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Вестник Челябинского государственного университета. 2013. № 25 (316). Физика. Вып. 18. С. 9-18.
ФИЗИКА МАГНИТНЫХ ЯВЛЕНИЙ
V D. Buchelnikov, M. A. Zagrebin, V V Sokolovskiy, I. A. Taranenko
INVESTIGATIONS OF CRYSTAL AND MAGNETIC PROPERTIES OF Fe-Mn-Al ALLOYS FROM FIRST PRINCIPLES CALCULATIONS
The composition and structure dependences of crystal lattice parameters, magnetic moments, magnetic exchange parameters and Curie temperatures in Fe2+xMn1-xAl Heusler alloys are investigated with the help of first principles . Our simulations have shown that the Fe-Fe nearest neighbors present a strong ferromagnetic coupling . Moreover, these exchange interactions are larger than other interactions . The substitution of Mn by Fe in Fe2+xMn1-xAl (0 < x < 1,0) leads to an increase in the Curie temperature . This tendency and the values of Curie temperatures are in agreement with the experimental results for Fe2+xMn1-xAl (x = 0 and 1,0) . The highest Curie temperature was observed for the Fe-rich alloy
Keywords: Heusler alloys, magnetic exchange interactions, first principles, density functional theory.
1. Introduction
Heusler alloys containing Ni and Mn atoms have attracted a lot of attention due to the possibility of observing shape memory effect, the giant magnetoresist-ence, magnetocaloric effect etc . [1-3] . In the last years, the novel Fe2+xMn1-xAl Heusler compounds have been intensively investigated by experimentalists and theoreticians [4-11] . These alloys are interesting magnetic materials considering its promising technological applications in view of their properties such as anomalous behaviors of optical, magnetic and transport properties The interesting properties are coupled to the competing ferromagnetic (FM) — antiferromagnetic (AF) exchange interactions between Fe atoms at the order-dis-order structural transition from A2 to B2 structure [5; 6] . The effect of variations of composition around stoichiometric on the structural and magnetic properties of Fe2+xMn1-xAl (x = 0 and ±0,1) has been studied experimentally by Paduani et al. [7] . As it has been shown, the Curie temperature and saturation magnetization increased with the Fe content. Recently, Omori et al. investigated the addition of Ni to Fe-Mn-Al alloys in order to improve their magnetic and structural properties . For one composition, Fe43 5Mn34Al15Ni7 5, the superelastic behavior was observed over a temperature range from 77 to 513 K [8] .It offers great promise as a candidate for large-scale applications such as space, automobile, and seismic technologies
From theoretical point of view, the magnetic, structural and electronic properties of Fe-Mn-Al compounds have been studied recently by means of first-principles calculations [9-11]. The role of the orbital magnetism in the half-metallic and full-metallic Heusler alloys by means of fully relativistic screened Korringa — Kohn — Rostoker (KKR) method is studied by Galanakis [9]. As it has been shown, the orbital moments are almost completely quenched and
they are negligible with respect to the spin moments Results of the investigations of the optical properties and of the calculations of the electronic structure of Fe2NiAl and Fe2MnAl are presented in Ref [10]. Authors have shown that the different behavior of the optical properties in these alloys can be explained by a significantly different character of the density of electronic state near the Fermi level The complex investigation of the structural, electronic and magnetic properties of Fe3-xMnxZ (Z = Al, Ge, Sb) Heusler alloys can be found in Ref [11]. Authors used the density functional theory (DFT) based on full-potential linearized augmented plane-wave (FP-LAPW) method It has been shown that a FM phase is stable for compositions with x < 1, whereas the compounds with x > 1 have a ferrimagnetic phase . As regards calculations of the couplings constants associated to the magnetic exchange interactions in Fe2+xMn1-xAl, then today there is not any information about this in the scientific works
In view of the aforesaid, in this work, we will present the ab initio calculations of both structural and magnetic properties of Fe2+xMn1-xAl compounds .
2. Calculation details
To perform calculations of electronic structure, we used two different ab initio codes: the open Source Package for Research in Electronic Structure, Simula-tion,andOptimization(QuantumESPRESSO)[12; 13] and the full-potential KKR Munich [14; 15] package . The Quantum ESPRESSO (QE) is an integrated suite of computer codes for electronic-structure calculations and materials modeling, based on DFT, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave) . The QE was used to determine the optimized lattice parameter of Fe2+xMn1-xAl (x = 0; 0,25; 0,5; 0,75 and 1,0)
cubic phase and da relation for Fe2MnAl and Fe3Al alloys . Structural optimizations are performed using the Broyden — Fletcher—Goldfarb — Shanno algorithm [12]. Pseudopotentials for structural optimization were modeled within the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) [16; 17]. Calculations of magnetic exchange couplings were carried out using the spin-polarized relativistic KKR (SPR-KKR) band structure code [14; 15]. This code is based on the KKR-Green’s function formalism that makes use of multiple-scattering theory, and the electronic structure is expressed in terms of the corresponding Green’s function as opposed to Blochwave functions and eigenvalues In this code, chemical disorder is treated through the coherent potential approximation (CPA) For the optimized lattice parameter, the self-consistent potential (SCF) was calculated . The lmax (the angular momentum expansion for the major component of the wave function) was restricted to two For SCF cycles, 6348 k points were generated by a k-mesh {45, 45, 45}. All calculations converged to 0,01 mRy of total energy To achieve the convergence, we have used the BROYDEN2 scheme [18] (an iterative quasi-Newton method to solve the system of nonlinear equations) with PBE exchange correlation potential [16; 17]. 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 calculate spin magnetic moment and the Heisenberg’s magnetic exchange coupling parameters using Spin-Polarized Scalar-Relativistic (SP-SREL) Hamiltonian with orbital momentum cutoff l = 2
max
on a grid of k-mesh {57, 57, 57} with 4495 k points . The Heisenberg exchange coupling parameters J.. within a real-space approach were calculated using an expression proposed by Liechtenstein et al. [19].
In our simulations, we used two structural phases: high-temperature austenite with space group of Fm3m (225) and low-temperature orthorhombic mar-tensite with space group of Fmmm (69) (see tab . 1) [20; 21]. We would like to point that in this work, we consider the orthorhombic martensite with a = b which corresponds to the tetragonal one
Table 1
Crystal and atomic parameters of Fe-Mn-Al system for austenite and martensite in our calculations
3. Calculation results
3.1. Crystal structure
In this section we present calculation results of equilibrium value of crystal lattice parameter obtained for austenite of Fe2MnAl by means of the QE package .
Our calculations of the optimized lattice parameter are shown that in Fe2+xMn1-xAl alloys the different magnetic states can exist Here we will consider only the FM state then spins of all Fe and Mn atoms are parallel (magnetic moment Al atom is negligible) .
The dependence of total energy (calculated for each formula unit) on lattice parameter for the cubic phase of stoichiometric Fe2MnAl alloy is shown in fig 1a We have found that the equilibrium lattice parameter (which has minimum of the total energy) is a = 5,786 A . This value is closely to the experimentally observed ones [7]. We have also calculated the equilibrium tetragonality (relation between lattice parameters c and a via c/a) for Fe2MnAl and Fe3Al alloys . The dependences of energy difference between martensite and austenite (calculated for each formula unit) on c/a tetragonality for Fe2MnAl and Fe3Al alloys are depicted in fig . 1b . It is clearly seen that there are three local minimums with follow tetragonalities for Fe2MnAl compound: c/a = 0,90, c/a = 1,17 and c/a = 1,53 . It means that three stable martensitic phases can be realized with c/a of 0,90, 1,17, and 1,53 . Contrary in Fe3Al compound, two local minimums at c/a = 0,94 and 1,12 can be founded whereas the minimum at c/a = 1,53 is disappeared . It should be noted that minimums are shifted towards c/a = 1,00 as compared to case of Fe2MnAl alloy.
We supposed that calculated values of equilibrium lattice constants in Fe Mn Al austenite could be
2+x 1-x
approximated by the linear dependence These values are listed in tab 2 The tetragonality and appropriate lattice constant a for martensitic state of Fe2MnAl and Fe3Al alloys are listed in tab 3 The lattice parameters a and c for martensite were obtained from the value of a in the cubic phase assuming that the unit cell volume does not change upon the structural phase transition
Table 2
Lattice parameter a of Fe2+xMn1-xAl austenite used in calculations
Fe concentration, x 0,00 0,25 0,50 0,75 1,00
a, A 5,786 5,779 5,771 5,772 5,761
Table 3
Lattice parameter a and tetragonality c/a of Fe2MnAl and Fe3Al martensites used in calculations
Phase Group of symmetry Wyckoff positions
Fe Mn Al
Austenite Fm3m 8c 4b 4a
Martensite Fmmm 8f 4b 4a
Alloy Fe2MnAl Fe3Al
c/a 0,9 1,17 1,53 0,94 1,12
a, A 5,993 5,687 4,936 5,881 5,663
Obtained lattice values we used for further calculations of magnetic properties of Fe2+xMn1-xAl alloys .
3.2. Magnetic moment calculations
The calculations of partial and total magnetic moments of Fe2+xMn1-xAl alloys have been done by means of the SPR-KKR package . For creation off-stoichiometric compositions, we assumed that the excess of Fe atoms are randomly distributed at the Mn sublattice As it is mentioned above, the chemical disorder was created in the framework of CPA . It should be noted that we will use follow designations such as Fe1 and Fe2 Here, Fe atoms, which are located at regular Fe sublattice, are denoted as Fe1, whereas
Fe2 designation corresponds to Fe atoms which randomly occupied the Mn sublattice
In fig 2a we present the composition dependences of partial moments for each sort of atoms, whereas the total magnetic moment as function of Fe excess x is depicted in fig . 2b . The value of total magnetic moment was obtained using Eq (1)
Vtotal = 2^ + x^2 +(1 - x>M + ^Al, (1)
where ^Fe1, ^Fe2, ^Mn and ^Al are the partial magnetic moment of Fe, Mn and Al respectively; x is the concentration of Fe .
An inessential varying the partial magnetic moments can be observed in fig . 2a. The magnetic moment of Fej
Fig . 1. Dependence of the equilibrium lattice parameter on Fe excess for the cubic phase of Fe2+xMn1-xAl alloys (a) and dependence of the energy difference AE between martensite and austenite (calculated for each formula unit) on tetragonality c/a (b)
Fig. 2 . Calculated spin magnetic moment of Fe2+xMn1-xAl as function of Fe excess (x): a — partial magnetic moment of Fe1; Fe2 and Mn atoms; b — total magnetic moment
increases from 1,76 (x = 0,0) up to 1,90 (x = 1,0), whereas the magnetic moment of Fe2 decreases from 2,42 (x = 0,0) up to 2,4 (x = 1,0) . Value of magnetic moment for Mn atoms increases from 2,15 (x = 0,0) up to 2,2 (x = 1,0) . As for the total magnetic moment, it increases with increasing of Fe excess (x) Obtained results are in a good agreement with theoretical calculations [11] .
The partial moments for each sort of atoms and total magnetic moment as functions on tetragonality c/a for Fe2MnAl and Fe3Al alloys are shown in fig 3
We can see from fig . 3a that in both Fe2MnAl and Fe3Al magnetic moments of Fe1, Fe2 and Mn atoms are positive and have values close to 2^B, whereas magnetic moments of Al are negative with values of a -0,2^B . Our calculations have shown that in case of Fe3Al, the magnetic moments of Fe2, Fe1 (Al) atoms increase (decrease) with an increase of c/a, respectively Moreover the magnetic moment of Fe1 has a minimum at c/a = 1 As for Fe2MnAl, the increase of c/a up to 1,17 leads to increase (decrease) magnetic moments of Fe1 and Mn (Al) atoms, respectively, and then the alternative behavior of magnetic moments is observed in range 1,17 < c/a < 1,53. The total magnetic moment of Fe2MnAl has maximum value of 7^B at c/a = 1,17 (see fig. 3b) .
3.3. Magnetic exchange couplings parameters
Fig 4 displays the magnetic exchange parameters for austenite and martensite of Fe2MnAl as a function of distance between atoms Here and further, the positive exchange constants (J. > 0) are presented a FM coupling, whereas the negative ones (J. < 0) indicate an AF coupling . The oscillating damped behavior of J. can be observed We can see that in the case of auste-nitic phase of Fe2MnAl (fig . 4a) the strongest FM interaction can be found between nearest-neighbor (NN) Fe1-Fe1 atoms . This interaction is six times greater the values of J for the Mn-Mn and Mn-Fe NN interac-
V 1
tions On the contrary, the second NN interaction for Fe1-Fe1 becomes strongly AF. Moreover, the AF coupling exists up to the fifth Fe1-Fe1 neighbors It clearly indicates on the competition behavior between the FM and AF interactions For martensitic state with c/a =
0,9 interaction between Fe1-Fe1 NN is the same as for austenite, while for the second neighbor is larger than in austenite . Interaction between Fe1-Fe1 NN for c/a = 1,17 is larger than for c/a = 1,0 and c/a = 1,53 . Interaction between Fe1-Mn NN is decreases with increasing of tetragonality and for c/a = 1,53 becomes AF Mn-Mn interaction increases with increasing of c/a .
The magnetic exchange parameters for austenite and martensite of Fe3Al as a function of distance between atoms are shown in fig 5 It is seen that in the case of Fe3Al alloy FM interactions between NN
Fe1-Fe2 atoms in the L21 structure are largest than other interactions in Fe3Al and Fe2MnAl alloys On the other side, in the case of Fe3Al the FM and AF couplings between Fe1-Fe1 neighbors are less than in the case of stoichiometric Fe2MnAl
The c/a dependences of magnetic exchange parameters between NN for each atom in Fe2MnAl and Fe3Al are displayed in fig 6
In fig 7 we present the magnetic exchange couplings for austenitic state of Fe2+xMn1-xAl as a function of distance between atoms, whereas the magnetic exchange parameters between NN for each atom in austenite state of Fe Mn Al as a function Fe
2+x 1-x
content (x) are depicted in fig 8 It can be observed from fig 7 that the substitution of Mn with Fe leads to an increase in the first NN Mn-Fe1, Fe1-Fe2 and Fe2-Fe2 interactions, and to a decrease in the first NN Mn-Mn and Fe1-Fe1 interactions as compared to Fe2MnAl alloy
In fig . 9 we show the composition dependence of the effective exchange parameters in the cubic structure of Fe2+xMn1-xAl alloys . The effective exchange parameter means the total J.. integral between atoms above all coordination shells It can be seen from fig 9a, when the Fe content (x) increases up to x = 0,5, the effective exchange parameter JMf-Mn becomes less compared to x = 0 by a factor of 2,5, whereas the further increase in Fe leads to practically constant values of JMn-Mn- The almost linear behavior can be observed for the JMn_Fe2 and Jfe2 exchange parameters that is slightly increased with increase in Fe content However, the composition behavior of Jfe1 is different compared to other behaviors of Jeff. When the Fe content (x) increases up to x = 0,3, the decline of Jie1-Fe1 can be observed, and the rise of Jfe1 is occurred after further increase in Fe up to x = 1,0 . Moreover, the values of Jie1-Fe1 for compositions with x = 0, and 1,0 are practically equal . The observed minimum of JFt[.Fe1 in the composition range (0,3 < x < 0,4) is related with a strong contribution of the AF interaction between Fe1-Fe1 atoms to the total effective exchange parameter We can see from fig 9b that all effective exchange parameters are increased with increase in Fe content. Moreover, the values of Jeff between the Mn-Fe1 and Fe1-Fe2 neighbors are more in several times than other values of Jeff illustrated in fig . 9a. The strongest effective coupling of all effective exchange parameters can be observed between Fe2 and Fe1 atoms It means that the Fe-Fe interactions are important as they contribute to the stabilization of the FM ground state of austenite .
In the Fig 10 we display dependence of effective exchange magnetic parameters on tetragonality c/a for Fe2MnAl and Fe3Al
c!a
Fig . 3. Calculated spin magnetic moment of Fe2MnAl (open symbols) and Fe3Al (closed symbols) as function of tetragonality c/a: a — partial magnetic moment of Fe1; Fe2, Mn and Al atoms; b — total magnetic moment
Fig . 4 . Exchange couplings parameters for Fe2MnAl as a function of a distance between pairs of atoms i and j (in units of the lattice constant a): a — austenite with c/a = 1,0; b — martensite with c/a = 0,9; c — martensite with
c/a = 1,17; d — martensite with c/a = 1,53
25
20
15
>
o
10
"V
0 12 3
d/a
Fig. 5. Exchange couplings parameters for Fe,Al as a function of a distance between pairs of atoms i and j (in units of the lattice constant a): a — austenite with c/a = 1,0; b — martensite with c/a = 0,94;
c — martensite with c/a = 1,12
20
15
10
"V
• (a)
- \ Fe2.25Mn0,75A1 “
\ da =1,0 -■-Fe^Fej ■
- \ Fe^Fe, "
■ \ ^^Fej-Mn '
\ \ -A- Fe2-Fe2
\ \ -v- Fe^-Mn
-o— Mn-Mn .
m'1
1/ ■ 1 . 1
d/a
d/a
Fig. 7. Exchange couplings parameters for Fe^JVhij vAl in austenite phase as a function of a distance between pairs of atoms i andj (in units of the lattice constant a): a — x = 0,25; b — x = 0,5; c — x = 0,75
Fig . 8 . Exchange couplings parameters in the first coordination sphere for Fe2+xMn1-xAl in austenitic phase
as a function of Fe excess (x)
Fe excess (x) Fe excess (.r)
Fig. 9. Effective exchange couplings parameters for Fc2 , IV[n| _ A1 in austenitic phase as a function of Fe excess (x)
da
da
Fig. 10. Effective exchange couplings parameters for Fe; vIV[ii A1 in austenite phase as a function of tetragonality
(c/a): a — Fe2MnAl; b — Fe3Al
1400
1200
1000
800
600
400
200
1 1 1 (a) 1 1 1 1 1 •
l;e2;Mnl A1 / " -X / -
_ c!a =1,0 /
•
• _
—Theory “ —Experiment '
- • 1 . 1
0.00 0,25 0,50 0,75
Fe excess (x)
1,00
Fig . 11. Curie temperatures for Fe2+xMn1-xAl: a — dependence on Fe excess (x) in austenite; b — dependence on tetragonality c/a for Fe2MnAl and Fe3Al . Experimental data have been taken from [7]
3.4. Curie temperature
The effective exchange parameters are necessary for calculation the Curie temperature TC by means of well-established Heisenberg Hamiltonian, in frame of mean field approximation [23]. The mean field solution Curie temperature TC is obtained by solving the system of coupled equations (2)-(4) [23; 24].
2 kBTSA = nJf Sa + nAnJl Sb + nAncJf Sc , (2)
| kBTSB = nAnBJf SA + njf SB + nBncJfc Sc, (3)
— kBTSc = nAncJcA SA + nBncJCB SB + ncJcC SC > (4)
In (2)-(4) A, B and C denotes Fe1, Mn and Fe2; nm is the concentration of appropriate atom in the non-stoichi-ometric Heusler alloys; Sm is the average z component of the spin In these equations we have neglected Al atom because exchange interaction between Al and other atoms is negligible . Equations (2)-(4) has non-trivial solution if its determinant is zero . In this case the largest eigenvalue gives the Curie temperature [22; 23]
The theoretical composition dependence of the Curie temperature in Fe2+xMn1-xAl alloys shown in fig . 11a.
From fig . 11a we can see that the Curie temperature increases with increasing of Fe content. The simulation value of the Curie temperature for compositions with x = 0 is close to experimental data [7] . It should be noted that the slight discrepancy between theoretical and experimental values depends on the initial parameters in ab initio simulations such as lattice constant, ex-change-correlation potential, number of k points etc The mean-field approximation has shown that Curie temperature of Fe3Al compound is six (approximately) times larger than the Curie temperature of Fe2MnAl The dependence of Curie temperature of Fe2MnAl and Fe3Al on tetragonality c/a is presented in fig . 11b . From this figure we can observe that the Curie temperature of Fe2MnAl in austenite is smaller than in martensite The largest value of Curie temperature for martensitic phase with c/a = 1,53 is 1400 K . The Curie temperature for Fe3Al is decreases with an increasing tetragonality c/a We suppose that proposed calculations, which are performed within the framework of the DFT, can predict the increase in the Curie temperature in Fe2+xMn1-xAl Heusler alloys at a deviation from stoichiometry
4. Conclusions
The composition and structure dependences of crystal lattice parameters, magnetic moments, magnetic exchange parameters and Curie temperatures in Fe2+xMn1-xAl Heusler alloys are investigated with the help of first principles using the PWSCF and SPR-KKR methods . As can be shown this system has both
austenitic and martensitic states Crystal lattice parameter is increased with Fe content (x) The strongest FM interaction is observed between Fe1-Fe2 neighbors, while the other interactions are weaken This FM coupling strengthens with an increase in Fe content and leads to the stabilization of the FM ground state of aus-tenite . The most competition of FM-AF interactions is observed in compounds with composition range (0,3 < x < 0,4) . The theoretical values of the Curie temperature are increased with an increase in Fe content due to the strengthening of exchange interactions between the Fe atoms . The simulation values of the Curie temperature are close to available experimental data It should be noted that more accurately values of the Curie temperature can be obtained from Monte Carlo simulations using the Heisenberg Hamiltonian with ab initio magnetic exchange parameters
Acknowledgements
This work was supported by RFBR grants 11-0200601, and RF President grant MK-6278 2012 2 SCF-CPA calculations were made on supercomputer of Chelyabinsk State University research centre and with the help of open program code
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