Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies

Journal of Magnetism and Magnetic Materials xxx (2017) xxx–xxx

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Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies Vladimir V. Sokolovskiy a,b,⇑, Yulia A. Sokolovskaya a, Mikhail A. Zagrebin a,c, Vasiliy D. Buchelnikov a, Alexey T. Zayak d a

Chelyabinsk State University, 454001 Chelyabinsk, Russia National University of Science and Technology ‘‘MIS&S”, 119049 Moscow, Russia National Research South Ural State University, 454080 Chelyabinsk, Russia d Bowling Green State University, OH 43403 Bowling Green, USA b c

a r t i c l e

i n f o

Article history: Received 20 August 2017 Received in revised form 9 November 2017 Accepted 14 November 2017 Available online xxxx Keywords: Heusler alloys Magnetic ground states Exchange interactions Curie temperature

a b s t r a c t A systematic ab initio calculations based on the DFT methodology and Spin-Polarized Relativistic Korringa-Kohn-Rostoker method are carried out to investigate magnetic properties of structuredisordered Ni-Mn-Ga Heusler alloys. In order to generate a complete set of disorder compositions, which will cover the whole area of the ternary diagram of Ni-Mn-Ga, the coherent potential approximation is used. Using our previously calculated lattice constants, the magnetic moments and exchange coupling constants are calculated for the optimized crystal structure of selected alloys. Obtained data from ab initio calculations are used as input for Monte Carlo simulations of Heisenberg Hamiltonian to compute the thermomagnetization curves. The Curie temperatures for the austenite structure of studied compositions are obtained and mapped onto the ternary diagram. The calculated data for Ni-Mn-Ga are in a good agreement with available experimental ones. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Among the Mn-based intermetallic compounds and alloys, NiMn-Ga Heusler systems have received significant attention regarding their magnetic shape memory properties for compositions close to stoichiometry and, in particular, their magnetocaloric properties at off-stoichiometric compositions [1–4]. The unique properties of Ni-Mn-Ga alloys resulting from the coupling between the magnetic and structural phase transitions. In general, Ni-MnGa alloys exhibit remarkable richness of physical properties, varying significantly with their chemical composition [4]. At present, the most of compositions studied are concentrated near the stoichiometric Ni2MnGa that is provided by its crystal structure stability. Evidently, the ability to obtain different compositional phase diagrams involving desired physical properties is a major challenge for improving Heusler-based technology. Large efforts are made experimentally to find optimal compositions with better properties. On the one hand, compositional phase diagrams can be obtained experimentally by using composition spreads that allow ⇑ Corresponding author at: Chelyabinsk State University, 454001 Chelyabinsk, Russia. E-mail address: [email protected] (V.V. Sokolovskiy).

for controlled simultaneous synthesis and characterization of large arrays of samples [5–7]. On the other hand, compositional diagrams can be obtained theoretically by means of first-principles and Monte Carlo approaches as example. At present, computational approaches on the level of density functional theory (DFT) have been quite successful in describing electronic, structural, magnetic and dynamical properties of Heusler alloys. From the latter point of view, numerous theoretical studies aimed to find ground state properties and to construct phase diagrams of Ni-Mn-Ga compounds have been recently performed by many research groups (for example, See Refs. [8–15]). These studies suggest that the complex magnetic order including ferromagnetic (FM), antiferromagnetic (AFM) or ferrimagnetic (FIM) as well as spin glass state in both austenite and martensite can be provided by the changes in the Mn content or the valence electrons per atom (e=a). In a previous recent work [15] the optimized structural and magnetic properties of Ni-Mn-Ga, which were obtained by zerotemperature ab initio calculations, have been projected onto the ternary Ni-Mn-Ga phase diagram. These results demonstrate the regions with a high and low magnetic moments together with a ground magnetic reference state, regions of stable compositions as well as the distribution of equilibrium lattice parameter of austenite.

https://doi.org/10.1016/j.jmmm.2017.11.055 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: V.V. Sokolovskiy et al., Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies, Journal of Magnetism and Magnetic Materials (2017), https://doi.org/10.1016/j.jmmm.2017.11.055

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V.V. Sokolovskiy et al. / Journal of Magnetism and Magnetic Materials xxx (2017) xxx–xxx

Based on the previous research, in the present paper we will study systematically composition-dependent magnetic exchange coupling constants to simulate the magnetization behavior as a function of temperature and to evaluated Curie temperatures of Ni-Mn-Ga alloys by Monte Carlo technique within the framework of Heisenberg Hamiltonian.

2. Methodology The ab initio calculations were performed using the SPR-KKR code based on the Korringa-Kohn-Rostoker Green’s function method [16]. The exchange correlation effects were treated by the generalized gradient approximation (GGA) in the PerdewBurke-Ernzerhof (PBE) formulation. In order to calculate the exchange coupling constants (J ij ), we invoked the optimized lattice parameters and magnetic reference states, which have been obtained in a previous work [15]. For both self-consistent field (SCF) and exchange parameter calculations, 4495 k points were generated by a k-mesh grid of 573. The angular momentum expansion (lmax ) was restricted to three. All calculations converged to 0.01 mRy of total energy. To achieve the better convergence, the Broyden2 scheme was taken into account. All calculations were performed with the help of the atomic sphere approximation (ASA) mode. The coherent potential approximation (CPA) was used to create off-stoichiometric Ni-Mn-Ga assuming four atoms in the unit cell. Notice that Ni-Mn-Ga Heusler alloys depending on a composition crystallize to become two possible L21 cubic structures: Cu2
MnAl-type (space group # 225, Fm3m) and Hg2CuTi-type (space
group # 216, F 43m). Moreover, these cubic structures undergo the martensitic transformation to the tetragonal structures during
! I4=mmm (# 139) and F 43m

cooling (Fm3m ! I4m2 (# 119)). Evidently, due to a shorter distance between Mn atoms the magnetic properties of martensite can be different from those for austenite. In our calculations we investigated features of offstoichiometric Ni-Mn-Ga with cubic L21 structure. Therefore, the calculations were performed for 36 compositions including 10 alloys with excess-Ni, Mn and Ga atoms, as shown in Fig. 1(a). The remaining six compositions had a structural disorder. The excess was taken to be an atomic concentration P 50%. As is shown in Ref. [15], the off-stoichiometric compositions are described by four kinds of atomic configurations in accordance with Wyckoff sites: (a) for Ni-excess alloys, Ni atoms are located at sites 8c (0.25, 0.25, 0.25), Mn atoms, at sites 4b (0.5, 0.5, 0.5), and Ga atoms, at sites 4a (0, 0, 0) (space group # 225); (b) for Ga-excess alloys, Ga atoms are located at sites 8c, Mn atoms, at sites 4b, and Ni atoms, at sites 4a (space group # 2250 ); (c) for Mn2-based alloys, Mn atoms are located at sites 8c, Ga atoms, at sites 4a, and Ni atoms, at sites 4b (space group # 22500 ); (d) for

Mn-excess atoms, Mn1 atoms are located at sites 4a (0, 0, 0), Mn2 atoms, at sites 4c (0.25, 0.25, 0.25), Ni atoms, at sites 4b (0.5, 0.5, 0.5), and Ga atoms, at sites 4d (0.75, 0.75, 0.75) (space group # 216). The corresponding crystal structures, which are derivated from L21 structure, are illustrated in Fig. 2. It is important to mention that the difference between space groups (# 225, # 2250 , and # 22500 ) is a matter of convention. In case of Mn-excess Ni-Mn-Ga alloys, we considered the stable magnetic reference states as follows: FM state with a parallel alignment between Mn magnetic moments on the Mn sublattice (Mn1), Mn on the Ga sublattice (Mn2), and Mn on the Ni sublattice (Mn3); FIM-1 state with antiparallel Mn3; FIM-3 state with antiparallel alignment of Mn at sites (4a); FIM-7 state with both antiparallel alignment Mn at sites (4a) and (4b). Note, that FIM-3 and FIM-7 orders can be realized in a cubic structure with

Fig. 2. Cubic cells of Ni-Mn-Ga alloys with L21 crystal structure. (a) Ni2MnGa


(Fm3m, # 225); (b) Ga2MnNi (Fm3m, # 2250 ); (c) Mn2NiGa (Fm3m, # 22500 ); (d)
Mn2NiGa (F 43m, # 216). The rows of atoms indicate the sequence of atoms in the unit cell. For off-stoichiometric compositions the respective excess atoms occupy sites with a lower concentration of parent atoms.

Fig. 1. Ternary Ni-Mn-Ga phase diagram. (a) Compositions under consideration. The intersection of three lines indicates stoichiometric Ni2MnGa. (b) Magnetic reference states in the austenitic phase that were obtained in ab initio calculations [15].

Please cite this article in press as: V.V. Sokolovskiy et al., Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies, Journal of Magnetism and Magnetic Materials (2017), https://doi.org/10.1016/j.jmmm.2017.11.055

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6

6

(a)

(b)

Ni2.4Mn0.8Ga0.8

Ni2.0Mn0.8Ga1.2 FM

FM 4

µMn = 3.501 µB µNi = 0.303 µB

Mn-Mn Mn-Ni1 Mn-Ni2 Mn-Ni3

1

µNi = 0.286 µB 2

µNi = 0.307 µB

2

3

FM

Jij (meV)

Jij (meV)

4

0

µMn = 3.58 µB µNi = 0.23 µB

2 FM

0 AFM

-2 0

AFM

Ni2 - Ni at Mn sites Ni3 - Ni at Ga sites

0.5

1

1.5

-2 0

2

0.5

1

1.5

d/a

2

d/a

4

8

(c)

(d)

Ni0.8Mn0.8Ga2.4 FM

Ni1.2Mn0.8Ga2.0 FM

6

FM

Mn-Mn Mn-Ni Ni-Ni

0

µMn = 3.107 µB

Jij (meV)

µMn = 3.38 µB µNi = 0.05 µB

2

Jij (meV)

Mn-Mn Mn-Ni Ni-Ni

Mn-Mn Mn-Ni1 Mn-Ni2

µNi = 0.061 µB

4

1

µNi = 0.107 µB 2

2 FM

AFM 0

-2

AFM

0

0.5

1

1.5

Ni2 - Ni at Mn sites

-2 0

2

0.5

d/a

1.5

(f)

Ni1.2Mn2.0Ga0.8

Ni1.2Mn2.4Ga0.4

10

Jij (meV)

2

0 AFM

µNi = 0.41 µB 1 µNi = 0.52 µB

Mn3-Ni1 Mn3-Ni2

0

Jij (meV)

Mn-Mn Mn-Ni1 Mn-Ni2

1

µNi = 0.843 µB

µMn = 3.19 µB

Space group # 216

FM

µNi = 0.581 µB

FM

Mn1-Mn1 Mn1-Mn2 Mn1-Mn3 Mn1-Ni1 Mn1-Ni2 Mn2-Mn2 Mn2-Mn3 Mn2-Ni1 Mn2-Ni2 Mn3-Mn3

FIM-3

FM µMn = 3.407 µB

10

AFM µMn = -2.67 µB 1

µMn = 3.24 µB

-10

2

3

-10

-20

2

Ni2 - Ni at Ga sites

-20 0

0.5

1

2

d/a

20

(e)

1

1.5

2

Mn3 - Mn at Ga sites Ni2 - Ni at Ga sites

-30 0

0.5

1

1.5

2

d/a

d/a

Fig. 3. Exchange coupling constants, Jij , as a function of distance (d=a) between atom pairs for (a, b) Ni2.4Mn0.8Ga0.8 and Ni2.0Mn0.8Ga1.2; (c, d) Ni0.8Mn0.8Ga2.4 and Ni1.2Mn0.8Ga2.0; (e, f) Ni1.2Mn2.0Ga0.8 and Ni1.2Mn2.4Ga0.4.

space group # 216. The more detailed information can be found in Ref. [15]. To obtain temperature dependent magnetic properties, we further performed the Monte Carlo (MC) simulations of the classical P three-dimensional Heisenberg model (H ¼  J ij Si Sj ) in zero magnetic field. Si is a classical three-dimensional spin vector of unit magnitude located at site i. The calculated exchange parameters (J ij ) and partial magnetic moments were taken as input parameters. Due to the long-range oscillation character of exchange interactions between different atoms, we restricted the J ij (r) coupling constants up to fifth coordination shell for all interaction pairs. The model lattice with periodic boundary conditions consists of 3925 atoms, and for Ni2MnGa, the it contains 1098 Mn, 1099 Ga

and 1728 Ni atoms. The MC simulations were carried out using the Metropolis algorithm. As time unit, we used one MC step consisting of N attempts to change the spin variables. For each temperature the properties (internal energy of the system hHi and magnetic order parameter hmi) were estimated allowing 106 MC steps and 105 thermalization steps. 3. Results and discussion 3.1. Magnetic exchange coupling constants In this subsection we present results of first-principles calculations the Heisenberg exchange parameters for a series of cubic

Please cite this article in press as: V.V. Sokolovskiy et al., Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies, Journal of Magnetism and Magnetic Materials (2017), https://doi.org/10.1016/j.jmmm.2017.11.055

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Magnetization (µB/f.u.)

8

(a)

Ni-Mn-Ga

6

Ni1.2Mn2.4Ga0.8 (# 216) Ni2Mn0.8Ga1.2 (# 225) Ni2.4Mn0.8Ga0.8 (# 225) Ni1.2Mn0.8Ga2.0 (# 225’) Ni0.8Mn0.8Ga2.4 (# 225’) Ni1.2Mn2.0Ga0.8 (# 225’’)

4

2

0 0

100

200

300

400

500

Temperature (K) Fig. 4. (a) Calculated temperature dependences of magnetization for a series of Ni-Mn-Ga. (b) Distribution of Curie temperature of austenite in the ternary phase diagram.

Ni-Mn-Ga alloys. We would like to remind that all calculations were performed for 36 compounds in accordance with equilibrium magnetic reference states. Since compositions have the similar disorder (Ni2+xMn1xGa, Ni2Mn1+xGa1x, Ni2+xMnGa1x etc.), we discuss here the magnetic exchange interactions for 6 compositions. The calculated J ij parameters between magnetic atoms in dependence of their distance within the cubic L21 structure are shown in Fig. 3. We considered alloys as follows: Ni-excess (Ni0.6Mn0.2Ga0.2 (Ni2.4Mn0.8Ga0.8) and Ni0.5Mn0.2Ga0.3 (Ni2.0Mn0.8Ga1.2)) (Fig. 3(a, b)), Ga-excess (Ni0.2Mn0.2Ga0.6 (Ni0.8Mn0.8Ga2.4) and Ni0.3Mn0.2Ga0.5 (Ni1.2Mn0.8Ga2.0)) (Fig. 3(c, d)), Mn-excess (Ni0.3Mn0.5 Ga0.2 (Ni1.2Mn2.0Ga0.8) and Ni0.3Mn0.6Ga0.1 (Ni1.2Mn2.4Ga0.4)) (Fig. 3(e, f)). The long-range damping oscillatory behavior can be clearly seen. One the one hand, the leading interactions stabilizing the ferromagnetism are the nearest-neighbor intra-sublattice couplings between Mn and Ni atoms. Evidently, the stronger intrasublattice exchange (J ij > 0), the higher Curie temperature will be expected for Ni-excess and Mn2-based alloys. On the other hand, the inter-and intrasublattice Mn-Mn interactions demonstrate the crossover between ferro- and antiferomagnetic couplings, which is observed within the first few coordination spheres for all alloys studied. Moreover, the strongest AFM between nearest Mn atoms can be seen due to the shorter distance between Mn atoms located at different sublattices compared to the Mn-Mn intra-sublattice distance. Evidently, large AFM contributions to the total exchange energy destabilizes the ferromagnetism in Mn-excess alloys with a structure having the space group # 216. Generally, the discussed behavior of the long-range exchange interactions for Ni-Mn-Ga alloys can be associated with three types of interactions: (1) the magnetic J sd interaction between the localized d and itinerant s-like electrons, (2) the magnetic J dd exchange interaction resulting from the interaction between localized and itinerant d-like electrons and (3) a super exchange-like interaction through the Ga-(sp) electrons [17]. 3.2. Thermomagnetization curves and Curie temperatures The knowledge of magnetic exchange couplings and magnetic moments allows us to simulate the temperature dependences of magnetization and to estimate the Curie temperatures of cubic Ni-Mn-Ga alloys within the framework of MC simulations. Note, MC simulations were performed for 36 compositions of Ni-MnGa. In view of the fact that for all cases the similar ferroparamagnetic transition behavior is realized, in Fig. 4(a) we display the thermomagnetization curves only for six compounds mentioned above. One can see that the magnetic phase transition observes clearly for Ni-excess and Mn2-based alloys, while it is suppressed for Ga- and Mn-excess alloys due to weak Mn-Ni and strong AFM Mn-Mn interactions.

Results of MC simulations of the magnetic ordering as a function of the composition are shown in Fig. 4(b). We see a good agreement with experimental data for the stoichiometric Ni2MnGa (T C  370 K [1]), and predict the behavior of the Curie temperature as the compositions start to deviate from the stoichiometric one. The diagram shows significant variations in the range of about 200 degrees. The Ni-rich area of the diagram consists of high values of Curie temperature as compared to Ga- and Mn-rich areas. Further increasing in Ni content leads to decrease in Curie temperature to small values. It is worth noting that the Ni with fcc structure is well known system for the ab initio and MC studies with account of Heisenberg Hamiltonian have difficulties in calculating the correct Curie temperature. For instance, as has been shown in Refs. [18,19], the calculated Curie temperature for fccNi is almost two times less than experimental one. In a case of Ga-rich area, the diagram is shown to be paramagnetic, while for Mn-rich area, the diagram is antiferromagnetic due to the strongest AFM exchange coupling constants between nearest Mn atoms. 4. Summary Using the first-principles SPR-KKR-CPA in combination with Monte Carlo technique, the magnetic properties of a series of NiMn-Ga alloys have been determined. For a set of optimized compositions we have performed calculations of exchange integrals and used those values in the Monte Carlo code to simulate the thermomagnetization curves and to create the map of Curie temperatures. The obtained data show exciting trends and can be directly tested by experiments. The present study is an example showing the important of the thermodynamic aspect, which cannot be addressed by DFT alone, but can be studied by a conjunction of DFT and Monte Carlo. Acknowledgment This work is supported by Russian Science Foundation No. 1772-20022/backslash17. References [1] [2] [3] [4]

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Please cite this article in press as: V.V. Sokolovskiy et al., Ternary diagrams of magnetic properties of Ni-Mn-Ga Heusler alloys from ab initio and Monte Carlo studies, Journal of Magnetism and Magnetic Materials (2017), https://doi.org/10.1016/j.jmmm.2017.11.055