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

Semiconductor nanocrystals (NCs) hold immense potential for various technologies, but their efficiency is hampered by nonradiative decay of electronically excited states, leading to thermal losses. A crucial process is hot exciton cooling, where a highly excited electron-hole pair relaxes nonradiatively to form a band-edge exciton. In bulk semiconductors, this cooling is efficient (~1 ps or less), facilitated by strong exciton-phonon coupling (EXPC) and continuous electronic and phonon state densities. However, quantum confinement in NCs alters EXPC and discretizes energy states, raising questions about the timescales and mechanisms of hot exciton cooling. The energy mismatch between electronic gaps and phonon frequencies led to the "phonon bottleneck" hypothesis, suggesting slow multiphonon relaxation. Conversely, enhanced electron-hole interactions propose ultrafast cooling. Experimental results on cooling timescales vary widely, spanning six orders of magnitude. This discrepancy arises from differences in NC size, material, and surface passivation. Some studies suggest slow relaxation in larger, self-assembled III-V NCs with many trap states, while others observe ultrafast cooling in colloidal II-VI NCs in strong confinement, attributed to Auger-assisted cooling. This mechanism involves a hot hole rapidly relaxing, followed by the hot electron re-exciting the hole in an Auger-like process. This mechanism is supported by faster relaxation in smaller NCs with stronger electron-hole correlations and increased Auger rates. However, the Auger model lacks the detailed physics of hole relaxation and may not be applicable to all systems. Computational challenges in accurately modeling exciton dynamics hinder the complete understanding of hot exciton cooling and its dependence on NC properties. This research develops an atomistic theory to address these challenges.

The literature surrounding hot exciton cooling in semiconductor nanocrystals reveals a significant divergence in experimental findings. Studies on larger, self-assembled III-V nanocrystals often report slow relaxation on timescales of 10 ps or longer, supporting the phonon bottleneck hypothesis. This slow relaxation is often associated with the presence of many trap states within the nanocrystal which act as non-radiative recombination centers. Conversely, experiments on colloidal II-VI nanocrystals, particularly those in strong confinement regimes, consistently show much faster relaxation within hundreds of femtoseconds. This discrepancy has led to various proposed mechanisms, including Auger-assisted cooling, multiphonon emission, and coupling to surface ligand vibrations. Auger-assisted cooling posits a process where the hot hole quickly relaxes, followed by Coulomb-mediated energy transfer from the hot electron to the hole, leading to rapid overall relaxation. While this mechanism explains some observations, it doesn't fully capture the complexities of the hole relaxation process and the role of multiphonon events. Other theories emphasize the importance of multiphonon emission or the influence of surface ligands in mediating rapid relaxation. The existing theoretical models often lack the complete treatment of exciton-phonon interactions and the complexity of the multiphonon processes, making the identification of a unifying mechanism challenging.

The authors developed an atomistic theory to describe hot exciton cooling in II-VI NCs. The theoretical framework considers phonon-mediated transitions between excitonic states, inherently including electron-hole correlations. Exciton-phonon couplings (EXPC) are accurately described, and multiphonon-mediated excitonic transitions are explicitly included. A master equation approach, assuming weak EXPC, is used to simulate exciton population dynamics. The Hamiltonian describes a manifold of excitonic states coupled to vibrational modes, with EXPC expanded to the lowest order in atomic displacements. Excitonic energies and states, as well as EXPC matrix elements, were calculated using the semiempirical pseudopotential method coupled with the Bethe-Salpeter equation. Phonon modes and frequencies were obtained by diagonalizing the dynamical matrix calculated using a parameterized atomic force field. The density of excitonic states, scaled by oscillator strengths, was calculated for a CdSe NC (3.9 nm diameter). The linear absorption spectrum showed distinct features consistent with experiments. Initially, the authors considered the single-phonon process limit, using Redfield equations to propagate the reduced density matrix via a quantum master equation perturbative to second order in EXPC. The phonon-mediated transition rate between excitonic states was calculated using the time-dependent golden rule. A kinetic master equation for populations, with rates from the golden rule, was constructed and used to simulate the dynamics. The average energy above equilibrium was then calculated. However, single-phonon simulations revealed a phonon bottleneck in smaller NCs. To incorporate multiphonon processes, a unitary polaron transformation was applied to the Hamiltonian. The golden rule transition rates were then recomputed, incorporating multiphonon-mediated transitions even in the lowest-order perturbation theory. The average energy above thermal equilibrium was calculated again with the modified rates. The transition rates were analyzed as a function of transition energy, revealing that multiphonon rates cover a wider range of energies than single-phonon rates. The authors then simulated hot exciton cooling for CdSe cores of different sizes and CdSe-CdS core-shell NCs. Energy loss rates were extracted by fitting the rise dynamics of the 1S peak in the simulated absorption spectrum change to an exponential function. These energy loss rates were compared with experimental measurements. Finally, the mechanism underlying ultrafast cooling was examined by analyzing the density of excitonic states scaled by the time-dependent population.

The simulations revealed that hot exciton cooling in CdSe NCs occurs on timescales of tens of femtoseconds, consistent with recent measurements. This ultrafast timescale is attributed to both electron-hole correlations and multiphonon emission processes, enabled by a quasi-continuous manifold of phonon states in NCs. The study demonstrated that the phonon bottleneck is circumvented through efficient multiphonon emission, where several phonons are emitted simultaneously to bridge the larger energy gaps between excitonic states. The simulations showed that smaller NCs relax more quickly than larger NCs due to stronger EXPC, despite having larger excitonic gaps and fewer phonon modes. The addition of a CdS shell to a CdSe core significantly slowed down the cooling process (five times slower) due to reduced EXPC. The analysis of transition rates showed that multiphonon rates are essential for bridging energy gaps larger than the highest phonon frequency and contribute significantly to the overall cooling process. The analysis of spectral densities further supported the importance of both acoustic and optical phonons in mediating the cooling process, highlighting the role of a quasi-continuous range of phonon frequencies. Calculated energy loss rates for CdSe NCs agreed with experimental measurements from two-dimensional electronic spectroscopy, which confirmed ultrafast cooling in CdSe cores and a significant slowing down in core-shell structures. The simulations also showed that energy loss rates increase linearly with temperature, with smaller NCs exhibiting a stronger temperature dependence. Finally, the analysis of the density of excitonic states scaled by time-dependent populations demonstrated that cooling occurs through a cascade of relaxation events across the manifold of excitonic states.

This research provides a comprehensive atomistic description of hot exciton cooling in semiconductor nanocrystals, resolving the long-standing discrepancies between experimental measurements and theoretical models. The study demonstrates the importance of multiphonon processes in overcoming the phonon bottleneck, highlighting the role of both electron-hole correlations and a wide range of phonon frequencies in achieving ultrafast cooling. The findings provide valuable insights into the design and optimization of NC-based devices by identifying key parameters, such as NC size and composition, that control the cooling timescale. The agreement between the simulations and recent experimental measurements from two-dimensional electronic spectroscopy validates the accuracy of the theoretical framework and its ability to capture the essential physics of hot exciton cooling. This work opens new avenues for designing NCs with tailored cooling properties, potentially leading to improved performance in optoelectronic and energy conversion applications.

This study presents a comprehensive atomistic model for hot exciton cooling in semiconductor nanocrystals, resolving inconsistencies in previous experimental and theoretical work. The model successfully predicts ultrafast cooling timescales (~30 fs) in CdSe nanocrystals, showing that multiphonon emission effectively circumvents the phonon bottleneck. Control over cooling timescales can be achieved by manipulating nanocrystal size and surface passivation, paving the way for optimized designs in nanocrystal-based technologies. Future work could explore the influence of higher-order terms in EXPC and investigate other material systems.

The study employs a master equation approach that assumes weak exciton-phonon coupling. While this approximation is valid for the systems studied, it may not hold for all nanocrystals, particularly those with strong coupling. Furthermore, the model includes EXPC only to lowest order, neglecting higher-order terms (Duschinsky rotations) that might become significant at higher temperatures. The simulations were primarily focused on CdSe and CdSe-CdS core-shell nanocrystals; extending the model to other material systems would provide broader applicability.