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51
Вестник Челябинского государственного университета. 2011. № 38 (253). Физика. Вып. 11. С. 18-21.
V. V. Sokolovskiy, V. D. Buchelnikov, M. A. Zagrebin, P. Entel
ab initio and monte-carlo investigations of the magnetic exchange and curie temperature
of Ni2Mn1+^Sn1-^ alloys1
The ab initio and Monte-Carlo calculations of exchange magnetic constants and the Curie temperatures of stoichiometric and non-stoichiometric Heusler Ni50Mn25+iSn25-i alloys have been performed. The Curie temperatures have been calculated by Monte-Carlo method using Heisenberg model. Our ab initio calculations have shown that the Curie temperatures for Ni-Mn-Sn alloys are in qualitative agreement with experimental transition temperatures.
Keywords: Heusler alloys, magnetic exchange constants, magnetic phase transition.
Introduction. The ferromagnetic (FM) shape memory alloys Ni-Mn-X (X = In, Sn, Sb) have drawn more and more attention recently [1]. Some interesting properties, such as shape memory effect, magnetic field induced strain, magnetoresistance, exchange bias effect (EB), and magnetocaloric effect (MCE) have been investigated in non-stoichiometric Heusler alloys.
In the case of Ni-Mn-Sn alloys, the stoichiometric Ni50Mn25Sn25 alloy has the L21 structure, in which the Sn atoms occupy the site (0, 0, 0), the Mn atoms occupy the (1/2, 1/2, 1/2) and the Ni atoms locate at the sites (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4). For each unit cell, it contains 16 atoms. For the case of non-stoichiometric Ni2Mn1+xSn1-x alloys, the excess of Mn atoms are occupied the Sn sites and these atoms interact antiferromag-netically (AF) with the surrounding Mn atoms on the regular Mn sites [1]. The FM behavior and L21 structure observe in the range 0 <x < 0,4. The alloys with 0,4 <x < 0,6 undergo a martensitic transition from the high temperature L21 structure to the 10M, 14M, L10 or 4O one depending on composition [2-4]. These alloys show a variety of magnetic transitions. For Ni-Mn-X alloys, after the martensitic transformation, the Mn-Mn distance decreases due to the twinnig of the martensitic state, leading to the enhancing of the AF exchange interaction. The coexistence of FM and AF phases in the martensitic phase is a result of EB and a inverse MCE [1].
From theoretical point of view, recently, Sasioglu et al. [5] performed the studies of the electronic structures and exchange interaction parameters for Ni2MnSn using the augmented spherical wave
1 This work was supported by RFBR (grants 10-02-96020-r-ural, 11-02-00601, and 10-02-92110), RF President RF Grant MK-1891.2010.2 and FSYS-03/11 of ChelSU.
method within the atomic-sphere approximation. It was found that the magnetic exchange interactions have oscillated behavior as a function of the interatomic spacing.
In this work we present ab initio and Monte-Carlo calculations of exchange magnetic constants and the Curie temperatures of stoichiometric and non-stoi-chiometric Heusler Ni2Mn1+xSn1-x alloys.
Ab initio simulations. In this section, we present the composition dependencies of magnetic exchange constants in the first coordination sphere for Ni2Mn1+xSn1-x alloys. Calculations have been carried out for the high-temperature austenitic L21 structure (the space group is Fm3m) and for the low-temper-ature martensitic 4O and L10 structures (the space group are Pmma, Fmmm). The magnetic exchange constants have been calculated for several alloys by the Munich SPR-KKR (Spin Polarized Relativistic Korringa-Kohn-Rostoker code) package [6] in combination with the single-site coherent potential approximation (KKR-CPA). In our simulations, we have used following lattice parameters which have taken from [2-3] and listed in table.
The ab initio results of the magnetic exchange constants are presented in fig. 1. From fig. 1 we can observe that in the case of the cubic state (L21 structure), excess of the Mn atoms leads to insignificantly decrease the Mn1-Mn1 and Mn1-Ni interactions (here, Mn1 and Mn2 denote atoms at regular Mn and Sn sites, respectively) in the composition range
0 < x < 0,27. Whereas the Mn1-Mn1 and Mn1-Ni interactions are neglible. Opposite, in the composition range 0,28 < x < 0,4, we can see the rapid change of magnetic interactions. For example, the Mn1-Mn1 (Mn1-Ni, and Mn2-Ni) interaction decreases (enhances) with an increasing of the Mn atoms, respectively. Moreover the Mn1—Mn2 interaction for alloys with x > 0,28 is the AF one. It should note that alloys with x > 0,28 are closed to alloys with the
Lattice parameters for Ni2Mn1+xSn1x alloys (in Â)
Structure L2i
X 0 0,1 0,2 0,27 0,3 0,33 0,37 0,4
a = b = c 6,046 6,034 6,024 6,009 6,005 6,002 5,998 5,995
Structure 4O LI0
X 0,43 0,48 0,52 0,55 0,59 0,8 0,85 0,9
a 8,5837 8,5834 8,5831 8,5828 8,5825 7,595 7,592 7,589
b 5,6021 5,6018 5,6015 5,6012 5,6009 7,595 7,592 7,589
c 4,3621 4,3618 4,3615 4,3612 4,3609 6,98 6,96 6,93
martensitic transformation. In the case of the or-thorhombic state (4O structure) we have found the largest values of the AF Mn1-Mn2 interaction. But for the case of tetragonal state (L10 structure) we can see small values of the AF Mn1-Mn2 interaction and the Mn1-Mn1 interaction is closed to zero. The large AF interactions are also responsible for the drop in the thermomagnetization curves of Ni-Mn-Sn al-
loys at the structural phase transition, EB and for the inverse MCE. It should note the magnetic exchange constants for Ni 2MnSn alloy are closed to the values which obtained by Sasioglu et al. [5].
Fig. 2 shows the experimental and theoretical concentration dependencies of the total magnetic moment per unit, ^tot, for Ni2Mn1+xSn1-x alloys (0,0 < x < 0,56).
0,8 1,0 d/a
7 6 5 » a Ni2Mn|: v\ . 52Sno.48 :
4
3 w*
2 1 \ V\
> 0
—j
^-22 j —#— Mn-Mn2 .
-24 —MiyNi
—T-Mn2-Ni .
-26 / —Mn2-Mn2 ’
-28
-30 0,
,0 0 ,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0
d/a
d/a
Mn excess, x
Fig. 1. (a-c) Ab initio magnetic exchange interactions of Ni2Mn1+xSn1-x (x = 0,3, 0,52 and 0,8) as a function of the distance between atoms. Here d/a is a distance between pairs of atoms i and j (in units of the lattice constant a). (d) Composition dependencies of magnetic exchange constants in the first coordination sphere for Ni2Mn1+xSn1-x alloys
Mn excess, x
Fig. 2. The experimental and theoretical concentration dependency of the magnetic moment of Ni2Mn1+xSn1_x alloys. Here, filled (open) symbols are ab initio (experimental) data
The experimental dependence has been taken from [4]. The theoretical values of ^Mn2, Xsn and Xn have calculated using SPR-KKR package. The total magnetic moment (xto) per formula unit of Ni2Mni+xSni_x alloys is given by xtot = 2x№ +
+ XMnl - XXMn2 + (1 - x)Xsn Xb As shoWn in fig.
2, the theoretical and experimental values of the magnetic moments decrease when the concentration x increases. Theoretical results show a drop of the magnetic moment in the composition interval 0,27 < x < 0,3. This drop is due to the strong AF interaction between Mni and Mn2 atoms (see fig. 1). We can assume that the martensitic transition appears in this composition interval. But experimental data show the structural transition occurs in the samples with x > 0,4. So, we can see from fig. 2 that the theoretical magnetic momenta are in a good agreement with the experimental data.
Heisenberg model. In this section, we present the results of Curie temperature simulations using Heisenberg model. The calculations have been performed on the three-dimensional lattice with the real unit cell of Heusler alloys with the help of Monte-Carlo simulations and Metropolis algorithm. In the proposed model, we considered only magnetic interactions between magnetic Mn and Ni atoms. The values of the magnetic exchange constants have been taken from ab initio calculations. The model Hamiltonian is defined by (eq. 1),
H = -X JijSiSj, (1)
<I ,J >
here, Jjj is the magnetic exchange parameter which may become positive and negative depending on the distance between the atoms; S. = {Sx/, S^., Sz} is a classical Heisenberg’s variables with S. = 1 .
The number of sites is N = L3, where L is the number of real cubic unit cells of Heusler alloys. We have used L = 7, and, for example, in the case of the Ni 2MnSn the simulation cell contains 1687 Mn1; 1688 Sn, and 2744 Ni atoms. The configurations of the excess Mn atoms (Mn 2) on the Mn sublattice are chosen randomly and its total number is fixed by the composition of Ni2Mni+xSni-x. For a given temperature, the number of MC steps at each site was taken as 105 .
In fig. 3a we present thermomagnetization curves for Ni2Mni+xSni_x alloys (x = 0-0,4, L2i cubic structure) in a zero magnetic field. Fig. 3b contains the experimental T-x phase diagram with theoretical values of Curie temperature. The experimental phase diagram has been taken from [3].
We can observe from fig. 3, that theoretical Curie temperatures decrease when the concentration x increases from 0 to 0,27. Because the interactions between Mn and Ni atoms are slowly reduced (See Fig, 1d). Opposite, we have found strong interactions in alloys with 0,28 < x < 0,4, and in this case the theoretical Curie temperatures are increased. But the experimental Curie temperatures are approximately constant in the interval (0 < x < 0,4). So, the classical Heisenberg model cannot reproduce the experimental Curie temperature in the composition interval (0,1 < x < 0,3). It should note we will able to obtain the more accurate theoretical Curie temperatures if we will take into account interactions of more than three coordination spheres.
Summary. In this work we have modeled magnetic properties of Ni2Mni+xSni-x alloys (0,0 < x < 0,9) alloys by means of first-principles and classical Monte-Carlo simulations. The magnetic exchange parameters have been determined by ab
100 200 300 400
Temperature, K
U.
600
Mn excess, x
Fig.
3. (a) The thermomagnetization curves for of Ni2Mn1+xSn1-x (x = 0-0,4) in zero magnetic field. (b) Experimental T-xphase diagram of Ni2Mn1+xSn1-x with theoretical Curie temperatures. Here, Lines with open (filled) symbols are experimental (theoretical) data
initio calculations. Our simulations have been found that there are strong AF interactions in the vicinity of the structural transformation and in the martensitic phase. The enhancement of AF correlations due to atomic disorder or distortion in the Heusler alloys will also enhance the inverse MCE and EB one. In summary we should note that theoretical data of magnetic moment and Curie temperatures are in qualitative agreement with experimental data.
References
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