Shape Memory Alloys: Properties, Technologies ...

Shape Memory Alloys: Properties, Technologies, Opportunities

Edited by Natalia Resnina Vasili Rubanik

Shape Memory Alloys: Properties, Technologies, Opportunities

Special topic volume with invited peer reviewed papers only.

Edited by

Natalia Resnina and Vasili Rubanik

Copyright  2015 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Churerstrasse 20 CH-8808 Pfaffikon Switzerland http://www.ttp.net

Volumes 81-82 of Materials Science Forum ISSN print 0255-5476 ISSN cd 1662-9760 ISSN web 1662-9752

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Table of Contents Preface

v

I. Theory and Modeling of Martensitic Transformation and Functional Properties Possible Wave Processes Controlling the Growth of Martensite Crystals at B2-B19, B2-B19' and B2-R Transformations M. Kashchenko and V. Chashchina ................................................................................................. 3 Modeling of Deformation and Functional Properties of Shape Memory Alloys Based on a Microstructural Approach A.E. Volkov, M.E. Evard and F.S. Belyaev .................................................................................. 20 Novel Achievements in the Research Field of Multifunctional Shape Memory Ni-Mn-In and Ni-Mn-In-Z Heusler Alloys V.V. Sokolovskiy, M.A. Zagrebin and V.D. Buchelnikov ............................................................ 38 Modeling of Thermomechanical Behavior of Shape Memory Alloys S. Pryakhin and V. Rubanik ........................................................................................................... 77

II. Martensitic Transformations and Shape Memory Effects Physics of Thermoelastic Martensitic Transformation in High-Strength Single Crystals Y.I. Chumlyakov, I.V. Kireeva, E.Y. Panchenko, E.E. Timofeeva, I.V. Kretinina and O.A. Kuts ....................................................................................................... 107 Thermoelastic Martensitic Transitions and Shape Memory Effects: Classification, Crystal and Structural Mechanisms of Transformations, Properties, Production and Application of Promising Alloys V.G. Pushin, N.N. Kuranova, E.B. Marchenkova, E.S. Belosludtseva, N.I. Kourov, T.E. Kuntsevich, A.V. Pushin and A.N. Uksusnikov .................................................................. 174 Some Physical Principles of High Temperature Shape Memory Alloys Design G. Firstov, Y. Koval, J. van Humbeeck, A. Timoshevskii, T. Kosorukova and P. Verhovlyuk............................................................................................... 207 Structural and Magnetic Properties of Ni-Mn-Al Heusler Alloys: A Review M.V. Lyange, E.S. Barmina and V.V. Khovaylo ........................................................................ 232

III. Controlling the Functional Properties of Shape Memory Alloys Mechanisms of Microstructure Evolution in TiNi-Based Alloys under Warm Deformation and its Effect on Martensite Transformations A. Lotkov, V. Grishkov, O. Kashin, A. Baturin, D. Zhapova and V. Timkin ............................. 245

viii

Shape Memory Alloys: Properties, Technologies, Opportunities

Thermomechanical Treatment of TiNi Intermetallic-Based Shape Memory Alloys S. Prokoshkin, V. Brailovski, K. Inaekyan, A. Korotitskiy and A. Kreitcberg ........................... 260 Thermomechanical Treatment of Ti-Nb Solid Solution Based SMA V. Brailovski, S. Prokoshkin, K. Inaekyan, M. Petrzhik, M. Filonov, Y. Pustov, S. Dubinskiy, Y. Zhukova, A. Korotitskiy and V. Sheremetyev ................................................. 342 Influence of Ultrasonic Vibrations on Shape Memory Effect V. Rubanik, V. Klubovich and V. Rubanik ................................................................................. 406 Martensitic Transformation and Shape Memory Effect in TiNi-Based Alloys during Neutron Irradiation S. Belyaev, V. Chekanov, G. Kolobanov, R. Konopleva, A. Nakin, I. Nazarkin, A. Razov, N. Resnina and A. Volkov .......................................................................................... 429 Thermo-Mechanical and Functional Properties of NiTi Shape Memory Alloy at High Strain Rate Loading A. Danilov and A. Razov ............................................................................................................. 457 Features of Deformation Behavior, Structure and Properties of TiNi Alloys Processed by Severe Rolling with Pulse Current V. Stolyarov and A. Potapova...................................................................................................... 480

IV. Shape Memory Alloys with Complex Structure TiNi Shape Memory Foams, Produced by Self-Propagating High-Temperature Synthesis N. Resnina, S. Belyaev and A. Voronkov .................................................................................... 499 Development of Two-Way Shape Memory Material on the Basis of Amorphous-Crystalline TiNiCu Melt-Spun Ribbons for Micromechanical Applications A. Shelyakov, N. Sitnikov, D. Sheyfer, K. Borodako, A. Menushenkov, V. Fominski, V. Koledov and R. Rizakhanov ................................................................................................... 532 Crystal-Chemical Aspects of the Stability of the Ordered Phase B2 in Volume Alloying of TiNi L. Meisner .................................................................................................................................... 554 High-Strength Precipitation-Hardening Austenitic Steels with Shape Memory Effect V.V. Sagaradze and S.V. Afanas’ev ............................................................................................ 575

V. Application of Shape Memory Alloys Application of Thermomechanically Treated Ti-Ni SMA I. Khmelevskaya, E. Ryklina and A. Korotitskiy......................................................................... 603 Keyword Index ............................................................................................................................... 639 Author Index .................................................................................................................................. 641

Novel achievements in the research field of multifunctional shape memory Ni-Mn-In and Ni-Mn-In-Z Heusler alloys V.V. Sokolovskiy1,2a, M.A. Zagrebin1,3b and V.D. Buchelnikov1,c 1

Chelyabinsk State University, Br. Kashirinykh Str. 129, Chelyabinsk 454001, Russia

2

National University of Science and technology "MIS&S", Leninskiy Prospekt 4, Moscow 119049, Russia

3

National Research South Ural State University, Prospekt Lenina 76, Chelyabinsk 454080, Russia a

[email protected], [email protected], [email protected]

Keywords: shape memory, magnetic shape memory alloys, magnetocaloric effect, Monte Carlo simulation, ab initio calculation.

magnetoresistance,

Abstract. Nowadays, ferromagnetic shape memory Heusler alloys are ones of famous multifunctional materials exhibiting many interesting features in the temperature interval of the martensitic transformation due to the strong interrelation between crystal structure and magnetic order. The multiferroic, magnetoresistive, martensitic and related magnetic shapememory behavior as well as magnetocaloric properties are examples of these unique features. Generally, tuning of both structural and magnetic transition temperatures can be useful to achieve better functional properties. Today, the optimization problem of Heusler compounds is of a great importance. In this chapter, we review the most important features of ternary and quaternary ferromagnetic shape memory Ni-Mn-In and Ni-Mn-In-Z materials, which are experimentally and theoretically obtained in the last three years. We discuss the experiments devoted to the study of phase diagrams, thermomagnetizations, magnetic field and stress induced strains, magnetoresistance and magnetocaloric effects. The theoretical investigations of magnetic and structural properties are reviewed in the framework of the phenomenological approach, first-principles and Monte Carlo methods. Introduction With the rapid development of science and technology, the demand for high-performance materials with multi-functional properties has been increasing. Modern technologies need transducing materials, also referred to as smart materials that undergo a substantial change in one or more properties in response to a change in external conditions. Distinctly from structural materials, smart materials possess physical and chemical properties that are sensitive to a change in the environment such as temperature, pressure, electric field, magnetic field, and humidity etc. All of the smart materials are transducer materials as they transform one form of energy into another, and hence they have wide applications as both actuators and sensors in various fields such as medical, defense, aerospace, and marine industries [1-5]. Nowadays, a new class of smart materials called ferromagnetic shape memory alloys (FSMAs), also referred to as magnetic shape memory alloys, has been developed. Ones of the famous multifunctional materials are the well-known Heusler alloys exhibiting many interesting features in the vicinity of a diffusionless and reversible martensitic transformation due to the strong interrelation between crystal structure and magnetic order. In spite of the fact that Heusler compounds with the general composition X2YZ were discovered at the beginning of the last century currently they are attracting great interest of both experimentalists and theoreticians due to their promising properties. The discovery of giant deformations in the Ni2MnGa single crystal, which are controlled by external magnetic fields, has been one of the key moments for intensive investigations of Heusler alloys. The measurements were

conducted in 1996 by Ullakko et al. [6]. Large magnetic and field-induced strains (as high as 9.5 %) resulted from the rearrangement of martensitic variants under applied magnetic fields. Thus they integrate the advantages of both thermally and magnetically controlled shape memory effect and fast dynamic response. Up to this moment, the "giant magnetostriction" was related with deformations of the order of ≈ 0.2 %, which have been observed early in TbxDy1-xFe2 (Terfenol-D) alloys [7]. Thereafter, experimental evidences of the field-induced thermoelastic martensitic transition with responce of ≈ 1 T/K, field-induced shape memory as well as evidences of the giant magnetocaloric effect (MCE) were observed in Ni-Mn-Ga based Heusler alloys. This resulted to a rise in potential for novel technological applications. At present, huge hopes are concerned with the family of Heusler alloys because of their technological applications as potential materials for actuators, sensors for expanding applications, as well as refrigerants in the magnetic refrigeration technology. A rich variety of scientific, materials engineering and applied works devoted to the complex study of Ni-MnGa Heusler alloys were stimulated by these hopes. Given works are discussed in detail in the following topical reviews [8-13]. The next advances in investigation of Heusler alloys were caused by the discovery of the new off-stoichiometric Ni-Mn-X (X = In, Sn, Sb) alloys exhibiting unusual and sometimes complex behaviors of magnetic, structural and metamagnetic phase transitions induced either by temperature, magnetic field or by hydrostatic pressure [13-19]. The various phase transitions are related to the complicated magnetic nature of martensite, in which a strong correlation of ferro- and antiferromagnetic exchange interactions between magnetic atoms exists. For example, the austenite-martensite transformation in Ni-Mn-X (X = In, Sn, Sb) can be accompanied by magnetic transformation from ferromagnetic (FM) to antiferromagnetic (AF), mixed FM-AF or paramagnetic (PM) states. In addition, a magnetic phase transition from the mixed FM-AF or PM martensite to a spin-glass martensite in off-stoichiometric Ni2Mn1+xX1-x alloys at low temperatures can occur. Such variety of low-temperature magnetic phases leads to new effects in comparison with Ni-Mn-Ga. The metamagnetic shape memory effect, giant inverse MCE, anomalous Hall effect, giant magnetoresistance, exchange bias effect are examples of these unique features. Among the most interesting alloys of Ni-Mn-X series are Ni2Mn1+xIn1-x compounds, in which the strong temperature dependence of the martensitic transition temperature (Tm) on an applied magnetic field (≈ 10 K/T) was observed since 2004 [16]. Moreover, the addition of the fourth component into ternary Ni-Mn-In alloy allows to change both structural and magnetic transition temperatures for achievement of better functional properties. As an example, the giant inverse MCE (ΔTad ≈ - 6 K) was observed recently in the Ni45Co5Mn37In13 polycrystalline sample [20]. We would like to point out that at present, this large value of ΔTad is the record value among all Heusler alloys. In this chapter, we review the most important features of ternary and quaternary ferromagnetic shape memory Ni-Mn-In and Ni-Mn-In-Z materials, which were experimentally and theoretically studied in the last three years. We discuss the experiments devoted to the study of phase diagrams, thermomagnetizations, magnetic field and stress induced strains, magnetoresistance and magnetocaloric effects. The theoretical investigations of magnetic and structural properties are reviewed in the framework of the phenomenological approach, first-principles and Monte Carlo methods. The complete analysis of available data about properties of Heusler alloys helps to reveal the fundamental problems pointed for improvement of the unique multifunctional properties. These properties of the novel materials are needed to solve large-scale applied problems such as a creation of magnetocaloric devices and new technology for actuators, which can be applied in an instrument engineering industry, micro and nanomechanics etc. Crystal structures and phase diagrams of Ni-Mn-In and Ni-Mn-In-Z Heusler alloys As it is well known, the Ni2MnIn Heusler alloys can crystallize to L21 cubic structure consisting of four face centered interpenetrating sublattices, in which the In atoms occupy the

positions (0, 0, 0), Mn occupy the positions (1/2, 1/2, 1/2), and Ni atoms are located at the sites (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4), respectively [21, 22]. The L21 unit cell belongs to a space group Fm3m, No. 225(Oh5) while the whole crystal has only tetragonal symmetry Td. The octahedral (tetragonal) sites with point symmetry m3m (43m) are occupied by the Ni (Mn and In) atoms, respectively. Heusler alloys also have disordered structures, referred to as the B2 and A2 structure. Simultaneously, the Heusler structure is bcc-like as it can be formed from the ordered combination of two binary B2 compounds NiMn and NiIn with CsCl structure. The B2 structure has 8-fold coordination and is the most frequent type of disorder for L21 structure. In this structure, the Mn and In atoms are disordered. The A2 structure has no ordered sublattice, i.e., both the Ni and Mn, In sublattices are disordered. During the experimental manufacturing process the specimen is cooled down from the completely disordered A2 phase at high temperatures. The Ni atoms occupy their own sublattices, while the Mn and In atoms still disordered as a result of the order-disorder transition. Depending on the rate of cooling, it may happen that the diffusion of the atoms is hindered and the high temperature disordered phase is frozen (quenching). The lattice parameter, a0, of stoichiometric Ni2MnIn alloy at room temperature is equal to 6.071 Å [21]. In the nonstoichiometric Ni2Mn1+xIn1-x alloys, the excess Mn atoms occupy In sites and interact antiferromagnetically with nearest surrounding Mn atoms from regular Mn sublattice. In this case, the shortest distance between Mn1-Mn2 (dMn1-Mn2 = 1/2a0) is realized in comparison with Mn1-Mn1 or Mn2-Mn2 distances (dMn1(2)-Mn1(2) = 2 / 2 a0). Here, “1” refers to the original Mn sublattice while “2” refers to the In sublattice. Both stoichiometric and off-stoichiometric structures are shown schematically in Fig. 1.

Fig. 1. The Heusler L21 structures for Ni2MnIn and Ni2Mn1+xIn1-x. For the standard stoichiometric Ni2MnIn, the Curie temperature (TC) of austenite is about 290 K in a low external magnetic field of 5 mT, whereas the martensitic transition is not observed over a whole temperature range. Upon deviation from the stoichiometric composition, Ni2Mn1+xIn1-x alloys undergo both the magnetic and structural phase transitions, and both critical temperatures TC and Tm can be sensitively tuned. Note that different combinations of TC and Tm result in different properties of the Ni-Mn-In alloys with various technological significances. As an example, the well studied classical phase diagram of Ni2Mn1+xIn1-x reported by Sutou et al. [14], Krenke et al. [21], Moya et al. [17], Kanomata et al. [22] is presented in Fig. 2.

Fig. 2. Phase diagram of Ni2Mn1+xIn1-x alloys. Full (a) and narrow (b) ranges of Mn concentration are shown. Here AFP, APP, MFP, and LMP are the austenitic ferromagnetic phase, austenitic paramagnetic phase, martensitic ferromagnetic phase, and low-magnetic martensitic phase, respectively. Figures were replotted from Refs. [22, 23]. The phase diagram is characterized by the three transition temperatures: TCM, Tm and TC. Where, TCM is the Curie temperature of martensite. Three regions can be clearly seen in Fig. 2. In case of stoichiometric and close to stoichiometric compositions (0 < x < 0.3), the magnetic phase transition from a PM phase to FM one is observed only in the hightemperature austenite. Moreover, the Curie temperature, TC ≈ 290 K, is practically unchanged in this range of composition. The further increase of the Mn excess atoms in a narrow interval (0.35 ÷ 0.44) leads to the appearance of the martensitic transformation. The Tm is increased from 100 K to 300 K, whereas the TCM is changed from 230 K to 110 K. Three types of phase transitions in Ni2Mn2+xIn1-x alloys take place in this composition range. The first phase transition is the PM – FM phase transition in the austenitic cubic phase. The second phase transition is a coupled magnetostructural phase transformation from the austenitic FM phase to the nonmagnetic martensitic phase. Finally, the third phase transition is the magnetic transition from nonmagnetic martensite to the FM martensite with competing ferromagnetic and antiferromagnetic exchange interactions. The strong AF correlations are dominantly realized between Mn1 and Mn2 atoms due to the shorter distance between these atoms. The sharp change in magnetization in the vicinity of the martensitic transformation occurs due to the low magnetic moment of nonmagnetic martensite in comparison with the FM austenite. The next interval (x > 0.44), contains only one phase transition from nonmagnetic martensite to the PM austenite. Finally, there is only one phase transition from nonmagnetic martensite to the PM austenite in the composition range (0.44 < x < 0.6). We can see here, the linear increase of the Tm from 300 K to 500 K. It should pointed that the narrow composition range (0.35 ÷ 0.44) of in Ni2Mn2+xIn1-x is of practical interest due to the availabilities of largest magnetization drop, inverse MCE, magnetoresistance etc. in the vicinity of the martensitic transformation. The temperature dependencies of magnetization in Ni2Mn1+xIn1-x at low magnetic fields are shown in Fig. 3. We can observe a typical magnetization behavior in the samples with x < 0.35 showing the FM-PM transition in austenite. In contrast, the samples with x from 0.35 up to 0.44 show M(T) curves significantly differ from the previous case because of pronounced abrupt changes in the magnetization. As it was mentioned above, we can observe here three phase transitions at TC, Tm and TCM, respectively.

Fig. 3. Experimental temperature dependences of magnetization in Ni2Mn1+xIn1-x alloys in 5 mT (a) [21] and 0.1 T (b) [24]. The PM-FM transition in the cubic austenite is observed at the Curie temperature TC. The second transition is the structural transition from the FM cubic phase (austenite) to the PM or AF tetragonal phase (martensite). The third transition is the magnetic phase transition in the tetragonal phase. The decrease in magnetization in the low-temperature FM martensitic phase is again the influence of AF exchange on the FM spin system. The experimental data in Fig. 3 have been taken from Refs. [21, 24]. Recently, Huang et al. [25] developed the composition map of Ni-Mn-In using empirical data derived from Tm and thermal hysteresis characterized by the difference between forward and reverse transformation temperatures, ΔTm. It is worth noting that for devices with high response, there is need to use an alloy with a narrow thermal hysteresis, which originates from the “friction” opposing the martensitic variants rearrangements during the field-induced martensitic transformation. Therefore, the search of compositions with smaller thermal hysteresis is the important problem. The contour map provided by Huang et al. [25] is helpful to design the Ni-Mn-In magnetic shape-memory alloys. The martensitic transformation temperature Tm and the thermal hysteresis in a wide composition range of Ni-Mn-In are summarized in Fig. 4.

Fig. 4. Contour maps showing (a) the martensitic transformation temperature and (b) the thermal hysteresis in Ni-Mn-In alloys in reference to the composition with a constant 50 % Ni (dashed lines) and room temperature represented by dotted-dashed line. It can be seen from Fig. 4a that Tm spreads over a very large temperature range from 122 to 768 K by varying the composition (Mn: 33 - 50 at.% and In: 4.9 - 16 at.%). In this map, one can find that alloys transforming around room temperature have a very small composition

range, as represented by the dot dash line. In Fig. 4b the contour map for the thermal hysteresis is plotted. Here, dot dash loops represent a smaller thermal hysteresis (below 10 K), as well as Tm near room temperature. Unfortunately, it is inferred that the composition range (Mn: 32 - 34 at.% and In: 15 - 16 at.%) of Ni-Mn-In alloys with smaller thermal hysteresis at room temperature is relatively small. On the other hand, a large amount of compositions having the small ΔTm can be used in devices operating at high temperatures (400 K < Tm < 650 K). Kaletina et al. [26, 27] have investigated the magnetic field influence on the martensitic transformation temperature in series of Ni47-xMn42+xIn11 alloys using the metallographic, magnetometric, and dilatometric methods. The concentration x of Mn excess was varied between 0 and 2 at the fixed In content. The magnetic measurements have been accomplished for three samples at low (5 mT) and high magnetic fields (5 T), respectively. In Fig. 5 we depict both the T-x phase diagram using experimental data and temperature dependences of the magnetization upon cooling and heating of samples in magnetic fields of 5 mT and 5 T. It can be seen from Fig. 5, the decrease of the Ni content, which is substituted by Mn, results in the shift of the martensitic transition towards low temperatures and to the slight increase of the TCM, whereas the Curie temperature of austenite is not changed. Note that in the case of the Ni47Mn42In11 alloy, the coupled magnetostructural transition at which the temperatures of magnetic and martensitic transformations coincide is observed. The magnetic measurements in the magnetic field of 5 T show a significant decrease in the martensitic transformation temperature upon the substitution of Ni atoms by Mn atoms. Moreover, the change in the Tm with a change in the magnetic field strength (ΔTm/ΔH) is of the greatest importance in the Ni45Mn44In11 alloy and it equals to -12.1 K/T.

Fig. 5. (a) Phase diagram of Ni47-xMn42+xIn11 alloys. Here, Tmf and Tms are the finish and start martensitic transition temperatures, respectively. (b) Low and high field magnetization curves in the Ni47-xMn42+xIn11 (x = 0, 1, 2). Data have been taken from Ref. [26]. Miyamoto et al. [28] have performed determinations of the phase equilibria at 973 K and 1123 K, critical temperatures of B2-L21 order-disorder transition by means of the diffusion triple method using a two-stage diffusion couple technique. The more detailed information about this technique may be found in Ref. [29]. The ternary phase diagrams at 973 K and 1123 K are shown in Fig. 6. The authors have confirmed that a single phase region of the β phase with B2 cubic structure exists in a wide composition range along the NiMn–NiIn section and that the L21 ordered phase region appears in the vicinity of the Ni2MnIn in the temperature region about 1123 K. It can be also seen that the liquid phase (L) is widely extended to the middle region of the ternary diagrams with increase in temperature. Moreover, three binary compounds Ni13In9 (B81: NiAs-type), NiIn (B35: CoSn-type), and Ni2In3 (D513: Al3Ni2-type) can exist at 973 K, and in contrast, they do not appear at 1123 K. It is worth noting that the authors have firstly found the Mn3Ni2In compound at 973 K. The analysis of

electron diffraction patterns has shown that the Mn3Ni2In has the cubic system with space group Fd3m. The prototype of this phase is conformed as the Mn3Ni2Si structure in which Mn, Ni, and Si (In) atoms occupy 48f, 32e, 16d sites, respectively.

Fig. 6. The phase equilibria at (a) 1123 K and (b) 973 K of the Ni-Mn-In ternary system determined by Miyamoto et al. The Figure has been taken from Ref. [28]. The recent systematic studies of quaternary Ni-Mn-In-Z alloys have shown that the substitution of Z element either for Ni, Mn, or In sites can change smoothly or sharply both the magnetic and structural phase transitions, to bring about an uncommon transformation sequence from low-magnetic martensite to ferromagnetic austenite, and to increase magnetic properties. The influence of Co addition into Ni-Mn-In alloys on magnetic and structural transition temperatures have been performed by Liu et al. [30], and Dubenko et al. [23]. Liu et al. have performed series of measurements in Ni50-xCoxMn37In13 (0 ≤ x ≤ 9 at.%) and Ni45+yCo5Mn37-yIn13 (0 ≤ y ≤ 1) alloys. The corresponding phase diagrams are presented in Fig. 7.

Fig. 7. Magnetic and structural transition temperatures as functions of Co and Mn contents for (a) Ni50-xCoxMn37In13 and (b) Ni45+yCo5Mn37-yIn13. Data have been taken from Ref. [30]. From this figure we notice that in the case of Ni50-xCoxMn37In13, where Ni is substituted by Co the martensitic transition temperature increases from 295 K up to 325 K with increasing of Co content for 0 ≤ x ≤ 3 at.%, and then, it decreases with further increasing of Co concentration from 3 to 9 at.%. As regards to the Curie temperatures of martensite and austenite, the TCM linearly decreases up to composition with x = 5.5 at.% whereas the TC linearly increases in the whole composition range (0 ≤ x ≤ 9 at.%). The authors have also

found the optimized composition range of 4.5 ÷ 5.5 at.%, in which the magnetically induced austenite can be realized. On the other hand, if Mn is replaced by Ni in the Ni45+yCo5Mn37yIn13, the structural transition temperature sharply increases with increasing Ni content. Additionally, for the Ni46Co5Mn36In13 compound, the martensitic transformation was not found in the whole temperature range. Dubenko et al. [23] have investigated the influence of Co and Si on structural and magnetic properties of the N2Mn1.4In0.6 alloy. In the Co-containing series of Ni-Mn-In alloys Co with concentrations (0.04 ≤ x ≤ 0.12 at.%) was added instead of Mn, while in the Si-containing samples In was replaced by Si with concentrations (0.04 ≤ x ≤ 0.2 at.%). In Fig. 8 we show the transition temperatures as a function of Co and Si contents for the Ni-Mn-In alloy family. The magnetic measurements have shown that critical transition temperatures (TCM, Tm, and TC) slightly change with Co increasing. In the case of the Si-containing alloys, both the Curie temperature of austenite and the martensitic transition temperature decrease with increasing the doping concentration and the Curie temperature of martensite slightly increases with x. It is evident that the area of FM austenite (PM martensite) is larger (less) in alloys with higher Si content in comparison with the Ni2Mn1.4In0.6 alloy, respectively. It can be concluded that the magnetization drop becomes less with Si increasing. It is worth noting that in order to obtain a large change of magnetization between the martensite and austenite, the criterion TCM < Tm < TC should be satisfied.

Fig. 8. Temperatures of magnetic and structural transitions as functions of Co and Si excess for (a) Ni2Mn1.4-xCoxIn0.6 and (b) Ni2Mn1.4In0.6-xSix. Data have been taken from Ref. [23]. The experimental investigation of Fe influence on the martensitic and magnetic phase transitions in the series of Ni50Mn37-xFexIn13 (x = 1 ÷ 4 at.%) alloys have been recently performed by Jing et al. [31]. The corresponding phase diagram and magnetization curves measured at the low magnetic field of 50 mT are shown in Fig. 9.

Fig. 9. (a) Temperatures of magnetic and structural transitions as a function of Fe in Ni50Mn37(b) Low field temperature - magnetization curves in Ni50Mn37-xFexIn13. Data have been taken from Ref. [31].

xFexIn13.

We emphasize that for the composition with x = 1 the martensitic transformation was not observed in the temperature range from 3 K up to 350 K. The room temperature X-ray diffraction pattern of Ni50Mn36Fe1In13 alloy has shown that this compound has 10M modulated martensitic structure. It is reasonable to suggest that the martensitic transformation will be occurred at higher temperatures. As one can see from Fig. 9b (case: x = 1), there is only one magnetic transition at TCM from the PM to low-magnetic martensite. Further increase of Fe substitution for Mn atoms results in a dramatic decrease of the martensitic transition temperature and to appear the FM austenitic phase. Note that according to the well-established rule, the martensitic transition temperature should be found to increase with the valence electron concentration e/a increasing. Therefore, it is expected that the Tm should be increased when Fe (3d64s2) substitutes for Mn (3d54s2). The authors have explained a decrease of the Tm as follows: with the increase of Fe content, the relative quantity of Mn atoms contributing to the hybridization decreases and weakens the hybridization between the Ni 3d orbitals and the 3d orbitals of excess Mn atoms at In sites. As a result, the martensitic transformation tends to occur at a lower temperature. The martensitic transformation combined with magnetic properties in series of Ni50Mn34In16-xGax (0 ≤ x ≤ 16 at.%) have been investigated recently by Liu et al. [32]. The authors have declared that the compositions with x ≤ 2 have the cubic L21 phase (c/a = 1); compositions with 2 < x ≤ 8 have the seven-layered modulated (14M) martensite (0.88 ≤ c/a ≤ 0.91); finally, compositions with 8 < x ≤ 16 have the non-modulated martensite (1.22 ≤ c/a ≤ 1.25) at room temperature. The lattice parameters as functions of the Ga content at room temperature are presented in Fig. 10a. Thus, when increasing Ga concentration the crystal structure at room temperature varies consecutively from L21 to 14M, then to L10. The composition behavior of the transition temperatures (Tm, TC) is depicted in Fig. 10b. The Tm increases linearly from 238 K to 486 K as the Ga content changes from x = 0 to 16. The authors have estimated the linearity coefficient of Tm curve, and it was equal to 15 K per at.% of Ga.

Fig. 10. (a) Effect of Ga addition on the lattice parameters of Ni50Mn34In16-xGax alloys. (b) Martensitic and magnetic transition temperatures Tm and TC, TCM of the Ni50Mn34In16-xGax alloys as a function of the Ga content. Data have been taken from Ref. [32] As we mentioned above, ternary and quaternary Ni-Mn-In-based Heusler alloys, can exhibit a spontaneous magnetization disappearance in the martensite due to magnetic transitions between FM austenitic phase and PM, AF, or spin glass martensitic phase. These transitions are provided by different competing exchange interactions and electron density in both phases leading to the different transition temperatures. The modification of electron density and exchange interaction due to the change of composition (the number of conducting electrons) can affect the martensitic and magnetic transitions. Accordingly, both the TC and Tm temperatures of these alloys are very sensitive to the exchange interactions and valence electron concentration e/a. Here the number of valence electrons is calculated as the number of 3d and 4s electrons of 3d-transition metals and the number of 5s and 5p electrons of the In atom. Note that the most suitable sequence of critical temperatures corresponds to the criterion TMC < Tm < TC. In this case, the large drop of magnetization can be achieved at the martensitic transformation, which is also sensitive to applied external magnetic fields. As a result, the giant magnetic field-induced strain, large inverse MCE and magnetoresistance, anomalous Hall effect, the field-induced transformation into austenite as well as the other interesting properties appear in Ni-Mn-In and Ni-Mn-In-Z alloys in the vicinity of the martensitic transition. In this connection it should be noted that more and more attention is paid to these unique functional features for promising applications. The problem of optimization of the magnetic effects in these alloys is of main interest. In the next section we will review the functional features of Ni-Mn-In and Ni-Mn-In-Z alloys. Magnetic, structural and magnetocaloric properties of Ni-Mn-In and Ni-Mn-In-Z Heusler alloys Magnetization behavior. As it follows from phase diagrams for Ni-Mn-In and Ni-Mn-In-Z alloys, they can exhibit various sequences of phase transformations in accordance with the degree of nonstoichiometry. Typically, these alloys transform from a FM austenite to a weak magnetic martensite. The temperature region between the TCM and the Tm shows low magnetization, and thus is commonly referred to as AF or PM-like phases. Note that the composition and the degree of long-range atomic order are the main parameters for controlling of both martensitic and magnetic transition temperatures. As we have mentioned above on cooling through the equilibrium ternary phase diagram, Ni-Mn-In alloys do not solidify directly from the melt to the L21 phase (ordered phase). Usually, the bcc CsCl-type B2 disordered structure, in which the Ni atoms occupy the centre of a cube whose vertices are occupied by Mn and In atoms randomly distributed, is the precursor of the L21 phase.

The critical temperature for order-disorder (L21-B2) transition is around 1000 K for the Ni2MnIn composition [33]. Evidently that the structural disorder will result in a decrease of the Curie temperature, since the exchange interactions between nearest Mn-Mn atoms will weaken due to substitution of Mn for In atoms. Recently, investigations of the atomic order in Ni-Mn-In alloys and its influence on magnetic and structural phase transitions have been presented in several studies [33-37]. Miyamoto et al. [33] have studied the influence of the degree of order and annealing conditions on the magnetic properties and magnetic phase diagram of Ni50Mn50-xInx (13 < x < 35 at.%). The polycrystalline samples were annealed at 1173 K for 1 day and subsequently quenched in water. Additional annealing was performed at various temperatures (1173 K for 1 day, 823 K for 1 day and 573 for 3 days). The X-ray diffraction patterns of Ni50Mn15In35 alloy have shown that the samples annealed at 573 K have an ordered L21high structure (high degree of order), whereas the samples annealed at 823 K possess a B2 structure. The weak L21 ordered structure was observed in the specimens annealed at 1173 K, therefore the authors have determined this structure as L21low (low degree of order). The low-field temperature dependences of magnetization in the Ni50Mn15In35 for different samples are presented in Fig. 11a. It is clearly seen that the Curie temperature is very sensitive to the degree of order and annealing temperature. The atomic disorder results in a shift of the TC towards low temperatures due to weakening of exchange coupling between Mn-Mn atoms in B2 structure. Therefore the compounds with a higher degree of order demonstrate higher TC. Concentration dependences of the Curie temperature in series of Ni50Mn50-xInx (13 < x < 35 at.%) alloys are presented in Fig. 11b. As illustrated in Fig. 11b, the samples annealed at 1173 K and 573 ÷ 673 K exhibit the similar behavior of TC, while the decrease of In content from 25 to 13 at.% results in the stabilization of the Curie temperatures, which are independent on x and close to 310 K. Note that the same values of the Curie temperature can be seen in Fig. 2. On the other hand, the increase of In content from 25 to 35 at.% leads to a decrease of TC for both cases from 310 to 150 ÷ 200 K.

Fig. 11. (a) Temperature dependences of magnetization M(T) measured at a magnetic field of 50 mT for the Ni50Mn15In35 samples annealed at 1773, 823, and 573 K. (b) The Curie temperature as a function of the In content in Ni50Mn50-xInx. Here, filled symbols are the Curie temperatures for the samples quenched from 1173 K, and open symbols are the results for the specimens annealed at 573 K with x = 32 and 35, and 673 K with the others. The data are taken from Ref. [38]. The influence of the atomic order and the effect of different quenched treatments on the magnetic and martensitic transition temperatures in Ni-Mn-In alloys have been studied by Recarte et al. [34, 35]. The authors have prepared two similar polycrystalline alloys, Ni50.2Mn33.4In16.4 (alloy 1) and Ni50.4Mn33.5In16.1 (alloy 2), by arc melting in a protective Ar atmosphere. In order to obtain different degrees of atomic order, 1 alloy was subjected to a 30

min annealing treatment at 1173, 823, and 723 K, followed by quenching into ice water. By this means the quenching temperatures were 1173, 823, and 723 K, respectively. Another piece of alloy 1 was slowly cooled from 1173 K down to the room temperature. In order to compare with quenched samples, a quenching temperature of 300 K was assigned to this slowly cooled sample (AQ300). Note that the slowly cooled process allows to obtain the high ordered L21 structure at the room temperature. As for the alloy 2, this sample was quenched at 1073 K. Here, we will discuss the properties of alloy 1. The calorimetric measurements for alloy 1 after quenching at different temperatures have shown that there was a strong dependence of the martensitic transition temperatures on the quenching temperature. As shown in Fig. 12a, the higher the quenching temperature, the higher the transition temperatures for both the forward and the reverse martensitic transformation. In regard to the Curie temperature, it exhibits the opposite behavior with a smaller dependence from the quenching temperature. Here, the values of critical transition temperatures were taken from the calorimetric and magnetic measurements. As noted above, Heusler alloys undergo an ordering process due to the order-disorder transition at high temperatures (900-1100 K). Although this transition cannot be suppressed by a quenching process, some atomic disorder can be retained after the sufficiently rapid quenching process. Therefore, the temperature dependence of the transition temperature on the quenching temperature can be related with the dependence on the degree of atomic order of the alloy. In Fig. 12b we present the temperature dependence of the low-field magnetization curves (0H = 10 mT) for alloy 1 after quenching at different temperatures. We can see that the decrease in the quenching temperatures lead to the shift of the martensitic transformation to lower temperatures. Moreover, the induced martensite becomes magnetically ordered [35].

Fig. 12. (a) Dependence of the Curie temperature of the austenite, TC, the forward martensite transformation temperature TFMT and the reverse martensite transformation temperature TRMT, on the annealing temperature in quenched Ni50.2Mn33.4In16.4 alloys. (b) Temperature dependence of the magnetization at 10 mT for Ni50.2Mn33.4In16.4 alloys after quenching at different temperatures. Data have been taken from Ref. [35]. Concerning the entropy change at the martensitic transition, the exponential dependence of ΔS (where ΔS = ΔQ/Tm) on the difference between TC and Tm for Ni50.2Mn33.4In16.4 alloy has been obtained from DSC measurements. It must be pointed out that high values of TC – Tm correspond to low values of the quenching temperature and high values of the atomic order. As a result, both the latent heat and the entropy change decrease with the atomic order (quenching temperature) increasing (decreasing). The maximal value of ΔS corresponded to the sample quenched at 1173 K with maximum degree of disorder, while the minimal value of ΔS corresponded to the ordered sample, which was slowly cooled from 1173 K to the room temperature. Recarte et al. have concluded that the increase of the total ΔS with disorder increasing was responsible for the magnetic contribution to the entropy change, ΔSmag =

Smag(aust) – Smag(mart), since the lattice contribution to the entropy change ΔSlat can be neglected. The latter follows from the fact that the austenitic and martensitic structures do not change with atomic order evolution. Barandiaran et al. [36, 37] have recently mentioned that the approach of constant lattice entropy change for all compositions is not complete because it does not take into account changes in the atomic order that also influence the structure, the unit cell volume, etc. [36]. They have proposed the ferroelastic model of martensitic transformation wherein the ΔSlat is directly-proportional to (1 – c/a)2dC′/dT, where C′ is the shear modulus, a and c are the lattice parameters of martensite. Evidently, these values are different for different alloy compositions and can be changed by applied magnetic fields. According to this point of view, the influence of the applied magnetic field on the transformation properties of the ordered and disordered Ni50Mn34.5In15.5 alloys have been recently studied by Barandiaran et al. [37]. In order to obtain the ordered sample, the specimen was heat-treated at 1070 K for 20 min and slow-cooled (SC) to room temperature to assure the L21 ordered structure in austenite [36, 37]. In contrast, to fabricate the disordered sample with B2 structure, the specimen was heat-treated at 1170 K and rapid quenched in ice water (WQ). The isofield magnetization curves as functions of temperature for both ordered and disordered Ni50Mn34.5In15.5 alloys are shown in Fig. 13.

Fig. 13. Thermo-magnetization curves at different magnetic fields for (a) ordered and (b) disordered Ni50Mn34.5In15.5 alloys. Data have been taken from Refs. [36, 37]. Comparative analysis of ordered and disordered alloys reveals that the SC alloy possesses higher Curie temperatures of austenite and martensite (TC = 311 K and TCM = 247 K) as compared to the WQ alloy (TC = 292 K and TCM = 185 K), whereas the martensitic transformation temperatures are approximately equal to Tm ≈ 270 K. We can also observe reducing of Tm for both cases under the application of an external magnetic field. Shifts of Tm are caused by different values of the saturation magnetization for the martensitic and austenitic states. Generally, the absolute value of dTm/0dH increases with increasing magnetic field. It ranges from about 3 to 6 K/T between 0 and 13 T, for the ordered alloy, and from 1 to 4.7 K/T for the disordered one. These values are high enough but stay below the maximum ones reported for Ni50Mn34In16 [39], which are as high as 11 K/T. We would like also to note that for the ordered (SC) alloy the magnetization change between the austenite and martensite, ΔM, remains almost constant with an increase of magnetic field, whereas in case of disordered (WQ) alloy, the ΔM steadily increases with the field. Therefore, according to the Clausius-Clapeyron equation (dTm/0dH = - ΔM/ΔS), the change of slope of the dTm/0dH curve can be related with a decrease of ΔS with the applied field. Moreover, the change in ΔS produced by disorder or magnetic field is always of magnetic origin, since it follows from extrapolating of ΔS to an infinite applied magnetic field. Baradiaran et al. [36, 37] have obtained a value of ΔS∞ ≈ 5 J/kgK for both ordered and disordered alloys. This value

of ΔS can be attributed to the lattice contribution of entropy change, ΔSlat, because of the magnetic part of the entropy change, ΔSmag, is negligible in the infinite magnetic field. Therefore, the origin of the decreasing ΔS behavior is attributed to the magneticallydisordered martensite having a large value of magnetic entropy, which is gradually ordered by the magnetic field, and, as a result, the decrease in both ΔSmag and ΔS takes place, respectively. This can be correlated to the effect of atomic order in the alloy. As it was previously mentioned, the change of atomic ordering varies the Mn-Mn exchange interactions, which can be FM or AF depending on the distance between atoms. Therefore atomic order reduces the magnetic disorder of the martensite to a larger degree than in austenite, and promotes a decrease of ΔS. Field- and thermo-induced strains. As pointed out above, the Ni-Mn-In alloys transform from a ferromagnetic austenite into a weak magnetic martensitic phase, thus the application of a magnetic field, stabilizing the high magnetization phase, can induce the reverse martensitic transformation. Namely, the metamagnetostructural transformation is occurred. The martensitic transformation temperature can be driven either by a temperature, or a magnetic field, or stress, which is a distinctive property of these multifunctional materials. The applying of an external influence on Ni-Mn-In-(Z) can modify the Mn-Mn distanse and affect the relative stability between the high temperature cubic phase and the low temperature martensitic phase, thereby affecting the magnetostructural transition. One of the first information about shape memory effect in Ni-Mn-In-(Z) alloys was reported by Kainuma et al. [16]. The authors have shown that the Ni45Co5Mn36.7In13.3 single-crystalline alloy exhibited almost 100 % recovery of the compressive strain about 7% in the vicinity of the reverse martensitic transformation. Moreover, a perfect shape memory effect could be also obtained by the application of an external magnetic field. At present, the studies of reversible magnetic-field-induced or stress-induced transformations in Ni-Mn-In(Z) still attract increasing interest both in fundamental and applied science. Recently, Feng et al. [40] have investigated the magnetic-field-induced or stress-induced transformations in Ni50Mn34In14Fe2 polycrystalline samples. It should be pointed that Fe is an important ferromagnetic transitional element and its substitution for In in Ni-Mn-In improves the ductility and reduces the alloy cost. From the measurements of AC susceptibility as function of a temperature, the martensitic transition temperature and the Curie temperature were found to coincide at Tm = TC = 303 K. Note that the TC is slightly increased after addition of Fe content (2 at.%) in comparison with the Ni50Mn34In16 alloy. This shift of TC is caused by the enhanced FM exchange interaction between the Mn-Fe atoms. The isothermal magnetic measurements in magnetic fields from 0 up to 8 T have shown that the magnetization of martensite is lower than that of austenite (See Fig. 14a). Moreover, from M(0H) curves at temperatures between 292 K and 302 K, the first order reversible metamagnetostructural phase transition has been induced within an applied field up to 8 T. With the decrease of magnetic field, the magnetization gradually reduces. In Fig. 14 we depict the relative length changes ΔL/L of the Ni50Mn34In14Fe2 as function of temperature in magnetic fields of 0 and 0.5 T.

Fig. 14. (a) Magnetization curves of the Ni50Mn34In14Fe2 alloy at different temperatures in the range 286–306 K. (b) Relative length changes ΔL/L of the Ni50Mn34In14Fe2 as function of temperature with and without magnetic field. Data have been taken from Ref. [40]. We can observe that the strain during the thermo-induced phase transformation is approximately close to that resulted from the magnetic-field-induced phase transformation. Hence, a reliable estimation of the magnetic-field-induced strain can be provided by Fig. 14. Note that the temperature of phase transition is changed by about 2 K under the magnetic field change from 0 up to 0.5 T. Using the compressive experiment, the authors have summarized that Fe addition in the Ni50Mn34In16 polycrystal leads to increase of the compressive strain up to 8 % by comparison with the undoped compound showing the compressive strain about 5 %. Note that the compressive stresses for the Ni50Mn34In16 and Ni50Mn34In14Fe2 were about 470 MPa and 370 MPa, respectively. The influence of Si content on temperature and field-induced strain in polycrystalline Ni50Mn35In15 has been recently investigated by Pathak et al. [41]. The authors have prepared samples with Si content up to 4 at.% substituting In for Si. As follows from the T-x phase diagram, the Si addition results in a decrease of the martensitic transformation temperature. For strain measurements, the authors have prepared two samples, which were cut (a) parallel and (b) perpendicular to the direction of sample solidification during arc melting. Here, we will observe results obtained for case (a), because both types of samples have the similar strain behavior. Temperature behavior of strains has been measured in the magnetic fields of 0, 2, and 5 T. The authors have found that in the case of the Ni50Mn35In15 and zero magnetic field, the strain increases with temperature and exhibits a sharp jump (≈ 0.6 %) in the vicinity of the structural transformation, whereas the jump in strain slightly increases with increasing magnetic field. Previously, Aksoy et al. [42] have also observed the similar behavior in the Ni50.3Mn33.7In16. On the other hand, the In substitution with Si resulted in negative volume anomalies for compositions with x = 1, 2, and 4, while a positive volume anomaly was found for x = 3. Note that the latter case is similar to the case of x = 0. Similar signs of strain changes (positive and negative) were also reported in Refs. [43, 44]. In Fig. 15a we present the magnetic field dependence of the strain for Ni50Mn35In15-xSix (0 ≤ x ≤ 4 at.%) meausered at T = 300 K, 274 K, 264 K, 246 K, and 229 K for compositions with x = 1, 2, 3, and 4, respectively.

Fig. 15. The field dependences of the strain measured parallel to the external magnetic field for Ni50Mn35In15-xSix. Data have been taken from Ref. [41]. It should be noted that these temperatures are close to Tm. As illustrated in Fig. 15, in the case of x = 0 (x = 1), the strain remains almost unchanged until the critical field, and after that a sharp increase (decrease) in strain of up to 0.4 % (- 0.2%) is observed. The subsequent reduction of magnetic field results in the recovery of the initial value of strain. As it can be seen from Fig. 15, negative volume anomalies (ε < 0) were observed for x = 1, 2, and 4, while a positive volume (ε > 0) anomaly was observed for x = 3, as in the case of parent compounds (x = 0). We would like to note that a significantly large strain with a maximum value of about ≈ 1 % is found for x = 2 at 0H = 5 T. Experimental measurements of reversible magnetic and thermo-induced strain in Ni-Mn-In alloys with the addition of Co have been carried out in Refs. [45, 46]. Monroe et al. [45] have performed direct measurements of strains on a single crystalline Ni45Co5Mn36.5In13.5 alloy, which were attained via magnetic-field-induced martensitic transformation under different stresses (75 and 125 MPa) and temperatures (50 - 250 K). Before measurements, the specimen was first to the desired stress (75 or 125 MPa) in a custom-built micro-magnetothermomechanical test frame at room temperature and cooled down to 50 K under zero field to ensure the material was fully martensitic and then heated to a measured temperature. For the compressive stress of 75 MPa and the measured temperature of 200 K, the transformation from martensite to austenite is fully finished above 15 T, resulting in a total magnetic-fieldinduced strain of 2.4 %. In Fig. 16 we display the magnetic field-induced strains and magnetic field hysteresis (0ΔH) as a functions of temperature, which were measured in 75 MPa and 125 MPa tests. The field induced strain values for the 75 MPa tests are ~ 2 % between 50 K and 180 K, and then jump to 2.39 % at 200 K. The increase of compressive stress up to 125 MPa results in the linearly increasing strain behavior from 2.68 % to 3.13 % between 100 K and 250 K. Concerning the magnetic field transformation hysteresis, it can be seen that 0ΔH slightly increases in the range between 2.2 and 4.1 T in the 0, 75, and 125 MPa tests with decreasing temperature from 250 to 100 K. At 100 K, the 0ΔH values start to converge to 4.5 T then increase exponentially with decreasing temperature for the 0 MPa and 75 MPa experiments. The maximum value of transformation hysteresis, ΔH ≈ 12.5 T, is found at low temperature of 4.2 K for both 75 MPa and 125 MPa tests.

Fig. 16. Magnetic-field-induced strains (MFIS) (filled symbols) and magnetic transformation hysteresis (open symbols) under 75 MPa and 125 MPa determined from the constant compressive stress tests and magnetic transformation hysteresis under 0 MPa extracted from the magnetization tests as a function of temperature in the Ni45Co5Mn36.5In13.5 single crystal oriented along the [100] direction. Data have been taken from Ref. [45]. The stress-induced transformation behavior in Co-doped Ni45Co5Mn36In14 alloy has been investigated by Niitsu et al. [46]. In order to investigate the stress hysteresis, σhys = σAf – σMs, and its temperature dependence in the range of 4.2 – 200 K, the authors have performed compression tests on a single-crystalline Ni45Co5Mn36In14 alloy, which does not exhibit a thermally induced martensitic transformation down to the lowest temperature of 4.2 K. Where, σAf is the reverse martensitic transformation finishing stress, whereas σMs is the forward martensitic transformation starting stress. The critical stresses, σAf and σMs, were defined as the crossing points of the lines linearly extrapolated from the stress-strain curve and the forward/reverse superelastic plateaus, respectively. It is found that the perfect superelasticity with almost complete shape recovery occurs for temperatures below 200 K. The σMs and σAf slowly decreased with decreasing temperature for temperatures above ~ 125 K and then started to gradually separate into upper (320 MPa) and lower (75 MPa) values upon further cooling. The authors also have shown that the width of stress-strain hysteresis loops decreases exponentially with increasing temperature. Magnetoristance effect. As it follows from the phase diagram, the Ni–Mn–In(Z) alloys undergo a magnetostructural transition from the austenitic phase to the martensitic phase on cooling and a reverse process on heating. Since the crystallographic symmetry changes through the martensitic transition, the band structure varies correspondingly, leading to different densities of state near the Fermi level and a sharp change in resistivity near the martensitic transformation temperatures. The change in resistivity across the structural transition becomes larger when an external magnetic field is applied to a magnetic compound. As a rule, the magnetoresistance effect (MRE) can be determined in the following way: MRE 

 ( 0 H , T ) -  (0, T ) ,  (0, T )

(1)

where, ρ(0, T) and ρ(0H, T) are temperature dependences of electrical resistance of a material, which are measured under magnetic fields of 0 and 0H, respectively. Recent investigations of MRE were accomplished in the quaternary Ni-Mn-In-Z (Z = Co, Cu, Ga, and Si) alloys which are potential technological materials for the development of magnetic actuators and sensors [47-52]. The effect of the partial Co substitution (with 5 at.%) for Ni on magnetoresistance properties of Ni-Mn-In alloys has recently been reported by Chen et al. [47] and Porcar et al.

[48]. Chen et al. have studied the influence of post-annealing on magnetic and transport properties in Ni45Co5Mn36.6In13.4 alloys which were annealed for 3h at 523 K and 573 K and quenched in ice water. As it was shown, the martensitic transition temperature shifts from 314 to 283 K with annealing temperature increasing. Note that the compound without postannealing treatment has the Tm of 314 K. This value is in good agreement with the phase diagram presented in Fig. 8. The authors concluded that the decrease in Tm may be related to the stress relaxation and atomic order modification. The formed stress during the quenching process was relaxed to some extent depending on the annealing temperature and duration, which may modify atom site/ordering, Mn-Mn distance, as well as lattice symmetry. Hence, the Mn-Mn exchange coupling, Fermi surface and the Brillouin zone may be changed leading to the decreased temperature Tm. Concerning transport resistance properties, the authors have reported that the maximal MRE under 5 T reaches about 67% at 306 K for the as-prepared sample, and 72%, 69% at 286 K, and 266 K for samples post-annealed at 523 K and 573 K. It should be mentioned that the MRE under an external magnetic field is negative. This is related with Tm shifting to lower temperature under a magnetic field. Moreover, the authors have also found that the magnetoresistance behavior is full reversible for all three samples. Namely, it can return to its origin value after a magnetic field cycle from 0 to 5 T, as indicated in Fig. 17a. On the other hand, Porcar et al. [48] have shown the irreversibility of the resistivity in the vicinity of martensitic transformation in the Ni45Co5Mn37.5In12.5 single crystal upon applied magnetic field and uniaxial stress. According to measurements of resistivity as functions of temperature, uniaxial stress, or magnetic field, the MRE strongly depends on the thermal history and on the magnetic field or pressure kinetic arrest of the phases. The authors have shown that for a temperature of 355 K, magnetic field of 7 T, and uniaxial stress of 180 MPa, the maximum value of MRE is close to 60 % upon heating, whereas 31.5 % of MRE is observed upon cooling. We would like to noted that the martensitic transformation temperatures, namely, the martensitic (austenitic) start (finish) is 359 (368) K, whereas the austenitic (martensitic) start (finish) is 356 (343) K, respectively. The resistivity as a function of magnetic field in the Ni45Co5Mn37.5In12.5 alloy is shown in Fig. 17b. Here, the initial temperature of 355 K is reached due to heating from the martensitic state (open symbols) and cooling from austenitic state (filled symbols). It is obvious from Fig. 17b that when the sample is heated to 355 K in zero magnetic field, the resistivity starts to decrease with increasing magnetic field up to 7 T. Then the field is decreased down to zero and the resistivity reaches a value (~0.32 mΩ cm), corresponding to the value of resistivity in zero field during cooling. On the other hand, when the sample is cooled to 355 K, the resistivity also decreases with increasing magnetic field and reaches the value of the heating curve resistivity under 7 T. Then it comes back to the initial value when the field decreases. Evidently, there is the irreversibility in the resistivity of sample heating initially to 355 K. The martensitic phase cannot fully recover its initial state because of the thermal hysteresis and the field-induced kinetic arrest of the austenitic phase, especially when the sample is rapidly cooled down. A comparison between the magnetoresistance properties of Ni45Co5Mn36.6In13.4 (Fig. 17a) and Ni45Co5Mn37.5In12.5 (Fig. 17b) allows us to conclude that Chen et al. [47] have measured the resistivity as a function of magnetic field for the sample, which was initially cooled down to measuring temperatures (245, 255, 265, and 275 K) at zero magnetic field. In this case, the magnetoresistance behavior is full reversible, as well as the magnetoresistance behavior for Ni45Co5Mn37.5In12.5 alloy, which was initially cooled to 355 K at zero field. We would like to point out that Chen et al. [47] have not indicated how the sample achieved the measuring temperature (heating from martensite or cooling from austenite). In general, the irreversibility of magnetoresistance (in Heusler-family alloys) may be caused by the fieldinduced kinetic arrest of austenite, the accumulation of defects and impurities or altering of atomic ordering, resulting in the appearance of thermal hysteresis.

Fig. 17. Resistivity as a function of magnetic field for (a) Ni45Co5Mn36.6In13.4 and (b) Ni45Co5Mn37.5In12.5 alloys. Data have been taken from Refs. [47, 48]. The electrical resistivity of the Cu-doped Ni50Mn34In16 alloys has recently been studied by Dincer et al. [49], Chattopadhyay et al. [50] and Sharma et al. [51]. Dincer et al. have presented an experimental study on the Ni50-xCuxMn34In16 alloys where Ni was replaced by 2.6 and 4.6 at.% of Cu. As one can expect from the effect of valence electron concentration on the martensitic transition temperature, the substitution of Cu content (e/a = 11) for Ni (e/a = 10) must shift the transition temperature toward high temperatures. However, the authors have reported that an increase of Cu content (x) leads to a decrease of Tm from 192 K (x = 0 at.%) to 148 K (x = 2.6 at.%). As a result, the martensitic transition disappears for the alloy with x = 4.6 at.%. On the other hand, it was recently shown that a small substitution of Ni by Cu in the Ni50Mn34In16 alloy takes the martensitic transition to higher temperatures [53]. Therefore, Dincer et al. [49] have reported results of magnetoresistance measurements for Ni47.5Cu2.6Mn34In15.9 alloy in zero and 2 T magnetic fields upon heating and cooling. The value of MRE is close to -68% at 145 K upon cooling, while -60% of MRE is found at 176 K upon heating. For comparison, the Ni50Mn34In16 alloy exhibits -64% of MRE at 220 K on the cooling branch for the magnetic field change of 5 T [54]. The authors have also reported that the zero field resistivity before and after the application of the magnetic field show an irreversibility about 50% at 160 K. The combined effect of hydrostatic pressure and magnetic field on the martensitic transition and electrical resistivity in Ni49CuMn34In16 alloy was studied in detail by Chattopadhyay et al. [50] and Sharma et al. [51]. Low-field magnetization measurements have shown three transition temperatures in the prepared alloy. Two of them are the Curie temperatures of the austenite (TC = 302 K) and of the martensite (TCM = 189 K) and the third is the martensitic transition temperature (Tm = 295 K). A small addition of Cu has resulted to a sufficient increase of Tm in the Ni49CuMn34In16 as compared with the Ni50Mn34In16 alloy. Note that the critical temperatures are sensitive to the external hydrostatic pressure. It was shown that all critical temperatures increase with increasing the external pressure. As an example, the rate of increase of TC (TCM) is close to 2.8 (1.2) K/kbar, respectively. These values are higher than in the parent Ni50Mn34In16 alloy. In Fig. 18 (a,b) we image the temperature dependences of electrical resistivity for the Ni49CuMn34In16 in the presence of magnetic field and external pressure. The typical behavior of resistivity of Ni-Mnbased FSMA is observed. Namely, a slight bend (a pronounced drop) is found around TCM (Tm) temperatures during heating, respectively. Further heating leads to an increase of resistivity in the austenite. It seems that the resistivity exhibits a semimetal-like behavior in the martensitic state, while it is metallic for the austenitic state. As we know, the first-order martensitic transformation in Ni–Mn-based FSMAs often develops with a region of metastability with austenite and martensite coexisting. With increasing temperature, especially around the Tm temperature, more and more martensitic fractions in the sample are transformed into the austenitic phase. Since the martensite phase has a higher resistivity as

compared with the austenite phase, the overall resistivity decreases with increasing temperature, showing the semimetal-like behavior around the Tm temperature despite the metallic nature of martensite and austenite states. In addition, the downshift of Tm with applied magnetic fields can be seen from Fig. 18 (a, b). The decrease of resistivity in the martensite upon application of the external pressure is another significant feature. The authors have demonstrated that such behavior does not depend on applied magnetic fields and it is strongly caused by external pressure due to a strain-disorder coupling in the vicinity of the structural transformation. This means that there is strong strain-mediated correlation between different regions of the alloy in the vicinity of the martensitic transformation, which probably influences the nucleation and growth of phases. Note that after the application of the hydrostatic pressure, the strain fields produced in the compound are different from what are obtained in ambient pressure [50]. Probably, the strain fields induced by pressure affect the scattering of conduction electrons and as a result the resistivity is strongly dependent on the pressure. The pressure dependence of the maximum value of MRE observed in magnetic fields 2 and 8 T is plotted in Fig. 18c. We can see that the maximum MRE measured at ambient pressure is closed to 40 % and 60 % for magnetic fields of 2 and 8 T, respectively. We would like to note that the MRE is observed at room temperature that makes the compound useful for technological applications. Moreover, the MRE exponentially decreases with increasing of the external pressure.

Fig. 18. (a) and (b) The temperature dependences of electrical resistivity for the Ni49CuMn34In16 in the presence of different values of magnetic field and external pressure. (c) MRE as a function of pressure under magnetic fields of 2 and 8 T. The results of transport studies of Ga-doped polycrystalline Ni2Mn1.32In0.68-xGax (x = 0, 0.04, 0.08, 0.16, 0.20, and 0.28) alloys were recently reported by Pramanick et al. [52]. The parent alloy with x = 0 does not show any structural instability and there is only one magnetic phase transition from FM to PM austenitic state. The Curie temperature is close to 308 K.

This value agrees well with previous results (for example, See T-x phase diagram depicted in Fig. 2). In contrast, the substitution of Ga (x > 0.04 at.%) for In results in the appearance of the martensitic transformation. Moreover, Tm increases linearly from 180 K (for x = 0.08 at.%) to 242 K (for x = 0.28 at.%) whereas TC is found to slightly decrease with increasing of Ga. The resistivity measurements as a function of temperature under magnetic fields from 0 up to 14 T in series of Ni2Mn1.32In0.68-xGax alloys have shown that the maximum MRE (about -70 %) is observed at 180 K for composition with x = 0.08 at.%. So far as the measurements have been done at heating and cooling, the irreversibility in resistivity behavior when the magnetic field is applied or removed is found. It is to be noted that for sample with x = 0.08 at.%, the initial application of 9 T produces about - 70% MRE, whereas for removed field the sample still shows - 22 % MRE. Note that this type of field arrested state is only restricted in the vicinity of the martensitic transformation. In case of the highest Ga containing sample (x = 0.28 at.%), the maximum of MRE observed at 241 K is about -27 %. This value is almost three times less in comparison for composition with x = 0.08 at.%. Magnetocaloric effect (MCE). In this subsection we will discuss the novel achievements in the field of experimental investigation of magnetocaloric properties in Ni-Mn-In-(Z) alloys. Nowadays, a study of MCE based on various magnetic materials is gaining worldwide attention due to its application in the magnetic refrigeration technology. This technology is environment friendly and energy efficient. We would like to remind that the MCE is the ability to absorb or release heat as the result of the application of external magnetic fields. Generally, the MCE can be characterized by two factors: isothermal magnetic entropy change (ΔS) and adiabatic temperature change (ΔTad). At present many materials with first-order magnetic phase transition, such as Gd5(Si,Ge)4, La(Fe,Si), MnFeP1−xAsx and Ni–Mn–Ga have been found to exhibit great MCE. There are several review articles focused on studies of the best magnetocaloric materials (For example, See Refs [13, 55-59]). Here, we mainly review a recent progress for Ni-Mn-In-(Z) alloys only. As it was mentioned above, the complex sequence of phase transitions can be found in these alloys due to the change of chemical composition. It arises from the AF coupling between nearest Mn1-Mn2 atoms located at different sublattices. Moreover, this AF interaction in martensite is several times stronger than in austenite. Evidently, the martensite has lower magnetization and higher magnetic part of entropy, while the austenite is vice versa. As a result, a considerable large difference of saturated magnetization appears and gives the magnetization drop at the martensitic transformation upon cooling. An applied magnetic field can drive a structural transition towards to low temperatures and cause to increase ΔM. According to different contributions of magnetic and structural entropies for austenite and martensite, the inverse or negative MCE (ΔS > 0, ΔTad < 0) is realized in the vicinity of the structural transformation, while the conventional or direct MCE (ΔS < 0, ΔTad > 0) appears at the magnetic transition from FM to PM austenitic phase. A detailed investigation of Ni-rich Heusler alloys Ni52Mn48-xInx (x = 15.5, 16, and 16.5) carried out by Liu et al. [60] indicated that the composition with x = 15.5 is the only one that exhibit the structural transformation from martensite to austenite without any magnetic transition and, respectively, without any MCE. The further increase of In content leads to an appearance of the magnetic transitions, namely, the FM-PM transition in austenite and the magnetostructural transition from FM austenite to PM or AF martensite upon cooling. Moreover, the decrease of Mn2 atoms which occupy the In sites results to stabilize the FM austenitic phase in a wider temperature region, whereas the Tm temperature is found to decrease. The inverse MCE near the martensitic transformation was calculated by means of well-known Maxwell equations using the series of isothermal magnetization M(0H) curves for compositions with x = 16 and 16.5. Using the Maxwell relation / / the magnetic entropy change can be calculated as:

S = S ( 0 H , T ) - S (0, T ) 

0 H

 0

 M     0 dH .  T  0 H

(2)

It should be noted that the type of MCE (direct or inverse) can be derived from the sign of (∂M/∂T). In case of classical FM-PM transition, this derivative is negative; therefore the conventional MCE (ΔS < 0) occurs. As concerns to the AF-FM transition, the (∂M/∂T) and ΔS are positive. This case corresponds to the inverse MCE. Therefore, authors have concluded that the giant inverse MCE is observed in composition with less Mn content (x = 16.5) and its maximum value is about ΔS ~ 23 J/kgK under magnetic field change from 0 to 1.5 T. The reported ΔS occurs around the Tm of 270 K. In regard to compound with x = 16, the ΔS value reaches ~ 12 J/kgK in the vicinity of room temperature. The effect of hydrostatic pressure on the direct and inverse MCEs in the Ni50Mn34In16 alloy has recently been reported in detail by Sharma et al. [61]. The critical temperatures of this composition at ambient pressure are well known and their values are close to TC ≈ 305 K and Tm ≈ 240 K. The combined effect of applied magnetic field and pressure on the martensitic transformation temperature has shown the following tendency: while the magnetic field shifts the Tm temperature towards lower temperatures, application of pressure shifts the Tm towards higher temperatures. The increase in Tm with increasing pressure can be explained by the decrease in volume at the martensitic transformation through the Clausius-Clapeyron equation, which is valid for a first-order phase transition. Moreover, the applied pressure affects the inter-atomic separation, and as a result the magnetic interactions can be modified. Therefore the magnetocaloric properties of the alloy are also expected to change with pressure. Using the Maxwell relation, Sharma et al. have estimated the isothermal entropy change from iso-field thermomagnetization curves under applied external pressures from 0 to 9.5 kbar. These dependences for field changes of 2 and 5 T are presented in Fig. 19. It is clearly seen that for both cases ΔS peaks in the vicinity of the martensitic transformation is found to increase with pressure increasing from 0 to 9.5 kbar. Moreover, ΔS calculated under 9.5 kbar pressure from isothermal M(0H) curves for a field change of 5 T comes out as 20 J/kgK near 276 K. This value is close to the ΔS value (≈ 19 J/kgK) observed for Gd5Si2Ge2 alloy in a magnetic field change of 5 T [56].

Fig. 19. Temperature dependence of the isothermal magnetic entropy change in the Ni50Mn34In16 alloy under various applied pressures estimated for a field change of (a) 2 T and (b) 5 T. The authors also have reported that the effective refrigerant capacity (RC), however, decreases slightly with increasing pressure. But it still continues to remain close to 150 J/kg for a field change of 5 T as the pressure is raised from ambient to 9.5 kbar. Here, we would like to point out that the effective RC is another important MCE parameter which is a measure

of the amount of heat that can be transferred between hot and cool sinks during one thermodynamic cycle. If there are two different magnetic refrigerants which differ only in their RC value, then the one with higher RC is expected to perform better because of its capability to transport greater amounts of heat in a real cycle. In general, the refrigerant capacity can be defined as: RC 

Thot



SdT ,

(3)

Tcold

where Tcold and Thot are corresponding temperatures at full width half maximum of ΔS peak. It can be computed by numerically integrating the area under the ΔS(T) curve between Tcold and Thot. Kazakov et al. [62] and Dubenko et al. [23] have reported the direct measurements of adiabatic temperature changes (ΔTad) in the vicinity of the Curie and martensitic transition temperatures of Ni50Mn35In15 and Ni50Mn35In14Z (Z = Al and Ge) Heusler alloys. The measurements have been done under magnetic field changes up to 1.8 T using the adiabatic magnetocalorimeter working in the temperature range of 250-350 K. It was found that in case of all samples under investigation, the maximum ΔTad corresponded to -2 and 2 K for the first order martensitic transformation and for the second order magnetic phase transition in the austenite, respectively. The authors have observed that the substitution of In atoms by the fourth element (Z = Al or Ge) with 1 at.% leads to a shift of the critical transition temperatures towards the higher temperatures by up to 11 K relative to the parent compound (Ni50Mn35In15 with Tm ≈ 298 K and TC ≈ 320 K). The direct measurements of adiabatic temperature changes (ΔTad) near the Curie temperature of Ni50Mn25In25 and Ni45Co5Mn36.5In13.5 Heusler alloys under magnetic field variation from 0 to 2 T have been carried out by Buchelnikov et al. [63]. The magnetocaloric measurements were performed with the use of the Magnetocaloric Measuring Setup produced by Advanced Magnetic Technologies and Consulting Ltd (AMT&C) [64]. As we mentioned above, there is only magnetic phase transition from FM to PM austenite in the stoichiometric Ni50Mn25In25 alloy. The authors have shown that the maximal ΔTad (direct MCE) at the Curie point is close to 1.51 K at 317 K. In contrast, the Co addition into off-stoichiometric Ni50Mn36.5In13.5 alloy has resulted in an increase of the Curie temperature up to 400 K due to the fact of the strong ferromagnetic nature of Co. In this case, the value of direct MCE was found about 0.55 K. The influence of excess Ni atoms on the magnetic refrigerant properties of quaternary Ni50+xMn35-xIn14Si1 (x = 0, 1, and 1.5) alloys has been studied by Das et al. [65]. Concerning the critical temperatures of the studied alloys, the composition with x = 0 has Tm ≈ 275 K and TC ≈ 317 K, whereas the composition with x = 1 (1.5) has Tm ≈ 282 (285) K and TC ≈ 294 (292) K, respectively. The inverse MCE (ΔS) was estimated from magnetization measurements using the Maxwell relation. The authors have reported that the maximum magnetic entropy change obtained for the magnetic field change of 0 – 1.8 T was 28, 31.3, and 4 J/kgK for the compositions with x = 0, 1, and 1.5, respectively. As it should be from refrigerant capacity calculations, the highest RC (80.5 J/kg) is found for composition with x = 0, while the increase of Ni excess results in a decrease of RC. Note that practical applications require large ΔS and RC near room temperature at relatively low magnetic field change. Therefore, the authors have concluded that the Ni51Mn34In14Si1 alloy at temperatures close to the room temperature is the most promising for magnetic refrigerant application (ΔS ≈ 31.3 J/kgK and RC ≈ 57 J/kg). Recently, the same group of the authors has systematically investigated the effect of particle size on the magnetocaloric properties of the Ni51Mn34In14Si1 alloy [66]. Thereto, the arc melted compound was milled to reduce the average particle size and then vacuum annealed. Particle size varied from 850-1180 μm up to 20 μm. As a result, the ΔS is found to decrease but the RC increased to a maximum value and then decreased with

decreasing particle size for an applied magnetic field change of 1.2 T. The highest RC (40.4 J/kg) was observed for particles in the size range of 600-710 μm, while for the same particles the ΔS was close to 15 J/kgK. In contrast, both the lowest RC (3.5 J/kg) and ΔS (0.4 J/kgK) were found for composition with particle size of 20 μm. Using the information about heat capacity measurements, the authors also calculated the adiabatic temperature change for composition with particle sizes 600-710 μm under magnetic field change from 0 to 1.2 T. The ΔTad value of -1.99 K was exhibited at 278 K. This value may be compared with ΔTad = 1.5 K and 1.99 K reported for Ni50Mn35In14Z (Z = Al and Ge) alloys for 0ΔH = 1.8 T [62]. Therefore, this comparison serves to highlight the potential of Ni51Mn34In14Si1 alloy for magnetic refrigerant application. Generally, the authors have considered that powdered particles show better magnetic refrigeration ability due to their larger RC compared with the bulk sample. Takeuchi et al. [67] have studied the influence of partial substitution of In by Ga atoms in the Ni50Mn34.5In15.5 alloy on its magnetocaloric properties. The authors have done isothermal magnetization measurements in polycrystalline samples doped with 1.5, 3, 5, and 7.5 at.% of Ga under magnetic field change from 0 to 5 T. As it was shown that Ga-doping shifts the martensitic phase transformation towards room temperature, whereas the Curie temperature of austenite remains practically unchanged (TC ≈ 305 K). As an example, the martensitic transformation temperature can be varied in the temperature range from 280 K (parent alloy) to 298 K (sample with 7.5 at.% of Ga). Concerning the magnetocaloric properties, the authors have estimated the temperature dependences of ΔS by means of the Maxwell relation from magnetization data. Specifically, the ΔS-peak value of 12 J/kgK (at 280 K) is found for the undoped sample (Ni50Mn34.5In15.5), whereas this value increases by 200% (36 J/kgK at 292 K) for the sample with 3 at.% of Ga. The authors also reported that the relative cooling power for the latter sample has the RC value (≈ 180 J/kg) comparable to that of the first prototype material (Gd) which showed an extremely large magnetocaloric effect [55, 56]. The magnetocaloric properties of the Ni50Mn34.8In14.2B alloy have recently been studied by Dubenko et al. [68, 69]. The authors have combined direct measurements of ΔTad(T, 0H) with indirect ones from magnetization (M(T, 0H)) and specific heat (C(T, 0H)) curves. It is worth mentioning that the substitution of B for In results in a large shift of both Tm and TCM above TC, providing the first order magnetostructural transition from the FM martensite to PM austenite at 320 K upon heating. Therefore, the jump of magnetization will be negative and a negative (positive) entropy (temperature) change (direct MCE) is expected at Tm, respectively. The same situation was observed in Ni2+xMn1-xGa Heusler alloys [10]. The magnetocaloric characteristics as a function of temperature are displayed in Fig. 20.

Fig. 20. The adiabatic temperature changes obtained from direct ΔTad(T, 0H) and indirect C(T, 0H) measurements (a), and the magnetic entropy changes from C(T, 0H) and M(T, 0H) measurements (b) in the vicinity of the magnetostructural phase transition for 0ΔH = 1.8 T. Data have been taken from Ref. [68].

The magnetocaloric parameters, i.e., ΔS (2.9-3.2 J/kgK) and ΔTad (1.3-1.52 K) determined from direct and indirect methods under magnetic field change of 1.8 T are consistent with each other and comparable to that of Gd [55, 56]. Recently, Jing et al. [31] have performed the investigation of inverse MCE, namely, isothermal entropy change in Ni50Mn37-xFexIn13 (x = 1 - 4 at.%) alloys. As we mentioned above, the martensitic phase transition temperature is found to decrease dramatically with an increase of Fe substitution for Mn from 325 K (2 at.% of Fe) up to 140 K (4 at.% of Fe). For favorable potential application of magnetic refrigeration it is necessary that a refrigerant should exhibit the considerable MCE around room temperature. Therefore, compositions with less fraction of Fe should be taken into account. In order to investigate the MCE, the authors have done measurements of the magnetization isotherms in the vicinity of the martensitic transformation under applied magnetic field change of 8 T. The magnetic entropy change was calculated using the Maxwell relation. The authors have reported that the maximal ΔS for the Ni50Mn33Fe4In13 obtained at 48K and Ni50Mn34Fe3In13 obtained at 266K is about 31.2 J/kgK and 23.6 J/kgK, respectively. By comparing the above results, the MCE for the Ni50Mn35Fe2In13 is most remarkable and the ΔS of 38.7 J/kgK was obtained at 325 K under magnetic field change of 8 T. On the other hand, Feng et al. [40] have done entropy change calculations from the Maxwell relation in the polycrystalline Ni50Mn34In14Fe2 alloy. The reported positive ΔS values (inverse MCE) were found 26.5 and 53.6 J/kgK at Tm ≈ 303 K in applied magnetic field 5 and 8 T, respectively. These values are competitive with the ΔS of 25 J/kgK of the Ni50Mn35In15 alloy under 5 T at 301 K [70]. An effective method for enhancing the MCE and potential of the Ni-Mn-In alloy system for near-room temperature applications has been proposed by Sharma et al. [71]. The authors have prepared a Ni50(Mn, 2%Cr)34In16 alloy by substituting Mn by Cr in a parent Ni50Mn34In16 alloy. We would like to note that the parent alloy undergoes the austenitemartensite phase transformation around 240 K. It has been shown that a partial substitution of small amount of Mn (up to 2at.%) by Cr induces the shift of Tm towards higher temperatures, even though it means a decrease in the electron concentration in the alloy. This peculiar fact has been attributed to the effect of Cr on the ferromagnetic interactions in these alloys, in particular to a decrease in the ferromagnetic exchange (and therefore of the saturation magnetization) as a result of the partial substitution of Mn by Cr. The authors have shown that the Tm is close to 270 (294) K for composition with 1 (2) at.% of Cr, respectively. In order to estimate a quantitative measure of MCE (ΔS) for field changes of 1, 2, and 5 T, the common method in form of Maxwell’s relation was used. Estimation of inverse MCE for Ni50(Mn, 2%Cr)34In16 alloy has shown that the peak values of ΔS are 4.3 (11.1, and 24.4) J/kgK for the magnetic field changes of 1, 2, and 5 T, respectively. For comparison, the ΔS-peak value of 11.1 J/kgK for the present alloy in 2 T of applied field is larger than that of Gd (ΔS ≈ 10 J/kgK observed around 295 K in 7 T magnetic field [72]), while the ΔS-peak value of 24.4 in 5 T is larger than that for some of the Gd5Si2Ge2 alloys (ΔS ≈ 19 J/kgK observed close to 276 K in 5 T [55, 56]). This ΔS-peak value for a field change of 5 T in the present alloy is considerably larger than the ΔS ≈ 19 J/kgK obtained in the parent Ni50Mn34In16 alloy around 240 K for the same field change [73]. In order to judge the potential of the Ni50(Mn, 2%Cr)34In16 as a magnetic refrigerant, the authors also have estimated the RC of the alloy. The value of RC was found 90 J/kg for a field change of 5 T. This RC value is slightly smaller as compared with ≈ 100 J/kg of the Ni50Mn34In16 alloy for the same field change. In spite of that, the authors have concluded that the Ni50(Mn, 2%Cr)34In16 alloy is a better potential candidate for practical applications since its working temperature regime is at ambient temperature. The alternative assumption was declared by Sanchez-Alacros et al. [74]. The authors have investigated the effect of higher amount of Cr on Ni50Mn33-xIn17Crx (x = 0, 2, 3, and 4) compounds. It was found that a Cr-rich second phase appears for all Cr-doped compositions. The authors have related this fact with the low solubility of Cr in Ni-Mn-In system. It is worth

mentioning that no second phase has been reported in works on Cr-doped Ni-Mn-In alloys [71, 75]. Moreover, the fraction and the morphology of the appearing second phase drastically evolve with the increasing amount of Cr. According to magnetization measurements under different applied magnetic fields up to 6 T, the shift of Tm induced by a field, dTm/0dH, was significantly lower in the alloy with 4 % of Cr, which shows indeed a lower ΔS because of the presence of the nontransforming second phase. Besides, the ΔM jump (32 Am2/kg) in the parent alloy at 6 T was found more than twice as compared with ΔM (14 Am2/kg) in the alloy with 4 % of Cr. This fact can be attributed to the reduction in the FM exchange as a consequence of the partial substitution of Mn by Cr. As it can be found from the classical Maxwell relation, the maximum values of the inverse MCE are ΔS ≈ 8.4 J/kgK at 275 K and ΔS ≈ 1.4 J/kgK at 250 K for the parent Ni50Mn33In17 and Ni50Mn29In17Cr4 alloys under a magnetic field variation from 0 to 6 T, respectively. Therefore, the authors have concluded that the addition of high amounts of chromium to Ni-Mn-In may be highly detrimental to the achievement of large MCE and large magnetically induced martensitic transformation shifts, and thus to the potential applicability of these alloys. Recently, a breakthrough in research field related with MCE in Ni-Mn-In Heusler system has been accomplished by Liu et. al. [20]. The authors have done the magnetic and magnetocaloric measurements in Co-doped Ni45.2Mn36.7In13Co5.1 compound. It turned out that the substitution of Ni by Co leads to a shift of the structural transformation temperature towards higher temperatures and to the appearance of the non-magnetic martensitic phase. This sample undergoes the martensitic transformation from the non-magnetic martensite to FM austenite in temperature range from 317 to 327 K at low field of 10 mT and on heating. Whereas the martensite starts to form at 319 K and finishes at 311 K during cooling. An application of the magnetic field induces the negative shift of Tm with a rate of dTm/0dH  5.5 K/T. In order to directly measure the MCE, the authors have employed a device dedicated to adiabatic temperature change measurements (AMT&C) producing a magnetic field up to 1.9 T. In Fig. 21 we present the magnetization and ΔTad curves as a function of temperature for the Ni45.2Mn36.7In13Co5.1 alloy under magnetic field change of 1.9 T.

Fig. 21. (a) Low- and high-field magnetization as a function of temperature. (b) Adiabatic temperature change as a function of temperature in a magnetic field. The insets show the magnetic field dependence of adiabatic temperature change. Data have been taken from Ref. [20]. There is the giant inverse MCE (ΔTad ≈ -6.2 K) in the vicinity of the structural phase transformation, as indicated in Fig. 21b. Note that ΔTad is significantly higher than that reported for other Heusler alloys and is due to the highly dominant structural contribution in this system. As an example, the reported data on ΔTad (-1 K for ΔH = 1.9 T in Ni50Mn37Sn13Co1 [76], -0.6 K for ΔH = 1.3 T in Ni50Mn34In16 [77], -3.5 K for ΔH = 1.9 T in Ni50Mn34In16 [20], - 5.2 K for ΔH = 1.9 T in Ni50Mn35In15 [20]) are much smaller than in

Ni45.2Mn36.7In13Co5.1 alloy. Note that the advantage of the Co-doped alloy is in the appearance of giant MCE around ambient temperatures. It is worth noting that the giant MCE was obtained only in the first application of the field and was strongly reduced on subsequent runs of the field. The inset in Fig. 21b clearly shows irreversible behavior of MCE. The first application of field up to 1.9 T causes a cooling effect of -6.2 K. During removal of the field, the sample heats only by 1.3 K. After the field is applied for the second time up to 1.9 T and removed down to 0 T, a small cyclic temperature change of 1.3 K is obtained. This large discrepancy by magnetic cycles can be caused by the typically occurring but detrimental hysteresis losses for first-order transition materials. An irreversible energy loss directly lowers the efficiency of the magnetic refrigeration. Therefore, determining pathways to reduce hysteresis is the primary challenge in Heusler alloys [20]. As it was shown by Liu et al. [20], one way of solving the problem with hysteresis is related with application of the external pressure. It turns out that if the sample is magnetized without bias stress but demagnetized under a low external hydrostatic pressure of 1.3 kbar, the magnetic hysteresis can be significantly reduced. Therefore, the relative cooling power can be increased by simultaneously and precisely varying the magnetic field and pressure, when compared with the cooling power achieved when only the magnetic field is varied. Guillou et al. [78] have performed the calorimetric investigation of the Ni45Co5Mn37.5In12.5 single crystal by combining differential scanning calorimetry, heat capacity and magnetic measurements. According to magnetization measurements as a function of temperature, the authors have observed the martensitic transformation about 370 K, which was accompanied by a magnetic transformation from the PM martensite to the FM austenite during heating. The change of magnetization ΔM in the vicinity of the martensitic transition was equal to 90 Am2/kg. Both the isothermal entropy change ΔS(T) and the adiabatic temperature change ΔTad(T) were derived from heat capacity curves recorded in a series of magnetic fields up to 7 T. As it was reported, maximum values of ΔTad (ΔS) resulting from the application of 2, 5 and 7 T were found to be equal to −2.3 K (≈ 9 J/kgK), −6.1 K (≈ 22 J/kgK), and −9.4 K (≈ 28 J/kgK), respectively. It is worth noting that the maximum values of ΔS were observed to be in accordance with those determined from the Maxwell equation, whereas the ΔTad values are smaller than those reported by Liu et al. [20]. Theoretical description of physical properties in Ni-Mn-In alloys Phenomenological approach. Phenomenological analyses on the basis of a Landau theory for the phase diagram and magnetic properties of the alloy system Ni2Mn1-xIn1-x were made by Kataoka et al. [79]. For analysis the next free energy is used F

1 2 1 1 1 1 c2 e3  A3e33  I 4 e34   1M 2  J1M 4 2 3 4 2 4

1 2 1    G2  G4  e32 M 2  Q1e32 M 4  e3 P  Mμ 0 H , 2 2 6 

(4)

where e3 and M are the structural and magnetic order parameters, c2, A3 and I4 are to be expressed in terms of various elastic constants, χ-1 and J1 originate from the spin exchange energy, G2, G4, and Q1 represent the magnitudes of interactions between e3 and M, P is the external pressure, and 0H is the external magnetic field. Minimization of this free energy gives four phases labeled as paramagnetic austenite (Para-A), paramagnetic martensite (ParaM), ferromagnetic austenite (Ferro-A) and ferromagnetic martensite (Ferro-M). The calculated phase T-x diagram (which is presented in Fig. 22) is in a good agreement with experimentally observed data [21, 22, 80].

Fig. 22. Phase diagram of Ni2Mn1+xIn1-x Heusler alloy in the coordinates of temperature (T) and Mn excess (x) from [79]. The theoretical phase diagram [79] is depicted by dashed lines. The symbols are the experimental data which were taken from [21, 22, 80]. In a recent work, another sequence of phase transitions for Ni50Mn35In15 has been proposed by Bennett et al. [81], namely, from a paramagnetic (PM) cubic phase to a ferromagnetic (FM) tetragonal phase, and after that a transition to a ferrimagnetic (FRM) tetragonal phase. The transitions at both critical temperatures are first-order. In a closely related compound Ni50Mn34.8In14.2B, direct measurements of the magnetization and the differential scanning calorimetry have shown that Ni50Mn34.8In14.2B exhibits a magnetostructural first-order phase transition from a ferromagnetic martensitic phase to a paramagnetic austenitic phase [68]. In another example, measurements of the adiabatic temperature change and the latent heat in Ni2MnIn alloy revealed a transition from a FM to a FRM state that is also of the first-order [82]. The phenomenological Ginzburg-Landau theory for the description of the experimentally observed sequences of phase transitions in Ni50Mn35In15 Heusler alloy is proposed by Zagrebin et al. [83]. With the help of this theory the equilibrium states of an antiferromagnet with two ferromagnetic sublattices were found. For calculation the following free energy is used 1 1 1 1 1 F    m14  m24    m12 m22   1  m12  m22    2 m1m2  1e3  m12  m22  4 4 2 2 2 1 1 1 1  2 e3 m1m2  ae32  be33  ce34 . 2 2 3 4

(5)

In the Eq. 5, α, β and γ1,2 are the exchange constants; ω1,2 are the magnetoelastic constants; a, b, c are the elastic moduli of the second-, third- and fourth order, respectively, e3 is the structural order parameter, m1,2 are the dimensionless sublattice magnetizations. The analytical minimization of the functional (Eq. 5) leads to eight equilibrium states. In a general case, four different magnetic phases can exist: paramagnetic (P), ferromagnetic (F), antiferromagnetic (AF) and ferrimagnetic (FRM). In the considered approximation the symmetry of the paramagnetic and magnetoordered phases can be either cubic (C) or tetragonal (T) only. The phase diagram in the coordinates of intersublattice exchange and elasticity module of the second order (which are linearly dependent on the temperature) has been calculated. It is shown that the type of the phase diagram strongly depends on the values and signs of the free energy parameters.

Fig. 23. The calculated phase diagram of Ni50Mn35In15 Heusler alloy in the coordinates of intersublattice exchange and elasticity module of the second order. The phase diagram has been taken from Ref. [83]. The phase diagram is presented in Fig. 23. From the phase diagram, it follows that the following seven phases can exist: the paramagnetic cubic (PC), the ferromagnetic cubic and tetragonal (FC and FT), the antiferromagnetic cubic and tetragonal (AFC and AFT), the ferromagnetic cubic and tetragonal (FRMC and FRMT). Altogether, there are ten phase transitions on the phase diagram. As it was shown in Ref. [83], the thermodynamical path with the following phase transformations: the PС to the FT phase transition, then the FT to the FRMT phase can be found. This sequence qualitatively is in a good agreement with the experimentally observed phase transitions in Ni50Mn35In15 Heusler alloy [81]. Monte Carlo approach. In the previous section we mentioned about macroscopic theoretical models which could explain and predict the complex sequence of phase transitions in Heusler Ni-Mn-In alloys. In recent years, the rapid development of computer technology the Monte Carlo methods have been widely used for solving the problem of phase transitions and critical phenomena. Monte Carlo method is usually called a numerical method in which the solution of a completely deterministic problem is replaced by an approximate solution based on the introduction of stochastic elements that are missed in the original problem. Note that the Monte Carlo method can give arbitrarily accurate results depending on the available computer time. Today the model results of investigation of real magnetic materials are in a good agreement with the data obtained by other methods and with the experimental data. Theoretical model for the description of the direct and inverse MCE in Ni-Mn-X (X = In, Sn, Sb) Heusler alloys is presented in the work [84]. In this work a simple five-state Potts model with competing FM and AF interactions on a three-dimensional hypercubic lattice has been proposed. The AF interactions are considered between the nearest neighbors (nn) whereas the FM interactions are considered between the next-nearest neighbors (nnn). The Hamiltonian of the system is written as H  J afm  δ Si ,S j  J fm  nn 



 nnn 

Si , S j

 gμ 0μ B H ext  δ Si ,S g ,

(6)

i

Here, Jafm and Jfm are the magnetic exchange integrals of AFM and FM interactions; Si is the magnetic degree of freedom at the i-th lattice site which takes on the discrete values 1, 2 ,.., 5;

δSi,Sj is the Kronecker symbol which restricts spin-spin interactions to the interactions between the spins Si with same spin states; Sg is the spin, whose direction is determined by the external magnetic field 0Hext; μB is Bohr’s magneton; g is the Lande factor. Sums are taken over nn and nnn pairs in the first and second coordination spheres. The numbers of sites in the first and second coordination spheres are 6 and 12, respectively. The considered system is always PM or disordered at high temperatures. For the case of low temperatures the system can exist in several magnetic phases. The first phase is a FM state, in which one spin state, for example, 1 is favored on one sublattice, while a different spin state, say, 2 is favored on the other sublattice. The FM order disappears at Jfm→ 0. The second phase is an AF phase. It is possible when we have strong AF interactions between nn pairs. The AF state occurs when q/2 of the spin states are favored on one sublattice and the remaining q/2 states are favored on the other one. Finally, the third phase is a phase with a broken sublattice symmetry (BSS) in which one spin state is favored on one sublattice and the remaining q – 1 spin states on the other one. The simple model proposed takes into account only the magnetic subsystem and it can be applied to description of the origin of direct and inverse MCE in systems with competing interactions, for example, Heusler alloys. Simulations have shown that for the case of the strong AF interaction a sequence of FM→BSS→PM phase transitions can occur in the system. The inverse MCE (∆Smag > 0, ∆T < 0) takes place at FM→BSS transition, while the direct MCE (∆Smag < 0, ∆T > 0) occurs at BSS→PM transitions. For the case of strong FM interaction only the FM→PM phase transition occurs with the positive MCE. The same sequences of MCE in Heusler Ni-Mn-X (X = In, Sn, Sb) alloys are experimentally observed [13]. The presence of competing interactions can explain an occurrence of negative MCE in Heusler Ni-Mn-X (X = In, Sn, Sb) alloys. The magnetocaloric properties of Ni-Mn-In alloys have been studied from the theoretical point of view by Buchelnikov et al. [85]. The authors have developed the realistic Monte Carlo models for description of the positive and negative MCE in Ni-Mn-In alloys. The whole system in both models can be represented as consisting of magnetic and structural subsystems that interact with each other. For the magnetic part, the mixed 3-5 state Potts model has been considered. This model allows to simulate the magnetic phase transitions. A Mn atom has the spin magnetic moment S = 2 and therefore 5 spin states are possible (2S+1), while a Ni atom has S = 1 with 3 spin states. The structural part is described by the degenerated 3 states Blume – Emmery – Griffiths (BEG) model allowing for a structural transformation from the cubic (austenitic) phase to the tetragonal (martensitic) phase. The generalized Hamiltonian of the system (7) includes magnetic Hm, elastic Hel parts and magnetoelastic interactions Hint:

Hm 

NN

J

i , j 

m i , j Si , S j

δ

 gμ 0μ B H ext  δ Si ,Sg ,

(7)

i

NN

NN

i , j 

i , j 

N

NN

i

i , j 

Hel   J  σiσ j  K  (1  σi2 )(1  σ2j )  kBT ln( p)(1  σi2 )  K1gμ0μ B Hext σ gσ j  σiσ j (8) i

NN 1  1  1 NN H int  2  U i , j δ Si , S j   σi2   σ 2j    U i , j δ Si , S j . 2  2  2 i , j  i , j 

(9)

Here J im, j are the exchange parameters of the magnetic subsystem which may become negative depending on the degree of tetragonal distortion or disorder. Si is a spin defined on the lattice site i = 1… N. The Kronecker symbol δSi,Sj restricts spin–spin interactions to the

interactions between the same qMn states for Mn atoms and qNi states for Ni atoms. The other Kronecker symbol, δSi,Sg, couples the spin system to the external magnetic field 0Hext. Sg is called the ghost spin; its impact is such that a positive 0Hext favors spins parallel to the ghost spin. μB is Bohr’s magneton, g is the Lande factor. J and K are the structural exchange constants for the tetragonal and cubic states, respectively. The variable i = 1, 0, −1 defines the deformation state near each lattice site, with i = 0 as undistorted (cubic) phase taken to be p-fold degenerate, which allows us to approximately account for the high entropy of the lattice vibrations in the cubic phase. p is a degeneracy factor that characterizes the number of structural variants. The states i = 1 and −1 represent the distorted (tetragonal) phase. K1 is the dimensionless magnetoelastic interaction constant. g is a ghost deformation state, characterized by the structural variant which is favoured in an external magnetic field. T is the temperature. kB is the Boltzmann constant. Uij is the magnetoelastic interaction parameter. Summation is taken over neighbour pairs for Mn atoms in the first, second and third coordination sphere and in the first and second coordination sphere for Ni atoms. The isothermal entropy and adiabatic temperature changes with varying external magnetic field can be obtained from [86] Smag (T ,μ 0 H ext )  Smag (T ,μ 0 H ext )  Smag (T ,0) , Tad (T ,μ 0 H ext )  T

Smag (T ,μ 0 H ext ) C (T ,μ 0 H ext )

.

(10) (11)

Here, Smag(T, 0Hext) and Smag(T, 0) denote the magnetic entropy in presence of a magnetic field 0Hext and in zero field, respectively. C is the total specific heat. The values of austenitic and martensitic exchange constants have been taken from the ab initio simulations of Ni50Mn34In16 alloys [85]. Figure 24 shows the theoretical magnetic field dependence of the isothermal magnetic entropy change for Ni50Mn34In16. We notice that the entropy change for direct and inverse MCE changes linearly with increasing magnetic field. The same trends of the MCE for Ni-Mn-In alloys have also been observed experimentally [13].

Fig. 24. The calculated magnetic field dependence of entropy change for direct and inverse MCE for Ni50Mn34In16 [85]. We can observe from Fig. 24 the positive (direct) MCE (ΔSmag < 0, ΔTad > 0) at the FM-PM transition temperature T  305 K (close to room temperature) in the cubic (austenitic) phase and the negative (inverse) MCE (ΔSmag > 0, ΔTad < 0) at the metamagnetostructural phase

transition at T  220 K from the austenite to the martensite. We can see a good agreement between simulation results and experimental MCE data. The optimization of the magnetocaloric properties in Ni-Mn-In alloys by means of Monte Carlo simulations has been performed by Sokolovskiy et al. [87, 88]. The authors have used the same microscopic model as in the previous work, which consists of q-state Potts model and the degenerated Blume-Emmery-Griffiths model allowing to simulate a combined metamagnetostructural phase transition of first order. Within the proposed model, the temperature dependences of the magnetization, tetragonal deformation, and adiabatic temperature changes for magnetic field variation are obtained. In the paper [87], the authors have investigated the influence of the strength of the magnetic exchange parameters on the inverse and conventional MCE by scaling all exchange parameters by the same factor. It has been shown that different exchange interactions lead to various magnetocaloric values. The simulations have shown that a decrease in the magnetic exchange interactions leads to increased values of the inverse MCE and to minor changes in the direct MCE. The authors supposed that a reduction of the exchange interactions (Mn1–Mn2, Mn1–Ni, and Mn2–Ni) in the Ni–Mn–In alloy can be realized by doping with nonmagnetic atoms such as B, Si, Zn, Cu, etc. In our opinion, the quaternary Ni–Mn–In–Y Heusler alloys (Y = B, Si, Zn, Cu, etc.) are good candidates for refrigerants of magnetic cooling technology. This assumption has been verified in Ref. [88]. The simulations have been performed for 10 and 25 at% nonmagnetic atoms Y replacing Mn, Ni or both. For the calculation of magnetic and magnetocaloric properties of Ni50Mn34In16Y, the authors have used the nearest neighbour magnetic exchange couplings and magnetoelastic parameters of pure Ni50Mn34In16 composition. It has been shown that the addition of nonmagnetic impurities to ternary Ni50Mn34In16 alloys changes the temperature behavior of magnetization, martensitic transition and Curie temperatures as well as the magnetocaloric properties. Three types of distributing the impurities over the lattice of Heusler alloy Ni50Mn34In16 have been considered. Namely, Mn1 substituted by the nonmagnetic impurities in the Mn1 sublattice (case 1: Ni50Mn34−zYzIn16), or Mn1/Ni replaced by the impurity atoms on both sublattices (case 2: Ni50−zYzMn34−vYvIn16) and, finally, Ni substituted by the impurity on the Ni sublattice (case 3: Ni50-zYzMn34In16). It was found that both the increase in impurity concentration and various configuration cases can lead to a rapid drop in the magnetization curve near the martensitic transformation temperature. In this case, a phase transition from FM austenite to mixed FM–AFM (or PM) martensite was observed upon cooling. It is worth noting that the difference in magnetization values ΔM between different magnetically ordered crystallographic phases was found to decrease with an increase in impurity concentration depending on the impurity distribution except for case 1 (when the impurity atoms replace Mn1). A sharp drop in the magnetization around Tm with increasing impurity concentration occurred. This drop is less pronounced when impurities replace Ni (case 3). For impurities replacing Mn1 the authors observe the largest increase in the inverse MCE. Finally, the Cu doping as nonmagnetic impurity in Ni50Mn34In16 has been proposed by Sokolovskiy et al. [88]. The enhancement of the inverse MCE by nearly a factor of 2 was found in comparison with non-doped Ni50Mn34In16 alloy. This enhancement is associated with the change in magnetic exchange parameters favoring competing ferromagnetic– antiferromagnetic interactions. While the magnetic exchange parameters have been calculated from first principles, the structural part of the Hamiltonian must be treated in future with the same accuracy. Systematic experimental work studying Heusler systems Ni–Mn–(Ga, In, Sn, Sb) with nonmagnetic impurities would be highly welcome. Ab initio approach. The development of computers during the last decades has led to remarkable achievements in solid state theory as well as in other areas of natural science. Ab initio methods have become of ultimate importance in modern solid state physics. Ab initio

means to perform the calculations of the properties of a system from first principles with no parametrisation. The main goal of these methods is to solve the Schrödinger equation, that is in principle a perfect approach in order to obtain any desired information on the system under investigation. However, one has to make various approximations in order to solve this problem. Among many successful approximate approaches which allow to solve the ab initio problem, Hartree-Fock, Density-Functional and Korringa-Kohn-Rostoker methods form the basis for almost all current electronic-structure methods. The results of ab initio investigation of microscopic properties (generally magnetic properties) of Ni-Mn-In Heusler alloys are presented in [85, 89, 90-95]. The results of ab initio calculations of magnetic exchange coupling constants for nonstoichiometric Ni-Mn-In alloys are presented in [85]. Calculations have been carried out for stoichiometric Ni2MnIn, and non-stoichiometric Ni2Mn1.36In0.64, respectively, using the Korringa-Kohn-Rostoker method in combination with the coherent potential approximation. Exchange parameters have been calculated using Liechtenstein’s formula for small rotations of the spin moments with frozen potentials, which is good for ideal ferromagnetic systems, an additional averaging is required in case of atomic disorder so that J ij 

1 4

EF





dE ImTr{ i ij  j ij } ,

(12)

where ∆i is the difference in the inverse single-site scattering t-matrices for spin-up and spindown states, ∆i =1/ti↑ – 1/ti↑, and τ is the scattering path operator, which gives the scattered wave from site j due to a wave incident upon site i with all scatterings, and the trace (Tr) is over the product of the corresponding matrices. For disordered systems, the scattering path operator in the single-site coherent potential approximation (CPA) expression,  ij  [1   c00 (ti1  tc1 )]1   cij [1  (t j 1  tc1 ) c00 ]1

substituted in Eq. (12). Here tc and τc00 are the effective scattering matrices. The calculations have been done for the cubic structure in each case and also for the tetragonal structure in the case of Ni2Mn1.36In0.64. Calculations of non-stoichiometric Ni-MnIn alloys show that in the martensitic state, the Mn-excess atoms occupying the In sites interact with the Mn atoms on the Mn sublattice sites antiferromagnetically (Fig. 25).

Fig. 25. The ab initio magnetic exchange parameters Jij of Ni2Mn1.36In0.64 (a) with c/a = 1 (cubic phase) and (b) c/a = 0.94 (tetragonal phase) as a function of the distance d/a between the atoms in units of the lattice constant a [85]. It should be noted that the AF interaction dominates in the martensitic state. This strong AF

interaction can explain the complex sequence of the phase transitions experimentally observed in Ni-Mn-In alloys. Moreover the strong AF interaction is a reason also of the inverse MCE and exchange bias effect. The basic physical properties of the stoichiometric Ni2MnIn alloy have been investigated with the help of using Vienna Ab initio Software Package (VASP) [89]. The equilibrium value of lattice parameter of austenite in ferromagnetic state calculated in this work are in fair agreement with experimental or other theoretical values reported in the literature. The calculation of tetragonality (c/a ratio) has shown that there is only one energy minimum at c/a=1.0 for the stoichiometric Ni2MnIn. DOS calculations shown that the contributions to the total density of states are mainly from Ni 3d and Mn 3d electronic states. The composition-dependent lattice parameters, crystal structure, elastic properties, magnetic moment, and electronic structure of Ni2Mn1+xIn1−x (0 ≤ x ≤0.6) have been studied by using first-principles calculations (EMTO-CPA method) [91]. The authors showed that the Mn atoms on Mn sublattice (MnMn) are ferromagnetically coupled with the Mn atoms on In sublattice (MnIn) for the L21 austenite, whereas for the L10 martensite they are antiferromagnetically coupled, indicating that the martensitic transition occurs with the accompanying FM-AFM transition. It has been also shown that the energy difference between the AFM L10 phase and the FM L21 phase decreases linearly with increasing x. With x smaller than 0.32, the FM L21 phase is more stable than the AFM L10 phase, and, therefore, the magnetic phase transition (MPT) cannot occur. With x larger than 0.32, the FM L21 phase is less stable than the AFM L10 phase so that the MPT is expected. The theoretical critical composition x of about 0.32 for the MPT is in good agreement with the experimental measurement. The Mn-In disordering leads to decreasing stability of the martensite relative to the austenite, which may depress the MPT. The theoretical total magnetic moment of the L21 phase with x < 0.32 is in good agreement with that determined by experiments. The trend of the magnetic moment of the L10 phase against x agrees well with the experimental result but with smaller absolute value due to the possible Mn-In atomic disordering in the sample used in the experiments. The calculated electronic structure shows that the covalent bonding between the minority spin states of atoms plays an important role in both the magnetic and structural stability. The nature of unique magnetic properties and magnetostructural transformation of Ni2Mn1+xIn1-x shape memory alloy by first-principles calculations has been studied in Ref. [92]. The uncommon magnetic properties and type of magnetic ordering occurring upon martensitic transformation originate from the change of Mn-Mn interatomic distances. There is a critical value of 0.3 nm for Mn-Mn interatomic distance that corresponds to the crossover between ferromagnetic and antiferromagnetic exchange interactions for MnIn-MnMn. The density of states (DOS) near the Fermi level is an important factor that triggers the occurrence of martensitic transformation. A decrease of the intensity of minority-spin Ni 3d states at the Fermi level upon martensitic transformation has been observed for x=0.5. Moreover, the hybridization between the Ni 3d minority-spin states and the 3d states of the antiferromagnetically coupled Mn in the In site plays an important role in establishing the magnetic properties and driving the martensitic transformation. Calculations of the formation energies of various kinds of defects (atomic exchange, antisite, and vacancy) and the detailed magnetic properties in off-stoichiometric Ni-Mn-In alloys have been performed based on the density functional theory (DFT), using VASP package in Ref. [93]. The lowest formation energy in the studied series is the Mn antisite on the In sublattice (MnIn), it means that this kind of defects is most likely to form in the parent cubic phase during the synthesizing process. The In antisite on the Mn sublattice (InMn) and the Ni antisite on the Mn sublattice (NiMn) have moderate defect formation energies. The InNi antisite possesses the highest formation energy among all the antisite defects. This

information is of great importance to guide composition regulation during the fabrication process of these alloys. The value of the Ni magnetic moment sensitively depends on the distance between Ni and In. The smaller is the distance, the larger will be the Ni moment. When the extra-Mn occupies a Ni site, most of the free electrons gather around the extra-Mn. But when the extra-Mn occupies an In position, the charges are regularly distributed between Ni and Mn atoms. Therefore, various arrangements of the free electrons cause a huge difference of the extra-Mn moment in the MnNi defect (1.07 µB) and MnIn defect (3.62 µB). The magnetic interactions in Ni2Mn1.4In0.6 have been studied by means of ab initio spin polarized relativistic SPR-KKR Green’s function calculations [94]. The authors also attempted to explain the origin of AFM interactions that coexist with ferromagnetic ones. They have shown that the origin of AFM interactions present in the martensitic phase of Ni2Mn1.4In0.6 lies in superexchange interactions between Mn atoms mediated by Ni. The Xray absorption spectroscopy at Ni L2,3 edges in Ni2MnIn and Ni2Mn1.4In0.6 indicates a substantial increase in hybridization between Ni and Mn atoms. This observation is further supported by spin polarized DOS calculated for the two compounds. As a result of increased hybridization, a redistribution of electrons takes place between the Ni 3d–Mn 3d, hinting that superexchange like interactions are at play. The spectrum simulations for the Ni48Co5Mn35In12 composition using a relativistic realspace full multiple scattering code (IFEFFIT 9.0) have been performed by Klaer et al. [95]. The ab initio calculations of the electronic band structure reveal an insight in the phase transition mechanism. From the corresponding partial density of states it is obvious that the tetragonal distortion of the system causes a change of the Ni d-states directly at the Fermi edge. It has been shown by Ayuela et al. [96] for Ni2MnGa that this shift corresponds to a splitting of the eg orbitals of Ni which are degenerated in the cubic phase. For a tetragonal distortion with c/a > 1 the dx2−y2 orbital is shifted into the unoccupied part of the DOS. The splitting of the DOS peak at the Fermi energy causes a decrease of the kinetic energy favoring the tetragonally distorted structure in the ground state (0 K). The structure with higher symmetry favors with increasing temperature. Summary

In this chapter, we have reviewed the most important features of ternary and quaternary ferromagnetic shape memory Ni-Mn-In and Ni-Mn-In-Z materials, which were experimentally and theoretically obtained in the last three years. Nowadays, ternary and quaternary ferromagnetic shape memory Ni-Mn-In and Ni-Mn-In-Z Heusler alloys are ones of famous multifunctional materials exhibiting many interesting features in the vicinity of a martensitic transformation due to the strong interrelation between crystal structure and magnetic order. The multiferroic, magnetoresistive, martensitic and related magnetic shapememory behavior as well as magnetocaloric properties are the examples these unique features. Generally, a tuning of both structural and magnetic transition temperatures can be useful to achieve better functional properties. The complete analysis of the available data about properties of Heusler alloys helps to reveal the fundamental problems of the achievement of the complex of multifunctional properties. These properties rendered to novel materials will provide new opportunities for solving large-scale applied problems such as a creation of magnetocaloric devices and new technologies for actuators, which can be applied in an instrument engineering industry, micro and nanomechanics etc. Today, the optimization problem of Heusler compounds is of a great importance.

Acknowledgments

This work is supported by RSF No. 14-12-00570 (Theoretical description: Monte Carlo and ab initio approach), Ministry of Science and Education RF No. 3.2021.2014/K (Theoretical description: phenomenological approach), RFBR Grant Nos. 14-02-01085, 14-02-31189, and the Creation and Development Program of NUST “MIS&S” References

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