Modulated quasi-two-dimensional antiferromagnetic structures in La$_{1-y}$Nd$_y$MnO$_{3+\delta}$ manganites

The structural, electronic, and magnetic phase transitions induced by the isovalent substitution of the rare-earth Nd$^{3+}$ ion for the La$^{3+}$ ion with a larger radius have been investigated in the system of self-doped manganites La$_{1-y}$Nd$_y$MnO$_{3+\delta}$ (0 $\le y\le$ 1; $\delta\sim$ 0.1). For the average radius of the ions in $A$-sites of the lattice $\langle r_A\rangle$ $<$ 1.19 $\mathring{\mathrm{A}}$ ($y>$ 0.5), the phenomena revealed in the manganites are as follows: the ordering of Mn $e_g$ orbitals, the transition from the pseudocubic $O^*$ phase to the orthorhombic $O'$ phase, the opening of the dielectric Jahn–Teller gap, the frustration of the collinear ferromagnetic (FM) state, and the transition from the lowtemperature canted FM to canted antiferromagnetic (AFM) state of Mn spins. It is assumed that, in samples with neodymium concentrations $y$ = 0.9 and 1.0 $\bigl(\langle r_A\rangle$ $\approx$ 1.16 $\mathring{\mathrm{A}}\bigr)$ at temperatures $T <$ 12 K, there coexist $A$- and $E$-type modulated AFM states similar to the sinusoidal and helical structures of Mn spins, which were previously studied in $R$MnO$_3$ multiferroics. The magnetic $T$–$H$ phase diagrams of these samples are characteristic of quasi-two-dimensional antiferromagnets with a very low (zero) magnetic anisotropy in the $ab$ planes. Under these conditions, the phase transition from the $A$-type AFM phase to the spin-flop state occurs in a relatively weak magnetic field. The AFM ordering of the Nd magnetic moments with a critical phase transition temperature $T_{\mathrm{Nd}}\cong$ K is induced in magnetic fields with a strength $H\ge$ 3.5 kOe. For the NdMnO$_{3 +\delta}$ manganite in a magnetic field $H$ = 10.7 kOe, the curves $M(T)$ are characterized by additional very narrow peaks near temperatures $T_1\cong$ 4.5 K and $T_2\cong$ 5 K. The additional features revealed for the first time in the magnetization near $T$ = 0 are assumed to be caused by the quantization of the spectrum of free holes in the $ab$ planes by a strong magnetic field.