Dielectric Tensor of CrSBr from Spectroscopic Imaging Ellipsometry

Abstract

Chromium sulfur bromide (CrSBr) is a magnetic van der Waals semiconductor with a direct bandgap and pronounced anisotropy in its electronic, optical, spin and lattice degrees of freedom. Here, we employ spectroscopic imaging ellipsometry (SIE) and Mueller-matrix analysis to determine the full dielectric tensor of paramagnetic CrSBr thin films. Our measurements reveal optical anisotropy, characterized by three distinct diagonal components of the dielectric tensor. The in-plane elements are dominated by prominent excitonic resonances polarized along the two main crystallographic axes. Two main excitonic bands (A and B excitons) centered around 1.3eV and 1.7eV, respectively, are identified; the A-exciton polarized along the b-crystallographic direction, whereas the B-exciton appears to consist of two nearly degenerate contributions polarized along two orthogonal in-plane crystal axes. These results provide fundamental insight into anisotropic light-matter interactions in CrSBr, relevant for future spin-optoelectronic and photonic applications.

Figures (5)

• Figure 1: (a) Schematic illustration of the in-plane crystallographic axes $a$ (red) , $b$ (black) and the out-of-plane $c$ axis (blue) for a CrSBr flake on a substrate. (b) Stick-and-ball model of the orthorhombic crystal structure of CrSBr, showing cuts through the ($a,b$)-, ($c,a$)-, and ($c,b$)-planes. (c) Atomic force microscopy (AFM) topography of a typical CrSBr thin film, with the line profile indicating a thickness of $65 \pm 5$ nm. (d) Linearly co-polarized Raman spectra of CrSBr ($\vec{e}\parallel a$ and $\vec{e}\parallel b$) at room temperature showing the active modes $A^{\text{1}}_{\text{g}}$, $A^{\text{2}}_{\text{g}}$, and $A^{\text{3}}_{\text{g}}$. The inset displays a polar plot of $A^{\text{2}}_{\text{g}}$ and $A^{\text{3}}_{\text{g}}$ mode intensities with respect to the crystallographic directions.
• Figure 2: Schematic overview of the spectroscopic imaging ellipsometry (SIE) measurement technique. The change in the polarization state of incident light in reflection from a CrSBr flake placed atop the substrate is measured ina polarizer (P), compensator (c), sample (S) and analyzer (A) geometry (PCSA). Employing a rotation stage enables the Mueller matrix approach. This methodology yields access to the complete dielctric tensor of CrSBr.
• Figure 3: Measured and modeled Mueller-matrix elements $M_{ij}(E)$ for CrSBr thin films on rutile substrates. Mueller-matrix spectra are shown for two angles of incidence (AOI = 50° and 55°) and two sample azimuths ($\theta$ = 0° and 75°), capturing the pronounced optical anisotropy of CrSBr and rutile. Experimental data (symbols) are compared to regression-based optical multilayer modeling (solid lines), demonstrating excellent agreement across all channels. Distinct spectral features corresponding to the A-exciton ( 1.3 eV) and B-exciton ( 1.7 eV) regimes are clearly resolved in multiple matrix elements, highlighting the axis-dependent excitonic response. The schematic inset illustrates the imaging ellipsometry setup, including the rotating sample stage. [CrSBr on rutile substrate at 300K].
• Figure 4: Generalized ellipsometry (GE) and Mueller-matrix analysis of CrSBr. (a, b) Ellipsometric spectra $\Delta(E)$ and $\Psi(E)$ for $\vec{e_p} \parallel a$ and $\vec{e_p} \parallel b$ and converged fit to the optical multilayer model (orange lines) [CrSBr on glass substrate]. (c, d) Extracted refractive index $n(E)$ and extinction coefficient $\kappa(E)$ for the principal optical axes $a$ (red), $b$ (black) and $c$ (blue), extracted from fits to the multilayer models by Mueller-matrix (solid lines) and GE (open circles) approach. Results from both models are in very good agreement. Shaded regions indicate the energy bands of the A- and B-exciton resonances centered around 1.3 eV and 1.7 eV, respectively, while asterisks denote additional polarization-dependent spectral resonances not yet assigned.
• Figure S1: Real and imaginary parts of the dielectric function of CrSBr $\varepsilon_1$ and $\varepsilon_2$ at room temperature ($T = 300\,\mathrm{K}$) for the principal crystallographic axes $a$ (red), $b$ (black), and $c$ (blue). The solid lines correspond to the dielectric tensor components extracted from the Mueller-matrix analysis, whereas the open circles show the independently obtained in-plane GE results for the $a$- and $b$- axis. Dielectric functions extracted from both methods are in good agreement. The spectra reveal strong optical anisotropies: the A-exciton spectral range around $1.3\,\mathrm{eV}$ is most pronounced along the $b$-axis, whereas the broader B-excitonic band of resonances around $1.7\,\mathrm{eV}$ contributes to both in-plane axes. The dielectric response along the out-of-plane $c$-axis is considerably weaker. Asterisks ($\ast$) indicate additional polarization-dependent spectral features that are not assigned yet.