A quantitative criterion for determining the order of magnetic phase transitions using the magnetocaloric effect

Abstract

The ideal magnetocaloric material would lay at the borderline of a first-order and a second-order phase transition. Hence, it is crucial to unambiguously determine the order of phase transitions for both applied magnetocaloric research as well as the characterization of other phase change materials. Although Ehrenfest provided a conceptually simple definition of the order of a phase transition, the known techniques for its determination based on magnetic measurements either provide erroneous results for specific cases or require extensive data analysis that depends on subjective appreciations of qualitative features of the data. Here we report a quantitative fingerprint of first-order thermomagnetic phase transitions: the exponent n from field dependence of magnetic entropy change presents a maximum of n > 2 only for first-order thermomagnetic phase transitions. This model-independent parameter allows evaluating the order of phase transition without any subjective interpretations, as we show for different types of materials and for the Bean-Rodbell model.

Fig. 1

DSC results and time dependence of sample temperatures. a Temperature dependence of the heat flow at a constant heating rate of 10 K min⁻¹ shows that Si 1.4 and Si 1.6 present FOPT while Si 1.8 undergoes a SOPT; b and c time (t) dependence of the sample temperatures, at a constant heating/cooling rate of sample holder of 0.4 K min⁻¹, show a temperature plateau for FOPT samples

Fig. 2

Arrott plots for the Si 1.4, Si 1.6 and Si 1.8 samples. According to Banerjee criterion, both Si 1.4 and Si 1.6 samples exhibit FOPT while Si 1.8 undergoes a SOPT

Fig. 3

Magnetic entropy change of the studied La₁Fe_13-xSi_x alloys. While Si 1.8 exhibits “caret” type behavior, Si 1.4 and Si 1.2 have the shape of a “cliff”. Si 1.6 apparently exhibits one behavior or the other, depending if we look at the low or high temperature parts of the surface

Fig. 4

Field and temperature dependence of the exponent n for La–Fe–Si alloys. The values of n > 2 for Si 1.2, 1.4, and 1.6 clearly indicate their FOPT behavior, while Si 1.8 is a SOPT

Fig. 5

Exponent n for the Bean and Rodbell model. Temperature dependence of the field exponent of the magnetic entropy change for cases ranging from SOPT (η ≤ 1) to FOPT (η > 1) shows that values above 2 correspond to FOPT

Fig. 6

Influence of magnetic field on the maximum exponent n. Materials with a FOPT (η > 1) exhibit an exponent value larger than 2 for all studied fields. Inset: higher resolution curve for a field of 0.2 T in the proximity of the critical point of the second-order phase transition, where the overshoot of n > 2 is already evident for $\eta \ge 1.03$

Fig. 7

The magnetic entropy change and its field exponent for Heusler-type alloy. The overshoot of exponent n occurs near the FOPT of Ni–Mn–In–Co Heusler alloy. Characteristic spikes shown in the shaded gray region indicate the switching of inverse to conventional MCE. The maximum applied field was 1.5 T

Fig. 8

Influence of distributed transition temperatures on exponent n. The materials used for this example consist of a single LaFeSi-type particle and different LaFeSi-based composite materials. In all cases, the overshoot of n > 2 is observed, in agreement with their FOPT nature

Fig. 9

Simulations with a Gaussian distribution of transition temperatures. Three different σ values of the Gaussian distribution of transition temperatures in the Bean and Rodbell model for η = 1.5 at maximum field of 2.25 T were used. Despite the smooth appearance of the magnetic entropy change (upper panel), which does not resemble FOPT, the overshoot of n > 2 near the transition is observed in all cases (lower panel)

Fig. 10

Inverse magnetocaloric response of GdBaCo₂O_6−δ for 1.5 T. The overshoot of n is observed near its first order antiferromagnetic–ferromagnetic transition. Characteristic spikes shown in the shaded gray region indicate the switching from inverse to conventional MCE