Electronic structure of chromium trihalides beyond density functional theory

We explore the electronic band structure of freestanding monolayers of chromium trihalides $\mathrm{Cr}{X}_3$, $X$ = Cl, Br, I, within an advanced ab initio theoretical approach based on the use of Green's function functionals. We compare the local density approximation with the quasiparticle self-consistent GW (QSGW) approximation and its self-consistent extension $\left(\mathrm{QS}G\widehat{W}\right)$ by solving the particle-hole ladder Bethe-Salpeter equations to improve the effective interaction $W$. We show that, at all levels of theory, the valence band consistently changes shape in the sequence $\mathrm{Cl}\to \mathrm{Br}\to \mathrm{I}$, and the valence band maximum shifts from the $M$ point to the $\mathrm{\Gamma }$ point. By analyzing the dynamic and momentum-dependent self-energy, we show that $\mathrm{QS}G\widehat{W}$ adds to the localization of the systems in comparison with QSGW, thereby leading to a narrower band and reduced amount of halogens in the valence band manifold. Further analysis shows that $X$ = Cl is most strongly correlated, and $X$ = I is least correlated (most bandlike) as the hybridization between Cr $d$ and $X$ $p$ enhances in the direction $\mathrm{Cl}\to \mathrm{Br}\to \mathrm{I}$. For ${\mathrm{CrBr}}_3$ and ${\mathrm{CrI}}_3$, we observe remarkable differences between the QSGW and $\mathrm{QS}G\widehat{W}$ valence band structures, while their eigenfunctions are very similar. We show that weak perturbations, like moderate strain, weak changes to the $d-p$ hybridization, and adding small $U$, can flip the valence band structures between these two solutions in these materials.

Figure 1

Left: Ball-and-stick model of the crystal structure of monolayer chromium trihalides $\mathrm{Cr}{X}_3$ ($X=\mathrm{Cl}$, Br, I). Right: Brillouin zone of the corresponding hexagonal lattice with the high-symmetry points indicated. ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ denote reciprocal lattice vectors.

Figure 2

Partial density of states within the local density approximation (LDA), projected onto the Cr $d$ and $X$ $p$ states for (a) $X$ = Cl, (b) Br, and (c) I

Figure 3

$\mathrm{Cr}{\mathrm{Cl}}_3$: From left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$) valence band structures (with spin-orbit coupling) are shown, respectively. The colors correspond to Cl $p_x+p_y$ (red), Cl $p_z$ (green), and Cr $d$ (blue). In the bottom panels are shown the density of states projected onto the Cr $d$ and Cl $p$ states at different levels of the theory.

Figure 4

$\mathrm{Cr}{\mathrm{Br}}_3$: From left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$) valence band structures (with spin-orbit coupling) are shown, respectively. The colors correspond to Br $p_x+p_y$ (red), Br $p_z$ (green), and Cr $d$ (blue). In the bottom panels are shown the density of states projected onto the Cr $d$ and Br $p$ states at different levels of the theory. The structure of the topmost valence band is similar within LDA and $\mathrm{QS}G\widehat{W}$ but is different in QSGW. The $\mathrm{QS}G\widehat{W}$ topmost valence band is much narrower than both $\mathrm{QS}G\widehat{W}$ and LDA and is split from the rest of the valence band manifold.

Figure 5

$\mathrm{Cr}{\mathrm{I}}_3$: From left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$) valence band structures (with spin-orbit coupling) are shown, respectively. The colors correspond to I $p_x+p_y$ (red), I $p_z$ (green), and Cr $d$ (blue). In the bottom panels are shown the density of states projected onto the Cr $d$ and I $p$ states at different levels of the theory. Note that, in the QSGW case, the valence band maximum is at a low-symmetry point not on the lines of the figure.

Figure 6

$\mathrm{Cr}{\mathrm{Br}}_3$: The quasiparticle self-consistent GW extension ($\mathrm{QS}G\widehat{W}$) valence band structures (with spin-orbit coupling) are shown under different perturbations. Weak perturbations can modify the Cr $d$ and Br $p$ hybridization and qualitatively modify the valence band structure. (a) and (f) The unperturbed $\mathrm{QS}G\widehat{W}$ and quasiparticle self-consistent GW (QSGW) valence band structures. (b) $\text{Cr}+0.1\text{Ry}$ is the $\mathrm{QS}G\widehat{W}$ solution after shifting the Cr $d$ band center up by 0.1 Ry. (c) $\text{Cr}-0.1\text{Ry}$ is the $\mathrm{QS}G\widehat{W}$ solution after shifting the Cr $d$ band center down by 0.1 Ry. (d) $\epsilon =7\%$ is the $\mathrm{QS}G\widehat{W}$ solution with 7% volume conserving strain along (001). (e) $\mathrm{QS}G\widehat{W}+U$ solution with $U=1$ eV on the Cr $d$. The colors correspond to I $p_x+p_y$ (red), I $p_z$ (green), and Cr $d$ (blue).

Figure 7

$\mathrm{Cr}{\mathrm{Br}}_3$: Square of wave function $\psi$, in real space, of the highest valence band state at the $M$ point. All the left panels pass through a Cr plane, and right panels pass through a Br plane. Cr and Br site positions are labeled in green in the plane where they reside and gray when they lie in a different plane. Top panels display constant-amplitude contours for local density approximation (LDA) eigenfunctions. Contours are taken in half-decade increments in ${\left|\psi \right|}^2$, with a factor of 300 between highest contour (red) and lowest (blue). In the Cr plane, the atomic $d_{xy}$ character centered at Cr nuclei stand in sharp relief; in the Br plane, the Br atomic $p$ character is evident. Middle panels show the change in ${\left|\psi \right|}^2$ passing from LDA to quasiparticle self-consistent GW (QSGW) eigenfunctions; bottom panels show the corresponding change passing from $\mathrm{QS}GW$ to its self-consistent extension ($\mathrm{QS}G\widehat{W}$) eigenfunctions. In the bottom four panels, blue $\to$ red has a similar meaning as in the top panels (increasing positive ${\delta \left|\psi \right|}^2$), while contours of negative ${\delta \left|\psi \right|}^2$ are depicted by increasing strength in the change blue $\to$ green. Note that the color patterns in the middle and bottom panels are mirror images. As a consequence of the softening in $W$ in the change $W\to \widehat{W}$, the shift in density $\mathrm{LDA}\to \mathrm{QS}GW$ is partially reversed by the addition of ladder diagrams.

Figure 8

$\mathrm{Cr}{\mathrm{X}}_3$: Real part of $\mathrm{\Sigma }(k,\omega )$ is analyzed to extract the quasiparticle renormalization factor $Z_k$ from the quasiparticle self-consistent GW (QSGW) and its self-consistent extension ($\mathrm{QS}G\widehat{W}$). (a) and (b) The weak momentum dependence of the $Z_k$ for the topmost valence band at the quasiparticle peaks at the $k$ points chosen along the high-symmetry directions of the first Brillouin zone for the up and down spin sectors, respectively. (c) and (d) The relative suppression of the $Z_k$ factor in $\mathrm{QS}G\widehat{W}$ compared with QSGW for the up and down spin sectors, respectively.

Figure 9

$\mathrm{Cr}{\mathrm{Br}}_3$ and $\mathrm{Cr}{\mathrm{Cl}}_3$ : Scaling of the one particle band gap from QSGW with vacuum size $L$.

Figure 10

$\mathrm{Cr}{\mathrm{Cl}}_3$: The colors correspond to Cl $p_x+p_y$ (red), Cl $p_z$ (green), and Cr $d$ (blue) [from left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$), respectively].

Figure 11

$\mathrm{Cr}{\mathrm{Br}}_3$: The colors correspond to Br $p_x+p_y$ (red), Br $p_z$ (green), and Cr $d$ (blue) [from left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$), respectively).

Figure 12

$\mathrm{Cr}{\mathrm{I}}_3$: The colors correspond to I $p_x+p_y$ (red), I $p_z$ (green), and Cr $d$ (blue) [from left to right: local density approximation (LDA), quasiparticle self-consistent GW (QSGW), and its self-consistent extension ($\mathrm{QS}G\widehat{W}$), respectively).