Circumventing the phonon bottleneck by multiphonon-mediated hot exciton cooling at the nanoscale

The study addresses how hot excitons (highly excited electron–hole pairs) in semiconductor nanocrystals relax nonradiatively to band-edge states, a process crucial for device efficiency. Quantum confinement discretizes electronic and phonon states, raising questions about whether a phonon bottleneck (due to large electron energy gaps exceeding phonon energies) slows cooling or whether strong electron–hole interactions enable ultrafast cooling. Experiments report widely varying timescales. The research aims to resolve the mechanism and timescale of hot exciton cooling by developing an atomistic exciton–phonon framework that includes electron–hole correlations and multiphonon processes, and to identify nanocrystal parameters that tune cooling.

Prior work in bulk semiconductors shows sub-ps cooling via Fröhlich and deformation potential interactions supported by continuous electronic and phonon spectra. In nanocrystals, discretization and altered coupling led to the phonon bottleneck hypothesis, especially for electrons near the conduction edge with large level spacings. Some experiments (notably in weakly confined III–V NCs) observed slow (>10 ps) relaxation, consistent with a bottleneck and possibly influenced by trap states. Other studies in strongly confined II–VI NCs reported sub-ps to few-hundred-fs cooling, attributed to Auger-assisted electron–hole energy exchange where a rapidly cooling hole assists electron relaxation. Faster relaxation in smaller NCs and sensitivity to hole-accepting ligands supported this Auger picture. Alternative proposals implicated efficient multiphonon emission or coupling to ligand vibrational modes. However, Auger-only models treat electron–hole interactions perturbatively, lack details on hole relaxation, and may predict a hole bottleneck in some systems when multiphonon effects are neglected. Computationally, treating excitons and phonon-mediated dynamics in realistic NC sizes is challenging, limiting mechanistic clarity and design rules.

The authors formulate an exciton–phonon Hamiltonian consisting of a manifold of excitonic states with energies En, harmonic phonon modes with frequencies ωα, and exciton–phonon coupling (EXPC) linear in atomic displacements. Excitonic energies, states |Ψn⟩, and EXPC matrix elements Vnm,α are computed using a semiempirical pseudopotential approach coupled to the Bethe–Salpeter equation. Phonon modes/frequencies are obtained by diagonalizing a dynamical matrix from a parameterized force field. First, single-phonon processes are treated using a Redfield (second-order perturbative) master equation for populations, with time-dependent golden-rule transition rates between excitonic states derived from phonon correlation functions. Simulations of CdSe NCs with diameters 2.2, 3.0, 3.9, and 4.7 nm reveal a phonon bottleneck in smaller NCs when only single-phonon transitions are allowed due to excitonic gaps exceeding the largest phonon energies (~32 meV). To include multiphonon effects while retaining a kinetic description, a unitary polaron transformation is applied to the Hamiltonian. Golden-rule transition rates between polaronic excitonic states are then computed, where the effective couplings include exponentials of phonon momenta, inherently capturing multiphonon-mediated transitions. All pairwise rates are assembled into a kinetic master equation to propagate population dynamics. The average exciton energy above thermal equilibrium is monitored to quantify cooling. Systems studied include wurtzite CdSe cores across sizes and CdSe/CdS core–shell NCs (e.g., 3.9 nm cores with 3 monolayers of CdS shell). Spectral densities (phonon density of states weighted by EXPC) are computed to analyze mode contributions. For experimental comparison, transient absorption observables are modeled by the change in absorption Δσ(ω,t), assuming weak-field conditions, and the rise dynamics of the 1S excitonic peak following initial excitation to 1P are fit to extract a characteristic time; dividing by the 1P–1S energy splitting yields an energy-loss rate. Temperature-dependent simulations assess phonon-mediated trends. The approach assumes weak EXPC (validating Redfield/population kinetics), uses linear EXPC (neglecting higher-order Duschinsky rotations), and treats multiphonon effects via the polaron framework.

- Single-phonon-only dynamics exhibit a phonon bottleneck in CdSe NCs below ~4.7 nm diameter, preventing full relaxation to the band edge due to excitonic gaps exceeding the largest phonon energy (~32 meV). - Including multiphonon processes via the polaron-transformed rates eliminates the bottleneck: all CdSe NCs fully relax to thermal equilibrium with average energy decaying within ~100 fs, and state-to-state 1P→1S cooling occurs on ~30 fs timescales, consistent with recent 2D electronic spectroscopy. - Multiphonon transition rates span a broad range of transition energies; for |ΔE| ≲ 100 meV, rates are consistently substantial (≈10^-3 to 10^2 ps^-1), whereas single-phonon rates vanish beyond the largest phonon energy. The broad quasi-continuous set of phonon modes (e.g., ~3000 modes for a 3.9 nm CdSe NC) enables many energy-conserving phonon combinations, greatly increasing available relaxation pathways. - Transition rates versus energy difference show a Gaussian-like dependence, indicating significant contributions from both acoustic and optical modes, rather than only the highest-frequency optical modes. - Size dependence: Smaller CdSe NCs cool faster despite larger energy gaps because EXPC is stronger in smaller NCs; they also have fewer phonon modes but stronger coupling leads to faster overall relaxation. - Core–shell effect: CdSe/CdS (3 monolayer shell) exhibits cooling approximately five times slower than bare CdSe cores of the same diameter due to reduced EXPC, particularly suppressed coupling to low-frequency surface and delocalized modes. Multiphonon rates for core–shell structures are 1–2 orders of magnitude smaller (≈10^-4 to 10 ps^-1 for |ΔE| ≲ 100 meV) compared with bare cores (≈10^-3 to 10^2 ps^-1). - Spectral density analysis shows bare CdSe cores couple strongly to low-frequency (≤4 THz) acoustic modes and 7.5–8 THz optical modes, whereas core–shell structures suppress low-frequency coupling but retain slightly stronger coupling to CdSe optical modes; efficient cooling requires coupling across a quasi-continuous frequency range. - Energy-loss rates inferred from simulated transient absorption agree in trends and magnitude with experiments: faster in smaller cores; addition of a CdS shell slows cooling by about an order of magnitude. - Temperature dependence: Energy-loss rates increase approximately linearly with temperature for both 2.2 nm and 3.9 nm CdSe, with a stronger temperature dependence for the smaller NC, consistent with the increasing importance of thermally assisted multiphonon transitions at larger excitonic gaps.

By explicitly including electron–hole correlations (excitonic states) and multiphonon-mediated transitions via a polaron framework, the study reconciles conflicting experimental observations: it shows that a strict phonon bottleneck is avoided in strongly confined NCs because relaxation proceeds through a cascade of excitonic states with smaller energy gaps, enabled by numerous multiphonon pathways across a quasi-continuous phonon spectrum. The results validate the essence of Auger-assisted interpretations—that electron–hole interactions accelerate cooling—while demonstrating that multiphonon emission is a coequal, essential mechanism. The Gaussian dependence of rates on transition energy and the importance of both acoustic and optical modes revise the common assumption that only high-frequency optical phonons govern cooling. Design implications follow: strengthening or weakening EXPC (via size, composition, and heterostructuring) and controlling phonon spectra can tune cooling timescales to match device needs, and the observed temperature scaling supports phonon-mediated control strategies.

The work introduces an atomistic exciton–phonon theory combining a master-equation treatment with a polaron transformation to capture multiphonon processes in realistically sized nanocrystals. It demonstrates ultrafast (~30 fs) hot exciton cooling in CdSe cores and substantially slower cooling in CdSe/CdS core–shells due to reduced exciton–phonon coupling. The phonon bottleneck is circumvented by a cascade of multiphonon-mediated transitions among excitonic states, enabled by coupling to a broad spectrum of phonon modes. These insights provide design handles—NC size, core–shell architecture, and temperature—to tune cooling for optoelectronic applications. Future work could incorporate higher-order exciton–phonon couplings (e.g., Duschinsky rotations), explore coherence effects beyond population kinetics, and extend to other materials and surface chemistries to map how ligand and interface vibrational modes influence cooling.

- The model includes exciton–phonon coupling only to first order in phonon coordinates, neglecting higher-order effects (e.g., Duschinsky rotations), which may become important at higher temperatures. - The Redfield/master-equation approach assumes weak coupling and treats only population dynamics, with negligible role of coherences; strong-coupling or non-Markovian effects are not explored. - Semiempirical pseudopotential and classical force-field phonons, while enabling large systems, may introduce approximations relative to fully ab initio treatments. - Experimental comparisons may be affected by finite pulse durations and sample-specific trap states not explicitly modeled.