Real-Time Electron-Electron Scattering Dynamics in Plasmonic Nanostructures

Abstract

Electron-electron scattering is one of the most important hot carrier relaxation pathways in plasmonic nanoparticles. Understanding the dynamics of this scattering process and the effects of this on excited state dephasing and relaxation is therefore essential for the design of plasmonic nanostructures, including optical properties and the dynamics of electrons in plasmon-driven catalytic reactions. In this work, we have developed a novel approach that incorporates real-time time-dependent density functional tight-binding (DFTB) simulations with the Lindblad quantum Boltzmann equation (LQBE) based on a screened electron-electron interaction that is determined by the random phase approximation (RPA). This approach enables a self-consistent description of electron-electron scattering effects that occur during and after plasmon excitation in clusters/nanoparticles with hundreds of atoms. With our RT-TDDFTB+LQBE method, we investigate the quasiparticle lifetime as well as population and coherence dynamics in silver, gold and aluminum nanoclusters with sizes between 1.5-2.6 nm. Our results show that the quasiparticle lifetimes and relaxation dynamics are highly energy dependent, becoming much faster at higher energies. For clusters less than 2 nm, quantum effects associated with discrete energy levels can lead to the fluctuating lifetimes and deviation of the dynamics from the typical thermalization process, while for larger nanoparticles, the transition to bulk metallic behavior is found. Decoherence of the initially excited plasmon resonance is observed with a timescale of 10 fs, much faster than population relaxation. For gold, we find that the 5d-band can significantly slow down the relaxation of energetic electrons though Auger scattering, and interband transitions can lead to a secondary decoherence process longer than 50 fs.

Figures (6)

• Figure 1: (a) Silver clusters used in the simulations. The gold and aluminum clusters are similar in size. (b)-(d): The absorption spectra of silver, gold and aluminum clusters, respectively, calculated by the standard RT-TDDFTB simulation with delta-kick perturbation. The approximate location of plasmon resonance frequency is marked by the dashed line.
• Figure 2: QBE lifetime calculated according to Eq. \ref{['eq:qbe_lifetime']}, compared with various existing experimental and theoretical results. (a) For silver, we compare with 2PPE measurements of nanopits (20 nm width, 2 nm height) on graphite merschdorf2004Collective, and a linear Muffin-tin orbital (LMTO) GW calculation for the bulk.zhukov2001Corrected (b) For gold, we compare with 2PPE measurements of nanorods (40 nm length, 10 nm diameter) pettine2023EnergyResolved and the LMTO-GW calculation for the bulk.zhukov2001Corrected (c) For aluminum, we compare with 2PPE measurements of the bulk phase bauer1998Electron and a linearized-augmented-plane-wave (LAPW) GW calculation for the bulk.ladstadter2004Firstprinciples The free electron density parameters used are $r_s[\text{Ag}]=2.0, r_s[\text{Au}]=1.87$ and $r_s[\text{Al}]=2.07$.bauer2015Hotpettine2023EnergyResolved
• Figure 3: The quasilogarithmic distribution $\phi(E)=-\log_{10}{(1/f(E)-1)}$ for RT-TDDFTB+LQBE simulations of (a) Ag_561 (b) Au_561 and (d) Al_561, and (c) the standard RT-TDDFTB simulation of Ag_561. The laser frequency is marked by the vertical dotted lines. The distributions at 1000 K and the terminal temperature are shown by dashed lines. In all systems, RT-TDDFTB+LQBE shows population relaxation as a result of e-e scattering, while in the standard RT-TDDFTB simulation, the distribution has no significant change after the laser pulse up to 300 fs.
• Figure 4: Energy distribution of hot carriers among different clusters at 10 fs, 30 fs and 100 fs after the laser pulse peak, along with the distribution in simulations without QBE terms at 100 fs. Note that we have scaled some of the distributions to make better comparison. The actual change is largely monotonic with time.
• Figure 5: (a-c) HE population in (a) Ag_561 (b) Au_561 and (c) Al_561 as a function time, grouped by energy relative to the Fermi surface. The high energy orbitals relax significantly faster than low energy ones. (d)(e) The effective HE relaxation time of clusters, calculated by exponentially fitting (d) the total HE population or (e) the 1-2 eV population. (f) The HE generation rate in Au_561 at different excitation energies according to Eq. \ref{['eq:scattering_rate']}, with the destination energy $E_D$ between 1-2 eV relative to $E_F$. For particle scattering, the source energy $E_S>0$, for sp-band hole scattering $\text{-1.6 eV}<E_S<0$, and for d-band hole scattering $E_S<\text{-1.6 eV}$. For $\hbar\omega_{\text{exc}}$=2.3 eV, hole scattering from the d-band is a major source of HE generation. (g) Schematic view of particle and hole scattering. In the hole scattering, a newly produced electron fills an existing hole (marked by white circle).