Optical and magneto-optical properties of ferromagnetic monolayer ${\mathrm{CrBr}}_3$: A first-principles $GW$ and $GW$ plus Bethe-Salpeter equation study

The discovery of atomically thin two-dimensional (2D) magnetic semiconductors has triggered enormous research interest recently. In this paper, we use first-principles many-body perturbation theory to study a prototypical 2D ferromagnetic semiconductor, monolayer chromium tribromide (${\mathrm{CrBr}}_3$). With broken time-reversal symmetry, spin-orbit coupling, and excitonic effects included through the full-spinor $GW$ and $GW$ plus Bethe-Salpeter equation ($GW$-BSE) methods, we compute the frequency-dependent layer polarizability tensor and dielectric function tensor that govern the optical and magneto-optical (MO) properties. In addition, we provide a detailed theoretical formalism for simulating magnetic circular dichroism, MO Kerr effect, and Faraday effect, demonstrating the approach with monolayer ${\mathrm{CrBr}}_3$. Due to reduced dielectric screening in 2D and the localized nature of the Cr $d$ orbitals, we find strong self-energy effects on the quasiparticle band structure of monolayer ${\mathrm{CrBr}}_3$ that give a 3.8 eV indirect bandgap. Also, excitonic effects dominate the low-energy optical and MO responses in monolayer ${\mathrm{CrBr}}_3$ where a large exciton binding energy of 2.3 eV is found for the lowest bright exciton state with excitation energy at 1.5 eV. We further find that the MO signals demonstrate strong dependence on the excitation frequency and substrate refractive index. Our theoretical framework for modeling optical and MO effects could serve as a powerful theoretical tool for future study of optoelectronic and spintronics devices consisting of van der Waals 2D magnets.

Figure 1

Crystal structure and electronic structure of ferromagnetic monolayer ${\mathrm{CrBr}}_3$. (a) Top view and (b) side view of the crystal structure of monolayer ${\mathrm{CrBr}}_3$. Cr atoms are in blue, while Br atoms are in brown. Red arrows denote the out-of-plane magnetization, which points along the $+z$ direction. (c) Schematic energy diagrams of Cr $d$ orbitals in the presence of an octahedral crystal field and magnetic exchange interaction. The horizontal dashed line denotes the Fermi level. Local-spin-density approximation with an onsite Hubbard potential (LSDA + $U$) band structure (left) and projected density of states (DOS) (right) of monolayer ${\mathrm{CrBr}}_3$ (d) without and (e) with spin-orbit coupling (SOC) effects. A rotationally invariant Hubbard potential is employed with $U=1.5\mathrm{eV}$ and $J=0.5\mathrm{eV}$ in the LSDA + $U$ calculation. Colors denote the magnitude of spin polarization along the out-of-plane direction. The red (blue) color denotes the majority-spin (minority-spin) polarization. The DOS (in units of states per electronvolt per unit cell) is decomposed into contributions from Cr majority-spin $3d$ orbitals (red curve), Cr minority-spin $3d$ orbitals (blue curve), Br majority-spin $4p$ orbitals (cyan curve), and Br minority-spin $4p$ orbitals (brown curve). The energy of the valence band maximum is set to zero. A Gaussian broadening of 50 meV is used for the projected DOS plots.

Figure 2

$G_0{W}_0$ band structure (left) and projected density of states (DOS) (right) of monolayer ${\mathrm{CrBr}}_3$ with spin-orbit coupling (SOC) effects. Computational parameters and the color scheme are the same as in Fig. 1.

Figure 3

(a) Calculated real part (solid lines) and imaginary part (dashed lines) of the diagonal $P_{xx}$ (red) and off-diagonal $P_{xy}$ (blue) layer polarizability of ferromagnetic monolayer ${\mathrm{CrBr}}_3$ without electron-hole interaction ($GW$-RPA). (b) Calculated real part (solid lines) and imaginary part (dashed lines) of the diagonal $P_{xx}$ (red) and off-diagonal $P_{xy}$ (blue) with electron-hole interaction ($GW$-BSE). (c) Absorbance spectrum of linearly polarized light with electron-hole interaction ($GW$-BSE, solid blue line) and without electron-hole interaction ($GW$-RPA, dashed red line). The amplitudes $<1.8\mathrm{eV}$ (indicated by black dashed line) are multiplied by 10 for better visibility. An 80 meV energy broadening is applied.

Figure 4

(a) Exciton energy levels of ferromagnetic monolayer ${\mathrm{CrBr}}_3$ calculated using the first-principles $GW$-BSE method. Energy levels of optically bright exciton states are in red, while those of dark states are in gray. The bright excitons have at least two orders of magnitude stronger oscillator strength than the dark ones. The free electron-hole continuum starts from 3.81 eV. (b)–(e) Top view and side view of electron distribution in real space with the hole fixed on a Cr atom for selected bright exciton states. The exciton excitation energies are: (b) 1.5 eV, (c) 2.1 eV, (d) 2.6 eV, and (e) 2.9 eV. Shown are isovalue surfaces with the value set at 1% of the maximum. Here, the dominant states (with the largest oscillator strength among the nearby bright states) are plotted.

Figure 5

(a) Absorbance spectrum of circularly polarized lights without electron-hole interaction ($GW$-RPA). (b) Absorbance spectrum of circularly polarized lights with electron-hole interaction ($GW$-BSE). The solid red (dashed blue) curve corresponds to the ${\sigma }^{+}$ (${\sigma }^{-}$) circularly polarized light. The amplitudes $<1.8\mathrm{eV}$ (black dashed line) are multiplied by 10 for better visibility in (b). (c) Frequency-dependent magnetic circular dichroism (MCD) signal of absorbance with (red, $GW$-BSE) and without (blue, $GW$-RPA) electron-hole interaction. The MCD signal is calculated as $\left(A_{+}-A_{-}\right)/\left(A_{+}+A_{-}\right)$, where $A_{\pm }$ denotes the absorbance for ${\sigma }^{\pm }$ circularly polarized lights in (a) and (b). To avoid numerical instability, a small imaginary part of $\eta =0.001i$ is added to the denominator and the real part is plotted.

Figure 6

(a) Polar setup of the magneto-optical (MO) effects with two interfaces (located at $z_0$ and $z_1$). Each layer is homogeneous in the $x\text{−}y$ plane. The left (zeroth) and right (second) media are semi-infinitely thick, and the middle (first) layer has a finite thickness $d=z_1-z_0$. The normal incident light coming from right to left is linearly polarized along the $x$ axis. The red arrow pointing along the $+z$ direction denotes the magnetization of layer 1. ${\mathbf{E}}^{\left(0\right)}$ and ${\mathbf{E}}^{\left(1\right)}$ denote the amplitudes of electric fields in the zeroth and first layer at the $z_0$ interface, respectively. ${\mathbf{E}}^{\left(2\right)}$ denotes the amplitude of electric field in the second layer at the $z_1$ interface. (b) The polarization plane of the transmitted light. ${\mathbf{E}}_{\leftarrow }^{\left(0\right)}$ denotes the electric field amplitude of the transmitted (left-moving) light. The polarization ellipse is oriented at a Faraday angle ${\theta }_{\mathrm{F}}$ with respect to the $x$ axis. The Faraday ellipticity is defined through the Faraday ellipticity angle ${\chi }_{\mathrm{F}}$. (b) The polarization plane of the reflected light. ${\mathbf{E}}_{\to }^{\left(2\right)}$ denotes the electric field amplitude of the reflected (right-moving) light. The polarization ellipse is oriented at a Kerr angle ${\theta }_{\mathrm{K}}$ with respect to the $x$ axis. The Kerr ellipticity is defined through the ellipticity angle ${\chi }_{\mathrm{K}}$. (d) Calculated Faraday angle and ellipticity in the setup with layers of semi-infinite vacuum, monolayer ${\mathrm{CrBr}}_3$, and semi-infinite vacuum. (e) Kerr angle and ellipticity in the setup with layers (from left to right) of semi-infinite sapphire, monolayer ${\mathrm{CrBr}}_3$, and semi-infinite vacuum. (f) Kerr angle and ellipticity in the setup with layers (from left to right) of semi-infinite fused silica, monolayer ${\mathrm{CrBr}}_3$, and semi-infinite vacuum. A monolayer thickness $d=6.07\AA$ is used for monolayer ${\mathrm{CrBr}}_3$.