Intervalley Excitonic Folding in 2D Semiconductors

• Intervalley excitonic band folding is a quantum many-body phenomenon where hybridization of electronic and excitonic states across distinct valleys generates new, folded electronic bands.
• This effect stems from symmetry-allowed interactions that couple carriers with excitonic complexes, leading to observable ARPES signatures and modifications in effective mass and spin–orbit splitting.
• The phenomenon offers a tunable platform for engineering correlated electronic phases and exciton-driven ordered states in atomically thin semiconductors.

Intervalley excitonic band folding is a quantum many-body phenomenon in two-dimensional (2D) semiconductors whereby the hybridization of electronic and excitonic states across distinct Brillouin zone valleys leads to the emergence of new, folded electronic bands. This effect arises from symmetry-allowed couplings between carriers and intervalley excitonic complexes and manifests through spectroscopic signatures such as new photoemission features, mass renormalization, spin–orbit splitting enhancements, and the opening of an excitonic gap. Intervalley excitonic band folding provides a direct link between exciton-mediated electronic reconstruction and emergent correlated ground states, including charge-density-wave (CDW)-like phenomena, in atomically thin semiconductors such as monolayer WSe₂ and twisted homobilayers of MoSe₂ (Mo et al., 2 Dec 2025, Rosa et al., 2024).

1. Minimal Hamiltonian and Intervalley Coupling

The foundational model for intervalley excitonic band folding in 2D semiconductors focuses on the two inequivalent valleys ($\tau = \pm K$) in the conduction and valence bands. The Hamiltonian $H_0$ comprises free electronic terms ($H_{el}$) and bare excitonic terms ($H_{ex}$): $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$

$$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$

where $X_{\tau,q}^\dagger$ creates an exciton at valley $\tau$ with center-of-mass momentum $q$. Intervalley coupling arises via a symmetry-allowed interaction term: $$H_{int} = g\sum_{k,q}[X_{+K,q}^\dagger c_{-K,k} v_{+K,k-q} + h.c.]$$ Integrating out the tightly bound hole degrees of freedom yields an effective CB electron–exciton hybridization: $H_0$0 Focusing on a specific momentum transfer $H_0$1 (with $H_0$2), the Hamiltonian in the reduced subspace is: $H_0$3 Diagonalization yields two hybridized bands with eigenenergies: $H_0$4 When $H_0$5, hybridization opens an indirect gap $H_0$6 at the Fermi momentum $H_0$7, producing replica ("folded") bands displaced by $H_0$8 in momentum space (Mo et al., 2 Dec 2025).

2. Experimental Signatures via ARPES and Micro-PL

Angle-resolved photoemission spectroscopy (ARPES) directly reveals intervalley excitonic band folding through several key features:

• The emergence of sidebands, e.g., a dark-exciton sideband $H_0$9 at $H_{el}$0 meV below the conduction band edge at Q, corresponding to $H_{el}$1 excitonic states.
• Opening of an excitonic gap at $H_{el}$2 as doping and exciton density increase, evidenced by the splitting of the symmetrized energy distribution curve (EDC) by $H_{el}$3.
• Observation of hole-like sideband replicas $H_{el}$4 at Q—mirror images of the shallow valence bands ($H_{el}$5) seen at K—exhibiting spectral weights of $H_{el}$6–$H_{el}$7 relative to the main bands.

In twisted homobilayers of MoSe₂, photoluminescence (PL) signatures depend on twist angle. For small $H_{el}$8 ($H_{el}$9–$H_{ex}$0), moiré-induced mini-Brillouin zone folding brings K and Q valleys into proximity, allowing formation and gate control of hybrid intervalley trions. For large $H_{ex}$1 ($H_{ex}$2), the effect vanishes, and selection rules restore monolayer-like emission (Rosa et al., 2024).

Observed Quantity	Pristine value	Under excitonic folding
Valence mass $H_{ex}$3	$H_{ex}$4	$H_{ex}$5 at high $H_{ex}$6
SOC splitting $H_{ex}$7	$H_{ex}$8	Up to $H_{ex}$9 (increase matches trion binding energy)
Excitonic gap $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$0	Not present	$$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$1

3. Microscopic Metrics and Quantitative Analysis

The presence and magnitude of intervalley excitonic band folding are characterized by several quantitative measures:

• Effective Mass Renormalization: Fitting $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$2 to the valence band top, the effective mass increases from $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$3 in pristine to $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$4 under high exciton density.
• Spin-Orbit Coupling Enhancement: The valence band spin–orbit splitting increases from $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$5 to $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$6 after formation of trion–exciton sidebands, with the enhancement equal to the trion binding energy $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$7.
• Excitonic Gap: The hybridization gap at the Fermi level reaches $$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$8 at high carrier densities, tracking the intensity of exciton-induced ARPES features. (Mo et al., 2 Dec 2025)

4. Brillouin Zone Folding in Moiré and Twisted Structures

Twisted bilayers exhibit moiré superlattice effects; for small twist angles, the enlarged supercell Brillouin zone folds the original K and Q valleys to the same reciprocal lattice points. This enables intervalley hybridization and activates nominally momentum-dark intervalley excitons and trions in photoluminescence via Brillouin zone backfolding. Density functional theory (DFT) calculations for MoSe₂ find that:

• CBM location shifts: For the RH$$H_{el} = \sum_{\tau = \pm K}\sum_{k} \epsilon_c(k)c_{\tau,k}^\dagger c_{\tau,k} + \sum_{\tau = \pm K}\sum_{k}\epsilon_v(k) v_{\tau,k}^\dagger v_{\tau,k}$$9 stacked bilayer ($$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$0), the conduction band minimum shifts to Q with energy splitting $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$1.
• Twist dependence: At intermediate twist ($$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$2), K and Q nearly degenerate, but hybridization is quenched.
• Intervalley trions: In small-angle twisted homobilayers, gate-dependent PL identifies the formation and electrical tunability of intervalley (Q–K–K) trions—enabled by the above folding mechanism (Rosa et al., 2024).

5. Physical Interpretation: CDW Analogy and Emergent Ordered Phases

The exciton-induced intervalley hybridization acts analogously to a charge density wave (CDW) order parameter $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$3, with excitons ("excitonic glue") instead of phonons mediating the interaction. Under quasi-steady excitation and carrier doping, the resulting condensate of long-lived dark excitons represents a nearly static periodic potential at wavevector Q, promoting nontrivial band topology and electronic reconstruction. These features—gap opening near $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$4, mass renormalization, and new folded bands—are experimental hallmarks of CDW-like reconstruction by excitonic means (Mo et al., 2 Dec 2025).

A plausible implication is that light intensity and carrier concentration offer tunable handles to engineer and stabilize novel correlated phases, including exciton-driven quantum ordered states in single-layer and heterostructure transition metal dichalcogenides (TMDCs).

6. Exciton Binding and Trion Physics: Underlying Many-Body Scales

The fundamental binding energies mediating these effects are set by the 2D dielectric environment. The ground-state exciton binding energy in TMDC monolayers is typically $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$5 (WSe₂) or $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$6 (MoSe₂), with trion binding energies following $$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$7–$$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$8–$$H_{ex} = \sum_{\tau = \pm K}\sum_q \hbar\Omega_q X_{\tau,q}^\dagger X_{\tau,q}$$9, as observed in both ARPES and PL. These many-body energy scales underlie the strong-coupling regime where intervalley excitonic band folding and its fingerprints arise. Twist-angle engineering and electric gating enable precise control of the ratio of these energy scales to single-particle band splittings—dictating the efficiency and visibility of excitonic band folding in spectroscopic experiments (Rosa et al., 2024, Mo et al., 2 Dec 2025).