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 Néel temperature similar to that of ${\mathrm{CrI}}_3$, whereas the ${\mathrm{CrCl}}_3$ films prefer antiparallel in-plane magnetization.

中文翻译：

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双层CrX3(X=Cl,Br,andI)中的磁交换相互作用：DFT+U方法的关键评估

我们用 $\mathrm{离散傅立叶变换}+你$ 三种三卤化铬的方法 $铬{\mathrm{一世}}_3, 铬{溴}_3$， 和 $铬{氯}_3$目的是确定磁交换相互作用和磁有序温度。进行了全面调查以评估哈伯德参数的作用$你$以及用于校正库仑相互作用的重复计数 (DC) 的方案。对于具有低温 (LT) 或高温 (HT) 堆叠的双层系统，在自旋波的随机相位近似内计算磁交换参数和排序温度。我们的结果表明，最常用的 DC 方案，即“完全局部化限制”（FLL）$\mathrm{离散傅立叶变换}+你$，错误地有利于双层结构的层之间的铁磁耦合，并产生超过实验值的居里温度。在超交换微扰模型的帮助下，我们能够将这个错误的根源追溯到 FLL 计算中多数自旋通道中的带隙太小。相比之下，当使用“环绕平均场”（AMF）DC 方案和实际值$你=1.7$ Cr-的 eV$3d$ 轨道，我们发现磁性层间耦合在所有三种材料中都是反铁磁性的（AFM），这与最近的实验一致 ${铬}_3$. 我们计算出 LT 和 HT 堆叠的反铁磁排序温度为 22 K，双层为 29 K${铬}_3$. 因此，根据我们的计算，可以独立于薄膜的晶体结构观察 AFM 顺序，而 LT 结构仍然是基态。双层${\mathrm{溴化铬}}_3$ 预计薄膜的尼尔温度与 ${铬}_3$，而 ${\mathrm{氯化铬}}_3$ 薄膜更喜欢反平行的面内磁化。