First-order reversal curve of the magnetostructural phase transition in FeTe

We apply the first-order reversal curve (FORC) method, adapted from studies of ferromagnetic materials, to the magnetostructural phase transition of ${\mathrm{Fe}}_{1+y}\mathrm{Te}$. FORC measurements reveal two features in the hysteretic phase transition, even in samples where traditional temperature measurements display only a single transition. For ${\mathrm{Fe}}_{1.13}\mathrm{Te}$, the influence of magnetic field suggests that the main feature is primarily structural while a smaller, slightly higher-temperature transition is magnetic in origin. By contrast, ${\mathrm{Fe}}_{1.03}\mathrm{Te}$ has a single transition which shows a uniform response to magnetic field, indicating a stronger coupling of the magnetic and structural phase transitions. We also introduce uniaxial stress, which spreads the distribution width without changing the underlying energy barrier of the transformation. The work shows how FORC can help disentangle the roles of the magnetic and structural phase transitions in FeTe.

Figure 1

(a) Illustrative diagram of the warming and cooling lines demonstrating hysteresis and thermal coercivity, taken from ${\mathrm{Fe}}_{1.03}\mathrm{Te}$ at ${\mu }_{0}H=0T$, $P=120$ Mpa. (b) The family of FORCs shown to fill the major loop, highlighting the FORC branch starting at $T_R=59\mathrm{K}$. (c) Calculated FORC distribution with $(T,T_R)$ and $(T_C,T_B)$ coordinate axes shown. (d) Schematic illustration of a hysteron associated with the thermal hysteresis.

Figure 2

(a)–(d) Major hysteresis loop (black) and family of FORCs for ${\mathrm{Fe}}_{1.13}\mathrm{Te}$ measured at $P=0$ Pa and different magnetic fields applied along the $c$ axis. (e)–(h) FORC distributions extracted from (a)–(d). (i), (j) Projections of (e)–(h) onto the $T_B$ and $T_C$ axes, normalized to their respective maxima, with the main peak and satellite feature indicated by a dot and arrow, respectively. Error bars are determined by the resistance and temperature sensitivity, and are smaller than the linewidth.

Figure 3

(a)–(d) Major hysteresis loop (black) and family of FORCs for ${\mathrm{Fe}}_{1.03}\mathrm{Te}$ measured at $P=0$ Pa and different magnetic fields applied along the $c$ axis. (e)–(h) FORC distributions extracted from (a)–(d). (i), (j) Projections of (e)–(h) onto the $T_B$ and $T_C$ axes, with the main peak and satellite feature indicated by a dot and an arrow, respectively. The curves in each frame are normalized by the height of the main peak. The 2 T data are shown only in the projections, since the FORCs and resulting distribution are nearly identical to those for zero field. Scaling and error bars are determined as in Fig. 2. Plots in (i) and (f) are sequentially offset by 0.5 to improve visibility.

Figure 4

(a)–(d) Major hysteresis loop (black) and family of FORCs for ${\mathrm{Fe}}_{1.03}\mathrm{Te}$ measured at ${\mu }_{0}H=0$ T, at (a) $P=70$ MPa, (b) $P=120$ MPa, (c) $P=160$ MPa, (d) $P=0$ (the order shows the measurement sequence). (e)–(h) FORC distributions extracted from (a)–(d). (i), (j) Projections of (e)–(h) onto the $T_B$ and $T_C$ axes, with the main peak and satellite feature indicated by a dot and an arrow, respectively. The curves in each frame are normalized by the height of the main peak. Error bars are determined by the resistance and temperature sensitivity, and are smaller than the linewidth. Scaling, offset, and error bars are determined as in Fig. 3.

Figure 5

(a) Thermal energy calculated from the FORC feature displacement, as observed in the $T_B$ projections. The linear fit is calculated from the ${\mu }_{0}H>2$ T data. (b) Full width at half maximum (FWHM) of FORC distribution under increasing pressure. Arrows indicate the measurement sequence. The linear fit is determined using all of the data. Error bars are determined by the error of the peak fit location and width for (a) and (b), respectively.

Figure 6

Comparison of (a), (b) the derivative of the cooling and warming “direct” measurements with (c), (d) the $T_R$ and $T$ FORC projections from Figs. 1–1 in the main text.