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УДК 538.911 DOI: 10.14529/mmph240208
MAGNETIC AND STRUCTURAL PROPERTIES OF ALL-D METAL MN-NI-TI HEUSLER ALLOYS
V.V. Sokolovskiy1, V.D. Buchelnikov1, D. Cong2
Chelyabinsk State University, Chelyabinsk, Russian Federation 2Beijing University of Science and Technology, Beijing, China E-mail: vsokolovsky84@mail.ru
Abstract. The paper presents a theoretical study of the effect of different atomic and magnetic orderings on the structural and magnetic properties of Mn2Ni1+xTi1-x alloys, which are composed entirely of transition metals. Using the density functional theory, we predict the structural ground states and magnetic reference states of compounds with x = 0, 0,25, 0,5, and 0,75 in both cubic austen-ite and tetragonal martensite phases. Partial substitution of Ti atoms with Ni leads to an increase in the energy barrier between structural phases, to a change from a layered atomic ordering to an alternating staggered order, and to a change from antiferromagnetic to ferromagnetic spin alignment in the cubic phase. All compounds with tetragonally distorted structures reveal the out-of-plane spin configuration and easy axis magnetocrystalline anisotropy except tetragonal L10 phase of Mn2NiTi. For the latter structure, easy-plane magnetic ani-sotropy is observed. The calculated values of anisotropy are comparable with those of tetragonal L10-FeNi.
Keywords: ab initio calculations; all-d metal Heusler alloys; atomic arrangement; magnetocrystalline anisotropy.
Introduction
In recent years, Ni-Mn ferromagnetic shape memory alloys have been extensively researched for commercial applications owing to their diverse multifunctional characteristics, including the magnetic shape memory effect, magnetocaloric effect, and magnetoresistance arising from temperature- or magnetic field-induced martensitic transformations [1-6]. However, challenges related to poor mechanical durability and significant hysteresis losses during phase transitions limit their practical application.
Recently, Heusler alloys have been developed that contain all d-transition metal elements based on nickel (Ni), manganese (Mn), and titanium (Ti) [7-9]. In these alloys, Ti replaces the elements from the p-block, and it has been shown that the d-block Ti metal provides stabilization of the Heusler structure through d-d hybridization with its nearest neighbor element [10]. These alloys exhibit an ordered L2i structure and an partially disordered B2 structure, and they undergo a phase transition from B2 to L21, which is accompanied by a change in structural order and a martensitic transformation at a lower temperature. In the B2 phase, some Mn atoms exchange positions with Ti atoms, leading to a strong antifer-romagnetic interaction between Mn atoms located at different crystallographic sites.
A distinguishing feature of all-d metal Heusler alloys is their large relative volume change during the martensitic transformation process, which is significantly greater than in any other ferromagnetic shape memory alloy. This characteristic indicates a notable sensitivity of the transformation to applied mechanical stresses, particularly hydrostatic pressure. This makes these materials well-suited for demonstrating significant barocaloric effects. For example, Wei et al. [11] showed that a compressive stress of 900 MPa results in a deformation of approximately 3,9 % and an adiabatic temperature change of 10,7 K in Ni50Mn32Ti18 alloy. Yan et al. [12] observed large deformations of approximately 13 % in the Ni50Mn3175Ti1825 alloy under a compressive stress of 1,1 GPa. Due to the weaker d-d orbital hybridization in transition metal elements, there is a notable enhancement in mechanical properties, particularly improved plasticity, compared to conventional Heusler alloys that contain main group elements in combination with a strong p-d orbital interaction.
In this paper, we paid attention to the theoretical study of the effect of atomic ordering on the structural and magnetic characteristics of Mn2Ni1+xTi1-x alloys within the framework of density functional theory.
Sokolovskiy V.V., Magnetic and Structural Properties Buchelnikov V.D., Cong D._of All-d Metal Mn-Ni-Ti Heusler Alloys
1. Details of calculations
The calculations presented in this paper were performed within the framework of density functional theory, using the projection-augmented wave (PAW) method, which is implemented in the VASP software package [13, 14]. A series of Mn2Ni1+xTi1-x Heusler alloys, consisting entirely of transition metals, were selected as the objects of study. The exchange-correlation energy approximation was realized using the generalized gradient approximation for electron density in the Perdew-Burke-Ernzerhof parametrization [15]. The following valence electron configurations were considered in the PAW potentials: 3s23p64s23d5 for Mn, 3p63ct4s2 for Ni, and 3s23p64s23d2 for Ti. The plane wave basis cutoff energy was set at 450 eV, and a Monkhorst-Pack grid with a size of 8^8x8 was used for integration over the Brillouin zone. Geometric optimization of the 16 atom cells with cubic (L21, Fm3m, No. 225, and XA, F-43m, No. 216) and tetragonal (Tp, P4/nmm, No. 129) symmetries was performed using atomic position relaxation at constant volume and cell shape, with convergence criteria for energy and force equal to 10-6 eV and 10-2 eV/À, respectively. Nonstoichiometric compounds were created by sequentially replacing Ti atoms with Ni. The following magnetic configurations were considered for the spin ordering of Mn atoms (see fig. 1): FM (ferromagnetic state), AFM (antiferromagnetic state with staggered arrangement of magnetic moments), and AFM* (antiferromagnetic layer-like arrangement of magnetic moments).
^ Mn Q N1 ^ Ti
Fig. 1. The crystal structures of M^NiTi Heusler alloy with cubic (L21, Fm3m, #225, and XA, F-43m, #216) and tetragonal (Tp, P4/nmm, #129) symmetry. The L21 and XA structures differ due to the placement of Mn atoms in equivalent and non-equivalent positions, respectively. The Tp structure is based on the XA structure and composed of layers of Mn
and Ni atoms alternating along the [001] direction [16, 17]
The magnetocrystalline anisotropy energy (MAE) was calculated by performing successive self-consistent calculations that took into account spin-orbit interaction for structures with magnetic moment orientations along [001] and [100]. The MAE was then determined using the wave functions obtained from the self-consistent calculations. To obtain the MAE, the difference between energies E100 and E001 was evaluated. Here E100 and E001 are the total energies of the compounds with the corresponding magnetic moment orientations. A negative MAE value indicates easy plane anisotropy, while a positive value corresponds to easy axis anisotropy.
2. Results and discussions
This section discusses the results obtained for a set of Mn2Ni1+xTi1-x alloys (x = 0, 0,25, 0,5 and 0,75). The procedure of geometrical optimization of Heusler crystal structures of L21, XA and Tp types was performed, taking into account their different magnetic ordering, to establish the energetically pref-
erable structure for each compound. We would like to note that the Tp structure has been originally proposed by Neibecker et al. [16].
Fig. 2 clearly demonstrates the variations in total energy of the cubic structures based on their degree of tetragonal distortion da. It should be noted that the presence of a stable martensitic tetragonal phase in the alloy is indicated by a global energy minimum on the curve E(c/a) at c/a ^ 1.
As can be seen from Fig. 2, a, for the stoichiometric compound Mn2NiTi, layered atomic ordering of Ni and Mn is energetically favorable, thereby pointing to the Tp structure as the ground state of the austenite phase (c/a ~ 1), despite the fact that each of the structures exhibits pseudo-martensitic behavior at c/a > 1. The magnetic reference state of Tp structure is characterized by AFM* (layer-by-layer) ordering of the Mn spin moments. The energy difference between the Tp and XA, L21 structures with AFM* ordering is about 12,8 and 24,9 meV/atom, respectively. The FM solution for Mn2NiTi is located signifi-
cantly higher on the energy scale, indicating its disadvantage. It should be noted that the XA and L21 structures have been experimentally observed in Heusler alloys, whereas the theoretical Tp structure has not been yet experimentally confirmed. Several theoretical papers [17-19] have reported the energetic favorability of the Tp structure for Fe- and Co-based Heusler alloys.
tetragonal ratio c/a tetragonal ratio c/a
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
tetragonal ratio c/a tetragonal ratio c/a
Fig. 2. Variation of the total energy of the cubic crystal structures L2-,, XA and Tp taking into account FM and AFM ordering as a function of the degree of their tetragonal distortion for Mn2Ni1+,Ti1-, alloys
Partial substitution of Ti atoms with excess Ni atoms results in a stable tetragonal phase with atomic ordering of the XA structure type (see Figs. 2, b-d). The layered antiferromagnetic ordering of Mn magnetic moments is energetically favorable in the martensitic phase for all non-stoichiometric compounds. In this case, the Mn atoms are located in two non-equivalent sublattices and are at the closest distance to the other atoms. It should be noted that an increase in the Ni content leads to a change in the ground state of the austenitic phase (c/a = 1). Thus, Fig. 2, b demonstrates that the Tp structure remains stable for the compound with x = 0,25, despite the smaller energy difference, AE « 6,42 meV/atom, between the AFM* Tp and AFM* XA structures compared to the value of AE « 12,8 meV/atom for Mn2NiTi. Further increase in Ni content (x > 0,25) results in the instability of Tp structure for austenite phase. Similar finding has been reported recently for FeNi2+xAl1-x alloys [17]. It is obvious from Fig. 2, c, d that an increase in x changes the atomic and magnetic ordering of the cubic austenite phase from AFM* Tp to FM L21 one. For the L21 structure, Mn atoms occupy equivalent positions at a greater distance (d = 0,5a0) compared to the case of the Tp structure (d = V3/4a0). An increase in the distance between Mn atoms causes a change in the nature of the magnetic interaction from AFM to FM.
The E(c/a) curves for the Mn2Ni1+xTi1-x compositions indicate that the martensitic transition temperature Tm will increase with increasing Ni content. This is a result of the heightened energy barrier between the cubic and tetragonal phases. Replacing Ti atoms with Ni atoms increases the concentration of valence electrons (e/a) per atom, resulting in a corresponding increase in the energy barrier with concentration x. This observation is consistent with the experimentally established linear dependence Tm(e/a)
[1-6]. The martensitic transition temperature can be roughly estimated using the relation: AE ~ kBTm, where kB is the Boltzmann constant. The temperature Tm for compositions with x = 0,25, 0,5 and 0,75 or with e/a = 7,375, 7,75 and 8,125, respectively, is calculated to be 245,9, 407,16, and 726,16 K, assuming AE of 21,2, 35,1, and 62,6 meV/atom, respectively.
Table 1 summarizes the equilibrium lattice parameters, total and elemental magnetic moments for the energetically favorable crystal structures of the compounds Mn2Nii+xTi1-x, taking into account a magnetic reference ordering. The table clearly demonstrates that the equilibrium parameter of crystal structures in the austenitic phase increases as the Ni content increases. This is due to the larger atomic radius of Ni atoms compared to Ti atoms. However, the tetragonal structure parameters of the martensitic phase with XA ordering exhibit an ambiguous behavior that is dependent on the Ni content.
Table 1
Equilibrium lattice parameters (a and c), tetragonal ratio (c/a), total and elemental magnetic moments (h™ and m) in [MB/f-u.] and [MB/at.], respectively, for Mn2Nii+xTii-x alloys in cubic austenite and tetragonal martensitic phases. All parameters are presented for energetically favorable structures and magnetic configurations: AFM* Tp (x = 0 and 0,25), FM L21 (x = 0,5 and 0,75) for austenite; AFM* XA (x = 0,25, 0,5 and 0,75) for martensite
Compound Parameters x = 0 x = 0,25 x = 0,50 x = 0,75
a 5,805 5,82 5,88 5,86
Mtot 3,275 2,657 6,87 7,817
austenite MMn1 -0,282 -0,842 3,10 3,323
(c/a =1) ЦМп2 2,975 2,992 3,10 3,323
Ц№ 0,475 0,434 0,796 0,821
a 5,130 5,136 5,113
c 7,490 7,396 7,464
martensite c/a 1,46 1,44 1,46
(c/a Ф1 ) Mtot 0,425 0,238 0,176
^Mn1 2,509 2,688 2,981
^Mn2 -2,217 -2,534 -2,865
MNi 0,027 0,032 0,009
The behavior of magnetic moments shows a clear trend: the total magnetization decreases for compositions with Tp structure and AFM* ordering of Mn and Mn2 atoms. This is a result of the increased magnetic moment of Mni atoms located in the (002) plane, while Mn2 atoms occupy tetrahedral positions in the Tp structure (see Fig. 1). Compounds with L21 ordering in the austenitic phase (x = 0,5 and 0,75) exhibit an increase in total magnetization. This is attributed to the growth of the element-by-element magnetic moments of Ni, Mn1, and Mn2 atoms. It is worth noting that the difference between Mn1 and Mn2 is purely conventional, as these atoms occupy equivalent positions in two interpenetrating FCC sublattices (refer to Fig. 1). The data clearly shows that as the Ni content increases, the total magnetization of the martensitic phase with XA ordering decreases significantly. This is due to the antifer-romagnetic orientation of the magnetic moments of Mn1 and Mn2, which are located in non-equivalent positions (0, 0, 0) and (0,25, 0,25, 0,25) respectively (see Fig. 1). It is also worth noting that the magnetic moment of Ni atoms is suppressed. The binary compound Ni-Mn, resulting from the substitution of Ti for Ni, exhibits antiferromagnetism with a Neel temperature of around 1000 K [3, 20].
To calculate MAE, we chose the magnetic states of tetragonally distorted structures XA, L10, and Tp of Mn2Ni1+xTi1-x compounds, as presented in Fig. 2, due to their close energy levels. Our selection of these structures was based on their energy levels and their suitability for accurate calculations. For the x = 0, we have considered the AFM* Tp (c/a = 0,96), and AFM* L10 (c/a = 1,44) structures. For the x = 0,25, we have taken the AFM* XA (c/a = 1,46), and AFM* Tp (c/a = 0,98) structures. Finally, for the x = 0,5 and x = 0,75, we have considered the AFM* XA (c/a = 1,44 and 1,46), and AFM* Tp (c/a = 1,44 and 1,42) structures, correspondingly.
Fig. 3 shows the histograms of MAE calculated for crystal structures mentioned above. Here, the negative sign of MAE indicates the energetically favorable in-plane spin direction and easy plane MAE, and vice versa the out-of-plane spin direction and easy axis MAE. For all compounds with tetragonally distorted AFM* XA and Tp structures, the easy-axis-type anisotropy is observed except Mn2NiTi with tetragonal L10 structure, where the easy-plane-type anisotropy is preferable. In a case of compounds with x = 0 and 0,25, Tp structure with nearly cubic parameters (c/a = 0,96 and 0,98) demonstrates a considerable MAE, which is comparable with that of compounds with tetragonal structures (x = 0,5 and
0,75). The largest MAE value is calculated to be 0,4 MJ/m3 for AFM* XA structure (c/a = 1,46) of Mn2NiU5Ti0,75. However, for compounds with x = 0,5 and 0,75, the Tp structure reveals a larger values of MAE as compared to the XA structure. We would like to mentioned that the calculated values of MAE are comparable to those of L10-FeNi (0,22 - 0,48 MJ/m3 [21-24]), which is the most intensely discussed candidate for high-performance rare-earth-free permanent magnets.
Crystal structures
Fig. 3. Magnetocrystalline anisotropy energy diagram for tetragonally distorted L10, XA, and Tp structures of Mn2Ni1+^Ti1.^ alloys (x = 0, 0,25, 0,5, and 0,75) with the favorable magnetic configurations
Conclusions
In summary, we have performed the ab initio investigations of the influence of atomic and magnetic ordering on the ground state energy and magnitude of magnetocrystalline anisotropy for Heusler alloys of the Mn-Ni-Ti family, which consist entirely of transition d metals. For stoichiometric compound Mn2NiTi, an energetically favorable structure Tp with nearly cubic lattice parameters and layer-by-layer ordering of Mn and Ni atoms is predicted. This structure is characterized by the antiferromagnetic spin alignment of Mn magnetic moments. Furthermore, this structure demonstrates an easy-axis magnetocrystalline anisotropy of about 0,2 MJ/m3, which is not typical of cubic phases. When Ti atoms are partially substituted with Ni, stable tetragonal structures Tp and XA emerge, which are ordered antiferromagnetically. An increase in Ni content results in a higher energy barrier between cubic and tetragonal phases, indicating a rise in the martensitic transition temperature. It was found that the Mn2Nii,25Ti0,75 compound in the tetragonal XA structure with c/a ratio of 1,46 exhibits the highest MAE value of 0,4 MJ/m3. Nevertheless, for compounds with x = 0,5 and 0,75, the tetragonal Tp structure also exhibits larger MAE values compared to the XA structure. Overall, the MAE values obtained are comparable to those of tetratenite L10-FeNi, which is an alternative to rare-earth-free permanent magnets.
This study was supported by the Russian Science Foundation (RSF), Project No. 24-12-20016.
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Received April 11, 2024
Information about the authors
Sokolovskiy Vladimir Vladimirovich is Dr. Sc. (Physics and Mathematics), Associate Professor, Chelyabinsk State University, Chelyabinsk, Russian Federation, e-mail: vsokolovsky84@mail.ru.
Buchelnikov Vasiliy Dmitrievich is Dr. Sc. (Physics and Mathematics), Professor, Chelyabinsk State University, Chelyabinsk, Russian Federation, e-mail: buche@csu.ru.
Cong Daoyong is Professor, Beijing University of Science and Technology, Beijing, People's Republic of China, e-mail: dycong@ustb.edu.cn.
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2024, vol. 16, no. 2, pp. 78-85
УДК 538.911 DOI: 10.14529/mmph240208
МАГНИТНЫЕ И СТРУКТУРНЫЕ СВОЙСТВА СПЛАВОВ ГЕЙСЛЕРА MN-NI-TI, ПОЛНОСТЬЮ СОСТОЯЩИХ ИЗ ПЕРЕХОДНЫХ МЕТАЛЛОВ
1 12 В.В. Соколовскии , В.Д. Бучельников , Д. Конг
1 Челябинский государственный университет, Челябинск, Российская Федерация
2 Пекинский научно-технический университет, Пекин, Китайская Народная Республика E-mail: vsokolovsky84@mail.ru
Аннотация. Представлены теоретические исследования влияния различного атомного и магнитного упорядочения на структурные и магнитные свойства сплавов Mn2Ni1+xTi1_x, полностью состоящих из переходных металлов. Используя теорию функционала плотности, мы предсказываем основные структурные и магнитные состояния соединений с x = 0, 0,25, 0,5 и 0,75 как в кубических аустенитных, так и в тетрагональных мартенситных фазах. Обнаружено, что частичное замещение атомов Ti на Ni приводит к увеличению энергетического барьера между структурными фазами, к смене слоевого атомного упорядочения на чередующееся шахматное, а также к переходу от антиферромагнитного к ферромагнитному упорядочению спинов в кубической фазе. Все соединения с тетрагонально искаженной структурой демонстрируют конфигурацию спинов вне плоскости и магнитокристаллическую анизотропию тппа «легкая ось», за исключением тетрагональной фазы L10 Mn2NiTi. Для последней структуры наблюдается магнитная анизотропия типа «легкая плоскость». Рассчитанные значения анизотропии сравнимы с таковыми для тетрагонального L10-FeNi.
Ключевые слова: ab initio вычисления; сплавы Гейслера полностью состоящие из переходных металлов; атомное расположение; магнитокристаллическая анизотропия
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Поступила в редакцию 11 апреля 2024 г.
Сведения об авторах
Соколовский Владимир Владимирович - доктор физико-математических наук, доцент, Челябинский государственный университет, г. Челябинск, Российская Федерация, e-mail: vsokolovsky84@mail.ru.
Бучельников Василий Дмитриевич - доктор физико-математических наук, профессор, Челябинский государственный университет, г. Челябинск, Российская Федерация, e-mail: buche@csu.ru.
Конг Даоюн - профессор, Пекинский научно-технический университет, г. Пекин, Китайская Народная Республика, e-mail: dycong@ustb.edu.cn.
