Probing magnetic orbitals and Berry curvature with circular dichroism in resonant inelastic X-ray scattering

www.nature.com/npjquantmats ARTICLE OPEN Probing magnetic orbitals and Berry curvature with circular dichroism in resonant inelastic X-ray scattering 1234567890():,; Michael Schüler 1,2,3 ✉ , Thorsten Schmitt 4 and Philipp Werner3 Resonant inelastic X-ray scattering (RIXS) can probe localized excitations at selected atoms in materials, including particle-hole transitions between the valence and conduction bands. These transitions are governed by fundamental properties of the corresponding Bloch wave functions, including orbital and magnetic degrees of freedom, and quantum geometric properties such as the Berry curvature. In particular, orbital angular momentum (OAM), which is closely linked to the Berry curvature, can exhibit a nontrivial momentum dependence. We demonstrate how information on such OAM textures can be extracted from the circular dichroism in RIXS. Based on accurate modeling with a first-principles treatment of the key ingredient—the light–matter interaction—we simulate dichroic RIXS spectra for the prototypical transition-metal dichalcogenide MoSe2 and the twodimensional topological insulator 1T0 -MoS2. Guided by an intuitive picture of the optical selection rules, we discuss how the momentum-dependent OAM manifests itself in the dichroic RIXS signal if one controls the momentum transfer. Our calculations are performed for typical experimental geometries and parameter regimes, and demonstrate the possibility of observing the predicted circular dichroism in forthcoming experiments. Thus, our work establishes a new avenue for observing Berry curvature and topological states in quantum materials. npj Quantum Materials (2023)8:6 ; https://doi.org/10.1038/s41535-023-00538-x INTRODUCTION The microscopic quantum geometry of Bloch electrons is one of the most remarkable properties of quantum materials, giving rise to fascinating macroscopic effects. Notable examples are topological insulators and superconductors1,2, where the integral over the Berry curvature yields an integer—the Chern number—that is fundamentally connected to edge modes and transport properties. More generally, the momentum-resolved Berry curvature plays a fundamental role in the optical properties of solids, including intriguing effects in high-harmonic generation3,4, the valley Hall effect5, and nonlinear Hall effect6,7. Measuring the Berry curvature texture has proven difficult. The most promising route up to now is angle-resolved photoemission spectroscopy (ARPES)8,9 with circularly polarized photons. For example, the circular dichroism (CD) provides insights into the chirality in graphene10 and topological insulators11 and into the Berry curvature in two-dimensional materials12–15 and in Weyl semimetals16. Circular photons couple to the texture of the orbital angular momentum (OAM) of the Bloch states17,18, which, in turn, is closely connected to the Berry curvature15,19, although the precise relation can be complicated16,20. The connection between the OAM texture and the measured CD is governed by the photoemission matrix elements, which depend on the details of the final states21,22, the specifics of the experimental geometry23, mixing of different orbital channels24, and the short escape depth of the photoelectrons25. Especially for bulk materials, the link between the CD signal and OAM is obscured by these complications, rendering the extraction of Berry curvature a difficult task. One of the most powerful techniques for revealing the magnetic properties of materials—including OAM—is resonant inelastic X-ray scattering (RIXS)26. This technique provides access to excitations in various degrees of freedom, including magnetic excitations such as magnons, as well as orbital excitations27. Controlling the polarization of the incoming X-ray photons allows to select or enhance certain excitations as governed by the selection rules. For d electron systems, this selectivity has been exploited to maximize spin-flip excitations28 to map out the magnetic degrees of freedom29. The quantum geometry of the Bloch states can impose further restrictions on the final states30. Based on this idea, ref. 31 has demonstrated how the chirality in Weyl semimetals leads to linear dichroism in RIXS. Thanks to recent advances in measuring RIXS with circularly polarized photons, it becomes possible to reveal OAM and thus measure local magnetic orbital excitations. Transitions between magnetic orbitals are a direct fingerprint of OAM localized at the targeted atoms. Hence, CD in RIXS (CD-RIXS) provides an unprecedented probe for the chirality of the electronic wave function and the breaking of time-reversal symmetry. CD-RIXS is conceptually similar to X-ray magnetic CD (XMCD) in X-ray absorption, and it has been used to study magnetic materials32–36, spin-flip processes37, or anti-ferromagnetic orbital order38. However, the link between the local OAM texture and CD-RIXS has not been explored. Because Berry curvature is typically tied to local OAM— especially in transition-metal dichalcogenides (TMCDs), which are a prominent class of two-dimensional materials39–41—CD-RIXS can reveal quantum geometric information beyond basic magnetic properties. This novel aspect is the focus of this work. We show how CD-RIXS provides a fingerprint of magnetic orbital excitations in paradigmatic TMDCs. Exploiting the momentum resolution of RIXS, we explore how the momentum-dependent OAM texture manifests itself in the CD-RIXS signal, using accurate modeling with first-principle input. We focus on monolayer MoSe2 1 Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. 2Laboratory for Materials Simulations, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. 3Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland. 4Photon Science Division, Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. ✉email: Published in partnership with Nanjing University M. Schüler et al. 2 1234567890():,; Fig. 1 Sketch of the setup and dichroic selection rules. a We investigate resonant inelastic X-ray scattering (RIXS) with circularly polarized photons (with energy ℏωi) impinging on a sample, while scattered photons (with energy ℏωf) are detected without resolving their polarization. This setup is sensitive to local magnetic orbital excitations, which are controlled by the circular polarization of the incoming light. b Sketch of d-d excitations in the RIXS process: upon absorbing a circularly polarized photon, an electron is promoted from the 2p core levels to an unoccupied state in the d manifold with magnetic quantum number mi. A filled d orbital (quantum number mf) relaxes and fills the core hole. Effectively, this corresponds to an excitation within the d orbital space. c Circular dichroism of the d-d transition probability P resolved with respect to the magnetic quantum number of the d orbitals. to demonstrate how the valley-dependent OAM can be mappe (...truncated)