Suppression of spinodal instability by disorder in athermal system
SSuppression of spinodal instability by disorder in athermal system
Tapas Bar, ∗ Arup Ghosh, and Anurag Banerjee Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, West Bengal 741246, India Indian Institute of Science Education and Research Pune, Pune 411008, Maharashtra, India Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS,F-91191 Gif-sur-Yvette, France (Dated: February 24, 2021)We observed asymmetric critical slowing down and asymmetric dynamical scaling exponent inthe superheating and supercooling kinetic processes during the thermally-induced metal-insulatortransition of MnNiSn based heusler alloy. During the transition to the insulator phase, the critical-like features get enhanced compared to the transition back to the metal phase. These experimentalfindings suggest that the metastable phase in the cooling branch of hysteresis has approached closeto the spinodal instability. On the other hand, the extended disorder, generated over and above theintrinsic crystal defects during heating, triggers the excess heterogeneous nucleation before reachingthe spinodal point. Zero temperature random field Ising model (ZTRFIM) simulation, inscribed forthe athermal martensitic transitions, support the argument that the disorder smears the spinodalinstabilities as the correlation length is bounded by the average distance between the disorder points.
Hysteretic transition is an exciting subclass of abruptthermodynamic phase transition (ATPT) where equilib-rium thermodynamics breaks down on account of thesystem has accessed the metastable phase [1–3]. Ther-mal hysteresis implies that there is a discontinuous jumpfrom one metastable minimum to another free energyminimum and often accompanied by rate-dependent ef-fect [4, 5], exchange bias [6], kinetic arrest [7]. Such non-ergodic behavior is generally believed to arise from an in-terplay of disorder, thermal fluctuations, and activationbarriers separating the two phases [3]. When thermalfluctuation is insignificant in the kinetics of phase trans-formation (athermal), the metastable phase of a systemcan persist right up to spinodals (mean-field concept)where the activation barrier against nucleation vanishes[8]. One would expect divergence of correlation lengthalong with divergence of the relaxation time scale oforder parameter due to the diffusive nature of dynam-ics [8]. This kind of athermal transition arises due tothe suppression of fluctuation by the long-range forceduring the magnetic and structural phase transition ofmany complex functional materials, including mangan-ites, transition metal oxides, etc. [9–12]. Near spinodal,the ramified nucleating droplet diverges in all directions,unlike classical nucleation, where a single droplet of sta-ble phase grows in a compact form [13]. When the sys-tem approaches spinodal, the nucleation rate becomesvery slow, indicate spinodal slowing down [14]. The nu-cleation rate decrease further as the range of interactionincreases and finally goes to zero when the range of inter-action becomes infinity [13], then the transition becomesmean-field like [8, 15]. The recent experiment on dy-namic hysteresis scaling supported by numerical analysissuggested the mean-field like spinodal instability existsin the correlated system [11, 16].However, the disorder is known to yield heterogeneousnucleation of athermal martensitic transformation [17] ∗ [email protected] characterized by jerky propagation related to avalanches[9, 18]. The disorder may reduce the free energy barrier ofnucleation before arriving at the spinodal [2, 19]. The in-fluence of disorder on the spinodal has not yet been fullyexplored except some model system [5, 20–23]. Althoughdisorder associated athermal transition is found in manycomplex functional materials undergoing hysteretic tran-sition [3], earthquake [24], social and economic systems[25].This article focuses on one such system where a ki-netic asymmetry in the supersaturated transition arisesfrom the extended disorder generated during the super-heating process. We experimentally observed that, evenat finite temperature, the transition is independent ofthermal fluctuation, i.e., athermal. One would expect tosee the footprints of diverging susceptibility in its tem-poral features as the fluctuation-less metastable phasecan survive up to the unstable singularity (spinodal) un-der phase kinetics [9–12]. However, extended disorderreduces the degree of superheating through nucleationon the kinetic-path before vanishing the activation bar-rier. In this article, through the dynamic hysteresis andcritical slowing down measurements supported by ZTR-FIM simulations, we first experimentally report that inan athermal system, the extended disorder overrules thespinodal instabilities via heterogeneous nucleation.Figure 1 (a) shows that a typical resistance measure-ment done as a function of temperature during the first-order phase transformation (FOPT) at 200 K in poly-crystalline heusler alloy [26]. There is a thermal hystere-sis (width ∼ ∼
197 K) during heatingand the martensitic transition ( ∼
192 K) during cool-ing. The electronic transition, coupled with a magneticand a structural transition, occurs through a series ofavalanches. Each avalanche is accompanying with a jumpin resistance [Fig. 1 (a) (inset)] and a corresponding jerkylatent heat [Fig. 1(b)] [18]. The latent heat of transitionsis evaluated from the differential thermal analysis (DTA)signal [Fig. 1(b) right inset] by numerically fitting an a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b
170 180 190 200 210 2201.82.02.22.42.62.8
193 194 1952.02.12.22.3
Cooling R e s i s t an c e ( m ) Temperature (K)
Heating (a)
185 190 195 200 205 210-0.15-0.10-0.050.000.050.10
160 180 200 220-0.50-0.250.00165 170 175 180 185-0.40-0.35-0.30 T e m pe r a t u r e r i s e ( K ) Temperature (K) (b)
180 190 200 210 2201.82.02.22.42.62.8
192 194 196 1982.42.52.6 R e s i s t an c e ( m ) Temperature (K) (d) (c)
FIG. 1. (a) The avalanche-like jump in resistance during thetransition. (b) Jerky like peak during the transition seen inDTA measurement (right inset). The off-transition DTA sig-nal is shown for comparison (left inset). (c) Return pointmemory for three partial loops. The inset shows returningpoints more clearly in sub-subloops. (d) Histogram of jerkylatent heat as a function of magnitude. The heat fluxes weremeasured in 1/7 sec interval, and the corresponding temper-ature scanning rate was 6 K/min. equivalent model for the setup. Fig. 1 (d) shows thedistribution of jerky heat as a function of their size. Thejumps during cooling are larger than the heating impliesthat the disorder linked with the austenite transition ismore compare to the martensitic transition [20, 26, 34].Such kinetic asymmetry emerges due to the heteroge-neous nucleations at two distinct disorder points: ex-tended disorder [35] (structural twin walls [17, 18, 40],surface-localized defects [41], dislocation [36], etc) whichappear during hating branch of hysteresis on top of in-trinsic disorder which remains in both branches [35, 40].Return point memory of hysteresis loop [Fig. 1 (c)] im-plies the existence of intense disorder (above the criticalvalue σ c ) associated with ATPT [20, 42].Thermal hysteresis implies discontinuity in free energyand discontinuity in entropy at the transition points.Therefore, this transition can be termed as ‘zeroth-order’[11, 43]. The analytic continuation of free energy beyondthe expected transition point (where the free energy oftwo phases intersect) enlarge the lifetime of metastablestate [1, 43]. Consequently, the metastable state act likean equilibrium state and the transition becomes ather-mal. Such continuation breaks at the limit of stabilitydue to the essential singularity of spinodal points. Atthese points, the relaxation time of the metastable stateexpects to diverge [44].However, the phase ordering relaxation time was mea-sured by the quench and hold experimental technique.Starting from an initial temperature of 120K (or 250K),which is far from transition regions, the sample is heatedup (or cool down) rapidly to a specific set temperature
170 180 190 200 210 2200.00.20.40.60.81.0
Heating
180 K 185 K 188 K 190 K 192 K 194 K 196 K 198 K 200 K 202 K 204 K 206 K 208 K 210 K 220 K I n s u l a t o r f r a c t i on ( f ) Temperature (K)
Cooling
170 K 175 K 180 K 182 K 184 K 186 K 188 K 190 K 192 K 194 K 196 K 198 K 200 K 203 K (c)
Heating I n s u l a t o r f r a c t i on ( f ) time (s)
180 K 185 K 188 K 190 K 192 K 194 K 196 K 198 K 200 K 202 K 204 K 206 K 208 K 210 K (a)
Cooling I n s u l a t o r f r a c t i on ( f ) time (s)
170 K 175 K 180 K 182 K 184 K 186 K 188 K 190 K 192 K 194 K 196 K 198 K 200 K 203 K (b)
FIG. 2. Order parameter as a function of time (a),(b), andtemperature (c) acquired from quench-and-wait experimentsfor different waiting temperatures. The order parameter fi-nally reaches the steady-state value ( • within c) for a givensufficient wait. The steady-state values coming out from dis-sipative phase ordering experiments are very close to the dy-namic hysteresis measurement of ramp rate 0.5 K/min (blue-line inside c). The temperature quench rate of the measure-ments was 40 K/min. ( T R ), then waits for sufficient time to equilibrate. Thetemporal evolution of resistance is represented in termsof insulator fraction (order parameter) [26, 33]. Fig. 2shows the insulator fraction for different set temperaturesfor heating and cooling quenches. After an adequate waittime ( ∼
60 s), the system had reached the quasi-staticvalue. After that, no temporal evolution of order pa-rameter is observed; suggest that the system, driven bythermal fluctuations, unable to jump the activation bar-rier and remains in the metastable local minima. Suchbehavior indicates an athermal transition [9, 45], wherethe thermal fluctuations are suppressed by long-rangedshear force [10] and the kinetics of transition is onlycontrol by external parameters. The fluctuation-relatedbarrier crossing becomes disabled, and the transition dy-namic should follow the deterministic track—eventually,the transition acts like the mean-field one. Spinodal slow-ing down has been observed under deep supersaturationin the athermal system [13, 15, 46]. This slowing downcan manifest as the much-discussed delay in the onsetof switching around the bifurcation points of the FOPT(i.e., finite time effects); the change in the hysteresis looparea A ( R ) (or shift in transition points) with R , the rateof change in the field H or temperature T can dynami-cally scale as: A ( R ) = A + aR Υ . (1)The quasi-static hysteresis loop area A must be nonzerofor such transitions [11, 23, 47–52]. While the valuesof Υ have been obtained analytically [47], numerically[23, 48, 51], and experimentally [11, 49, 50] in differ-ent systems, for the field as well as temperature-drivenFOPTs. The mean-field exponent Υ = 2 /
185 190 195 200 205246810
Cooling Heating R e l a x a t i on t i m e , t ( s ) Temperature (K) (a)
180 190 200 2102.02.22.42.62.8 R e s i s t an c e ( m W ) Temperature (K) (b) g = 0.93 g = 0.85 T e m pe r a t u r e s h i ft [ D T = | T - T | ] ( K ) Ramp Rate (K/min) (c)
FIG. 3. (a) Temperature dependent phase ordering relax-ation time constant. The time constant has been extractedby exponential curve fitting in the temporal evolution of or-der parameter of quench and hold experiments [Fig. 2]. Thevertical lines corresponds to the quasi-static transition tem-peratures for cooling (green) and heating (orange). (b) Tem-perature dependent resistance for different ramp rates. (c)log-log plot of shift in transition temperatures with tempera-ture scanning rates. The power laws fitting exponent comingout to be Υ ≈ .
93 for heating and Υ ≈ .
85 for cooling.
However, the signature of spinodal slowing down re-flects both in quench-and-hold and dynamic hysteresismeasurements [Fig. 3]. The phase ordering relaxationtime constant peaks at the transition points, but thereis a clear qualitative difference between the heating andcooling branch of hysteresis [Fig. 3(a)]. The peaks aresignificantly milder in the heating branch, hints non-diverging time scale of relaxation. Such non-robust criti-cal behavior in the athermal system arises from local fluc-tuations induced by the impurities, which acts as a het-erogeneous nucleating site [53]. The extended disorder,along with crystal impurities, is present during the tran-sition to the metal phase. The physics is controlled by itsdisorder-induced fluctuations leading to a large roundingof the time scale divergence. On the other hand, thetransition back to the insulator phase has no extendeddisorder, and a comparatively sharp peak is observed.Besides, we also observe the asymmetric finite time effectof the transitions, the shift in transition temperature withthe ramp rates R [Fig. 3(b)]. Although the dynamicalrenormalized shifts ∆ T both for heating and cooling ful-fil a scaling relationship ∆ T ( R ) = | T i − T iobs ( R ) | ∝ R Υ , i = heat or cool around two decades of ramp rates [3 (c)],the exponent are found to be dissimilar; Υ = 0 . ± . . ± .
05 for cooling. The expo-nents are inconsistent with the mean-field spinodal value(Υ = 2 / > / φ theory for field-driven transitions [48]. Therefore, eventhough the experimental system is temperature driven in-stead of field-driven, the ZTRFIM captures the essentialnature of athermal transition, including random disorder[20, 58]. The Hamiltonian of the model given by H = − J (cid:88) s i s j − (cid:88) i [ H ( t ) + h i ] s i , (2)where Ising spins s i = ±
1, placed on the 3-dimensionallattice, coupled with nearest-neighbor pairs through cou-pling strength J . H ( t ) is the external field, and h i rep-resents the random field disorder, taken from a Gaussiandistribution of zero mean and variance σ . σ representsthe disorder strength.The quench-and-hold experiments were performed inthe ZTRFIM system on a cubic lattice of size L underperiodic boundary conditions with disorder σ > σ c , σ c isthe critical disorder [20, 42]. We choose a fully polarizedinitial state, and then the magnetic field is abruptly set toa specific value close to the transition point as the systemis allowed to equilibrate at T = 0.[26]. The equilibrationtime (relaxation time) sharply peaks at the transition, al-though the peak height decreases with the increasing dis-order strength [Fig. 4]. The relaxation time peak growsslowly with system size, showing a mild finite-size effect,unlike power-law growth in the case of classical critical-ity followed by spinodal transition with critical disorder[26]. This manifests the non-robust critical behavior ofspinodal transition.To quantify the role played by disorder on spinodaltransition, we study the dynamic hysteresis in the ZTR-FIM. The systematic shift in coercive fields H c ( R ), thefields at which the transition takes place, with the fieldrates R fulfill a scaling relationship similar to the ex-periment [Eq. 1] where the steady-state coercive field -3 -2 -1 005001000150020002500 0 1 2 3
100 150 200 250 300 35050050001001000100002002000 s = 2.25 s = 2.50 s = 3.00 s = 3.50 (c) (b)
Decreasing field R e l a x a t i on t i m e , t Field, H s = 2.25 s = 2.50 s = 3.00 s = 3.50
L = 128 (a)
Increasing field
Field, H R e l a x a t i on t i m e , t System size (L)
FIG. 4. Phase ordering relaxation time constant during waitafter rapid increasing (a) and decreasing (b) fields drawn fromZTRFIM simulations of size 128 for disorder strength σ =2.25, 2.5, 3, and 3.5. (c) The time constant peaks for increas-ing ( • ) and decreasing ( (cid:50) ) fields as a function of system sizefor different disorder strength. Note that a little mismatchof peak height during field increasing and decreasing for aparticular disorder strength depends upon how close one canreach the transition points. H c (0) is no longer a free parameter (see SM for details).Here no excess disorder (unlike experiments) is inducedeither during increasing or decreasing the field; conse-quently, the asymmetric exponents are not expected andobserved. Most importantly, the exponent Υ increaseswith the increasing disorder, and the scaling does notform when the disorder crosses a threshold level, σ th =3.30 [34] [Fig 5]. At this disorder strength, disorder-induced nonperturbative athermal fluctuations destroythe spinodal singularity, and transition occurs throughpercolation [59]. The experimentally observed exponentsemphasize that the cooling cycle is more critical thanthe heating. Based on the RFIM calculations, we ar-gue that for the systems with low disorder the transi-tion is governed by a small number of relatively largemacroscopic jumps. In contrast, systems with large dis-order display smooth transitions with a large number ofsmall-size avalanches, and the lopsided disorder gives riseto asymmetric avalanches in the superheating and su-percooling kinetic processes. Heterogeneous nucleatingdroplets emerge from disorder-induced local fluctuationon the kinetic-path before reaching the spinodal insta-bility. However, the thermal growth of the droplets sup-pressed by a long-range interacting force leads to a non-mean field spinodal slowing down: relaxation times are peaking but not diverging and show mixed order (first or-der and continuous) fracture at the transitions [60, 61].The crossover from the robust MF spinodal, i.e., with di-verging correlation length and susceptibility, to the non-robust mixed transition, exhibiting nonuniversal dynam-ical exponents, takes place due to the finite correlationlength whose growth is bounded by the average distance ** D y na m i c a l e x ponen t, Disorder strength, s
ZTRFIM ZTMF Expt.
FIG. 5. Dynamical hysteresis scaling exponent (Υ) versusdisorder strength ( σ ) yield in the 3D-ZTRFIM simulations.( ∗ ) represent the experimentally obtained exponent valuesfor cooling (green) and heating (orange). ( (cid:50) ) stand for zerotemperature mean field dynamical exponent, Υ ≈ . ± . between the disorder points [53]. As disorder increases,the heterogeneous nucleation increases, which leads tothe suppression of spinodal more and more, and finallyreaches a threshold level where a distinct crossover takesplace from spinodal-like to percolation behavior [60].Finally, we conclude by saying that when the thermalfluctuation is irrelevant, disorder-induced local fluctua-tion smeared the spinodal instabilities, and our resultsestablished the existence of spinodal-singularity beyondMF. Similar behavior has been seen in simulations oflow-disorder RFIM [21], 2D-Ising model [62], and Kob-Andersen model [63].It is a pleasure to thank Bhavtosh Bansal for providingaccess to the lab equipment and useful discussions. Wealso thank Prabodh Shukla, Varsha Banerjee, Fan Zhongfor helpful comments and suggestions. T.B. acknowl-edges IISER-Kolkata for a doctoral fellowship. A.B. ac-knowledges postdoctoral funding from ERC, under grantagreement AdG694651-CHAMPAGNE.A.G. prepared the sample. A.B. carried out the simu-lations. T.B. performed the measurements, analyzed thedata, conceived the problem, and wrote the paper. [1] K. Binder, Theory of first-order phase transitions, Rep.Prog. Phys. , 783 (1987).[2] P. G. Debenedetti, Metastable Liquids: Concepts andPrinciples (Princeton University Press, 1996). [3] T. Kakeshita, T. Fukuda, A. Saxena, and A. Planes,
Dis-order and Strain-Induced Complexity in Functional Ma-terials (Springer Berlin Heidelberg, 2012).[4] F. J. P´erez-Reche, B. Tadic, L. Ma˜nosa, A. Planes, and E.
Vives, Driving Rate Effects in Avalanche-Mediated First-Order Phase Transitions, Phys. Rev. Lett. , 195701(2004).[5] C. Liu, E. E. Ferrero, F. Puosi, Jean-Louis Barrat, and K.Martens, Driving Rate Dependence of Avalanche Statis-tics and Shapes at the Yielding Transition, Phys. Rev.Lett. , 065501 (2016).[6] S Giri, M Patra and S Majumdar, Exchange bias effectin alloys and compounds, Journal of Physics: CondensedMatter , 073201 (2011).[7] W. Ito, K. Ito, R. Y. Umetsu, and R. Kainumaa, Kineticarrest of martensitic transformation in the NiCoMnInmetamagnetic shape memory alloy, Appl. Phys. Lett. ,021908 (2008).[8] K. Binder, Nucleation barriers, spinodals, and theGinzburg criterion, Phys. Rev. A. , 341 (1984).[9] F. J. P´erez-Reche, E. Vives, L. Ma˜nosa, and A. Planes,Athermal Character of Structural Phase Transitions,Phys. Rev. Lett. , 195701 (2001).[10] M. Zacharias, L. Bartosch, and M. Garst, Mott Metal-Insulator Transition on Compressible Lattices, Phys.Rev. Lett. , 176401 (2012).[11] T. Bar, S. K. Choudhary, Md. A. Ashraf, K. S. Sujith,S. Puri, S. Raj, and B. Bansal, Kinetic Spinodal Insta-bilities in the Mott Transition in V O : Evidence fromHysteresis Scaling and Dissipative Phase Ordering, Phys.Rev. Lett. , 045701 (2018).[12] G. Parisi, I. Procaccia, C. Rainone, and M. Singh, Shearbands as manifestation of a criticality in yielding amor-phous solids, Proc. Natl. Acad. Sci. USA , 5577(2017).[13] C. Unger, and W. Klein, Nucleation theory near the clas-sical spinodal, Phys. Rev. B , 2698 (1984) for a review,see L. Monette, Spinodal Nucleation, Int. J. Mod. Phys.B , 1417 (1994).[14] S. Kundu, T. Bar, R. K. Nayak, and B. Bansal, Crit-ical slowing down at the abrupt mott transition: whenthe first-order phase transition becomes zeroth order andlooks like second order, Phys. Rev. Lett. , 095703(2020).[15] N. Gulbahce, H. Gould, and W. Klein, Zeros of the par-tition function and pseudospinodals in long-range Isingmodels, Phys. Rev. E , 036119 (2004).[16] K. W. Post, A. S. McLeod, M. Hepting, M. Bluschke, Y.Wang, G. Cristiani, G. Logvenov, A. Charnukha, G. X.Ni, P. Radhakrishnan, M. Minola, A. Pasupathy, A. V.Boris, E. Benckiser, K. A. Dahmen, E. W. Carlson, B.Keimer, and D. N. Basov , Coexisting first- and second-order electronic phase transitions in a correlated oxide,Nat. Phys. , 1056 (2018).[17] W. Cao, J. A. Krumhansl, and Robert J. Gooding,Defect-induced heterogeneous transformations and ther-mal growth in athermal martensite, Phys. Rev. B ,11319 (1990).[18] L. Z. T´oth, S. Szab´o, L. Dar´oczi, and D. L. Beke, Calori-metric and acoustic emission study of martensitic trans-formation in single-crystalline Ni MnGa alloys, Phys.Rev. B , 224103 (2014).[19] H. Wang, H. Gould, and W. Klein, Homogeneous andheterogeneous nucleation of Lennard-Jones liquids, Phys.Rev. E , 031604 (2007).[20] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B.W. Roberts, and J. D. Shore, Hysteresis and hierarchies:Dynamics of disorder-driven first-order phase transfor- mations, Phys. Rev. Lett. , 3347 (1993).[21] S. K. Nandi, G. Biroli, and G. Tarjus, Spinodals with Dis-order: From Avalanches in Random Magnets to GlassyDynamics, Phys. Rev. Lett. , 145701 (2016).[22] S. Zapperi, P. Ray, H. E. Stanley, and A. Vespignani,First-order transition in the breakdown of disordered me-dia, Phys. Rev. Lett., , 1408 (1997). Avalanches inbreakdown and fracture processes, Phys. Rev. E , 5049(1999).[23] G. P. Zheng, and M. Li, Influence of impurities on dy-namic hysteresis of magnetization reversal, Phys. Rev. B, , 054406 (2002). While this paper considers the effectof quenched disorder in the finite temperature random-field Ising model, the average over the random field con-figurations is taken before the dynamical equation issolved (i.e., mean field estimation). Not surprisingly, theyget Υ = 2 / , 031114 (2007).[25] J. P. Bouchaud, Crises and collective socio-economic phe-nomena: simple models and challenges, J. Stat. Phys. , 567 (2013).[26] Supplemental Material at which includes Refs. [27-34],for the details of experiments, simulation and data anal-ysis.[27] L. Ma, S. Q. Wang, Y. Z. Li, C. M. Zhen, D. L. Hou, W.H. Wang, J. L. Chen, and G. H. Wu, Martensitic andmagnetic transformation in Mn Ni − x Sn x ferromag-netic shape memory alloys, J. Appl. Phys. , 083902(2012). Q. Tao, Z. D. Han, J. J. Wang, B. Qian, P. Zhang,X. F. Jiang, D. H. Wang, and Y. W. Du, Phase stabilityand magnetic-field-induced martensitic transformation inMn-rich NiMnSn alloys, AIP Adv. , 042181 (2012).[28] A. Ghosh, P. Sen, and K. Mandal, Measurement proto-col dependent magnetocaloric properties in a Si-dopedMn-rich Mn-Ni-Sn-Si off-stoichiometric Heusler alloy, J.Appl. Phys. , 183902 (2016).[29] A. Ghosh and K. Mandal, Effect of structural disorder onthe magnetocaloric properties of Ni-Mn-Sn alloy, Appl.Phys. Lett. , 031905 (2014).[30] B. Ravel, J. O. Cross, M. P. Raphael, V. G. Har-ris, R. Ramesh, and L. V. Saraf, Atomic disorder inHeusler Co MnGe measured by anomalous x-ray diffrac-tion, Appl. Phys. Lett. , 2812 (2002).[31] A. Sharoni, J. G. Ram´ırez, and I. K. Schuller, Multi-ple Avalanches across the Metal-Insulator Transition ofVanadium Oxide Nanoscaled Junctions, Phys. Rev. Lett. , 026404 (2008).[32] J. D. Valle, N. Ghazikhanian, Y. Kalcheim, J. Trastoy,M. H. Lee, M. J. Rozenberg, and I. K. Schuller, Resis-tive asymmetry due to spatial confinement in first-orderphase transitions, Phys. Rev. B , 045123 (2018).[33] D. Kumar, K. P. Rajeev, J. A. Alonso, and M. J.Mart´ınez-Lope, Journal of Physics: Condensed Matter , 185402 (2009). K. H. Kim, M. Uehara, C. Hess,P. A. Sharma and S.-W. Cheong, Thermal and Elec-tronic Transport Properties and Two-Phase Mixturesin La / − x Pr x Ca / MnO , Phys. Rev. Lett. , 2961(2000).[34] O. Perkovi´c, K. Dahmen, and J. P. Sethna, Avalanches,Barkhausen Noise, and Plain Old Criticality, Phys. Rev.Lett. , 4528 (1995). [35] However, at classical critical point, the type of extendeddisorder, whether quench [36–38] or anneal [39], is con-troversial for fine-tuning the criticality. Fortunately, weare apart from the classical critical point, and we alsoobserved twining even after a fair amount of training thesample, and thus we would consider the quenched char-acter of disorder for the modeling purposes.[36] F. J. Perez-Reche, L. Truskinovsky, and G. Zanzotto,Training-Induced Criticality in Martensites, Phys. Rev.Lett. , 075501 (2007).[37] B. Cerrut, and E. Vives, Random-field Potts model withdipolarlike interactions: Hysteresis, avalanches, and mi-crostructure, Phys. Rev. B , 064114 (2008).[38] E. Vives, J. Goicoechea, J. Ort´ın, and A. Planes, Univer-sality in models for disorder-induced phase transitions,Phys. Rev. E R5 (1995).[39] F. J. Perez-Reche, C. Triguero, G. Zanzotto, and L.Truskinovsky, Origin of scale-free intermittency in struc-tural first-order phase transitions, Phys. Rev. B ,144102 (2016).[40] W. Fan, J. Cao, J. Seidel, Y. Gu, J. W. Yim, C. Barrett,K. M. Yu, J. Ji, R. Ramesh, L. Q. Chen, and J. Wu,Large kinetic asymmetry in the metal-insulator transi-tion nucleated at localized and extended defects, Phys.Rev. B , 235102 (2011).[41] L. Kang, L. Xie, Z. Chen, Y. Gao, X. Liu, Y. Yang,and W. Liang, Asymmetrically modulating the insulator-metal transition of thermochromic VO films upon heat-ing and cooling by mild surface-etching, Appl. Surf. Sci. , 676 (2014).[42] Return point memory has been observed above the criti-cal disorder bellow that it is completely absent, see M. S.Pierce, C. R. Buechler, L. B. Sorensen, S. D. Kevan, E. A.Jagla, J. M. Deutsch, T. Mai, O. Narayan, J. E. Davies,K. Liu, G. T. Zimanyi, H. G. Katzgraber, O. Hellwig, E.E. Fullerton, P. Fischer, and J. B. Kortright, Disorder-induced magnetic memory: Experiments and theories,Phys. Rev. B , 144406 (2007).[43] R. Gilmore, Catastrophe Theory for Scientists and Engi-neers (Dover, New York, 1981).[44] X. An, D. Mesterh´azy, and M. A. Stephanov, On spinodalpoints and Lee-Yang edge singularities, J. Stat. Mech.(2018) 033207.[45] H. Zheng, W. Wang, D. Wu, S. Xue, Q. Zhai, J. Frenzel,and Z. Luo, Athermal nature of the martensitic transfor-mation in Heusler alloy Ni-Mn-Sn, Intermetallics, , 90(2013).[46] P. Bhimalapuram, S. Chakrabarty, and B. Bagchi, Eluci-dating the Mechanism of Nucleation near the Gas-LiquidSpinodal, Phys. Rev. Lett. , 206104 (2007).[47] M. Rao, H. R. Krishnamurthy, and R. Pandit, Magnetichysteresis in two model spin systems, Phys. Rev. B 42,856 (1990). M. Rao and R. Pandit, Magnetic and ther-mal hysteresis in the O(N)-symmetric (Φ ) model, Phys.Rev. B 43, 3373 (1991). M. Rao, Comment on “Scalinglaw for dynamical hysteresis”, Phys. Rev. Lett. , 1436(1992).[48] F. Zhong, and J. Zhang, Renormalization Group Theoryof Hysteresis, Phys. Rev. Lett. , 2027 (1995). F. Zhongand Q. Chen, Theory of the Dynamics of First-OrderPhase Transitions: Unstable Fixed Points, Exponents,and Dynamical Scaling, Phys. Rev. Lett. , 175701(2005). N. Liang and F. Zhong, Renormalization grouptheory for temperature-driven first-order phase transi- tions in scalar models, Front. Phys. , 126403 (2017); F.Zhong, Renormalization-group theory of first-order phasetransition dynamics in field-driven scalar model, Front.Phys. , 126402 (2017).[49] P. Jung, G. Gray, R. Roy, and P. Mandel, Scaling law fordynamical hysteresis, Phys. Rev. Lett. , 1873 (1990).G. P. Zheng and J. X. Zhang, Thermal hysteresis scal-ing for first-order phase transitions, J. Phys.: Condens.Matter , 275 (1998). W. Lee, J.-H. Kim, J. G. Hwang,H.-R. Noh and W. Jhe, Scaling of thermal hysteretic be-havior in a parametrically modulated cold atomic system,Phys. Rev. E , 032141 (2016).[50] Y. Z. Wang, Y. Li, and J. X. Zhang, Scaling of the hys-teresis in the glass transition of glycerol with the temper-ature scanning rate, J. Chem. Phys. , 114510 (2011).[51] P. Shukla, Hysteresis in the Ising model with Glauberdynamics, Phys. Rev. E , 062127 (2018).[52] S. Pal, K. Kumar, and A. Banerjee, Universal scalingof charge-order melting in the magnetic field–pressure-temperature landscape, Phys. Rev. B , 201109(R)(2020).[53] Y. Imry, and M. Wortis, Influence of quenched impuritieson first-order phase transitions, Phys. Rev. B , 3580(1979).[54] F. Zhong and J. X. Zhang, Scaling of thermal hysteresiswith temperature scanning rate, Phys. Rev. E (4),2898(1995).[55] A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein,F. Broner, and S. H. Strogatz, Scaling Laws for Dynami-cal Hysteresis in a Multidimensional Laser System, Phys.Rev. Lett. , 2220 (1995).[56] L. Berthier, P. Charbonneau , and J. Kundu, Finite Di-mensional Vestige of Spinodal Criticality above the Dy-namical Glass Transition, Phys. Rev. Lett. , 108001(2020).[57] A. Pelissetto and E. Vicari, Dynamic Off-EquilibriumTransition in Systems Slowly Driven across ThermalFirst-Order Phase Transitions, Phys. Rev. Lett. ,030602 (2017).[58] The mean-field spnodal cannot in general survive inshort-range low-dimensional (d = 2, 3) systems [22] dueto the existence of high enough thermal fluctuation butthe long-range physics can be studied through RFIM atzero temperature [21] although the notion of spinodal in-stability has recently been observed in finite dimensionalshort-range interacting system [56] even at finite temper-ature [57].[59] A. A. Moreira, C. L. N. Oliveira, A. Hansen, N. A. M.Ara´ujo, H. J. Herrmann, and J. S. Andrade, Jr., Fractur-ing Highly Disordered Materials, Phys. Rev. Lett. ,255701 (2012).[60] A. Shekhawat, S. Zapperi, and J. P. Sethna, From Dam-age Percolation to Crack Nucleation Through Finite SizeCriticality, Phys. Rev. Lett. , 185505 (2013).[61] T. Rizzo, Fate of the Hybrid Transition of Bootstrap Per-colation in Physical Dimension, Phys. Rev. Lett. ,108301 (2019).[62] B. Scheifele, I. Saika-Voivod, R. K. Bowles, and P.H. Poole, Heterogeneous nucleation in the low-barrierregime, Phys. Rev. E , 042407 (2013).[63] B. P. Bhowmik, S. Karmakar , I. Procaccia, and C. Rain-one, Particle pinning suppresses spinodal criticality inthe shear-banding instability, Phys. Rev. E , 052110(2019). Supplemental Material
In the Supplemental Material, we discuss (a) sam-ple preparation and characterization, (b) rate-dependentavalanche, (c) extraction of phase ordering time, (d) dy-namic hysteresis measurements through DTA technique,(e) simulation details of zero temperature random fieldIsing model and finally (f) the power-law fitting and errorcalculation.
I. SAMPLE PREPARATION ANDCHARACTERIZATION
The Mn Ni Sn off-stoichiometric Heusler alloy wasprepared using an arc-melting technique inside a 4N pu-rity argon atmosphere. The purity of the used elements(Mn, Ni, and Sn) is at least 99.9 %. The as-prepared in-got was sealed separately in evacuated quartz ampoulesand annealed at 1173 K for 96 h. Afterward, the am-poule was quenched in ice water directly from the an-nealing temperature. Room temperature powder X-raydiffraction pattern was recorded using CuK α radiation toconfirm the crystallographic parent phase. The composi-tional analysis was carried out in energy dispersive analy-sis of X-ray attached with a field emission scanning elec-tron microscope. All the magnetic measurements wereperformed in a superconducting quantum interference de-vice (Quantum Design).Structural and magnetic properties: Figure 6 showsthe room temperature XRD pattern for off-stoichiometricMn Ni Sn Heusler alloy, which confirms the pres-ence of cubic austenite (Hg CuTi- type) phase at thattemperature [27]. One can notice the co-existence of asmall amount of tetragonal martensite phase also. Thezero-field cooled (ZFC) and field cooled cooling (FCC)temperature-dependent magnetization (MT-curves) of
20 30 40 50 60 70 80 90 I n t e n s it y ( a . u . ) M ( ) A ( ) M ( )
2q (degree) A ( ) A ( ) A ( ) A ( ) A ( ) A ( ) Mn Ni Sn (A) ~ Austenite(M) ~ Martensite FIG. 6. Room temperature XRD of Mn Ni Sn Heusleralloy using CuK α radiation
50 150 2500 100 200 300123012 2.32.82.02.5 M agne t i z a t i on ( A m / k g ) Temperature(K)
MT(Zero field cooled warming) MT (Field cooled cooling) R e s i s t an c e ( m W ) RT (Cooling) RT (Heating)
FIG. 7. Magnetization versus temperature and resistance ver-sus temperature of a polycrystalline Mn-Ni-Sn based Heusleralloy. the sample are recorded under 100 Oe fields within thetemperatures between 5 K and 300 K and plotted in Fig-ure 7. It shows a magnetic transition around 270 K,the ferro-para transition at Curie temperature (TCA) inthe austenite phase. At a temperature around 200 K,the magnetization of the sample rapidly decreases oncooling. Here, the material transforms from a highlymagnetic cubic austenite structure to a weakly mag-netic tetragonal martensite structure. This transition in-volves thermal hysteresis and thus can be identified as afirst-order magneto-structural transition (FOMST). Be-low this temperature, the magnetization again increasesdue to the presence of Curie point in the martensitephase. The temperatures below 125 K, a glassy phase,may be present, which might lead to the existence of theExchange Bias (EB) effect [28]. However, we are notdiscussing EB in detail in this report.Recalling the XRD data, we have found that themartensite phase exists near room temperature. It isclear from magnetic measurements that the structuraltransition resides near 200 K. It is unusual to observe theXRD peaks of a low-temperature structure at a tempera-ture 100 degrees above the transition. It is only possibleif the sample has a significant amount of disorder andstrain [29, 30]. In that case, some part of the sampledoes not transform and exhibits different behavior typescompared to ordered materials.
II. RATE DEPENDENT AVALANCHE
In the DTA experiments corresponding to Fig. 1(b)and 1(d) (main text), we obtain jerky latent heat peaksrelated to the formation of new phase nuclei at the defectpoints, and the amount of jerky heat released in coolingis more compare to the heating. The DTA curve plottedin the main text corresponds to the temperature scan-ning of rate 6 K/min. In the athermal ATPT, The sys-tem has to drive above some minimum temperature orfield rate, bellow that no avalanches have not been de-tected due to insignificance excitations in the metastablestates [4]. On the other hand, the transition becomescontinuous rather than a series of avalanches above somedriven rate, where the time scale of each avalanche issmaller than the data acquisition time [31]. In between0.5 K/min and 10 K/min temperature scanning rates,we were able to measure the MIT through a series ofavalanches. It is essential to mention that the driving-rate dependence of the critical exponents of the avalanchedistribution in first-order phase transitions had been ob-served previously [4, 5]. However, the asymmetry in thesuperheating and supercooling kinetic processes are in-dependent of the sweep rate [32].
Heating
FIG. 8. Avalanche distribution of jerky latent heat as afunction of magnitude, measured in 1/7 sec interval. Thetwo curves correspond to two different temperature scanningrates: 4 K/min (left) and 8 K/min (right).
Figure 8 shows the latent heat jump distribution fortwo different temperature scanning rate of 4 K/min and 8K/min. Both the curves manifest that the large jump ap-pears in the cooling branch of the hysteresis. The drivingrate-dependent results indicate that an apparent asym-metry has always been observed for an individual ratewithin our measurement capabilities.
III. PHASE ORDERING TIME
In Fig 2 [Main Text], we have plotted the insulatorfraction or order parameter ( φ ) as a function of timeand temperature. The quench-and-hold measurements’temperature starts from a perfectly insulating state (ormetallic state). The sample is heated up (or cool down)rapidly to a specific set temperature, then waits to equili-brate the system. Simultaneously, the sample’s resistance was recorded and converted to the insulator fraction (or-der parameter φ ) using a percolation model based onMcLachlan’s general effective medium theory [33]. Themodel reads φ ( σ /tI − σ /tE )( σ /tI + Aσ /tE ) + (1 − φ ) ( σ /tM − σ /tE )( σ /tM + Aσ /tE ) = 0 , (3)where σ M and σ I are the conductivities of metallic andinsulating phases. A define as (1 − φ c ) /φ c where φ c is theinsulator fraction at the percolation threshold. φ c = 0 . t = 2 for three dimensionalsystem.Figure 9 shows the first 50 seconds of phase evolu-tion after reaching the set temperature, and the phaseordering relaxation times τ were inferred by the fittingequation given below φ ( t ) = [ φ t =0 − φ eq ] exp ( − t/τ ) + φ eq (4)where φ eq stand for quasi-static value of the order pa-rameter. IV. DYNAMIC HYSTERESIS IN DTAMEASUREMENTS
Figure 10 shows that the dynamic hysteresis scalinghas been observed in an independent thermodynamicmeasurement using the DTA technique. There is hugeambiguity in figuring out the transition temperaturesfrom the DTA curve of a lower rate due to the multi-ple minor peaks. It is worth noting that the lower valueof the data dominates the straight-line fits in a log-loggraph. Therefore we narrowly consider the smoothed (de-gree 100) data with ramp 4 K/min and above to achievescaling exponent from DTA measurements. Nevertheless, I n s u l a t o r f r a c t i on ( f ) time (s) Cooling 203 K 200 K 198 K 196 K 194 K 192 K 190 K 188 K 186 K 184 K time (s)
Heating 185 K 188 K 190 K 192 K 194 K 196 K 198 K 200 K 202 K 204 K 206 K 208 K
FIG. 9. Temporal evolution of order parameter (insulatorfraction) during wait after shock heating (a) and cooling (b).Solid lines are exponential fits. the exponent values remain consistent with the preced-ing values extracted from resistance data. The transitiontemperatures, hence the exponent values, differ a bit de-pending upon the degrees of smoothness. That is why wemay regard these measurements as rough. Remark thatthe dynamical exponent in the heating branch is higherthan the cooling branch of hysteresis. Despite havingsome limitations of the technique, the qualitative find-ings are somewhat supreme as the results obtained fromthe heat measurement only.
160 180 200 220-2.0-1.5-1.0-0.50.00.51.01.5 5 10 300.5135 T e m pe r a t u r e r i s e ( K ) Temperature (K) T e m pe r a t u r e s h i ft [ D T = | T - T | ] ( K ) Ramp rate (K/min)
Heating Cooling g = 0.97 g = 0.88
FIG. 10. (a) DTA signal as a function of temperature fordifferent ramp rates. (c) The shift in transition temperaturesflow scaling with temperature scanning rates through expo-nents Υ = 0 .
88 (cooling) and Υ = 0 .
97 (heating).
V. ZERO TEMPERATURE RANDOM FIELDISING MDOEL
The algorithm for simulating the zero temperature ran-dom field Ising model (ZTRFIM) is presented in this sec-tion. The Hamiltonian for the random field Ising modelis defined as, H = − J (cid:88) (cid:104) i,j (cid:105) s i s j − (cid:88) i [ H ( t ) + h i ] s i , (5)The Ising-like spins, s i = ± L with periodicboundary conditions. The nearest neighbor spins inter-act ferromagnetically with interaction strength denotedby J . We set J = 1 for this work. All the spins on the lat-tice experience an external time-dependent uniform field, H ( t ). The form of H ( t ) depends on the real situation wewould like to simulate and is discussed in detail below.Additionally, the spins feel a random site-dependent buta time-independent magnetic field, h i . The random local h i is chosen from a random Gaussian distribution, V ( h )given by V ( h ) = 1 √ πσ e − h / (2 σ ) , (6) where σ is the width of the distribution and denotes thedisorder strength in this model. We average all the phys-ical quantities using 50-100 independent disorder realiza-tions. Here the random field Ising model is simulated atthe zero-temperature, and therefore, there are no ther-mal fluctuations. The spin-flip is entirely determined bythe sign change of the local field, E i = J (cid:88) j s j + h i + H, (7)where the sum over j is carried out for the neighbor sitesof i . A. Dynamic hysteresis
For a linear ramping of the field, H ( t ) = H + Rt where R is the rate of change of the magnetic field, and H isthe initial external field strength at t = 0. The algo-rithm to perform the simulation for the linear ramping ispresented below.1. Set the values of spin for every site to s i = −
1. Setthe field to H = − H m , where at H = H m all thespin is polarized.2. Increase the field by R .3. Check all the sites if the local field defined in Eq. (7)changes sign on any of the sites.4. Go to step 2 or stop if H = H m .For a quasi-static process R →
0, the system is allowedto equilibrate before increasing the value of the field. Toperform the equilibration, if a spin flips in step (3), all thespins are rechecked to see if the initial flip triggers flippingof the neighboring spin. This procedure continues untilnone of the spins flips sign during a check, after whichthe magnetic field is increased further. This equilibrationprocedure is bypassed in the linear ramping to capturethe non-equilibrium nature of the physical quantities. InFig. 11, we show that for a prolonged ramping, the mag-netization curve indeed approaches the quasi-static value.Note that hysteresis is symmetric to the origin [Fig. 11]and the width of the hysteresis loop reduces with disor-der [Fig. 12(a)]. The shift in coercive fields H c ( R ), thefields at which the magnetization changes the sign, fromthe steady-state coercive field H c ( R = 0) fulfill a scalingrelation with fields rate R:∆ H c ( R ) = | H c ( R ) − H c (0) | ∝ R Υ . (8)Figure 12(b) shows the power-law fitting for the dif-ferent disorders, and the fitting deteriorates with the in-creasing disorder from the critical disorder. The scalingexponent versus disorder graph has been shown in Fig.5 (main text). It is worth noting that when the disorderstrength higher than a threshold value σ th ≈ .
30, thefitting with a single exponent will not be feasible (seesection VI).0 -4 -2 0 2 4-1.0-0.50.00.51.0 -4 -2 0 2 4 s = 2.25 M agne t i s a t i on , f Field, H
Ramp Rate (fi 0) 1·10 -4 -4 -3 -3 -3 -3 -2 -2 -2 -2 -1 -1 s = 3.25 Field, H
FIG. 11. Magnetization φ (order parameter) versus exter-nal field H for different ramp rate yield in the 3D-ZTRFIMsimulations of system size 256 under periodic boundary con-ditions for disorder strength σ = 2.25, and 3.25. Ramp rate( →
0) indicate the steady state hysteresis -4 -3 -2 -1 -3 -2 -1 H c ( R = ) Disorder strength, s
Increasing Decreasing
Hysteresis width (a)(b) s = 2.25 g = 0.49 s = 2.75 g = 0.76 s = 3.25 g = 0.90 D H c ( R ) = | H c ( R )- H c ( ) | Rate, R
FIG. 12. (a) The absolute value of the steady-state coercivefield, H c (0), decreases with increasing disorder. (b) The shiftin coercive fields H c ( R ) from the steady-state coercive fields H c (0) follow a power law with ramp rate. (—) represent linearfitting with exponent Υ. Zero temperature mean field
Furthermore, we also performed a zero temperaturemean-field analysis of same model. In order to do so, thespin flip is determined by, E i = Jzm + h i + H, (9)where z is the average nearest neighbor and m is theaverage magnetization of the system m = (cid:80) Ni =1 s i . Notethat here the local field is determined by the averagemagnetization of the whole system. Using the spin-flipprotocol of equation 9 we perform the same algorithm presented in the previous section. B. Time constant
We also perform a simulation to find the time requiredfor a fully polarized system to relax to a steady stateif the external field is suddenly quenched to a certainvalue, mimicking the quench-and-hold experiment. Thealgorithm for the same is given below:1. Set the values of spin for every site to s i = − H = H f
3. Check all the sites if the local field defined in Eq. (7)changes sign on any of the sites.4. If there is even a single flip increase the unit of time,and go to step 3.5. When there is no flip exit, the process and the timeduring the exit is the relaxation time τ . -1.0-0.50.00.51.0 -2.0 -1.5 -1.0 -0.50200040006000 (b) M agne t i s a t i on , f Increasing FieldDecreasing Field (a) t H M agne t i s a t i on , f time, t t H FIG. 13. Temporal evolution of order parameter φ in phase-ordering simulation on ZTRFIM of system size 256 for σ =2 .
25. The individual color represents the distinct quenched-field shown in the equilibration time (relaxation time) vs.quenched-field graph (inset) for decreasing (a) and increas-ing (b) the fields.
The relaxation time results are presented in the maintext Fig. 4 (a) and 4 (b). This time-scale arises becausethe system relaxes to a steady state by series of smallavalanches [20, 34]. Fig. 13 shows the time evolution ofthe magnetization when the field is quenched to a partic-ular value, and the magnetization φ finally reached thesteady-state after time τ . Equilibration time τ peaks atthe transition points (coercive fields) [Fig. 4 (main text)and Fig. 13(inset)], demonstrates that dynamics slowdown considerably near the spinodal point.1 VI. POWER LAW FITTING AND ERROR
The the dynamical renormalized shifts ∆ T ( | T i − T iobs ( R ) | , i = heat or cool ) follows scaling with temper-ature rates R such as | T i − T iobs ( R ) | = aR Υ . (10)The exponent Υ was extracted by fitting the above equa-tion [Eq. 10] with the experimentally and numericallyobtained data points. We analyzed the data set twiceusing the best straight-line fitting method and statisticaldistribution of the non-linear fitting method. The best straight-line fitting:
In equation 10, thereare three unknown parameters, T i the quasistatic tran-sition temperature, Υ and the constant a. The constanta could be bypass by fitting a straight line in the log-log graph, where the slope of the straight line wouldbe the Υ. We varied the T i within the acceptable re-gion to achieve the best straight line fit. The accept-able values of T i stand bellow (or above) T iR , the ob-served transition temperature under the lowest heating(or cooling) rate. Since ∆ T is a monotonic increas-ing function of R, the acceptable region (let say δT )is bounded by transition temperature difference undertwo low rates ( | T iR − T iR | ) whose difference is largerthan the lowest rate (i.e., R − R > R T R − δT ) < T < T R and ( T R + δT ) > T > T R arethe allowed quasi-static temperatures region for heatingand cooling.Figure 14 shows that the least square error in the slopeof the straight line as a function of the exponent. Eachpoint of the graph corresponds to a hand-picked qua-sistatic transition temperature T i . The exponent cor-responding to the minimum of least square error couldbe thought of conclusive exponent. However, the lowervalues of the data dominate in the straight line fittingon the log-log graph. Small changes in T i , worsen thelower-rate transition temperature shift ∆ T , lead to a sig-nificant change in the exponent value. For these reasons,we come up with more logical data fitting kits where ev-ery data point contributes equally to extract the scalingexponent. Statistical distributions of non-linear fitting:
The sim-plest technique is to pick out any four data points out ofthe whole set of data and calculate the scaling exponentfor all possibilities. The exponent’s acceptance can bejudged according to the quasi-static temperature, whichhas been calculated using that exponent. The histogramof all accepted exponents will follow a normal distribu-tion where the mean value can be considered the finalscaling exponent.Let us consider T i and T j be the observed transitiontemperatures under ramp rate R i and R j . We assumethat shift in transition temperature from the quasi-staticlimits T under temperature scanning rate will follow apower law: T i = T + aR Υ i , T j = T + aR Υ j (11) E rr o r i n s l ope ( D g ) Exponent ( g ) Cooling Heating
FIG. 14. Least square fitting error vs the exponent Υ fordifferent values of the quasi-static transition temperature T .The minimum error are corresponding to Υ = 0 .
93 for heatingand Υ = 0 .
85 for cooling. C oun t s Exponent (g)
HeatingMean = 0.93Std. = 0.13
Exponent (g)
CoolingMean = 0.85Std. = 0.07
FIG. 15. Distribution of fitting exponent Υ. The exponentswere calculated using four independent points out of the wholedata set, and every data point has an equal impact for extract-ing the fitting exponent. where the unknown constant a could be negative or pos-itive depending upon the cooling or heating. The impactof quasi-static temperature to determine the exponentcan be taken off by subtracting the above equations,( T i − T j ) = a ( R Υ i − R Υ j ) . (12)If N be the total total number of data points we canpick up two points in a N C (say N
1) possible ways. Weeliminate the unknown constant a by divide the equation12 with the same equation for another pair of data { k,l } , i.e. ( T i − T j )( T k − T l ) = ( R Υ i − R Υ j )( R Υ k − R Υ l ) (13)Iterative numerical solutions of the above equation givesus N C numbers of Υ.2 C oun t s (a) s = 2.25 s = 2.75(b) s = 3.25(c) C oun t s Exponent ( g ) s = 3.50(d) Exponent ( g ) FIG. 16. Distribution of exponents for numerically evaluateddynamic hysteresis in ZTRFIM for disorder strength σ = 2.25(a), 2.75 (b), 3.25 (c), and 3.50 (d). The exponents (Υ) are ex-tracted from iterative numerical solutions of the equation 13.The distribution broadened as the disorder strength increased(a to c) and finally above the threshold value ( σ th ≈ .
30) thedistribution is no longer Gaussian (d).
For each Υ there are two T T { i,j } = T i − ( R i R j ) Υ T j − ( R i R j ) Υ , T { k,l } = T k − ( R k R l ) Υ T l − ( R k R l ) Υ (14)There is no restriction imposed on the value of T which is not acceptable for equation 11 itself. The values of Υ forwhich the inferred T lying inside the acceptable regions[( T R − δT ) < T heat < T R and ( T R + δT ) > T cool >T R ] are allowed to draw the statistical distribution ofexponent.The mean of the distribution [Fig. 15], Υ mean = 0 . mean = 0 .
85 for cooling, are compeer tothe best straight-line fitting exponents [Fig. 14]. The his-togram’s standard deviation, larger than the best straightline fitting error, can be considered the maximum errorof the results. Note that the distribution in the coolingbranch is quite sharp compared to the heating, and theerror in heating (stander deviation) is twice the error incooling.
Error in simulated exponent: