Hot-Carrier Generation in Plasmonic Nanoparticles: The Importance of Atomic Structure

Metal nanoparticles are attractive for plasmon-enhanced generation of hot carriers, which may be harnessed in photochemical reactions. In this work, we analyze the coherent femtosecond dynamics of photon absorption, plasmon formation, and subsequent hot-carrier generation through plasmon dephasing using first-principles simulations. We predict the energetic and spatial hot-carrier distributions in small metal nanoparticles and show that the distribution of hot electrons is very sensitive to the local structure. Our results show that surface sites exhibit enhanced hot-electron generation in comparison to the bulk of the nanoparticle. Although the details of the distribution depend on particle size and shape, as a general trend, lower-coordinated surface sites such as corners, edges, and {100} facets exhibit a higher proportion of hot electrons than higher-coordinated surface sites such as {111} facets or the core sites. The present results thereby demonstrate how hot carriers could be tailored by careful design of atomic-scale structures in nanoscale systems.


Supplementary Figures
: Distribution of the stored energy over electron-hole transition energies in Ag 561 . Distribution of the stored energy E ia (t) (Supplementary Note S1) over electron-hole transition energies ω ia , i.e., E(ω, t) = ia E ia (t)δ(ω − ω ia ) with a Gaussian smoothing over ω is shown. The color scale is logarithmic. The pulse is overlaid at its frequency ω 0 and the pulse width σ = √ 2/τ 0 is indicated as dotted lines at ω 0 ± 2σ. The contribution from resonant transitions in Fig. 1e in the main text comprises of transitions with energy ω ia between ω 0 ± 2σ, and that from the non-resonant transitions comprises of the plasmonic low-energy transitions (ω ia < ω 0 − 2σ) and the screening high-energy transitions (ω ia > ω 0 + 2σ).     Fig. 1 in main text) is further decomposed to contributions from the real and imaginary parts of density matrix, i.e., from q ia (t) and p ia (t) defined in Eq. (33), respectively. Specifically, terms with q ia (t) are 1 2 ω ia q 2 ia (t) + 1 2 q ia (t) jb K ia,jb q jb (t) and terms with p ia (t) are 1 2 ω ia p 2 ia (t), summing up to the energy E ia (t) in Eq. (35). The upper panel illustrates that plasmon can be thought as a classical harmonic oscillator with energy oscillating between density (q ia or "position coordinate") and current (p ia or "momentum coordinate").  The data for Ag 55 has been multiplied by 0.5. For each shape, the plasmon resonance show a redshift with increasing particle size, following the shift in the d-band onset with respect to the Fermi level apart from the small Ag 55 . Ag 309 Cub-Oh is an exception with broader and more redshifted plasmon peak than expected based on Ag 147 and Ag 561 , with a corresponding difference seen in the d-band onset.
Energy ( Ag 586 RTO ( 0 = 3.62 eV) Supplementary Figure S8: Hot-carrier distributions in silver nanoparticles. Occupation probabilities of hole and electron states after plasmon decay in silver nanoparticles of icosahedral (Ih), cuboctahedral (Cub-Oh), and regularly truncated octahedral (RTO) shapes. Occupation probabilities at different atomic sites (core, facets, edges, and corners) are also shown. The distributions are per atom (the number of atoms in each set is indicated in parenthesis). The percentages adjacent to the −1 eV and 1 eV dotted lines indicate the amount of holes and electrons with energy < −1 eV and > 1 eV, respectively, in comparison to the total amount in each set. Scale is the same in each plot.

Supplementary Notes
Supplementary Note S1: Perturbation expansions of time-dependent quantities.
Wave function. The expansion of the time-dependent wave function up to second order in perturbation is (notation: |i(t)⟩ = ψ i (t) and |i⟩ = ψ which gives the projection ⟨k|i(t)⟩ = e −iϵ k t δ ik + C (1) ki (t) . Consider the norm up to second order where the denoted first and second order terms are required to vanish in order to have unitary evolution, i.e., ⟨j(t)|i(t)⟩ = δ ji . Density matrix. The expansion of the time-dependent Kohn-Sham density matrix is By invoking the conditions from unitary evolution from Eq. (2), the first-order contribution simplifies to Note that in the first order the electron-electron and hole-hole parts of the density matrix are zero: ρ  The leading second-order term of the electron-electron and hole-hole parts is obtained by setting f m = f n and using Eqs.
(2) and (5) as the first and second terms of which constitute the electron-electron and hole-hole parts of the density matrix. In particular, the diagonal can be simplified by noting that |ρ (10) By defining transition probability Eq. (10) reads where P e n (t) and P h n (t) corresponds to induced occupations of electrons and holes on state n, respectively. Energy. In the Kohn-Sham density-functional theory, the total energy is composed of kinetic, Hartree, and XC contributions, and of the external potential energy (including the potential created by nuclei v ext and first-order light pulse v pulse ), respectively. In the basis of KS states, these energy terms are (assuming adiabatic XC kernel) and where ρ ij (t) = ρ ij . The perturbation expansions of the energy contributions are (note that here the summations over ij and kl run over all indices, including both the electron-hole and hole-electron spaces and the diagonal).
The first order gives where the first term vanishes as the ground-state Hamiltonian H ii (t) = 0. The second order gives By using Eq. (12) and assuming real-valued ground-state KS wave functions and frequency-independent XC kernel, simplifying the Hartree-XC term, [1] Eq. (28) can be written as a sum over electron-hole space only (1) where the Hartree-XC term is used to estimate the Coulomb energy E C ia (t). In RPA, the XC is neglected and K Hxc is replaced by the Hartree kernel K H (r, r ′ ) = |r − r ′ | −1 .
By collecting all the terms, the energy up to the second order is where dots denote time derivatives. By defining the auxiliary quantities Eq. (31) can be written in convenient form as     q These equations are identical to the equations of motion of a collection of coupled classical harmonic oscillators when q ia (t) and p ia (t) are identified as position and momentum coordinates. In this notation, the electron-hole decomposition of energy of Eq. (29) is and the Coulomb energy contribution from Eq. (29) is