Competition of L21 and L10 ordering in Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe Heusler alloys

Results in Physics 12 (2019) 1398–1404

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Competition of L21 and L10 ordering in Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe Heusler alloys

T

Mengxin Wua,1, Yilin Hana,1, Liying Wangb, Tie Yanga, Minquan Kuanga, Xuebin Chenc,d, ⁎ Xiaotian Wanga,c,d, a

School of Physical Science and Technology, Southwest University, Chongqing 400715, PR China Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Faculty of Science, Tianjin University, Tianjin 300350, PR China c Hebei Key Laboratory of Data Science and Application, North China University of Science and Technology, Tangshan 063009, PR China d Tangshan Key Laboratory of Data Science, North China University of Science and Technology, Tangshan 063009, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: L21 and L10 ordering Tetragonal ground state Elastic constant

Based on first-principles method, for Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe, we investigated the competition between cubic L21-type and their tetragonal L10-type ordering. Firstly, low-energy tetragonal ground states are found in these alloys during the tetragonal deformation. By regulating the c/a ratio under the fixed volume of equilibrium cubic phase, we obtain two minimums in total energy-c/a curve: a shallow one in c/a < 1 and the deeper one in c/a > 1. Meanwhile the uniform strain is also taken into consideration in this work, and we found that ΔEM and Vopt + X%Vopt are negatively correlated, that is, with the volume increases from Vopt − 3%Vopt to Vopt + 3%Vopt, the absolute value of ΔEM decreases accordingly. Secondly, the origin of the tetragonal ground states of Pd2 [Fe/Mn][Se/Te] are also be explained by the density of states (DOS) figures. From the valley-andpeak structure, we can find the DOS at or in the vicinity of the Fermi level is much broader and shallower in tetragonal L10 state than cubic L21 state, indicating the phase stability of tetragonal L10 phases of these Pd2based alloys. Moreover, to further study the competition of L21 and L10 ordering in these alloys and verify the stability of the tetragonal L10 state, elastic constants were introduced in this work, C11, C12 and C44 for cubic state while C11, C12, C13, C33, C44, C66 for tetragonal L10-type structure. According to the judgment basis, all alloys exhibit L10-stable, satisfying our conclusions about the stability of tetragonal states. We hope our work can provide a guidance for researchers to further explore and study new magnetic functional tetragonal materials among the full-Heusler alloys.

Introduction Heusler alloys have attracted more and more research interest due to their numerous excellent properties and potential for many applications in various technical fields since the first Heusler alloys Cu2MnX (X = Al, In, Sn, Sb, Bi) were proposed in 1903 [1]. They have high spin polarization and tunable electronic structure, and thus they can be seen as promising candidates for spin-gapless semiconductors (SGSs) [2–6], thermoelectric materials [7–9], shape memory alloys (SMAs) [10,11], superconductors [12,13], and topological insulators [14]. Furthermore, with the deepening of the research, some enhanced performance in theoretical and experimental were found continuously, so ongoing investigations of Heusler alloys are quite active. There are generally three types of Heusler structures, i.e., full-Heusler [15], half-Heusler and

quaternary-Heusler [16], having stoichiometric compositions of X2YZ, XYZ and XYMZ, respectively. Typically, the X, Y and M atoms are transition elements and the atom Z is a main group element. FullHeusler alloys were widely studied due to their various magnetic properties. It became popular with the archetype Cu2MnAl, a remarkable compound which shows ferromagnetic ordering despite the absence of any ferromagnetic element. Reviewing the past study of Pd2-based full-Heusler alloys, lots of superconductivities in their cubic system were investigated in detail. In 1985, Kierstead et al. [17] studied the cubic state of the first Heusler alloy containing rare earths, Pd2YbSn. Through the heat capacity, resistivity, and magnetic susceptibility, they found the superconductivity and magnetic states are coexist in Pd2YbSn, as well as this alloy goes through a superconducting transition with Tc = 2.36 K and a magnetic

⁎

Corresponding author at: School of Physical Science and Technology, Southwest University, Chongqing 400715, PR China. E-mail address: [email protected] (X. Wang). 1 These authors contributed equally to this work. https://doi.org/10.1016/j.rinp.2019.01.030 Received 29 November 2018; Received in revised form 8 January 2019; Accepted 11 January 2019 Available online 14 January 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

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transition occurs at TM = 0.23 K, resulting in a coexisting state of superconducting and magnetic. Winterlik et al. [18] investigated the superconductivity of Pd2ZrAl and Pd2HfAl also in their cubic system in 2008. The transition temperatures of the two alloys are 3.2 K and 3.4 K respectively, making a demonstration for magnetism-controlled Josephson junctions for applications. Then in 2009, Winterlik et al. [19] further investigated the superconductivity of Pd2ZrAl, Pd2HfAl, Pd2ZrIn, and Pd2HfIn, and found they all have the prospect of becoming superconducting according to the van Hove scenario. In addition, the pressure influence in the cubic Pd2-based full-Heusler alloys has been studied in detail by Wiendlocha et al. [20] both in experiment and theory. They tested out the linear thermal expansion coefficient α = 1.40(3) × 10−5 K−1 and the estimated bulk modulus for HfPd2Al at room temperature is B0 = 97(2) GPa, and dTc/ dp = −0.13(1) K GPa−1. Indeed many researches of Pd2-based fullHeusler alloys have been applied in their cubic system and found some excellent properties, but there are few studies focusing on the tetragonal deformation and the competition between the cubic and tetragonal states. However, the tetragonal phases have many excellent properties, like large magneto-crystalline anisotropy [21,22], large intrinsic exchange bias behavior [23,24], and high Curie temperature. In addition, it is reported that the tetragonal Heusler alloys have a large perpendicular magnetic anisotropy (PMA), which is the key to spintransfer torque (STT) devices [25]. Thus, in this work, we focus on the competition between the cubic L21 structure and tetragonal L10 structure of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe and prove their tetragonal states are the stable state by the elastic constants.

Fig. 1. Crystal structures of L21 and L10 Pd2[Fe/Mn][Se/Te].

difference ΔE = E (c / a) − E (c / a = 1.0) as functions of the c/a ratio of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe as shown in Figs. 1(b) and 2. The zero point of the total energy is set as the L21 cubic phase (c/ a = 1). There are two obvious minimums in the curve of total energy in all of these four alloys, with one shallower in c/a < 1 called metastable state and a deeper one in c/a > 1 called steady state. A larger ΔEM which is the absolute energy difference value between the cubic L21 phase and the tetragonal L10 phase is needed for the possible tetragonal distortion. According to the classical tetragonal Heusler alloys Mn3Ga [32] and Mn2FeGa [33], the value of ΔEM is about 0.14 eV/f.u. and 0.12 eV/f.u., respectively, which provides a standard to judge whether alloys can overcome the energy barriers between cubic and tetragonal states and possess possible tetragonal transformations. We can clearly see from the Fig. 2 and Table 2, the values of ΔEM are 0.244, 0.217, 0.132, and 0.203, correspondingly to Pd2MnSe, Pd2FeSe, Pd2MnTe, and Pd2FeTe, all almost two times that in Mn2FeGa, indicating that these four alloys may probably have tetragonal deformations. Additionally, the larger value of ΔEM, the more likely phase transition will occur, and the more stable corresponding tetragonal structure will be. Meanwhile, the c/a ratios of Pd2MnSe, Pd2FeSe, Pd2MnTe, and Pd2FeTe about 1.40 are reasonable shown in Table 2 according to the classical tetragonal Heusler alloys, 1.30, 1.40, 1.41 for Mn3Ga [32], Mn2FeGa [33], and Zn2RuMn [34]. Then, we come to the competition between cubic L21 and tetragonal L10 ordering. In the precious discussion, all the tetragonal states of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe are stable state due to less total energy than the cubic state. Meanwhile, the band structures of these alloys have also been studied by the example of Pd2MnSe and Pd2FeTe, plotted in Fig. 3. These band structures in tetragonal state are more order-less than that in cubic state. And the high symmetry points change from W-L-G-X-W-K to Z-G-X-P-N-G as the state alter from cubic phase to tetragonal phase. But whether in cubic L21-type or tetragonal L10-type, it both exhibits metallic properties explained by the definite value at the EF in both majority and minority channels.

Computational methods First-principles calculations were carried out via the plane-wave pseudo-potential method [26] using CASTEP code in the framework of density functional theory (DFT) [27]. The Perdew-Burke-Ernzerhof (PBE) functional of the generalized gradient approximation (GGA) [28] and ultra-soft [29] pseudo-potential were used to describe the interaction between electron-exchange-related energy and nucleus and valence electrons, respectively. A Monkhorst-Pack special k-point mesh of 12 × 12 × 12 was used in the Brillouin zone integrations with a cutoff energy of 450 eV and a self-consistent field tolerance of 10−6 eV of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe. The quality of the k-point separation for the band structure calculation is 0.01 Å. The phonon energy calculation of Pd2MnTe alloy is performed in the NanoAcademic Device Calculator (Nanodcal) [30]. Results and discussion Possible tetragonal distortion Stable tetragonal phases and possible tetragonal transformations are important for investigating Heusler alloys in spintronic applications. For example, compared to the zero perpendicular magnetic anisotropy (PMA) in the cubic of full-Heusler alloys, some tetragonal Heusler alloys, like Mn3-xGa [25] and Mn3Ge [31], whose thin films show a large PMA for films grown epitaxially on single-crystal substrates, allows for reduced currents to switch the magnetization of the electrode using spin torque. We plotted the crystal structures of the cubic stable L21 state and the tetragonal L10 state in Fig. 1. From the Fig. 1(a), Pd atoms with the most valence electrons tends to occupy the Wyckoff sites A (0, 0, 0) and C (0.5, 0.5, 0.5), and Mn/Fe atoms carrying the relative more valence electrons prefer the site B (0.25, 0.25, 0.25), while the atoms Se/Te having the least valence electrons incline to the site D (0.75, 0.75, 0.75), meeting the famous Slater-Pauling rule. 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 stable L10-type structures and curves of the total energy

Effect of uniform strain In the above discussion, we keep the volume Vequilibrium = a3 unchanged to investigate the possible tetragonal transformations. Furthermore, we now applied uniform strain to Pd2MnSe, Pd2FeSe, Pd2MnTe, and Pd2FeTe to study the influence of volume to tetragonal distortion. We selected Pd2MnSe as examples and plotted its curves of total energies as functions of the c/a ratio with expansion/contraction of the unit cell volume in Fig. 4(a) and (b). We can clearly see that when the alloy expands from optimized volume (Vopt) to Vopt + 3%Vopt, the absolute value of total energy of tetragonal phase gets smaller gradually and when the alloy contract from optimized volume to Vopt − 3%Vopt, the absolute value of total energy increases. That is, the absolute value of ΔEM and Vopt + X%Vopt are negatively correlated. Same situation occurs in other three alloys Pd2FeSe, Pd2MnTe, and Pd2FeTe as shown 1399

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Fig. 2. Relationship between the total energy difference ΔE = E (c / a) − E (c / a = 1.0) and c/a ratio.

Fig. 3. Calculated band structures of (a) L21-Pd2MnSe, (b) L10-Pd2MnSe, (c) L21-Pd2FeTe and (d) L10-Pd2FeTe. 1400

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Fig. 4. (a) and (b) Total energies as functions of the c/a ratio for Pd2MnSe with expansion/contraction of the unit cell volume. (c)–(f) |ΔEM | as functions of the (1 + X %) Vopt (x = −3, −2, −1, 0, 1, 2, 3) for Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe, respectively.

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Fig. 5. The total and atomic density of states (DOS) in L21 and L10 of Pd2FeSe and Pd2FeTe. Table 1 Equilibrium lattice constants and total and atomic magnetic moments of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe Heusler alloys with L21 structure. Alloy

a (Å)

Mt (μB)

MMn/Fe (μB)

MZ (μB)

MPd (μB)

Table 2 ΔEM values, c/a ratios, and total and atomic magnetic moments of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe Heusler alloys with L10 structure.

MPd (μB)

Alloy

ΔEM (eV)

c/a ratio

Mt (μB)

MMn/ (μB)

MZ (μB)

MZn (μB)

MCd (μB)

3.92 3.17 3.96 2.94

0.09 0.15 0.11 0.08

0.27 0.36 0.22 0.23

0.27 0.36 0.22 0.23

Fe

Pd2MnSe Pd2FeSe Pd2MnTe Pd2FeTe

6.21 6.25 6.43 6.38

4.54 3.50 4.46 3.38

4 3.13 4.02 3.05

0.06 0.01 0.09 0.05

0.24 0.18 0.18 0.14

0.24 0.18 0.18 0.14

Pd2MnSe Pd2FeSe Pd2MnTe Pd2FeTe

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0.244 0.217 0.132 0.203

1.41 1.43 1.36 1.42

4.55 4.04 4.51 3.48

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Table 3 Elastic constants and bulk modulus (B) for Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe with type with L21 and L10 ordering. Alloy

Structure

C11

C12

C13

C33

C44

C66

Bv

Stability

Pd2MnSe

L21 L10

138.74 173.03

149.57 170.95

– 151.40

– 184.33

29.94 33.97

– 31.21

145.96 164.14

No Yes

Pd2FeSe

L21 L10

119.30 141.85

125.13 122.25

– 102.24

– 121.95

52.17 21.86

– 42.10

123.19 114.11

No Yes

Pd2MnTe

L21 L10

115.06 171.89

128.38 125.04

– 126.99

– 156.58

52.75 45.63

– 24.93

123.94 139.43

No Yes

Pd2FeTe

L21 L10

131.82 162.92

138.94 140.81

– 126.25

– 133.99

15.84 27.08

– 45.47

136.56 135.34

No Yes

Elastic constants

in Fig. 4(d)–(f). Moreover, we also found that anyhow the volume changes, the c/a ratio of one certain alloy keep constant.

To further discover the structure stability and prove the mechanical properties of Pd2MnSe, Pd2FeSe, Pd2MnTe and Pd2FeTe, the elastic constants are also taken into consideration. There are three independent elastic constants in the cubic L21 state, C11, C12 and C44, obtained by adding various small strain stress tensors to the ground state structures, while the L10 tetragonal state have six independent elastic constants, C11, C12, C13, C33, C44, C66. In this part, we computed the elastic constants of these four alloys to advance the structural stability competition between the cubic L21 structure and the tetragonal L10 structure and the results are shown in Table 3. As we all know, the structural stability criteria for cubic crystal structure L21 are as follows:

Origin of L21 and L10 ordering competition Density of states (DOS) can interpret the origin of the cubic L21 and tetragonal L10 ordering competition. We chose Pd2FeSe and Pd2FeTe as examples to illustrate. First, we found that metallic property is exhibited in both L21- and L10-type due to the definite value at EF in both spin channels. From Fig. 5, because of the strong exchange splitting around the Fermi level of Mn/Fe atom, the total magnetic moment mainly stems from the d states of Mn/Fe atoms as shown in Tables 1 and 2. It is worth noting that there is only one line of two Pd’s magnetic moment due to the surrounding environment of two Pd atoms is consistent based on the symmetry and periodicity of structures so that two lines to be recombined into one. And for Te and Se atoms, the PDOSs of them are nearly zero near the Fermi level, making almost no contribution to the total magnetic moment. The almost-symmetry PDOSs of Pd atom also contributes little to the total magnetic moment. As we all know, the valence electrons at or in the vicinity of Fermi level have a significant influence of phase stability of full-Heusler alloys. Typically, near the Fermi level, the total DOS values for the two spin directions in the tetragonal L10-state are less than the cubic L21state to varying degrees. Fermi energy is used as a sensor for the peakto-valley DOS structure [33]. And a sharp peak near EF always gives rise to structure instability due to the increasing energy of the system. Specifically, as Pd2FeSe showed in the Fig. 5(a),(b), in the spin-up channel, the EF locates quite next to a peak around 0.2 eV in the cubic–type structure, resulting in a high DOS valuing 1.00 states/eV. While the high DOS shift to high energy and the total DOS become smoother, making a deep valley falls on Fermi level valuing only 0.43 states/eV, which lowers the total energy and stables the alloy. Meanwhile, in the minority of the total DOS, a 4.49 states/eV peak situating at EF in the L21 state moves into high energy and merges with another peak existing in the cubic state into a loftier peak at around 0.3 eV energy, so the absolute total DOS attenuate dramatically to 3.84 states/ eV. The lower DOS in both majority and minority channels leads to a lower total energy so that the structural stability of Pd2FeSe is enhanced through tetragonal deformation. Similar to the Pd2FeSe, in the case of majority DOS of Pd2FeTe, a relatively sharp peak valuing 0.86 states/eV becomes a relatively gentle peak valuing 0.71 states/eV through tetragonal distortion and a quite high peak locating at about 0.1 eV in the minority of DOS becomes so much flatter and shifts to high energy so that the absolute value at the EF decreases from 4.98 states/eV to 3.81 states/eV, which lower the total energy of Pd2FeTe and stabilize the structure, as shown in Fig. 5(c),(d). To conclude, during the tetragonal deformation, the symmetry of the d-state is destroyed, resulting in a much broader and shallower DOS structure at or near the EF, or even disappearing, with the help of valley-and-peak structure. According to the well-known Jahn-Teller effect, this can reduce EF (called N (EF)) and increase phase stability.

C11 〉 |C12 | C11 + 2C12 〉 0 C44 〉 0

(1)

And the structural stability criterion of the tetragonal L10 crystal structure is:

C11 〉 0, C33 〉 0, C44 〉 0, C66 〉 0 C11 − C12 〉 0 C11 + C33 − 2C13 〉 0 2(C11 + C12) + C33 + 4C13 〉 0

(2)

From Table 3, we can see the elastic constants of these four alloys are not satisfy the first formula C11 > |C12|, which indicates the cubic L21 structures are not stable for Pd2MnSe, Pd2FeSe, Pd2MnTe, and Pd2FeTe. While these elastic constants meet all the conditions of tetragonal L10-type Heusler alloys in c/a ratios equal 1.41, 1.43, 1.36, and 1.42, hinting the tetragonal L10 state is the most likely form of existence due to its stability of these alloys. Through the discussion of elastic constants, we more proved the tetragonal distortion stabilize the phase and the tetragonal L10 structure is stable state of these four alloys, Pd2MnSe, Pd2FeSe, Pd2MnTe, and Pd2FeTe. In order to further validate

Fig. 6. Calculated phonon dispersion curves for L10-Pd2MnTe. 1403

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the stability of our predicted L10 structures, as a special example, we choose Pd2MnTe to study its calculated phonon dispersion curve, as shown in Fig. 6. Obviously, L10-Pd2MnTe has no imaginary frequencies, indicating the dynamical stability of this material from Fig. 6.

[8]

[9]

Conclusions [10]

We investigated the competition between the cubic L21 and tetragonal L10 ordering and the possible tetragonal deformation for Pd2[Mn/Fe][Se/Te] alloys. Through regulating the c/a ratio during the tetragonal deformation, we found that the tetragonal L10 ground state was more stable than the cubic L21 state due to its lower energy. Meanwhile the uniform strain was also taken into consideration, and we found that as the volume increases from Vopt − 3%Vopt to Vopt + 3%Vopt, the absolute value of ΔEM decreases accordingly. Secondly, the origin of the tetragonal ground states of Pd2[Fe/Mn][Se/Te] are also be explained by the valley-and-peak structure in the density of states (DOS). It was found that the DOS at or in the vicinity of the Fermi level is much broader and shallower in tetragonal L10 state than cubic L21 state, which explains the stability of tetragonal L10 phases of these Pd2-based alloys. Moreover, in order to further study the orderly competition of L21 and L10 in these alloys and verify the stability of the tetragonal L10 state, elastic constants are also introduced in this work, C11, C12 and C44 for cubic state and C11, C12, C13, C33, C44, C66 for tetragonal L10-type structure. Based on the judgment basis, all four alloys exhibited L10stable, in line with our results for tetragonal stability.

[11]

[12]

[13]

[14]

[15]

[16]

[17] [18] [19]

Funding

[20]

This research was funded by the Program for Basic Research and Frontier Exploration of Chongqing City (No. cstc2018jcyjA0765), National Natural Science Foundation of China (No. 51801163), and Doctoral Fund Project of Southwest University (No. SWU117041, SWU117037), Research Funds for the Central Universities (No. XDJK2018C078).

[21]

[22]

[23] [24]

Conflicts of interest

[25]

The authors declare no conflict of interest. [26]

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