Multiscale origin of the magnetocaloric effect in Ni-Mn-Ga shape-memory alloys

We have analyzed magnetization measurements in a series of composition-related Ni-Mn-Ga shape-memory alloys. It is shown that the magnetocaloric effect in the vicinity of the martensitic transition mainly originates from two different contributions: ~i! magnetostructural coupling on the mesoscopic scale between the magnetic moments and the martensitic variants, which is also responsible for the magnetic shape-memory effect and ~ii ! the microscopic spin-phonon coupling which gives rise to the shift of the transition temperature with the applied magnetic field. The relative importance of these two contributions has been shown to vary with composition, which is suitably expressed through the average number of valence electrons per atom /a. In alloys with a large difference between the Curie and martensitic transition temperatures ( e/a.7.5), mesoscopic coupling is dominant and a negative giant magnetocaloric effect ~increase of temperature by adiabatic demagnetization ! is induced at moderate applied fields. In contrast, in alloys when these temperatures are very close to one another ( e/a.7.7), the microscopic coupling is the most relevant contribution and gives rise to a positive giant effect.


I. INTRODUCTION
Ni-Mn-Ga is a ferromagnetic alloy with a Curie temperature slightly above room temperature ͑weakly dependent on composition͒ that undergoes a martensitic transition at a temperature T M which is strongly sensitive to composition. 1 This structural transition is responsible for the shape-memory properties 2 displayed by these materials including the magnetic shape-memory effect. 3,4Such a magneto-structural effect refers to the possibility of inducing giant deformations ͑in some cases more than 5%͒ by the application of a moderate magnetic field ͑less than 10 kOe͒ in the martensitic phase.Interestingly, this exotic property has been observed not only in ferromagnetic, but also in antiferromagnetic materials. 5t has recently been reported that, in the vicinity of the martensitic transition, the application of a magnetic field induces a large change of entropy. 6,7Such a magnetocaloric effect 8 is of great technological interest for cryogenic applications and has been intensively studied in recent years, especially after the discovery of giant magnetocaloric materials such as Gd 5 (Si x Ge 1Ϫx ) 4 ͑Ref.9͒ and MnAs-based compounds. 10,11In nearly stoichiometric Ni 2 MnGa alloys, the magnetocaloric effect has been shown 12 to be strongly dependent on the changes of the domain mesostructure of the system induced by the application of a magnetic field.These changes are known to be controlled by the cross-correlation between structural and magnetic domains and is also responsible for shape-memory properties.This effect is, however, expected to lose importance as the martensitic transition approaches the Curie temperature.With this idea in mind, in the present paper we have analyzed magnetization data recently reported in a series of composition-related Ni-Mn-Ga alloys with T M ՇT c that have compositions with an average number of valence electrons per atom e/aՇ7.7. 13The evolution of the magnetostructural properties as T M approaches T c shows the multiscale origin of the magnetocaloric effect.Results are expected to be of interest in order to select the appropriate materials with optimal magnetocaloric performances for technological applications.

II. MODELING AND RESULTS
Magnetization measurements at different temperatures through the martensitic transition have recently been reported 6,7,12,14 for five alloys with different compositions.The composition and characteristic temperatures of these alloys are listed in Table I.From the reported magnetization data, the field-induced entropy change ͑as the field is increased from 0 to H) can be obtained as In systems transforming martensitically, the temperature dependence of ⌬S(T,H) for a given field is strongly influenced by discontinuities in the transformation path which are caused by ͑unavoidable͒ defects, composition inhomogeneities, etc.This discontinuous behavior gives rise to a ⌬S(T,H) that shows peaks over a temperature range ⌬T(H). 12This behavior is illustrated in Fig. 1.In order to compare ⌬S(T,H) for different samples it is then convenient to calculate, for each field, the average of the field-induced entropy change over ⌬T(H) as This integral is performed numerically by taking a suitable base line which enables elimination of the contribution to ͗⌬S(H)͘ arising from any possible temperature variation of the magnetization outside the transformation region.The method of integration is also illustrated in Fig. 1.The obtained values of ͗⌬S(H)͘ as a function of the applied field for the five analyzed alloys are shown in Fig. 2.This entropy change first increases with H, then reaches a positive maximum, and linearly decreases for high fields.Except for alloys 1 and 2, the initial increase of ͗⌬S(H)͘ is very weak and is not even observed for sample 5.It is worth noting that the maximum value of ⌬S(T,H), ⌬S max (H), follows the same qualitative behavior as ͗⌬S(H)͘.
In order to gain a deeper understanding of the results presented in Fig. 2, it is interesting to analyze the behavior in detail of the magnetization as a function of temperature, at given applied fields, in the vicinity of the martensitic transformation.The most relevant feature is the abrupt change, ⌬M ͑magnetization difference between martensite and parent phases͒, which occurs at the martensitic transition.In Fig. 3, we have plotted ⌬M vs H for all the analyzed samples.It is observed that ⌬M significantly depends on the  II.applied magnetic field.It first decreases with H for very small fields ͑this is not observed for sample 5͒, shows a negative minimum at a given value of the field (H*), and finally increases and reaches a positive saturation value ⌬M sat which renders the difference in the saturating magnetic moments of martensite and parent phases.
Recently reported optical observations of the magnetic and structural domain mesostructure formed at the martensitic transition under selected applied fields 15,16 suggested the actual mechanism responsible for the behavior of the magnetization.These mesostructures are a consequence of the magnetostructural coupling between martensitic variants and magnetic moments, which originates from the strong uniaxial magnetic anisotropy ͑along the c axis͒ of the tetragonal martensitic phase.When the martensitic transition takes place at zero field, the nucleation gives rise to martensitic plates formed by parallel strips of twin-related variants.This mesostructure ensures that parent-martensite interfaces satisfy the invariant plane strain condition 17 which minimizes the elastic strain energy arising from the crystal lattice misfit along the interfacial boundaries. 17,18Magnetic domains are formed within each variant in such a way that the magnetization alternates between two opposite values along the corresponding easy axis.When the system is cooled through the transition under a magnetic field larger than the saturating field of the high-temperature phase ͑which is expected to be close to H*), the twin-related variants are magnetized.As the field is increased, due to the strong uniaxial anisotropy of the tetragonal phase, the Zeeman energy difference between neighboring variants is minimized by increasing the fraction of those variants with their easy-magnetization axis forming a smaller angle with the applied field.Finally, transformation under high enough fields results in a magnetically saturated single variant martensitic crystal.It is worth noting that such a mechanism is controlled by the same magnetostructural coupling that accounts for the magnetic shape-memory effect. 3,19ithin the framework of the preceding scenario, for H ϾH*, the fraction f of variants favored by the application of a magnetic field is determined by the balance between elastic and magnetic energies. 3At a given temperature f is obtained as the fraction minimizing the free energy where the elastic energy is given by E elas ϭE 0 ⌽( f ) (E 0 is the elastic energy at zero applied field and ⌽ is a dimensionless function accounting for the dependence on the transformed fraction͒.In this expression ⌽ϭ1 for HрH*.We will assume, for the sake of simplicity, that the system contains two twin-related variants 1 and 2. Thus, for HрH*, f ϭ1/2.For HуH*, the magnetization can be expressed as where M ជ 1 and M ជ 2 are the magnetization of variants 1 and 2 along the corresponding easy axis (c axis͒.Let be the angle between M ជ 1 and H ជ and the angle between M ជ 1 and , where M M is the saturation magnetization of the martensitic phase.Thus, minimization of G( f ) with respect to f yields where the characteristic field H c is H c is the field that characterizes the response of the magnetization associated with twin-boundary motion against the elastic energy.Note that the angle for twin-related variants is approximately /2 and, in general, is small.Therefore, cos Ϫcos(ϩ)Ӎ1.The first value is estimated from the numerical integration procedure used to obtain ͗⌬S͘ from magnetiza- tion data; the second is the value used to fit Eq. ͑11͒ to the experimental values.
b H* is the field corresponding to the minimum of the ⌬M (H) curve, and it approximately corresponds to the saturating field of the parent phase.c H c is the field that characterizes the response of the magnetization associated with twin boundary motion against the elastic energy ͓see Eq. ͑5͔͒, and it is close to the saturation field of the martensitic phase.
Ϫ 1 2 e Ϫ(HϪH*)/H c .For HрH* it is reasonable to assume that ⌬M varies linearly with H. Therefore, for HрH*, where ⌬M sat is the saturation value of ⌬M , and ϭM M /2⌬M sat .H c and ⌬M sat /M M can be estimated from experimental data by fitting an exponential function to the measured values of ⌬M ͑in the range HуH*).The fits for the different analyzed alloys are plotted in Fig. 3, and the obtained values of the parameters are listed in Table II.The fact that an exponential function yields a good fit to the data ͑it reproduces the fast saturation of ⌬M ) confirms the suitability of the selected function ⌽( f ) in establishing the f dependence of the elastic strain energy.Moreover, it is worth noting that the obtained values of H c clearly decrease with increasing e/a.That is, on reducing the temperature range between Curie and martensitic transition temperatures T c ϪT M , which in turn varies linearly with e/a.The contribution arising from the martensitic transition to the magnetization along the direction of the applied magnetic field is taken in the form M (T,H)ϭM P (H)ϩ⌬M (H)F͕͓T ϪT M (H)͔/⌬T(H)͖.In this expression, F͕͓TϪT M (H)͔/ ⌬T(H)͖ is a monotonously decreasing function of width ⌬T such that F→0 for TӷT M (H) and F→1 for TӶT M (H) ͑for ⌬T→0, F is the Heaviside function͒.⌬T(H)(Ͻ0) is the temperature range over which the transition spreads when the field changes from 0→H, and T M an estimation of the martensitic transition temperature.M P (H) is a temperatureindependent quantity.From Eqs. ͑1͒ and ͑2͒, the averaged field-induced entropy change in the vicinity of the martensitic transition is obtained as where we have taken into account the fact that In the preceding expression ⌬T(H) is given by where ⌬T t (H) is the temperature range over which the transition spreads for a given field H and (dT M /dH)H is a measure of the shift of the transition temperature induced by the field.From the magnetization measurements it is obtained that for moderate fields ⌬T t is almost independent of H. 20 Moreover, since dT M /dH is small in Ni-Mn-Ga alloys ͑see Actually, this is a suitable approximation in order to analyze the behavior of the field-induced entropy change in Ni-Mn-Ga in a broad range of fields. 21By using the quantities H*, H c , , and ⌬M sat given in Table II, it is possible to fit Eq. ͑11͒ to the experimental data shown in Fig. 2. In such a fit, ⌬T is the only free parameter, which permits the adjustment of the vertical scale.The fitted values of ⌬T are also given in Table II, where they can be compared with the corresponding values estimated from the integration procedure used to calculate ͗⌬S(H)͘.The agreement between model and experiments is very satisfactory for all studied alloys ͑the maximum discrepancy between the two estimated values of ⌬T is less than 15%͒.The last term in Eq. ͑11͒ can be expressed in the following form: where ⌬T M ϭT M (H)ϪT M (0) is the shift in the transition temperature due to the magnetic field and ⌬S t is the whole entropy change taking place at the martensitic transition which is known to be independent of H. 12 Equation ͑12͒ is obtained taking into account the Clausius-Clapeyron equation HϭϪ(⌬S t /⌬M sat )͓T M (H)ϪT M (0)͔.For small H, ⌬T M Ӷ⌬T t so that ⌬T M /(⌬T t ϩ⌬T M )Ӎ⌬T M /⌬T t Ӎ␣, where ␣ is the transformed fraction of martensite induced by the application of the magnetic field.͑iii͒ For large enough fields such that H Ͼ͉⌬T t ͉(dT M /dH) Ϫ1 ϾH c (ϾH*) the whole transition is induced (␣→1), and the first two terms in Eq. ͑11͒ have reached their saturation values ( f →1), therefore, Notice that in some cases the term (⌬M sat /⌬T)͓(1 ϩ)H*/2ϩH c ͔ can be very small compared to ⌬S t .For instance, the ratio between these two quantities is ϳ5% for alloys 3 and 4, and less than 2% for alloy 5.In the limit of very high fields, ⌬T t Ӷ⌬T M , then ͗⌬S(H)͘→⌬S t .The transition entropy change thus represents the maximum ͑in absolute value͒ reachable value of the averaged field induced entropy change.
It is now interesting to analyze separately the three contributions to ͗⌬S(H)͘ in Eq. ͑11͒.The first term on the right- hand side of this equation is related to the initial decrease of ⌬M (H).This term is proportional to H*, which is usually a weak field compared with H c ͑see Table II͒, and therefore this term leads to a small contribution to the magnetocaloric effect.The second term provides the contribution which arises from the magnetostructural coupling between martensitic variants and magnetic moments.This contribution is positive and is dominant for magnetic fields less than H c ͑see Fig. 2͒.Interestingly, its magnitude is proportional to the saturation magnetization of the martensitic phase M M ϭ2⌬M sat , and to H c .From our analysis both H c and M M ͑Ref.22͒ are maximum close to the stoichiometric Ni 2 MnGa composition (e/aϭ7.5).This dependence of the magnetization saturation on e/a is in agreement with recently reported data. 23,24This explains that the maximum positive contribution to the field-induced entropy change is obtained for alloys 1 and 2 with e/a closer to 7.5.The third contribution to ⌬S max is negative and is the expected contribution to the magnetocaloric effect in the vicinity of a magnetostructural transition. 25Figure 2 shows that this contribution to ͗⌬S(H)͘ depends linearly on the applied field.Moreover, the relative importance of this term increases with increasing ⌬T M which, for a given field, depends on dT M /dH.This derivative can be obtained from the Clausius-Clapeyron equation as the ratio between ⌬M sat and ⌬S t .Notice that this derivative provides an estimation of the strength of the microscopic spin-phonon coupling in the system.In our case we see that it strongly increases with e/a ͑see Table II͒.
A quantitative comparison of the three contributions to the magnetocaloric effect for all the studied samples can be performed by evaluating them at HϭH c .Results are listed in Table III.

III. SUMMARY AND CONCLUDING REMARKS
From a general viewpoint, the magnetocaloric effect is an interesting property in magnetic materials that depends sen-sitively on the complex magnetic structures that develop at different length scales during magnetization processes.The magnetocaloric effect is defined as the adiabatic temperature change arising from the application/removal of a magnetic field H.A positive magnetocaloric effect corresponds to an adiabatic increase ͑decrease͒ of temperature when H is applied ͑removed͒, and it is associated to a negative ͑positive͒ entropy change on isothermal application ͑removal͒ of magnetic field.
In the vicinity of the martensitic transition of Ni-Mn-Ga, the magnetocaloric effect is controlled by the interplay between magnetic and structural degrees of freedom.The phenomenon occurs simultaneously at multiple length scales from the microscopic atomic scale to mesoscopic scales, which range from the scale of magnetic domains inside martensitic variants to the scale of the twin variants.In the present work, we have evaluated the contribution from these different scales to the magnetocaloric effect.
We have seen that the dependence of the magnetic properties of Ni-Mn-Ga on e/a leads to a magnetocaloric effect which also shows a strong e/a dependence.Two characteristic magnetic fields are relevant in order to evaluate the relative importance of each contribution to the magnetocaloric effect, namely H* and H c , which approximately correspond to the saturating fields in the parent and martensitic phases, respectively.We have also found that both H* and  H c decrease as e/a increases, that is, as T c ϪT M goes to zero ͑see Tables I and II͒.
For fields in the range between H* and H c ͑low fields͒, magnetization occurs throughout a twin reorientation process in the transition region due to the high anisotropy of the martensitic phase ͑tetragonal symmetry͒.This mechanism mainly controls the magnetocaloric effect in such a range of fields, and it gives ⌬SϾ0, which means that the sample increases its temperature by adiabatic demagnetization.The maximum value for this contribution is achieved for alloys with a composition close to the Heusler composition ͑which transforms below room temperature͒.In contrast, for high fields (HϾH c ) the microscopic coupling, responsible for the change of intrinsic magnetic properties at the phase transition, is at the origin of the magnetocaloric effect.This effect is adequately accounted for by the derivative dT M /dH which gives the rate of change of the martensitic transition temperature with an applied magnetic field and can be obtained from the Clausius-Clapeyron equation.This contribution yields ⌬SϽ0, and is maximum in absolute value for alloys with the martensitic and Curie transition temperatures ͑above room temperature͒ close to each other.In this case, the magnetocaloric properties are comparable to those reported for Gd 5 (Si x Ge 1Ϫx ) 4 ͑Ref.9͒ or MnAs-based compounds, 10,11 which display the giant magnetocaloric effect in the vicinity of a magnetostructural transition, at which the crystallographic change is accompanied by a discontinuity of magnetic order which changes from paramagnetic to ferromagnetic.The maximum values of the field-induced en-tropy in the Ni-Mn-Ga alloy with T M ϳT c are approximately -30 J/K kg for a field of 15 kOe, 14 which is even larger than maximum values reported for Gd 5 (Si x Ge 1Ϫx ) 4 and MnAsbased materials, which are of the order of -15 J/K kg for a field of 20 kOe.
Figure 4 summarizes the obtained dependence on e/a of the magnetocaloric effect.The figure shows the averaged field-induced entropy change at selected values of the applied field, as a function of e/a and of T c ϪT M .These values are compared with the values of the transition entropy change which provide, for each value of e/a, the limiting ͑high fields͒ value of ͗⌬S(H)͘.⌬S t as a function of e/a is shown in the inset of Fig. 4. The values are taken from Refs. 12, 26, 27.The values of ⌬S t given in Table II correspond to interpolated values obtained from a least square fit ͑continuous line in the figure͒ to experimental data.Notice that in all cases the field-induced entropy change is far from the value of the transition entropy change which represents the maximum absolute reachable value.This means that the magnetic field is not high enough to enable the transformation of the whole sample.

FIG. 1 .
FIG. 1. Example of the entropy change obtained from magnetization curves using the Maxwell relation, which illustrates the procedure to compute the average values of the field-induced entropy change over the transformation region ⌬T.

FIG. 4 .
FIG. 4. Average field-induced entropy change ͗⌬S͘ as a func- tion of e/a and T c ϪT M at selected values of the inducing field H ϭ5 kOe (᭺), 7 kOe (), 10 kOe ( ), 15 kOe ( ), and 20 kOe (᭹).Inset: Transition entropy change ⌬S t as a function of e/a " from Ref. 12, ᮀ from Ref. 26 ͓⌬S t is estimated as the ratio between the latent heat and T 0 ϭ(M s ϩA f )/2 with M s and A f the temperatures of the starting and finishing temperatures of the forward ͑cooling͒ and reverse ͑heating͒ transitions͔, and from Ref. 27….The continuous line is a linear fit to these data.

TABLE I .
Atomic composition and transition temperatures of the alloys analyzed.Alloy No. a Ref. at. % Ni at.% Mn at.% Ga e/a T M ͑K͒ T c ͑K͒ a The lattice parameters determined for alloy 1 are aϭbϭc ϭ5.817 Å for the parent phase (Tϭ295 K) and aϭbϭ5.92Å, cϭ5.57Å for the martensitic phase (Tϭ4.2K), while those corresponding to alloy 3 ͑taken from Ref. 7͒ are aϭbϭcϭ5.828Å for the parent phase (Tϭ350 K) and aϭbϭ5.923Å, c ϭ5.556 Å for the martensitic phase (Tϭ250 K).

TABLE II .
Magnetic and thermodynamic quantities for the alloys analyzed.

Table II͒ ,
for fields that are not too high it is reasonable to assume that ⌬T is independent of H.͑i͒ For HрH*, from the preceding Eqs.͑6͒ and ͑7͒, a quadratic dependence of ͗⌬S(H)͘ on the field is obtained in

TABLE III .
Contributions to the magnetocaloric effect as given by Eq. ͑11͒, evaluated at HϭH c for all the studied alloys.All data are given in J/K mol.