Exploring the possible origin of the spin reorientation transition in ${\mathrm{NdCrO}}_3$

Spin reorientation transitions and other related magnetic phenomena, which owe their origin to the complex interplay between multiple magnetic sublattices, have long attracted scientific attention both from the perspective of fundamental curiosity and technological applications. In this study, combining first principles calculations together with finite temperature Monte Carlo simulations, we explore the possible origins of reorientation transition of Cr spins in ${\mathrm{NdCrO}}_3$. We construct a ${\mathrm{NdCrO}}_3$ specific magnetic model, consisting of symmetric superexchange interactions between magnetic ions, as well as their magnetic anisotropy. We show that the observed spin reorientation in ${\mathrm{NdCrO}}_3$ arises out of a delicate balance between $\mathrm{Nd}\text{−}\mathrm{Cr}$ magnetic exchange interactions, single ion anisotropy of Nd spins, and single ion anisotropy of Cr spins. Moreover, though our model does not take into consideration the effect of antisymmetric and anisotropic-symmetric magnetic exchanges, the qualitative as well as quantitative agreement of the theoretically derived and the experimentally observed spin-reorientation transition in ${\mathrm{NdCrO}}_3$ confirms the merit of our proposed microscopic model. Our results also propose a hitherto unobserved collective magnetic ordering in the Nd sublattice, which is challenging to detect as it is an extreme low temperature phenomena, which therefore calls for further investigations.

Figure 1

Orthorhombic $Pbnm$ crystal structure of ${\mathrm{NdCrO}}_3$. Crystal structure showing the first nearest-neighbor isotropic M-M exchange interactions, in plane ($J_{M-M}^p$) (a) and out of plane ($J_{M-M}^{ap}$) (b). Crystal structure plot exhibiting four in-equivalent NN M-Nd exchange interactions, denoted as $J_{M-Nd}^b$ and $J_{M-Nd}^{b\prime }$ (in the crystallographic $\mathit{bc}$ plane) (c) and $J_{M-Nd}^a$ and $J_{M-Nd}^{a\prime }$ (in the crystallographic $\mathit{ac}$ plane) (d). The arrows indicate the antiferro off-centric displacements of Nd ions.

Figure 2

(a),(c) Calculated density of states (DOS) for ${\mathrm{NdCrO}}_3$ and ${\mathrm{NdFeO}}_3$ considering collinear $G$-type antiferromagnetic order in the Cr sublattice as well as the Nd sublattice. The Nd-Cr and Nd-Fe hybridization are highlighted in zoomed plots (b) and (d). We considered $U=2.2\mathrm{eV}$ and $J_H=0.3\mathrm{eV}$ for Cr $3d$ states and $U=4.5\mathrm{eV}$ and $J_H=0.3\mathrm{eV}$ for Fe $3d$ states. A higher $U$ value of 5.5 eV and $J_H=0.3\mathrm{eV}$ for Nd $4f$ states was used.

Figure 3

Observed collective spin ordering in the Cr sublattice at 5 K (a) and 75 K (b) as a function of $E_{Cr}$ and $D_{Cr}$ for $|E_{Nd}|=|D_{Nd}|=0$. We performed MC simulations considering $E_{Cr}\ne 0$ and $D_{Cr}\ne 0$, as the $E_{Cr}=D_{Cr}=0$ case leads to huge degeneracy. The same for $\mathrm{GGA}+U$ estimated values of $E_{Nd}$ and $D_{Nd}$ at 5 K (c) and 75 K (d). $G_x, G_y$, and $G_z$ denote orientation of Cr spins along the crystallographic $a, b$, and $c$ axes, respectively. Different phases are shaded with different colors. The color coding is the same between 5 K and 75 K, if there is no change in symmetry of spin ordering of Cr spins and shaded differently when there is a change in symmetry of spin ordering of Cr spins between 5 K and 75 K. The values of isotropic magnetic exchanges are kept fixed at $\mathrm{GGA}+U$ estimated values obtained with the choice of $J_H=0.3\mathrm{eV}$. In the parameter region shaded with light gray, none of the following phases, $G_x, G_y$, and $G_z$, are formed.

Figure 4

(a) Calculated temperature dependence of specific heat for a choice of $\gamma$ values. We used $E_{Cr}=0.02\mathrm{meV}$ and $D_{Cr}=-0.05$ meV. Though the nature of magnetic phase transitions strongly depends on the magnetic anisotropy of Nd and Cr ions, the value of transition temperatures shows weak dependence. (b) Magnetic phase diagram in $T\text{−}\gamma$ plane, demarcating the paramagnetic, $G_z$ and $G_y$ spin ordered phases of the Cr sublattice. The calculated Néel temperature $T_N$ values, plotted in solid circles, denote paramagnetic to $G$-type AFM ordering temperature, while the second transition temperature is plotted in stars. For $\gamma \sim 0.35$–0.62, stars denote $G_z\to G_y$ SR transition.

Figure 5

(a) Experimentally proposed $C$-type magnetic ordering in the Nd sublattice [18] having $Pbnm$ magnetic symmetry. Nd spins (denoted by red arrow) orient along the $z$ axis. The Nd spins order in AFM and FM patterns in the (001) plane and along the [001] axis, respectively. We denote this magnetic ordering as $C_{001}$. (b) Proposed $C$-type magnetic ordering in the Nd sublattice through the MC simulations. Nd spins are oriented along the $x$ axis. The Nd spins form AFM ordering along the [001] and [$1-10$] axes and FM ordering along the [110] axis. We denote this magnetic ordering as $C_{110}$. The Cr spins (denoted by orange arrow) are ordered in $G_y$ pattern. The strong isotropic exchange interactions between two magnetic sublattices create a $C_{110}$ type canted spin component in the Cr sublattice.