Electronic-cluster model of yttrium iron garnets

We review electronic and magnetic properties of yttrium iron garnets, considered as a superlattice of quantum dots of two different types. Those are the tetrahedral and octahedral iron-oxygen clusters characterized by their respective localized energy levels with the orbital eigenstates, specified by the clusters’ respective point symmetries. We discuss two-dot mechanisms of the superexchange coupling between spins, attached to the orbital states, via the $p\text{−}d$ tunneling, only (between clusters of different types), or via the $p\text{−}d$ tunneling combined with the next-nearest neighbor $p\text{−}p$ hopping (between clusters of the same type). The electronic density of states of a superlattice of the clusters is found and analyzed both for pure and valence-uncompensated garnet systems. A scheme of control of the superexchange coupling due to an external magnetic field, applied to the system is proposed.

Figure 1

The lowest-lying and first excited energy levels of the octahedral ($a$) cluster as a function of the $p\text{−}d$ hybridization parameter, $V$. Solid lines show the orbital doublets, mutually conjugate linear combinations of ${\left(e_g^{S_6}\right)}^{*}$ and $e_u^{S_6}$; dotted lines show the orbital singlets, linear combinations of $a_g^{S_6}$ and $a_u^{S_6}$; dashed lines show the orbital doublets, mutually conjugate linear combinations of $e_g^{S_6}$ and $e_u^{S_6}$. Values of the relevant model parameters are given inside the figure.

Figure 2

The lowest-lying and first excited energy levels of the tetrahedral ($d$) cluster as a function of the $p\text{−}d$ hybridization parameter, $V$. Dotted lines show the orbital sextet, all the mutually conjugate linear combinations of $b_{3d}^{S_4}$ with $e_{2p}^{S_4}$ and $e_{3d}^{S_4}$ with $e_{2p}^{S_4}$; dashed-dotted lines show the orbital singlets, linear combinations of $a_{3d}^{S_4}$ with $b_{2p}^{S_4}$; dashed lines show the orbital doublets, mutually conjugate linear combinations of $b_{3d}^{S_4}$ with $e_{2p}^{S_4}$; solid lines show the orbital doublets, mutually conjugate linear combinations of $a_{3d}^{S_4}$ with $e_{2p}^{S_4}$. Values of the model parameters are given inside the figure.

Figure 3

All four energy levels of the sets with the respective ground states of the octahedral (dashed lines) and tetrahedral (solid lines) clusters, respectively, versus the $p\text{−}d$ hybridization parameter, $V$. The remaining model parameters take the following values: $U=8\mathrm{eV}$, $E_a=-1.5\mathrm{eV}$, $E_d=-0.44\mathrm{eV}$, $E_p=2.5\mathrm{eV}$.

Figure 4

Probability of localization at the iron sites for the lowest-lying and corresponding opposite-spin energy levels of the octahedral and tetrahedral clusters, respectively, vs the $p\text{−}d$ hybridization parameter, $V$, with $\left(B=0.5\mathrm{eV}\right)$ and without $\left(B=0\right)$ the external magnetic field.

Figure 5

Effect of the external magnetic field, applied along the $⟨111⟩$ axis on the doublet eigenvalues of the octahedral cluster. The eigenstates are the same as those given in Fig. 3. The solid and dashed lines show both eigenenergies of each doublet. Values of the relevant model parameters are given inside the figure.

Figure 6

Effect of the external magnetic field, applied along the $⟨111⟩$ axis, on three of the set of the four eigenvalues of the tetrahedral cluster. The eigenstates are the same as those in Fig. 3. Values of the relevant model parameters are indicated inside the figure.

Figure 7

Probability of localization at the iron sites for lowest-lying (the upper curves) and the corresponding opposite-spin (the lower curves) energy levels of the octahedral (the solid lines) and tetrahedral (the dashed lines) clusters, respectively, vs strength of the applied magnetic field. Again, values of the model parameters are given inside in the figure.

Figure 8

The superexchange integral $J_{ad}$ between spins attached to the ground orbital eigenstates of the $a$ and $d$ clusters vs the hybridization parameter $V$, without the external magnetic field $B$, for pure (the solid line) and doped (the dotted curve) system. And, the spin triplet-singlet energy difference $D_{ad}$ with the external magnetic field $B=0.2\mathrm{eV}$ again for pure (the dashed curve) and doped (the dotted-dashed curve) system. The relevant model parameters take the following values: $U=8\mathrm{eV}$, $E_a=-1.5\mathrm{eV}$, $E_d=-0.44\mathrm{eV}$, and $E_p=2.5\mathrm{eV}$.

Figure 9

The spin triplet-singlet difference in energy $D_{ad}$ vs the applied magnetic field for an undoped system. Values of the relevant model parameters are $U=8\mathrm{eV}$, $E_a=-1.5\mathrm{eV}$, $E_d=-0.44\mathrm{eV}$, and $E_p=2.5\mathrm{eV}$.

Figure 10

The exchange integrals $J_{aa}$ (the solid line) and $J_{dd}$ (the dashed line) between spins, attached to the orbital ground states of the $\mathrm{NN}$ octahedral and tetrahedral clusters, respectively, vs the $p\text{−}p$ tunneling parameter $t_2$, for the pure system and without the external magnetic field. Values of the system parameters: $U=8\mathrm{eV}$, $E_a=-1.5\mathrm{eV}$, $E_d=-0.44\mathrm{eV}$, $E_p=2.5\mathrm{eV}$.

Figure 11

The exchange integrals $J_{aa}$ (the solid line) and $J_{dd}$ (the dotted line) between spins attached to the orbital ground states of the $\mathrm{NN}$ tetrahedral clusters, respectively, vs the $p\text{−}p$ tunneling parameter $t_2$, in the doped-system, without the magnetic field. Values of the microscopic parameters are like those in Fig. 10.

Figure 12

The EDOS of $\mathrm{YIG}$ with an antiferromagnetic (the dashed line) and ferromagnetic (the solid line) mutual orientation of the ground-state spins of the nearest-neighbor $a$ and $d$ clusters, for $V=0.5\mathrm{eV}$, $t_2=0.25\mathrm{eV}$.

Figure 13

The EDOS of $\mathrm{YIG}$ with mutually antiparallel (dashed line) and parallel (solid line) spins of the orbital ground states of the $a$ and $d$ clusters in the external magnetic field ($B=0.5\mathrm{eV}$ and the same values of $V$ and $t_2$ like those in Fig. 12).