Investigation of magnetic properties of Ni-Mn-Ga Heusler alloys with the help of ab initio calculations Текст научной статьи по специальности «Физика»

Вестник Челябинского государственного университета. 2011. № 38 (253).

Физика. Вып. 11. С. 22-30.

M. A. Zagrebin, V. V. Sokolovskiy, V. D. Buchelnikov

INVESTIGATION OF MAGNETIC PROPERTIES OF Ni-Mn-Ga HEUSLER ALLOYS WITH THE HELP OF AB INITIO CALCULATIONS1

In this work the structural, electronic and magnetic properties of Ni-Mn-Ga Heusler alloys with the help of ab initio calculations have been performed. Calculations have been carried out for the stoichiometric and non-stoichiometric compositions. Composition dependences of the exchange coupling constants, magnetic moments, density of states, the lattice constant and calculation parameters have been obtained. Composition dependence of magnetic moments is in a good agreement with experimental data.

Keywords: Heusler alloys, ab initio calculations, phase transitions, exchange interactions, density of states.

1. Introduction. In the ferromagnetic Heusler alloys with a magnetically controlled shape memory effect, a thermoelastic structural (martensitic) transition occurs from the high temperature cubic phase (austenite) to the tetragonal (martensite) phase in the ferromagnetic state [1-4]. This transition can occur in the ferromagnetically ordered state, in this case the Curie temperature TC is higher than the martensitic transformation temperature, Tm. According this fact the Ni-Mn-Ga Heusler alloys have attracted much attention in view of their unique properties such as shape memory effect, giant magnetocaloric effect (MCE), large magnetore-sistance, exchange bias effect, etc. [4]. Recent experimental studies of non-stoichiometric Ni2+xMn1_ xGa have revealed that both the transition temperatures are sensitive to the chemical composition and with an increase of the Ni excess x the structural transition temperature Tm increases whereas the Curie temperature TC shows a tendency to decrease [1-5]. Both the structural and magnetic transitions are merged at the range of compositions 0,18 < x < 0,27. For these alloys, the large magnetocaloric effect at the point of first order magnetostructural phase transition (MST) has been reported [5]. From the literature there are series of electronic structure calculations of Ni-Mn-Ga, for example, see Refs. [2; 4; 6-8]. They show the importance of a band Jahn-Teller effect for the martensitic transformation and the existence of a band gap around the Fermi energy, Ef, in the minority-spin channel in these alloys. This fact plays a dominant role in the spin-tronics. In this paper, we present results of studying

1 This work was done under support of RFBR (grants 10-02-96020-r-ural, 11-02-00601) and FSYS-03/11 of Chelyabinsk State University. SCF-CPA calculations were made on supercomputer of Chelyabinsk State University research centre and with the help of open program code.

of the electronic and magnetic properties of Ni-Mn-Ga Heusler alloys, based on ab initio calculations. Obtained results allow us to explain existent experimental results.

2. Calculation details. In this study the calculations were performed within two different ab initio codes: the opEn Source Package for Research in Electronic Structure, Simulation, and Optimization (Quantum ESPRESSO) [9-10] and the full-potential Korringa-Kohn-Rostoker Munich SPR-KKR [1113] package. QUANTUM ESPRESSO is an integrated suite of computer codes for electronic-struc-ture calculations and materials modeling, based on density-functional theory, plane waves, and pseudopotentials (norm-conserving, ultrasoft, and projector-augmented wave). Quantum ESPRESSO was used to determine the optimized lattice parameter of Ni2MnGa cubic phase. Structural optimizations are performed using the Broyden-Fletcher-Goldfarb-Shanno algorithm [10]. Electronic structure calculations are carried out using the spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) band structure code [12-13]. 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, configurational disorder is treated through the coherent potential approximation (CPA). The exchange-correlation potential was modeled within the generalized gradient approximation of Perdew, Burke, and Ernzerhof. For the optimized lattice parameter, the self-consistent potential (SCF) is calculated. 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) [12]. 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 the Heisenberg’s magnetic exchange coupling parameters using Spin-Polarized Scalar-Relativistic (SP-SREL) Hamiltonian with orbital momentum cutoff lmax = 2 on a grid of k-mesh {57, 57, 57} with 4495 k points. This new self-consistent potential is used to calculate density of states (DOS), spin magnetic moment and the Heisenberg magnetic exchange coupling parameters J j using Spin-Polarized Scalar-Relativistic (SP-SREL) Hamiltonian with orbital momentum cutoff lmax = 2 on a grid of k-mesh {45, 45, 45} with 2300 k points. The Heisenberg exchange coupling parameters J j within a real-space approach were calculated using an expression proposed by Liechtenstein et al [14]. DOS calculations were performed using a spin-polarized scalar-relativistic Hamiltonian with full potential.

In our study we use two structural phases: austenite and martensite phases (tab. 1).

Experimental studies of crystal structure shows that high-temperature austenite phase has L21 cubic structure of Fm3m (225) space group. Low-temperature martensite phase has L10 tetragonal structure of/4/mmm (139) space group [15]. Wedel et al. at [16] shown that martensite phase more correctly to describe by orthorhombic Fmmm (69) space group with a = b. So we will use this approach in our calculations and name low temperature phase as tetragonal.

Experiments show that in non-stoichiometric case of Ni2+xMn1-xGa the Ni excess is placed to the Mn positions [5]. Atomic positions for this case which we used in our calculations listed in table 2.

The tetragonality of the martensite state is c/a = 1,25. Values of lattice parameters a and с were obtained from the value a in the cubic phase and an assumption that a volume is constant at the structural phase transition.

3. Results. 3.1. Equilibrium lattice parameter. Fig. 1 shows the dependence of the total energy on lattice parameter for the cubic phase of stoichiometric Ni2MnGa alloy. We have found that the equilibrium lattice parameter (which has minimum of the total energy) is a = 5,82 А. This value is the same as experimentally observed [15].

Obtained value we used for further calculation of properties in non-stoichiometric Ni2+xMn1xGa alloys.

Table 1

Crystal and atomic parameters of Ni2MnGa alloy for austenite and martensite in calculations

Phase Austenite Martensite

Group of symmetry c/a Wyckoff positions Ni Mn Ga Fm 3m 1,0 Sc: 0,25; 0,25; 0,25 4b: 0,5; 0,5; 0,5 4a: 0; 0; 0 Fmmm 1,25 Sf: 0,25; 0,25; 0,25 4b: 0; 0; 0,5 4a: 0; 0; 0

Table 2

Atomic positions in austenite and martensite of Ni2+xMn1xGa alloys

Element Austenite Martensite

Nia Sc: 0,25; 0,25; 0,25 Sf: 0,25; 0,25; 0,25

Ni4 4b: 0,5; 0,5; 0,5 4b: 0; 0; 0,5

Mn 4b: 0,5; 0,5; 0,5 4b: 0; 0; 0,5

Ga 4a: 0; 0; 0 4a: 0; 0; 0

3.2. Magnetic moments and densities of states. From SPR-KKR calculations the concentration dependence of magnetic moment was obtained. Fig. 2 displays dependences partial magnetic moments of the Ni and Mn and total magnetic moment of Ni2+xMn1-xGa alloys. Calculations show that magnetic moments of the Ni and Mn atoms are decreased with increasing of the Ni excess, x. The magnetic moment of the Mn and Ni atoms in Ni2MnGa is 3,38 iB and 0,29 jub, respectively. In Ni2 39Mn061Ga the magnetic moment of the Mn and Ni atoms is 3,33 iB and 0,24 ji, B respectively (fig. 2a). The total magnetic moment is also decreased with increasing of the Ni excess from 4,08 iB in stoichiometric Ni2MnGa to 2,58 iB in Ni2 39Mn061Ga (fig. 2b). The theoretical

behavior of the total magnetic moment in Ni2+xMn1_ xGa is in a good agreement with the experimental dependence observed by Khovailo et al [5].

The densities of states for cubic and tetragonal phases of Ni2+xMn1-xGa (x = 0; 0,1; 0,2; 0,3) alloys are presented in fig. 3. In these figures the zero of energy denotes the position of the Fermi level. Let us consider DOS for cubic phase at fig. 3a. It can be seen that for minority-spin DOS below and above Ef has peaks. With increasing of Ni component x these peaks are decreased. For x = 0,3 peak above Fermi level is vanished. The second peak below EF for this case is increased. For tetragonal phase (fig. 3b) calculations DOS for Ni2+xMn1-xGa show presence of peak in the minority-spin DOS above EF and

Fig. 1. Dependence of the total energy on lattice parameter for the cubic phase of stoichiometric Ni^MnGa alloy

3,40

cq 3,35

=L

<L>

I 3,30J S

G

SP

s

0,25

1 1 1 1 1 a) Ni Mn, Ga 7 ■ ~ 2+X 1 -JE

* -■-Ni ^>Mn ■

, i.i,

0,0 0,1 0,2 0,3

Ni excess (x)

0,4

4,0

3,5

G 3,0

W)

as

2,5

%

b)

Ni Mn, Ga

2+x l-x

- Experiment at 4,2 K

- Calculation

'V

0,0 0,1 0,2 0,3

Ni excess (x)

0,4

Fig. 2. (a) Composition dependences of the Ni and Mn magnetic moments in Ni2+xMn1xGa alloys.

Here lines with open (filled) symbols are magnetic moments of the Mn (Ni) atoms, respectively.

(b) The theoretical and experimental compposition dependences of the total magnetic moment in Ni2+xMn1-xGa alloys. Here lines with open (filled) symbols are theoretical (experimental) results, respectively.

The experimental data have been taken from [5]

this peak is decreased with increasing of Ni component x .

In the fig. 4 we present the composition dependence of the spin-minority DOS at EF for cubic and tetragonal phases. DOS for cubic phase is bigger than for tetragonal one. From figure can be seen that DOS at Fermi level both for cubic and tetragonal phase are increased with increasing of the Ni excess.

3.3. Magnetic exchange coupling parameters. The decrease of TC observed in Ni2+xMn1-xGa with increasing the Ni excess atoms is due to several reasons. It is well known that in stoichiometric Ni2MnGa the magnetic moment of ~4 iB is largely confined to the Mn atoms while the contribution of the Ni atoms to the total magnetic moment is much smaller, ~0,3 iB [15]. Hence, the decrease of TC may be explained by the dilution of the magnetic Mn-subsystem. This is

a reasonable assumption because the magnetic Compton-scattering study of Ni2+xMn hxGa has shown that the small magnetic moment of the excess Ni atoms on the Mn sublattice remains essentially unchanged [17]. In spite of the fact that Ni atoms have a small magnetic moment, the ab initio calculations of the magnetic exchange parameters of Ni-Mn-Ga have shown that the Nia -Mn exchange interaction is positive and the largest of all exchange integrals (Mn-Mn, Nia-Mn, Ni b -Mn, respectively). The Nia-Mn interaction is ferromagnetic and mainly determines the magnitude of the Curie temperature. The magnetic interactions of Ni and Mn with the Ga atoms are small and negligible.

Fig. 5 shows exchange couplings parameters for Ni2MnGa austenite, Ni218Mn0 82Ga austenite and for Ni218Mn0 82Ga martensite.

o

o

E - Ef, eV

o

o

E - Ef, eV

Fig. 3. Calculated DOS for cubic (a) and tetragonal (b) phases of Ni2+xMn1_xGa

Ni excess (x)

Fig. 4. Dependences of DOS of Ni2+xMnlxGa at EF level for cubic and tetragonal phases on Ni excess x

Fig. 6 shows the composition dependencies of the magnetic exchange integrals Ni2+xMn1-xGa in the first, second and third coordination spheres for the cubic L2j and tetragonal L10 structures. As it should be seen from fig. 6, the Mn-Mn, Mn-Nia and Mn-Nib interactions in the martensitic state are stronger than in the austenitic state. We can also see that with increasing dilution of the magnetic Mn-sublattice with increasing x the exchange parameters J j decrease systematically. This leads to a decrease of TC in the composition region 0,0 < x < 0,18. For the compositions with x > 0,18, the experimentally measured TC the first increases in for 0,18 < x < 0,22 and then decreases with further increase in x [5]. The increase of TC in the region 0,18 < x < 0,22 can probably be explained by enhanced Mn-Mn interactions (d/a ~ 0,7) in the martensitic state (see fig. 6c). The

ab initio calculations show that for the non-cubic phase the change of Mn-Mn distances leads to the appearance of antiferromagnetic (AF) contributions to exchange parameters. There are two AF interactions in martensite. The first AF interaction is between second nearest Mn atoms (d/a = 1,0) located in the (a, b) plane and the second AF interaction is between nearest Mn and Nib atoms (d/a ~ 0,7) which are also located in the (a, b) plane (see fig. 6d, e). We can see that with increasing x the AF correlations decrease and vanish in the austenite phase (fig. 5).

We studied the dependence of exchange coupling constant J j on k-points number in Brillouin zone. In fig. 7 the dependences of exchange coupling parameter between Mn (see fig. 7a, b) and Ni (see fig. 7c. d) atoms on k-points in Brillouin zone in the cubic phase of Ni2,18Mn0,82Ga alloy are presented.

Fig. 5. Exchange couplings parameters for NiMnGa austenite (a), Ni218Mn0 82 Ga austenite (b) and for Ni218Mn0 82Ga martensite (c) as functions 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)

Fig. 6. Composition dependences of exchange coupling parameters of Ni2+xMnlxGa alloys for cubic (a, b)

and tetragonal (c, d, e) phases from Ni excess

From fig. 7 we can see that Jtj for the Mn-Mn interaction with values of k-points until to 4 000 significant depend on magnitude, and after they saturate, then values of Mn-Ni exchange parameters do not have strong dependence on number of k-points. So we can conclude that calculation should be carried out with values not small than 4 000 k-points.

In the fig. 8 we present dependences of exchange coupling parameters on lattice constant in cubic phase of Ni218Mn0 82Ga alloy. From fig. 8 we can observe that the Mn-Mn exchange parameters in the first coordination sphere increased until

a = 5,81 A where it has maximum and after they are decreased. Exchange parameters for the second sphere are decreased with increasing of the lattice constant. Exchange parameters for Nia-Mn are decreased and the J tj for Nib-Mn are increased with increasing of the lattice constant.

In the fig. 9 we present dependences of the sum of all exchange coupling parameters on the Ni excess of Ni2+xMn1-xGa alloys for cubic and tetragonal phases. From fig. 9 we can observe that the exchange parameters decreased with increasing of Ni excess x. For cubic phase the Mn-Mn and Mn-Nia

Fig. 7. Dependences of exchange coupling parameter between Mn (a, b) and Ni (c, d) atoms on k-points in Brillouin zone in cubic phase of Ni21SMng S2Ga alloy

Fig. 8. Dependences of exchange coupling parameters on lattice constant in the cubic phase

of Ni218Mn0 82Ga alloy

exchange parameters are crossed approximately for Ni2 09Mn091Ga composition. Our calculations show that values of exchange integrals in tetragonal phase are larger then the same interactions in cubic phase. Considered the Mn-Nia interaction in tetragonal phase we can see that this interaction is predominate in the composition range 0 < x < 0,39. The same behavior of the sum of all exchange coupling parameters is observed in [18].

In the fig. 10 we present dependence of Curie temperature TC which is calculated using mean-field approximation with the help of method from [19]. Experimental results have taken from [5]. For the comparison with mean-field method we have shown also our theoretical Curie temperatures obtained from Monte Carlo simulations using Heisenberg model with ab initio magnetic exchange couplings [20]. From figure we can observe that mean-field

36

33

a 30

S-

27

24

1 1 1 1 1 1 1 1 - Ni Mn Ga {da =1,0) \ 2+x l-x v 5 J 1 a) ' 50 45 . i 1 i _ • 1 l 1 l 1 l -• • • • . # ^ b)-

| > 40 - Ni^Mn^Ga {c/a = 1,25) -

- V 0> a 35 ; ;

”'»x _ • Mn-Mn S- 30 -

Mn-Ni • ■ ° 25 - Mn-Mn - Mn-Ni

I , I , I , I , i 20 a , i.i.i

0,0

0,1

0,2

Ni excess (x)

0,3

0,4

0,0

0,1

0,2

Ni excess (x)

0,3

0,4

Fig. 9. Dependences of effective exchange coupling parameters of Ni2+xMnlxGa alloys for cubic (a)

and tetragonal (b) phases from Ni excess

Fig. 10. Dependences of Curie temperature (TC) of Ni2+xMnl x Ga alloys calculated using mean-field approximation (triangles), Monte Carlo method (circles) from [19] and experimental data from [5]

approximation gives us overestimated values of the Curie temperature in compositions with x < 0,2 . Whereas for compositions with x > 0,2 in the tetragonal phase the values of the Curie temperature obtained from mean-field approximation are in a good agreement with experimental results. Considered the Monte Carlo results we can suppose that this method is more accuracy than the mean-field approximation.

4. Conclusion. In this work we have investigated electronic and magnetic properties of Ni2+xMn1-xGa Heusler alloys with the help of ab initio calculations. Calculations have been performed for different compositions x. The results for full magnetic moment show that it is decreased with increasing of Ni excess. This dependence is in a good agreement with experimental data for Ni-Mn-Ga alloys in wide range of composition [5]. DOS calculations shows that spin polarization at Fermi level increased with increasing of Ni excess. The exchange

parameters between the Mn atoms in cubic phase in the first coordination sphere changed sign from ferromagnetic to antiferromagnetic with increasing of the Ni excess. The exchange parameters between the Mn atoms in tetragonal phase are the antiferromagnetic. The coupling between the Mn and Ni atoms in the Mn site is also antiferromagnetic. Other couplings are ferromagnetic. Calculations show that values of exchange integrals are depended on number of k-points mesh in Brillouin zone until 4000. Also our calculations show that exchange coupling parameters are depended on the lattice constant. Calculated Curie temperature by mean field approximation is in a good agreement with experimental data [5] and with existed theoretical results [18; 20].

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