Overview No. 145 Metamagnetic transitions, phase coexistence and metastability in functional magnetic materials

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Acta Materialia 56 (2008) 5895–5906 www.elsevier.com/locate/actamat

Overview No. 145 Metamagnetic transitions, phase coexistence and metastability in functional magnetic materials S.B. Roy a, P. Chaddah a,c, V.K. Pecharsky b, K.A. Gschneidner Jr. b,* b

a Magnetic and Superconducting Materials Section, Raja Ramanna Centre for Advanced Technology, Indore 452 013, India Ames Laboratory of the US Department of Energy and Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011-3020, USA c UGC-DAE Consortium for Scientific Research, Indore 452001, India

Received 21 July 2008; accepted 21 August 2008 Available online 14 October 2008

Abstract Magnetic field-induced transitions (metamagnetic transitions) play an important role in defining functionality of various classes of magnetic materials. Rare earth manganites showing colossal magnetoresistance and Gd5(Ge1xSix)4 alloys showing a giant magnetocaloric effect are typical examples that are of interest to the solid-state physics, chemistry and materials science communities. The key features of the metamagnetic transitions occurring in these systems are phase coexistence and metastability. This generality is highlighted by comparing experimental results characterizing three different classes of magnetic materials. A generalized framework of disorder-influenced first-order phase transition is introduced to understand the experimental data, which have considerable bearing on the functionality of these model materials. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetism; Phase coexistence; Phase transformations; Metamagnetism; Magnetostructural transformations

1. Introduction This review is concerned with magnetic materials where temperature- and magnetic field-induced first-order magneto-structural phase transitions lead to interesting phenomena that are of both fundamental and technological interest, such as the giant and colossal magnetoresistance, giant magnetocaloric effect and colossal magnetostriction. According to the Ehrenfest classification, a first-order phase transition (FOPT) is accompanied by discontinuities of the first derivatives of the free energy taken with respect to the thermodynamic variables that trigger the phase transformation. When control variables are temperature (T) and magnetic field (H) at atmospheric pressure, one can state that an FOPT occurs across a TC(H) boundary, where TC is the critical transformation temperature if one observes a discontinuous change in entropy, S, or a discon*

Corresponding author. Tel.: +1 515 294 8220; fax: +1 515 294 9579. E-mail address: [email protected] (K.A. Gschneidner Jr.).

tinuous change in magnetization, M. The FOPT would then be confirmed if the latent heat and the magnetization jump satisfy the Clausius–Clapeyron relation. These stringent requirements pose experimental difficulties when either the latent heat or the magnetization discontinuities are small. In the absence of disorder, which is rare, the FOPT may occur at a sharply defined (TC, HC) boundary [1]. Most of the functional magnetic materials of interest, however, are multicomponent systems whose properties become more interesting with substitutions, and thus have intrinsic frozen disorder. Early theoretical arguments of Imry and Wortis [2] showed that this would result in a rounding of the FOPT. There would be a landscape of free energy densities, and the sharp (TC, HC) boundaries would now be defined over individual regions having length scales of the order of the correlation length. This, in turn, would lead to a band of (TC, HC) boundaries, with each of them corresponding to a different local region. The transition between two magnetic phases would then occur over a

1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.08.040

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range of T (or H), and the two phases would grow (or diminish) as this range is traversed. The state of each local region is dictated by how far away from the ambient T and H the TC and HC of that region are. The coexisting phases would collapse into a single homogeneous phase as this broadened (TC, HC) band is exited. For such a broadened FOPT, it becomes difficult to distinguish the latent heat from a peak in specific heat [3]. In such cases the characteristic feature of hysteresis in a physical property, which varies sharply between the two phases, is used to identify an FOPT [3]. This hysteresis is usually observed with cycling of T (or H) across the (TC, HC) band, and is due to supercooling or superheating associated with an FOPT [4]. Hysteresis in a broad transition indicates that phase coexistence and metastability are present across an FOPT. While examples of phase coexistence and metastability across a temperature-induced FOPT are abundant in nature (e.g. a water–ice mixture, and supercooling and superheating of water, respectively), the same features associated with a magnetic field-induced FOPT are not uncommon either. In fact, field-induced antiferromagnetic to ferromagnetic transitions, commonly known as metamagnetic transitions [4], and the associated phase coexistence and metastability in various classes of magnetic materials have recently been drawing much attention. It is now recognized that metamagnetism, which is accompanied by a structural transition, plays an important role in defining properties of at least two prominent classes of magnetic materials, namely manganite compounds, which are well known for colossal magnetoresistance (CMR), and Gd5(GexSi1x)4 compounds, best known for the giant magnetocaloric effect (GMCE). In this review, we shall highlight the experimental evidence of phase coexistence and metastability associated with FOPT in these two classes of magnetic materials. Many common features found in these otherwise quite distinct types of materials systems – CMR manganites and GMCE materials – point to a common underlying physics at least at the phenomenological level. We believe that the disorder-influenced first-order phase transition provides a fundamental framework towards understanding a wide variety of experimental results in different classes of magnetic materials. The described phenomenology is not necessarily limited to these two classes of materials only, and can be observed in any system with disorder-influenced FOPT irrespective of the electronic mechanisms responsible for their magnetic states. To underscore this generality, we shall start the discussion with the experimental evidence of phase coexistence and metastability in a test bed system, namely doped CeFe2 alloys, which belong to an entirely different class of magnetic materials, and argue how their physical properties can be understood within the framework of a disorder-influenced FOPT. We shall then review the H–T dependencies of various physical properties across an FOPT in CMR manganites and a representative GMCE material – Gd5Ge4 – highlighting the commonalities between different systems, and how these properties influ-

ence the interesting functionalities (e.g. CMR and GMCE) in these materials. Finally, we show that, under certain temperature and field conditions, the process of FOPT may become kinetically arrested, giving rise to a low-temperature non-ergodic magnetic state with glass-like dynamics. This magnetic state is quite different from the wellknown spin-glass state. We also point out that the ease with which one can change magnetic field and temperature in these unconventional magnetic glasses (in comparison with pressure and temperature in structural glasses), can make these systems ideal models for studying the physics of glass-transformation in a two-parameter H–T phase space. 2. Metamagnetic transition and phase coexistence in systems with a low-temperature antiferromagnetic state 2.1. Doped CeFe2 alloys – a test bed materials system CeFe2 with the cubic C15 Laves phase structure is a ferromagnet with a Curie temperature TC = 230 K [5]. Small (3–6%) substitutions of certain elements such as Co, Al, Ru, Ir, Os and Re induce (stabilize) a low-temperature antiferromagnetic (AFM) state in this otherwise ferromagnetic compound [6,7]. The ferromagnetic (FM) to AFM transition in these alloys is accompanied by a cubic to rhombohedral structural distortion and a discontinuous change of the unit cell volume [8]. The application of an external magnetic field to the low-temperature AFM state causes the sample to reset to the original FM state, while at the same time eliminating the structural distortion and recovering the original cubic structure. This clearly implies that the FM–AFM transition temperature will shift in the presence of a magnetic field. Fig. 1 shows the real components of the ac magnetic susceptibility (v0 ) for Ir- and Ru-doped CeFe2 samples as a function of temperature [9]. A sharp increase in v0 at the Curie temperature, TC, with decreasing T marks the onset of the paramagnetic (PM) to FM transition. Below TC, the susceptibility becomes nearly constant before decreasing sharply at a lower temperature, TN, which is indicative of an FM–AFM transition. Thermal cycling has no effect on the PM–FM transition, and this is in accord with the second order nature of this transformation. Within the same experimental resolution, however, distinct thermal hysteresis is observed across the FM–AFM transition [9]. The hysteresis is a signature of an FOPT. The phase coexistence across this FOPT was studied in detail using the minor hysteresis loop (MHL) technique [9]. This technique was originally developed to study a first-order vortex matter phase transition in type-II superconductors [10,11] and has also been used later to study austenite to martensite phase transition in shape memory alloys [12]. Fig. 2 shows magnetization vs. temperature plot for a 4% Ru-doped CeFe2 sample measured in an applied field of 20 kOe [13]. Three different measurement protocols were used: zero-field cooled (ZFC) heating, field-cooled cooling

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Fig. 1. The in-phase ac-susceptibility (v0 ) vs. temperature (T) plots for Ir- and Ru-doped CeFe2 alloys (from Ref. [9]).

temperature can be reduced by the application of external field. This, in turn, gives rise to a giant magnetoresistance effect, which is clearly visible in Fig. 3. A schematic H–T phase diagram based on the magnetization measurements for this 4% Ru-doped CeFe2 alloy is

Fig. 2. Magnetization (M) vs. temperature (T) plots for Ce(Fe0.96Ru0.04)2 alloy (from Ref. [13]).

(FCC) and field-cooled warming (FCW). A rapid rise of M with decreasing T below 210 K indicates the onset of PM–FM transition and it is thermally reversible. The FM–AFM transition is marked by a sharp drop in M below 50 K, and it shows a substantial thermal hysteresis, which once again signifies an FOPT. It should be noted that the FCC curve does not merge with the ZFC curve down to the lowest measured temperature of 5 K. Similar measurements with applied fields varying between 100 Oe and 30 kOe show that thermal hysteresis broadens with increasing H, and when the magnetic field exceeds 15 kOe, the MFCC(T) and MZFC(T) curves do not merge. We note that for applied fields below 15 kOe MFCC(T) does overlap with MZFC(T). The onset of the FM–AFM transition gives rise to a marked increase in the electrical resistivity (see inset of Fig. 3). The FM–AFM transition

Fig. 3. Resistivity ratio vs. temperature (T) plots for Ce(Fe0.96Ru0.04)2 alloy (from Ref. [13]).

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shown in Fig. 4, where TNW and TNC are the temperatures where the beginning of the sharp rise and fall of M are observed in the ZFC and FCC cycles, respectively, as marked in the inset of Fig. 4. The boundary labeled T* is the low-temperature boundary where MZFC(T) and MFC(T) merge, and T** is the high-temperature counterpart (T** and TNC are nearly coincident in the magnetization measurements, and therefore only TNC is marked in Fig. 4). A similar H–T phase diagram can be obtained from resistivity measurements, and in such measurements (under the same experimental protocol) T** and TNC can be clearly distinguished [13]. Note that TNW(H) < TNC(H), i.e. the onset of nucleation of the AFM state on cooling occurs at a higher temperature than does nucleation of the FM phase during warming. This feature is a signature of a disorder-broadened FOPT. Such an influence of disorder was also observed in the same sample in the fieldinduced AFM–FM transition [14]. Fig. 5 is a schematic representation of the free energy density (f) expressed as a function of an order parameter S as f(T, S) = (r/2)S2  wS3 + uS4 for a first-order transition, where w and u are positive temperature-independent constants. At T = TN, the high- and low-temperature phases coexist. The standard treatment [4] assumes that r(T) = a[T – T*], where a is positive and temperature independent, and where d2f/dS2 at S = 0 vanishes at T = T*. The limit of metastability on cooling is reached at T* = TN – w2/(2ua). The height of the barrier which separates the stable and metastable states at the transition temperature (see curve a in Fig. 5) decreases with decreasing temperature (see curves b and c in Fig. 5) and finally disappears at T* (see curve d in Fig. 5). Finite energy fluctuations, however, can destroy the supercooled state in the temperature regime T* < T < TN. Similarly, T** is the limit of metastability on warming. In the low-temperature AFM regime a transition from the zero-field AFM state to the FM state can be induced by application of H. As in the case of temperature variation, this field-induced first-order AFM–FM transition or metamagnetic transition [4] is also

Fig. 4. H–T phase diagram for Ce(Fe0.96Ru0.04)2 alloy.

Fig. 5. Schematic representation of free energy for T* < T < TN. The high-temperature state remains in a local minimum and is stable against infinitesimal energy fluctuations in this temperature regime. The height of the barrier between the minima corresponding to the high and lowtemperature phases at TN (curve a) decreases with decreasing T and then disappears at T* (curve d). Curves b and c represent the free energy at two intermediate temperatures between TN and T* with Tb > Tc.

marked by distinct hysteresis and phase coexistence, and, accordingly, the limits of metastability H* and H** can be defined. The H–T phase diagram in Fig. 4 shows that it is possible to retain residual FM state in this 4% Ru-doped CeFe2 alloy down to the lowest temperature of measurements by following the FCC path with applied H P 15 kOe. This is exactly what is known as field annealing of the FM state in the manganese oxide systems showing CMR effect [15]. It has been shown that this residual FM obtained by the FCC path is quite metastable in nature, and can be converted into a stable AFM state by field cycling [13,16]. All of these observations can be rationalized in terms of supercooling of the FM state. While cooling across the first-order FM–AFM transition, some of the FM state will supercool into the temperature regime well below the transition boundary. It is clear from Fig. 4 that with the applied H < 15 kOe the supercooled FM state will cease to exist below a finite T and one can reach the stable AFM state. This is indicated by the merger of the FC and ZFC magnetization (see Fig. 2). With H P 15 kOe some of the supercooled FM state remains in the system down to the lowest T of measurement. The region between TNC(H) and T*(H) boundaries in Fig. 4 marks the phase coexistence region formed during the cooling path. This region consists of mixtures of AFM and FM clusters, and is metastable in nature. A new concept, namely the ‘‘lack of end point memory effect” [17], has been used to study the metastable nature of this phase coexistence regime in detail [13,16]. Any field cycling in this metastable phase coexistence region (obtained via an FCC path) introduces energy fluctuations, which drive the clusters of the metastable FM state to the stable AFM state. The metastable nature of the phase coexistence region is highlighted further through the strong relaxation effects observed in magnetization

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(Fig. 6) and resistivity studies. For a more detailed description and analysis of these experimental results, readers are referred to Refs. [13,14]. The actual composition in any alloy or doped compound varies around some average composition simply due to disorder that is frozen in as the solid crystallizes from the melt [18]. It was proposed earlier [2] that such static, quenched-in, purely statistical compositional disorder can under certain circumstances introduce a landscape of transition temperatures in a system undergoing FOPT. The effect of quenched disorder in an FOPT was known to the metallurgists for ages, but a systematic study of the effect of disorder on an FOPT process started only in late 1970s [2]. Detailed computational studies confirmed the applicability of disorder-influenced FOPT in CMR manganese oxide compounds and further emphasized that phase coexistence can occur in any system in the presence of quenched chemical disorder whenever two states are in competition through an FOPT [19,20]. Such an intrinsic disorder-induced landscape of transition temperatures/ fields has actually been observed across the vortex solid melting transition in a high-temperature superconductor BSCCO [21]. The applicability of such a picture in the AFM–FM transition of the doped CeFe2 alloys has now been pointed out through an imaging study of the AFM–

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FM transition using a micro-Hall probing technique [16]. It was observed that, in both T and H variation measurements in doped CeFe2 alloys, the FM clusters of various sizes appear in random positions of the sample at the onset of the AFM–FM transition. As the temperature or field is increased, newer FM clusters appear until the whole sample is converted to the FM state. This is clearly indicative of the local variation of the AFM–FM transition temperature (TN) or field (HM) leading to a rough TN/HM landscape. This distribution of TN or HM gives rise to the impression of a global rounding of the transition in bulk magnetization measurements (or other physical property measurements). The imaging study also provided a visual proof of supercooling of the FM state across the AFM–FM transition and that such supercooled state can easily be destabilized with small energy fluctuations [16]. In less disordered samples, the growth process of the clusters is relatively fast, with a smaller number of nucleating clusters, which suggests that different disorder landscapes can control nucleation and growth with the key point that if the growth is slow enough, percolation will occur over an observable T or H interval before phase coexistence collapses. Such percolating behavior can be controlled by subtle changes in sample doping, and the ramifications to tuning functionality of the CMR manganese oxide systems, for example, are obvious [16,22]. Ru-doped CeFe2 alloys show an inverse magnetocaloric effect (MCE) associated with the FM–AFM transition, apart from a conventional MCE at the higher-temperature PM–FM transition. As opposed to the conventional MCE, the inverse MCE is cooling rather than heating by adiabatic magnetization. This requires an increase in configurational entropy by applied magnetic field. The inverse MCE has been reported for several other materials exhibiting first-order magnetic transitions, such as FeRh [23]. Such magnetic transition leads to the occurrence of a magnetically inhomogeneous state in the vicinity of a transition point. As mentioned above, doped CeFe2 alloys indeed show such inhomogeneous phase coexistence across the magneto-structural AFM–FM transition [13,16]. In a recent study, Pecharsky et al. [24] suggested that, in systems with first-order magnetostructural transitions, the large MCE is the sum of the magnetic entropy-driven contribution plus the difference in entropies of the two crystallographic structures associated with the phase transition process. Accordingly it can be argued that a substantial portion of the entropy change at the ferromagnetic to antiferromagnetic transition in Ru-doped CeFe2 alloys originates from the associated rhombohedral to cubic structural distortion [8]. 2.2. Manganese-oxide compounds showing CMR

Fig. 6. Normalized magnetization of Ce(Fe0.96Ru0.04)2 (M/M0) measured as function of time (t) at various fixed temperatures along the FCC path. The solid lines represent the fits using the KWW exponential function. The inset shows the characteristic relaxation time s obtained from the fitting using the KWW function as a function of T (from Ref. [69]).

One of the earliest reports of a transition from a metallic FM state to an insulating charge-ordered AFM state has been published for the manganese oxide system La0.5Ca0.5MnO3 [25]. The charge-ordered state is charac-

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terized by a real space ordering of nominally Mn3+ and Mn4+ ions. When temperature decreases, magnetization drops sharply at the FM–AFM transition temperature (TN) and the resistivity (q) changes by several orders of magnitude. Both M(T) and q(T) show thermal hysteresis across TN, and these were taken as an indication of the first-order nature of this transition [26]. In the vicinity of the AFM–FM transition in La0.5Ca0.5MnO3, a drastic change of lattice constants was also observed [26]. A similar first-order magneto-structural ferromagnetic to antiferromagnetic transition has also been observed in Nd0.5Sr0.5MnO3 (NSMO) [27,28]. Inside the AFM state, an AFM–FM transformation can be induced by the application of an external magnetic field. This field-induced metamagnetic transition is again clearly marked with sharp changes in various physical properties and accompanying hysteresis [27,28]. An H–T phase diagram of the NSMO compound highlights a hysteretic region around the TC(H) boundary, which broadens with temperature decrease [27]. This FM metallic to AFM charge-ordered insulator transition in various manganese oxide systems has been a subject of extensive experimental scrutiny, which amongst other things has also revealed the electric field-induced change of the charge-ordered state. Detailed discussion of all these interesting results, however, is beyond the scope of this review, and only a few key experimental observations concerning the magneto structural AFM–FM transition in various manganese oxide systems are enumerated below. For detailed information on CMR manganese oxides the readers are referred to the excellent monograph by Dagotto [29] and recent reviews by Goodenough [30] and Tokura [31]: (1) The AFM–FM transition is manifested by sharp changes in various physical properties (e.g. magnetization and resistivity) both in T and H varying measurements, and is accompanied by a pronounced hysteresis [32]. There exists a clear evidence of phase coexistence across the transition and this will be elaborated below. (2) The extent of hysteresis in the temperature-dependent measurements broadens with the increasing applied magnetic field [32]. (3) Detailed investigation of the first-order AFM insulator to FM metal transition in Nd0.5Sr0.5MnO3, carried out by resistivity and magnetization measurements [28] reveals a hysteretic region that expands with decreasing temperature (see Fig. 7). Isothermal resistivity vs. H plot finally shows an open hysteresis loop at the lowest temperatures (see Fig. 7(a)) [28]. (4) In various manganese oxide systems, it is possible to retain the FM state (at least in part) in the temperature regime well below TN when cooling across the AFM ordering temperature in the presence of an applied magnetic field. This phenomenon is known as field annealing of the FM state [15]. There are

Fig. 7. Resistivity and magnetization of Nd0.5Sr0.5MnO3 as functions of magnetic field (from Ref. [28]).

some general results indicating that cooling in a certain non-zero field followed by warming in a lower non-zero field leads to an increased concentration of the FM metallic component [33–35]. By following different measurement protocols, one can control or tune the ratio of coexisting FM metallic (FMM) and AFM insulating (AFI) phases at low temperatures [33–35]. Tunability of coexisting FMM and AFI phases provides additional degree of control over the functionality in these manganese oxide compounds. (5) After a closer inspection, the finite width of the transition is clearly visible even in single crystals. In the field-induced transition at low temperatures, the magnetization changes via distinct steps [36]. (6) There is a structural change accompanying the FM– AFM transition in Nd0.5Sr0.5MnO3 [27]. While the phenomenology of the CMR effect in manganese oxide systems can be explained within the framework of double exchange interaction, this mechanism alone is insufficient to explain the observed effects quantitatively [29]. A promising approach in this regard is a picture including the formation of a percolation path involving the metallic FM phase in the phase coexistence regime across the AFM–FM transition, which can be manipulated by the applied magnetic field [29]. Hence, this AFM–FM transition region has been a subject of much scrutiny in the recent past using various techniques, including microscopic imaging with electron microscopy [37] and electron holography [38]. Distinct phase coexistence on a micrometer scale has been reported [37,38], leading to the visualization of a percolating path [21]. Such phase coexistence on the micrometer scale has been rationalized within the framework of a disorder-influenced first-order phase tran-

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sition [19,29]. In addition, the lattice distortions and longrange strain are known to be important in manganese oxides [39], and the intrinsic complexity of systems with strong coupling between electronic and elastic degrees of freedom introduces further interesting features in the phase coexistence states [40,41]. Similar resistivity and magnetization measurements have been made on Pr0.5Sr0.5MnO3 (PSMO), which also displays an FM–AFM transition with decreasing temperature [35]. Here again one can control or tune the ratio of coexisting FM metallic and AF insulating phases at low temperatures by following non-conventional pathways in the H–T phase space. In these pathways the sample is cooled to a specific temperature (say 5 K) in one value of the field (HC), and magnetization or resistance are measured as the sample is warmed in a different magnetic field (HW). Data can also be collected when the field is varied isothermally at the lowest temperature. This is a generalization of the so-called ZFC measurement protocol, where the sample is cooled in a zero field and warmed in the measurement field. For both NSMO and PSMO, and in both q(T) and M(T) measurements, it was found that the behavior during warming depends qualitatively on the sign of (HC – HW) [35]. Two sharp changes of opposite sign are observed during warming when (HC – HW) is positive and only one sharp change when it is negative (see Fig. 8). Several CMR Mn oxide compounds show a substantial MCE associated with the first-order AFM–FM transition [42–45]. A few cases are discussed here in the context of this review, and for a general survey of the MCE in manganites and other materials the readers are referred to reviews by Gschneidner et al. [46] and Phan and Yu [47]. A large MCE has been observed in the vicinity of the first-order transition from AFM insulator phase to FM metallic phase taking place in Nd0.5Sr0.5MnO3 [42]. Here, the magnetocaloric effect is about three times larger than that observed near the PM to FM transition in the same compound

Fig. 8. Magnetization of PSMO measured during warming from 5 K in a magnetic field of 20 kOe. The three curves, as labeled, correspond to the sample being cooled from 300 K in fields of 30, 20 and 10 kOe. In two cases (30 and 10 kOe), the field is changed to 20 kOe isothermally at 5 K. The cooling history dependence is striking (from Ref. [35]).

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[42]. A similar correlation between large MCE and firstorder AFM–FM transitions has also been observed in Pr1xCaxMnO3 [43], Eu0.55Sr0.45MnO3 and La0.27Nd0.47Ca0.33MnO3 [44], and La1xSrxMnO3 [45]. It is quite clear that the magnetic inhomogeneity arising from the AFM– FM phase coexistence across the disorder-broadened AFM–FM first-order phase transition, as well as additional configurational entropy associated with the structural transition contribute to the observed large MCE in these CMR Mn oxide compounds. 3. Metamagnetic transition and phase coexistence in systems with low-temperature ferromagnetic state 3.1. Giant MCE material Gd5Ge4 Magneto-structural transitions are at the heart of various interesting functional properties in Gd5(GexSi1x)4 compounds. We elaborate on this topic below using one of the parent compounds of this series – Gd5Ge4 – as an example. This alloy (and many other Gd5(GexSi1x)4 materials) has a complex Sm5Ge4-type orthorhombic structure at room temperature (Fig. 9a). It may be described in terms of strongly interacting monolayers of atoms forming tightly bound, nearly two dimensional, 0.7-nm-thick slabs [48]. Because of the strong bonding between the monolayers that form the slabs, the latter are remarkably rigid – the slabs undergo negligible changes upon transition from one layered Gd5(GexSi1x)4 structure to another as, for example, when their chemical composition (x) varies [49]. On the other hand, interactions between the slabs vary widely, and therefore they may be stacked one upon another with different lateral displacements along the aaxis. This unique layering supports a surprisingly flexible crystallography, intriguing physics and, in many instances, displacive, martensitic-like structural changes that take place when the temperature, magnetic field or pressure varies [50]. The Gd5Ge4 compound orders antiferromagnetically at TN = 128 K [51]. As follows from bulk magnetization measurements of a single crystal [52] and microscopic data obtained from X-ray magnetic resonant scattering [53], the individual slabs are ordered ferromagnetically with the net FM moment along the c-axis, although the coupling between the slabs is AFM. This indicates that the intraslab exchange interactions in the room temperature polymorph of Gd5Ge4 are strong, while the interslab exchange interactions are weak. The latter may be related to weak chemical [-Gd-Ge-Ge-Gd-] interslab interactions, and to the resulting lack of 5d–4p hybridization [54]. In magnetic fields lower than 10 kOe, this AFM order is sustained at least down to 2 K [55]. Under an applied H exceeding 10 kOe (the precise field value is temperature dependent), Gd5Ge4 shows an interesting AFM–FM transition that can be driven both by T and H, and a detailed H–T phase diagram for Gd5Ge4 has been constructed through a series of magnetization and heat capacity measurements [55]. High-reso-

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Fig. 9. Two polymorphic modifications of Gd5Ge4: (a) the Sm5Ge4-type, which supports either the paramagnetic or antiferromagnetic states; and (b) the Gd5Si4-type, which the system adopts when it orders ferromagnetically. Both are layered structures built from nearly identical slabs (one of the slabs is marked by a bracket in (a)) and are related to one another by shear displacements of the neighboring slabs in the directions shown by short arrows near each of the slabs. The numbers above the arrows indicate the magnitude of the displacements. The most drastic difference is observed in the interslab Ge– ˚ , as marked (from Ref. [24]). Ge interactions, with corresponding interatomic distances that vary from 3.62 to 2.62 A

lution X-ray powder diffraction experiments performed in situ under applied magnetic fields of up to 35 kOe have shown that the AFM–FM transition in Gd5Ge4 is coupled to a structural transition in which the low H–high T Sm5Ge4-type structure transforms to the high H–low T Gd5Si4-type orthorhombic structure, which is illustrated in Fig. 9b [24]. The difference between the former and the latter is in the interslab bonding, and it appears that the strong interslab bonds in the Gd5Si4-type structure support strong interslab exchange interactions via strong spindependent Ge 4p–Gd 5 d hybridization [54]. Various other experimental features enumerated above for Mn oxides are observed in Gd5Ge4 and the crucial role of the first-order magneto-structural AFM–FM transition in the observed giant magnetocaloric effect in this compound is now well recognized [24]. Magneto-thermal hysteresis is one of the numerous evidences pointing towards the first-order nature of this transition [52,56]; its firstorder nature has been also confirmed from first principles [57]. Step-like features in the magnetization measurements across the field-induced AFM–FM transition have been observed and the commonality with the Mn oxide systems has been noted [58]. The AFM–FM transition in Gd5Ge4 has been imaged with a scanning micro-Hall probe, revealing the coexistence of FM and AFM phase in micrometer scale in a finite H–T regime around the transition point [59]. Further, this imaging study in combination with dimensional analysis and numerical simulation point out how this AFM–FM phase coexistence and the associated dynamics (nucleation and growth) can provide useful information for tuning the functionalities of the material [60].

3.2. La0.215Pr0.41Ca0.375MnO3 (LPCMO) and Al-doped Pr0.5Ca0.5MnO3 (PCMO) LPCMO has been studied as the prototype Mn oxide compound showing phase coexistence across the AFM– FM transition on length scales ranging from tens of nanometers to micrometers. It has been shown that this compound has an FM metallic ground state; even though resistance and magnetization measurements as temperature is reduced suggest that it may be an AFM insulator state [61]. Various groups have recognized that this observation reflects a glassy or ‘‘blocked” state [62,63]. As the temperature rises, the FM fraction increases considerably around 23 K, a characteristic temperature related to the unblocking of the low-temperature frozen state [63]. It has been argued that the observed phase coexistence is due to frozen metastable states and the actual ground state is homogeneous FM-metallic. More detailed studies were performed to determine the minimum disturbance of the robust charge-ordered PCMO, which can be driven toward the FM ground state by 2.5% Al substitution on the Mn sites. In both of these cases, as in NSMO and PSMO, one can control or tune the ratio of coexisting FM metallic and AF insulator phases at low temperatures by following specific non-conventional pathways in the H–T space [64]. In both materials, cooling in higher magnetic fields results in a larger fraction of the FM metallic phase. It was shown that if the sample is warmed in a field HW, which is different from the cooling (or annealing) field HC, then the M vs. T shows a sharp rise corresponding to a de-arrest of the AF insulating fraction when (HC – HW) is negative. This de-

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Fig. 10. Magnetization of Al-doped PCMO measured while warming from 5 K in a magnetic field of 40 kOe. The three curves, as labeled, correspond to the sample being cooled from 300 K to the lowest temperature in fields of 60, 40 and 30 kOe. In two cases (60 and 30 kOe), the field is changed to 40 kOe isothermally at 5 K. The cooling history dependence is striking (from Ref. [35]).

arrest is not observed when (HC – HW) is positive (see Fig. 10). 4. Kinetic arrest of first-order antiferromagnetic (ferromagnetic) to ferromagnetic (antiferromagnetic) phase transition An important aspect of disorder-influenced FOPT is that some liquids, called ‘‘glass-formers”, experience a viscous retardation of crystallization in their supercooled state [65]. Within the experimental time scale, such supercooled liquid ceases to be ergodic and enters a ‘‘glassy state”. The dynamical picture of a glass is a liquid where the atomic or molecular motions (or kinetics) are arrested [66]. It should be noted that the behavior typical of glass formation is not necessarily restricted to materials that are positionally disordered [66]. Even for conventional glasses, other than the general definition that ‘‘glass is a noncrystalline solid material which yields broad nearly featureless diffraction pattern”, there exists another widely acceptable picture of glass as a liquid where the atomic or molecular motions are arrested. Within this latter dynamical framework, ‘‘glass is time held still” [66]. It appears reasonable to assume that the viscous retardation of crystallization or kinetic arrest on experimental time scale would occur below some temperature TK. In the case where the glass is formed in a state that is metastable (in terms of free energy), then glass formation can be caused by slow cooling, as in O-terphenyl [66]. If the state in which the glass is formed is unstable, then rapid quenching is essential so that the kinetics is frozen before it can effect a structural change, as in metallic glasses. In this picture, what matters is whether TK < T* (metallic-glass) or TK > T* (O-terphenyl). In the latter case, even at T*, the system is already trapped in a deep valley in the potential energy landscape corresponding to a glass structure, even though it has reached the spinoidal point in the free energy

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configuration, so that any infinitesimal energy fluctuation should bring the system to a stable free energy minimum (see curve d in Fig. 5). For a graphical illustration and a detailed discussion of the potential energy landscape, readers are referred to a review by Debenedetti and Stillinger [65]. This valley is well separated from the potential energy minimum corresponding to a crystalline structure, and within a finite experimental time with finite energy fluctuations a glassy state does not go across such potential barriers, and therefore does not get transformed into a crystalline state. These concepts of glass formation can actually be applied to the magneto-structural FOPT observed in doped CeFe2 alloys, Gd5Ge4 and various manganese oxide compounds. It has recently been shown that in certain regions of the H–T phase diagrams of these different classes of magnetic materials there is a viscous retardation of nucleation and growth of the low-temperature phase across the FOPT. The low-temperature state attains a configuration consisting of a small fraction of transformed low-T equilibrium phase (AFM in doped CeFe2 alloys, NSMO and PSMO, and FM in Gd5Ge4 and LPCMO) in an untransformed non-equilibrium matrix of the high-T phase (FM in doped CeFe2 alloys, NSMO and PSMO and AFM in Gd5Ge4 and LPCMO). The micrometer-scale regions with long-range FM and AFM order are frozen randomly in space and time. The onsets of these non-trivial low-T magnetic-glass states are characterized by recognizable dynamical features, which are usually considered to be the hallmark of a structural glass transformation [65,66]. This kind of magnetic-glass state is distinctly different from a conventional spin-glass state, where the spin configuration is frozen at random on a much shorter, microscopic scale [67], or from a spin ice state, where disorder of the magnetic moments at low temperatures is analogous to the proton disorder in water ice [68]. To elucidate on this novel magnetic-glass behavior, the magnetization relaxation on various points on the FCC leg of the M–T curve of a 4% Ru-doped CeFe2 alloy is discussed here. It has already been observed that in the FCC path the FM phase persists at temperatures well below TN as a supercooled, non-equilibrium state [69]. This FM phase is highly metastable and any energy fluctuations tend to convert it into the equilibrium AFM state. As the system is cooled below TN and approaches the limit of supercooling T*, the free energy barrier height between the metastable (FM) and equilibrium (AFM) states decreases and hence a decrease of M with higher relaxation rate is expected. This is actually observed in the temperature range from 25 to 40 K [69]. However, some marked change in relaxation takes place below approximately 23 K (see Fig. 6). The relaxation rate decreases drastically and below 15 K the relaxation of M is very small, even though the metastable FM state persists. Relaxation data below 23 K can be fitted well with the Kohlrausch–Williams–Watt (KWW) stretched exponential function (// exp[–(t/s)b]) (see the inset of Fig. 6), where s is the characteristic relax-

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ation time and b is a shape parameter between 0.6 and 0.9. The characteristic relaxation time s tends to diverge below 15 K. This behavior is typical of what has been observed in many glass-formers in the T-regime of glass formation [65]. The non-Arrhenius behavior of s(T) resembles that of a fragile glass-former like O-terphenyl [65]. A similar relaxation in magnetization following the KWW stretched exponential function has been observed across the FM–AFM transition in LCMO and LPCMO [70], suggesting the glass-like dynamical response of the low-temperature magnetic state in LPCMO [71]. The signature of the kinetic arrest of the FM–AFM transition is also clearly visible in the isothermal fielddependent measurements [72]. This is in the form of a striking feature of the ZFC virgin M–H curve lying outside of the envelope M–H curve obtained by subsequent field cycling between ±Hmax, where Hmax  HM. A similar behavior has been observed in CMR manganese oxide compounds [28,35]. The ZFC virgin curve lying outside the envelope hysteresis curve has been reported in low-temperature M vs. H measurements in LPCMO [61], in Aldoped PCMO [35,64], in PSMO [35] and in NSMO [28]. The same behavior has been also detected in q vs. H measurements in Al-doped PCMO [35,64] and in NSMO [28]. As mentioned earlier, one can control or tune the ratio of coexisting FMM and AFI phases at low temperatures by following nonconventional pathways in the H–T space. One of these two phases is the equilibrium state (FM for LPCMO and Al-doped PCMO, AFM for NSMO and PSMO) while the other is the glassy phase, which is frozen in time. These features have been extensively studied in recent papers [34,64]. The ratio of equilibrium and frozen phases can be changed continuously and controllably in these manganite systems by following different paths in H–T space. Continuously varying values of magnetization and resistivity were obtained at the same measuring field and temperature, and this was shown for various values of the measuring field [35]. Since only one of the many values obtained corresponds to the equilibrium value, the remaining must correspond to metastable states. As mentioned earlier, the majority of systems discussed above have intrinsic substitutional disorder (the only exception is binary germanide – Gd5Ge4 – where Gd for Ge substitutions and vice versa are extremely unlikely due to geometrical and electronic reasons). Imry and Wortis [2] showed that such quenched disorder would give rise to a landscape of free energies, leading to a broad transition with a spatial distribution or landscape of the phase transition boundary (HC, TC) across the sample. Similar phase separations observed in many other systems of current interest arise from the landscape of free energy densities and a spread of local (HC, TC) values across the sample. The large number of (HC, TC) boundaries would thus form a band. The spinodal lines corresponding to the limits of supercooling (H*, T*) and superheating (H**, T**) would also be broadened into bands for samples with quenched disorder [73–75]. Each of these bands

corresponds to a quasi-continuum of lines, each line representing a specific region of the disordered sample. It is believed that in various materials the kinetics is actually arrested on experimental timescales and a glass is formed. The FOPT can be fully or partially arrested at low temperatures; the arrest occurs as one cools, and this frozen state gets de-arrested over a range of temperatures, which depends on the magnetic field as one warms [73–75]. If such a kinetic arrest were to occur below an (HK, TK) boundary in a pure system analogous to the glass transition line Tg for structural glasses, the disordered system would have an (HK, TK) band formed out of the quasi-continuum of (HK, TK) lines. Each line in this (HK, TK) band representing a local region of the sample would have its conjugate in the (H*, T*) band. A correlation has been established between the ordering of these quasi-continuum of lines in the (H*, T*) and (HK, TK) bands from magnetization measurements, and it was shown that the type of correlation has a profound effect on the behavior of magnetization for two divergent systems. The most intriguing outcome of this study is the unified observation of an anti-correlation, implying that the region having high T* has low TK for a polycrystalline sample of Os-doped CeFe2 having an AFM ground state as well as for a single crystal of the charge-ordered CMR manganese oxide compound LPCMO having an FM ground state. A similar anti-correlation between supercooling and kinetic arrest temperatures has been seen in NSMO and in Gd5Ge4. For the experimental details concerning NSMO and Gd5Ge4, readers are referred to Ref. [28] and [75], respectively. This magnetic-glass state can be devitrified by following specific paths in the (H, T) phase space. Such a process of devitrification (or recrystallization) is, of course, a wellknown phenomenon in structural and metallic glasses [76]. Devitrification has actually been studied in Gd5Ge4 [75] and some CMR-manganite compounds [34,35]. As in structural glasses, the process of devitrification of the magnetic-glass state is also quite different from the process of glass formation. 5. Outlook In this review we have highlighted that the phase coexistence and metastability associated with magneto-structural transition in various classes of magnetic materials are actually universal characteristics of a disorder-influenced firstorder phase transition. Under specific circumstances this disorder-influenced FOPT can be kinetically arrested, and this event gives rise to a metastable magnetic state whose dynamical features resemble those of a structural glass. The role of this FOPT and the associated phenomena in the functionality of the various classes of magnetic materials are also recognized. This generality concerning FOPT may even be extended to encompass ferroelectric materials [77], which in turn is likely to have interesting implications in multiferroic materials which exhibit interplay between magnetism and ferroelectricity [78]. Furthermore, the

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emerging results from a few families of functional magnetic materials considered above may have a considerable influence by leading to a better understanding of the FOPT processes in general. These processes are now more than purely academic, especially when it is becoming quite apparent that FOPTs are involved in many of the practical issues in our surroundings [79]. For example, the FOPT process involving the water–ice transition has tremendous implications for biological as well as ecological systems, and this continues to be a very active area of research [79]. Some of the magnetic systems discussed may actually be used as test bed materials to study an FOPT process in a two-parameter magnetic field–temperature phase space. Similar exploration in a more conventional pressure–temperature phase space is relatively difficult. Only very recently has it been discovered that liquid germanium can undergo a glassy transition in the presence of applied pressure [80]. Knowledge gathered in exploring H–T phase space in magnetic materials can also be exported to other areas of technological interest, namely vortex matter phases of type-II superconductors, understanding of which is absolutely necessary for tuning the dissipationless current-carrying capacity of a type-II superconductor. Acknowledgements The authors would like to thank numerous colleagues, discussions with whom have helped to refine ideas and views expressed in this article. Work at the Ames Laboratory is supported by the Office of Science, Office of Basic Energy Siences of the US Department of Energy under contract No. DE-AC02-07CH11358 with Iowa State University of Science and Technology. References [1] Pecharsky VK, Gschneidner KA, Fort D. Scripta Mater 1996;35:843. [2] Imry Y, Wortis M. Phys Rev 1979;B19:3580. [3] White RM, Geballe TH. Long range order in solids: solid state physics, supplement 15 (solid state physics supplement). New York: Academic Press; 1979. [4] Chaikin PM, Lubensky TC. Principles of condensed matter physics. Cambridge: Cambridge University Press; 1995. [5] Paolasini L, Ouladdiaf B, Berenhoeft N, Sanchez JP, Vulliet P, Lander GH, et al. Phys Rev Lett 2001;90:057201. [6] Roy SB, Coles BR. J Phys Condens Mat 1989;1:419. [7] Roy SB, Coles BR. Phys Rev B 1990;39:9360. [8] Kennedy SJ, Coles BR. J Phys Condens Mat 1990;2:1213. [9] Manekar MA, Roy SB, Chaddah P. J Phys Condens Mat 2000;12:L409. [10] Roy SB, Chaddah P. Physica 1997;C279:70. [11] Roy SB, Chaddah P. J Phys Condens Mat 1997;9:L625. [12] Majumdar S, Sharma VK, Manekar M, Kaul R, Sokhey KJS, Roy SB, et al. Solid State Commun 2005;136:85. [13] Sokhey KJS, Chattopadhyay MK, Nigam AK, Roy SB, Chaddah P. Solid State Commun 2003;129:19. [14] Chattopadhyay MK, Roy SB, Nigam AK, Chaddah P. Phys Rev B 2003;68:174404. [15] Kimura T, Tomioka Y, Kumai R, Okimoto Y, Tokura Y. Phys Rev Lett 1999;83:3940.

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