Keywords: shape – memory Heusler alloys, structural and magnetic phase transitions.
Abstract. On the basis of Monte Carlo simulations, we investigate the temperature dependence of magnetization of Ni-Mn-Sb Heusler alloys, in which the part of the Mn atoms interact antiferromagnetically. It is shown that this antiferromagnetic exchange is responsible for existence of several magnetic phase transitions in the Heusler alloys. For a certain set of parameters of the model Hamiltonian we obtain coupled martensitic and magnetic phase transitions. Results of simulations agree in an excellent way with the experimental magnetization curves.
Introduction
In recent years, Ni-Mn-X (X =In, Sn, Sb) alloys has been attracting attention in view of their unique properties, such as the shape memory effect, the magnetocaloric effect, the magnetoresistence and other properties are associated with the martensitic transition (MT) [1, 2]. These properties are applicable in developing actuator materials and materials for magnetic refrigeration.
Recent experiments have shown that in Heusler Ni-Mn-X (X = In, Sn, Sb) alloys, the structural phase transition from the paramagnetic (PM) high-temperature cubic (austenitic) phase to the ferromagnetic (FM) low-temperature tetragonal (martensitic) phase can occur [3]. It is well known, that in stoichiometric Ni50Mn25X25 alloys all Mn atoms on its regular sublattice site interact FM. However, recent experiments on non-stoichiometric Ni50Mn25+xX25-x alloys revealed that in the martensite, the excess of Mn atoms occupying the sites of X atoms, interact with Mn atoms on the Mn sublattice sites antiferromagnetically (AF) [4]. Furthermore, the AF interaction is enhanced due to the MT, which is also proved by the drop in thermomagnetization curves [4, 5]. Thus, in the martensitic phase of Ni50Mn25+xX25-x alloys the FM and the AFM interactions coexist. Therefore, in these compounds the phase transition from the mixed AFM – FM martensite to the FM austenite and from FM austenite to PM austenite with increasing temperature is experimentally observed [3 - 5].
From a theoretical point of view, it is a demanding task to try to obtain all phases in self-consistent way. The ab initio calculations for the austenitic and martensitic phase and the resulting magnetic exchange constants of Ni2MnGa, Ni2MnIn and non- stoichiometric Ni50Mn34In16 and Ni50Mn36.25In13.75, using KKR-CPA method [6] clearly show the competition of FM and AF interactions. For example, for Ni50Mn34In16 the AF exchange constant is -6.0 meV, for Ni-Mn-Sn the values of magnetic exchange constants (5.0 meV for the FM interaction and -5.0 meV for the AF interaction) for nearest neighbors, which was obtained from ab initio calculations [7]. Performing Monte Carlo simulations for the underlying Heisenberg model using the ab initio magnetic constants [6] shows however that it is difficult to obtain the sequence of structural and magnetic 1st-order-like phase transitions as observed in experiment [7].
A more refined Heisenberg-like model, which has been used for the description of magnetocaloric properties of rare-earth based and Heusler compounds, was introduced in Ref. [8, 9]. In addition to the model, which allows only a magnetic transition of 2nd-order without taking into account the MT, authors of [10] used a model for the description of the pre-martensitic phase transformation in Ni2MnGa. For the description of the magnetic and structural part, the authors used the Ising model and the degenerated Blume-Emery-Griffiths (DBEG) model [10], respectively.
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Another question is whether the physical properties of Ni-Mn-X would require considering also the softening of phonons associated with pre-martensitic or MT observed in some of the alloys close to stoichiometric composition [11]. In the most pronounced cases, the phonon softening involves energy changes of the order of 1-2 meV. In DBEG model [10], authors considered that the structural part the exchange constant is proportional to the soft phonon energy at the MT. In our Monte Carlo simulations of magnetic properties of Ni-Mn-Sb alloys using an extended BEG model, which allows for coupled magnetostructural phase transitions from the mixed AF – FM martensite to the FM austenite and the magnetic phase transition from FM austenite to PM austenite.
The Theoretical Model
In presence of the MT, the martensitic phase may exhibit several low-temperature variants. These variants can be described by lattice distortions (compression or expansion) during the MT along the x, y, z axes, which for the cubic phase yields six structural variants. The austenitic phase corresponds to a phase with zero displacement, i.e. x = y = z = 0. Thus, during cooling, the austenite may choose any of the six variants, and can be thought of as having a high degeneracy. In our model we consider two variants of martensite with the lattice deformation along ±x or ±y or ±z axes.
In the model we use a cubic lattice with periodic boundary conditions. Since the magnetic moments of the Ni atoms are small compared to the Mn moments and since the X atoms not have spin moment, we omit the contributions from the Ni and X atoms. Thus, in our model all lattice sites are occupied by Mn atoms. For Ni2Mn1+xX1-x alloys model we assume that a certain part of the Mn atoms on the lattice interact AF with each other and others Mn atoms. The initial configuration of these Mn atoms is randomly chosen and was determined from experimental compositions.
The system can be representing as two interacting magnetic and structural parts. For the magnetic part, we use the q-state Potts model which allows simulating the 1st order magnetic phase transition from FM to PM phases [12]. Here q is number of spin states. In our case we consider q=5 spin states because in Ni-Mn-X the spin state of Mn corresponds to S =4/2 with 2S+1 possible spin projections. Hence, we have q = (1 ,.., 5), a five-states Potts model. The structural part is described by the DBEG model which allows a structural transition from the cubic phase to the tetragonal one [10].
The generalized Hamiltonian (Eq. 1) include of three parts: magnetic part (Eq. 2), elastic part (Eq. 3) and the magnetoelastic interaction (Eq. 4).
* * * *intm elH H H H= + + , (1)
* ** * * *
, , ,*, ,
i j i j i g
�� �� �afm
m fm S S afm S S S Si j i j iafm
T TH J J h
T< > < >
−= − δ − δ − δ∑ ∑ ∑ , (2)
* * 2 2 * 2
, ,
(1 )(1 ) ln( )(1 )�� ��
el i j i j ii j i j
H K T p< > < >
= − σ σ − − σ − σ − − σ∑ ∑ , (3)
* * 2 2 *int , ,
, ,
1 1 12 ( )( )
2 2 2i j i j
�� ��
S S i j S Si j i j
H U U< > < >
= δ − σ − σ − δ∑ ∑ , (4)
* * * * * * */ , / , / , / , / , / , /fm fm B afm B afm B extH H J J J J K K J U U J T k T J T k T J h g H Jµ= = = = = = = . (5)
Here, Jfm and Jafm are the FM and AF exchange constants; J, K are exchange constants in the structural subsystem; T is the actual temperature; Tafm is the temperature at which the AF interaction change its sign, see second term in (Eq. 2); δSi,Sj is the Kronecker symbol which limits the spin-spin interaction to interactions between the same q-states only; Sg is kind of a ghost-spin, whose
140 Smart Materials for Smart Devices and Structures
direction is that of the external magnetic field, a spin parallel to the ghost spin is favored by positive Hext; g is the Lande factor; µB is the Bohr’s magneton; kB is the Boltzmann constant; U is the constant of magnetoelastic interaction and p is the degeneracy factor. The variable σi = 1, 0, -1 represents the deformation state on each sites on the lattice, where σi = 0 denotes the undistorted phase and states σi = ±1 the distorted phases. Sums are performed over all nearest-neighbor pairs.
The AF interaction term in (Eq. 2) limits the AF interactions to the low-temperature phase. For T >Tafm this interaction term will change sign resulting in additional FM interactions. Accordingly, above this temperature, enhance FM between all Mn atoms occur. Since, this means that there will be no transition from FM austenite to PM austenite, we allow the the maximal value of the factor (T*-T*
afm)/T*afm to be unity, i.e. for temperatures T*>2Tafm
* the prefactor of the second term (Eq. 2) is assumed not depend on temperature any more and is set equal to –Jafm
*. By the help of this an admission we can describe several magnetic transitions from the PM phase to different magneto-ordered phases with decreasing temperature like the FM, AF phases or mixed FM-AF phases.
In the elastic part (Eq. 3), the first (second) term characterizes the interaction between single strains σi in the martensite (austenite), respectively. Large values of K* will stabilize the pure cubic phase in which σi = 0 holds. Moreover, the parameter K* determine and control the order of the transition. In case of low values of K*, the cubic phase will be stabilized in the high temperature region where we find an equal population of the strain variables σi = 0 and σi = ±1, this case corresponds to an average cubic phase [10]. The last term in (Eq. 3) characterizes a temperature-dependent degeneracy factor p for the cubic phase [10].
In the q-state Potts model the magnetization of the system is presented by (Eq. 6).
3max1
1q� L
m� q
−=
−, (6)
where �max is the maximal number of same magnetic states in sites of the lattice. As in DBEG model [10] the strain order parameter is calculated by (Eq. 7).
∑ σ=ε�
ii�
1, (7)
For ε = 0 in the DBEG model we have two cubic states (pure cubic or average cubic) depending on the values of strain variables σi. For ε = 1, we find the marteniste for one of variants σi=1 or σi= –1.
�umerical Results
Following we discuss numerical results for Ni50Mn25+xSb25-x alloys obtained by the Monte Carlo method. The corresponding simulations have been carried out using a standard Metropolis algorithm [13]. The changes in the spin states q and σi are treated independently and are accepted or rejected according to the single-site transition probability W=min{1, e-∆H*/T*}. For simulations of the cubic lattice, we used periodic border conditions and 6 near-neighbors for each site. The number of sites in the lattice was equal to �=153. The time unit is one Monte Carlo step, which consists of � attempts to change the q and σi variables. For given the temperature the number of Monte Carlo steps on each site is 105. We start the Monte Carlo simulations from the FM martensite with q=1 and σi =1. The various quantities are averaged over 400 configurations taken every 100 Monte Carlo steps and discarding the first 104 Monte Carlo steps for equilibrium. The degeneracy factor and the Lande factor were taken as p=2, g=2. In order to find the spin state at each lattice site, we choose a random number 0 < r < 1 and fix the value of q according to the scheme: if 0 ≤ r ≤ 1/5, then q = 1 if 1/5 < r ≤ 2/5, then q = 2; if 2/5 < r ≤ 3/5, then q = 3; if 3/5 < r ≤ 4/5, then q = 4; if 4/5 < r ≤ 5/5, then q = 5.
In the Monte Carlo simulations we adopted the parameter values listed in Table I in order to describe the magnetic and structural properties of the Ni-Mn-X alloys in an optimal way.
Table 1. Model parameters for Ni50Mn25+xSb25-x alloys. Jfm
The information about the value of the structural exchange interaction J was taken from
experimental data of the phonon dispersion curves of the Ni-Mn-Ga. As it known from [11] that the exchange constant J is proportional to the energy of the soft phonon ħω, and for Ni-Mn-Ga value of J at the MT is equal 2 meV approximately. Values of our FM and AF exchange constants Jfm and Jafm (see Table I) nearly coincide with values of the magnetic constants for Ni-Mn-X obtained from ab initio calculations [7]. In order to determine values of parameters K, which characterizes the MT, and values of U, we used following conditions: U<0 and K<0 [10]. It is necessary to note that for values of K>0 by the help of the BEG model it is possible to obtain the martensitic, pre-martensitic and magnetic phases transitions at temperatures Tm<TI<TC [10]. Since the pre-martensitic transition in Ni-Mn-X has not been observed experimentally, so we consider that values of K < 0. In our model for determining the distance between Tm and TC we used different values of U, K and Tafm.
The results of the simulations are presented in Figs. 1-3. Fig. 1 and 2 show the temperatures dependences of the magnetizations and strain order parameters (tetragonal distortion ε) for Ni50Mn25+xSb25-x alloys. Here, the filled circles and solid lines are results of our Monte Carlo simulations and the open circles are experimental results which have been taken from [4].
Fig.1a. Temperature dependences of the magnetization of Ni50Mn32Sb18 alloy in a magnetic field 0.01T.
Fig.1b. Thermomagnetization curves and tetragonal distortion of Ni50Mn37.5Sb12.5 alloy is absent of a magnetic field.
Fig. 1a shows the temperature dependences of the magnetizations for the Ni50Mn32Sb18 in a
magnetic field 0.01 T. For this composition the structural transformation from the cubic state to the tetragonal one is absences and the Ni50Mn32Sb18 alloy has the austenitic phase in whole a temperature region [4, 5]. Therefore, in our calculations we have taken value of parameter K, which determines the temperature of the structural transition, as equal 0 (Table 1). So, in Fig. 1a, we show the magnetic transition from the FM austenite to PM austenite at 360 K.
Fig. 1b represented the magnetization and the strain as function of a temperature for Ni50Mn37.5Sb12.5 alloy is absent of a magnetic field. The two magnetic phase transitions are now at 350 K and at 280 K, respectively. We obtain from the simulations a PM-FM transition in cubic state
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at 350 K. The second transition at 280 K describes the FM to mixed FM-AF phase transition which is again accompanied by a tetragonal transformation.
Fig.2a. Temperature dependences of the magnetization and tetragonal distortion of Ni50Mn38Sb12 alloy is absent of a magnetic field.
Fig.2b. Thermomagnetization curves and tetragonal distortion of Ni50Mn38.5Sb11.5 alloy is absent of a magnetic field.
In Fig. 2a, we show theoretical and experimental results for Ni50Mn38Sb12 alloy is absent of a magnetic field. We observe here two phase transitions, at 350 K and 300 K. At 350 K, we find the PM-FM transition in the austenite. The second transition is the coupled magnetostructural transition from the FM austenitic state to the mixed FM-AF martensitic state.
Figure 2b shows thermomagnetization curves and tetragonal distortion of Ni50Mn38.5Sb11.5 alloy is absent of a magnetic field. As seen from figure, we may observe four phase transitions, at 360 K, at 340, at 320 K and at 270 K, respectively. The first transition at 360 K is the structural transition from the PM austenite to PM martensite; the second transition is the magnetic transition from the PM martensite to FM martensite; the third transition from FM martensite to PM or AF martensite, and the fourth transition from the PM or AF martensite to the mixed FM-AF martensitic state.
In order to test the predictability of our model, we have tried to describe the change of measured magnetization curves with composition for one system, only by allowing K to vary in order to obtain the experimental values for the structural transition, but leaving the other parameters unchanged. Values of model parameters: Jfm=5.1 meV, Jafm=4.5 meV, U=-1.8 meV, Tafm=1.34 meV. The parameter K was taken to be composition dependent: x=12, K=-0.6 meV; x=13.5, K=-3 meV; x=14, K=-6.2 meV; x=15, K=-31 meV; x=18, K=-150 meV. The resulting composition dependent theoretical and experimental magnetization curves are shown in Fig.3a and 3b. A comparison of theoretical and experimental results shows that the Hamiltonian (Eq. 1) allows to reproduces qualitatively the experimental trends [5].
Fig.3a. Theoretical variation of the magnetization as function of temperature of Ni50Mn25+xSb25-x for different compositions in a magnetic field 0.1 T as obtained from the Monte Carlo simulations.
Fig.3b. Experimental variation of the magnetization as function of temperature of Ni50Mn25+xSb25-x for different compositions in a magnetic field 0.1 T, data from [5].
In this work we have investigated the temperature dependences of the magnetization and strain of Heusler Ni50Mn25+xSb25-x alloys using Monte Carlo simulations. Part of Mn atoms on simple lattice are assumed to interact only AF, whereby the concentration of these Mn atoms is determined by the experimental compositions. In addition to the magnetic interactions, the model Hamiltonian contains structural degrees of freedom allowing for a tetragonal deformation. The magnetic subsystem is described by q-state Potts model. For the structural subsystem we have used the DBEG model. The resulting model allows for a coupled 1st order magnetostructural phase transition from FM martensite to PM austenite. The AF interaction term in the model leads to several magnetic phase transitions from high temperature PM phase to different magnetoordered phases with decreasing of the temperature like the FM phase, the AF phase or the mixed FM - AF phase. The comparison of results of simulations and experiment is in some case satisfying.
Work was supported by grants RF and CRDF Y2-P-05-19, RFBR 05-08-50341, 06-02-16266, 07-02-96029-r-ural, 06-02-39030-NNSF, 05-02-19935-YaF_a RFBR and JSPS.
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Smart Materials for Smart Devices and Structures doi:10.4028/www.scientific.net/SSP.154 Monte Carlo Study of Magnetostructural Phase Transitions inNi<sub>50</sub>Mn<sub>25+x</sub>Sb<sub>25-x</sub> Heusler Alloys doi:10.4028/www.scientific.net/SSP.154.139
