Landau model for the phase diagrams of the orthorhombic rare-earth manganites $R{\text{MnO}}_3$ ($R=\text{Eu}$, Gd, Tb, Dy, Ho)

The present work aims to describe, within a single phenomenological approach, the specific sequence of phase transitions observed in the rare-earth manganites $R{\text{MnO}}_3$ at zero magnetic field. It is shown that a single integrated description of the temperature versus composition phase diagrams of these compounds and related solid solutions can be obtained within the scope of Landau theory by adopting the so-called type-II description of the modulated phases.

Figure 1

(Color online) The ionic radius of the rare-earth elements (from Pr to Lu) plotted as a function of the modulation wave number immediately below the stability limit of the paramagnetic phase. The straight line from Ho to Eu is a linear fit. The points marked by arrows at the ionic radii 101.77 and 113.19 pm were obtained by extrapolation of that best fit.

Figure 2

(a) Experimental (dots) and simulated (lines) temperature dependences of the modulation wave numbers for the $R{\text{MnO}}_3$ compounds ($R=\text{Gd}$, Tb, Dy, and Ho). For ${\text{EuMnO}}_3$ only the simulation is shown (see text).

Figure 3

(Color online) The dependence of the parameters (a) $\mu$ and (b) $T_0$ on the ionic radius of the rare-earth ion $R$ (from Eu to Ho). The inset in (b) shows the variation in $T_0$ over an extended range that includes $R=\text{La}$, Sm Pr, Nd, Yb, and Lu (see text).

Figure 4

(Color online) Polarization, modulation wave vector, and free energy of the phases L-INC, Cycl-XY, and $A$-AFM as functions of temperature for [(a)–(c)] ${\text{TbMnO}}_3$ and [(d)–(f)] ${\text{DyMnO}}_3$. The model parameters are given in Table and the reference energy corresponds to that of the paramagnetic phase.

Figure 5

Experimental (dots) and simulated (lines) of the temperature dependence of the wave numbers of the magnetic modulation in the ${\text{Eu}}_{1-x}{\text{Y}}_x$ ${\text{MnO}}_3$ mixed system for the compositions $x=0.2$, $x=0.3$, and $x=0.4$. The experimental data was taken from Ref. 62. The predicted modulation wave number for pure ${\text{EuMnO}}_3$ is also shown for comparison.

Figure 6

(a) Detail of the experimental (dots) and fitted (line) temperature dependence of the modulation wave number of the spin wave for $x=0.4$; (b) the electric polarizations $P_y$ and $P_z$ as functions of the temperature. The polarization rotation observed at the transition between the Cycl-XY and the Cycl-XZ phases is well reproduced by the model.

Figure 7

(a) Simulated and (b) experimental temperature versus composition phase diagram of the ${\text{Eu}}_{1-x}{\text{Y}}_x$ ${\text{MnO}}_3$ solid solution. The experimental phase diagram was taken from Ref. 14. The inset shows the concentration dependence of the adjustable parameter $T_2$ (see text).