Biquadratic and ring exchange interactions in orthorhombic perovskite manganites

We use ab initio electronic structure calculations within the generalized gradient approximation (GGA+U) to density functional theory to determine the microscopic exchange interactions in the series of orthorhombic rare-earth manganites, o-$R{\mathrm{MnO}}_3$. Our motivation is to construct a model Hamiltonian (excluding effects due to spin-orbit coupling), which can provide an accurate description of the magnetism in these materials. First, we consider ${\mathrm{TbMnO}}_3$, which exhibits a spiral magnetic order at low temperatures. We map the exchange couplings in this compound onto a Heisenberg Hamiltonian and observe a clear deviation from the Heisenberg-like behavior. We consider first the coupling between magnetic and orbital degrees of freedom as a potential source of non-Heisenberg behavior in ${\mathrm{TbMnO}}_3$, but conclude that it does not explain the observed deviation. We find that higher order magnetic interactions (biquadratic and four-spin ring couplings) should be taken into account for a proper treatment of the magnetism in ${\mathrm{TbMnO}}_3$ as well as in the other representatives of the o-$R{\mathrm{MnO}}_3$ series with small radii of the $R$ cation.

Figure 1

Crystal structure of o-$R{\mathrm{MnO}}_3$: (a) side view; (b) top view $(R$ ions are not shown). (c) and (d) Representation of the normal modes of Jahn-Teller distortion $Q_2$ and $Q_3$, respectively.

Figure 2

Collinear magnetic orderings (in Wollan-Koehler notation [6]) of the Mn spins within the perovskite unit cell: (a) A-AFM, (b) G-AFM, (c) C-AFM, and (d) FM. $J_c$ and $J_{ab}$ indicate the NN exchange couplings along the $c$ axis and within the $ab$ planes, respectively.

Figure 3

Heisenberg interactions in ${\mathrm{TbMnO}}_3$, which are considered in this work. Mn atoms within the 40-atom supercell are highlighted with dark purple. Light purple circles indicate Mn ions which belong to the neighboring supercells. NN exchanges are indicated in red, second NN in blue, and third NN in green.

Figure 4

Forty atom supercell of ${\mathrm{TbMnO}}_3$ (side view) with magnetic orders which were used to calculate the exchange parameter $J_c$. Tb and O ions are not shown.

Figure 5

Dependence of the energy $E$ (relative to the energy of A-AFM order) of ${\mathrm{TbMnO}}_3$ on the rotation angle $\alpha$ of spins from the A-AFM to G-AFM state calculated using $U=2$ eV and $J_H=0$ eV. The results of DFT calculations are shown by dots and the fitting to the Heisenberg model [Eq. (11)] by the blue line. The red line indicates the fitting to a Hamiltonian which includes bilinear and higher order exchange couplings [Eq. (15)], $B_2$ and $C_2$ are the fitting parameters, which define the sizes of the bilinear and higher order terms, respectively.

Figure 6

(a) Ideal cubic perovskite structure; (b) purely JT distorted structure (tetragonal); (c) fully JT+GFO distorted structure (orthorhombic) of ${\mathrm{TbMnO}}_3$.

Figure 7

Orbital mixing angle as a function of the amplitudes of JT and ${\mathrm{GdFeO}}_3$-type distortions for G-AFM and A-AFM magnetic orderings in ${\mathrm{TbMnO}}_3$. In the part of the graph highlighted with violet (cyan), only the amplitude of the JT (GFO) distortion is varied. $\phi$ is determined using Eq. (3).

Figure 8

Dependence of the energy $E$ (relative to the energy of A-AFM order) on the rotation angle $\alpha$ of spins from A-AFM to G-AFM state for the structures of ${\mathrm{TbMnO}}_3$ with different amplitudes of JT and GFO distortions. Dots indicate the results of DFT calculations; lines show the fittings to the Heisenberg Hamiltonian [Eq. (11)]. (a) and (b) show the values for the structures with 40% and 100% JT distortion, respectively, without octahedral tiltings; (c) and (d) show the values for the structures with the full JT distortion and 60% and 100% GFO distortion, respectively. Plot (d) was obtained using the crystal structure which unlike the experimental one does not include the antiferroelectric displacements of $R$ cations, thus it is not identical to the one shown in Fig. 5.

Figure 9

Dependence of the energy $E$ (relative to the energy of A-AFM order) on the rotation angle $\alpha$ of spins from the A-AFM to G-AFM state for compounds without orbital ordering: (a) ${\mathrm{TbCrO}}_3$ and (b) ${\mathrm{TbFeO}}_3$. Dots correspond to the results of DFT calculations; lines show the fittings to the Heisenberg Hamiltonian [Eq. (11)].

Figure 10

Energies of the 80 atom supercell of ${\mathrm{TbMnO}}_3$ with 54 inequivalent collinear magnetic configurations (referred to the energy of the A-AFM state) predicted by (a) the pure Heisenberg Hamiltonian and (b) the Hamiltonian, which includes bilinear and four-spin ring couplings, and plotted versus the energies of corresponding states calculated using DFT. Ideally, the model and DFT energies should be equal and points should lie on the dashed line. Insets show the deviations of the model energies from those calculated using DFT. Each bar corresponds to one considered magnetic configuration.

Figure 11

Magnetic orderings which are used to extract the biquadratic exchange interactions: (a) $j_{ab}$ in the $ab$ planes $(G$ and $K$ indicate four-spin ring exchange couplings in plaquettes of Mn spins confined in the $ab$ planes and those containing pairs of spins from neighboring $ab$ planes, respectively); (b) $j_c$ along the $c$ axis (violet dashed rectangle indicates a rotation plane of spin 4).

Figure 12

Dependencies of the energy $E$ (relative to the energy of A-AFM order) of ${\mathrm{TbMnO}}_3$ on the rotation angle $\alpha$ of spins from A-AFM to G-AFM state calculated using (a) $U=4$ eV and $J_H=0$ eV; (b) $U=8$ eV and $J_H=0$ eV; (c) $U=3$ eV and $J_H=1$ eV. The results of DFT calculations are shown by dots. The blue line indicates the fitting to the Heisenberg model [Eq. (11)] and the red line—fitting to a Hamiltonian which includes bilinear and higher order exchange couplings [Eq. (15)]. $B_2$ and $C_2$ are the fitting parameters of Eq. (15), which define sizes of the bilinear and higher order terms, respectively.

Figure 13

Energies of the 80 atom supercells of ${\mathrm{PrMnO}}_3$ (a) and (b), ${\mathrm{TbMnO}}_3$ (c) and (d), and ${\mathrm{LuMnO}}_3$ (e) and (f) with more than 30 inequivalent magnetic configurations (referred to that of the lowest-energy state) predicted by the pure Heisenberg Hamiltonian (blue dots) and the Hamiltonian which includes bilinear and four-spin ring exchanges (red dots) and plotted versus the energies of corresponding states calculated using DFT.