Magnetic anisotropy from interligand hopping in strongly correlated insulators: application to the magnon spectrum of CrI$_3$

Abstract

Spin-orbit coupling (SOC) gives rise to complex magnetic states such as spin liquids, skyrmion crystals, and topological spin-wave excitations. We consider exchange interactions in multi-orbital Mott insulators where SOC is strong on ligand ions. SOC on the ligands enables electron hopping accompanied by spin flips and fluctuations in the orbital state of the ligand hole. These processes generate anisotropic exchange interactions and greatly increase the number of possible exchange paths. The number grows further with the inclusion of hopping between ligands, which mediates interactions between more distant spins. We propose an effective method to calculate exchange interactions at arbitrary separations between spins. Applying it to monolayer CrI$_3$, we obtain anisotropic interactions between nearest-neighbor and next-nearest-neighbor Cr spins, as well as single-ion anisotropy induced by long-range hopping. In this material, magnetic anisotropy stabilizes long-range ferromagnetic order and opens a magnon gap at the Dirac points, which defines a nontrivial magnon band topology. Using Hubbard model parameters from first-principles calculations, the resulting spectrum agrees well with the spin-wave dispersion observed experimentally in bulk CrI$_3$, except that the calculated Dirac gap is much smaller.

Figures (5)

• Figure 1: CrI$_{3}$ monolayer crystal and electronic structure. (a) The white (black) spheres show Cr$^{3+}$ (I$^{-}$) ions. Blue lines indicate the two-fold rotational symmetry axes of the CrI$_3$ monolayer, $2_{[110]}$ and $2_{[011]}$ (we use the Cartesian ${ x}, { y}$ and ${ z}$ axes shown by yellow arrows). The red point indicates the inversion symmetry center of an ideal iodine lattice, important for exchange interactions between the Cr ions $A$ and $C$. (b) Symmetries of an ideal Cr-I plaquette in the $xy$-plane: two-fold rotation axes, $2_{[110]}$ and $2_{[001]}$, and inversion around the center of the plaquette. (c) $t_{pd\pi}$ is the amplitude of hopping between the $d_{yz}$ orbital of the metal ion $A$ and the $p_z$ orbital of the ligand ion 1; $t_{pd\sigma}$ describes the hopping between the $p_x$ and $d_{3x^2 - r^2}$ orbitals. (d), (e) The amplitudes $t_{pp\pi}$ and $t_{pp\sigma}$ describe the hopping between $p$-orbitals of neighboring ligand ions.
• Figure 2: Effective $dd$-hopping. Isotropic Heisenberg exchange interactions from electron hopping between (a) $d_{yz}$ and $d_{xz}$ orbitals via $p_z$ ligand orbital and (b) $d_{xy}$ and $d_{3x^2 -r^2}$ orbitals via $p_x$ orbital. White (black) spheres show Cr$^{3+}$ (I$^{-}$) ions and the hopping direction is indicated by arrows. An electron hops with spin flip (c) between the $d_{yz}$ and $d_{xy}$ orbitals via the SOC-entangled $p_z$ and $p_y$ ligand orbitals, and (d) between the $d_{yz}$ and $d_{3x^2 -r^2}$ orbitals via the SOC-entangled $p_z$ and $p_x$ ligand orbitals. (e) An electron hops from the $p_z$ orbital of ligand 6 to the empty $d_{3z^2-r^2}$ orbital of Cr-$B$; the ligand hole at site 6 is filled by the electron from the ligand 7 $p_z$ orbital; finally, the electron occupied the $d_{yz}$ orbital of Cr-$A$ hops into the $p_z$ orbital of ligand 7 via the $p_y$ orbital with the spin flip due to the SOC. (f) Hopping from the $d_{xy}$ orbital on site $A$ to the $d_{3z^2-r^2}$ orbital on site $C$ due to the tunneling between the $p_z$ orbitals of ligand ions 2 and 5. It is preceded by a spin flip on site 2, followed by a change of the orbital state from $p_y$ to $p_z$. The electron can hop back to site $A$ via the $p_y$ orbitals of ligand ions 2 and 6.
• Figure 3: (a) The iodine sublattice energy band structure, shown for high-symmetry lines. Zero energy corresponds to the energy of isolated iodine's $p$ orbital state. Hopping amplitudes are $t_{pp\pi}=0.15$ eV and $t_{pp\sigma}=0.7$ eV; SOC is $\lambda = 0.63$ eV. (b) Illustration of electron virtual hopping from fully-occupied $p$ orbital I bands to localized $d$ Cr states.
• Figure 4: Calculated spin model parameters. (a,b) SOC dependence of the single-ion anisotropy $A_c$ and $JK\Gamma$ parameters for (a) nearest- and (b) next-nearest-neighbor Cr ions; $t_{pp\sigma}=0.7$ eV. (c,d) $pp$-hopping amplitude dependence of $A_c$ and $JK\Gamma$ for (c) nearest- and (d) next-nearest-neighbor Cr ions; $\lambda=0.63$ eV. Other parameters are: $t_{pp\pi}=t_{pp\sigma}/4$, $t_{pd\sigma}=1$ eV, $t_{pd\pi}=0.5$ eV $U=3$ eV, $J_{\rm H} = 0.25 U$, $\Delta_c = 1.1$ eV and $U' = U - 2J_{\rm H}$.
• Figure 5: Magnon energy spectrum (blue lines) along the high symmetry lines $\Gamma -K-M$ in the honeycomb Brillouin zone. It is obtained using spinW Toth2014. Color encodes the unpolarized neutron scattering cross-section. The insets show the spectrum details near the $\Gamma$ and $K$ points. The model parameters are: $J_1=-2.12$ meV, $J_2=-0.2$ meV, $J_3=0.005$ meV, $A_c = -0.1$ meV and $K_1=0.04$ meV, and other are set to zero.