Bright and Dark Excitons in CrSBr: Local Ligand-Field Character and Band-Coherent Optical Selection Rules

Technical Summary: Bright and Dark Excitons in CrSBr

Problem Statement
Magnetic van der Waals semiconductors, specifically CrSBr, host a complex excitonic landscape where charge, spin, orbital, and lattice degrees of freedom interact at comparable energy scales. While recent many-body calculations (QSGc$^W$) have successfully established the electronic band gap, binding energies, and orbital character of the primary bright excitons ($X_A$ near 1.34 eV and $X_B$ near 1.8 eV), a significant mystery remains: why do these intense bright states coexist within a few tens of meV of "companion" states that are several orders of magnitude darker? These dark states draw from essentially the same single-particle transition manifold as the bright ones, yet standard interpretive frameworks (Frenkel vs. Wannier-Mott, ligand-field vs. charge-transfer) fail to explain the drastic disparity in oscillator strength. Furthermore, the optical response of CrSBr is often obscured by magnon, phonon, and photon couplings; a clear understanding of the "bare" excitonic spectrum is required to disentangle these effects.

Methodology
The study employs a quasiparticle self-consistent $GW$ theory augmented by electron-hole ladder vertex corrections (QSGc$^W$) to generate the electronic band structure and screened Coulomb interaction. This framework is used to solve the Bethe-Salpeter equation (BSE) for the excitonic spectrum of bulk CrSBr in its antiferromagnetic ground state.

• Computational Setup: Calculations utilize a 12-atom orthorhombic primitive cell ($Pmmn$) with a $10 \times 7 \times 2$ $k$-mesh. The BSE includes 26 valence and 9 conduction bands per $k$-point.
• Analysis Framework: The authors decompose the BSE eigenfunctions ($A^S_{kvc}$) and the underlying single-particle transition-density vectors ($\rho^\alpha_{kvc}$) to calculate the optical amplitude $D^\alpha_S$ and oscillator strength $f^\alpha_S$.
• Interference Metric: A key diagnostic introduced is the "interference figure of merit" ($I^\alpha_S$), defined as the ratio of the squared coherent sum of transition channels to the incoherent sum of their magnitudes. This metric quantifies whether transition dipoles within an exciton add constructively (bright) or destructively (dark).

Key Contributions and Results

1. Mechanism of Brightness/Darkness: The paper demonstrates that the brightness of an exciton in CrSBr is a band-coherent property determined by symmetry-adapted interference, not merely by the local orbital character (e.g., onsite $d$-$d$ weight) or spatial extent (Frenkel vs. Wannier).

   • Bright States: Arise from sublattice-symmetric ("diagonal") superpositions of Bloch transitions across the two Cr atoms in the primitive cell (e.g., $v_0 \to c_0 + v_1 \to c_1$). In these states, transition dipoles add constructively.
   • Dark States: Arise from sublattice-antisymmetric ("cross") superpositions (e.g., $v_0 \to c_1 + v_1 \to c_0$). Here, transition dipoles carry opposite signs and cancel destructively, resulting in vanishing oscillator strength despite the underlying transition amplitudes being comparable to bright states.

2. Quantitative Verification:

   • The $X_A$ bright exciton (1.33 eV) and its dark partner (1.38 eV) share nearly identical $k$-space distributions (overlap > 0.99). The brightness difference is purely an internal phase effect: the bright state has a diagonal weight ($D_2$) of ~0.50 and an interference metric $I \sim 40$, while the dark state has a cross weight ($X_2$) of ~0.40 and $I \sim 10^{-4}$.
   • Similarly, the $X_B$ bright exciton (1.84 eV) and its dark partner (1.82 eV) exhibit the same dichotomy, with the bright state being predominantly diagonal ($D_2 \sim 0.81$) and the dark state predominantly cross ($X_2 \sim 0.83$).
   • The study identifies additional dark states (e.g., at 1.50 eV and 1.72 eV) that are "interference-dark," consistent with experimental observations from resonant inelastic x-ray scattering (RIXS) and transient reflectivity, which detect spectral weight in regions where steady-state optical probes are silent.

3. Refutation of Rydberg and Local-Only Models:

   • The results refute the early assignment of $X_A$ and $X_B$ as members of a single Rydberg series (e.g., 1s and 2s states). The bright $X_A$ is built overwhelmingly from a single channel ($v_0 \to c_0$), whereas the bright $X_B$ populates all four channels of the active block ($v_{0,1} \to c_{0,1}$) constructively. They are structurally distinct band-pair compositions.
   • The study clarifies that high onsite $d$-$d$ weight does not guarantee brightness. The 1.33 eV state has significant onsite character but is bright due to coherent addition of allowed intersite/ligand-assisted components. Conversely, the 1.84 eV state has low onsite weight but is bright due to constructive interference of its allowed components.

4. Structural Origin of Pairing: The orthorhombic crystal field ($Pmmn$) fully lifts the degeneracy of $d$-orbitals, leaving only the two-Cr sublattice degree of freedom as the source of doubling. This creates a minimal active space of two valence and two conduction branches, which naturally splits into one bright (diagonal) and one dark (cross) pair. Triplets or other multiplicities are structurally forbidden in this lattice.

Significance
The paper establishes that the "bare" excitonic spectrum of CrSBr is defined by a symmetry-adapted interference rule between transition dipoles across the sublattice. This insight provides the fundamental baseline required to interpret complex optical responses where magnons, phonons, and photons layer onto the excitonic manifold. By showing that bright and dark partners are nearly degenerate, share the same $k$-space footprint, and differ only in their internal phase coherence, the work explains why dark states are invisible in equilibrium optical spectroscopy but accessible via probes that bypass optical matrix elements (like RIXS) or transiently relax selection rules (like pump-probe reflectivity). This reframes the understanding of CrSBr not as a simple molecular ligand-field system or a conventional Wannier-Mott exciton, but as a magnetic band insulator where ligand-field-like excitations acquire optical strength through band hybridization and symmetry-adapted coherence.