Mechanisms and origins of half-metallic ferromagnetism in ${\mathrm{CrO}}_2$

Using a realistic low-energy model, derived from the first-principles electronic structure calculations, we investigate the behavior of interatomic exchange interactions in ${\mathrm{CrO}}_2$, which is regarded to be one of the canonical half-metallic (HM) ferromagnetics. For these purposes we employ the dynamical mean-field theory (DMFT), based on the exact diagonalization of the effective Anderson impurity Hamiltonian, which was further supplemented with the theory of infinitesimal spin rotations for the exchange interactions. In order to elucidate the relative roles played by static and dynamic electron correlations, we compare the obtained results with several static techniques, including the unrestricted Hartree-Fock (HF) approximation, static DMFT (corresponding to the infinite frequency limit for the self-energy), and optimized effective potential method for treating the correlation interactions in the random-phase approximation. Our results demonstrate that the origin of the HM ferromagnetism in ${\mathrm{CrO}}_2$ is highly nontrivial. As far as the interactions in the neighboring coordination spheres are concerned, HF and DMFT methods produce very similar results, due to the partial cancellation of ferromagnetic (FM) double-exchange and antiferromagnetic (AFM) superexchange contributions, which represent two leading terms in the ${\left(\mathrm{\Delta }\widehat{\mathrm{\Sigma }}\right)}^{-1}$ expansion for the exchange interactions ($\mathrm{\Delta }\widehat{\mathrm{\Sigma }}$ being the intra-atomic spin splitting). Both contributions are weaker in the HF approximation due to, respectively, additional orbital polarization of the $t_{2g}$ states and neglect of dynamic correlations. The role of higher-order terms in the ${\left(\mathrm{\Delta }\widehat{\mathrm{\Sigma }}\right)}^{-1}$ expansion is twofold. On the one hand, they give rise to additional FM contributions to the neighboring exchange interactions, which tend to stabilize the FM state. On the other hand, they produce AFM long-range interactions, which make the FM state unstable in the single-site DMFT calculations for the minimal model, consisting of the $t_{2g}$ bands. Thus, the robust ferromagnetism in the minimal model, which can be easily obtained using static approximations, is fortuitous and this picture is largely revised at the level of more rigorous DMFT approach. We argue that the main ingredients, which are missing in the minimal model, are the direct exchange interactions and the magnetic polarization of the oxygen $2p$ band. We evaluate these contributions in the local-spin-density approximation and argue that they play a very important role in stability of the FM ground state in ${\mathrm{CrO}}_2$.

Figure 1

Total and partial densities of states of ${\mathrm{CrO}}_2$ in the LDA. The shaded light (blue) area shows the contribution of the Cr $3d$ states. The positions of the main bands are indicated by symbols. The Fermi level is at zero energy (shown by dot-dashed line).

Figure 2

Atomic electron densities, explaining relative positions of Cr $t_{2g}$ orbitals at the sites 1, $1^{\prime }$, and 2. The oxygen atoms are indicated by the green circles.

Figure 3

Partial densities of states as obtained in the unrestricted HF approach (left) and the DMFT (right) for the FM state. The Fermi level is at zero energy (shown by a dot-dashed line).

Figure 4

(Left) Distance dependence of interatomic exchange interactions as obtained in the theory of infinitesimal spin rotations for the minimal $t_{2g}$ model, supplemented with different types of approximations for treating the electron correlations: local-spin density approximation (LSDA), unrestricted Hartree-Fock approximation (HF), dynamical mean-field theory (DMFT), and static limit for the DMFT self-energy $\widehat{\mathrm{\Sigma }}(\omega \to \infty )$ (SDMFT). (Right) Lattice of Cr sites with the notation of interatomic exchange interactions.

Figure 5

Results of the ${\left(\mathrm{\Delta }\widehat{\mathrm{\Sigma }}\right)}^{-1}$ expansion for the NN (top) and next-NN (bottom) exchange interactions. The individual contributions, $J_i^{\left(n\right)}$, are shown on the left panel and their sum is shown on the right panel. The asymptotic values of $J_1$ and $J_2$ are shown by the dash-dotted lines.

Figure 6

Frequency dependence of the intraatomic spin splitting $\mathrm{\Delta }\widehat{\mathrm{\Sigma }}\left(\omega \right)$ in DMFT. The static limit $\mathrm{\Delta }\widehat{\mathrm{\Sigma }}(\omega \to \infty )$ is shown by dashed lines. The Fermi level is at zero energy (shown by a dot-dashed line).

Figure 7

Partial densities of states as obtained in the unrestricted HF approach (left) and the DMFT (right) for the antiferromagnetic state. The Fermi level is at zero energy (shown by a dot-dashed line).

Figure 8

Results of calculations of the spin-wave dispersion with the parameters, obtained in the theory of infinitesimal spin rotations in the cases of HF, SDMFT, and DMFT techniques. Notations of the high-symmetry points of the Brillouin zone are taken from [55].

Figure 9

Results of calculations of the spin-wave dispersion with the DMFT parameters obtained for the isolated $t_{2g}$ band (solid line) and after taking into account the additional FM contribution $\mathrm{\Delta }{J}_2=17.81$ meV, arising from magnetic polarization of the oxygen band and direct exchange interactions in the $t_{2g}$ band (dotted line). Notations of the high-symmetry points of the BZ are taken from [55].

Figure 10

Electronic band structure of ${\mathrm{CrO}}_2$, obtained in the OEP approach: (Left) Total and partial densities of states of three $t_{2g}$ orbitals and (right) band dispersion along high-symmetry directions of the BZ (notations of the high-symmetry points are taken from [55]) The Fermi level is at zero energy.

Figure 11

Results of energy minimization in the OEP method versus the splitting in the potential between atomic levels 3 and 2: the HF part of the energy $E_{\mathrm{HF}}=E_{\mathrm{kin}}+E_{\mathrm{C}}+E_{\mathrm{X}}$, the correlation energy $E_{\mathrm{corr}}$, and the total energy $E=E_{\mathrm{HF}}+E_{\mathrm{corr}}$.