Thermodynamics around the first-order ferromagnetic phase transition of ${\mathrm{Fe}}_2\text{P}$ single crystals

The specific heat and thermodynamics of ${\mathrm{Fe}}_2\mathrm{P}$ single crystals around the first-order paramagnetic to ferromagnetic (FM) phase transition at $T_{\mathrm{C}}\simeq 217\mathrm{K}$ are empirically investigated. The magnitude and direction of the magnetic field relative to the crystal axes govern the derived $H\text{−}T$ phase diagram. Strikingly different phase contours are obtained for fields applied parallel and perpendicular to the $c$ axis of the crystal. In parallel fields, the FM state is stabilized, while in perpendicular fields the phase transition is split into two sections, with an intermediate FM phase where there is no spontaneous magnetization along the $c$ axis. The zero-field transition displays a textbook example of a first-order transition with different phase stability limits on heating and cooling. The results have special significance since ${\mathrm{Fe}}_2\mathrm{P}$ is the parent material to a family of compounds with outstanding magnetocaloric properties.

Figure 1

Temperature dependence of the specific heat of an ${\mathrm{Fe}}_2\mathrm{P}$ crystal. (a) Specific heat in zero field. Inset: The sample mounted on the nanocalorimeter membrane cell. (b) Specific heat with magnetic field along the $c$ axis (0, 0.1, 0.25, 0.5, 1, and 2 T). (c) Specific heat with field perpendicular to the $c$ axis (0, 0.5, 0.75, 1, 1.5, and 2 T). Inset: The magnetic contribution $C_{\mathrm{m}}$ at high fields (1, 1.5, and 2 T).

Figure 2

(a) Specific heat of an ${\mathrm{Fe}}_2\mathrm{P}$ single crystal around the zero-field transition. The paramagnetic state supercools to a temperature $T_0$ while the ferromagnetic state superheats to $T_1$. The specific heat diverges at these corresponding temperatures rather than at $T_{\mathrm{C}}$, following $C=C_0+C_{1}T+C_{\mathrm{m}}$ as shown by the solid curves. (b) Differential temperature $\Delta T=T_{\mathrm{sample}}-T_{\mathrm{ref}}$ as a function of reference temperature on cooling in zero field. A sudden temperature change of the sample is seen when the latent heat is released during the transition. (c) Persistent thermometer resistance change of the calorimeter due to the thermal expansion at the magnetoelastic transition of the sample for fields $\mathbf{H}\perp \mathbf{c}$.

Figure 3

Magnetization along the applied field of an ${\mathrm{Fe}}_2\mathrm{P}$ crystal. (a) Magnetization as a function of field applied along the $c$ axis (solid curves) and perpendicular to the $c$ axis (dashed curves) for three different temperatures: $188\mathrm{K},216\mathrm{K}\approx T_{\mathrm{C}}$, and $276\mathrm{K}$. (b) Temperature dependence of the magnetization for fields along (filled symbols) and perpendicular to (open symbols) the $c$ axis $(\circ$ = 0.01 T, $\square$ = 0.1 T, and $⊳$ = 1 T). Inset: The low-field curves for $\mathbf{H}\perp \mathbf{c}$. $T_{\mathrm{C}}\left(0\mathrm{T}\right)$ is indicated by vertical dashed lines.

Figure 4

(a) Magnetic entropy $S_{\mathrm{m}}$ around $T_{\mathrm{C}}$ calculated from specific-heat data in $0\mathrm{T}$ (black curve) and $1\mathrm{T}$ with field applied along (blue curve) and perpendicular to the $c$ axis (red curve). (b) The magnetic entropy change $\Delta S_{\mathrm{m}}=S_{\mathrm{m}}\left(H\right)-S_{\mathrm{m}}\left(0\right)$ obtained from specific heat (solid curves) and magnetic measurements (symbols), with field applied along (blue) and perpendicular to the $c$ axis (red).

Figure 5

Magnetic phase diagram for ${\mathrm{Fe}}_2\mathrm{P}$ obtained from specific heat (filled symbols) and magnetization measurements (open symbols), complemented with published results [15]. The upper panel shows the phase diagram for magnetic field applied along the $c$ axis, while the lower panel shows the case for $\mathbf{H}\perp \mathbf{c}$ (with reversed ordinate). The direction of $M_{\mathrm{s}}$ is indicated for each ferromagnetic phase. The transition is first order for fields below the directional-dependent critical point, marked CP.