Magnetic states of Ni2MnZ and Ni2CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys

Accepted Manuscript Magnetic states of Ni2MnZ and Ni2CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys Vasiliy D. Buchelnikov, Mikhail A. Zagrebin, Vladimir V. Sokolovskiy PII: DOI: Reference:

S0304-8853(17)32153-4 https://doi.org/10.1016/j.jmmm.2017.12.018 MAGMA 63487

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Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

10 July 2017 30 October 2017 4 December 2017

Please cite this article as: V.D. Buchelnikov, M.A. Zagrebin, V.V. Sokolovskiy, Magnetic states of Ni2MnZ and Ni2CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys, Journal of Magnetism and Magnetic Materials (2017), doi: https://doi.org/10.1016/j.jmmm.2017.12.018

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Magnetic states of Ni2 MnZ and Ni2 CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys Vasiliy D. Buchelnikova,∗, Mikhail A. Zagrebina,b , Vladimir V. Sokolovskiya,c a Chelyabinsk

State University, 454001 Chelyabinsk, Russia Research South Ural State University, 454080 Chelyabinsk, Russia c National University of Science and Technology ”MIS&S”, 119991 Moscow, Russia b National

Abstract In this work we study the magnetic states of ternary Ni2 CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys in comparison with Ni2 MnZ ones by ab initio methods. It is shown that the Ni-Mn based alloys are ferromagnetic. Contrary, the Ni-Cr based alloys with Z = Ga, In, Tl, Si, Ge, Sn, Pb are compensated aniferromagnets and with Z = Al, P, As, Sb, Bi are ferromagnets. The formation energy of alloys studied is calculated and stable compounds are predicted. The possible martensitic transitions in Ni-Mn and Ni-Cr based Heusler alloys are estimated. Within the Heisenberg model and Monte Carlo tecnique, the Curie temperatures are obtained. Theoretical results are compared with other theoretical end experimental results. Keywords: Heusler alloys, ab initio methods, martensitic transformations, magnetic states

1. Introduction The physical effects such as magnetically and thermally induced shape memory effect (SME), the large magnetoresistance and giant magnetocaloric effect (MCE) are promise properties of magnetic Heusler alloys [1, 2, 3]. Ones of the ∗ Corresponding

author Email address: [email protected] (Vasiliy D. Buchelnikov)

Preprint submitted to Journal of Magnetism and Magnetic Materials

December 8, 2017

most studied and well-known Heusler alloys are Ni-based materials. The most known from these alloys is Ni2 MnGa. It is ferromagnetic (FM) in austenite and martensite states. It was shown that the properties of ternary Heusler alloys can be enhanced by addition of fourth or fifth element such as Cr, Co, and etc (see, for example, [4, 5]). For instance, an optimization of MCE in Heusler alloys can be achieved by Cr addition. The ab initio and Monte Carlo (MC) calculations have shown that the Cr addition leads to a large change in magnetization at magnetostructural transition from magnetically weak or antiferromagnetic (AFM) martensite to FM austenite [5]. It is interesting to investigate the magnetic states of Cr-based ternary Heusler alloys. In this work, we study the structure and magnetic properties of ternary Ni2 Mn-based and Ni2 Cr-based Heusler alloys with different Z elements of Group III, IV, and by ab initio and MC methods.

2. Details of calculations Ground state calculations were performed within the framework of quantumchemical modeling in the software package VASP 5.4.1 (Vienna Ab-initio Simulation Package) [6, 7] in the framework of the density functional theory (DFT) [8, 9] using a basis of plane waves and PAW (projector augmented-wave method) formalism [10]. The calculations were carried out within the framework of the generalized gradient approximation (GGA), where Perdew-BurkeErnzerhof (PBE) [11] parametrization was used for the exchange-correlation potential.

Geometric optimization of the crystal structure was carried out

within 16-atom supercell. For this, the first Brillouin zone was divided into an 8 × 8 × 8 grid according to the Morhorst-Pack scheme [12]. The energy of cutting of plane waves in the calculations was assumed equal to 400-750 eV. For all alloys, the geometry optimization was carried out to the value of forces acting on atoms equal to 0.01 meV/˚ A. All calculations were carried out for the total energy with an accuracy of 10−8 eV. The PAW PBE potentials were used with the electronic configurations as follows: Ni(3d8 3p6 4s2 ),

2

Mn(3p6 3d5 4s2 ), Cr(3p6 3d5 4s1 ), Al(3p1 3s2 ), Ga(3d10 4p1 4s2 ), In(4d10 5p1 5s2 ), Tl(5d10 6p1 6s2 ), Si(3p2 3s2 ), Ge(3d10 4p2 4s2 ), Sn(4d10 5p2 5s2 ), Pb(5d1 06p2 6s2 ), P(3p3 3s2 ), As(3d10 4p3 4s2 ), Sb(5p3 5s2 ), Bi(5d1 05p6 6p3 6s2 ). The 16-atom supercell possesses the f cc L21 -type of structure (the space group F m¯ 3m, No. 225) with Cu2 MnAl as the prototype. This structure with the general formula X2 Y Z consists of four mutually penetrating f cc sublattices, in which two X atoms are located in 8c Wyckoff positions ((1/4, 1/4, 1/4) and (3/4, 3/4, 3/4)), while the Z and Y atoms occupy positions 4a and 4b ((0, 0, 0) and (1/2, 1/2, 1/2)), respectively. Note, the ground-state calculations were performed for FM and AFM orderings of Mn (Cr) atoms, as shown in Fig. 1.

figs-pb/Fig_1a.png

figs-pb/Fig_1b.png

figs-pb/Fig_1c.png

Figure 1: Considered magnetic states in 16-atom supercell of Ni2 Mn(Cr)Z alloys: (a) ferromagnetic (FM) and b) antiferromagnetic (AFM) spin configurations of Mn(Cr) atoms.

To calculate the exchange coupling constants for alloys studied, the SPRKKR package within the GGA-PBE scheme was used [13]. For this, we used optimized lattice parameters obtained from the geometric optimization procedure within the VASP code. The knowledge of exchange integrals was allowed us to estimate the Curie temperature of austenite for alloys studied using classical Heisenberg Hamiltonian and MC technique. The MC simulations were performed on the three-dimensional lattice (≈ 4000 atoms) with a real cell and periodic boundary conditions using the standard Metropolis algorithm.

3

3. Results and discussions 3.1. Geometric optimization of crystal structure In this subsection we compare the calculation results of optimized lattice parameter and magnetic reference state for stoichiometric Ni2 Mn-based and Ni2 Cr-based Heusler alloys. Figure 2 presents the dependences of energy differences on the lattice parameter in Ni2 (Mn, Cr)Z alloys with respect to the states with the lowest energy. Here, we plotted separately results for the compounds

Figure 2: Total energy differences as a function of lattice parameter for (top) Ni2 MnZ and (bottom) Ni2 CrZ alloys with FM (filled symbols) and AFM (open symbols) spin orderings. Where Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb, As, P, Sb, Bi.

with Group III, IV, and V Z elements. Evidently, alloys mentioned in Figures 2 (a), (b), and (c) are characterized by the number of valence electron per atom, e/a, as follows: 7.5, 7.75, and 8, respectively. On the one side, it is clearly seen that for all Ni2 Mn-based alloys the FM order is more stable than AFM one. On the other side, for Ni2 Cr-based alloys, the non-uniqueness of magnetic order may be seen. In the cases of compounds with e/a = 7.5 and 7.75 except for Ni2 CrAl, the AFM state is energetically more favorable. But, for the compounds with e/a = 8, we can observe the stable FM spin ordering. In general, it is seen that for each series of Ni2 (Mn, Cr)Z alloys, the optimized lattice parameter is found to increase with increasing atomic number of Z element. The equilibrium lattice parameters and magnetic moments for Ni2 (Mn, Cr)Z alloys are given in Table 1. Here the equilibrium lattice parameters and magnetic moments taken from other theoretical and experimental works are also listed. It should be noted that other theoretical works for Ni2 CrZ alloys were carried out only for ferromagnetic ordering. Generally, it is seen that our data agree well with other calculations and experimental ones. Moreover, zero value of a magnetic moment clearly demostrates the AFM stable order. We suggest 4

that the low experimental value of magnetic moment for Ni2 CrAl alloy may be explained by a presense of structural or chemical disorder. Table 1: Equilibrium lattice parameter a0 (in ˚ A) and magnetic moment µ0 (in µB ) for FM and AFM states of Ni2 Y Z (Y =Mn, Cr; Z =Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) Heusler alloys. For comparison, the available theoretical and experimental data denoted as acalc (µcalc ) and aexp (µexp 0 0 0 0 ), are also listed here.

Ni2 MnZ

Al

Ga

In

Tl

Si

Ge

Sn

Pb

P

As

Sb

Bi

a0

5.80

5.81

6.06

6.14

5.70

5.80

6.06

6.19

5.70

5.85

6.06

6.26

µ0

4.00

3.87

4.07

4.08

3.62

3.69

3.87

3.97

3.77

3.86

3.72

4.04

5.78

5.80

6.09

6.13

5.70

5.82

6.07

6.18

-

-

6.07

6.26

4.02

4.01

4.41

4.22

4.03

4.01

4.06

4.17

-

-

4.00

4.29

aexp 0

5.82

5.83

6.07

-

-

5.82

6.05

-

-

-

5.98

-

µexp 0

4.19

4.17

4.04

-

-

3.81

4.01

-

-

-

3.22

-

Al

Ga

In

Tl

Si

Ge

Sn

Pb

P

As

Sb

Bi

a0

5.80

5.82

6.05

6.16

5.70

5.82

6.07

6.20

5.69

5.86

6.07

6.26

µ0

3.24

0.00

0.00

0.00

0.00

0.00

0.00

0.00

2.91

3.00

3.21

3.45

5.82

5.81

6.10

6.14

-

5.82

6.06

6.19

-

-

6.08

6.26

3.50

3.14

3.57

3.68

-

3.14

3.46

3.58

-

-

3.42

3.70

aexp 0

5.74

-

-

-

-

-

-

-

-

-

-

-

µexp 0

0.13

-

-

-

-

-

-

-

-

-

-

-

Other. calc.a acalc 0 µcalc 0 b

Exp.

Ni2 CrZ

Other. calc.c acalc 0 µcalc 0 d

Exp.

a

Theoretical data for Ni2 MnZ alloys taken from [14, 15, 16, 17, 18, 19]

b

Experimental data for Ni2 MnZ alloys taken from [20, 21, 22]

c

Theoretical data for FM order of Ni2 CrZ alloys taken from [16, 19, 23]

d

Experimental data Ni2 CrZ alloys taken from [20] To check the phase stability of considered alloys, we calculated the formation

energy as difference between the total energy and partial total energies of the 5

pure elements, as follows: Ef orm = Etot (Ni8 (Mn, Cr)4 Z4 ) −


4 2ENi + E(Mn,Cr) + EZ , 16

(1)

Here, Etot is the total energy per formula cell, ENi , EMn , ECr , and EZ are the total energy per atom of pure Ni, Mn, Cr, and Z unit cells, respectively. In Figure 3 we show the calculation results of the formation energy for studied Ni2 Mn-based and Ni2 Cr-based Heusler alloys. As can be seen from the Figure 3 that the following compositions are not stable: Ni2 MnZ (Z = Tl, Pb, and Bi) and Ni2 CrZ (Z = In, Tl, Pb, As, and Bi). Therefore, the further results will be discussed for stable compounds only.

Figure 3: Formation energy for Ni2 (Mn, Cr)Z Heusler alloys with L21 cubic structure.

To determine the possibility of martensitic transformation in alloys studied, in Figure 4 we present results of total energy difference calculations between cubic and tetragonal structure as a function of the tetragonal distortion (c/a) of cubic lattice along c axis. Firstly, let us discuss results for Ni2 Mn-based alloys presented in Figures 4(a-c). It is seen that for compositions with e/a = 7.5 (Figure 4(a)), the austenite-martensite phase transformation takes place only for Ni2 MnGa, where the energy minimums in the E(c/a) curve are found at c/a ratio of 0.94 and 1.25. This result completely agrees with the results of previous papers (See, for example, [14]). For the second case (e/a = 7.75, See

Figure 4: The variation of total energy relative to FM L21 phase of Ni2 (Mn, Cr)Z alloys from the tetragonal c/a ratio. (top) Ni2 MnZ (Z = Al, Ga, In, Si, Ge, Sn, P, As, Sb); (bottom) Ni2 CrZ (Z = Al, Ga, Si, Ge, Sn, P, Sb). Here, filled (open) symbols denote FM (AFM) solution.

Figure 4(b)), there is no martensitic transition for all alloys with the exception of Ni2 MnGe. For the latter, the minimum with small energy difference may be seen at c/a = 1.15. However, other calculations [15] and experiment [22] 6

have pointed to the absent of structural transformation in Ni2 MnGe. This discrepancy between calculations can be explained by their different accuracy. In the case of alloys with e/a = 8 (Figure 4(c)), we can see that the martensitic phase with c/a ≈ 0.85 and c/a ≈ 1.4 is possible only for Ni2 MnP and Ni2 MnAs. It should be noted that for both alloys, the similar trend in E(c/a) curve can be seen except Ni2 MnP. For the latter, the martensite with c/a < 1 (c/a > 1) has FM (AFM) order, respectively, while for Ni2 MnAs, both martensitic states possess the FM spin ordering. Let us consider E(c/a) curves for Ni2 CrZ alloys, as shown in Figure 4(d-f). We would like to note that we present here the calculation results obtained only for stable compositions (Ef orm < 0). As Figure 4(d) suggests, for Ni2 CrGa, the AFM martensite can be realized while for Ni2 CrAl, a structural transition takes place from the FM austenite to the AFM martensite with c/a ≈ 1.2. For alloys with e/a = 7.50, we can observe that tetragonal state with AFM spin ordering takes place at c/a ≈ 1.2 (Z = Ge), 1.2 (Z = Si), and 1.1 (Z = Sn). Finally, in a case of alloys with e/a = 7.75, the martensitic FM and AFM states with c/a ≈ 0.85 and c/a ≈ 1.45, respectively, can be realized for Ni2 CrP. 3.2. Magnetic exchange interaction constants and Curie temperatures In this subsection we discuss the exchange interactions for the stable Ni2 MnZ and Ni2 CrZ alloys with respective Z group. In Figure 5, the exchange coupling constants, which were calculated within the SPR-KKR package considering the optimized lattice parameters, are presented. In regard to Ni2 MnZ alloys with FM order (See Figures 5(a-c)), the Mn-Ni and Mn-Mn exchange constants show the almost similar behavior within Group III, IV, and V Z elements with increasing Z atomic number. On the one hand, for all Ni2 Mn-based alloys, MnNi nearest-interaction decreases slightly with increasing e/a ratio. This fact is related to a decrease of Ni magnetic moment and an increase of the Mn-Ni distance. On the other hand, Mn-Mn interactions change sufficiently and show long-range oscillation behavior. Namely, the Mn-Mn nearest-neighbor FM interactions increase with increasing e/a ratio, while, the AFM Mn-Mn couplings 7

Figure 5: Heisenberg exchange integrals, Jij , as a function of interatomic distance d/a for Ni2 (Mn, Cr)Z alloys. (top) Ni2 MnZ (Z = Al, Ga, In, Si, Ge, Sn, P, As, Sb); (bottom) Ni2 CrZ (Z = Al, Ga, Si, Ge, Sn, P, Sb).

enhance and shift from sixth shell to second one for alloys with e/a = 7.5 and e/a = 8, respectively. Let us consider exchange parameters for a series Ni2 CrZ alloys, which are displayed in Figures 5(d-f). We would like to remind that all presented compounds except for Ni2 Cr(Al, P, Sb) have the AFM spin ordering in austenite phase. The AFM interactions between Cr1 -Cr1 , Cr1 -Cr2 , Cr2 -Cr2 pairs can be clearly seen from Figures 5(d, e). Here, Cr1 (Cr2 ) atoms have a parallel (opposite) direction of magnetic moment, respectively. Besides, the Cr1 (Cr2 )-Ni interactions are essentially zero. In regard to Ni2 Cr(Al, P, Sb) compounds with the FM favorable ordering, we can observe that the Cr-Ni interaction decreases while the Cr-Cr interaction enhances with increasing e/a ratio. To estimate Curie temperatures, we performed the classical MC simulations of Heisenberg Hamiltonian in the absence of magnetic and anisotropy fields. Note, the long-range interactions were taken into account. Namely, we truncated the Jij interaction constants up to eight coordination shells (d/a = 2). In Figure 6, we display the distribution of Curie temperatures of the austenitic phase for stable Ni2 MnZ and Ni2 CrZ alloys. It should be noted that values of Curie point are shown only for compositions with favorable FM spin ordering. It is seen that the largest Curie temperatures are found for Ni2 Mn(Al, Ga, In). Besides, Curie temperature decreases with increasing e/a ratio. The our theoretical data agree with experimental ones exept Ni2 MnSn and Ni2 MnSb. We suppose that to reduce this descrepency it is necessary to accomplish the Jij integrals more carefully and with a higher degree of accuracy.

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Figure 6: Calculated Curie temperatures for the cubic austenite of Ni2 (Mn, Cr)Z as a function of Z element and experimental data taken from [20, 21, 22].

4. Summary The equilibrium structure and magnetic ground states as well as magnetic properties of Heusler Ni2 MnZ and Ni2 CrZ (Z = Al, As, Bi, Ga, Ge, In, P, Pb, Sb, Si, Sn, Tl) alloys are determined using the first-principles calculations with Monte Carlo simulations. Using the formation energy calculations, it is shown that the following compounds are not stable: Ni2 MnZ (Z = Tl, Pb, and Bi) and Ni2 CrZ (Z = In, Tl, Pb, As, and Bi). On the one hand in a case of cubic L21 Ni2 MnZ alloys, the FM spin ordering is found to be stable. On the other hand in a case of cubic L21 -Ni2 CrZ alloys, AFM state is realized for compositions with Z = Ga, Ge, Si, Sn while the FM state is favorable for compounds with Z = Al, P, Sb. The calculations of a L21 -cubic distortion have shown that martensitic transformation is possible for Ni2 MnGa and Ni2 MnGe in the FM state. Concerning the Ni2 MnP, the stable martensite with FM (AFM) spin ordering is realized at c/a ratio of 1.4 (0.85), respectively, while for Ni2 MnAs the stable martensite with FM spin ordering is realized at c/a ratio of 1.4 and 0.85. With respect to Ni2 CrZ (Z = Ga, Ge, Sn) alloys, the AFM ordering is favorable configuration in martensite while for Ni2 Cr(Al, Si) a structural transition is realized from FM austenite to AFM martensite. Finally, for Ni2 CrP, two possible martensitic states with FM and AFM spin orderings are found at c/a ratio of 0.85 and 1.4, respectively. Using the exchange coupling constants as a function of interatomic distance, the Curie temperatures are calculated within the Monte Carlo simulations of Heisenberg model. It is shown that for Ni2 Mn(Al, Ga, In), the largest values of Curie temperature are observed while an increase in e/a ratio results in a decrease in Curie temperature. In a summary, we can consider that discussed alloys could be interesting for 9

further theoretical investigations.

Acknowledgments This work is supported by RSF-Russian Science Foundation No. 17-7220022\17. VS acknowledges the financial support from the Ministry of Education and Science of the RF in the framework of increase Competitiveness Program of NUST MISIS, implemented by a governmental decree dated 16th of March 2013, No. 211. MZ acknowledges Act 211 Government of the Russian Federation, contract No. 02.A03.21.0011 and advanced research foundation of the ChelSU.

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