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

Abstract

The photoluminescence (PL) of momentum-indirect $\rm Γ- K$ excitons in monolayer WS$_2$ under biaxial strain was recently observed by Blundo et al. [Phys. Rev. Lett. 129, 067402 (2022)], yet its microscopic origin remains elusive. Here we develop a unified framework that reproduces the measured PL and reveals its fundamental excitonic mechanism. We reveal that: (i) the PL originates from genuinely hybridized direct-indirect excitonic eigenstates, rather than nominally mixed species with fixed dominant character; (ii) the direct exciton converts into the indirect one via a previously unrecognized two-step pathway -- exchange-interaction-driven exciton transfer followed by a spin flip; and (iii) a higher-energy indirect exciton, absent from prior studies, acts as a crucial intermediate mediating this conversion. Beyond explaining experiment, our theory predicts a striking strain-driven "sinusoidal'' valley response, furnishing a continuously tunable valley dial that far exceeds binary control schemes. This unified picture of strain-engineered direct-indirect exciton dynamics introduces a new paradigm for manipulating long-lived valley degrees of freedom, opening a pathway toward programmable valley pseudospin engineering and next-generation valleytronic quantum technologies.

Figures (3)

• Figure 1: (Color online) (a) Spin-valley configurations of the direct exciton ($\mathrm{X_b}$) and momentum-indirect excitons ($\mathrm{X_i}^{\Gamma\mathrm{K}}$, $\mathrm{X_i}^{\Gamma\mathrm{K'}}$), involving the spin-split CB branches ($c_1, c_2$), the upper VB ($v$), and the $\Gamma$-valley VB ($v_\Gamma$). (b) Schematic hybridization between $\mathrm{X_b}$ and $\mathrm{X_i}$ forming the eigenstates $|\pm\rangle$. (c) Strain dependence of mixing coefficients $C_\pm^{\mathrm b}$ and $C_\pm^{\mathrm i}$, with the dominant component of $|\pm\rangle$ interchanging near $\varepsilon \approx 2\%$. (d) Strain evolution of excitonic energies. (e–g) PL intensity vs strain: (e) full model (red/blue) compared with experimental data [shaded (purple, green, pink) regions] from Ref. PRL; (f) model without the intermediate state; (g) perturbative approach, with the shaded (gray) region [referring to Regime II in (c)] marking where the perturbation breaks down.
• Figure 2: (Color online) (a) Upper: Transition pathway from the direct exciton $\mathrm{X_b}$ in the K valley to the indirect exciton $\mathrm{X_i^{\Gamma K'}}$ in the K$^\prime$ valley, mediated by exchange interaction footnote-exchange and spin flip, with $\mathrm{X_{m}^{\Gamma K'}}$ as the intermediate state. Lower: time-reversed counterpart of the upper panel. (b) Intra- (solid arrows, 7–10) and intervalley (dashed arrows, 1–6) relaxation channels among hybridized ($|+\rangle$, $|-\rangle$) and non-hybridized ($\mathrm{X_m}$, $\mathrm{X_d}$) excitonic states under linearly polarized excitation. The hybridized states contain direct (yellow) and indirect (gray) components. Double arrows denote reversible up- and down-conversion processes, with the latter energetically favored.
• Figure 3: (Color online) (a), (b) Valley polarization of $|\pm\rangle$ versus strain at $T=100$, 200, 300 K. (c), (d) Energy separation $\Delta E_{\pm, m}^{\tau}=E_{|\pm\rangle}^{\rm K}-E_{\rm X_m}^{\rm \tau}$ between $|\pm\rangle$ in the K valley and the intermediate state $\rm X_m^{\Gamma K}$ (K valley) and $\rm X_m^{\Gamma K^\prime}$ (K$^\prime$ valley) as functions of strain.