Magnetic Properties of Spin-Orbital Polarons in Lightly Doped Cobaltates

We present a numerical treatment of a spin-orbital-polaron model for ${\mathrm{Na}}_x{\mathrm{CoO}}_2$ at small hole concentration ($0.7<x<1$). We demonstrate how the polarons account for the peculiar magnetic properties of this layered compound: They explain the large susceptibility; their internal degrees of freedom lead both to a negative Curie-Weiss temperature and yet to a ferromagnetic intralayer interaction, thereby resolving a puzzling contradiction between these observations. We make specific predictions on the momentum and energy location of excitations resulting from the internal degrees of freedom of the polaron, and discuss their impact on spin-wave damping.

Figure 1

(a) Cubic symmetry for the ${\mathrm{Co}}^{3+}$ ion is removed next to a hole (${\mathrm{Co}}^{4+}$ ion); this reduces the energy splitting between $t_{2g}$ and $e_g$ levels of ${\mathrm{Co}}^{3+}$. Hund’s rule coupling $j_H$ then stabilizes $S=1$ instead of $S=0$. (b) Virtual $t_{2g}\mathrm{\text{−}}{e}_g$ hopping leading to antiferromagnetic coupling $J_{ij}$ between two ${\mathrm{Co}}^{3+}$ ions. (c) Spin structure and interactions within the polaron: ${\mathrm{Co}}^{4+}$ with $s=1/2$ in the center, induced spin $S=1$ on the six adjacent ${\mathrm{Co}}^{3+}$ sites. (${\mathrm{Co}}^{3+}$ ions further away have $S=0$.) Parameters: $J$ couples nearest neighbor (NN) spins with $S=1$ on the ring; $J_{\mathrm{diag}}$ couples them across the diagonal; both are antiferromagnetic. $J^{\prime }$ couples the central $s=1/2$ to the ring.

Figure 2

(a) Inverse susceptibility and (b) susceptibility of a polaron [see Fig. 1c] for some parameter values. For comparison, the susceptibility resulting from a single $s=1/2$ spin is included in (b) (solid black line without symbols).

Figure 3

Three possible configurations of interacting polarons. $J_{ab}$ gives the bare ferromagnetic coupling between $s_a$ and $s_b$; remaining couplings and symbols are as in Fig. 1c.

Figure 4

Ground-state spin correlations $⟨{\overrightarrow{s}}_a{\overrightarrow{s}}_b⟩$ between the two ${\mathrm{Co}}^{4+}$ sites for the interacting-polaron configurations depending on $J^{\prime }$. (a), (b), and (c) indicate the cluster as depicted in Fig. 3. $J_{\mathrm{diag}}=J,J_{ab}=0$.

Figure 5

Energies of the lowest states with $S_{\mathrm{tot}}^z=0$, 1, 2 for the bipolaron Fig. 3a depending on $J^{\prime }$ for $J_{ab}=0$ (main panel) and depending on $J_{ab}$ for $J^{\prime }=-J$ (inset). $J_{\mathrm{diag}}=J$.

Figure 6

Singlet-triplet gap depending on $J^{\prime }$ for all three clusters: crosses for the bipolaron Fig. 3a, triangles for Fig. 3b, and circles for Fig. 3c. $J_{\mathrm{diag}}=J$.