Simulating spin waves in entropy stabilized oxides

The entropy stabilized oxide ${\mathrm{Mg}}_{0.2}{\mathrm{Co}}_{0.2}{\mathrm{Ni}}_{0.2}{\mathrm{Cu}}_{0.2}{\mathrm{Zn}}_{0.2}\mathrm{O}$ exhibits antiferromagnetic order and magnetic excitations, as revealed by recent neutron scattering experiments. This observation raises the question of the nature of spin-wave excitations in such disordered systems. Here, we investigate theoretically the magnetic ground state and the spin-wave excitations using linear spin-wave theory in combination with the supercell approximation to take into account the extreme disorder in this magnetic system. We find that the experimentally observed antiferromagnetic structure can be stabilized by a rhombohedral distortion together with large second-nearest-neighbor interactions. Our calculations show that the spin-wave spectrum consists of a well-defined low energy coherent spectrum in the background of an incoherent continuum that extends to higher energies.

Figure 1

Spins on a face centered cubic lattice arranged in the antiferromagnetic order of the second kind with ordering wave vector $q=(1/2,1/2,1/2)$ in which the spins are within/between the (111) planes and aligned ferromagnetically/antiferromagnetically. Under the rhombohedral distortion, the nearest-neighbor exchange coupling is split into $J_1$/$J_1^{\prime }$ between spins within/between the (111) planes, whereas the second-nearest-neighbor exchange coupling $J_2$ is not split.

Figure 2

Magnetic ground states for a ${\mathrm{Co}}_{52}{\mathrm{Ni}}_{52}{\mathrm{Cu}}_{52}{\mathrm{Mg}}_{52}{\mathrm{Zn}}_{48}$ supercell in the case of isotropic interactions ($K=0$) with (a) $J_2=-J_1$, $\mathrm{\Delta }=0$ and (b) $J_2=-15J_1$, $\mathrm{\Delta }=0.5$.

Figure 3

Spin-wave spectra of ordered CoO (left), NiO (middle), and CuO (right) fcc lattice spin models with $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$.

Figure 4

Spin-wave spectrum of the ${\mathrm{Mg}}_{0.2}{\mathrm{Co}}_{0.2}{\mathrm{Ni}}_{0.2}{\mathrm{Cu}}_{0.2}{\mathrm{Zn}}_{0.2}\mathrm{O}$ fcc lattice spin model with moment size and spin vacancy disorder, obtained from averaging over 100 nonorthogonal supercells with 250 fcc lattice sites on average and $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$.

Figure 5

Spin-wave spectrum of the ${\mathrm{Co}}_{0.33}{\mathrm{Ni}}_{0.33}{\mathrm{Cu}}_{0.33}\mathrm{O}$ fcc lattice spin model with moment size disorder and without spin vacancy disorder, obtained from averaging over 100 nonorthogonal supercells with 250 fcc lattice sites on average and $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$.

Figure 6

Spin-wave spectrum of the ${\mathrm{Ni}}_{0.6}{\mathrm{Mg}}_{0.4}\mathrm{O}$ fcc lattice spin model without moment size disorder and with spin vacancy disorder obtained from averaging over 100 nonorthogonal supercells with 250 fcc lattice sites on average and $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$.

Figure 7

Primitive cell of the $q=(1/2,1/2,1/2)$ AFM configuration spanned by $a_1$, $a_2$, and $a_3$ imbedded in the corresponding orthogonal conventional cell spanned by $A_1=3a_1-a_2-a_3$, $A_2=-a_1+3a_2-a_3$, and $A_3=-a_1-a_2+3a_3$

Figure 8

One of the nonorthogonal supercells used for computing the disorder-configuration averaged spin-wave spectra.

Figure 9

Energy distribution curves of the spin-wave spectrum of the disordered ${\mathrm{Mg}}_{0.2}{\mathrm{Co}}_{0.2}{\mathrm{Ni}}_{0.2}{\mathrm{Cu}}_{0.2}{\mathrm{Zn}}_{0.2}\mathrm{O}$ fcc lattice spin model obtained from averaging over 100 nonorthogonal supercells with 250 fcc lattice sites on average and $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$.

Figure 10

Spin-wave spectrum of the ${\mathrm{Ni}}_{0.6}{\mathrm{Mg}}_{0.4}\mathrm{O}$ fcc lattice spin model without moment size disorder and with spin vacancy disorder obtained from averaging over 100 nonorthogonal supercells with 250 fcc lattice sites on average and $J_2=-15J_1$, $\mathrm{\Delta }=0.5$, and $K=0$, projected on spins that have two (upper panel) and three (lower panel) next-nearest-neighbor spin vacancies.