Circumventing the Phonon Bottleneck by Multiphonon-Mediated Hot Exciton Cooling at the Nanoscale

We adopt the following Hamiltonian to describe a manifold of excitonic states coupled to vibrational modes, with EXPC expanded to lowest order in the atomic displacements:⁴³

The excitonic energies, $E_{n}$, and states, $|\psi_{n}\rangle$, as well as the EXPC matrix elements, $V_{n,m}^{\alpha}$, were calculated using the semiempirical pseudopotential method coupled with the Bethe-Salpeter equation (see Jasrasaria et al. ^{43, 14} and the SI for more details). Phonon modes and frequencies, $\omega_{\alpha}$, were obtained by diagonalizing the dynamical matrix computed using a previously-parameterized atomic force field.⁴⁵

In Fig. 2a we show the density of excitonic states scaled by the oscillator strengths for a typical CdSe NC with a diameter of 3.9 nm as well as the corresponding linear absorption spectrum. We find that the underlying density of excitonic states is relatively high due to the dense spectrum of hole states. Some of these excitonic states correspond to bright transitions from the ground state with large oscillator strengths while others correspond to dim transitions for which the oscillator strengths are small. Note that we only show the bright/dim states, and the dark (forbidden) transitions are not shown. The linear absorption spectrum shows several distinct features, in agreement with experiments,^{26, 46} that are governed by a few excitonic transitions with large oscillator strengths. We label the main transitions as $1S$ and $1P$, following the literature convention. With the dense manifold of excitonic states, the relaxation from the $1P$ excitonic state, in which the exciton electron is primarily composed of $p$-like, single-particle electron states, to the $1S$ ground excitonic state, in which both the exciton electron and hole are primarily comprised of band edge single-particle states, should occur through a cascade of phonon-mediated transitions, where both bright and dim excitonic states are involved.

We first consider the limit of single-phonon processes. We focused on the population dynamics and computed the phonon-mediated transition rates between excitonic states $n$ and $m$ using Fermi’s golden rule, which assumes weak system-bath coupling and employs the Markovian approximation:

where $\langle\dots\rangle_{\text{eq}}$ denotes an equilibrium average over bath coordinates. We computed these rates for all excitonic transitions and used them to build a kinetic master equation and propagate phonon-mediated exciton dynamics. We then calculated the average energy above equilibrium, $\langle\Delta E(t)\rangle=\sum_{n}E_{n}\big{(}p_{n}(t)-p_{n,\text{eq}}\big{)}$, where $p_{n}(t)$ is the population of state $n$ at time $t$ and $p_{n,\text{eq}}$ is the population of state $n$ at thermal equilibrium.

Because Eq. (1) describes the EXPC to first order in the phonon mode coordinates, the Fermi’s golden rule rates given by Eq. (2) only account for excitonic transitions that occur via the absorption or emission of a single phonon. While the largest energy gaps between excitonic states are an order of magnitude smaller than those between single-particle, electron states, they can still be larger than the phonon frequencies. Thus, transitions between those excitonic states would require the simultaneous emission of multiple phonons. Indeed, single-phonon-mediated cooling simulations for CdSe NCs of different sizes shows a phonon bottleneck (Fig. 2b). This phonon bottleneck is especially significant in smaller NCs for which confinement results in larger excitonic energy gaps, particularly at low excitonic energies, preventing the hot exciton from fully relaxing to the band edge through single-phonon emission.

To account for multiphonon processes and maintain the simplicity of the master equation, we performed a unitary polaron transformation ^{47, 48, 49, 50} to the Hamiltonian in Eq. (1):

where

and $p_{\alpha}$ is the momentum of phonon mode $\alpha$. A detailed derivation and description of the polaron-transformed Hamiltonian and its consequences are given in the Supporting Information.

With respect to the polaron-transformed Hamiltonian, Fermi’s golden rule transition rates can be computed as

where $\varepsilon_{n}\equiv E_{n}-\lambda_{n}$ is the energy of exciton $n$ scaled by its reorganization energy, and $g_{n,m}\equiv\sum_{\alpha}\tilde{V}_{n,m}^{\alpha}q_{\alpha}-\tilde{\lambda}_{nm}$ is the coupling between the polaronic states $n$ and $m$ (see the SI for more details). The Markovian approximation is not necessarily valid for this polaron-transformed Hamiltonian, as described in the Methods, so we compute the time-dependent, non-Markovian rates. Again, we computed all transition rates to build a kinetic master equation and propagate dynamics. Within this framework, we calculate the average energy above thermal equilibrium as before; namely, as $\langle\Delta\varepsilon(t)\rangle=\sum_{n}\varepsilon_{n}\big{(}p_{n}(t)-p_{n,\text{eq}}\big{)}$.

Note that $g_{n,m}$ includes exponential functions of the phonon momenta, so multiphonon-mediated transitions are accounted for even in the lowest-order perturbation theory rate given by Eq. (5). Including multiphonon processes enables transitions between excitonic states that have energy differences that are larger than the highest-frequency phonons. Multiphonon-mediated cooling simulations show that all NC systems fully relax to thermal equilibrium (Fig. 3a), indicating that multiphonon transitions involving a few phonon modes are essential to the cooling mechanism. Furthermore, the average energy relaxes within 100 fs, much faster within the single-phonon scheme.

Examining the transition rate as a function of transition energy for a 3.9 nm CdSe NC, illustrated in Fig. 3b, demonstrates that the single-phonon rates are larger for low-energy transitions, but there are no single-phonon transitions between excitonic states that have energy differences greater than $\sim$32 meV (i.e., greater than the largest phonon energy). The multiphonon rates, however, cover the full range of transition energies. Importantly, multiphonon relaxation between excitonic states with energy differences of 100 meV or less consistently have relatively high rates (ranging from $10^{-3}-10^{2}$ ps^-1 for multiphonon transitions as compared to $0-10^{2}$ ps^-1 for single-phonon transitions). This difference makes accessible many more relaxation channels and leads to a cooling timescale that is an order of magnitude faster than that resulting from single-phonon processes alone. The fast multiphonon relaxation is a result of the large number of phonon modes (approximately 3000 modes for a 3.9 nm CdSe NC) that quasi-continuously span a wide frequency range and that are all coupled, to some degree, to excitonic transitions. Thus, many phonon combinations satisfy the energy conservation requirement for phonon-mediated exciton transitions, leading to efficient relaxation via the emission of multiple phonons.

The asymmetry in rates about 0 meV transition energy reflects detailed balance (see Methods for more details). Furthermore, Fig. 3b shows a Gaussian relationship between the transition energy and the transition rate instead of the exponential dependence of the rate on the energy that results from the assumption that only the highest-frequency modes participate in nonradiative transitions.⁵¹ This result indicates that lower-frequency acoustic and optical modes are important to this cooling process, extending previous expectations that only high-frequency optical modes would be responsible for cooling.^{15, 28}

Fig. 3a also demonstrates that smaller NCs relax more quickly to thermal equilibrium than larger NCs. Due to stronger quantum confinement, smaller NCs have more energy to dissipate during the cooling process. Smaller NCs also have larger excitonic gaps and a smaller number of phonon modes. However, smaller NCs have stronger EXPC than larger NCs,⁴³ resulting in overall faster cooling timescales for smaller NCs.

The longer cooling timescales for larger CdSe NCs suggests that controlling the magnitude of EXPC may allow for tuning of the hot exciton cooling timescale. CdSe-CdS core-shell NCs have EXPC that is about five times smaller than that of bare cores due to suppression of exciton coupling to lower-frequency surface modes.⁴³ Fig. 4a compares simulations of the cooling process for a 3.9 nm CdSe core and a 3.9 nm CdSe core with 3 monolayers of CdS shell. Because of the quasi-type II band alignment in CdSe-CdS core-shell NCs, the exciton hole is confined to the CdSe core while the electron somewhat delocalizes into the CdS shell.⁵² This decreased quantum confinement leads to a smaller $1P$-$1S$ excitonic gap in core-shell NCs, so hot excitons have less energy to dissipate in core-shell NCs than in bare cores. However, cooling still takes about five times longer in the core-shell NC as a result of the weaker EXPC.

The multiphonon transition rates for both systems are shown in Fig. 4b, demonstrating that rates are one or more orders of magnitude smaller in the core-shell NC than in the bare core. For transitions with energies of 100 meV for less, the multiphonon relaxation rates range from $10^{-3}-10^{2}$ ps^-1 for the CdSe NC and from $10^{-2}-10^{1}$ ps^-1 for the CdSe-CdS core-shell NC.

We can further understand the role of EXPC in the cooling mechanism by examining the spectral densities, which describe the phonon densities of states weighted by the EXPC, of the core and core-shell NCs (Fig. 4c). The spectral density of the CdSe core shows significant EXPC to acoustic modes with frequencies of 4.0 THz or less as well as optical modes between 7.5$-$8.0 THz.^{43, 44} The core-shell NC, however, has negligible EXPC at lower phonon frequencies, as the presence of the CdS shell prevents coupling to delocalized and surface-localized modes at those frequencies, but it has slightly stronger coupling to the CdSe optical modes. These results provide further evidence that both acoustic and optical modes play an essential role in the cooling process. Exciton coupling to phonons with a quasi-continuous frequency range allows for resonance conditions to be more easily satisfied; for every excitonic energy gap, a set of phonons with the corresponding energy is easily found. However, exciton coupling to phonons within a narrow energy range restrict the set of phonons that would satisfy the necessary resonance conditions.

To allow for more meaningful comparison between our calculations and experimental measurements, we simulate changes in the absorption spectrum of a system initially excited to the $1P$ excitonic state as it relaxes to the $1S$ ground excitonic state. Assuming that the electric field, $\mathcal{E}$, is weak, such that the population of the ground state remains approximately 1 and that the population of the excited state is proportional to $\mathcal{E}^{2}$, the change in absorption is given by

where $\bm{\mu}_{n}$ is the transition dipole moment from the ground state to excitonic state $n$ and $p_{n}(t)$ is the population of excitonic state $n$ at time $t$.

The change in absorption, $-\Delta\sigma(\omega,t)$, shows a fast decay of the $1P$ excitonic peak and a slower rise of the $1S$ ground excitonic peak (Fig. S1). The dynamics of the rise of the $1S$ peak reflect those of hot exciton cooling. For each system, the rise dynamics were fit to an exponential function, and the extracted timescale was divided by the energy difference between the $1P$ and $1S$ excitonic peaks to yield an energy loss rate. The calculated energy loss rates are illustrated in Fig. 5a along with those measured experimentally using transient absorption spectroscopy²⁶ and state-resolved pump-probe spectroscopy³⁰ on wurtzite CdSe NCs. In agreement with experiment, our simulations show faster energy loss rates for smaller CdSe NCs. Smaller NCs have larger excitonic gaps due to quantum confinement and a smaller number of phonon modes, but they have stronger EXPC than larger NCs. Similarly, core-shell NCs, which have significantly weaker EXPC to lower-frequency acoustic modes,⁴³ show energy loss rates that are an order of magnitude slower than those of bare cores.

While Fig. 5a initially suggests that the calculated energy loss rates for CdSe NCs are larger than the measured values (but within the experimental error bars), those experiments use $\sim$100 fs pulses that obscure the observation of dynamics between states with spectral overlap,⁵³ like those measured here, and they measure NCs with very low photoluminescence quantum yields of around 1%, where carrier trapping may lead to dynamics that complicate the hot exciton cooling process. Two-dimensional electronic spectroscopy (ES) measurements, which are able to clearly resolve the features corresponding to excitonic relaxation, on 3.5 nm CdSe NCs show that hot exciton cooling from the higher-energy $1S$ excitonic peak to the ground excitonic peak occurs within $\sim$30 fs,⁵³ which is consistent with our findings. Furthermore, more recent two-dimensional ES experiments observe that cooling slows by an order of magnitude with the addition of a shell to a CdSe core,⁵⁴ in agreement with our calculated results.

While changing the size and composition of NCs is one avenue for tuning the cooling timescale, changing the temperature is another. The energy loss rates for 2.2 nm CdSe and 3.9 nm CdSe NCs simulated at different temperatures are illustrated in Fig. 5b. For both NCs, the energy loss rates are $\sim 0.1$ eV/ps at 10 K, and they monotonically increase with temperature, as expected for phonon-mediated processes. While both systems show a linear relationship between energy loss rate and temperature, the rate for 2.2 nm NC shows a stronger dependence on temperature than that of the 3.9 nm NC. This steeper scaling with increasing temperature may be a result of the stronger quantum confinement in the 2.2 nm NC, which leads to larger energy gaps between excitonic states. Thus, multiphonon processes at larger transition energies are more important. As those transition rates are very sensitive to temperature (Fig. S2), the overall cooling process in smaller NCs has a stronger temperature dependence. Interestingly, as shown in Fig. S3, the single-phonon-mediated cooling process also depends on temperature, but the dynamics converge above a threshold temperature. The threshold temperature is $\sim$300 K for the 2.2 nm CdSe NC while it is $\sim$50 K for the 3.9 nm CdSe. Again, this result may be due to the larger excitonic gaps in the smaller NC system. Note that our model given by Eq. (1) includes EXPC to lowest order in the phonon mode coordinates and ignores higher-order terms (Duschinsky rotations), which may influence the hot exciton cooling process at higher temperatures.⁵⁵

Finally, we investigate the mechanism underlying this ultrafast hot exciton cooling process. We calculated the density of excitonic states for a 3.9 nm CdSe NC and scaled it by the time-dependent population, as illustrated in Fig. 6a. We see that cooling occurs via a cascade of relaxation events through the manifold of excitonic states, as opposed to being dominated by a single or a few, higher-energy non-radiative transitions, as expected previously.^{16, 15, 28}

We wanted to understand the relationship between this multiphonon-mediated, hot exciton cooling mechanism (Fig. 1c) and the Auger-assisted cooling mechanism (Fig. 1b), which was first proposed to explain the breaking of the phonon bottleneck.³³ To this end, we projected our simulated exciton cooling dynamics for a 3.9 nm CdSe NC onto a single-particle picture of non-interacting electron-hole pair states, shown in Fig. 6b, and find that the Auger cooling mechanism emerges naturally from our excitonic dynamics. The hole quickly relaxes to the band edge via multiphonon emission followed by electron relaxation by $\sim$400 meV that results in hole re-excitation, and then the hole once again relaxes to the band edge by multiphonon emission. This result indicates that both Coulomb-mediated electron-hole correlations, which are inherent in our formalism, and multiphonon-mediated excitonic transitions are required to circumvent the phonon bottleneck and lead to ultrafast timescales of hot exciton cooling. These mechanistic insights are consistent for core-shell NCs, as illustrated in Fig. S4.