J. & Dames, C. (2017). Appl. Phys. Rev. 4, 041304.]; Lue et al., 2007[Lue, C. S., Chen, C. F., Lin, J. Y., Yu, Y. T. & Kuo, Y. K. (2007). Phys. Rev. B, 75, 064204.]; Lue & Kuo, 2002[Lue, C. S. & Kuo, Y. K. (2002). Phys. Rev. B, 66, 085121.]), shape memory alloys (Li et al., 2018[Li, Z., Jiang, Y., Li, Z., Sánchez Valdés, C. F., Sánchez Llamazares, J. L., Yang, B., Zhang, Y., Esling, C., Zhao, X. & Zuo, L. (2018). IUCrJ, 5, 54-66.]; Aksoy et al., 2009[Aksoy, S., Acet, M., Deen, P. P., Mañosa, L. & Planes, A. (2009). Phys. Rev. B, 79, 212401.]), superconductors (Nakajima et al., 2015[Nakajima, Y., Hu, R., Kirshenbaum, K., Hughes, A., Syers, P., Wang, X., Wang, K., Wang, R., Saha, S. R., Pratt, D., Lynn, J. W. & Paglione, J. (2015). Sci. Adv. 1, e1500242.]; Sprungmann et al., 2010[Sprungmann, D., Westerholt, K., Zabel, H., Weides, M. & Kohlstedt, H. (2010). Phys. Rev. B, 82, 060505.]; Shigeta et al., 2018[Shigeta, I., Kubota, T., Sakuraba, Y., Kimura, S., Awaji, S., Takanashi, K. & Hiroi, M. (2018). Physica B, 536, 310-313.]) and topological insulators (Hou et al., 2015[Hou, Z., Wang, W., Xu, G., Zhang, X., Wei, Z., Shen, S., Liu, E., Yao, Y., Chai, Y., Sun, Y., Xi, X., Wang, W., Liu, Z., Wu, G. & Zhang, X. (2015). Phys. Rev. B, 92, 235134.]; Lin et al., 2015[Lin, S. Y., Chen, M., Yang, X. B., Zhao, Y. J., Wu, S. C., Felser, C. & Yan, B. (2015). Phys. Rev. B, 91, 094107.]). Therefore, ongoing investigations of Heusler alloys are quite active and are set to continue due to predictions of their enhanced performance through theoretical design and experimental synthesis. Heusler alloys normally have three types of structure, full-Heusler, half-Heusler and quaternary Heusler, with stoichiometric compositions of X2YZ, XYZ and XYMZ, respectively. Usually, the X, Y and M atoms are transition elements and the Z atom is a main group element.

As a classic type of Heusler alloy, full-Heusler alloys have been attracting much interest from researchers; however, the innovative properties of full-Heusler alloys depend strongly on their highly ordered structure. So, ignoring other factors, we now only consider this highly ordered structure to yield two possible atomic ordering figurations: the XA type [or the Hg2CuTi/inverse type, with space group F43m (No. 216)] and the L21 type [or Cu2MnAl type, with space group Fm3m (No. 225)], represented stoichiometrically by XXYZ and XYXZ, respectively. According to the general site preference rule, for full-Heusler alloys represented by X2YZ, if the valence electrons of X are more numerous than those of Y, X tends to occupy Wyckoff sites a (0, 0, 0) and c (0.5, 0.5, 0.5), but prefers sites a (0, 0, 0) and b (0.25, 0.25, 0.25) when Y possesses more valence electrons than X. However, some counter-examples were found: when X represents a low-valence metal in particular, X tends to occupy the a and c positions, forming an L21 structure such as in Ti2CrZ, Ti2CuZ and Ti2ZnZ (Wang, Cheng, Yuan & Khenata, 2017[Wang, X., Cheng, Z., Yuan, H. & Khenata, R. (2017). J. Mater. Chem. C. 5, 11559-11564.]). However, we found that the cubic competitive mechanism of XA and L21 for X2YZ alloys mostly focuses on the X elements with fewer valence electrons such as Ti and Sc. The occupation of the atomic position has been confirmed to have a great influence on the properties of the Heusler alloy (Qin et al., 2017[Qin, G., Wu, W., Hu, S., Tao, Y., Yan, X., Jing, C. & Ren, W. (2017). IUCrJ, 4, 506-511.]), so it is necessary to study the positioning in the X2YZ Heusler alloys for X with numerous valence electrons such as in Cu, Ni and Pd.

On the other hand, a review of recent studies of Heusler compounds suggests that researchers are more interested in cubic structures than tetragonal structures; the progress made in finding better tetragonal phases is limited. However, the tetragonal phases have some excellent properties such as large magneto-crystalline anisotropy (Salazar et al., 2018[Salazar, D., Martín-Cid, A., Garitaonandia, J. S., Hansen, T. C., Barandiaran, J. M. & Hadjipanayis, G. C. (2018). J. Alloys Compd. 766, 291-296.]; Matsushita et al., 2017[Matsushita, Y. I., Madjarova, G., Dewhurst, J. K., Shallcross, S., Felser, C., Sharma, S. & Gross, E. K. (2017). J. Phys. D Appl. Phys. 50, 095002.]), large intrinsic exchange-bias behavior (Felser et al., 2013[Felser, C., Alijani, V., Winterlik, J., Chadov, S. & Nayak, A. K. (2013). Phys. Rev. Lett. 110, 127204.]; Nayak et al., 2012[Nayak, A. K., Shekhar, C., Winterlik, J., Gupta, A. & Felser, C. (2012). Appl. Phys. Lett. 100, 152404.]), high Curie temperature and low magnetic moment. In addition, it is reported that the tetragonal Heusler alloys have a large perpendicular magnetic anisotropy, which is the key to spin-transfer torque devices (Balke et al., 2007[Balke, B., Fecher, G. H., Winterlik, J. & Felser, C. (2007). Appl. Phys. Lett. 90, 152504.]). Therefore, searching for new, better tetragonal phases and studying their possible tetragonal transformations from the cubic phase are essential and fundamental.

In this work, a series of full-Heusler alloys, Pd2YZ (Y = Co, Fe, Mn; Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb), were chosen in order to study their atomic ordering competition between two cubic-type structures: the XA type and the L21 type. Furthermore, the tetragonal transformation and phase stability of the above-mentioned alloys were also investigated by means of first principles. The origin of the tetragonal ground state of Pd2YZ alloys is explained with the help of density of states (DOS).

2. Computational methods

First-principle band calculations were carried out via the plane-wave pseudo-potential method (Troullier & Martins, 1991[Troullier, N. & Martins, J. L. (1991). Phys. Rev. B, 43, 1993-2006.]) using CASTEP code in the framework of density functional theory (Becke, 1993[Becke, A. D. (1993). J. Chem. Phys. 98, 5648-5652.]). The Perdew–Burke–Ernzerhof functional of the generalized gradient approximation (Perdew et al., 1996[Perdew, J. P., Burke, K. & Ernzerhof, M. (1996). Phys. Rev. Lett. 77, 3865-3868.]) and ultra-soft (Al-Douri et al., 2008[Al-Douri, Y., Feng, Y. P. & Huan, A. C. H. (2008). Solid State Commun. 148, 521-524.]) pseudo-potential were used to describe the interaction between electron-exchange-related energy and the nucleus and valence electrons, respectively. The integration over the first Brillouin zone was performed with a k-mesh grid of 12 × 12 × 12 for the cubic structure and 12 × 12 × 15 for the tetragonal structure, using a Monkhorst–Pack grid with a cut-off energy of 450 eV and a self-consistent field tolerance of 10−6 eV. The spin-polarization was also considered in the total energy calculation.

For the calculations of the phonon spectrum of Pd2-based Heusler alloys, we used the finite displacement method implemented in the CASTEP code. During the phonon spectrum calculations, the k-mesh grids of 12 × 12 × 12 and 12 × 12 × 15 in the Brillouin zone integration are used for cubic and tetragonal Heusler alloys, respectively.

3. Results and discussion

3.1. The site ordering competition between XA and L21 structures in the cubic phases of Pd2YZ full-Heusler alloys

Since not all the full-Heusler alloys obey the site preference rule (Zhang et al., 2016[Zhang, X. J., Liu, Z. H., Zhang, Y. J., Liu, H. Y., Liu, G. D., Cui, Y. T. & Ma, X. Q. (2016). Intermetallics, 73, 26-30.]; Lukashev et al., 2016[Lukashev, P., Kharel, P., Gilbert, S., Staten, B., Hurley, N., Fuglsby, R., Huh, Y., Valloppilly, S., Zhang, W., Yang, K., Skomski, R. & Sellmyer, D. J. (2016). Appl. Phys. Lett. 108, 141901.]; Meng et al., 2017[Meng, F., Hao, H., Ma, Y., Guo, X. & Luo, H. (2017). J. Alloys Compd. 695, 2995-3001.]; Wang, Cheng & Wang, 2017[Wang, X., Cheng, Z. & Wang, W. (2017). Materials, 10, 1200.]), clarifying the preferable atomic ordering of these alloys is necessary. Two structural configurations of Pd2MnAl are given as examples in Fig. 1[link]. The first is the L21 structure, where Pd atoms carrying more valence electrons than Mn and Al atoms occupy Wyckoff sites a (0, 0, 0) and c (0.5, 0.5, 0.5), while the Mn atom is at site b (0.25, 0.25, 0.25) and the Al atom is located at site d (0.75, 0.75, 0.75); this meets the well known site-preference rule (Bagot et al., 2017[Bagot, P. A. J., Silk, O. B. W., Douglas, J. O., Pedrazzini, S., Crudden, D. J., Martin, T. L., Hardy, M. C., Moody, M. P. & Reed, R. C. (2017). Acta Mater. 125, 156-165.]; Burch et al., 1974[Burch, T. J., Litrenta, T. & Budnick, J. I. (1974). Phys. Rev. Lett. 33, 421-424.]). The second is the XA type, where Pd elements are at Wyckoff sites a and b. To clarify which is the favorable atomic ordering of Pd2YZ, we calculated and plotted EL21EXA as a function of different alloys in Fig. 2[link]. When the value of the difference is negative, EXA is larger than EL21, indicating that the L21 phase is more stable than the XA phase due to the lower total energy. The inverse is also true. We can clearly see that most of these alloys prefer the L21 phase, except for Pd2CoZ (Z = As, Sb, P, Pb) and Pd2FeZ (Z = As, Sb, P) from Fig. 2[link]. The positive difference values in Pd2CoAs, Pd2CoSb and Pd2FeSb imply that the XA state with lower energy is the most stable phase for the three alloys. We note that the difference values for Pd2CoP, Pd2CoPb, Pd2FeP and Pd2FeAs are around zero, which shows that these alloys have no obvious preferred steady state, and it is likely to be the state where the XA type and the L21 type coexist. Moreover, the larger the absolute values of the difference, the more stable the steady state of the corresponding substance. In addition, as the atomic radius of Y atoms increases from Co to Fe to Mn, the absolute value of the difference mostly becomes larger, indicating that the stability of these alloys is enhanced.

[Figure 1]
Figure 1
Crystal structures of L21- and XA-type full-Heusler alloys of Pd2MnAl.
[Figure 2]
Figure 2
The difference in total energy for L21 and XA types in Pd2-based alloys.

In order to further elucidate the dynamic stability of the alloys of interest, as an example we calculated the phonon dispersion curves of Pd2MnAl along the W-L-Γ-X-W-K directions for L21- and XA-type structures in the Brillouin zone (displayed in Fig. 3[link]). There are four atoms in a primitive cell of Pd2MnAl, resulting in 3 × 4 = 12 branches in its phonon dispersion curves, and each branch corresponds to a mode of vibration. Among these, the three low-frequency branches correspond to acoustic phonon curves, while the other nine high-frequency branches correspond to optical phonon curves. From Fig. 3[link] we can see that the phonon dispersion spectra for the L21-Pd2MnAl compound have no imaginary frequencies, whereas Pd2MnAl in the XA-type structure has an imaginary frequency, which further proves that Pd2MnAl is stable in L21 and unstable in the XA phase.

[Figure 3]
Figure 3
Calculated phonon dispersion curves for (a) L21-Pd2MnAl and (b) XA-Pd2MnAl.

In order to further explain the favorability of the atomic ordering, the DOS curves are given in Fig. 4[link]. Pd2MnAl and Pd2CoSb were selected as examples; we found that whether in XA or L21 type, the total magnetic moment arises mainly from the Y element – here, Mn and Co atoms – due to their strong exchange splitting (Zhao et al., 2017[Zhao, C., Norden, T., Zhang, P., Zhao, P., Cheng, Y., Sun, F., Parry, J. P., Taheri, P., Wang, J., Yang, Y., Scrace, T., Kang, K., Yang, S., Miao, G. X., Sabirianov, R., Kioseoglou, G., Huang, W., Petrou, A. & Zeng, H. (2017). Nat. Nanotechnol. 12, 757-762.]) in the vicinity of EF. The magnetic moments of Al and Sb atoms are quite small, so they can be ignored. The almost-symmetry of the PDOSs of Pd in the spin-down channel makes Pd have a very small magnetic moment, so it also makes very little contribution to the total magnetic moment. Note that in the L21-type structures, there is only one line of the two Pd atoms' magnetic moments. In the L21 states, two Pd atoms occupy sites a (0, 0, 0) and c (0.5, 0.5, 0.5); thus, the surrounding environments of the Pd atoms are the same based on the symmetry and periodicity of the structures, causing the two lines to be recombined into one. This situation does not exist in the XA structures. The valence electrons at, or in the vicinity of, the Fermi level mostly determine the magnetic and electronic structures of these full-Heusler alloys. In Figs. 4[link](a) and 4(b), the L21-type structure of Pd2MnAl has lower energy in both the majority and minority spin channels than the XA type at EF, with 0.17 and 0.51 states per eV, respectively, indicating that Pd2MnAl is more stable in the L21 type than in the XA type. Meanwhile, the situation is different in Pd2CoSb, with the value of DOS at or around EF in the XA-type structure, being less than that in the L21 type (1.18 and 0.28 states per eV) separately in the spin-up and spin-down channels, respectively – making it more stable in XA structures. These two alloys correspondingly exhibit L21- and XA-type structures, respectively. Through calculation, we found that Pd2YZ alloys mostly exhibit L21-type structures.

[Figure 4]
Figure 4
The total and atomic density of states (DOSs) of Pd2MnAl and Pd2CoSb in L21 and XA structures, respectively.

3.2. Magnetic and Slater–Pauling rules of cubic-type Pd2YZ

To investigate the magnetic properties of Pd2YZ, we plotted the total magnetic moment per formula unit as a function of different alloys in two cubic phases, the XA and L21 states (Fig. 5[link]). It is clear that all magnetic moments in the L21 phases are larger than those in the XA type in certain alloys. Secondly, the magnetic moments of Pd2MnZ (Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb) – about 4 µB – are the largest, followed by Pd2FeZ (Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb), while Pd2CoZ (Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb) alloys have the smallest magnetic moments at around 1.5 µB. We also computed the total and atomic magnetic moments of the equilibrium lattice constants in the XA- and L21-type structures of Pd2YZ; these are listed in Tables S1 and S2 of the supporting information and show that the total magnetic moments mainly come from the Y elements owing to their large strong exchange splitting.

[Figure 5]
Figure 5
The total magnetic moment per formula unit as a function of the different Pd2-based alloys in both L21 and XA structures.

Moreover, the sums of the valence electrons of the two Pd and Y atoms are already 27, 28 and 29, corresponding to the three types of alloy, Pd2MnZ, Pd2FeZ and Pd2CoZ. So, if these alloys meet the famous Slater–Pauling rule (Galanakis et al., 2014[Galanakis, I., Şaşıoğlu, E., Blügel, S. & Özdoğan, K. (2014). Phys. Rev. B, 90, 064408.]; Faleev et al., 2017a[Faleev, S. V., Ferrante, Y., Jeong, J., Samant, M. G., Jones, B. & Parkin, S. S. (2017a). Phys. Rev. B, 95, 045140.]; Skaftouros et al., 2013b[Skaftouros, S., Özdoğan, K., Şaşıoğlu, E. & Galanakis, I. (2013b). Phys. Rev. B, 87, 024420.]), they should obey the rule of Mt = Zt − 28. These alloys should have a larger magnetic moment than current magnetic moments according to the Slater–Pauling rule, which suggests that this rule does not apply for Pd2YZ. Furthermore, we should point out that all the Pd2YZ alloys in this study are not half-metallic materials or spin-gapless semiconductors, and even the majority of Pd2-based alloys do not have half-metallic or spin-gapless semiconducting behaviors. The well known Slater–Pauling rule is a method of predicting half-metallic or spin-gapless semiconductor materials, and thus the Pd2YZ (or even Pd2-based Heusler) alloys do not obey the Slater–Pauling rule.

3.3. Possible tetragonal transformations in Pd2YZ compounds

Stable tetragonal phases and possible tetragonal transformations are important for investigating Heusler alloys. Thus, we now discuss the possible tetragonal transformations in Pd2YZ. Because most of these alloys are L21-type stable, we applied tetragonal deformation and uniform strain to search for the tetragonal phases and possible tetragonal transformations in only L21-type structures. We should point out here, for Pd2CoSb, Pd2CoAs and Pd2FeSb, the XA structure is much more stable than L21; we also studied the possible tetragonal transformations in XA-type structures of these three alloys (see Fig. S1 of the supporting information).

By maintaining the volume of the tetragonal unit cell Vtetragonal = a × b × c (a = b) as equal to the equilibrium cubic volume Vequilibrium = a3 while changing the c/a ratio, we obtain the L10-type structures as shown in Fig. 6[link]. We assume that the volume for the equilibrium state does not change with tetragonal distortions. During the tetragonal deformation, there are two important parameters: ΔEM and the c/a ratio. ΔEM is the difference in energy between the most stable cubic state and the most stable tetragonal phase; the total energy is set to zero at a c/a ratio of 1, which represents the most stable L21-type cubic phase. By relaxing the c/a ratios, the minimum of the total energy can be obtained in the tetragonal distortion, which corresponds to the most stable tetragonal phase. It can be seen from Fig. 7[link] that almost all of the Pd2YZ alloys [except for Pd2Mn(Al/In/Si/Ge/Sn/Pb)] can undergo tetragonal deformation and form a tetragonal Heusler L10 structure.

[Figure 6]
Figure 6
Crystal structures of (a) cubic L21 and (b) tetragonal L10 Pd2MnGa.
[Figure 7]
Figure 7
EM per formula unit as a function of the different Pd2-based alloys.

According to the classic tetragonal Heusler alloys, for the occurrence of a stable tetragonal phase, a relatively large ΔEM is needed. Generally, an absolute value of ΔEM ≥ 0.1 eV per formula unit (f.u.) is required for Mn2-based Heusler alloys; for example, the absolute values of ΔEM for Mn3Ga (Liu et al., 2018[Liu, Z. H., Tang, Z. J., Tan, J. G., Zhang, Y. J., Wu, Z. G., Wang, X. T., Liu, G. D. & Ma, X. Q. (2018). IUCrJ, 5, 794-800.]) and Mn2FeGa (Faleev et al., 2017b[Faleev, S. V., Ferrante, Y., Jeong, J., Samant, M. G., Jones, B. & Parkin, S. S. (2017b). Phys. Rev. Appl. 7, 034022.]) are about 0.14 and 0.12 eV per f.u., respectively. We were excited to find that the vast majority of absolute values of ΔEM for Pd2CoZ alloys are larger than 0.1 per f.u., which hints that for most Pd2CoZ alloys, we may not observe a cubic state for them in the experiment. We note that the maximum value of ΔEM in these alloys appears in Pd2CoTi and is about 0.225 eV per f.u., almost two times that of Mn2FeGa. We also found that the stable tetragonal structures of Mn3Ga (Liu et al., 2018[Liu, Z. H., Tang, Z. J., Tan, J. G., Zhang, Y. J., Wu, Z. G., Wang, X. T., Liu, G. D. & Ma, X. Q. (2018). IUCrJ, 5, 794-800.]), Mn2FeGa (Faleev et al., 2017b[Faleev, S. V., Ferrante, Y., Jeong, J., Samant, M. G., Jones, B. & Parkin, S. S. (2017b). Phys. Rev. Appl. 7, 034022.]) and Zn2RuMn (Han et al., 2019[Han, Y., Bouhemadou, A., Khenata, R., Cheng, Z., Yang, T. & Wang, X. (2019). J. Magn. Magn. Mater. 471, 49-55.]) occur at c/a = 1.30, 1.40 and 1.41, respectively; this indicates that the stable tetragonal phases of these Pd-based alloys occur in the reasonable c/a range of from 1.23 to 1.42, as shown in Fig. 8[link]. However, there are also cases where the tetragonal transformation occurs at c/a < 1, such as for Pd2FeZ (Z = Si, Ge, Pb). The curves of the tetragonal deformation of each alloy can be seen in Figs. S2, S3 and S4.

[Figure 8]
Figure 8
The c/a ratio as a function of the different Pd2-based alloys.

Uniform strain has also been taken into consideration to study the possible tetragonal transformations. To facilitate the study of all the alloys, we take Pd2MnGa and Pd2FeGa as examples. Change in volume can influence the value of ΔEM as shown in the inset of Fig. 9[link]. ΔEM and Vopt + X%Vopt are negatively correlated; that is, when X changes from −3 to +3, the absolute value of the lowest energy corresponding to the alloy is lower, resulting in a decrease in the absolute value of ΔEM. This proves that the L10 phases become increasingly stable with the contraction of the optimized volume. However, regardless of any change in volume, the c/a ratio remains stable: 1.29 for Pd2MnGa and 1.3 for Pd2FeGa. Furthermore, there is only one minimum located at c/a > 1 during the tetragonal deformation of Pd2MnGa, but two minima for Pd2FeGa, with the shallow minimum located at c/a < 1 and the deeper minimum at c/a > 1. The stable tetragonal phases of Pd2YZ are the states with the lowest energy.

[Figure 9]
Figure 9
Total energies as functions of the c/a ratio for Pd2MnGa and Pd2FeGa with contraction/expansion of the unit-cell volume. The zero point of the total energy was set to that of the most stable L21 cubic phase (c/a = 1).

In order to further validate the stability of our predicted L10 structures, as a special example, we choose Pd2MnGa to study its calculated phonon dispersion curves and phonon DOS, as shown in Fig. 10[link]. It is clear from Fig. 10[link](a) that L10-Pd2MnGa has no imaginary frequencies, indicating the dynamical stability of this material. Furthermore, via analysis of phonon DOS, from which the phonon dispersion originates in Fig. 10[link](b), we easily found that the three low-frequency (0–4 THz) acoustic phonon curves are mainly attributed to Pd atoms, while the three relatively high-frequency (4–6 THz) optical phonon curves come from Ga atoms and the remaining six high-frequency (6–8 THz) optical phonon curves originate from the Mn atom.

[Figure 10]
Figure 10
Calculated (a) phonon dispersion curves and (b) phonon DOS for L10-Pd2MnGa.

It is clear that whether the cubic L21-type or the tetragonal L10-type structures exhibit metallic properties is explained by the definite value at the EF in both majority and minority DOSs. The total DOSs are both almost entirely contributed by the Mn/Fe atoms due to their strong exchange splitting around the Fermi level in these two types. First, we take Pd2MnGa as an example: the origin of the tetragonal ground states of these Pd2YZ alloys can also be explained based on the DOS structures. It is noted that in the work by Faleev et al. (2017[Faleev, S. V., Ferrante, Y., Jeong, J., Samant, M. G., Jones, B. & Parkin, S. S. (2017b). Phys. Rev. Appl. 7, 034022.]b), one of the contributions to the total energy was the band energy Eband = ∫EminEFdEDOS(E)E, a reduction of the DOS near the EF in a tetragonal phase, in conjunction with conservation of the integral for the number of valence electrons NV = ∫EminEFdEDOS(E), often leads to a lower total energy for the tetragonal phase than for the cubic phase (Emin here is the minimum energy of the valence bands). As shown in Figs. 11[link](a) and 11(b), in the spin-up channel, the comparatively high total DOS value of 0.65 states per eV at EF in the cubic L21 phase becomes a valley DOS structure, having a lower energy of 0.54 states eV−1 in the tetragonal L10 phases. Also, a shallow valley turns into a deeper valley in the spin-down channel during the tetragonal deformation, from an absolute value of 1.14 to 0.68 states eV−1. Thus, the tetragonal state is more stable than the cubic states for Pd2MnGa. We then consider Pd2FeGa: a high peak located at EF in the minority of the total DOS can be clearly seen in Fig. 11[link](c). We were excited to find that this high peak with absolute value of 2.48 states per eV shifted to a lower energy, resulting in an apparent valley of only 2.31 states per eV occurring at the EF in the tetragonal L10 phase in the minority DOS in Fig. 11[link](d). Also, the higher DOS moves to a lower energy in the majority of the total DOS, to an extent of 0.26 states per eV, which effectively increases the phase stability of the tetragonal structures. Fermi energy is used as a sensor for the peak-to-valley DOS structure. As a result of the tetragonal deformation, the symmetries of the alloys are destroyed, resulting in a much broader and more shallow DOS structure, or even its disappearance at the EF; this increases the phase stability.

[Figure 11]
Figure 11
The total and atomic DOS in L21 and the stable tetragonal phases of Pd2MnGa and Pd2FeGa.

4. Conclusions

We investigated the atomic ordering competition between XA and L21 types, tetragonal transformation, and phase stability of full-Heusler alloys of Pd2YZ. We found that most of these alloys favor crystallization in an L21 structure as opposed to an XA structure, meeting the well known site preference rule. Tetragonal geometric optimization of Pd2YZ under the equilibrium cubic type phase indicated that the total energy of the tetragonal L10 phases is lower than that of the cubic L21 phase; thus, a phase transition from cubic to tetragonal is likely to occur in these full-Heusler alloys. We found that the valley-and-peak structure in the vicinity or at the Fermi level in the minority/majority spin channels can be mainly attributed to the tetragonal ground state occurring. Most Pd2CoZ alloys can overcome the energy barrier between the cubic and tetragonal ground states and possess possible tetragonal transformations as indicated by their large ΔEM values, such as the ΔEM value of 0.225 eV per f.u. of Pd2CoTi which is almost twice that of Mn2FeGa. Moreover, the uniform strain can also tune the tetragonal transformation: as the lattice constant increases, ΔEM values for Pd2YZ decrease. Additionally, these alloys are metallic materials in both cubic and L10 states, and the total magnetic moment mainly originates from the Y atoms.

5. Outlook

In this work, we investigated the competition between L21 and L10 structures for 36 palladium-rich Heusler alloys, and we found that 30 of the alloys have a possible phase transition from cubic to tetragonal states, implying that the tetragonal structure is the ground state for these alloys. Moreover, for most of the Pd2CoZ alloys, the energy difference between the cubic and tetragonal structures is larger than 0.1 per f.u.; that is, only the tetragonal Heusler structure may be observed in these Pd2CoZ alloys.

To the best of our knowledge, to date there has been little research into the topic of palladium-rich Heusler alloys. Some articles (Winterlik et al., 2008[Winterlik, J., Fecher, G. H. & Felser, C. (2008). Solid State Commun. 145, 475-478.], 2009[Winterlik, J., Fecher, G. H., Thomas, A. & Felser, C. (2009). Phys. Rev. B, 79, 064508.]) have reported a few cubic-type Pd2-based Heusler alloys, such as Pd2ZrAl, Pd2HfAl, Pd2ZrIn and Pd2HfIn, and found that these exhibit excellent superconducting properties. However, based on our current study, much importance should also be attached to the tetragonal-type palladium-rich Heusler alloys, and the experimental preparation of tetragonal-type palladium-rich Heusler alloys is imminent.

Furthermore, Pd3−xCoxZ, Pd3−xFexZ and Pd3−xMnxZ alloys may also be investigated theoretically and experimentally in future work. Because of their tunable crystal structures, Pd3−xYxZ (Y = Co, Fe, Mn) alloys can display a wide range of multifunctionalities.

Supporting information


Footnotes

These authors contributed equally to this work

Funding information

This research was funded by the Program for Basic Research and Frontier Exploration of Chongqing City (grant No. cstc2018jcyjA0765), National Natural Science Foundation of China (grant No. 51801163), Natural Science Foundation of Chongqing (grant No. cstc-2017jcyjBX0035) and the Doctoral Fund Project of Southwest University (grant No. 117041).

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IUCrJ
Volume 6| Part 2| March 2019| Pages 218-225
ISSN: 2052-2525