A GINZBURG-LANDAU THEORY FOR Ni-Mn-Ga

Transcription

1 A GINZBURG-LANDAU THEORY FOR Ni-Mn-Ga A.T. ZAYAK a, V.D. BUCHELNIKOV b and P. ENTEL a a Theoretische Tieftemperaturphysik, Gerhard-Mercator-Universität, Duisburg, Germany b Physics Department, Chelyabinsk State University, Chelyabinsk, Russia (Received...) Abstract We present an effective Ginzburg-Landau theory for ferromagnetic cubic Ni-Mn-Ga in which structural and magnetic phase transitions take place. The general problem of a cubic ferromagnet is taken into account by fitting the parameters of the Ginzburg-Landau functional to experimental data. This enables us to obtain a realistic phase diagram for Ni-Mn-Ga. Keywords: Phenomenological theory, Ni-Mn-Ga, Phase transitions, 1. INTRODUCTION During the last years there has been growing interest in alloys which exhibit the shape memory effect. Of particular interest is the ferromagnetic shape memory Heusler compound Ni 2 MnGa (Webster, Ziebeck, Town and Peak, 1984). It is well known that such materials undergo reversible, crystallographic, thermoelastic martensitic transformations, whereby the lattice of the low-temperature product phase is coherent with respect to its hightemperature parent phase. Upon cooling the martensitic transformation starts at M S, the martensite start temperature. Above this temperature we have no martensite texture in the austenite matrix. Below this temperature, formation of the martensite texture starts inside the austenite matrix. The transformation is complete at M F, the martensite final temperature. By Corresponding author, tel , fax: , alexei@thp.uni-duisburg.de 1

2 the similar way, the reverse transformation occurs upon heating starting at A S and the transformation is complete at A F, the final reverse martensite transformation temperature. For Ni 2 MnGa the differences M S M F and A F A S are below 3 degrees while the hysteresis of the martensitic transformation is about 8 degrees (Ullakko, Huang, Kantner, O Handley et al., 1996; Tickle and James, 1999; Bozhko, Vasil ev, Khovailo, Dikshtein et al., 1999). The smallness of M S M F and A F A S allows to work (in the frame of a macroscopic theory) with only two temperatures by assuming M S M F and A S A F. The two temperatures used below are T AM and T MA (Bozhko, Vasil ev, Khovailo, Dikshtein et al., 1999), where T AM is the temperature for which the Austenite-Martensite transformation upon cooling occurs while T MA is the corresponding one for which the Martensite-Austenite transformation upon heating takes place. A characteristic feature of Ni 2 MnGa is that the Curie temperature T C is higher than the temperature for the martensitic transition, T M (Webster, Ziebeck, Town and Peak, 1984). Therefore, the influence of magnetic order on the martensitic transformation is of particular interest. Experimental data and approximate theoretical estimations show that the magnetic order in Ni- Mn-Ga alloys (with a composition close to the stoichiometric one) allows to induce the martensitic transformation by means of an external magnetic field (Bozhko, Vasil ev, Khovailo, Dikshtein et al., 1999). Another feature of Ni- Mn-Ga is the existence of an intermediate modulated structure between the parent and the product phases. Castan et al. (1999) has presented a theoretical model for the premartensitic transition with an attempt to consider the magneto-elastic interactions to be at the origin of the premartensitic phase. In many experimental studies the interesting premartensitic phenomena in Ni 2 MnGa have been investigated (Planes, Obrado, Gonzales-Comas and Manosa, 1997; Worgull, Petti and Trivisonno, 1996; Stuhr, Vorderwisch, Kokorin and Lindgard, 1997; Zuo, Su and Wu, 1998; Zheludev, Shapiro, Wochner and Tanner, 1996). However, a satisfying picture of the electronic origin of the intermediate phase of Ni-Mn-Ga has not emerged. In this paper we discuss a phenomenological Ginzburg-Landau model (Fradkin, 1994) for the formation of the intermediate phase as well as for the martensitic transformations. In order to be as realistic as possible, experimental input has been used to fix the parameters of the theory. It is emphasized that a careful analysis of the magneto-structural coupling is needed in order to understand the influence of magnetic order on the martensitic transition and the intermediate phase. 2. THE THEORETICAL MODEL A Ginzburg-Landau-type of theory allows to describe a cubic ferromagnet of O h symmetry which undergoes a structural phase transformation from a high symmetry cubic phase to the low symmetry tetragonal D 4h phase. The 2

3 symmetry group of the latter phase is a subgroup of the cubic one. In this case, the components of a macroscopic strain tensor e ik canbeconsideredas order parameters for the structural transformation. The magnetic state is described by the magnetization M. A free-energy expansion is formulated in such a way that the functional is invariant under the actions Ô of the space group of the fcc phase of Ni-Mn-Ga (Ô(F ) F, Ô O h ), leading to F = 1 2 (c 11 +2c 12 )e a(e2 2 + e2 3 )+1 2 c 44(e e2 5 + e2 6 )+1 3 be 3(e 2 3 3e2 2 ) c(e2 2 + e2 3 )2 + 1 [ 1 B 1 e 1 m 2 + B 2 2 e 2 (m 2 x 3 m2 y )+ 1 ] e 3 (3m 2 z 6 m2 ) +B 3 (e 4 m x m y + e 5 m y m z + e 6 m z m x )+K 1 (m 2 x m2 y + m2 y m2 z + m2 x m2 z ) αm δ 1m 4 M 0 H 0, (1) where the e i are linear combinations of the strain tensor components e ik, e 1 = (e xx + e yy + e zz )/ 3, e 2 = (e xx e yy )/ 2, e 3 = (2e zz e xx e yy )/ 6, e 4 = e xy,e 5 = e yz,e 6 = e zx. (2) In (1), a, b and c are linear combinations of the components of the second, third and fourth order elasticity moduli, respectively, with a = c 11 c 12, b =(c 111 3c c 123 )/6 6andc =(c c c c 1123 )/48 (Fradkin, 1994); m = M/M 0 is the unit vector of the magnetization and M 0 saturation magnetization; B i are magnetostriction constants; K 1 is the first cubic anisotropy constant; α 1 and δ 1 are exchange parameters. It is difficult to minimize the free energy (1) analytically for all possible parameter values. In order to obtain analytical results, we may, however, consider the functional at low temperature. A minimization of (1) with respect to the deformation tensor components e 1, e 4, e 5 and e 6,whichare not responsible for the considered kind of structural transformation leads to a renormalization of the free energy parameters, δ = δ 1 B1/(6(c c 12 )), K = K 1 B3 2/2c 44, F 0 = B1 2/[6(c 11 +2c 12 )]. Here we concentrate on the case K<0, B 2 > 0andc>0. For T T C and M M 0 the exchange terms can be omitted, which leads to F = F a(e2 2 + e2 3 )+1 3 be 3(e 2 3 3e2 2 )+1 4 c(e2 2 + e2 3 )2 + [ 1 B 2 2 e 2 sin 2 θ cos 2φ + 1 ] e 3 (3 cos 2 θ 1) K(sin4 θ sin 2 2φ +sin 2 2θ), (3) where θ and φ are the polar and the azimuth angles of the magnetization vector. 3

4 A phase diagram with the second and third elastic moduli, a and b, as parameters was obtained from an analytical minimization of (3), see Fig. 1 (Buchelnikov, Romanov and Zayak, 1999). In this paper we discuss a detailed analysis of the phase diagram. Each phase in Fig. 1 is labeled with respect to the direction of the magnetic easy axis. Solid lines in the diagram correspond to first order phase transitions; dashed lines are boundaries of stability areas. Solid and dashed lines, which are close to each other, mark second-order phase transitions. According to Landau theory we may use a = a 0 (T T M )witht M as critical temperature for the martensitic transformation. The parameter a 0 can be determined from the experimental data. In Fig. 1 [111] denotes the only cubic high-temperature phase with the magnetic easy axis pointing along [111]. Cooling of this phase can then be described by decreasing the quantity a. Below the dashed line H 2 K 2 G 2 S 2 A 1 S 1 G 1 K 1 H 1 a phase with a strained structure exists while the cubic phase exists above the dashed line E 2 E 1.In the area between H 2 K 2 G 2 S 2 A 1 S 1 G 1 K 1 H 1 and E 2 E 1 two equilibrium states, cubic and strained, coexist. As a phase transition we consider the line C 2 C 1 for which the free energies of both phases are equal. With respect to the temperatures T AM and T MA defined in the introduction, we may say that T AM corresponds to the line E 2 E 1 and T MA to the line H 2 K 2 G 2 S 2 A 1 S 1 G 1 K 1 H 1. The interval between these two lines corresponds to the hysteresis associated with the martensitic transformation. Depending on sign and magnitude of b on cooling we obtain different low temperature phases. The parameter b is of special interest. The strains e i are roughly proportional to b/c (Buchelnikov, Romanov and Zayak, 1999) (we keep c constant). Consequently, decreasing b leads to smaller tetragonal distortions in the product phase. The line C 2 B 2 marks a transition from the phase [111] to the tetragonal phase [110]. The line B 2 A 1 is a transition from [111] to the angle phase [uuv] with magnetic easy axis rotating from [111] to [110] upon cooling and vice versa when heating. The line A 1 B 1 represents a transition of the latter type but to the phase [vvu]. In this phase the magnetization easy axis rotates from [111] to [001] 1 upon cooling and vice versa upon heating. Rotation of the magnetic easy axes in the angular phases happens due to a competition of cubic anisotropy and magnetostriction. This kind of interaction is important for the case K < 0. The line B 1 C 1 describes the phase transition from the cubic [111] to the tetragonal phase [001] 1. There is a difference between the cases b>0andb<0. In the first case, the cubic phase undergoes distortions which leads to a reduced value for the lattice constant in [001] 1 direction and an increase of the lattice constants in the (001) plane. In the other case, when b<0, the strains act in opposite directions. Note also that a change of sign of B 2 leads to a symmetric reflection of the diagram with respect to the vertical axis a. On the lines B 2 A 2 and B 1 A 3 rotations of the magnetic easy axes in the phases [uuv] and [vvu] are complete and second-order spin reorientation phase transitions to the phases with fixed directions of magnetic easy axes [110] and [001] 1, 4

5 respectively, take place. Inside the area H 2 F 2 G 2 two equilibrium tetragonal solutions for the martensitic phase [110] exist. The line K 2 F 2 is a first-order phase transition between two tetragonal states. Above the point T 2 this transition occurs between two metastable states, while below this point it occurs between two stable tetragonal martensitic phases. The same situation is found for the case b>0in G 1 H 1 F 1. The existence as well as the sizes of both, H 2 F 2 G 2 and G 1 H 1 F 1, depend on the ratio of elastic constants and magnetostriction. The first order phase transitions on K 2 F 2 and K 1 F 1 are of iso-structural nature. From a mathematical point of view this is possible because inside H 2 F 2 G 2 and G 1 H 1 F 1 we have three solutions. As soon as both [001] 1 and [110] are described by cubic equations of state, inside these areas the discriminants of the equations become negative. We may also say that these iso-structural transitions occur because of the anharmonicity term in the free-energy expression (1). The phase [uv0] inside A 2 A 3 A 4 A 2 is an intermediate one between the phases [110] and [001] 1. We also find in this diagram a splitting of the firstorder phase transition line, A 4 A 5, into two second-order phase transitions lines, A 2 A 4 and A 3 A 4 with a critical point at A 4. The phase [110] exists on the left of the dashed line H 2 K 2 G 2 S 2 B 2 A 2 A 4 LM while [001] 1 is found right of H 1 K 1 G 1 S 1 B 1 A 3 A 4 NA 5 A 7. A 5 LA 7 corresponds to a second-order phase transition between the phases [001] 1 and [001] 2. The latter one has another type of symmetry. The phase [001] 2 exists below the QPNA 5 LA 7. A 5 A 6 describes a first-order phase transition between [110] and [001] 2 ;A 5 A 7 is a second order phase transition line separating [001] 1 and [001] 2. According to the experimental data (Vasil ev, Bozhko, Khovailo, Dikshtein et al., 1999) the approximation T T C is valid for the case of stoichiometric composition, Ni 2 MnGa. We are not allowed to use this approximation for other compositions because the increase of Ni atoms causes both T C and T M to approach each other. In this case we may not use the low-temperature approximation in the free energy expansion (1). An analytical solution of this problem was obtained by Vasil ev et al. (1999). However, the situation is more complicated if we take into account, in addition, the magnetic field or pressure because these fields lower the symmetry of the problem. The calculation for T T C leads to the conclusion that because of the symmetry breaking and magneto-elastic coupling, the martensitic transformation in a ferromagnet can be accompanied by a magnetic spin-reorientation. 3. NUMERICAL MINIMIZATION Further treatment of (1) can only be done by means of numerical methods. The problem to be solved is to find all local minima of the functional (1) in parameter space. However, employing general minimization methods does not guarantee that all local minima will be found. It is necessary to 5

6 construct a special scheme for the numerical minimization of (1). For the magnetization vector we are able to estimate its magnitude, but it will be difficult to determine the direction. We know the exact directions for a few solutions like [111], [001], [110] and so on. But there can be conditions for which the magnetization vector will not have a fixed direction, for instance, in phases like [uuv] and [vvu] of Fig. 1. In order to find such solutions one has to scan all possible directions θ (0, 2π)andφ (0,π). Figure 2 shows stereographic projection of the cubic lattice. It is cumbersome to analyze all of these directions and to find some initial points for the minimization process. Fortunately, in the present case, the symmetry of the expansion (1) allows us to find all possible solutions. Since we constrained the free energy expansion to be invariant under cubic symmetry transformations, any equilibrium solution for the cubic phase belongs to some elements of cubic symmetry. The same statement is valid for the tetragonal phase. Figure 3 shows the stereographic projection of all directions which satisfy the foregoing statement for the cubic phase. In the following we will consider only the cubic phases because the tetragonal symmetry group is a subgroup of the cubic one. Let us concentrate on Fig. 3 and consider two types of elements. The first type is related to fixed directions like [111], [001], [110] etc. while the second one is related to sections like the set of points between directions [001] and [111]. We introduce a useful notation for those sections. Let us consider two integer indices u and v and assume u > v. In this case, the section between [001] and [111] can be represented as [vvu] for all possible positive integer values u and v. The section between [111] and [110] is represented by [uuv]. In the same way all directions in Fig. 3 can be represented by different elements directions and sections. We may further reduce the space to 1/8 part of it and consider only {u 0; v 0}. The latter restriction is valid if the vector of the possible external force belongs to this part of space. In that case solutions in other parts of phase space are metastable. For the following directions and sections we are able to find the minima of the free energy (1), [111], [100], [010], [001], [110], [101], [011], [vvu], [uuv], [uvv], [vuu], [vuv], [uvu], [uv0], [vu0], [u0v], [v0u], [0vu], [0uv]. (4) For further calculations we will assume that each direction or section mentioned above corresponds to only one minimum of the free energy. There can be conditions when this is not true, but we suppose that it is an easier problem to separate two or three solutions in a small region than many solutions in the whole space. The same technique can be used for the case of a deformed lattice. This is possible because in the minimization process we consider only diagonal 6

7 elements of the deformation tensor. Therefore, we can consider some vector of deformation like e =(e x,e y,e z ), which has hardly any physical meaning, and apply to it the same technique as for the magnetization vector. For the vectors m and e we have to find their directions compared to the crystallographic directions sketched in Fig. 3. If m [vvu], we have to find e which minimizes the energy for a given m. The relationship between m and e can be determined from the free energy expansion (1). For example, consider the term [ 1 F B2 = B 2 2 e 2 (m 2 x m2 y )+ 1 ] e 3 (3m 2 z 6 m2 ) (5) and the corresponding deformations e 2 and e 3 in (2). For B 2 > 0 we assume that m lies in section [vvu]. In this case m x = m y and e x = e y holds while the quantity e 2 does not play any role. The expression (3m 2 z m2 ) is positive which means that e 3 has to be negative in order to obtain a minimum for the energy. The expression e 3 =(2e z e x e y )/ 6 is negative for e z <e x and e x = e y (for m [vvu] e [uuv]). The same procedure can be used for all other cases. For the energy expansion (1) minima are obtained for m and e belonging to sections having opposite indices. The case when m [001] and alike cases can be discussed in the same way. Since the direction [001] can be considered to belong to [vvu] for u > v, the relationship for m [vvu] can be used for m [001]. We then obtain conditions which determine the location of all possible extrema of the free energy. If we concentrate on Fig. 1, then all solutions can be found according to the following relationships: [111] : m [uuu]; e [000], [vvu] : m [vvu]; e [uuv], [001] : m [001]; e [uuv], [uuv] : m [uuv]; e [vvu], [110] : m [110]; e [vvu], [uv0] : m [uv0]; e [vu0]. For each section or fixed direction of magnetization m and corresponding sections for e, we use the simple minimization method outlined above. This technique guarantees that all local minima will be found. In addition it is necessary to check whether the obtained solution belongs to the same conditions for m and e, from which the procedure of minimization was started. If this is not the case the solution will not be accepted. This technique is also valid in the presence of an external force (field) pointing in any direction in Fig. 3. 7

8 4. THE TEMPERATURE CONCENTRATION PHASE DIAGRAM Since we have a reliable method for minimizing (1), we are able to find all equilibrium states of Ni-Mn-Ga. Let us first consider the temperature range which includes the vicinity of the Curie point. In order to obtain results which can be compared to experimental data, we will deal with the temperature T and concentration x as main parameters for Ni 2+x Mn 1 x Ga. Close to the critical points T C and T M the parameters a and α can be expressed as a = a 0 (T T M ), α = α 0 (T T C ), T C = T C0 γx, T M = T M0 + σx. (6) Here a 0, α 0, γ, σ are parameters which can be determined from the available experimental (T x) phase diagram obtained by Vasil ev et al. (1999). T M0 and T C0 are the critical temperatures for Ni 2 MnGa (i.e. for x =0). Figure 4 shows the calculated phase diagram of Ni 2+x Mn 1 x Ga obtained for the following set parameters, b = erg/cm 3, c = erg/cm 3, K = 10 5 erg/cm 3, B 2 =10 7 erg/cm 3, δ =10 9 erg/cm 3, H 0 =0,andwith respect to (6), a 0 = erg/cm 3, α 0 = 10 9 erg/cm 3, T M0 = 200 K, T C0 = 375 K, γ = 175 K, σ = 700 K. The phase diagram shows five different states. Near the point where the lines AA and CC intersect, we find a rather complicated merging of phase transitions in the rectangular part shown enlarged in Fig. 5. In the following we briefly discuss the different phases in Figs. 4 and 5. A high-temperature paramagnetic cubic phase (PC) exists above the line AMSRNN E, while a ferromagnetic cubic phase with magnetic easy axis [111] (FC[111]) is on the left side of AMM TN NRE. The paramagnetic tetragonal phase (PT) has an area of stability right of H M MSRNN A. FT[vvu] denotes a ferromagnetic phase with magnetic easy axis rotating from [111] to [001] upon cooling and vice versa when heating, which is stable in the region HFG. FT[001] is a low-temperature ferromagnetic tetragonal phase with magnetic easy axis [001] in the area defined by HMM TN A. The structural phase transition (dotted line CC ) is a hysteretic one. For x<0.186 the structural phase transition occurs in the ferromagnetic state while for x>0.186 in the paramagnetic state. The magnetic phase transition at the Curie temperature marked by the solid line AA, is of second order. The line FG marks a secondorder spin reorientation phase transition between FT[vvu] and FT[001]. In MM NN M in Fig. 5 coupled structural and magnetic phase transitions occur which show hysteresis characteristic for first-order transitions. Inside this area paramagnetic and ferromagnetic solutions coexist with each other. On the line ST a first order magnetic phase transition occurs simultaneously with the structural phase transition of first order. This unique situation is caused by the magnetostriction and anisotropy. The length of the line ST is proportional to the value of magnetostriction, B 2. For Ni 2 MnGa B 2 is of 8

9 the order of 10 7 erg/cm 3 (Bozhko, Vasil ev, Khovailo, Dikshtein et al., 1999). The concentration range of the combined magneto-structural transition ST in Fig. 5 is very narrow. However, it is of interest for the case that one wants to control the martensitic transformation by an external magnetic field. 5. THE PREMARTENSITIC TRANSFORMATION Besides the martensitic transition, there is a premartensitic transformation of the high temperature phase into an intermediate phase preceding the transformation into the modulated martensite structure. In this intermediate phase the parent cubic phase is modulated by transverse waves with wave vector q [110] and polarization i [110] (TA 2 acoustic phonon branch) (Zheludev, Shapiro, Wochner and Tanner, 1996). Among the different wavelengths there is a dominating 1/3[110] mode. The transformation from austenite into the intermediate phase is of weakly first-order. According to an early theoretical proposition by Gooding and Krumhansl (1988) we consider the displacements of atoms according to u(r) = ψ i sin(kr+ φ), (7) with k =1/3(1, 1, 0), i =(1, 1, 0); ψ and φ as new order parameters. The expression for the free energy has to be completed by additional terms like F ψ (ψ), F ψm (ψ, M) andf ψe (ψ,e i ) defined by F ψ = 1 2 A ψ B0 ψ C 0 ψ C 1[ψ 6 +(ψ ) 6 ], (8) [ 1 F ψm = 3 N 1 0 m 2 + N 2 (m 2z 13 ) ] m2 + N 03 m x m y ψ 2, (9) ( 1 F ψe = 3 D 1e ) D 2 e 3 + D 3 e 4 ψ 2. (10) 6 In expression (8) the term containing C 1 is minimized with respect to the phase φ. There is a minimum of (8) for φ = ±π/6, ±π/2, ±5π/6 ifc 1 > 0 and φ =0, ±π/3, ±2π/3,π if C 1 < 0. We will assume that C 1 is positive, B negative, A can be expressed as A = A 0 (T T ψ ), where T ψ is the critical temperature for the premartensitic transition. The full free energy expansion is then of the form, F = F 0 + F e (e i )+F ψ (ψ)+f ψe (ψ,e i )+F ψm (ψ,m i )+ F me (m i,e j )+F a (m i )+F ψm (m i,ψ)+f ex (m i ). (11) An analytical minimization of (11) with respect to the components of the deformation tensor, e 1 and e 4 6, leads to a renormalization of the parameters, i.e. for Eqs. 8, 9 and 10 we obtain B = B 0 2D 2 3/c 44 2D 2 1/3(c 11 +2c 12 ), 9

10 N 1 = N1 0 D 1B 1 /6(c 11 +2c 12 ), N 3 = N3 0 B 3D 3 /c 44. For other coefficients the renormalization is the same as in Section 2. N 1, N 2, N 3, D 2,whichare responsible for the interaction of the modulation order parameter with other order parameters, can be estimated by a fit to experimental data. We have now to minimize a functional with six order parameters. The additional parameter ψ does not complicate the minimization procedure. The calculations were performed by assuming that ψ is of the order of or less than u, whichmeans ψ 10 8 cm. Consequently, the coefficients in (8) are A =10 23 erg/cm 3, B = erg/cm 3, C = C 0 + C 1 =10 55 erg/cm 3. These parameters are valid for the case that F e and F ψ in (11) are of the same order of magnitude. The phase diagram obtained on the basis of Eq. (11) is shown in Fig. 6. In this diagram a dotted line LL shows a first- order phase transition to the phase with modulated structure. The line LF corresponds to the premartensitic phase transition. The existence of this transition is supported by experiment. The line FL marks the transition to the modulated structure after the martensitic transition has occurred. However, there is no experimental evidence for this kind of transformation. There is little chance to find the theoretical line FL in experiments because a realistic martensitic structure with different variants might not allow for this phase transition. For an enlarged intersection of lines AA and CC see Fig. 5. The phases PC and PT are the same as in Fig. 4. The phase FC[111] exists inside the area AMSRVUXP. The area of stability for FT[001] is marked by P UXWMM T- N A. The phase FCM[vvu] is stable in the regions KWVUE and FTM[001] below the line HXWVK. The intermediate phase FCM[vvu] shows the modulation defined in (7). The coupling between the modulation and the tetragonal deformations (10) leads to a quasi-cubic lattice. This small tetragonal distortion depends on the sign of the parameter D 2. If b and D 2 have the same sign then the intermediate premartensitic and martensitic phases show the same type of tetragonal distortion. This case corresponds to the diagram in Fig. 6, where the intermediate phase is labeled as FCM[vvu]. If these parameters have different signs then the tetragonal distortions have a complicated character. In the latter case the martensitic transformation shows a larger hysteresis and the intermediate phase must be labeled as FCM[uuv] (u >v). Moreover, if the interaction defined by the coefficient D 2 is quite large, an additional magnetic phase transition occurs: A second-order transition from FCM[uuv] to FCM[110] and, after that the martensitic transformation, from FCM[110] to FTM[001]. In order to emphasize the difference between the phases with [001] and [110] directions for the magnetization vector, we refer to the a, b phase diagram of Section 2. Thus we conclude that, because of the magnetostriction, the weak tetragonal distortion changes the magnetic order in the intermediate phase, whereby details depend on the values of D 2 and N 1,2,3. The phases in Fig. 6 have been obtained by using the parameters: D 2 = 10 3 erg/cm 3, N 1 =10 3 erg/cm 3, N 2 = 10 2 erg/cm 3, N 3 = 10 2 erg/cm 3. 10

11 Figure 7 shows the dependence of the components of the magnetization on the temperature along YY in Fig. 6. Along this line we find the Curie temperature (T C = 375 K), the temperature for the transition to the modulated structure (T ψ = 260 K), the structural transition to the tetragonal phase (T M = 220 K). For T > 375 K the paramagnetic state (PC) is stable, for 260 K <T <375 K the ferromagnetic state with M [111]. For 220 K <T <260 K the magnetic order changes because of deformations in the premartensitic phase, while for T<220 K the magnetic order is that of the tetragonal phase FTM[001]. The dependence of the magnetic order on the structural transformations in the premartensitic phase is uniquely defined by the parameters N 1,2,3. Other sets of parameters N 1,2,3 lead to modifications of the magnetic order. The parameter N 1 changes the magnitude of the exchange parameter α by α = α+n 1 ψ 2. The parameter N 2 is a magnetostriction-like parameter. The sign of N 2 determines which kind of magnetic phase exists, m [vvu] or m [uuv] according to (u > v). If the parameter N 2 is large enough, it can cause an additional magnetic spin-reorientation transition inside the intermediate phase. N 3 is also a very interesting parameter: It is responsible for the special type of magnetic modulation. For N 3 < 0 the sign of m x is equal to that of m y causing a longitudinal modulation along the direction [110]. For N 3 > 0 a transverse magnetic modulation occurs. A detailed analysis of all possible configurations in the T x diagram requires much more efforts. In this paper we have discussed a few results which have been obtained on the basis of a Ginzburg-Landau formulation. An extension of the present treatment will have to take into account external fields. Also according to O Handley (1998), one can consider simple structural and magnetic domains within the framework of the Ginzburg-Landau theory. This would allow to describe the large magnetostrictive deformations occurring in Ni-Mn-Ga (Murray, Marioni, Kukla, Robinson et al., 2000). 6. SUMMARY The theory presented here allows to describe structural, magnetic and coupled magneto-structural transitions of first- and second-order in a cubic ferromagnet like Ni-Mn-Ga. Magneto-structural interactions lead to a variety of different equilibrium states. If the Curie temperature happens to lie inside the hysteresis of the martensitic transformation, then the magnetic disorderorder transition is of first order. For Ni 2+x Mn 1 x Ga this situation occurs only for a very narrow range of compositions. The size of this range depends on the magnetostriction. A phenomenological description of the premartensitic transition is also possible, however, there are no experimental data available for the determination of the Landau-Ginzburg parameters, D 2 and N 1,2,3, which leads to some ambiguity when fixing the equilibrium conditions. One possible way to improve would be to analyse the behavior of the magnetiza- 11

12 tion in the intermediate phase. The Ginzburg-Landau theory presented here allows to understand many macroscopic effects in solids. Acknowledgments Financial supports by the Graduate School Structure and Dynamics of Heterogeneous Systems (Duisburg, Germany), the Russian Foundation for Basic Researches (# ), the Russian Ministry of Education (# ) and the program University of Russia are acknowledged. References Bozhko, A., A. Vasil ev, V. Khovailo, I. Dikshtein et al. (1999). Magnetic and structural phase transitions in the shape-memory ferromagnetic alloys Ni 2+X Mn 1 X Ga. JETP, 88, 957. Buchelnikov, V., V. Romanov and A. Zayak (1999). Structural phase transitions in cubic ferromagnets. J. Magn. Magn. Mater., 191, 203. Castan, T., E. Vives and P.-A. Lindgard (1999). Modeling premartensitic effects in Ni 2 MnGa: A mean-field and monte carlo simulation study. Phys. Rev. B 60, Fradkin, M. (1994). External field in the landau theory of a weakly discontinius phase transition: Pressure effect in the martensitic transitions. Phys. Rev. B, 50, Gooding, R. and J. Krumhansl (1988). Theory of the bcc-to-9r structural phase transformation of Li. Phys. Rev. B 38, Murray, S., M. Marioni, A. Kukla, J. Robinson et al. (2000). Large field induced strain in single crystalline Ni-Mn-Ga ferromagnetic shape memory alloy. J. Appl. Phys., 87, O Handley, R. (1998). Model for strain and magnetization in magnetic shapememory alloys. J. Appl. Phys., 83, Planes, A., E. Obrado, A. Gonzales-Comas and L. Manosa (1997). Premartensitic transition driven by magnetoelastic interaction in bcc ferromagnetic Ni 2 MnGa. Phys. Rev. Lett., 79, Stuhr, U., P. Vorderwisch, V. Kokorin and P.-A. Lindgard (1997). Premartensitic phenomena in the ferro- and paramagnetic phases of Ni 2 MnGa. Phys. Rev. B, 56, Tickle, R. and R. James (1999). Magnetic and magnetomechanikal properties of Ni 2 MnGa. J. Magn. Magn. Mater., 195, 627. Ullakko,K.,J.Huang,C.Kantner,R.O Handley et al. (1996). Large magneticfield-induced strains in Ni 2 MnGa single crystals. Appl. Phys. Lett. 69,

13 Vasil ev, A., A. Bozhko, V. Khovailo, I. Dikshtein et al. (1999). Structural and magnetic phase transitions in shape-memory alloys Ni 2+X Mn 1 X Ga. Phys. Rev. B, 59, Webster, P., K. Ziebeck, S. Town and M. Peak (1984). Magnetic order and phase transformation in Ni 2 MnGa. Phil. Mag., 49, 295. Worgull, J., E. Petti and J. Trivisonno (1996). Behavior of the elastic properties near an intermediate phase transition in Ni 2 MnGa. Phys. Rev. B, 54, Zheludev, A., S. Shapiro, P. Wochner and L. Tanner (1996). Precursor effects and premartensitic transformation in Ni 2 MnGa. Phys. Rev. B, 54, Zuo, F., X. Su and K. Wu (1998). Magnetic properties of the premartensitic transition in Ni 2 MnGa. Phys. Rev. B, 58,

14 C H2 K 2 G2 S2 2 T 2 E 2 B 2 [uuv] A1 [vvu] A [110] Q P N F 2 [uv0] A 5 A 4 L a A 2 3 [111] B 1 [001] H 1 K 1 G 1 S 1 F 1 1 T 1 C 1 E 1 b [001] 2 A 6 MA 7 Figure 1: Schematic phase diagram of the cubic ferromagnet in the (a, b) plane. The solid curves represent phase transitions; the dashed curves are stability boundaries. Dashed and solid lines approaching each other mark a second-order phase transition. 14

15 [010] [110] [011] [111] [001] [101] [100] Figure 2: The stereographic projection of the primitive vectors of a cubic crystal. [010] [vu0] [0uv] [110] [vuv] [uuv] [011][vuu] [111] [uv0] [0vu][vvu][uvu] [uvv] [001][v0u][101] [u0v] [100] Figure 3: The stereographic projection of the primitive vectors of a cubic crystal showing the cubic symmetry elements. 15

16 400 A PC H C T (K) 300 FC[111] F PT E A H FT[001] 200 C E G FT[vvu] Concentration x Figure 4: The phase diagram of the ferromagnet Ni 2+x Mn 1 x Ga in the (T,x) plane. The dotted line represents a first-order phase transition; dash-dotted lines are stability area boundaries; solid lines represent phase transitions of second-order. The phases are labeled as P: paramagnetic state, C: cubic lattice, F: ferromagnetic state and T: tetragonal lattice. 350 H C T (K) 345 A 340 H C M M T S N RN E A E Concentration x Figure 5: Full (T,x) phase diagram of Ni 2+x Mn 1 x Ga showing the region of T C and T M intersection. The dotted line corresponds to first-order phase transitions; dashed and dash-dotted lines mark the stability boundaries; the solid line represents phase transitions of second order. 16

17 T (K) 400 A 300 H Y L K P FCM[vvu] 200 C E Y FC[111] F W X U V PC FTM[001] FT[001] PT H C E A K L P Concentration x Figure 6: The phase diagram of the ferromagnet Ni 2+x Mn 1 x Ga in the (T,x) plane for the case that a the modulated structure has been taken into account. The dotted line represents first-order phase transitions; dash-dotted lines are the stability boundaries; the solid line represents phase transitions of second order. The phases are labeled as in Fig. 4; M refers to the modulated structure m x, m y m z 0.6 m = M/M T (K) Figure 7: The dependence of the dimensionless magnetization components on the temperature for Ni 2+x Mn 1 x Ga with x =