Nature of the magnetic interactions in ${\mathrm{Sr}}_3{\mathrm{NiIrO}}_6$

Iridates abound with interesting magnetic behaviors because of their strong spin-orbit coupling. ${\mathrm{Sr}}_3{\mathrm{NiIrO}}_6$ brings together the spin-orbital entanglement of the ${\mathrm{Ir}}^{4+}$ ion with a $3d$ Ni cation and a one-dimensional crystal structure. It has a ferrimagnetic ground state with a 55-T coercive field. We perform a theoretical study of the magnetic interactions in this compound, and elucidate the role of anisotropic symmetric exchange as the source of its strong magnetic anisotropy. Our first-principles calculations reproduce the magnon spectra of this compound and predict a signature in the cross sections that can differentiate the anisotropic exchange from single-ion anisotropy.

Figure 1

(a) Crystal structure of ${\mathrm{Sr}}_3{\mathrm{NiIrO}}_6$. Green, grey, yellow, and red spheres represent Sr, Ni, Ir, and O ions respectively. (b) The local axes used for the Ir ions. (c) The $|3z^2-r^2\rangle$ orbital on an Ir ion.

Figure 2

(a) Densities of states of the Ir ion projected onto the $d$ orbitals in the FiM state with magnetic moments along the $z$ axis (magnetic ground state). Red curve is the DOS projected onto the $|3z^2-r^2\rangle$ orbital, and the blue curve is the sum of the projected DOS's onto the other four $d$ orbitals. (b) Same quantity as in (a), but in the FM state with magnetic moments along the $x$ axis. (c) Energy resolved expectation value of the $z$ component of spin $\langle S_z\rangle$ for the $d$ orbitals of the Ir ion in the FiM state with magnetic moments along the $z$ axis. (d) Same quantity as in (c), but in the FM state with magnetic moments along the $x$ axis.

Figure 3

(a) Ferrimagnetic state with moments along the $z$ axis, which is the lowest-energy state (FiM-$z$: ${\varphi }_{\mathrm{Ni}\text{−}\mathrm{Ir}}={180}^{\circ },{\varphi }_{\mathrm{Ni}}=0^{\circ },{\varphi }_{\mathrm{Ir}}={180}^{\circ })$. (b) Ferromagnetic state with moments along the $x$ axis (FM-$x$: ${\varphi }_{\mathrm{Ni}\text{−}\mathrm{Ir}}=0^{\circ },{\varphi }_{\mathrm{Ni}}={90}^{\circ },{\varphi }_{\mathrm{Ir}}={90}^{\circ })$. (c) Possible ferrimagnetic intermediate state $\left({\varphi }_{\mathrm{Ni}\text{−}\mathrm{Ir}}={180}^{\circ }\right)$. (d) Observed, canted ferrimagnetic intermediate state $\left({\varphi }_{\mathrm{Ni}\text{−}\mathrm{Ir}}<{180}^{\circ }\right)$. (e) Definitions of ${\varphi }_{\mathrm{Ni}}$ and ${\varphi }_{\mathrm{Ir}}$.

Figure 4

First-principles results for the intrachain magnetic interaction between Ni and Ir atoms. (a) Energy as a function of the angle between the magnetic moments of nearest-neighbor atoms. Blue squares are the energy calculated from DFT, and yellow circles are the energy obtained from the fitted anisotropic exchange model (AEM). (b) Same as in (a), but yellow circles are the energy obtained from the fitted model with anisotropic exchange and SIA on the Ni ion. (c) The angle that the Ir and Ni magnetic moments make with the [001] axis as a function of the angle between the magnetic moments of nearest-neighbor atoms.

Figure 5

Magnon spectra of ${\mathrm{Sr}}_3{\mathrm{NiIrO}}_6$ for wave vectors along the [001] direction in the magnetically ordered phase, obtained from the anisotropic exchange model.