Theoretical Modeling of Magnetocaloric Effect in Heusler Ni-Mn-In Alloy by Monte Carlo Study

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1 Materials Science Foru Online: ISSN: , Vol. 635, pp doi: / Trans Tech Publications, Switzerland Theoretical Modeling of Magnetocaloric Effect in Heusler Ni-Mn-In Alloy by Monte Carlo Study V.D. Buchelnikov 1,a, V.V. Sokolovskiy 1,b, S.V. Taskaev 1,c and P. Entel 2,d 1 Chelyabinsk State University, Chelyabinsk, Russia 2 University of Duisburg-Essen, Duisburg, Gerany a buche@csu.ru, b vsokolovsky84@ail.ru, c tsv@csu.ru Keywords: shape eory Heusler alloys, structural and agnetic phase transitions, agnetocaloric effect. Abstract. In this paper we present a theoretical odel for calculation of the (positive and negative) agnetocaloric effects and agnetic properties of the Heusler Ni 50 Mn 34 In 16 alloy by the classical Monte Carlo study. By the help of the proposed odel the teperature dependences of the agnetization, tetragonal deforation, heat capacity, positive and negative isotheral agnetic entropy changes for agnetic field variation are obtained. All quantities are in good qualitative agreeent with the available experiental data. Introduction The agnetic cooling technology is based on the ability of any agnetic aterial to change its teperature and entropy under the influence of a agnetic field [1]. Such effect is called the agnetocaloric effect (MCE). Recent experiental researches have shown a presence of the large MCE in Gd 5 (Si 2 Ge 2 ), MnFeP 1-x As x and Ni 2+x Mn 1-x Ga alloys [1]. The reason of large MCE in these alloys is the coupled agnetostructural (MS) transition. In these alloys an applied agnetic field causes the alignent of agnetic oents under adiabatic conditions. Due to this alignent the agnetic part of the total entropy is reduced. In order to copensate this reduction, other coponents of the total entropy (electronic and lattice parts) are increased. It leads to heating of alloys (the positive MCE). However, there are certain aterials where the applied agnetic field leads to further spin disorder, causing an increase in the agnetic part and a decrease in the electronic and lattice parts of the total entropy. In that case the copounds are cooling (the negative MCE). In these aterials the 1 st order agnetic transition fro antiferroagnetic (AF) state to ferroagnetic (FM) one takes place (e.g. Fe 0.49 Rh 0.51 [2]). Recent experiental researches have shown the possibility of using of Heusler Ni-Mn-X (X = In, Sn, Sb) alloys as refrigerants in the technology of agnetic cooling. [3]. In these copounds two types of large MCE (positive and negative) are observed experientally. The reason of the positive MCE in these alloys is following. Recent ab initio siulations of Ni 50 Mn 25+x X 25-x for a artensite and austenite phases clearly have shown the copetition of AF and FM interactions in both states. The AF interaction is the interaction between of the excess of Mn 2 atos occupying the X sublattice sites and Mn 1 atos on the Mn sublattice sites i.e. Mn 1 -Mn 2 and Mn 2 -Mn 2 interactions. At the sae tie both austenitic and artensitic phases also have the following ferroagnetic interactions: Mn 1 - Mn 1, Mn 1 Ni and Mn 2 Ni [4]. This fact leads to coplex phase transitions with decreasing teperature such as the cubic paraagnetic cubic FM transition near Curie teperature T C and cubic FM ixed tetragonal AF-FM transition near the structural transition teperature T. The negative MCE occurs at coupled MS transition fro the ixed AF-FM artensite to the FM austenite and the positive MCE takes place at agnetic phase transition fro the FM austenite to the PM austenite after application of the agnetic field [5]. In this work we theoretically investigate the (positive and negative) MCE of the Heusler Ni 50 Mn 34 In 16 alloy by the Monte Carlo siulations. This is an open access article under the CC-BY 4.0 license (

2 138 Ferroagnetic Shape Meory Alloys II Theoretical Model In our odel we use a three-diensional lattice with periodic boundary conditions and with real unit cells of Heusler Ni-Mn-X alloys. The first unit cell ay be considered as four interpenetrating fcc sublattices with the ato of Mn at site (1/2, 1/2, 1/2), the ato of Ga at site (0, 0, 0) and atos of Ni at sites (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4), respectively (Fig. 1a). This unit cell corresponds to the high-teperature parent cubic austenite in which the lattice distortions (copression or expansion) are absent along the x, y and z axes. During cooling the austenite transfors to the lowteperature tetragonal artensite with tetragonal unit cell (Fig. 1b). The artensitic phase ay exhibit several low-teperature variants and in our odel we consider two variants of artensite with the lattice deforation along ±x axes. So in the case of the austenite we consider all interactions between nearest-neighbor atos within cubic unit cell (Fig. 1a) and in the case of the artensite we propose interactions between nearest atos within tetragonal unit cell (Fig. 1b). Fig. 1: (Color online) Left panel: The cubic L2 1 structure of the Heusler Ni 2 MnIn alloy. Right panel: The tetragonal bct (c/a=0.94) structure of Ni 2 MnIn. Here blue, red and black sybols are Mn, In and Ni atos, respectively. Black bold solid line indicates the tetragonal unit cell. In the proposed odel for foration of Ni 50 Mn 34 In 16 alloy the excess of the Mn 2 atos is taken as corresponding to noinal copositions whereas configuration of the Mn 2 atos in the In sublattice is set randoly. Crystallographic sites of the lattice occupied by Mn 1, Mn 2 and Ni atos are ascribed with agnetic and structural degree of freedo whereas ones occupied by In atos having only structural degree of freedo. The agnetic subsyste is describes by a ixed q-state Potts odel for FM-PM phase transition [4]. Here q is the nuber of spin states. Since the Ni-Mn interaction in Ni-Mn-X alloy plays iportant role in the aking of ferroagnetis we should take into account spin agnetic oents S of the Mn and Ni atos. The spin agnetic oents S of the Mn and Ni atos are different and for Mn atos S is 4/2 and therefore 5 spin projections are possible and hence q Mn =5, opposite the Ni atos have S=1 with following 3 spin projections and q i =3. Therefore in our odel we consider the odel with three - five spin states Potts odel. The structural subsyste is described by a degenerated three state Blue-Eery-Griffiths (BEG) odel for structural transforations fro the austenite to the artensite [4]. The generalized Hailtonian (Eq. 1) consists of three parts: agnetic part (Eq. 2), elastic part (Eq. 3) and the agnetoelastic interaction (Eq. 4) [4]. H = H + Hel + Hint, (1) = i, jδsi, S µ j B ext δsi, Sg < i, j> i, (2) H J g H el = ( + 1 µ B ext σgσ j ) σiσ j (1 σi )(1 σ j ) B ln( )(1 σi ) i < i, j> < i, j>, (3) H J U g H K k T p

3 Materials Science Foru Vol H U U 2 2 int = 2 i, jδs, ( )( ) i S σ j i σ j i, jδsi, S j < i, j> < i, j>, (4) Here J i,j is the exchange constant of the agnetic subsyste, J and K are the exchange constants of the structural subsyste, U i,j and U 1 are the agnetoelastic interaction constants, T is the teperature, H ext is the external agnetic field, δ Si,Sj is the Kronecker sybol which restricts spinspin interactions to the interactions between the sae q states, S i is a spin defined on the lattice site i=1,..,, S g is a ghost spin, whose direction is deterined by the external agnetic field (positive H ext favors spins parallel to the ghost spin), k B is the Boltzann constant, µ B is the Bohr s agneton, g is the Lande factor, p is the degeneracy factor, σ i = 1, 0, -1 represents the deforation state of each site of the lattice (σ i = 0 corresponds to the undistorted state whereas σ i = ±1 represents distorted states), σ g is a ghost deforation state, whose value is that of a structural variant in the external agnetic field (positive H ext favors deforation states coinciding with the ghost deforation state). Suing up is taken over all nearest neighbor pairs. In the elastic part of the Hailtonian (Eq. 3) the first ter describes the interaction between single strains σ i in the tetragonal (artensitic) state. The second ter shows the favorable orientation dependence of the artensitic variant in the external agnetic field. The third ter defines the interaction between single strains σ i in the cubic (austenitic) phase. The last ter characterizes a teperature-dependent crystal field [4]. In the proposed odel the teperature dependencies of a agnetization and a strain order paraeter (Eq. 5), a specific heat and entropy of a syste (Eq. 6) are presented by: i Mn 1 q q = + q i 1 q Mn 1 i ax i Mn ax Mn 1, ε = σ i i, (5) 2 2 < H > < H > C( T, H ext ) =, 2 k T B T2 C( T, Hext ) S( T, H ext ) = dt. (6) T T1 Where is the total nuber of Ni and Mn atos, q i and q Mn are the nubers of agnetic states of Ni and Mn atos, i ax and Mn ax are the axial nubers of identical agnetic states on the lattice, i and Mn are the nubers of Ni and Mn atos on the lattice, respectively. For ε = 0 in the DBEG odel we have the cubic state. In the case of ε = 1 the artensitic state for one of variants with σ i =1 or σ i = 1 takes place. uerical Results In this section we present the nuerical results of our odel for description of the MCE of the Ni 50 Mn 34 In 16 alloy using Monte Carlo siulation techniques [4]. The siulation was carried out using following Metropolis algorith: (1) Generate the initial spin configuration (the ferroagnetically ordered state) and the initial strain configuration (the tetragonal state, one of the artensitic variants). (2) Choice the equilibriu strain configuration on the lattice with tetragonal or cubic unit cell. (2.1) Randoly select a particular site i of the lattice: if σ i =1 or σ i =-1 then calculate the initial elastic energy H 1el (Eq. 3) on the tetragonal unit cell; if σ i =0 then calculate the initial elastic energy H 1el (Eq. 3) on the cubic unit cell. (2.2) Randoly change the values of the strain σ i on this particular site i and calculate the energy for this new configuration H 2el : if σ i =1 or σ i =-1 then calculate the initial elastic energy H 2el (Eq. 3) on the tetragonal unit cell; if σ i =0 then calculate the initial elastic energy H 2el (Eq. 3) on the cubic unit cell. (2.3) If H 2el < H 1el, accept the new configuration with energy H 2el and go to step (3). (2.4) If H 2el > H 1el, calculate the probability factor exp(- H el /k B T): generate a rando nuber r such that 0 < r <1; if r < exp(- H el /k B T), accept the

4 140 Ferroagnetic Shape Meory Alloys II new configuration with energy H 2el, else preserve the old configuration of strain and return to step (3). (3) Choice the equilibriu spin configuration on the lattice with selected unit cell. (3.1) Calculate of the full energy with the selected values of the strain on the particular site i: if σ i =1 or σ i =-1 then calculate the initial full energy H 1 (Eq. 1) on tetragonal unit cell; if σ i =0 then calculate the initial full energy H 1 (Eq. 1) on the cubic unit cell. (3.2) Randoly change the values of the spin state q on this particular site i and calculate the energy for this new configuration H 2 : if σ i =1 or σ i =- 1 then calculate the initial full energy H 2 (Eq. 1) on tetragonal unit cell; if σ i =0 then calculate the initial full energy H 2 (Eq. 1) on the cubic unit cell. (3.3) If H 2 < H 1, accept the new configuration with energy H 2 and go to step (4). (3.4) If H 2 > H 1, calculate the probability factor exp(- H/k B T): generate a rando nuber r such that 0 < r <1; if r < exp(- H/k B T), accept the new configuration with energy H 2, else preserve the old configuration of spin and return to step (2.1). (4) Move the next site of the lattice (2.1). (5) Repeat the entire process until all the lattice sites are swept. Since we have used real lattice, the coordination nuber of nearest-neighbor atos has taken various values for each ato of the cubic and tetragonal unit cells. For the case of agnetic subsyste in the artensitic state each Mn ato has 8 nearest-neighbor Mn 1, 2 Mn 2, and 8 Ni atos, each Ni ato has 4 nearest-neighbor Mn 1 and Mn 2 atos; in the austenitic state each Mn ato has 12 Mn 1, 6 Mn 2, 8 Ni atos and each Ni ato has 4 nearest-neighbor Mn 1 and Mn 2 atos. For the case of the structural subsyste in the artensitic state each ato Mn 1 (In or Mn 2 ) has 8 Mn 1 (In or Mn 2 ) atos, 2 In or Mn 2 (Mn 1 ) atos and 8 Ni atos, respectively, and each Ni ato has 4 Mn 1, In or Mn 2 atos and 6 nearest Ni atos; in the austenitic state each ato Mn 1 (In or Mn 2 ) has 12 nearest Mn 1 (In or Mn 2 ) atos, 6 nearest In or Mn 2 (Mn 1 ) atos and 8 Ni atos, respectively, and each Ni ato has 4 Mn 1, In or Mn 2 atos and 6 nearest Ni atos. In our siulations we have used the lattice in which 1098 Mn 1, 396 Mn 2, 1728 Ni and 703 In atos, respectively. As the tie unit, we used one Monte Carlo step consisting of attepts to change q i, q Mn and σ i variables. For a given teperature, nuber of the Monte Carlo steps on each site was taken The siulation started fro the ferroagnetic artensitic phase. The internal energy of the syste H and the order paraeters and ε were averaged over 400 configurations for each 100 Monte Carlo steps. In order to obtain equilibriu values of H, and ε, the first 10 4 Monte Carlo steps were discarded. The degeneracy factor p and the Lande factor g were taken as p = 2 and g = 2. The value of diensionless agnetoelastic interaction U 1 =-1.5 has been chosen that the agnetic and structural transitions are coinciding in an external agnetic field. The agnitude of spin states (i.e. the q i and q Mn variable) were taken as corresponding to a rando nuber r such that 0< r < 1 and fix the values of q i and q Mn according to the schee: if 0 r l/3 then q i = l, l = 1 3 and 0 r l/5 then q Mn = l, l = 1 5. For the calculation of MCE in the Ni 50 Mn 34 In 16 alloy, the following values of constants were used (Table 1). Table 1: Model paraeters in [ev] for Ni 50 Mn 34 In 16 alloy. J J U K J J Mn Mn Mn1 Mn J 2 Mn2 Mn J 2 Mn1 i Mn2 i 1 1 Martensite Austenite The values of agnetic austenitic and artensitic exchange constants for Ni 50 Mn 34 In 16 alloy have been taken fro results of ab initio siulations [4]. The value of the structural exchange interaction J has been obtained fro Monte Carlo siulations taking into account the experiental teperature of the MS transition T s and reduced odeled teperature of the MS transition T * s fro the following relation J=k B T s /T * s. This value of J is equal approxiately with experiental exchange constant J ~ 2 ev for Ni-Mn-Ga, which obtained fro experiental data of the phonon dispersion curves of the Ni-Mn-Ga [6].

5 Materials Science Foru Vol Results of our Monte Carlo siulations of the agnetic and agnetocaloric properties of Ni 50 Mn 34 In 16 alloy are presented in Figures 2 4. In Figure 2 we present theoretical and experiental results of teperature dependencies of the agnetization and strain order paraeter for Ni 50 Mn 34 In 16 alloy in agnetic fields of 0 and 5 T. Therodynaic teperature T will be able to calculate fro the expression for the reduced teperature T * =k B T/J using value of J = 3.06 ev (Table 1). We observe here two phase transitions at 305 K (T * C~8.5) and at 215 K (T * s~6.05), respectively. At T * C~8.5 we find the PM-FM transition in cubic (austenitic) state. The second transition is the agnetostructural transition fro the FM cubic state to the ixed AF-FM tetragonal phase. The behavior of the strain order paraeter ε shows the onset of the structural phase deforation at 215 K (T * s~6.05). The experiental teperature dependence of agnetization was taken fro Ref. [5]. Fig. 2: (Color online) Left panel: The theoretical teperature dependencies of agnetizations curves and strain deforations of Ni 50 Mn 34 In 16 alloy in agnetic fields of 0 and 5 T. Here, filled circles and the line are results obtained for zero agnetic field and filled triangle sybols, dash line are siulations in agnetic field of 5 T. Right panel: The experiental agnetization curve of Ni 50 Mn 34 In 16 alloy in agnetic field of 5 T. Fig. 3: (Color online): Theoretical teperature dependencies of the specific heat of Ni 50 Mn 34 In 16 alloy in agnetic fields of 0 and 5 T. As can we see fro Fig. 3 there are two picks (at T * C~8.5 and T * s~6.05) on teperature dependencies of the specific heat, which correspond to the PM-FM transition in cubic state and the MS transition fro the FM cubic state to the ixed AF-FM tetragonal phase. The external agnetic field shifts the MS transition teperature T * s in a low-teperature region and the Curie teperature in a high-teperature region T * C, respectively. Fig. 4 presents theoretical and experiental isotheral agnetic entropy changes in Ni 50 Mn 34 In 16 alloy upon variation of the agnetic field fro 0 to 5 T. Here we find the positive MCE at the FM-PM transition teperature (near the roo teperature) and the negative MCE at

6 142 Ferroagnetic Shape Meory Alloys II the coupled MS phase transition teperature. For decreasing of a discrepancy between the values of the theoretical and experiental positive MCE it should be to carry out ore accurate siulation (e.g. an increase in the nuber of the Monte Carlo steps but this increase leads to a rise of the coputer coputation tie). The experiental result for S ag has been obtained fro isotheral agnetization easureents with the help of the Maxwell relation and was taken fro Ref. [5]. Fig. 4: (Color online): Theoretical (left panel) and experiental (right panel) S ag in Ni 50 Mn 34 In 16 alloy at agnetic field of 5 T. Suary In this work the agnetic properties and the positive and negative MCE of Ni 50 Mn 34 In 16 alloy upon various variation of the agnetic field fro 0 to 5 T have been studied by the Monte Carlo siulations. It is shown that the results of the calculations are in good qualitative agreeent with available experiental data. The agnetic subsyste is described by the q-state Potts odel. For the structural subsyste, we have used the Blue-Eery-Griffiths odel. Using the ab initio copetitive ferro-antiferroagnetic exchange constants and the change-over the agnetic and structural interactions fro the tetragonal to the cubic real unit cell of Heusler alloys have allowed us to obtain the coplex phase transitions with decreasing teperature such as the austenite PM - FM and austenite FM ixed artensite AF-FM and to calculate the MCE at both phase transition teperatures. It is significant that the Heisenberg Hailtonian with using the ab initio exchange integrals and real unit cell of Ni 50 Mn 34 In 16 alloy do not reproduce that coplex trend of phase transitions [4]. In spite of that we can deterine exactly the Curie teperature of Heusler alloys by the help both Potts odel and Heisenberg one using the ab initio integrals and real unit cell. Acknowledgeents This work was supported by RFBR grants NNSF, r-ural, OFI_ts, and References [1] K.A. Gschneidner Jr. et al: Rep. Prog. Phys. Vol. 68 (2005), p [2] M.P. Annaorazov et al.: J. Appl. Phys. Vol. 79 (3) (1996), p [3] T. Krenke et al.: Nature Mater. Vol. 4 (2005), p [4] V.D. Buchelnikov et al.: Phys. Rev. B Vol. 78 (2008), p [5] S. Aksoy et al.: J. Appl. Phys. Lett. Vol. 91 (2007), p [6] U. Stuhr et al.: Phys. Rev. B Vol. 56 (1997), p