Role of nonlocality in exchange correlation for magnetic two-dimensional van der Waals materials

To obtain accurate independent-particle descriptions for ferromagnetic two-dimensional van der Waals materials, we apply the quasiparticle self-consistent $GW$ (QSGW) method to ${\mathrm{VI}}_3, {\mathrm{CrI}}_3, {\mathrm{CrGeTe}}_3$, and ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$. QSGW provides a description of the nonlocal exchange-correlation term in the one-particle Hamiltonian. The nonlocal term is important not only as the $U$ of density functional theory $\left(\mathrm{DFT}\right)+U$ but also for differentiating occupied and unoccupied states in semiconductors. We show the limitations of $\mathrm{DFT}+U$ in mimicking QSGW.

Figure 1

Total and atom-resolved partial density of states calculated using QSGW in ${\text{Fe}}_3{\text{GeTe}}_2$. For comparison, DOS obtained by DFT is shown (shaded area). Spin-orbit coupling is not included.

Figure 2

The partial density of states projected on Cr-$3d$ and I-$5p$ states in ${\mathrm{CrI}}_3$ calculated within (a) DFT and (b) QSGW. (c) QSGW band structures of ${\text{CrI}}_3$ calculated with (red) and without (blue) SOC.

Figure 3

The partial density of states projected on the V $3d$ states in ${\text{VI}}_3$ within DFT (green shaded), $\mathrm{DFT}+U$, and QSGW. $\mathrm{DFT}+U$ calculation is performed using the AMF scheme. $U=2.7\mathrm{eV}$ is used so that the majority-spin V-$3d$ states peak at similar positions as in QSGW. SOC is not included.

Figure 4

$E_{\mathrm{g}}$ as a function of $U$ in ${\text{CrI}}_3, {\text{VI}}_3$, and $\text{Cr}\text{Ge}{\text{Te}}_3$, calculated using the (a) fully-localized-limit scheme (FLL) and (b) around-the-mean-field (AMF) scheme. The lower bound (open circles) and upper bound (open squares) of the shaded areas correspond to calculations with and without SOC, respectively. $\mathrm{QSGW}+\mathrm{SOC}$ results are included to compare.

Figure 5

The CBM, VBM, and centers of $3d$ states in m2Dv in both spin channels calculated in $\mathrm{DFT}+U$ and QSGW. Band centers are denoted by solid circles $(\mathrm{DFT}+U)$ or stars (QSGW) while CBM and VBM are denoted by open circles $(\mathrm{DFT}+U)$ or stars (QSGW). The small, medium, and large circles represent $U=0$, 1.4, and $4.1\mathrm{eV}$, respectively. The AMF scheme is used for $\mathrm{DFT}+U$ calculation. SOC is not included.