Bi-Self-Trapped and Superfluorescent STEs

• The paper demonstrates that bi-self-trapped excitons form a cooperative, mirror-symmetric state via strong exciton–phonon coupling, resulting in lower energy configurations and enabling Dicke-type superfluorescence.
• It outlines how material design—through optimized lattice connectivity, cation choice, and phonon characteristics—critically influences self-trapping and efficient emission in hybrid perovskites.
• Experimental findings reveal that superfluorescent bursts exhibit a distinct narrow emission profile and quadratic intensity scaling with exciton density, contrasting with conventional broad STE emissions.

Bi-self-trapped excitons (bi-STE) and their connection to superfluorescent emission represent a cooperative regime of exciton-phonon physics, where two excitons are jointly localized by a long-lived optical phonon mode and their subsequent radiative decay exhibits Dicke-type superradiant features. These phenomena are central for understanding and engineering advanced optoelectronic properties—including superfluorescence—in hybrid perovskites and related materials. The interplay of local lattice structure, exciton-phonon coupling, and electronic configuration fundamentally determines whether materials support efficient self-trapping, bi-self-trapping, and cooperative emission regimes.

1. Theoretical Framework: Exciton–Phonon Coupling and Self-Trapping

A microscopic theory of self-trapped excitons is provided by the Holstein-type Hamiltonian for Wannier–Mott excitons ($B_{\bm q}^{(\dagger)}$) interacting with a dispersionless longitudinal optical (LO) phonon branch ($b_{\bm k}^{(\dagger)}$):

$$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$

(Osipov et al., 28 Jan 2025).

Here, $W_q$ is the exciton dispersion, $D(\bm k)$ is the Fröhlich-type matrix element, $\mathcal{H}_{bath}$ introduces phonon damping at rate $\gamma$, and $\mathcal{H}'$ describes exciton–photon coupling. The multiconfiguration Hartree (Davydov) ansatz employs coherent-state expansions for the phonon field, leading to effective equations of motion for the coupled exciton-phonon system.

A single self-trapped exciton (STE$’$) forms via strong coupling to a single phonon mode, resulting in a stable, phase-locked solution for the coherent phonon amplitude. The key criterion for self-trapping at high temperature is expressed via a Lyapunov-type energy functional $H^{(1)}(J_a)$, with a local minimum only when $b_{\bm k}^{(\dagger)}$0. Here, $b_{\bm k}^{(\dagger)}$1 encodes the occupation of the exciton mode and $b_{\bm k}^{(\dagger)}$2 parametrizes the energy detuning. The self-trapped state features finite polarization and is stable against perturbations (Osipov et al., 28 Jan 2025).

2. Bi-Self-Trapped Excitons: Formation and Energetics

At elevated exciton concentrations, two excitons can become simultaneously localized by the same long-lived phonon, entering a bi-self-trapped regime. This induces a unique “mirror-symmetric” configuration characterized by phase-locked exciton pairs ($b_{\bm k}^{(\dagger)}$3, $b_{\bm k}^{(\dagger)}$4) and a global phase $b_{\bm k}^{(\dagger)}$5. The total energy is:

$b_{\bm k}^{(\dagger)}$6

(Osipov et al., 28 Jan 2025).

Minimization with $b_{\bm k}^{(\dagger)}$7 yields a cooperative, mirror-symmetric arrangement with lower energy than two independent STE$b_{\bm k}^{(\dagger)}$8s: $b_{\bm k}^{(\dagger)}$9. There is a threshold for the exciton concentration $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$0 above which bi-self-trapping dominates, captured phenomenologically by rate equations.

Table: Key Energy Relations in Bi-STE vs. STE$$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$1 | Configuration | Energy Functional | Minimum Condition | |-----------------------|-------------------------|-----------------------| | Single STE$$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$2 | $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$3 | $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$4 | | Bi-STE (mirror sym.) | $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$5 | $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$6 |

3. Superfluorescence and Dicke-Type Cooperative Emission

In the bi-STE mirror-symmetric state, the coupled exciton-photon system realizes a Dicke-type Hamiltonian:

$$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$7

(Osipov et al., 28 Jan 2025).

The symmetric (superradiant) and antisymmetric (subradiant) Dicke states emerge, but only the symmetric state couples strongly to the photon mode, yielding a collective, cooperative emission process. The superfluorescent burst (a $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$8-like line at $$\mathcal{H} = \omega_{ph}\sum_{\bm k}b_{\bm k}^\dagger b_{\bm k} + \sum_{\bm q}W_{q}B_{\bm q}^\dagger B_{\bm q} + \sum_{\bm k,\bm q}D(\bm k)B_{\bm q}^\dagger B_{\bm q-\bm k} (b_{-\bm k}^\dagger + b_{\bm k}) + \mathcal{H}_{bath}+\mathcal{H}'$$9) exhibits a characteristic $W_q$0 time profile and intensity scaling as $W_q$1, with pulse delays of 10–15 ps as observed in methylammonium lead iodide and phenethylammonium perovskite experiments.

Single STE$W_q$2s contribute a broad Gaussian emission centered at $W_q$3 eV with a width $W_q$4 eV, while the bi-STE superfluorescence is narrowly centered at $W_q$5 eV (Osipov et al., 28 Jan 2025).

4. Materials Design Principles and Structure–Property Relationships

The emergence of (bi-)self-trapped excitons and superfluorescence is highly sensitive to the underlying crystal structure and composition. Studies in bismuth and antimony halides highlight the following principles (Klement et al., 24 Sep 2025):

• Connectivity: Edge-sharing [Bi₂Cl₁₀]⁴⁻ motifs enable maximal STE localization via strong transient lattice distortion, compared to less-trapping chain or isolated octahedral architectures.
• Halogen Identity: Cl-substituted phases favor higher-energy phonon modes ($W_q$6 meV) and larger Stokes shifts than Br analogues, promoting efficient STE formation and emission.
• Ground-State Distortion: Optimal static distortions ($W_q$7–$W_q$8, $W_q$9–$D(\bm k)$0 deg$D(\bm k)$1) are linked to enhanced trapping but avoid non-radiative loss mechanisms.
• Organic Cation Tuning: The choice and packing of organic cations modulate lattice rigidity, shifting the Huang–Rhys parameter $D(\bm k)$2 (ideally $D(\bm k)$3) and thereby optimizing STE emission. For example, benzylammonium (BZA) templating produces STE-bright phases, while cyclohexylmethylammonium (CMA) leads to STE-dark or nonradiative behavior unless under strong pumping.
• Electron–Phonon Coupling: Intermediate coupling strengths ($D(\bm k)$4–0.5; $D(\bm k)$5–$D(\bm k)$6 for key Bi chlorides) balance strong localization with radiative efficiency.

Table: STE Parameters in Representative Bismuth Halides (Klement et al., 24 Sep 2025) | Compound | $D(\bm k)$7 (eV) | $D(\bm k)$8 (eV) | $D(\bm k)$9 | $\mathcal{H}_{bath}$0 (meV) | |-----------------------------|------------|---------------|----------|-----------------------| | [BZA]₄[Bi₂Cl₁₀] | 3.21 | 1.83 | 4.9 | 120 | | [BZA]₃[BiCl₅]Cl | 3.30 | 1.91 | 6.9 | 90 | | [CMA]₃[BiBr₅]Br | 2.77 | 1.94 | 13 | 71 |

5. Experimental Manifestations: Photoluminescence and Cooperative Effects

Broadband STE photoluminescence (PL) is characterized by:

• Wide PL Bands: Peaks at $\mathcal{H}_{bath}$1–$\mathcal{H}_{bath}$2 eV and FWHM $\mathcal{H}_{bath}$3–$\mathcal{H}_{bath}$4 eV, ideal for white-light emission.
• Large Stokes Shifts: Up to $\mathcal{H}_{bath}$5 eV, suppressing reabsorption and conferring efficient emission.
• Material Dependence: Chlorides show high relative PL intensity, while bromides are orders of magnitude dimmer.
• No Mid-Gap Trap Emission: PLE drops sharply below $\mathcal{H}_{bath}$6.

In hybrid perovskites with strong, long-lived LO phonon modes and high exciton densities, superfluorescent bursts emerge above a distinct pumping threshold ($\mathcal{H}_{bath}$7 µJ cm$\mathcal{H}_{bath}$8 at low temperature, $\mathcal{H}_{bath}$9 µJ cm$\gamma$0 at room temperature) (Osipov et al., 28 Jan 2025). In contrast, bismuth halides as synthesized in (Klement et al., 24 Sep 2025) did not display transient superfluorescent features—likely due to insufficiently high excitation densities, suboptimal coupling regimes, or excess disorder and defects. A plausible implication is that rigorous control over lattice rigidity, defect densities, and $\gamma$1 can tune materials between standard STE emission and superfluorescence.

6. Kinetic Models and Scaling Laws

Population dynamics of STE and bi-STE states are captured by kinetic rate equations (Osipov et al., 28 Jan 2025):

$\gamma$2

These equations reproduce the observed $\gamma$3 scaling, quadratic intensity growth with $\gamma$4, and temporal delays matching experimental superfluorescent pulses. The transition from ordinary self-trapping to cooperative emission proceeds as the exciton density crosses the threshold for bi-STE occupation.

7. Prospects and Opportunities for Material Design

Guidelines for realizing room-temperature bi-self-trapped and superfluorescent STE emission (Osipov et al., 28 Jan 2025, Klement et al., 24 Sep 2025):

• A long-lived optical phonon mode with $\gamma$5 and $\gamma$6;
• Strong exciton–phonon coupling ($\gamma$7), maximal for exciton-phonon overlap at $\gamma$8;
• An edge-sharing, chloride-rich framework to maximize localization and minimize nonradiative loss;
• Moderately rigid host lattice, minimized disorder, with cation selection to tune $\gamma$9 and phonon frequency;
• High-enough exciton density to support cooperative bi-self-trapping and sustain superfluorescent emission.

While superfluorescence has been confirmed in hybrid lead-halide perovskites under strong pumping (Osipov et al., 28 Jan 2025), it has not yet been observed in analogous bismuth systems under the conditions explored in (Klement et al., 24 Sep 2025). A plausible implication is that future synthetic efforts toward ultra-rigid, defect-controlled, edge-connected halide dimers with tuned $\mathcal{H}'$0 and phonon energies may realize cooperative emission regimes in lead-free compounds.

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Key references:

• Osipov et al., "Bi-self-trapping of excitons via the long-living phonon mode and their superfluorescent markers" (Osipov et al., 28 Jan 2025)
• Baker et al., "Enhanced White-Light Emission from Self-Trapped Excitons in Antimony and Bismuth Halides through Structural Design" (Klement et al., 24 Sep 2025)