Optimizing Hubbard U parameters for Enhanced Description of Electronic and Magnetic Properties in CrI$_3$ Monolayers and Bilayers

TL;DR

The paper addresses the challenge of accurately describing electronic structure and magnetism in CrI3 monolayers and bilayers by calibrating DFT+U corrections on Cr 3d and I 5p orbitals to reproduce hybrid-functional results from HSE06. By benchmarking U configurations against HSE06 DOS using Pearson correlation, the authors identify optimal values of UCr=3.5 eV and UI=2.0 eV, with SOC further tightening the agreement to P≈0.98. This calibration improves predictions of geometry, magnetic ordering, magnetic anisotropy energy, and interlayer coupling in both monolayer and bilayer CrI3, and enables reliable study of stacking-dependent magnetism in the HT and LT phases, albeit within the limitation that HSE06 cannot be combined with non-local vdW corrections in the same calculation. The approach provides a computationally efficient, transferable framework for studying correlated 2D magnets and related trihalide systems, with potential applicability to other layered magnetic materials used in spintronics.

Abstract

The magnetic properties of CrI$_3$ monolayers, which were recently measured, have been investigated considering electronic repulsion and localization effects in Cr 3d orbitals. In this study, we propose a DFT approach using Hubbard U corrections to improve accuracy. We compare the valence density-of-states using the HSE06 hybrid functional and the DFT+U approach, which includes U parameters for both Cr 3d and I 5p orbitals. The results of our study indicate that the optimal values for U(Cr$_{3d}$) and U(I$_{5p}$) are 3.5 eV and 2.0 eV, respectively. This approach is further applied to improve calculations of electronic and magnetic properties in CrI$_3$ monolayers and, more importantly, in bilayers combined with van der Waals functionals. These refinements hold promise for further studies of complex CrI$_3$ nanostructures, and may also be of interest for other trihalide few-layer systems.

Figures (11)

• Figure 1: Schematic of the atomic structure of a CrI$_3$ monolayer. (a) The top view along the c-direction shows the hexagonal lattice with the unit cell highlighted in green and the lattice constant $a$. (b) Side view along the a-direction, indicating key structural parameters: the Cr–I bond length ($d_{\text{Cr–I}}$), I–I distance ($d_{\text{I–I}}$), Cr–Cr distance ($d_{\text{Cr–Cr}}$), and the Cr–I–Cr bond angle ($\alpha$). Chromium and iodine atoms are depicted in blue and red, respectively.
• Figure 2: Density of states of CrI$_3$ monolayer within different models. The calculation using the HSE06 hybrid functional is used as a reference and is shown as a shadow region. The raw DFT+U density of states is represented by a dashed red line, and the energy-scaled DFT+U is represented by a black line. DOS results are given using (a) $U_{Cr,I}=\left\lbrace 0.0\,,0.0 \right\rbrace$ ($\mathcal{P}=0.78$), (b) $U_{Cr,I}=\left\lbrace 3.5\,,0.0 \right\rbrace$, ($\mathcal{P}=0.93$), and (c) $U_{Cr,I}=\left\lbrace 3.5\,,2.0 \right\rbrace$ ($\mathcal{P}=0.95$).
• Figure 3: CrI$_3$ band center energies as a function of the Hubbard U values on d-Cr and p-I orbitals. The red lines depict the Cr d-band centers in spin-up ($\uparrow$) and spin-down ($\downarrow$) states, while the blue lines represent the I p-band centers for the same spin states. (a) The variation of band centers with respect to $U_{d-Cr}$ in the range [1.0, 6.0] eV while keeping $U_{p-I}$ fixed at 0.0 eV. (b) The variation of band centers as a function of $U_{p-I}$ in the range [3.0, 5.5] eV for $U_{d-Cr}$ fixed at 3.5 eV. The yellow highlighted region in (b) marks the point of best correlation between HSE06-hybrid and DFT+U, corresponding to $U_{d-Cr} = 3.5$ eV and $U_{p-I} = 2.0$ eV.
• Figure 4: Density of states of CrI$_3$ with different settings of Hubbard parameters including spin-orbit coupling. The color scale and the plot distribution is similar to Fig. \ref{['fig:DOSCrI3-escalado']}. In panel (a), results for $U_{Cr,I}=\left\lbrace 0.0\,,0.0 \right\rbrace$ ($\mathcal{P}=0.84$), in (b) $U_{Cr,I}=\left\lbrace 3.5\,,0.0 \right\rbrace$, ($\mathcal{P}=0.96$), and (c) $U_{Cr,I}=\left\lbrace 3.5\,,2.0 \right\rbrace$ ($\mathcal{P}=0.98$).
• Figure 5: Density of states of CrI$_3$ bilayer in the HT-AF stacking within different models. The hybrid HSE06 functional is our reference, shown as a shadow region. The raw DFT+U density of states is represented by a dashed red line, and the energy-scaled DFT+U is represented by a black line. In panel (a), results for $U_{Cr,I}=\left\lbrace 0.0\,,0.0 \right\rbrace$, in (b) $U_{Cr,I}=\left\lbrace 3.5\,,0.0 \right\rbrace$, and (c) $U_{Cr,I}=\left\lbrace 3.5\,,2.0 \right\rbrace$.