Spin-lattice couplings in two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>CrI</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msub></mml:math> from first-principles computations

Abstract

Since thermal fluctuations become more important as dimensions shrink, it is expected that low-dimensional magnets are more sensitive to atomic displacement and phonons than bulk systems are. Here we present a fully relativistic first-principles study on the spin-lattice coupling, i.e., how the magnetic interactions depend on atomic displacement, of the prototypical two-dimensional ferromagnet ${\mathrm{CrI}}_{3}$. We extract an effective measure of the spin-lattice coupling in ${\mathrm{CrI}}_{3}$, which is up to ten times larger than what is found for bcc Fe. The magnetic exchange interactions, including Heisenberg and relativistic Dzyaloshinskii-Moriya interactions, are sensitive both to the in-plane motion of Cr atoms and out-of-plane motion of ligand atoms. We find that significant magnetic pair interactions change sign from ferromagnetic (FM) to antiferromagnetic (AFM) for atomic displacements larger than 0.16 (0.18) \AA{} for Cr (I) atoms. We explain the observed strong spin-lattice coupling by analyzing the orbital decomposition of isotropic exchange interactions, involving different crystal-field-split $\mathrm{Cr}\text{\ensuremath{-}}3d$ orbitals. The competition between the AFM ${t}_{2g}\text{\ensuremath{-}}{t}_{2g}$ and FM ${t}_{2g}\text{\ensuremath{-}}{e}_{g}$ contributions depends on the bond angle formed by Cr and I atoms as well as Cr-Cr distance. In particular, if a Cr atom is displaced, the FM-AFM sign changes when the I-Cr-I bond angle approaches ${90}^{\ensuremath{\circ}}$. The obtained spin-lattice coupling constants, along with the microscopic orbital analysis, can act as a guiding principle for further studies of the thermodynamic properties and combined magnon-phonon excitations in two-dimensional magnets.

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