Temperature hysteresis of magnetic phase transitions in Tb$_{1-x}$Ce$_x$Mn$_2$O$_5$ ($x$ = 0, 0.20, 0.25)

The temperature evolution of the magnetic structure of multiferroics Tb$_{1-x}$Ce$_x$Mn$_2$O$_5$ ($x$ = 0, 0.20, 0.25) has been investigated using the neutron scattering methods. It has been found that, despite the qualitative similarity of the magnetic states and the series of phase transitions for pure TbMn$_2$O$_5$ (TMO) and doped crystals Tb$_{1-x}$Ce$_x$Mn$_2$O$_5$ (TCMO, $x$ = 0.20 and 0.25), there are significant differences in their properties. In contrast to TMO, where there are three magnetic phases, TCMO can include two magnetic phases that coexist in a wide temperature range and exhibit a rather wide temperature hysteresis. One of these phases with wave vector $\mathbf{k}_1$ = (0.5, 0, $k_{z1}$), $k_{z1}$ = 0.25, is commensurate and arises at temperatures below $T_\mathrm{N}\sim$ 39 K (for $x$ = 0.2) and $T_\mathrm{N}\sim$ 38 K ($x$ = 0.25). The second phase is incommensurate with wave vector $\mathbf{k}_2$ = (1/2, 0, $k_{z2}$), $k_{z2}$ = 0.256(2), and appears upon cooling at $T$ = 21 K ($x$ = 0.2) and $T$ = 19 K ($x$ = 0.25). Upon further cooling to 16 K, the component $k_{z2}$ increases to 0.292(2) and then remains constant. The component $k_{z1}$ increases to the value of 0.280(2) upon cooling in the range from 15 to 10 K and then remains constant down to 1.5 K. With an increase in the temperature, the components $k_{z1}$ and $k_{z2}$ undergo reverse changes to their initial values, but these changes occur at temperatures 7 K higher than those observed with a decrease in the temperature. For TMO, two phases also coexist, but the temperature hysteresis in this case is considerably smaller than for TCMO. This is explained by different densities of domain walls and different sizes of domains in pure and doped crystals.