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Вестник Челябинского государственного университета. 2012. № 14 (268). Физика. Вып. 13. С. 78-87.
ФИЗИКА МАГНИТНЫХ ЯВЛЕНИЙ
V. V. Sokolovskiy, V. D. Buchelnikov, M. A. Zagrebin, P. Entel
AB INITio AND MONTE CARLO INVESTIGATION OF DISORDER IN Ni, Mn Sn HEUSLER ALLOYS
2 1+* 1—*;
We present ab initio calculations of magnetic exchange parameters of stoichiometric Heusler compound N2MnSn and a few non-stoichiometric Ni2MnHxSn1x cases . Use of the exchange parameters in subsequent Monte Carlo simulations allowsto evaluate the magnetizations curves as a function of temperature and composition as well as the critical temperatures of the magnetic phase transitions . The latter are compared to those obtained from a mean-field approximation using the Heisenberg model . We find that the variation of the experimental Curie temperatures of non-stoichiometric alloys can be explained theoretically if we assume that the main impact of disorder is the intermixing of manganese and tin on their corresponding sublattices and the simultaneous appearance of strong antiferromangetic trends which originate from the nearest neighbor Mn-Mn interactions on different sublattices . The Curie temperatures of the Ni-Mn-Sn alloys which have been obtained from the Monte Carlo simulations, are in qualitative agreement with the experimental transition temperatures
Keywords: Heusler alloys, magnetic exchange constants, magnetic phase transition.
Introduction. The ferromagnetic (FM) shape memory materials of type Ni-Mn-X (X = In, Sn, Sb) have led to an exiting field of research over the last decade because of the possible use of the diverse functionality of the alloys in different fields of technological applications [1-5] . Some interesting properties, such as magnetic shape memory effect (MSME), magnetic field-induced strain (FIS), magnetoresistance (MR), exchange bias effect (EBE), and magnetocaloric effect (MCE) have been investigated for the non-stoichiometric Heusler alloys . As we know, the magnetic properties of these alloys are very sensitive to the content of Mn because the Mn excess atoms substitute for X on the X-sublattice of the Heusler structure .
In the case of Ni-Mn-Sn alloys, the stoichiometric N2MnSn compound orders in the L21 structure, in which the Sn atoms occupy the sites (0, 0, 0), Mn occupy the (1/2, 1/2, 1/2) one and the Ni atoms locate at the sites (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4) with 16 atoms per unit cell [6] . In the non-stoichiometric case, Ni2Mn1+xSn1x, we may assume that the excess Mn atoms occupy sites of the Sn-sublattice and that these atoms can interact antiferromagneti-cally (AFM) with the surrounding Mn atoms on the regular Mn-sublattice because of the much shorter distance between Mn1-Mn2 compared to the Mn1-Mn2 and Mn2-Mn2 distances (“1” refefers to the original Mn-sublattice while “2” refers to the Sn-sublattice) [6] . Ferromagnetic behavior in the L22 structure is observed for the range 0 < x < 0 .4, and the Curie temperature, TC ~ 350 K, is practically
unchanged in this range of composition [1-3] . (For simplicity, we use here “L21” instead of B2 do designate the cubic austenitic Heusler structure and “L10” for the corresponding tetragonally distorted structure regardless of composition and c/a-ratio .) The alloys with 0 . 4 <x < 1 . 0 undergo a martensitic transition from the high temperature L21 structure to the 10M, 14M, L10 or 4O structure depending on the composition [5; 7-8] . These alloys show a variety of magnetic transitions For Ni-Mn-X alloys, after the martensitic transformation, the Mn-Mn distance may decrease further due to the twinning in the martensitic phase which leads to enhanced antiferromagnetic (AFM) exchange interactions . The coexistence of FM and AFM interactions in the martensitic phase is also responsible for the EBE and inverse MCE [9-14] .
The importance of the magnetic exchange interactions for the physical properties of the disordered Heusler materials was recently highlighted by Sasioglu et al . [15] and Entel et al . [16] . The dependence of the electronic structures, magnetic exchange parameters and Curie temperatures of N2MnSn (X = Ga, In, Sn, and Sb) was investigated using different implementations of density functional theory (DFT) (augmented spherical wave method within the atomic-sphere approximation [15], plane wave method as implemented in the Vienna ab initio simulation package [17] and the SPR-KKR-CPA method [18]) . It was found that the magnetic exchange parameters show RKKY-like oscillatory behavior as a function of the interatomic spacing
which however gets more and more disturbed with increasing amount of structural disorder For example, in case of N2MnSn the Mn-Mn exchange interactions increase within the first atomic shell compared to N2MnGa and N2MnIn while the exchange parameters within the third and fourth shells are small and strongly negative, respectively Not only structural disorder is important but the addition of a quartenary transition element which has recently been discussed by Siewert et al . [19] for the (Pt, Ni)-Mn-Z alloys with Z = Ga, Sn using DFT and Monte Carlo (MC) simulations The theoretical martensitic transition temperatures have been obtained from free energy calculations involving phonons and DFT total energies, whereas the Curie temperatures have been calculated from MC simulations using the Heisenberg model with the ab initio exchange parameters
The influence of configurational order and disorder in Heusler alloys based on the Co2MnGa system was investigated theoretically by Singh et al [20], Arroyave et al . [21] and Siewert et al . [22] using DFT calculations . The authors suggested that the austenitic and martensitic phases have disordered B2 structure . Ghosh and Sanyal discussed the influence of structural disorder in Ni-Mn-Ga alloys also using first-principles calculations [23] The discussion of the different magnetic phases which emerge from the magnetic exchange interactions, show that structural disorder and increasing valence electron concentrations leads to competing FM and AFM interactions which are present at all temperatures . However below the martensitic transformation temperature, the influence of the AFM interactions become overwhelmingly large . This is the characteristic feature of all alloys which emerge from the series Ni2Mn1+x(Ga, In, Sn, Sb)1-x . This behavior is decisive since it influences mostly the various kinds of func -tional properties mentioned above Another quite general observation is that the Ni-Mn exchange interaction is usually FM and helps to stabilize a FM ground state although the Mn-Mn interaction can be larger and AFM . Details of first-principles and MC simulations show that the actual magnetic spin configuration depends on composition, temperature and whether the system is in the austenitic or mar-tensitic phase [16; 24] .
In this paper we report ab initio calculations of the magnetic exchange parameters and Monte Carlos simulations of the Curie temperatures of non-stoichiometrically disordered Heusler Ni2Mn1+xSn1x alloys For some non-stoichiometrically disordered
alloys we consider different degrees of disorder from 5 to 50 percent between Mn and Sn atoms . For example, 5 % of disorder in Ni2MnSn alloy means that 5 % of the Sn atoms on the regular Sn-sublattice are randomly replaced by 5 % of Mn atoms, whereas 5 % of Mn atoms at the regular Mn sublattice are randomly replaced by 5 % of Sn atoms . Henceforth, we call this “structural disorder” to be distinguished from the “chemical disorder” in case of Mn excess where the Mn-excess atoms substitute Sn on the Sn sublattice (denoted as “non-stoichiometrically ordered”) . We show that both types of disorder are required to explain the magnetic trends of the Ni2Mn1+xSn1x alloys in order to reproduce the experimental trends The magnetic exchange parameters are calculated for disordered L21 structure and in which partial disorder between Mn and Sn atoms exists For the disordered cubic structure, we calculate the magnetization curves and plot the magnetic T-x diagram of Ni2Mn1+xSn1x alloys as it is obtained from the MC simulations and mean-field approximation (MFA) using the ab initio magnetic exchange parameters Also, the MC simulations of the magnetization curves for the disordered L22 structure allow us to plot the composition-disorder phase diagram displaying ferromagnetic and antiferromagnetic phases The paper is organized as follows . In Section II we discuss the results of ab initio calculations of the magnetic properties of ordered and disordered Ni-Mn-Sn systems . In section III we present the results of Monte Carlo simulations Concluding remarks are listed in section IV Ab initio calculation of the magnetic properties of Ni2Mn1+*Sn1* alloys. In this section, we present computational details of the calculation of magnetic exchange parameteres, magnetic moments and densities of states (DOS) curves for the disordered Ni2Mn1+xSn1x alloys. The calculations have been carried out for the high-temperature austenitic L22 structure (space group Fm3m). For the electronic structure calculations and evaluation of exchange parameters we used the spin polarized relativistic Korringa—Kohn—Rostoker (SPR—KKR) code [18] . Here, the magnetic exchange parameters are calculated using Liechtenstein’s formula [25] where the exchange interaction between a pair of spins is projected onto the classical Heisenberg Hamiltonian Since within this method the exchange parameters are computed from the total energy variation due to small rotations of a pair
of spins causing a perturbation in spin density, it is obvious that structural disorder as well as changes of distance between the atoms due to martensitic transformations will greatly affect the magnetic exchange parameters, J.. .
In this paper, we include the effect of chemical disorder on the J.. using the single-site coherent potential approximation (CPA) . The maximum number of CPA iterations and the CPA tolerance were set to 20 and 0 . 01 mRy, respectively. The first step in these calculations is to calculate the self-consistent potential (SCF) . The lmax (the angular momentum expansion for the major component of the wave function) was restricted to two . For SCF cycles, 834 k points were generated by a k-mesh {22, 22, 22} . All calculations converged to 0 . 01 mRy of total energy. To achieve the convergence.
we have used the BROYDEN2 scheme (an iterative quasi-Newton method to solve the system of nonlinear equations) with exchange-correlation potential of Vosko—Wilk—Nusair (VWN) [26] . In order to achieve faster convergence, the SCF mixing parameter was set to 0 20 The maximum number of SCF iterations was taken to 200. After the self-consistent potential is calculated it is used to simulate the Heisenberg’s magnetic exchange coupling parameters using Spin-Polarized Scalar-Relativistic (SP-SREL) Hamiltonian with orbital momentum cutoff l = 2 on a grid of k-mesh {57,
max 0
57, 57} with 4495 k points .
For the lattice parameters we have used the values form Refs . [5; 7-8] which are listed in table .
In fig . 1 we show the dependence of the exchange parameters on the concentration of Mn excess and
Lattice parameters for Ni2Mn1+xSn1-x alloys (in A)
Structure L21
x 0 0.1 0. 2 0. 27 0. 3 0.33 0. 37 0.4
a l 0 1 c 6 .046 6 .034 6 .024 6 .009 6 .005 6 .002 5. 998 5 .995
Fig. 1. The dependence of the Ab initio magnetic exchange parameters of the first coordination shell of NiMn1+Sn1x on composition and disorder between Mn and Sn atoms. (a) shows the variation of the Mn—Mnj exchange parameters as a function ofx for different degrees of disorder (y) ranging from zero to 50 % while (b) shows the Mn—Mn2 and (c) the Mn—Ni (filled symbols) andMn2-Ni
(open symbols) exchange parameters
the disorder between Mn and Sn atoms . It should be noted that, for instance, a disorder of 10 % means that 10 % of regular Mn (Sn) sites are now occupied by the Sn (Mn) atoms which corresponds to a partially disordered B2 phase . Clearly, the disorder has strong influence on the magnetic exchange parameters, see fig . 1 . We notice that first, the compositional dependence is rather smooth and second, the compositional range of rapid change of magnetic exchange parameters decreases with increasing disorder. For example, the Mn1-Mn1 interaction in fig . 1 (a) is reduced linearly due to increasing disorder. For 50 % disorder and composition Ni2Mn1 4Sn0 6 the Mn1-Mn1 interaction is approximately zero . This means that disorder destabilizes ferromagnetic order in cubic austenite of Ni-Mn-Sn and favors the appearance of strong AFM exchange between Mn1-Mn2 atoms in whole concentration range of austenite (0 < x < 0.4). Further increase of disorder leads not to a large decrease of the AFM exchange parameters . On the other hand, it is obvious from fig . 1(c) that the Mn1(Mn2)-Ni exchange increases linearly with increasing structural disorder (y) and that the Mn1-Ni and Mn2-Ni exchange parameters are of same magnitude for the disordered systems We notice that for the case of ordered structures the Mn1-Ni exchange does practically not change in the compositional range from 0 to 0 . 27, whereas the Mn2-Ni interaction has practically zero value . However, for the range 0 . 27 < x < 0 .4 these interactions rise sharply The drop is due to the strong AFM exchange between Mn1 and Mn2, see Ref. [27] . If we compare the exchange of Mn1(Mn2)-Ni for ordered and disordered cases, we notice that the exchange parameters of the ordered systems and disordered structures have the same slope in the range 0 . 27 < x < 0 .4 .
In fig 2 (a) we show the electronic density of states curves of Ni2Mn1 31Sn0 69 for different degrees of disorder (from 0 to 50 %) . We notice that for the ordered case the DOS curve is shifted as a whole to lower energies by ~ 0 . 4 eV. We also observe that with increasing degree of disorder all peaks in the DOS curves are diminished . The DOS at the Fermi level of spin-down electrons is increased compared to the ordered case . For spin-up electrons the DOS at Ef does not depend much on the degree of disorder. Fig . 2(b) shows the DOS curves for Ni2Mn1 21Sn0 79. In this case the DOS curves for the ordered alloy are not shifted Note that the DOS curves for 10 % and for 20 % disorder have a peak at -8 eV. For x = 0 . 21, only the alloy with 10 % disorder shows this peak .
The value of spin-down DOS at Ep of Ni2Mn1+xSn1x (x = 0-0.4) is plotted in fig. 3 for different degrees of disorder (from 0 to 50 %) The DOS value for the ordered alloy (0 %) increases with increasing concentration and in the concentration range 0 . 27 < x < 0 . 29 shows a jump followed by a constant value If we allow for disorder then the jump is shifted to lower concentration . For 10 % disorder the jump has vanished and the DOS value increases monotonously
Fig 4 shows the theoretical dependence of the magnetic moment on excess Mn and disorder between Mn and Sn atoms in NLMn^ Sn, . The be-
2 1+x 1-x
havior of p.t0t is presented in fig . 4 (a) while ^Mn1 and ^Mn2 are plotted in fig . 4 (b) . In this case $ot per formula unit of the disordered Ni2Mn1+xSn1-x alloys is
given by : tf0t = 2^Ni + (1-J)^Mn1 - (y +xKn2 W ■ Where y denotes the degree of structural disorder.
With increasing degree of disorder the total magnetic moment decreases smoothly with increasing x. For a large degree of disorder (more than 40 %) the magnetic moment can even become zero or negative . This means that with increasing degree of disorder
Fig. 2. The total electronic DOS curves for the cubic structure of (a) Ni^n13ISng 69 and (b) Ni,MnI 2ISng 79
with different degrees of disorder (y)
the alloy makes a transition to an antiferromagnetic or a ferrimagnetic state . This crossover has been discussed before . It is associated with the competition between ferromagnetic M^-Mnp M^-Ni, and Mn2-Ni exchange interactions and the antiferromagnetic Mn1-Mn2 ones . It is well known that Mn which substitutes Sn becomes antiferromagnetically coupled to the surrounding Mn atoms which sit on the regular Mn sublattice . This allows us to describe the physics in terms of where the first one is the Mn1 sublattice and the second one Mn2 sublattice . With increasing degree of disorder the number of Mn atoms on the second sublattice increases but the magnetic moment (^Mn2) of this sublattice decreases, see fig . 4(b) . The opposite situation is observed for the Mn1 sublattice, here, the number of Mn atoms decreases with increasing disorder but the magnetic moment ^Mn1 of this sublattice increases . At some degree of disorder the magnetic moments of both sublattices are equal and the alloy transforms to an antiferromagnetic state (for example, for composi-
tion Ni2Mn1 2Sn0 8 with a degree of disorder of 40 %) . With further increase of disorder a ferrimagnetic state can be achieved with ^Mni > |uMn^ while the full magnetic moment of the alloy will be negative . This information may be used to derive from the MC simulations compositional-disorder phase diagram for the Ni2Mn1+xSn1-x alloys which is presented in the next section . We note that a similar tendency of the overall behavior the magnetic moments has been observed experimentally and theoretically for the quaternary Heusler compound Mn2-xCoxVAl with B2 order [28] . In this work the increase of Co content leads to a negative total magnetic moment and hence to the antiferromagnetic and ferrimagnetic order.
Evaluation of the composition dependent Curie temperatures. In this section, we discuss the Curie temperatures of ordered and disordered Ni2Mn1+xSn1-x alloys using the MFA and the MC simulations by employing the Heisenberg model . The data for ordered cubic structures we have
Fig. 3. The compositional dependence of the spin-down DOS at the Fermi level of'NiMn + Sn as a function of x and different degrees of disorder
Fig. 4. Theoretical magnetic moments per formula unit ofNi2Mn1+xSn1x as a function of the composition x and structural disorder (y) showing the (a) dependence of the total magnetic moment (^tot) and (b) dependence of the magnetic moments juMni and ^Mn.. Lines with filled (open) symbols mark juMn1 and ^Mn2, respectively
taken from work [27] . The mean-field solution of the Heisenberg model has been obtained by diagonalizing corresponding matrices for the magnetic exchange parameters which can be obtained from the coupled equations for a multisublattice material [15; 20; 29] . We like to remind that the crystal structure of L21 Ni-Mn-Sn austenite consists of four interpenetrating fcc lattices A, B and C with origin (A) at (0 . 5, 0 . 5, 0 . 5), B at (0.25, 0.25, 0.25), C at (0.75, 0.75, 0.75) and D at (0, 0, 0), respectively. Mn and Sn occupy the A and D and Ni occupies the B and C sites . Since the excess Mn atoms are distributed randomly on the Sn sublattice, and Sn is assumed to be nonmagnetic, we consider a multisublattice Heusler system where Mn1 (Mn2) are located on sublattices (A) and (D) and two Ni sublattices (B) and (C) . The system of coupled equations (1) listed below has non-trivial solutions if the corresponding determinant is zero whereby the largest eigenvalue of the determinant determines the Curie temperature
2 kBTSA nAJ AASA + nAJABSB + nAJACSC + nAHDJADSD ;
^ kBTSB nAJBASA + JBBSB + JBCSC + nDJ BDS D ;
(1)
2 kBTSC nAJCASA + JCBSB + JCCSC + nDJCDSD ;
3
2 kBTSD = nAnDJ DASA + nDJDBSB + n DJ DCSC + nDJ DDS D ,
here, the J (m, n = A, B, C, D) represent the total
? mn V ’ 5 5 5 / f
sum of the ab initio exchange interactions between the m and n sublattices; n is the concentration of
’ m
each atom in the non-stoichiometric Heusler alloys;
Sm is the average z component of the spin .
For the Ni2Mn1+xSn1-x alloys we have used the following parameters: « = 1, n = x (where x = 0,
..., 0 . 4) . The summed values J (where m, n = Mn,,
5 / mn v ’ 1'
Mn2, Ni) have been obtained from the SPR-KKR calculations . In order to check this method for non-stoichiometric Heusler alloys, we have simulated the concentration dependence of the Curie temperature of Ni2+xMn1-xGa alloys for the cubic and tetragonal structure [30] as exemplary cases . We find that our Curie temperatures are in good agreement with other theoretical values which were obtained by Li et al . [29] . For example, in the case of x = 0 (0 . 25) we obtain for the Curie temperature of austenite (TCA) and martensite (TCM, c/a = 1 .25) TA = 435 K and TCM = 615 K (TCA = 286 K, TCM = 378 K) . Approximately, similar values have been presented by Li et al . [29] (x = 0: TCA = 462 K, TCM = 658 K, and x = 0 . 25:
TCA = 263 K, TCM = 362 K) . The small differences in Curie temperatures can be associated with the different methods when calculating the magnetic exchange parameters . In this context, we would like to point out that if we take into account two sublattices A and B, for example, for stoichiometic Ni2MnGa (Ni 2MnSn), we obtain Curie temperatures very close to the experimental values, TCA = 389 K (Ga) (TCA = 352 K (Sn)), respectively (compare also Ref [15] . However, this is not completely accurate since we have to take into account the Ni sublattices as well (as is done here)
The Monte Carlo simulations have been performed for the real three-dimensional Heusler lattice using the Metropolis algorithm [31] . In its simplest version, some new, random spin direction is chosen and the energy change which would result if this new spin orientation is kept, is then calculated . The number of sites is N = L3, where L is the number of cubic unit cells of the Heusler alloys . We have used L = 7 which in case of Ni2MnSn leads to a simulation cell which contains 1687 Mn1, 1688 Sn, and 2744 Ni atoms . The configurations of excess Mn2 atoms on the Sn sublattice are chosen randomly and the total number of Mn2 atoms is fixed by the composition Ni2Mn1+xSn1-x . In our MC simulations, we considered only magnetic interactions between magnetic Mn1, Mn2 and Ni atoms and have taken into account interactions within three coordination spheres For example, each Mn1 atom interacts with 42 Mn1 atoms, 38 Mn atoms and 56 Ni atoms
The model Hamiltonian is written as
h = -! JjSSj,
< I ,J >
(2)
where the Jj are the magnetic exchange parameters (positive in case of FM interactions and negative in case of AFM interactions depending on the distance between the atoms); S. = (Sx, Sy, Sz) is a classical Heisenberg spin variable |S.| = 1 . The values of the magnetic exchange constants have been taken from our ab initio calculations
As time unit, we have used one Monte Carlo step consisting of N attempts to change the St variables A new spin direction can be chosen by randomly choosing new spin components The spin components are chosen in the following manner [31] . Two random numbers r1 and r2 are chosen from the interval [0, 1] to produce a vector with two components Z = 1 - 2r1 and Z2 = 1 - 2r2 . The length of the vector is determined by Z2 = Z12 + Z22 and
if Z2 < 1 a new spin vector is then computed with components
S„ = 2 Zi(1 -Z2)1'2,
Sy = 2Z2(1 -Z2)1'2, Sz =1 -2Z2. (3)
For a given temperature, the number of MC steps at each site was taken as 105 . The simulation started from the ferromagnetic phase with S iz = 1 . In order to obtain equilibrium values of the internal energy and order parameter, the first 104 MC steps were discarded . The internal energy of the system and the order parameter were averaged over 225 configurations for each 400 MC steps
The order parameter is defined in the following way
1 N I---------------------------
Xm=—lylCsnf+Csnf+CsfY, (4) N i=i
here x denotes the Mnp Mn2 and Ni atoms . N is a number of x atoms . i runs over the corresponding lattice sites
The total magnetization for non-stoichiometric disordered (y ^ 0) Ni 2Mn1+xSn1-x alloys is calculated as
M = 2|iNi Nlm +
+VMni Mni m(l - y) -^ ^Mn'1 m(y + x). (5)
As we have noticed above considering the dependencies of the magnetic moments (^Mni, and ^Mn2) on the composition and structural disorder, we have calculated the composition — disorder phase diagram of Ni2Mn1xSn1-x alloys (See fig . 5) using the Heisenberg model and MC method .
The dependence of the magnetic moments (^Mni, and ^Mn2) on the composition and structural disorder allows to derive the composition-disorder phase diagram of Ni2Mn1+xSn1-x alloys (see fig . 5) using the Heisenberg model and MC method
The grey area in fig 5 marks the region of stable FM austenite with non-vanishing total magnetization given by eq . (5) and finite Curie temperature, respectively With increasing degree of disorder, i . e . with increasing intermixing of Mn and Sn on the corresponding Mn and Sn sublattices with concentration y (which can be described by NLMn.^ Sn. = NLMn. Sn Mn Sn. , where x
2 1+x 1-x 2 1-y y x + y 1-x-y
is the Mn excess concentration and y the degree of disorder), an AFM or ferromagnetic austenitic phase appears (white area) . It is interesting to note that for large disorder (y) and deviation from stoichiometric composition (x) we obtain ^ ^
(compare fig . 4(b)), although the number of Mn2 atoms may become larger compared to the number of Mn1 atoms . This can lead to zero or negative values of the total magnetization although the sublattice magnetizations ^ and ^ are non-zero emphasizing the existence of an AFM or ferrimag-netic phase
Fig . 6(a) and 6(b) show the magnetization curves of Ni2Mn1+xSn1-x alloys for the cubic L21 structure in zero magnetic field with degree of disorder (y) between Mn and Sn atoms ranging from 0 to 25 % . Finally, fig . 6(c) displays the experimental (T, x) phase diagram taken from [8] to which we have added the theoretical Curie temperatures
Fig. 5. Calculated composition-disorder phase diagram of' Ni2Mn1 SnMnx + Snlx . Here FM marks the region of the ferromagnetic phase while AFM denotes the antiferromagnetic or ferrimagnetic one
Fig. 6. (a) Magnetization curves ofNi2MnI+SnI_ (0 < x < 0.4) as a function of temperature in zero magnetic field and vanishing disorder (y = 0) [27]. (b) Magnetization curves of Ni2Mn1+xSn1x (0 < x < 0.31) in zero magnetic field for 25 % disorder. (c) Experimental (T, x) phase diagram ofNidMnI+xSn1_ to which the theoretical Curie temperatures have been added. The curves with open (filled) symbols mark the experimental (theoretical) data where the experimental data have been taken from ref. [8]
x
Obviously, the experimental Curie temperatures (open blue circles) do not vary much with Mn excess up to x = 0 . 5 . This is in contrast to the theoretical TC values (filled circles) which decrease between
0 < x < 0 . 27 in case of vanishing disorder, y = 0 . This can be related to a slow reduction of the magnetic exchange interaction between Mn and Ni for this range of composition (See ref. [27]) . For the subsequent range of compositions, 0. 28 < x < 0. 4, we observe stronger interactions leading to an increase of TC(x). The same trend is found when using the mean field approximation (filled diamonds). However, if we allow for disorder (y = 25 at %) we can reproduce the approximately constant behavior of the experimental Curie temperatures in the compositional range 0 < x < 0. 4 (filled stars) which underlines the importance of intermixing effects in the Mn and Sn sublattices
We have investigated the influence of disorder between Mn and Sn atoms on the Curie temperature more systematically. The Monte Carlo simulations of the Heisenberg model show that already for a disorder of 5 % the Curie temperature is of the order of 400 K for all compositions from the interval 0 . 1
< x < 0 . 3 . Further increase of the degree of disorder leads to a small reduction of TC. For y = 0 .5-0 . 25 TC does practically not change for compositions with 0
< x < 0. 4 . Best agreement between theoretical and experimental results is obtained for a disorder of
25 %, see fig . 6(c) . It should be noted we have also performed ab initio calculations of the magnetic exchange parameters and subsequent Monte Carlo simulations in order to obtain the Curie temperature for compositions with different disorder between Ni and Mn atoms . It seems that this type of intermixing is not realistic and does not reflect the actual composition since in this case the calculated Curie temperatures deviate largely from the experimental ones We tentatively conclude that structural disorder in Ni-Mn-Sn alloys involves primarily Mn and Sn on the corresponding two fcc sublattices .
Summary. We have investigated the effect of structural disorder on the magnetic properties of Ni2Mn1+xSn1-x alloys in the concentration range (0
< x <0 . 4) on the basis of density functional theory calculation and Monte Carlo simulations of the classical Heisenberg model . The magnetic exchange parameters and magnetic moments as well as the
electronic structure and density of states curves have been determined by the ab initio calculations using the SPR-KKR package The calculations reveal that strong AFM exchange interaction exist between nearest neighbor Mn1-Mn2 atoms with Mn2 located on the original Mn-sublattice and Mn2 on the Sn-sublattice . This AFM interaction is important i . e . particularly strong in the region 0 . 27 < x < 0 . 4 . The calculations also show the important role of additional degree of disorder between Mn and Sn (y) on the magnetic properties of Ni2Mn1+xSn1-x . This structural disorder in the L21 structure leads to strong AFM interactions between Mn1 and Mn2 in the composition range 0 < x < 0 . 4 . Moreover, with increasing x and y an AFM or ferrimagnetic austenitic phase is stabilized .
The Curie temperatures have been obtained from the MC simulations using ab initio magnetic exchange parameters as input . The numerical results for the Curie temperatures in case of structural disorder (y = 0 .25) agree well with the experimental data
We would like to point out that when taking into account the partially disordered B2 austenitic phase (y 4 0), we find a stabilizing effect on the Curie temperature in the compositional range 0 < x < 0 . 4 . This is clearly in contrast to the case of an ordered L21 structure (y = 0) where both the MC simulations and MFA lead to drastic changes of the Curie temperature with increasing x This means that partial structural disorder is one way to control the Curie temperature in some compositional range . We expect that this fact may be used to improve functional properties like the magnetic shape-memory effect by, for example, appropriate alloying . On the other hand, the increase of AFM correlations may be used to enhance the magnetocaloric and exchange bias effects of the magnetic Heusler alloys .
Acknowledgments. This work was supported by RFBR (grants 10-02-96020-r-ural, and 11-0200601) and RF President grant MK-6278 .2012 .2 . P. E . acknowledges funding by the Deutsche Forschungsgemeinschaft (SPP 1239) .
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