Structural, magnetic and electronic properties of Ni-Mn-Ga-Cr Heusler alloys: ab initio and Monte Carlo studies

Structural, magnetic and electronic properties of Ni-Mn-Ga-Cr Heusler alloys: ab initio and Monte Carlo studies

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Available online at www.sciencedirect.com

ScienceDirect Materials Today: Proceedings 4 (2017) 4621–4625

www.materialstoday.com/proceedings

SMA 2016

Structural, magnetic and electronic properties of Ni-Mn-Ga-Cr Heusler alloys: ab initio and Monte Carlo studies Elizaveta E. Smolyakovaa,1, Mikhail A. Zagrebina,b, Vladimir V. Sokolovskiya,c and Vasiliy D. Buchelnikova aChelyabinsk StateUniversity, 129 Brat'ev Kashirinykh Str., 454001Chelyabinsk, Russia National Research South Ural State University,76 Lenin prospekt, 454080 Chelyabinsk, Russia cNationalUniversity of Science and Technology ”MIS&S”, 4 Leninskiy prospect 119991 Moscow, Russia b

Abstract A series of Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) Heusler alloys have been theoretically investigated by density functional theory as well as finite-temperature Monte Carlo simulations to obtain Curie temperatures. By using ab initio calculations it was found that the most energetically favorable magnetic state for both austenite and martensite phases in Ni2Mn1-xCrxGa is a ferromagnetic state. Besides, an increase in energy difference between austenite and martensite with increasing Cr content was observed. With respect to the exchange coupling constants, it was shown that Ni-Mn(Cr) pairs show the strongest ferromagnetic exchange interaction between nearest neighbor atoms in austenite of all alloys considered. It was demonstrated that the most part of total spin up and spin down density of states is caused by Ni atoms. Estimated Curie temperatures were found to be less than experimental ones approximately by 80 K, but the TС(x) function repeats pattern of experimental one. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”. Keywords: Heusler alloys; first principles calculations; density of states; magnetic exchange interactions; Curie temperature

1.Introduction A wide range of impressive physical effects such as magnetically and thermally induced shape memory effect, large magnetoresistance and giant magnetocaloric effect is a distinguishing feature of ferromagnetic (FM) Heusler alloys [1-3]. Ones of the most studied and well-investigated Heusler alloy are Ni-Mn-Ga-based materials [1-3]. At present, it is well known that the addition of fourth element (Fe, Co, Cu, Pt e.g.) into Ni-Mn-Ga can strongly affect the Curie temperature (TС) and martensitic transformation temperature (Tm) [1-5]. Recent experimental studies are * Corresponding author. Tel.: +7-351-799-7117; fax: +7-351-742-0925. E-mail address: [email protected] 2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”.

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focused on magnetic and structural properties of Cr-doped Ni2MnGa [6, 7]. Adachi et al. [6] presented the temperature vs. Cr concentration phase diagram obtained from electrical resistivity measurements for Ni2Mn1-xCrxGa (x ≤ 0.25). Khan et al. [7] reported that the FM martensite-austenite transformation occurs at higher temperatures, while the ferro-paramagnetic transitionis observed at lower temperatures with increasing Cr concentration (x > 0.5). The aim of this work is theoretical investigation of structural, magnetic and electronic properties of Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) Heusler alloys and comparison with available experimental data. 2.Calculation details In this study, we used density functional formalism (DFT) as implemented in the Vienna ab-initio simulation program (VASP) [6] and spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) package [8] jointly with the classical Monte-Carlo simulations. The generalized gradient approximation (GGA) in Perdew, Burke and Ernzerhoff parameterization [9] was considered in both VASP and SPR-KKR calculations. To describe the austenitic phase of stoichiometric Ni2MnGa, we took into account the L21 structure with space group of Fm3m (225), which can be represented as four interpenetrating face-centered cubic sublattices. Where Ga and Mn atoms occupy the positions (0; 0; 0) and (1/2; 1/2; 1/2), while the Ni atoms locate the sites (1/4; 1/4; 1/4) and (3/4; 3/4; 3/4). Additional Cr atoms were assumed to be at Mn-site. To create non-stoichiometric compositions, the 32-atom supercell approach was used. As the result, Ni16Mn8-xCrxGa8 compositions (x = 1, 2, 3, and 4) were taken into consideration. We note that it is enough to cover experimental range of Cr concentration in Cr-doped Ni2MnGa alloys. The latter compositions correspond to the Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) ones. The SPR-KKR calculations were made for the full potential mode in the atomic sphere approximation with additional empty spheres inserted at the interstitial sites and coherent potential approximation (CPA) to model chemical disorder [10]. Temperature dependences were obtained from Monte-Carlo simulations using the three-dimensional Heisenberg model without anisotropy and magnetic field, where the Hamiltonian takes the form: H = -∑JijSiSj [11]. Here, Si is the spin of the unit length (|Si| = 1) placed on a lattice site and Jij isthe exchange coupling parameter obtained from ab initio calculations for austenite phase [9]. 3.Results 3.1.Structural properties To determine an optimized lattice parameter and magnetic reference state, we performed the lattice relaxation calculations of 32-atoms supercell as a function of a lattice parameter. In our calculations, we considered only two types of magnetic states. The first one is the ferromagnetic(FM) state, where magnetic moments of all atoms are parallel. The second one is the ferrimagnetic (FIM) state, where magnetic moments of Ni and Mn atoms are parallel and magnetic moment of Cr atoms is antiparallel. We note that magnetic moment of Ga atomis negligible. Energy curves for different magnetic reference states (FM and FIM) of Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) were computed using VASP. The structural optimization of cubic phase for all compositions studied shows that the FM spin configuration is more favorable than the FIM one. The equilibrium lattice parameter is closed to 5.81 Å and weakly depends on Cr excess. In order to investigate the possibility of martensitic transformation, we carried out the calculations of total energy as a function of tetragonal distortion c/a for both FM and FIM spin configurations. It should be noted that a martensite phase could be predicted at c/a ratio if an energy minimum in ΔE(c/a) function is realized at c/a ≠ 1. Fig. 1 shows the calculated total energy difference with respect to the cubic phase as a function of tetragonal ratio c/a for two magnetic reference states and different compositions of Ni2Mn1-xCrxGa.

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Fig. 1. (a) the variation of the total energy as functions of tetragonalityc/a for Ni2Mn0.875Cr0.125Ga for different magnetic configurations; (b) The variation of the total energy as functions of tetragonalityc/a for Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) calculated for FM state. The energy difference is taken with respect to the FM austenite.

As it is observed in Fig. 1(a), the FM spin configuration is energetically stable than FIM one in both austenite and martensite for Ni2Mn0.875Cr0.125Ga. We can see that the energy difference is found to increase with increasing Cr content, as shown in Fig. 1(b). Moreover, the c/a ratio practically does not change with increasing Cr excess. It should be pointed out that these results are presented for the stable FM reference state. We further discuss the results obtained for the FM spin configuration. 3.2.Magnetic exchange parameters and density of states In this subsection we discuss the calculation results of magnetic exchange couplings and density of states (DOS). All calculations were done using the SPR-KKR-CPA approach. Figure 2 displays the magnetic exchange parameters as a function of distance between pairs of atoms in austenite phase of Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) alloys.

Fig. 2. (a) Calculated exchange couplings parameters for Ni2Mn0.875Cr0.125Ga in austenite phase; (b) calculated exchange couplings parameters for Ni2Mn0.75Cr0.25Ga in austenite phase; (c) calculated exchange couplings parameters for Ni2Mn0.625Cr0.375Ga in austenite phase; (d) calculated exchange couplings parameters for Ni2Mn0.5Cr0.5Ga in austenite phase.

It is worth mention that positive values of Jij correspond to the FM coupling, whereas the negative ones mark the AF coupling. On the one side, we can see that Ni-Mn(Cr) pairs show the strongest FM exchange interaction between nearest neighbor atoms among all interactions for all compounds studied. On the other side, the strongest AFM interaction is observed between nearest Cr-Cr pair. Generally, despite the fact that the Cr-Cr pair clearly demonstrates the strongest AFM interaction, the Mn-Cr interactions show crossover from AFM to FM interaction for next nearest neighbors. It is obvious that the FM character of Mn-Cr interaction pairs are slightly weaker compared with Mn-Mn interaction pairs.

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Fig. 3 shows the total and partial density of states (DOS) curves for the FM austenitic phase of Ni2Mn1-xCrxGa alloys (x = 0.125, x = 0.5). We can see that in both cases the most part of total spin up and spin down bands in DOS curves is caused by Ni atoms, while the Cr atoms give less contribution into total spin up band than Mn atoms. As the result, Ni2Mn0.5Cr0.5Ga demonstrates lower total spin up band than Ni2Mn0.875Cr0.125Ga. Moreover, in both alloys, the spin up and spin down bands show the metallic behavior.

Fig. 3. (a) calculated total and partial DOS curves for cubic phase of Ni2Mn0.875Cr0.125Ga; (b) calculated total and partial DOS curves for cubic phase of Ni2Mn0.5Cr0.5Ga.

3.3.Finite temperature calculations Figure 4(a) illustrates temperature dependences of total magnetization for Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) obtained by Monte Carlo simulations. Alloys studied demonstrate the ferromagnetic to paramagnetic phase transition in austenitic phase during heating. As can be seen, Curie temperature decreases with increasing Cr excess and takes less values (by 80 K) as compared with experimental trend (Fig. 4(b)). That fact can be explained by the calculation assumption: exchange interactions were limited up to the third coordination shell, but accumulated effect of interactions with larger radius can be valuable and should be estimated in future. In spite of difference between calculated and experimental results, decreasing behavior of Curie temperature is observed in both cases.

Fig. 4. (a) The temperature dependences of total magnetization and (b) concentration dependences of Curie temperature for Ni2Mn1-xCrxGa (0.125 ≤ x ≤ 0.5) alloys, obtained by means of Monte Carlo simulations in the absence of magnetic field. For comparison, the available experimental data taken from [7] are also depicted here.

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4.Conclusions In summary, the structural and magnetic properties of Cr-doped Ni2Mn1-xCrxGa Heusler alloys (0.125 ≤ x ≤ 0.5) have been theoretically investigated by using first-principles calculations and Monte Carlo simulations.We have shown that the FM spin configuration is more energetically stable in both austenite (c/a = 1) and martensite (c/a = 1.25) for all compositions studied. On the one hand, the addition of Cr practically does not change the c/a ratio of martesite and slightly increases the energy difference between austenite and martensite. The latter fact clearly indicate on the rise in the martensitic transition temperature with increasing Cr content. On the other hand, the Curie temperature of austenite, which was estimated from Heisenberg model and Monte Carlo technique, is found to linearly decrease with increasing Cr concentration. This trend is related with the strong AFM contribution between Cr-Cr nearest pair to the total exchange interaction energy. In general, the theoretical behavior of Curie temperature for austenitic phase repeats qualitative the experimental trend. We consider that an account of longrange interactions in the Heisenberg model may result in better quantitative agreement with experimental data. Acknowledgements This work is supported by RSF-Russian Science Foundation No. 14-12-00570\14 (VASP calculations), Ministry of Education and Science RF No. 3.2021.2014/K (SPR-KKR calculations), President RF Grant MK-8480.2016.2 (MC simulations) and RFBR No. 14-02-01085. References [1] V.D. Buchelnikov and V.V. Sokolovskiy, Phys. Met. Metallogr. 112 (2011) 633-665. [2] P. Entel, M. Siewert, M.E. Gruner et al., Eur. Phys. J. B 86 (2013) 65. [3] P. Entel, M. Siewert, M.E. Gruner et al., J. Alloys Compd. 577 (2013) S107–S112. [4] S. Singh, S.W.D’Souza, J.Nayak et al., Phys. Rev. B 93 (2016) 134102. [5] H.F. Wang, J.M. Wang, C.B. Jiang et al., Rare Met. 33 (2014) 547–551. [6] Y. Adachi, R. Kouta, M. Fujio et al. Phys. Procedia 75 (2015) 1187–1191 [7] M. Khan, J. Brock, and I. Sugerman, Phys. Rev. B 93 (2016) 054419. [8] G. Kresse and J. Furthmuller, Phys. Rev. B 54 (1996) 11169-11186. [9] H. Ebert, D. Ködderitzsch, J. Minár, Rep. Prog. Phys. 74 (2011) 096501. [10] J.P. Perdew, K. Burke, M. Enzerhof, Phys. Rev. Lett. 77 (1996) 3865-3868. [11] D.P. Landau and K. Binder, A guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge 2005.