Magnetic exchange interactions in bilayer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Cr</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo> </mml:mo><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mtext>=Cl</mml:mtext><mml:mo>,</mml:mo><mml:mi>Br</mml:mi><mml:mo>,</mml:mo><mml:mtext>and</mml:mtext><mml:mo> </mml:mo><mml:mi mathvariant="normal">I</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>: A critical assessment of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>DFT</mml:mi><mml:mo>+</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math> approach
Soumyajit Sarkar, Peter Kratzer
Abstract
We perform calculations with the $\mathrm{DFT}+U$ approach for the three Cr trihalides $\mathrm{Cr}{\mathrm{I}}_{3}, \mathrm{Cr}{\mathrm{Br}}_{3}$, and $\mathrm{Cr}{\mathrm{Cl}}_{3}$ with the aim to determine magnetic exchange interactions and magnetic ordering temperatures. A comprehensive investigation is carried out to assess the role of the Hubbard parameter $U$ as well as of the schemes used to correct for the double counting (DC) of the Coulomb interaction. For the bilayer systems, both with low-temperature (LT) or high-temperature (HT) stacking, magnetic exchange parameters and ordering temperatures are calculated within the random-phase approximation for spin waves. Our results show that the most commonly used DC scheme, the ``fully localized limit'' (FLL) of $\mathrm{DFT}+U$, erroneously favors ferromagnetic coupling between the layers of a bilayer structure, and yields Curie temperatures in excess of the experimental values. With the help of a perturbative model for superexchange, we are able to trace the source of this error back to the too small band gap in the majority spin channel in the FLL calculations. In contrast, when using the ``around-mean-field'' (AMF) DC scheme and a realistic value of $U=1.7$ eV for the Cr-$3d$ orbitals, we find that the magnetic interlayer coupling is antiferromagnetic (AFM) in all three materials, in agreement with recent experiments for ${\mathrm{CrI}}_{3}$. We calculate the antiferromagnetic ordering temperature for both LT and HT stacking to be 22 K and 29 K for bilayer ${\mathrm{CrI}}_{3}$. Thus, according to our calculations, the AFM order can be observed independently of the crystal structure of the film, while the LT structure remains to be the ground state. Bilayer ${\mathrm{CrBr}}_{3}$ films are predicted to have a N\'eel temperature similar to that of ${\mathrm{CrI}}_{3}$, whereas the ${\mathrm{CrCl}}_{3}$ films prefer antiparallel in-plane magnetization.