Electronic excitations and spin interactions in chromium trihalides from embedded many-body wavefunctions

Although chromium trihalides are widely regarded as a promising class of two-dimensional magnets for next-generation devices, an accurate description of their electronic structure and magnetic interactions has proven challenging to achieve. Here, we quantify electronic excitations and spin interactions in Cr<i>X</i> ₃ (<i>X</i> = Cl, Br, I) using embedded many-body wavefunction calculations and fully generalized spin Hamiltonians. We find that the three trihalides feature comparable <i>d</i>-shell excitations, consisting of a high-spin ⁴ <i>A</i> ₂ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>g</mml:mi></mml:mrow> <mml:mrow><mml:mn>3</mml:mn></mml:mrow> </mml:msubsup> <mml:msubsup><mml:mrow><mml:mi>e</mml:mi></mml:mrow> <mml:mrow><mml:mi>g</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msubsup> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> ground state lying 1.5-1.7 eV below the first excited state ⁴ <i>T</i> ₂ ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>g</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msubsup> <mml:msubsup><mml:mrow><mml:mi>e</mml:mi></mml:mrow> <mml:mrow><mml:mi>g</mml:mi></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> ). CrCl₃ exhibits a single-ion anisotropy <i>A</i> _sia = - 0.02 meV, while the Cr spin-3/2 moments are ferromagnetically coupled through bilinear and biquadratic exchange interactions of <i>J</i> ₁ = - 0.97 meV and <i>J</i> ₂ = - 0.05 meV, respectively. The corresponding values for CrBr₃ and CrI₃ increase to <i>A</i> _sia = -0.08 meV and <i>A</i> _sia= - 0.12 meV for the single-ion anisotropy, <i>J</i> ₁ = -1.21 meV, <i>J</i> ₂ = -0.05 meV and <i>J</i> ₁ = -1.38 meV, <i>J</i> ₂ = -0.06 meV for the exchange couplings, respectively. We find that the overall magnetic anisotropy is defined by the interplay between <i>A</i> _sia and <i>A</i> _dip due to magnetic dipole-dipole interaction that favors in-plane orientation of magnetic moments in ferromagnetic monolayers and bulk layered magnets. The competition between the two contributions sets CrCl₃ and CrI₃ as the easy-plane (<i>A</i> _sia + <i>A</i> _dip >0) and easy-axis (<i>A</i> _sia + <i>A</i> _dip <0) ferromagnets, respectively. The differences between the magnets trace back to the atomic radii of the halogen ligands and the magnitude of spin-orbit coupling. Our findings are in excellent agreement with recent experiments, thus providing reference values for the fundamental interactions in chromium trihalides.