Unified understanding of the first-order nature of the transition in ${\mathrm{TbCo}}_2$

The determination of the nature of phase transitions, especially those that involve two symmetry-breaking order parameters, is a fundamental issue in condensed matter physics. For the Laves-phase rare-earth–transition-metal intermetallic compounds, their phase transitions involve both magnetic ordering and structural ordering. As a typical material of the Laves-phase intermetallics, ${\mathrm{TbCo}}_2$ has been studied extensively for its transition around 230 K. However, the understanding on the nature of this transition has remained controversial (first/second order) for decades. Here, in this paper, based on the criteria that determine first-order and second-order transitions for magnetic materials, which are (1) latent heat, (2) thermal hysteresis, (3) coexistence of phases, and (4) the Banerjee criterion, we show direct evidence to reveal the first-order nature of the transition in ${\mathrm{TbCo}}_2$, which is further interpreted by a Landau theory based phenomenological approach. Our work reconciles the lasting arguments on the transition of ${\mathrm{TbCo}}_2$ and may pave the way for deepening the understanding on the transitions of magnetic materials that involve both magnetic and structural transitions.

Figure 1

(a) $M\text{−}T$ curve and (b) $\chi \text{−}T$ curve of ${\mathrm{TbCo}}_2$. The derivative of magnetization over temperature and fitted $1/{\chi }^{\prime }\text{−}T$ curve are shown in the insets of (a) and (b), respectively.

Figure 2

(a) The refined XRD patterns for ${\mathrm{TbCo}}_2$ at 130 K. The background and calculated Bragg peak positions are shown below the observed (plus) and calculated (red line) intensities, and the difference is shown by the blue line at the bottom. (b1) Characteristic XRD reflections $\left\{222\right\}$ and $\left\{800\right\}$ at 260, 230, 220, and 200 K, respectively, with the red, green, and blue lines denoting the cubic fit, rhombohedral fits, and full sum, respectively. (b2) Lattice parameters and lattice strain within the temperature range 130–260 K.

Figure 3

Heat flow of ${\mathrm{TbCo}}_2$ on heating and cooling processes at a rate of 6 K/min. The inset shows the ramp rate dependence (linear fitting) of exothermic and endothermic peak temperatures.

Figure 4

(a) The magnetization isotherms measured at selected temperatures near $T_C$ under an applied field of 2 T. (b) The isotherms of $H/M\text{−}{\mathit{M}}^2$: The inset shows the curve of the low magnetization region.

Figure 5

(a) $H$ vs $M$ curve and nonlinear fitting. (b) Lattice strain $\varepsilon$ vs ${Ms}^2$ and linear fitting. In (b), the values of $Ms$ and $\varepsilon$ are obtained from $M\text{−}H$ loops and the synchrotron XRD patterns measured at 180, 200, 220, and 230 K; the value of $K$ (71.2 GPa) is taken from Ref. [20].