Coupling between phase transitions and glassy magnetic behaviour in Heusler alloy Ni50Mn34In8Ga8

The transition sequence in the Heusler alloy Ni50Mn34In8Ga8 has been determined from measurements of elasticity, heat flow and magnetism to be paramagnetic austenite → paramagnetic martensite → ferromagnetic martensite at ∼335 and ∼260 K, respectively, during cooling. The overall pattern of elastic stiffening/softening and acoustic loss is typical of a system with bilinear coupling between symmetry breaking strain and the driving structural/electronic order parameter, and a temperature interval below the transition point in which ferroelastic twin walls remain mobile under the influence of external stress. Divergence between zero-field-cooling and field-cooling determinations of DC magnetisation below ∼220 K indicates that a frustrated magnetic glass develops in the ferromagnetic martensite. An AC magnetic anomaly which shows Vogel–Fulcher dynamics in the vicinity of ∼160 K is evidence of a further glassy freezing process. This coincides with an acoustic loss peak and slight elastic stiffening that is typical of the outcome of freezing of ferroelastic twin walls. The results suggest that local strain variations associated with the ferroelastic twin walls couple with local moments to induce glassy magnetic behaviour.


Introduction
Phase transformations in ferroic materials, i.e. ferroelectrics, ferroelastics and ferromagnets, have been intensively studied in recent years because of their particular structure/property relationships which arise as a consequence of order parameter instabilities in response to conjugate elds [1][2][3]. As an important group of ferroic materials, Heusler ferromagnetic shape memory alloys have attracted particular interest due to their giant magnetostrain, ∼3%-9.5%, which can be induced by the phase transformation between austenite and martensite [4] or the reorientation of ferromagnetic martensitic variants under applied eld [5]. In addition, a large inverse magnetocaloric effect of ∼−7 K has been reported in the Ni-Mn-In system under a magnetic eld of 6 T [6].
These promising multifunctional properties essentially depend on the nature and strength of strain/order parameter coupling and the mobility of ferroelastic domain walls [1]. The temperature dependence of elastic constants, magnetization and elastic damping properties of Ni-Mn-In and Ni-Mn-Ga based Heusler alloys have already been studied to some extent [7,8], but those of polycrystalline NiMnIn-xNiMnGa alloys have not and should be much more complex than the two end members. Insights into strain-related phenomena at phase transitions of materials are provided by the associated variations of elastic constants. Resonant ultrasound spectroscopy (RUS) has turned out to provide a convenient tool for characterizing both the strength of strain/order parameter coupling, and the relaxation dynamics of ferroelastic domain walls or other defects at frequencies in the vicinity of ∼1 MHz [1,[7][8][9][10][11][12][13]. Integration of AC and DC magnetization data with results from RUS measurements then allows strain relaxation and magnetic dynamics to be characterized properly. This combined experimental approach should shed light on the full physical picture of magnetoelastic transitions in Heulser alloys, in particular for the NiMnIn-xNiMnGa alloys.
Diverse magnetic states have been reported for the martensite phase of Ni-Mn-X (X = In, Sn, Sb) based Heusler alloys, such as paramagnetic, superparamagnetic, antiferromagnetic, ferrimagnetic, ferromagnetic (FM) phases, two-phase mixtures and spin glass states [14][15][16][17][18]. This raises the interesting question of how the magnetic and ferroelastic properties interact. The origin of divergence between zero-eld-cooling (ZFC) and eld-cooling (FC) curves slightly below the Curie temperature of martensite state in almost all Ni-Mn-X (X = In, Sn, Sb) Heusler alloys has not been completely understood, though a number of possible mechanisms were proposed. [7,15,[18][19][20][21] For instance, Gigla et al [22] and Ma et al [20] attributed splitting of magnetic ZFC/FC curves to spin disorder induced by excess magnetic atoms, e.g. Ni in martensite. The results of other work [18,23] have implied that the excess magnetic atoms such as Co in Ni-(Co)-Mn-X (X = In, Sn, Sb) did not necessarily lead to the divergence between ZFC and FC curves, however. An alternative key factor might be the presence of ferroelastic twin walls of the martensite phase.
In this work, data from RUS, DC and AC magnetization and differential scanning calorimetry rstly show that Ni 50 Mn 34 In 8 Ga 8 alloy undergoes phase transitions in the sequence, paramagnetic austenite → paramagnetic martensite → ferromagnetic martensite at ∼335 and ∼260 K, during cooling. Divergence between ZFC and FC curves below ∼220 K can be explained in terms of the development of a frustrated magnetic glass state in the ferromagnetic martensite. Correlation of a peak in acoustic loss with a peak in magnetic loss at ∼160 K is due to the effect of freezing of ferroelastic twin walls in giving rise to a further glassy magnetic transition. In addition, disrupting the martensitic structure by deforming the sample with a uniaxial force of 10 kN at room temperature can dilute the observed frustrated magnetic glass at 160-220 K in unstrained samples to a spin glass state.

Experimental methods
Ni 50 Mn 34 In 8 Ga 8 (NiMnInGa8 hereafter) ingots were prepared by arc melting of high-purity Ni(99.9%, T-metals), Mn(99.9%, T-metals), In(99.9%, T-metals) and Ga(99.9%, T-metals) with electromagnetic stirring in an argon atmosphere. They were then sealed in an evacuated silica tube and annealed at 1073 K for 48 h in order to improve chemical homogeneity, followed by quenching in air. A differential scanning calorimeter (DSC, TA Instruments Q-200) was used to detect the structural transition. A physical properties measurement system (PPMS Quantum Design) was used for vibrating sample magnetometer and AC magnetic susceptibility tests. Details of the RUS technique have been given in references [25][26][27]. Selected peaks in the RUS spectra were tted with an asymmetric Lorentzian function using the software package Igor (WaveMetrics) to determine the resonance peak frequency, f , and width at half maximum height, ∆f . The elastic constants scale with f 2 and the inverse mechanical quality factor, taken here as Q −1 = ∆f /f , is a measure of acoustic loss. In a polycrystalline sample, f 2 scales primarily with the shear modulus because the predominant resonance motions involve shearing. Only a relatively small contribution is due to breathing motion, and hence to the bulk modulus, for most resonances. Figure 1(a) shows a stack of RUS spectra collected in the temperature range 10-300 K and frequency range 0.1-1.2 MHz. Individual spectra, with amplitude in volts, have been displaced up the y-axis in proportion to the temperature at which they were collected, and the axis then labelled as temperature. Temperature dependent f 2 and Q −1 data obtained by tting of the resonance peak near 0.46 MHz at room temperature are given for the entire temperature range, 10-450 K, as shown in gure 1(b). With falling temperature from ∼450 K, f 2 (∝shear modulus) rst softened and then stiffened, resulting in a dip near 335 K. There is a hysteresis of ∼5 K between cooling and heating for the minimum point of f 2 (∼335 K during cooling, ∼340 K during heating). Further con rmation of this being the martensitic transition is provided by the sharp peak in heat ow (blue curve in gure 1(d)). Steep non-linear softening of the shear modulus as the transition is approached from below and above is characteristic of a pseudoproper ferroelastic transition, which has bilinear coupling between a symmetry-breaking shear strain e and the driving order parameter Q. The formalities of how this coupling leads to the observed pattern have been set out in full in the literature (see, for example, [28]). The structural/electronic order parameter, Q E , has the symmetry properties of irreducible representation Γ + 3 of the parent L2 1 cubic state with space group Fm3m and couples bilinearly with tetragonal and orthorhombic shear strains, i.e., as λeQ [7,8,29]. Here, softening and stiffening of the combination of elastic constants C 11 -C 12 arises by bilinear coupling between Q E and the symmetry breaking shear strain of DO 22 -type tetragonal state (space group I4/mmm, con rmed by the diffraction pattern shown in gure S1). This is different from terms for strain coupling with the ferromagnetic order parameter, M, that are linear/quadratic, λeM 2 , and biquadratic, λe 2 M 2 [8,24], and would give different patterns of elastic softening.

Experimental results and discussion
With respect to anelastic dissipation, the austenitic phase at T > T M gave low values of Q −1 , but displayed a steep increase from ∼10-20 K ahead of the martensitic transition ( gure 1(b)). The increase is related to the precursor dynamics but would also be affected by the mobility of interfaces between coexisting austenite and martensite near T M . In the stable martensitic state (T < T M ), a plateau of high anelastic loss, Q −1 ≈ 0.008, persisted until the peak in Q −1 near 160 K, below which the Q −1 values returned to those of the parent austenite. This mirrors the classic pattern below a ferroelastic phase transition, in which high loss is due to the mobility of twin walls in an effectively viscous medium followed by Debye freezing or pinning of the walls at lower temperature, as marked by a peak in Q −1 and a small increase in elastic stiffness. For example, it is closely similar to the pattern shown by Ni 50 Mn 35 In 15 [7].
Conventional M-H loops in gure 1(c) show that the austenite phase exhibits near-zero hysteresis at 380 K, with low remnant and saturation magnetizations. It is thus considered to be paramagnetic.  sitic transition was observed in the austenitic phase above T M , which is different from previously reported Ni-Mn-Ga alloys [30][31][32][33]. The phase transition sequence in NiMnInGa8 is therefore paramagnetic austenite (L2 1 , space group Fm3m) → paramagnetic martensite (DO 22 , space group I4/mmm) → ferromagnetic martensite. Diffraction data show persistence of some of the parent austenite down to room temperature (supplementary material, gure S1).
ZFC/FC splitting behaviour observed in the temperature range 10-220 K under different magnetic elds is shown in gures 1(d) and 2. The ZFC measurements were made during heating from 10 to 390 K in a eld of 100 Oe, 1 kOe or 1 T following cooling in zero eld. FC measurements were made during cooling back to 10 K with the same eld as had been applied during heating, i.e. 100 Oe, 1 kOe or 1 T. There was a weak anomaly in the magnetic moment at T M ∼335 K, the amplitude of which increased with increasing eld. The transition temperature did not overtly shift between 100 Oe and 1 T. Divergence between the ZFC and FC curves at 100 Oe occurred at T f2 ≈ 220 K, indicating the start of irreversibility for the magnetic state of the martensite phase below the ferromagnetic transition temperature of martensite T c M = 260 K. The divergence became smoothed out when the measuring eld was increased to 1000 Oe and 1 T and the divergence temperature of ZFC/FC reduced with increasing eld ( gure 2). These two features are attributed to the development of a frustrated magnetic glass state in ferromagnetic martensite [15,20,23,34,35]. There is then another weak anomaly near 160 K in the low eld data, labelled here as T f1 . Figure 3(a) shows the temperature dependence of real, χ ′ , and imaginary, χ ′′ , parts of the AC susceptibility measured at a single frequency of 111 Hz with the DC eld set at 100 Oe and the amplitude of the AC eld set at 5 Oe. A weak shoulder appearing at ∼260 K in the χ ′ -T curve is ascribed to the magnetic transition of martensite (T c M ). The sharp peak in χ ′ at ∼220 K (T f2 ) was accompanied by a sharp peak in χ ′′ . There is then a broad, much weaker peak in both χ ′ and χ ′′ at ∼160 K (T f1 ). A close analogue, in terms of structure and chemistry, is Ni 50 Mn 35 In 15 , in which a paramagnetic to ferromagnetic transition in the martensitic phase starts at ∼200 K, followed by bifurcation of FC and ZFC DC magnetization curves at a slightly lower temperature attributed to the presence of a spin-glass-like state [7]. Khovaylo et al [15] referred to the glassy magnetic state which develops below T M c in Ni 50 Mn 34.8 In 15.2 as a cluster glass. Neither of these two Heusler alloys showed the additional weak anomaly reported here at T f1 for NiMnInGa8, however. This seems to be more like the magnetic behaviour shown by Ni 48 Co 2 Mn 38 Sb 12 which has a glassy magnetic freezing temperature at ∼130 K (T f1 ) which is well below the FC/ZFC bifurcation temperature (T f2 ) of ∼254 K. Figures 3(b) and (c) show the results of AC magnetic measurements in a 100 Oe DC eld with a 5 Oe AC eld at frequencies of 9, 19, 199 and 999 Hz, before and after the sample was deformed in a hydropress. The unstrained sample, with mass 64.6 mg and thickness ∼0.8 mm, was subject to a uniaxial force of 10 kN until its thickness was reduced to ∼0.48 mm, corresponding to a strain of ∼40%. The force was released prior to new magnetic measurements being made. Further DSC data and M-T data (supplementary material, gures S2 and S3) con rmed that the martensitic transition still occurred at ∼340 K in the deformed sample. Values of the tted empirical parameter Φ = ∆T f /(T f ∆log 10 f ) at T f1 and T f2 of unstrained samples were found to be ∼0.004 and ∼0.005, respectively, which are closer to the reported values of magnetic (spin) glass, 0.005-0.02, than to the value of ∼0.1 for the blocked superparamagnetic state [18,20]. This further supports the arguments for glassy magnetic behaviour at T f1 and T f2 in NiMnInGa8. The temperature-dependent AC magnetic properties of deformed samples ( gure 3(c)) were markedly different from those of the as-grown, unstrained samples ( gure 3(b)). In particular, the χ ′ and χ ′′ peaks at T f2 of unstrained samples almost disappeared from measurements on the deformed sample. The T f1 peak became stronger for strained NiMnInGa8 alloy and shifted to higher (∼175 K) and lower (∼150 K) temperature on χ ′ -T and χ ′′ -T curves, respectively. The value of Φ around the frequency-dependent T f1 peak of strained samples is ∼0.014, indicative of a slowing-down spin glass state.
Following Nayak et al [23], the frequency dispersion in χ ′ -T curves through T f1 of the strained NiMnInGa8 sample was analysed using the empirical Vogel-Fulcher law ω = ω 0 exp −E a /k B T g (ω) − T 0 , as shown in the gure 3(d). E a is an activation energy, k B is the Boltzmann constant, T g (ω) is the glass freezing temperature at measuring frequency ω, and T 0 is the glass freezing temperature for ω = 0 Hz [36,37]. The best t to the data gave T 0 ∼ 169 K, ω 0 ∼10 5 s −1 and T g /T 0 ∼ 1.05, which is viewed as additional evidence for a spin-glass-like transition [36,38].
The complex glassy magnetic behaviour at T f1 and T f2 in the ferromagnetic martensite phase of NiMnInGa8 provides an illuminating comparison with the acoustic relaxation data from RUS. Although the measuring frequencies were different, freezing of the ferroelastic domain walls appears to be more or less coincident with the glassy magnetic freezing behaviour seen at T f1 , implying that the two processes are interdependent. As suggested in previous work [14][15][16][17][18]20], some ferroelastic variants of martensite in Ni-Mn-X (X = In, Sn, Sb) Heusler alloys couple with magnetic clusters below T c M . If there is any magnetoelastic coupling, local strain variations associated with ferroelastic interaction between randomly oriented martensite variants below T c M must in uence the mobility of the coupled magnetic domain walls to some extent. Freezing of the ferroelastic domain walls could then give rise to freezing of the magnetic domain walls or clusters and would imply that the glassy magnetic behaviour at T f1 relates speci cally to glassy freezing of magnetic moments within or close to the ferroelastic twin walls.
Although the resulting microstructure of the deformed sample was not characterized, it is well understood that plastic deformation of martensite is accommodated via dislocations and deformation twinning. Magnetic ordering in the deformed sample will therefore have been strongly in uenced by a high density of dislocations and pinned/interacting twin walls, which would account for the extra spin disorder. An additional contribution to spin disorder might have come from local changes to Mn(4b)-Mn(4b) and Mn(4b)-Mn(4d) distances, which are believed to in uence magnetic exchange interactions in Ni-Mn-Fe-In alloys [15].

Conclusions
In summary, the transition sequence of the NiMnInGa8 alloy during cooling has been determined by resonant ultrasound spectroscopy, DC and AC magnetization and differential scanning calorimetry. The divergence of ZFC/FC curves is ascribed to magnetic frustration/disorder in the martensite phase. Correlation between acoustic and AC magnetic measurements suggests that magnetic glassy freezing are coupled with strain variations associated with the presence and relaxation dynamics of ferroelastic twin walls at lower temperature or external elds.