Spin-orbit excitons in CoO

CoO has an odd number of electrons in its unit cell, and therefore is expected to be metallic. Yet, CoO is strongly insulating owing to significant electronic correlations, thus classifying it as a Mott insulator. We investigate the magnetic fluctuations in CoO using neutron spectroscopy. The strong and spatially far-reaching exchange constants reported recently [P. M. Sarte et al., Phys. Rev. B 98, 024415 (2018)], combined with the single-ion spin-orbit coupling of similar magnitude [R. A. Cowley et al., Phys. Rev. B 88, 205117 (2013)] results in significant mixing between $j{}_{\mathrm{eff}}$ spin-orbit levels in the low-temperature magnetically ordered phase. The high degree of entanglement, combined with the structural domains originating from the Jahn-Teller structural distortion at $\sim 300\mathrm{K}$, make the magnetic excitation spectrum highly structured in both energy and momentum. We extend previous theoretical work on ${\mathrm{PrTl}}_3$ [W. J. L. Buyers et al., Phys. Rev. B 11, 266 (1975)] to construct a mean-field and multilevel spin-orbit exciton model employing the aforementioned spin exchange and spin-orbit coupling parameters for coupled ${\mathrm{Co}}^{2+}$ ions lying on a rocksalt lattice. This parametrization, based on a tetragonally distorted type-II antiferromagnetic unit cell, captures both the sharp low-energy excitations at the magnetic zone center, and the energy broadened peaks at the zone boundary. However, the model fails to describe the momentum dependence of the excitations at high-energy transfers, where the neutron response decays faster with momentum than the ${\mathrm{Co}}^{2+}$ form factor. We discuss such a failure in terms of a possible breakdown of localized spin-orbit excitons at high-energy transfers.

Figure 1

(a) First four coordination shells of the high-temperature CoO rocksalt structure. (b) Tanabe-Sugano diagram for $d^7{\mathrm{Co}}^{2+}$ in octahedral coordination calculated by Cowley et al. [54]. Shaded rectangles correspond to experimentally measured excitations for cubic CoO at room temperature with heights and the width corresponding to experimental errors in energy and the statistical error of the refined value for $10Dq/J\left(dd\right)$, respectively. The dashed red line at $10Dq/J\left(dd\right)\sim 2.5$ denotes the spin crossover from (left) high-spin $S=\frac{3}{2}$, ${}^{4}T_1$ to (right) low-spin $S=\frac{1}{2}$, ${}^{2}E$. (c) Calculated normalized energy variation as a function of the tetragonal distortion (${\widehat{\mathcal{H}}}_{\mathrm{dis}}$) and the magnetic molecular field (${\widehat{\mathcal{H}}}_{\text{MF}}$) perturbations to the $j{}_{\mathrm{eff}}$ manifolds from the ground-state crystal-field triplet ${}^{4}T_1$ of ${\mathrm{Co}}^{2+}$ in octahedral coordination. Both the energy eigenvalues and individual parameters are presented to scale.

Figure 2

(a) Background (pure and nonmagnetic MgO) subtracted powder-averaged neutron-scattering intensity maps of ${\mathrm{Mg}}_{0.97}{\mathrm{Co}}_{0.03}\mathrm{O}$ measured on (top) mari at 5 K with an $E_i=30\mathrm{meV}$, (middle) mari at 5 K with an $E_i=10\mathrm{meV}$, and (bottom) IRIS at 11 K with an $E_f$ of 1.84 meV revealing seven low-energy bands of dispersionless magnetic excitations. (b) Relevant energy scales for the effective pair Hamiltonian. (c) Calculated difference in the ground-state manifold's energy eigenstates obtained from the diagonalization of the effective pair Hamiltonian (black line). The nonlinearity is in contrast with the behavior predicted by the projection theorem (gray line). (Inset) The mechanism for antiferromagnetism (top) and weaker ferromagnetism (bottom) is a result of a combination of the ${90}^{\circ }{\mathrm{Co}}^{2+}\text{−}{\mathrm{O}}^{2-}\text{−}{\mathrm{Co}}^{2+}$ exchange pathway and the orbital degree of freedom in the $t_{2g}$ channel on each ${\mathrm{Co}}^{2+}$, in agreement with the predictions of the Goodenough-Kanamori-Anderson rules. Yellow arrows denote local $t_{2g}$ spin configurations and teal arrows denote total spin configurations on each ${\mathrm{Co}}^{2+}$.

Figure 3

Isometric view of all ${\mathrm{Co}}^{2+}$ cations located in the (a) first ($m=1$), (b) second ($m=2$), (c) third ($m=3$), and (d) fourth ($m=4$) coordination shells of the CoO rocksalt structure. For the purposes of reference, all (111) planes are labeled as either $s$ and $d$ planes with respect to the reference ${\mathrm{Co}}^{2+}$ (central black site). All displacement vectors ${\mathbf{d}}_{m,ij}$ are listed in Table SI in the Supplemental Material [60] and used to calculate $J_s\left(\mathbf{Q}\right)$ and $J_d\left(\mathbf{Q}\right)$ discussed in the text.

Figure 4

$(\mathbf{Q},E)$ slices of CoO measured on merlin at 5 K with an $E_i$ of (a) 110 meV, (b) 75 meV, and (c) 45 meV. All $(\mathbf{Q},E)$ slices have been folded along [001]. A comparison of $\mathbf{Q}$-integrated cuts of (c) and (b) with previous measurements in the literature at the (d) magnetic zone centers, and (e) magnetic zone boundaries, respectively. Solid lines in (e) indicate the location of excitations previously determined by IR spectroscopy [106]. Horizontal bars indicate instrumental resolution.

Figure 5

Comparison of $(\mathbf{Q},E)$ slices and corresponding $\mathbf{Q}$-integrated cuts for CoO measured on merlin at 5 K and calculated with a mean-field multilevel spin-orbit exciton model employing the refined parameters listed in Tables 3 and 2 for an $E_i$ of [(a), (d), (g)] 45 meV, [(b), (e), (h)] 110 meV, and [(c), (f), (i)] 75 meV. Horizontal bars in $\mathbf{Q}$-integrated cuts indicate experimental resolution. All $(\mathbf{Q},E)$ slices have been folded along [001]. Individual contributions for each $xAxx\gamma$ orbital configuration to the $\mathbf{Q}$-integrated cuts presented in (g) and (h) are illustrated in Fig. S1 [60].

Figure 6

Comparison of (a) measured ($E_i=110$ meV, 5 K) and (b) calculated $(\mathbf{Q},E)$ slices at high-energy transfers illustrating the individual contributions of the $G^{-+}$ and $G^{zz}$ components to the total neutron response. The $(\mathbf{Q},E)$ slice presented in (a) has been folded along [001].

Figure 7

A comparison of the calculated $(\mathbf{Q},E)$ slices along $(1.5,1.5,L)$ for (a) $G^{+-}$, (b) $G^{-+}$, (c) $G^{zz}$, and (d) $G^{++}$ components of (e) the total response function $G$ and (f) the corresponding slice measured on merlin at $T=5$ K with an $E_i=110$ meV, with (g), (h) the same comparison for an $E_i=45$ meV. The calculated model presented here only includes the AAAA orbital configuration. The model's parameters' values were fixed to those initial values listed in Table 1. The $(\mathbf{Q},E)$ slices presented in (f) and (h) have been folded along [001]. The intensity modulation observed in the calculated response for high-resolution measurements presented in (g) is an artifact of the steep dispersion of the excitation, manifesting itself as singularities in $g^{\alpha \beta }$ and creating numerical difficulties with sampling the $G^{+-}$ mode.

Figure 8

A comparison of $(\mathbf{Q},E)$ slices (a) measured at 5 K on merlin and (b) calculated using spin-orbit exciton model for an $E_i=110$ meV. (c) Comparison between (a) and (b) in a corresponding $\mathbf{Q}$-integrated cut. The success of the spin-orbit exciton model to reproduce the $G^{+-}$ component is emphasized. $(\mathbf{Q},E)$ slices presented in (a) and (b) have been folded along [001].

Figure 9

(a) Influence of the spin-orbit exchange constant on the splitting of the $j{}_{\mathrm{eff}}=\frac{1}{2}$ manifold as a function of $\left|J\right|$ for the pair Hamiltonian ${\widehat{\mathcal{H}}}_{\mathrm{pair}}$ [Eq. (11)]. ${\mathrm{Co}}^{2+}$ pair excitations previously measured on mari are shown explicitly with the corresponding value of $J_2$ (inset) particularly influenced by the nonlinearity at high-energy transfers. Calculated $(\mathbf{Q},E)$ slices for $E_i=110$ and 45 meV with corresponding $\mathbf{Q}$-integrated cuts at the zone boundary and center, respectively, illustrating the influences of (b) the spin-orbit coupling constant $\lambda$, (c) the mean molecular field $H_{\text{MF}}$, and (d) the tetragonal distortion $\mathrm{\Gamma }$ parameters. Solid and dashed lines denote AAAA and AAAF orbital configurations, respectively. The prominent $G^{-+}$ and much weaker $G^{+-}$ have been labeled explicitly for reference. Unless otherwise stated, the values of $\lambda$, $H_{\text{MF}}$, $\mathrm{\Gamma }$, and magnetic exchange constants $J$ were set to their initial values described in the main text and listed in Table 1.

Figure 10

A comparison between (a), (c) the phonon scattering at large momentum transfers centered about the nuclear zone boundaries and (b) the modes located near the magnetic zone boundary along the (2,2,0) direction that are unaccounted by our spin-orbit exciton model. The overlap of the energy transfers range between the (b) low-$\mathbf{Q}$ scattering near the (2,2,0) magnetic zone boundary and both gapped optical phonon modes centered near the (a) (5.5,3.5,4.5) and (c) (1,1,6) nuclear zone boundaries with energy transfers above $\sim 30$ and $\sim 60$ meV, respectively, are emphasized with dashed lines. All three $(\mathbf{Q},E)$ slices have been folded along [001] and have been renormalized to share a common relative intensity scale.

Figure 11

(a) Energy-integrated ($E=[70,105]$ meV) slice measured on merlin at 5 K with an $E_i$ of 110 meV. (b) $(\mathbf{Q},E)$ slice folded along [001] measured on merlin at 5 K with an $E_i$ of 75 meV. $\mathbf{Q}$-integrated ($\xi ,\xi ,L$) cuts measured on merlin at 5 K with an $E_i$ of (c) 110 meV, (d) 75 meV, and (e) 45 meV. Solid and dashed lines in $\mathbf{Q}$-integrated cuts correspond to the ${\mathrm{Co}}^{2+}$ magnetic form factor $f^2\left(Q\right)$ that includes and excludes the intensity at (1.5,1.5,$\pm 0.5$), respectively. Both $(\mathbf{Q},E)$ slices presented in (a) and (b) have been folded along [001]. Arrows in (a), (b), and (c) indicate fluctuations exhibiting potential itinerantlike behavior. For the purposes of comparison, the specific region in $(\mathbf{Q},E)$ space (blue-green) identified in (c) as possibly containing itinerantlike fluctuations has been labeled explicitly in Fig. 12.

Figure 12

Comparison of (a) $(\mathbf{Q},E)$ slice measured on merlin with $E_i=110$ meV and the corresponding (b) calculated kinematically permissable $(\mathbf{Q},E)$ region for the 2-magnon continuum. The specific region in $(\mathbf{Q},E)$ space that was previously identified in Fig. 11 being labeled explicitly. The $(\mathbf{Q},E)$ slice presented in (a) has been folded along [001].