Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Magnetic relaxation dynamics driven by the first-order character of magnetocaloric La(Fe,Mn,Si)13


    Here, we study the temporal evolution of the magnetic field-driven paramagnetic to ferromagnetic transition in the La(Fe,Mn,Si)13 material family. Three compositions are chosen that show varying strengths of the first-order character of the transition, as determined by the relative magnitude of their magnetic hysteresis and temperature separation between the zero-field transition temperature Tc and the temperature Tcrit, where the transition becomes continuous. Systematic variations in the fixed field, isothermal rate of relaxation are observed as a function of temperature and as a function of the degree of first-order character. The relaxation rate is reduced in more weakly first-order compositions and is also reduced as the temperature is increased towards Tcrit. At temperatures above Tcrit, the metastability of the transition vanishes along with its associated temporal dynamics.

    This article is part of the themed issue ‘Taking the temperature of phase transitions in cool materials’.

    1. Introduction

    The La(Fe,Si)13 family of materials are of interest for use in magnetocaloric effect (MCE) solid-state cooling, due to substantial adiabatic temperature change (ΔTad) and isothermal entropy change (ΔSM) when driven in temperature T or field H between their paramagnetic (PM) and itinerant ferromagnetic (FM) states [1,2]. La(Fe,Si)13 with low Si content has a sharp, first-order transition [3] at the Curie temperature, TC, allowing the MCE to be exploited in small magnetic fields within the range of permanent magnets, with small hysteresis greatly reducing hysteretic losses when cycling. In addition, the compound’s working temperature, which for applied fields of up to 2 T (for permanent magnets) corresponds to a range of between 0 and 10 K above TC, can be easily tuned over a significant range (approx. 100–340 K) by substitution on the Fe sites with Si, Co, Mn, etc. [35], or by introduction of interstitial hydrogen [6], making it very attractive for room temperature application. Substitution however also impacts the properties of the MCE, in particular Mn doping moves the first-order PM–FM transition towards continuous behaviour while decreasing TC [5,7], increased Si also moves the first-order transition towards continuous behaviour but increases TC. For room temperature applications, a combination of hydrogenation, which contributes a significant increase in TC [5,6,8], and suitable Mn/Si substitution is desired [9].

    The La(Fe,Si)13 material family is also of fundamental interest in showing a zero-field thermally induced FM–PM transition at TC and a phase line of the first-order itinerant electron metamagnetic transitions at fields and temperatures above TC, ending at a temperature Tcrit above which the magnetic transition is continuous [1,6]. The dynamics of the field-driven first-order transition [10] and the associated magnetic relaxation behaviour have been a subject of some interest [11], partly because of the insight provided in terms of the performance of the magnetic refrigerator cycle where magnetocaloric materials are subjected to field changes of the order of 1 T s−1. The evolution of the zero-field, thermally induced transition in macroscopic bulk samples has been well described by the Johnson–Mehl–Avrami model of nucleation and growth [12], showing a cooling rate dependence, although more recent work in small fragments studied in vacuum found an athermal rate-independent character to features associated with avalanche-like behaviour in the zero (or very low)-field transition [13]. The temporal evolution of the isothermal transition has also been studied previously [11] and found to evolve from PM to FM over surprisingly long time scales of the order of tens to hundreds of seconds, affected by the demagnetization (shape) of the sample and the magnetic dipole coupling. Previously, we have shown that thermal linkage across the sample plays an equally important role in the temporal evolution of the transition [14]. From these studies, a description of the evolution of the magnetic transition in La(Fe,Si)13 emerges as a very local process of nucleation, latent heat release, magnetic coupling and growth of the minority phase in the host phase [12].

    In the present work, we revisit the temporal dynamics in samples of the same geometric shape, but with different degrees of first-order character. We find that, in all samples, the transition evolves in time from PM to the high-field FM state, and that the magnitude of the relaxation rate varies in approximate proportionality with the total magnetization change in each sample. Moreover, we find that the relaxation rate decreases with increasing temperature up to the temperature Tcrit.

    2. Experimental methods

    LaFexMnySiz alloys with variable Mn content were prepared by powder metallurgy techniques as described in [5]. Master alloys were prepared by vacuum induction melting followed by mechanical milling steps to produce fine powders. The composition of each alloy was adjusted by blending master alloys with elemental powers. Compaction of the powder blends was performed by cold isostatic pressing. The green bodies were vacuum sintered at around 1100°C followed by an annealing treatment at 1050°C [4].

    Three compositions were investigated: LaFe11.33Mn0.37Si1.30, LaFe11.57Mn0.18Si1.25 and LaFe11.74Mn0.06Si1.20. Samples had similar size and shape, with masses of 3.4 mg, 3.4 mg and 3.2 mg, respectively, and dimensions 2×1 mm with a thickness of 0.4 mm. For clarity, the samples will be referred to by their Mn content, x=0.37, 0.18 and 0.06.

    Magnetometry measurements were conducted using vibrating sample magnetometry (VSM) with a 4 T maximum magnetic field. Samples were temperature controlled via nitrogen exchange gas. Addition VSM measurements were done using a Quantum Design PPMS with a 9 T maximum field with temperature control via helium exchange gas. Field was applied parallel to the axis of the longest sample dimension (i.e. the smallest demagnetizing factor), which yields the maximum magnetic relaxation rate as has been previously shown [11]. For the magnetic relaxation measurements, the field sweep rate to reach the target field was set at 1.7 or 8.3 mT s−1, as described in the text. We estimate that the time constants for intrinsic sample thermalization and sample to bath equilibration were generally both less than 1 s, but note that the strong peak in heat capacity near the transition will extend these times considerably.

    Microcalorimetry measurements were performed on a commercial Xensor SiN membrane chip (TCG-3880) adapted to operate either as an AC calorimeter to obtain the heat capacity [15] or as a quasi-adiabatic temperature probe for the latent heat [16] in a cryostat with temperature range 5–293 K and an external magnetic field up to 8 T. The latent heat could also be estimated using an indirect method, and agreed with direct measurements within the experimental uncertainties [17]. The sample is a fragment of the bulk, typically approximately 100 μm, with mass of the order of a few micrograms. For an accurate determination of mass, the fragments were measured in the magnetometer and the saturation magnetization of the FM state was compared with the bulk.

    3. Results and discussion

    Figure 1 shows the isothermal dependence of magnetization on field M(H) at temperatures above TC for the three LaFe13−xyMnxSiy compositions investigated. It can be seen that, with increasing Mn content, the transition is broadened and the jump in magnetization at the transition ΔM as well as the hysteresis width ΔH are reduced. The sharpness of the ΔM jump and the magnitude of the ΔHhyst are the two commonly used indicators of the degree of first-order character; although this is not a useful comparison between different material families [18], it can serve as a useful guide in a single systematic series of samples as we consider here. For each sample, ΔHhyst reduces as the temperature is increased, up to a critical point in the phase diagram at the temperature Tcrit where ΔHhyst drops to zero. Indeed the disappearance of hysteresis is usually used to define the critical point, and as we have shown before it is also the temperature where the latent heat drops to zero simultaneously [19,20].

    Figure 1.

    Figure 1. LaFe13−xyMnxSiy magnetization loops at selected temperatures for (a) x=0.37, y=1.30, (b) x=0.18, y=1.25 and (c) x=0.06, y=1.20. The field sweep rate for the curves shown was 1.7 mT s−1. Inset to (a): an example of the extraction of the rate-independent HC↑ and HC↓ and therefore ΔHhyst at μ0 dH/dt=0 T s−1 (represented by the crosses) for x=0.18 at 174 K.

    ΔM has been determined for the PM–FM transition using the onset of hysteretic character of the M(H) loops as shown in figure 1; however, this becomes increasingly uncertain for weaker transitions. It can be seen that, based on the sharpness of the transition and the magnitude of the hysteresis, the sample with the least amount of Mn is the most strongly first order in character. Increasing the Mn weakens the order of the transition [7]. In addition, the two most first-order compositions x=0.06 and 0.18 were seen to show a fracturing of the transition upon initial field-cycling due to permanent microstructural changes of the sample caused by the large and sudden magnetovolume changes creating a range of distinct effective HC values [21], after which the transition loop was always reproducible. This is particularly clear for x=0.06 (figure 1c), where several small steps have formed in the loop shape. Therefore, all samples were cycled across their transitions multiple times before measuring.

    The magnetic hysteresis width ΔHhyst is defined as the difference between the critical field for field increasing, HC↑, and field decreasing, HC↓, at the mid-point of each transition, as shown in figure 1a,b. La(Fe,Si)13 shows a dependence of the hysteresis width on the magnetic field ramp rate, dH/dt, due to the onset of non-isothermal conditions [10,14,22]. However, the intrinsic (i.e. field rate independent) hysteresis can be extracted by measuring M(H) at different field sweep rates and extrapolating to dH/dt=0, as shown in figure 1a, inset.

    The HT phase diagram for the three compositions, extracted for intrinsic HC↑ and HC↓ using the method described above, is shown in figure 2. The critical point Tcrit is defined as the temperature where the hysteresis goes to zero [3], as shown in figure 2, inset (with associated uncertainty arising from the lack of resolution once the hysteresis decreases below 0.01 T). Another measure of the degree of first-order character within the La(Fe,Si)13 family is the separation between Tcrit and TC which, as can be seen in figure 2, diminishes as the Mn content increases.

    Figure 2.

    Figure 2. HC values determined from magnetometry as the mid-point of transition for field increase, HC↑ (closed), and field decrease, HC↓ (open), in temperature for LaFe13−xyMnxSiy, with x=0.06, 0.18 and 0.37. Inset: magnetic hysteresis calculated as the difference between HC↑ and HC↓; Tcrit is defined (with uncertainty) where μ0ΔHhyst=0 T. This is also depicted in the HT plane in the main figure as the grey oval.

    Temporal relaxation, whereby the first-order transition from one (superheated or supercooled) state to the other can proceed in constant H and T, is strongly dependent on sample shape, size and field orientation, as well as other experimental factors such as the thermal contact with the bath. Figure 3a shows the magnetic relaxation for x=0.18 at several fixed magnetic fields specified in the lower inset. The lowest field at which the transition occurs, 0.64 T in this case, is identified with the onset of the PM–FM transition, which we denote as HC1. Owing to the tendency for the strongly first-order compositions to crack after first cycling (again first cycling not shown), the post-virgin relaxation curves for both x=0.18 and 0.06 show several distinct stages of relaxation. For example, for x=0.18 in figure 3a as the period of varying dM/dt (such as from 0 to 15 s for the data with H paused at 0.64 T) before a stage of nearly constant dM/dt takes place after 15 s. This latter stage, commencing at a time we denote t0, can be easily identified in each measurement and so was used to define the relaxation rate dM/dt via a linear fit. Figure 3a and the upper inset show that the relaxation rate dM/dt at the moment the field is paused, t=0, increases linearly with increased paused field value Hpause/HC1 (as also shown in [14]). Previously, the nucleation and growth process across the thermally driven PM–FM transition in zero field was analysed within the Johnson–Mehl–Avrami model of nucleation and growth [12]. That model assumes that the growth rate of the new phase with temperature is not dependent on the fraction of phase already transformed. As set out in [12], the situation for the field-driven metamagnetic transformation is more complicated, partly due to the magnetic dipole coupling of the transformed phase influencing the ease of transformation of the remainder of the parent phase. Our observations are consistent with this latter scenario. A plateau in dM/dt is reached when the fixed field is high enough for the transition to complete before H is paused (figure 3a, upper inset).

    Figure 3.

    Figure 3. For LaFe13−xyMnxSiy with x=0.18 (a) temporal evolution of the magnetization at 160 K as the PM–FM transition is approached in field, at a field sweep rate of μ0 dH/dt=8.3 mT s−1 and then held constant at t=0 s at several fields along the magnetization loop (see bottom inset) and allowed to relax through the transition in constant H and T. Between each measurement the field is reset to 0 T. The labelled field values are: H1=0.60 T, H2=0.64 T, H3= 0.66 T, H4=0.68 T, H5=0.70 T, H6=0.74 T and H7=0.78 T. Top inset: dM/dt extracted from a linear fit to the principal stage of the transition in the main figure, as a function of the pause field value normalized to HC1 (see text for details). (b) Temporal evolution of the magnetization at fixed field HC1 for different temperatures after approaching HC1 at 1.7 mT s−1. The data have been offset in both M and t to the point where the principal relaxation across the transition begins for each temperature. For all the data, the field has already been paused before tt0=0.

    Figure 3b shows an example of the temporal evolution of the magnetization at fixed magnetic field HC1, as a function of temperature for x=0.18. The temperature dependence of the relaxation rate dM/dt is shown in figure 4a for all three samples, extracted from the M(t) data for x=0.06 and 0.37 following the method described in figure 3 for x=0.18. To make a direct comparison, the temperature axis has been normalized to T/TC. The relaxation rate of all three compositions decreases as T increases, with an exponential-like decay (particularly clear for x=0.18 and 0.06), as marked on the graph. The data are consistent with the relaxation rate reducing to zero at Tcrit, which, for x=0.37, 0.18 and 0.06, Tcrit/TC is 1.12, 1.20 and 1.22, respectively. Figure 4b plots the absolute magnetization jump, ΔM(T), for each sample, and figure 4c shows the relaxation rates normalized by ΔM(T). The three sets of data approximately collapse onto a single curve.

    Figure 4.

    Figure 4. (a) Absolute relaxation rate, dM/dt, as a function of reduced temperature, T/TC, for the three samples of LaFe13−xyMnxSiy. The curves are the fit to an exponential form. The temperature and applied field for each relaxation curve is labelled individually. (b) Measured jump in magnetization of the PM–FM transition, ΔM(T), calculated from figure 1; the curves in this case are a guide-to-the-eye only. (c) The temperature dependence of the normalized relaxation rate.

    To explore the origin of the temperature dependence of the relaxation rate, we examine the temperature dependence of the latent heat and the heat capacity. Figure 5a shows the increase in the heat capacity and figure 5b the decrease in the latent heat spikes (shown as the variation of the thermopile voltage), as a function of increasing field and temperature in a 100 μm fragment taken from the x=0.06 sample. Figure 6a,b compares the evolution of the latent heat (taking one readily identifiable main spike) and the inverse specific heat (proportional to the thermal diffusivity) as a function of temperature, in two different compositions. The magnetic relaxation (figure 4), the latent heat and the inverse specific heat all decrease as the temperature approaches Tcrit, although in detail there are differences in their functional form making direct comparison difficult. Nevertheless, the trend in the data is similar for both samples and consistent with a picture of spatially localized behaviour of the evolution of the transition in terms of nucleation, latent heat release, thermalization, time for the system to re-equilibrate with the bath (both times will increase as the heat capacity increases) and then growth of the high-field phase.

    Figure 5.

    Figure 5. (a) Heat capacity and (b) latent heat (thermopile voltage signal Vth) measured for LaFe13−xyMnxSiy, with x=0.06, at different temperatures as a function of applied field.

    Figure 6.

    Figure 6. Normalized temperature dependence of the main latent heat spike Vth (left axis) and inverse of the specific heat at the peak 1/c (right axis) taken on a fragment of the sample for LaFe13−xyMnxSiy, with (a) x=0.18 and (b) x=0.06.

    The magnetization relaxation data have been acquired on bulk samples in nitrogen exchange gas to ensure good thermal linkage of the sample with the bath. The latent heat data, shown in figures 5 and 6, have been taken on a 100 μm size fragment, acquired in vacuum to ensure weak thermal linkage to the bath. It is interesting to acknowledge that different thermal environments may change the observed dynamics of the transition [2224]. Precisely how the thermal environment influences the measurements described here will be the subject of future studies.

    4. Conclusion

    Here, we investigate the magnetic relaxation rate as a function of temperature in a set of La(Fe,Mn,Si)13 samples with varying degrees of first-order character. We show that the absolute value of the relaxation rate is reduced in samples with weaker first-order character, and that the relaxation rates vary in approximate proportionality with the total magnetization change at the transition in each sample. We find that for each sample the relaxation rate also reduces as the temperature is increased from TC towards Tcrit and becomes zero once the sample shows a continuous PM–FM transition. We have extracted the latent heat and the inverse heat capacity change in fragments taken from each sample and show that these also decrease to zero as Tcrit is approached, demonstrating the role played by local thermal linkage in the dynamics of the evolution of the transition. The local nature of the magnetic transition in these materials can be probed also with imaging techniques such as Hall probe imaging [14,25], and can be numerically simulated [26]. We have pointed out the important role of the thermal environment on the observed dynamics in these materials.

    Data accessibility

    Data can be accessed by emailing .

    Authors' contributions

    L.F.C. and E.L. were jointly responsible for the conception and design of the experiment; M.B. and E.L. were jointly responsible for acquiring data; E.L., L.F.C. and A.D.C. were responsible for drafting and revising the manuscript; A.B. and M.K. were responsible for supply of samples and critically reading the manuscript; L.G. was responsible for key discussions, and critically reading and revising the manuscript.

    Competing interests

    We have no competing interests.


    L.F.C. and M.B. have received funding from the European Community’s Seventh Framework Programme (FP7/2997-2003) under grant agreement no. 310748 ‘DRREAM’. E.L. is supported by an EPSRC DTA Studentship. L.G. is supported by FAPERJ, CNPq and CAPES (Science without Borders programme).


    One contribution of 16 to a discussion meeting issue ‘Taking the temperature of phase transitions in cool materials’.

    Published by the Royal Society. All rights reserved.