Strain-Driven "Sinusoidal" Valley Control of Hybridized $Γ-\mathrm{K}$ Excitons

Problem Statement
Recent experiments by Blundo et al. [Phys. Rev. Lett. 129, 067402 (2022)] observed photoluminescence (PL) from momentum-indirect $\Gamma-K$ excitons in monolayer $WS_2$ under biaxial strain. However, the microscopic origin of this emission remained elusive. Existing interpretations, which treated the system as nominally mixed direct or indirect excitons where each state retains a fixed dominant character, failed to reproduce the observed peak behavior and its specific strain dependence. Furthermore, the microscopic pathway facilitating population transfer between direct and indirect species was unresolved. A unified theoretical model capable of incorporating strain-mediated PL, valley polarization (VP), and the full set of intra- and intervalley relaxation channels was needed to explain these phenomena and enable strain-programmable valley pseudospin functionalities.

Methodology
The authors developed a unified theoretical framework combining first-principles insights with many-body physics:

1. Band Structure and Exciton Energies: The study utilized a two-band model (Xiao et al.) for single-particle band edges, refined by an 11-band tight-binding model including second-nearest-neighbor hopping. Excitonic binding energies were calculated by solving the Bethe-Salpeter equation (BSE).
2. Hamiltonian Construction: A multiexcitonic Hamiltonian was constructed to describe the hybridization between direct bright excitons ($X_b$) and momentum-indirect excitons ($X_i$). This model accounts for spin-orbit coupling, valley-Zeeman shifts (calculated via orbital angular momentum from first-principles), and strain-induced energy shifts.
3. Identification of Intermediate States: The authors introduced a previously unrecognized higher-energy indirect exciton ($X_m$), where the electron resides in the upper conduction band branch ($c_2$), acting as a crucial intermediate state.
4. Dynamics Simulation: The authors formulated coupled valley dynamic equations to model intra- and intervalley relaxation channels. These equations included exchange-interaction (EI) driven scattering and spin-flip (SF) processes, accounting for phonon-assisted scattering and magnetic field effects.

Key Contributions and Results
The study resolves the experimental observations through three primary revelations:

1. Hybridized Eigenstates: The observed PL originates from genuinely hybridized direct-indirect excitonic eigenstates ($|\pm\rangle$), rather than nominally mixed species. The dominant character of these eigenstates continuously interchanges between direct ($X_b$) and indirect ($X_i$) as strain varies, particularly near the crossover point ($\varepsilon \approx 2\%$).
2. Two-Step Conversion Mechanism: The transfer of population from direct to indirect excitons occurs via a previously unrecognized two-step pathway: (i) exchange-interaction-driven intervalley scattering to the higher-energy intermediate state $X_m$, followed by (ii) an intravalley spin-flip relaxation into the indirect state. The intermediate state $X_m$ acts as a bottleneck that selectively blocks population flow, stabilizing the occupation of the upper hybridized state at higher strains.
3. "Sinusoidal" Valley Control: The model predicts a robust, non-monotonic "sinusoidal" valley polarization response in the upper hybridized branch ($|+\rangle$). This response arises from the interplay between direct-indirect hybridization, intervalley pathways, and the intermediate state. The valley polarization can be continuously tuned, reversing sign at specific strain values (e.g., around $\varepsilon \sim 1.3\%$ to $1.7\%$), offering an analog-like control mechanism that exceeds binary switching schemes.

Significance
The paper establishes a unified picture of strain-engineered direct-indirect exciton dynamics, providing the first theoretical framework that quantitatively reproduces the strain-tunable PL observed in experiments. By clarifying the microscopic origin of indirect PL and identifying the critical role of the intermediate state $X_m$, the work resolves fundamental obstacles in understanding momentum-indirect $\Gamma-K$ excitons.

The authors propose the concept of "sinusoidal" valley control as a new paradigm for manipulating long-lived valley degrees of freedom. This mechanism offers a continuously tunable valley dial, enabling programmable valley pseudospin engineering. The findings open a pathway toward next-generation valleytronic quantum technologies, specifically for logic operations and information processing utilizing long-lived indirect excitons. The study also provides concrete predictions for future experiments, particularly regarding strain-controlled valley polarization dynamics.