Surface-assisted carrier excitation in plasmonic nanostructures
Tigran V. Shahbazyan
We present a quantum-mechanical model for surface-assisted carrier excitation by optical fields in plasmonic nanostructures of arbitrary shape. We derive an explicit expression, in terms of local fields inside the metal structure, for surface absorbed power and surface scattering rate that determine the enhancement of carrier excitation efficiency near the metal-dielectric interface. We show that surface scattering is highly sensitive to the local field polarization, and can be incorporated into metal dielectric function along with phonon and impurity scattering. We also show that the obtained surface scattering rate describes surface-assisted plasmon decay (Landau damping) in nanostructures larger than the nonlocality scale. Our model can be used for calculations of plasmon-assisted hot carrier generation rates in photovoltaics and photochemistry applications.
ld-carier.tex
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\begin{document}
\title{Surface-assisted carrier excitation in plasmonic nanostructures}
%\title{Surface-assisted carrier excitation in plasmonic nanostructures}
\author{Tigran V. Shahbazyan}
\affiliation{
Department of Physics, Jackson State University, Jackson, MS 39217 USA
}

\begin{abstract}
We present a quantum-mechanical model for surface-assisted carrier excitation by optical fields in plasmonic nanostructures of arbitrary shape. We derive an explicit expression, in terms of  local fields inside the metal structure, for surface absorbed power and surface scattering rate that determine  the  enhancement of carrier excitation efficiency near the metal-dielectric interface. We show that surface scattering is highly sensitive to the local field  polarization, and can be incorporated into metal dielectric function along with  phonon and impurity scattering.  We also show that the obtained surface scattering rate describes  surface-assisted plasmon decay (Landau damping) in nanostructures larger than the nonlocality scale. Our model can be used for calculations of plasmon-assisted hot carrier generation rates in photovoltaics and photochemistry applications.
\end{abstract}
\maketitle

\section{Introduction}

Plasmon-assisted hot carrier excitation and transfer across the interfaces \cite{nordlander-nn15} has recently attracted intense interest due to wide-ranging applications in  photovoltaics \cite{park-nl11,melosh-nl11,halas-science11,halas-nc13,lian-nl13,halas-nl13-2,clavero-np14,brongersma-nl14,atwater-nc14,halas-nc15} and photochemistry \cite{brongersma-nl11,moskovits-nl12,halas-nl13,moskovits-nn13,halas-jacs14}. In metal nanostructures with characteristic size $L$ below the diffraction limit,  i.e., $L<c/\omega$, where $c$ and $\omega$ are, respectively, the light speed and frequency, the light scattering is relatively weak, and the absorption is dominated by resonant excitation of surface plasmons, which subsequently decay into electron-hole (\textit{e-h}) pairs through several decay mechanisms depending on the size and shape of a plasmonic system \cite{halperin-rmp86,kresin-pr92,vallee-jpcb01,schatz-jpcb03,noguez-jpcc07}.  While in relatively large systems,  excitation of \textit{e-h} pairs with   optical energy $\hbar \omega$ is accompanied by momentum relaxation due to phonon and impurity scattering, for systems with characteristic size $L \sim 20$ or smaller, the dominant momentum relaxation channel is surface scattering that leads, in particular, to size and shape dependence of the plasmon linewidth \cite{klar-prl98,mulvaney-prl02,halas-prb02,klar-nl04,vallee-prl04,halas-nl04,hartland-pccp06,vallee-nl09,vanduyne-jpcc12,vallee-nl13,schatz-nl15}. Calculations of surface-assisted decay rate have been performed for some simple shapes (mostly spherical) within random phase approximation (RPA) \cite{kawabata-jpsj66,lushnikov-zp74,schatz-jcp83,barma-pcm89,yannouleas-ap92,eto-srl96,uskov-plasmonics13,khurgin-oe15,kirakosyan-prb16} or time-dependent local density approximation (TDDFT) \cite{jalabert-prb02,jalabert-prb05,yuan-ss08,vallee-jpcl10,lerme-jpcc11,li-njp13,nordlander-acsnano14} approaches, while for general shape systems, the classical scattering  model \cite{kreibig-zp75,ruppin-pss76,schatz-cpl83,schatz-jcp03,moroz-jpcc08} suggested the surface scattering rate in the form $\gamma_{cs}=A  v_{F}/L_{s}$, where $L_{s}$ is the ballistic electron scattering length, and the  constant $A$ includes the effects of surface potential, electron spillover, and dielectric environment \cite{kreibig-book}. Note, however, that  recent measurements of plasmon linewidth in nanostructures of various shapes revealed  significant discrepancies with a simple $L^{-1}$ dependence\cite{halas-nl04,vanduyne-jpcc12,vallee-nl13,schatz-nl15}. 



Here we present a quantum-mechanical model for surface-assisted excitation of \textit{e-h} pairs by alternating local electric field of the form $\textbf{E}(\bm{r})e^{-i\omega t}$ created in the metal either by excitation of a plasmon or as a response to monochromatic external field. We show that surface contribution to the absorbed power due to energy transfer to the excited carriers is given by 
%
\begin{equation}
\label{power-surface}
Q_{s}=\frac{e^{2}}{2\pi^{2} \hbar} \frac{E_{F}^{2}}{ (\hbar\omega)^{2}} \! \int \!  dS  |E_{n}|^{2},
\end{equation}
%
where integration is taken over the metal surface, $E_{n}$ is the local field  component \textit{normal} to the interface, and $E_{F}$ is the Fermi energy in the metal. The above expression, which is derived within RPA approach, is valid for systems of arbitrary shape that are significantly larger than the nonlocality scale $v_{F}/\omega$ \cite{mortensen-pn13,mortensen-nc14}, i.e., for systems at least several nm large. We compute the surface enhancement of the carrier excitation efficiency for some common nanostructures and show that it is highly sensitive to the local field polarization and  system geometry.


%


\section{Theory}
We consider a metal nanostructure characterized by complex dielectric function $\varepsilon(\omega)=\varepsilon'(\omega)+i\varepsilon''(\omega)$ in a medium with dielectric constant $\varepsilon_{d}$, and, for simplicity,  restrict ourselves by  systems with a single metal-dielectric interface. The standard expression for absorbed power has the form \cite{landau}, 
%
\begin{equation}
\label{power-bulk}
Q=\frac{\omega\varepsilon''(\omega)}{8\pi} \! \int \! dV  |\textbf{E}|^{2}.
\end{equation}
%
where integration is carried over the metal volume, while the local field $\textbf{E}(\bm{r})$ is determined,  in the quasistatic limit,  by the Gauss's law $\bm{\nabla} [ \varepsilon' (\omega,\bm{r})\textbf{E}(\bm{r})]=0$ [here $\varepsilon (\omega,\bm{r})$ equals $\varepsilon(\omega)$ and $\varepsilon_{d}$  in the metal and dielectric regions, respectively]. The surface contribution to the absorbed power is $Q_{s}=\hbar\omega/\tau$, where $1/\tau$ is the first-order transition probability rate, leading to the standard RPA expression \cite{kirakosyan-prb16} 
%
\begin{equation}
\label{power-rpa}
Q_{s}=\pi \omega\sum_{\alpha\beta}|M_{\alpha\beta}|^{2}
%|\langle \beta|\Phi_{n}|\alpha\rangle|^{2}
\left [f(\epsilon_{\alpha})-f(\epsilon_{\beta})\right ]
\delta(\epsilon_{\alpha}-\epsilon_{\beta}+\hbar\omega).
\end{equation}
%
Here, $M_{\alpha\beta}=\int dV \psi_{\alpha}^{*}\Phi\psi_{\beta}$ is the matrix element of local potential $\Phi(\bm{r})$ defined as $e\textbf{E}=-\bm{\nabla} \Phi$ ($e$ is the electron charge),  $\psi_{\alpha}(\bm{r})$ and $\psi_{\beta}(\bm{r})$ are wave-functions for electron states with energies $\epsilon_{\alpha}$ and $\epsilon_{\beta}$, respectively, separated by $\hbar\omega$,  and $f(\epsilon)$ is the Fermi distribution function. For nanostructures of arbitrary shape, numerical evaluation of $M_{\alpha\beta}$ is a highly complicated task  due to the complexity of electron wave functions. However, as we show below, for the hard-wall confining potential, $Q_{s}$ can be derived in a closed form for any system significantly larger than $v_{F}/\omega$ (but still smaller than $c/\omega$).


Excitation of an \textit{e-h} pair with a large, compared to the electron level spacing, energy $\hbar\omega$ requires momentum transfer to the cavity boundary. We note that the boundary contribution  to $M_{\alpha\beta}$ can be presented as an integral over  the metal surface \cite{sm},
%
\begin{equation}
\label{matrix}
M_{\alpha\beta}^{s}= \frac{-e\hbar^{4}}{2m^{2} \epsilon_{\alpha\beta}^{2}} \! \int \! dS [\nabla_{n}\psi_{\alpha}(\bm{s})]^{*}E_{n}(\bm{s}) \nabla_{n}\psi_{\beta}(\bm{s}),
\end{equation}  
%
where $\nabla_{n}\psi_{ \alpha}(\bm{s})$ is the wave function derivative normal to the surface, $E_{n}(\bm{s})$  is the corresponding normal field component, $\epsilon_{\alpha\beta}=\epsilon_{\alpha}-\epsilon_{\beta}$ is the \textit{e-h} pair excitation energy, and $m$ is the electron mass.  Using this matrix element,  Eq.~(\ref{power-rpa})  takes the form
%
\begin{equation}
\label{Q-surface}
Q_{s}=\frac{e^{2}\hbar^{4}}{4\pi m^{4} \omega^{3}} \! \int \! \! \int \!  dS dS' E_{n}(\bm{s})E_{n'}^{\ast}(\bm{s}') F_{\omega}(\bm{s},\bm{s}'),
\end{equation}
%
where $F_{\omega}(\bm{s},\bm{s}')$ is the \textit{e-h} surface correlation function,
%
\begin{equation}
\label{F}
F_{\omega}(\bm{s},\bm{s}')=\! \int \! d\epsilon   f_{\omega}(\epsilon) \rho_{nn'}(\epsilon;\bm{s},\bm{s}')\rho_{n'n}(\epsilon+\hbar\omega;\bm{s}',\bm{s}),
\end{equation}
%
defined in terms of normal derivative of the electron cross density of states, $\rho(\epsilon;\bm{s},\bm{s}')=\text{Im} G(\epsilon;\bm{s},\bm{s}')$, at surface points: $\rho_{nn'}(\epsilon;\bm{s},\bm{s}')=\nabla_{n}\nabla'_{n'}\text{Im} G(\epsilon;\bm{s},\bm{s}')$. Here $G(\epsilon;\bm{s},\bm{s}')$ is the confined electron Green function, and the function $f_{\omega}(\epsilon)=  f(\epsilon)-f(\epsilon+\hbar\omega)$ restricts the initial energy of promoted electron to the interval $\hbar\omega$ below $E_{F}$.
 
 
To evaluate $Q_{s}$, we note that excitation of an \textit{e-h} pair with energy $\hbar\omega$ is accompanied by momentum transfer $\sim \hbar\omega/v_{F}$ and, hence, takes place in  a region of size $\sim v_{F}/\omega$, so that the \textit{e-h} correlation function $F_{\omega}(\bm{s},\bm{s}')$ peaks in the region $|\bm{s}-\bm{s}'|\lesssim v_{F}/\omega$ and rapidly oscillates outside of it (see below). At the same time, the local fields significantly change  on a much larger scale $\sim L$. Therefore,  for $L\gg v_{F}/\omega$,  the main contribution to the integral in Eq.~(\ref{Q-surface}) comes from the regions with $E_{n}(\bm{s})\approx E_{n}(\bm{s}')$,   so that $Q_{s}$ takes the form
%
\begin{equation}
\label{Q-surface1}
Q_{s}=\frac{e^{2}\hbar^{4}}{4\pi m^{4} \omega^{3}} \! \int \!  dS  |E_{n}(\bm{s})|^{2} \bar{F}_{\omega}(\bm{s}),
\end{equation}
%
where $\bar{F}_{\omega}(\bm{s})=\! \int \!  dS'   F_{\omega}(\bm{s},\bm{s}')$. Evaluation of $\bar{F}_{\omega}$ is based upon multiple-reflection expansion for the electron Green function $G(\epsilon;\bm{s},\bm{s}')$ in a hard-wall cavity \cite{balian-ap70}.  For system's characteristic size $L\gg \lambda_{F}$, where $\lambda_{F}$ is the Fermi wavelength, the main contribution comes from the direct and singly-reflected paths, while   the  higher-order reflections are suppressed as powers of $\lambda_{F}/L$. In the leading  order, we obtain $G(\epsilon;\bm{s},\bm{s}')=2G_{0}(\epsilon,\bm{s}-\bm{s}')$, where $G_{0}(\epsilon,r)=(m/2\pi \hbar^{2})e^{ik_{\epsilon}r}/r$, with $k_{\epsilon}=\sqrt{2m\epsilon}/\hbar$, is  the free electron Green function and factor 2 comes from equal contributions of the direct and singly-reflected paths at a surface point \cite{sm}.  It is now easy to see that the integrand of Eq.~(\ref{F}) peaks in the region  $|\bm{s}-\bm{s}'|\lesssim (k_{\epsilon+\hbar\omega}-k_{\epsilon})^{-1}$ and rapidly oscillates outside of it. For $\epsilon\sim E_{F}$ and $\hbar\omega/E_{F}\ll 1$, this sets the  scale $|\bm{s}-\bm{s}'|\sim v_{F}/\omega$  for the \textit{e-h} correlation  function $F_{\omega}(\bm{s},\bm{s}')$ in Eq.~(\ref{Q-surface}). Finally, for $L\gg v_{F}/\omega$, after computing normal derivatives in $\rho_{nn'}(\epsilon;\bm{s},\bm{s}')=2\nabla_{n}\nabla'_{n}\text{Im}G_{0}(\epsilon,\bm{s}-\bm{s}')$ relative to the tangent plane at a surface point \cite{sm}, we obtain $\bar{F}_{\omega}=(2m^{4}E_{F}^{2}/\pi \hbar^{8})\hbar\omega$, yielding Eq.~(\ref{power-surface}).

The surface contribution $Q_{s}$ to the full absorbed power $Q$  should be considered in conjunction with its bulk counterpart $Q_{0}$, i.e., $Q=Q_{0}+Q_{s}$. The bulk contribution is given by the standard expression (\ref{power-bulk}), where the metal dielectric function $\varepsilon(\omega)$ includes only the bulk processes. In the following, we adopt the Drude  dielectric function $\varepsilon(\omega)  = \varepsilon_{i}(\omega)-\omega_{p}^{2}/\omega(\omega+i\gamma)$, where $\varepsilon_{i}(\omega)$ describes interband transitions, $\omega_{p}$ is the plasma frequency, and $\gamma$ is the scattering rate. In fact, both bulk and surface contributions can be combined, in a natural way, within the general expression (\ref{power-bulk}). Indeed, using the relation $\omega_{p}^{2}=16e^{2}E_{F}^{2}/3\pi \hbar^{3}v_{F}$, the surface absorbed power (\ref{power-surface}) can be recast as
%
\begin{equation}
\label{Q-surface2}
Q_{s}
%=\frac{e^{2}}{2\pi^{2} \hbar} \frac{E_{F}^{2}}{ (\hbar\omega)^{2}} \! \int \!  dS  |E_{n}|^{2},
=\frac{3v_{F}}{32\pi}\frac{\omega_{p}^{2}}{\omega^{2}}\! \int \!  dS  |E_{n}|^{2}.
\end{equation}
%
Then, it is easy to see that the surface contribution can be incorporated into general expression (\ref{power-bulk}) for the absorbed power by modifying the Drude scattering rate as $\gamma=\gamma_{0}+\gamma_{s}$, where $\gamma_{0}$ is the usual bulk scattering rate and
%
\begin{equation}
\label{rate-ld}
\gamma_{s}=
\frac{3 v_{F}}{4}
 \frac{\int \! dS |E_{n}|^{2}}{\int \! dV |\textbf{E}|^{2}},
\end{equation}
%
is the surface scattering rate. Indeed, after this modification,  $Q_{s}$ can be  obtained from Eq.~(\ref{power-bulk})  as the first-order expansion term  in $\gamma_{s}$, indicating that the surface scattering mechanism should be treated on par with  phonon and impurity scattering. Note that  $\gamma_{s}$ is independent of the local field overall strength  but highly sensitive to its polarization relative to the metal-dielectric interface.




Turning to the surface-assisted  plasmon decay (Landau damping), the plasmon decay rate in any metal-dielectric structure is given by general expression \cite{shahbazyan-prl16} $\Gamma =  Q/U$, where $U$ is the plasmon energy \cite{landau},
%
\begin{align}
\label{energy-LL}
U 
%=\! \int \! \frac{dV_{t}}{16\pi} \frac{\partial (\omega\varepsilon')}{\partial \omega}  |\textbf{E}|^{2}
=\frac{\omega}{16\pi} \frac{\partial \varepsilon'(\omega)}{\partial \omega}\! \int\! dV |\textbf{E} |^{2}.
\end{align}
%
Using Eq.~(\ref{power-bulk}) for the absorbed power,  the decay rate has the standard form,
%
\begin{equation}
\label{ld}
\Gamma = 2\varepsilon''(\omega)\left [\frac{\partial \varepsilon'(\omega)}{\partial \omega}\right ]^{-1}.
\end{equation}
%
Let us show that the full plasmon decay rate that includes both bulk and surface contributions is given by Eq.~(\ref{ld}), but  with  $\varepsilon(\omega)$ modified according to Eq.~(\ref{rate-ld}). Indeed, using Eq.~(\ref{Q-surface2}), the surface contribution to $\Gamma$ takes the form 
%
\begin{equation}
\label{ld1}
\Gamma_{s} =  \frac{Q_{s}}{U}=  \frac{2\omega_{p}^{2}\gamma_{s}}{\omega^{3}}\left [\frac{\partial \varepsilon'(\omega)}{\partial \omega}\right ]^{-1} ,
\end{equation}
%
where $\gamma_{s}$ is given by Eq.~(\ref{rate-ld}). The same expression is obtained, in the first order in $\gamma_{s}$, from Eq.~(\ref{ld}) with surface-modified $\varepsilon(\omega)$.  Note that for $\omega$ well below the interband transitions onset, the plasmon decay rate and  scattering rate coincide, $\Gamma\approx\gamma=\gamma_{0}+\gamma_{s}$.

\section{Applications}

Let us now discuss  the effect of field polarization and system geometry on the carrier excitation efficiency. The surface enhancement factor of the absorbed power is given by the ratio of full ($Q$) to bulk ($Q_{0}$) absorbed power, $M=Q/Q_{0}$, which, within RPA, takes a simple form
%
\begin{equation}
\label{enh}
M=1+\frac{\gamma_{s}}{\gamma_{0}}.
\end{equation}
%
Evaluation of surface scattering rate $\gamma_{s}$ is made more convenient by noting that the Gauss's law reduces the volume integral in Eq.~(\ref{rate-ld}) to the surface term, so that  
%
\begin{equation}
\label{rate-surface1}
\gamma_{s}=A\, v_{F} \, \frac{\int \! dS |\nabla_{n}\Phi |^{2}}{\int \! dS \Phi^{*} \nabla_{n}\Phi},
\end{equation}
%
where real part of the denominator is implied. This form  of $\gamma_{s}$ as the ratio of two surface integrals reflects the fact that  \textit{e-h} pairs are excited in a close proximity (within $v_{F}/\omega$) to the interface. The constant $A$, which equals $A=3/4$ for the hard-core confining potential, can be determined by matching Eq.~(\ref{rate-surface1}) with the corresponding rate for a spherical particle with radius $a$. A straightforward evaluation of Eq.~(\ref{rate-surface1}) recovers the standard result for a sphere $\gamma_{sp}=Av_{F}/a$, while in the recent TDLDA calculations for relatively large (up to $a=10$ nm)  spherical particles \cite{vallee-jpcl10,lerme-jpcc11}  the value $A\approx 0.32$ was obtained. Note that for systems, whose geometry permits separation of variables, the form (\ref{rate-surface1}) of $\gamma_{s}$ is especially useful since it leads to \textit{analytical} results  for some common structures that so far eluded attempts of any quantum-mechanical evaluation, as we illustrate below for metal nanorods and nanodisks.




%%\textit{\textbf{Discussion}}.--- 
%Our results can be used for calculations of surface-assisted carrier excitation rates directly from the local fields.   Let us first apply Eq.~(\ref{rate-surface1}) to the simplest case of a sphere of radius $a$. In the long-wave limit, the potentials inside the sphere are given by regular solutions of the Laplace equation, $\Phi\propto  r^{l}Y_{lm}(\hat{\textbf{r}})$, where $r$ is the radius and $Y_{lm}(\hat{\textbf{r}})$ are the spherical harmonics ($l$ and $m$ are, respectively, the polar and azimuthal numbers). Then, a direct evaluation of Eq. (\ref{rate-surface1})  recovers the known $l$th mode's rate, $\gamma_{sp}=3lv_{F}/4a$. The same result is obtained for the $l$th  transverse  mode in a cylindrical nanowire.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[tb]
%\centering
\begin{center}
\includegraphics[width=0.9\columnwidth]{rate-prolate}
\caption{\label{fig1}
Normalized rates for longitudinal and transverse dipole modes in prolate spheroidal particles (nanorods) relative to the spherical  particle rate are shown with changing aspect ratio $b/a$ Inset: Schematics for the mode polarizations. 
  }
\end{center}
\vspace{-6mm}
\end{figure}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[tb]
%\centering
\begin{center}
\includegraphics[width=0.9\columnwidth]{rate-oblate}
\caption{\label{fig2}
Normalized rates for longitudinal and transverse dipole modes in oblate spheroidal particles (nanodisks) relative to the spherical  particle rate are shown with changing aspect ratio $b/a$ Inset: Schematics for the mode polarizations. 
  }
\end{center}
\vspace{-6mm}
\end{figure}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%


In Figs.~\ref{fig1} and \ref{fig2}, we show calculated surface scattering rates for nanorodes and nanodisks, which are modeled here by prolate and oblate spheroidal nanoparticles, respectively. These structures support longitudinal and transverse plasmon modes oscillating along the symmetry axis (semi-axis $a$) and within the symmetry plane (semi-axis $b$).  Using  Eq.~(\ref{rate-surface1}), $\gamma_{s}$ for all modes can be found in an analytical form \cite{sm}, but here only the results for the  dipole mode  are presented.  For a  nanorod  (prolate spheroid) with the aspect ratio $b/a<1$,  we obtain $\gamma_{s} =\gamma_{sp}f_{L,T}$, where 
%
\begin{align}
\label{rates-spheroid}
f_{L}
=\frac{3}{2\tan^{2}\!\alpha}\left [\frac{2\alpha}{\sin 2\alpha}-1
\right ],
~
f_{T}
=\frac{3}{4\sin^{2}\!\alpha}\left [1-\frac{2\alpha}{\tan 2\alpha}
\right ],
\end{align}
%
are the normalized rates for longitudinal and transverse modes  relative to the spherical particle rate, and $\alpha=\arccos (b/a)$ is the angular eccentricity. For a nanodisk (oblate spheroid) with    $b/a>1$, the normalized rates have the same form (\ref{rates-spheroid}) but with $\alpha=i\arccosh (b/a)$.  In Fig.~\ref{fig1}, we show the normalized rates in a prolate spheroid ($b/a<1$) for both longitudinal (PL) and transverse (PT) modes as the system shape evolves from a needle to a sphere. With changing aspect ratio $b/a$, the rates exhibit a dramatic difference in behavior depending  on the local field polarization.  As  nanorods become thinner,  the normalized rate  decreases for the PL mode but increases for the PT modes. These trends are reversed  for  nanodisks ($b/a>1$), shown in Fig.~\ref{fig2}: the normalized rates increase for the longitudinal (OL) mode and decrease for the transverse OT mode as the nanodisk radius increases (at fixed height). 


The dominant factor that determines the surface enhancement of carrier excitation efficiency is the  local field polarization relative to the metal-dielectric interface. In a given system, the surface-assisted carrier excitation rate can be  manipulated in a wide range by choosing the electric field  orientation.   Note that recent measurements \cite{schatz-nl15} of plasmon spectra in cylinder-shaped nanorods and nanodisks revealed strong polarization dependence of   plasmon linewidth.

\section{Conclusions}

In summary, we developed a quantum-mechanical model for surface-assisted carrier excitation in plasmonic  nanostructures  of arbitrary shape. We derived explicit expressions  for surface absorbed power and scattering  rate that are highly sensitive to the local field polarization relative to the metal-dielectric interface. Our results can be used for calculations of  hot  carrier generation rates in photovoltaics and photochemistry  applications \cite{nordlander-nn15}. 

%%%%%%%%%%%%%%%%%%%% ACKNOWLEDGMENTS     %%%%%%%%%%%%%%%%%%%%

%\suppinfo
%Derivation of the matrix element (\ref{matrix}), evaluation of the \textit{e-h} correlation function (\ref{F}), and technical detail involving the scattering rates (\ref{rates-spheroid}).

\acknowledgments

This work was supported in part by the National Science Foundation under grants  No. DMR-1610427 and No. HRD-1547754.


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%\textit{An Autonomous Photosynthetic Device in Which All Charge Carriers Derive from Surface Plasmons},
Nat. Nanotechnol. \textbf{8}, 247 (2013).

\bibitem{halas-jacs14} S. Mukherjee, L. Zhou, A. M. Goodman, N. Large, C. Ayala-Orozco, Y. Zhang, P. Nordlander, and N. J. Halas,
%\textit{Hot-electron-induced dissociation of H$_{2}$ on gold nanoparticles supported on SiO$_{2}$},
J. Am. Chem. Soc. \textbf{136}, 64  (2014).



%%%%%%%%%%%%%Size-shape reviews

\bibitem{halperin-rmp86} W. P. Halperin, 
\textit{Quantum size effects in metal particles}, 
Rev. Mod. Phys. \textbf{58}, 533 (1986).

\bibitem{kresin-pr92} V. V. Kresin, 
%\textit{Collective resonances and response properties of electrons in metal clusters},
Phys. Rep. \textbf{220},  1 (1992).


\bibitem{vallee-jpcb01}  C. Voisin, N. Del Fatti, D. Christofilos, and F. Vall\'{e}e,
%\textit{Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles}, 
J. Phys. Chem. B \textbf{105}, 2264  (2001).

\bibitem{schatz-jpcb03} K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, 
%\textit{The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment}, 
J. Phys. Chem. B \textbf{107} 668  (2003).


\bibitem{noguez-jpcc07} C. Noguez, 
%\textit{Surface plasmons on metal nanoparticles: The inﬂuence of shape and physical environment}, 
J. Phys. Chem. C \textbf{111}  3806 (2007).

%%%%%%%%%%LD Exp

\bibitem{klar-prl98} T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann,
%\textit{Surface-Plasmon Resonances in Single Metallic Nanoparticles},
Phys. Rev. Lett. \textbf{80}, 4249 (1998).

%%%%%%%%%% Nanorods

\bibitem{mulvaney-prl02}C. S\"{o}nnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. V. Wilson, and P. Mulvaney,
%\textit{Drastic reduction of plasmon damping in gold nanorods},
Phys. Rev. Lett. \textbf{88}, 077402 (2002).

%%%%%%%%%%%%%%%Nanoshells
\bibitem{halas-prb02}S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas, 
%\textit{Relative contributions to the plasmon line shape of metal nanoshells},
Phys. Rev. B \textbf{66}, 155431 (2002).

\bibitem{klar-nl04}G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. A Klar, and J. Feldmann, 
%\textit{Gold nanoshells Improve Single Nanoparticle Molecular Sensors},
Nano Lett. \textbf{4}, 1853 (2004).

%%%%%%%%%%%%%%%%%% LD single NP

\bibitem{vallee-prl04} A. Arbouet, D. Christofilos, N. Del Fatti, F. Vall\"{e}e, J. R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, 
%\textit{Direct Measurement of the Single-Metal-Cluster Optical Absorption},
Phys. Rev. Lett. \textbf{93}, 127401 (2004).

\bibitem{halas-nl04}C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J. Halas, and J. H. Hafner, 
%\textit{Scattering Spectra of Single Gold Nanoshells},
Nano Lett. \textbf{4}, 2355 (2004).

\bibitem{hartland-pccp06} C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang,   H.Petrova,   M. Reismann,   P. Mulvaney, and G. V. Hartland,  
%\textit{Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study},
Phys. Chem. Chem. Phys. \textbf{8}, 3540 (2006).


\bibitem{vallee-nl09} H. Baida, P. Billaud, S. Marhaba, D. Christofilos, E. Cottancin, A. Crut, J. Lerm\'{e}, P. Maioli, M. Pellarin, M. Broyer, N. Del Fatti, and F. Vall\'{e}e,
%\textit{Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO$_2$ Nanoparticles},
Nano Lett. \textbf{9}, 3463 (2009).



\bibitem{vanduyne-jpcc12} M. G. Blaber, A.-I. Henry, J. M. Bingham, G. C. Schatz, and R. P. Van Duyne,
%\textit{LSPR Imaging of Silver Triangular Nanoprisms: Correlating Scattering with Structure Using Electrodynamics for Plasmon Lifetime Analysis},
J. Phys. Chem. C \textbf{116}, 393 (2012).

\bibitem{vallee-nl13} V. Juv\'{e}, M. F. Cardinal, A. Lombardi, A. Crut, P. Maioli, J. P\"{e}rez-Juste, L. M. Liz-Marz\'{a}n, N. Del Fatti, and F. Vall\'{e}e,
%\textit{Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods},
Nano Lett.  \textbf{13}, 2234 (2013).

\bibitem{schatz-nl15} M. N. O'Brien, M. R. Jones, K. L. Kohlstedt, G. C. Schatz, and C. A.
Mirkin, 
%\textit{Uniform Circular Disks With Synthetically Tailorable Diameters: Two-Dimensional Nanoparticles for Plasmonics},
Nano Lett. \textbf{15}, 1012 (2015).



%%%%%%%%%%%%%%%%% LD Semi %%%%%%%%%%%%%%%%%%%%

\bibitem{kawabata-jpsj66} A. Kawabata and R. Kubo,  
%\textit{Electronic properties of fine metallic particles. II. Plasma resonance absorption}, 
J. Phys. Soc. Jpn. \textbf{21}, 1765 (1966).

\bibitem{lushnikov-zp74} A. A. Lushnikov and A. J. Simonov, 
%\textit{Surface plasmons in small metal particles},
Z. Physik \textbf{270}, 17 (1974).


\bibitem{schatz-jcp83} W. A. Kraus and G. C. Schatz,
%\textit{Plasmon resonance broadening in small metal particles},
J. Chem. Phys. \textbf{79}, 6130 (1983).

\bibitem{barma-pcm89} M. Barma and V. J. Subrahmanyam,
%\textit{Optical absorption in small metal particles},
J. Phys.: Cond. Mat. \textbf{1}, 7681 (1989).

\bibitem{yannouleas-ap92} C. Yannouleas and R. A. Broglia,  
%\textit{Landau damping and wall dissipation in large metal clusters},
Ann. Phys. \textbf{217}, 105 (1992).

\bibitem{eto-srl96} M. Eto and K. Kawamura,
%\textit{Theory of optical absorption in a small spherical shell: Iinvestigation of quantum-size effect},
Surf. Rev. Lett. \textbf{3}, 151 (1996).

\bibitem{uskov-plasmonics13} A. V. Uskov,  I. E. Protsenko, N. A. Mortensen, and E. P. O'Reilly,
%\textit{Broadening of Plasmonic Resonance Due to Electron Collisions with Nanoparticle Boundary: a Quantum Mechanical Consideration}, 
Plasmonics \textbf{9}, 185 (2013).

\bibitem{khurgin-oe15} J. B. Khurgin and G. Sun,
%\textit{Impact of surface collisions on enhancement and quenching of the luminescence near the metal nanoparticles}, 
Opt. Exp. \textbf{23}, 250905 (2015).

\bibitem{kirakosyan-prb16} A. S. Kirakosyan, M. I. Stockman, and T. V. Shahbazyan,  
%\textit{Surface plasmon lifetime in metal nanoshells},
Phys. Rev. B \textbf{94}, 155429 (2016).

%%%%%%%%%%%DFT
\bibitem{jalabert-prb02} R. A. Molina, D. Weinmann, and R. A. Jalabert,
%\textit{Oscillatory size dependence of the surface plasmon linewidth in metallic nanoparticles}, 
Phys. Rev. B \textbf{65}, 155427 (2002).

\bibitem{jalabert-prb05} G. Weick, R. A. Molina, D. Weinmann, and R. A. Jalabert,
%\textit{Lifetime of the first and second collective excitations in metallic nanoparticles},
Phys. Rev. B \textbf{72}, 115410 (2005).

\bibitem{yuan-ss08} Z. Yuan and S. Gao,
%\textit{Landau damping and lifetime oscillation of surface plasmons in metallic thin films studied in a jellium slab model}, 
Surf. Sci.  \textbf{602}, 440 (2008).

\bibitem{vallee-jpcl10} J. Lerm\'{e}, H. Baida, C. Bonnet, M. Broyer, E. Cottancin, A. Crut, P. Maioli, N. Del Fatti, F. Vall\'{e}e, and M. Pellarin,
%\textit{Size Dependence of the Surface Plasmon ResonanceDamping in Metal Nanospheres},
J. Phys. Chem. Lett. \textbf{1}, 2922 (2010).

\bibitem{lerme-jpcc11} J. Lerm\'{e},
%\textit{Size Evolution of the Surface Plasmon Resonance Damping in Silver Nanoparticles: Confinement and Dielectric Effects},
J. Phys. Chem. C \textbf{115}, 14098 (2011).


\bibitem{li-njp13} X. Li, Di Xiao, and Z. Zhang,
%\textit{Landau damping of quantum plasmons in metal nanostructures},
New J. Phys. \textbf{15}, 023011 (2013).


\bibitem{nordlander-acsnano14} A. Manjavacas, J. G. Liu, V. Kulkarni, and P. Nordlander,
%\textit{Plasmon-Induced Hot Carriers in Metallic Nanoparticles},
ACS Nano \textbf{8}, 7630 (2014).


%%%%%%%%%%%Scattering model 

\bibitem{kreibig-zp75} L. Genzel, T. P. Martin, and U. Kreibig, 
%\textit{Dielectric function and plasma  resonances of small metal particles},
Z. Phys. B \textbf{21}, 339 (1975).

\bibitem{ruppin-pss76} R. Ruppin and H. Yatom,
%\textit{Size and Shape Effects on the Broadening of the Plasma Resonance Absorption in Metals},
Phys. Status Solidi \textbf{74}, 647 (1976).

\bibitem{schatz-cpl83} W. A. Krauss and G. C. Schatz, 
%\textit{Plasmon resonance broadening in spheroidal metal particles},
Chem. Phys. Lett. \textbf{99}, 353 (1983).

\bibitem{schatz-jcp03}
E. A. Coronado and G. C. Schatz,
%\textit{Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach},
 J. Chem. Phys. \textbf{119}, 3926 (2003).

\bibitem{moroz-jpcc08} A. Moroz, 
%\textit{Electron Mean Free Path in a Spherical Shell Geometry},
J. Phys. Chem. C \textbf{112}, 10641 (2008).

\bibitem{kreibig-book} U. Kreibig and M. Vollmer,
{\em Optical Properties of Metal Clusters}  (Springer, Berlin, 1995).

\bibitem{mortensen-pn13} N. A. Mortensen, 
%\textit{Nonlocal formalism for nanoplasmonics: Phenomenological and semi-classical considerations},
Photonic. Nanostruct. \textbf{11}, 303 (2013).

\bibitem{mortensen-nc14} N. A. Mortensen,	S. Raza,	M. Wubs,	T. Sondergaard, and S. I. Bozhevolnyi, 
%\textit{A generalized non-local optical response theory for plasmonic nanostructures},
Nat. Commun. \textbf{5}, 3809 (2014).

\bibitem{landau}  L. D. Landau and E. M. Lifshitz, {\it Electrodynamics of Continuous Media} (Elsevier, Amsterdam, 2004). 

\bibitem{sm} T. V. Shahbazyan, 
%Landau damping of surface plasmons in metal nanostructures.
Phys. Rev. B \textbf{94}, 235431 (2016).

\bibitem{balian-ap70} R. Balian and C. Bloch, 
%\textit{Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surface},
Ann. Phys. \textbf{60}, 401 (1970).

\bibitem{shahbazyan-prl16} T. V. Shahbazyan, 
%\textit{Local Density of States for Nanoplasmonics}, 
Phys. Rev. Lett. \textbf{117}, 207401 (2016).

\end{thebibliography}



\end{document}

1703.05874.extracted_bib
{nordlander-nn15 M. L. Brongersma, N. J. Halas, and P. Nordlander, %Plasmon-induced hot carrier science and technology, Nat. Nanotechnol. 10, 25 (2015). %%%%%%%%%%%%% Hot Carriers
1703.05874.extracted_bib
{park-nl11 Y. K. Lee, C. H. Jung, J.Park, H. Seo, G. A. Somorjai, and J. Y. Park, %Surface plasmon-driven hot electron flow probed with metal-semiconductor nanodiodes, Nano Lett. 11, 4251 (2011).
1703.05874.extracted_bib
{melosh-nl11 F. Wang and N. A. Melosh, %Plasmonic energy collection through hot carrier extraction, Nano Lett. 11, 5426 (2011).
1703.05874.extracted_bib
{halas-science11 M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, %Photodetection with active optical antennas, Science 332, 702 (2011).
1703.05874.extracted_bib
{halas-nc13 A. Sobhani, M. W. Knight, Y. Wang, B. Zheng, N. S. King, L. V. Brown, Z. Fang, P. Nordlander, and N. J. Halas, %Narrowband photodetection in the near-infrared with a plasmon-induced hot electron device, Nat. Commun. 4, 1643 (2013).
1703.05874.extracted_bib
{lian-nl13 K. Wu, W. E. Rodriguez-Cordoba, Y. Yang, and T. Lian, %Plasmon-induced hot electron transfer from the Au tip to CdS rod in CdS-Au nanoheterostructures, Nano Lett. 13, 5255 (2013).
1703.05874.extracted_bib
{halas-nl13-2 M. W. Knight, Y. Wang, A. S. Urban, A. Sobhani, B. Y. Zheng, P. Nordlander, and N. J. Halas, %Embedding Plasmonic Nanostructure Diodes Enhances Hot Electron Emission, Nano Lett. 13, 1687 (2013). %%%%% Polarization dependence
1703.05874.extracted_bib
{clavero-np14 C. Clavero, %Plasmon-induced hot-electron generation at nanoparticle/metal-oxide interfaces for photovoltaic and photocatalytic devices, Nat. Photonics 8, 95 (2014).
1703.05874.extracted_bib
{brongersma-nl14 H. Chalabi, D. Schoen, M. L. Brongersma, %Hot-electron photodetection with a plasmonic nanostripe antenna, Nano Lett. 14, 1374 (2014).
1703.05874.extracted_bib
{atwater-nc14 R. Sundararaman, P. Narang, A. S. Jermyn, W. A. Goddard III, and H. A. Atwater, %Theoretical predictions for hot-carrier generation from surface plasmon decay, Nat. Commun. 5, 5788 (2014).
1703.05874.extracted_bib
{halas-nc15 B. Y. Zheng, H. Zhao, A. Manjavacas, M. McClain, P. Nordlander, and N. J. Halas, %Distinguishing between plasmon-induced and photoexcited carriers in a device geometry, Nat. Commun. 6, 7797 (2015). %%%Reactions
1703.05874.extracted_bib
{brongersma-nl11 I. Thomann, B. A. Pinaud, Z. Chen, B. M. Clemens, T. F. Jaramillo, and M. L. Brongersma, %Plasmon Enhanced Solar-to-Fuel Energy Conversion, Nano Lett. 11, 3440 (2011).
1703.05874.extracted_bib
{moskovits-nl12 J. Lee, S. Mubeen, X. Ji, G. D. Stucky, and M. Moskovits, %Plasmonic Photoanodes for Solar Water Splitting with Visible Light, Nano Lett. 12, 5014 (2012).
1703.05874.extracted_bib
{halas-nl13 S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V. Brown, J. Cheng, J. B. Lassiter, E. A. Carter, P. Nordlander, and N. J. Halas, %Hot Electrons Do the Impossible: Plasmon-Induced Dissociation of H$_{2$ on Au, Nano Lett. 13, 240 (2013).
1703.05874.extracted_bib
{moskovits-nn13 S. Mubeen, J. Lee, N. Singh, S. Kr\"{amer, G. D. Stucky, and M. Moskovits, %An Autonomous Photosynthetic Device in Which All Charge Carriers Derive from Surface Plasmons, Nat. Nanotechnol. 8, 247 (2013).
1703.05874.extracted_bib
{halas-jacs14 S. Mukherjee, L. Zhou, A. M. Goodman, N. Large, C. Ayala-Orozco, Y. Zhang, P. Nordlander, and N. J. Halas, %Hot-electron-induced dissociation of H$_{2$ on gold nanoparticles supported on SiO$_{2$, J. Am. Chem. Soc. 136, 64 (2014). %%%%%%%%%%%%%Size-shape reviews
1703.05874.extracted_bib
{halperin-rmp86 W. P. Halperin, Quantum size effects in metal particles, Rev. Mod. Phys. 58, 533 (1986).
1703.05874.extracted_bib
{kresin-pr92 V. V. Kresin, %Collective resonances and response properties of electrons in metal clusters, Phys. Rep. 220, 1 (1992).
1703.05874.extracted_bib
{vallee-jpcb01 C. Voisin, N. Del Fatti, D. Christofilos, and F. Vall\'{ee, %Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles, J. Phys. Chem. B 105, 2264 (2001).
1703.05874.extracted_bib
{schatz-jpcb03 K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, %The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment, J. Phys. Chem. B 107 668 (2003).
1703.05874.extracted_bib
{noguez-jpcc07 C. Noguez, %Surface plasmons on metal nanoparticles: The inﬂuence of shape and physical environment, J. Phys. Chem. C 111 3806 (2007). %%%%%%%%%%LD Exp
1703.05874.extracted_bib
{klar-prl98 T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, %Surface-Plasmon Resonances in Single Metallic Nanoparticles, Phys. Rev. Lett. 80, 4249 (1998). %%%%%%%%%% Nanorods
1703.05874.extracted_bib
{mulvaney-prl02C. S\"{onnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. V. Wilson, and P. Mulvaney, %Drastic reduction of plasmon damping in gold nanorods, Phys. Rev. Lett. 88, 077402 (2002). %%%%%%%%%%%%%%%Nanoshells
1703.05874.extracted_bib
{halas-prb02S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas, %Relative contributions to the plasmon line shape of metal nanoshells, Phys. Rev. B 66, 155431 (2002).
1703.05874.extracted_bib
{klar-nl04G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. A Klar, and J. Feldmann, %Gold nanoshells Improve Single Nanoparticle Molecular Sensors, Nano Lett. 4, 1853 (2004). %%%%%%%%%%%%%%%%%% LD single NP
1703.05874.extracted_bib
{vallee-prl04 A. Arbouet, D. Christofilos, N. Del Fatti, F. Vall\"{ee, J. R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, %Direct Measurement of the Single-Metal-Cluster Optical Absorption, Phys. Rev. Lett. 93, 127401 (2004).
1703.05874.extracted_bib
{halas-nl04C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J. Halas, and J. H. Hafner, %Scattering Spectra of Single Gold Nanoshells, Nano Lett. 4, 2355 (2004).
1703.05874.extracted_bib
{hartland-pccp06 C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H.Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, %Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study, Phys. Chem. Chem. Phys. 8, 3540 (2006).
1703.05874.extracted_bib
{vallee-nl09 H. Baida, P. Billaud, S. Marhaba, D. Christofilos, E. Cottancin, A. Crut, J. Lerm\'{e, P. Maioli, M. Pellarin, M. Broyer, N. Del Fatti, and F. Vall\'{ee, %Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO$_2$ Nanoparticles, Nano Lett. 9, 3463 (2009).
1703.05874.extracted_bib
{vanduyne-jpcc12 M. G. Blaber, A.-I. Henry, J. M. Bingham, G. C. Schatz, and R. P. Van Duyne, %LSPR Imaging of Silver Triangular Nanoprisms: Correlating Scattering with Structure Using Electrodynamics for Plasmon Lifetime Analysis, J. Phys. Chem. C 116, 393 (2012).
1703.05874.extracted_bib
{vallee-nl13 V. Juv\'{e, M. F. Cardinal, A. Lombardi, A. Crut, P. Maioli, J. P\"{erez-Juste, L. M. Liz-Marz\'{an, N. Del Fatti, and F. Vall\'{ee, %Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods, Nano Lett. 13, 2234 (2013).
1703.05874.extracted_bib
{schatz-nl15 M. N. O'Brien, M. R. Jones, K. L. Kohlstedt, G. C. Schatz, and C. A. Mirkin, %Uniform Circular Disks With Synthetically Tailorable Diameters: Two-Dimensional Nanoparticles for Plasmonics, Nano Lett. 15, 1012 (2015). %%%%%%%%%%%%%%%%% LD Semi %%%%%%%%%%%%%%%%%%%%
1703.05874.extracted_bib
{kawabata-jpsj66 A. Kawabata and R. Kubo, %Electronic properties of fine metallic particles. II. Plasma resonance absorption, J. Phys. Soc. Jpn. 21, 1765 (1966).
1703.05874.extracted_bib
{lushnikov-zp74 A. A. Lushnikov and A. J. Simonov, %Surface plasmons in small metal particles, Z. Physik 270, 17 (1974).
1703.05874.extracted_bib
{schatz-jcp83 W. A. Kraus and G. C. Schatz, %Plasmon resonance broadening in small metal particles, J. Chem. Phys. 79, 6130 (1983).
1703.05874.extracted_bib
{barma-pcm89 M. Barma and V. J. Subrahmanyam, %Optical absorption in small metal particles, J. Phys.: Cond. Mat. 1, 7681 (1989).
1703.05874.extracted_bib
{yannouleas-ap92 C. Yannouleas and R. A. Broglia, %Landau damping and wall dissipation in large metal clusters, Ann. Phys. 217, 105 (1992).
1703.05874.extracted_bib
{eto-srl96 M. Eto and K. Kawamura, %Theory of optical absorption in a small spherical shell: Iinvestigation of quantum-size effect, Surf. Rev. Lett. 3, 151 (1996).
1703.05874.extracted_bib
{uskov-plasmonics13 A. V. Uskov, I. E. Protsenko, N. A. Mortensen, and E. P. O'Reilly, %Broadening of Plasmonic Resonance Due to Electron Collisions with Nanoparticle Boundary: a Quantum Mechanical Consideration, Plasmonics 9, 185 (2013).
1703.05874.extracted_bib
{khurgin-oe15 J. B. Khurgin and G. Sun, %Impact of surface collisions on enhancement and quenching of the luminescence near the metal nanoparticles, Opt. Exp. 23, 250905 (2015).
1703.05874.extracted_bib
{kirakosyan-prb16 A. S. Kirakosyan, M. I. Stockman, and T. V. Shahbazyan, %Surface plasmon lifetime in metal nanoshells, Phys. Rev. B 94, 155429 (2016). %%%%%%%%%%%DFT
1703.05874.extracted_bib
{jalabert-prb02 R. A. Molina, D. Weinmann, and R. A. Jalabert, %Oscillatory size dependence of the surface plasmon linewidth in metallic nanoparticles, Phys. Rev. B 65, 155427 (2002).
1703.05874.extracted_bib
{jalabert-prb05 G. Weick, R. A. Molina, D. Weinmann, and R. A. Jalabert, %Lifetime of the first and second collective excitations in metallic nanoparticles, Phys. Rev. B 72, 115410 (2005).
1703.05874.extracted_bib
{yuan-ss08 Z. Yuan and S. Gao, %Landau damping and lifetime oscillation of surface plasmons in metallic thin films studied in a jellium slab model, Surf. Sci. 602, 440 (2008).
1703.05874.extracted_bib
{vallee-jpcl10 J. Lerm\'{e, H. Baida, C. Bonnet, M. Broyer, E. Cottancin, A. Crut, P. Maioli, N. Del Fatti, F. Vall\'{ee, and M. Pellarin, %Size Dependence of the Surface Plasmon ResonanceDamping in Metal Nanospheres, J. Phys. Chem. Lett. 1, 2922 (2010).
1703.05874.extracted_bib
{lerme-jpcc11 J. Lerm\'{e, %Size Evolution of the Surface Plasmon Resonance Damping in Silver Nanoparticles: Confinement and Dielectric Effects, J. Phys. Chem. C 115, 14098 (2011).
1703.05874.extracted_bib
{li-njp13 X. Li, Di Xiao, and Z. Zhang, %Landau damping of quantum plasmons in metal nanostructures, New J. Phys. 15, 023011 (2013).
1703.05874.extracted_bib
{nordlander-acsnano14 A. Manjavacas, J. G. Liu, V. Kulkarni, and P. Nordlander, %Plasmon-Induced Hot Carriers in Metallic Nanoparticles, ACS Nano 8, 7630 (2014). %%%%%%%%%%%Scattering model
1703.05874.extracted_bib
{kreibig-zp75 L. Genzel, T. P. Martin, and U. Kreibig, %Dielectric function and plasma resonances of small metal particles, Z. Phys. B 21, 339 (1975).
1703.05874.extracted_bib
{ruppin-pss76 R. Ruppin and H. Yatom, %Size and Shape Effects on the Broadening of the Plasma Resonance Absorption in Metals, Phys. Status Solidi 74, 647 (1976).
1703.05874.extracted_bib
{schatz-cpl83 W. A. Krauss and G. C. Schatz, %Plasmon resonance broadening in spheroidal metal particles, Chem. Phys. Lett. 99, 353 (1983).
1703.05874.extracted_bib
{schatz-jcp03 E. A. Coronado and G. C. Schatz, %Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach, J. Chem. Phys. 119, 3926 (2003).
1703.05874.extracted_bib
{moroz-jpcc08 A. Moroz, %Electron Mean Free Path in a Spherical Shell Geometry, J. Phys. Chem. C 112, 10641 (2008).
1703.05874.extracted_bib
{kreibig-book U. Kreibig and M. Vollmer, {\em Optical Properties of Metal Clusters (Springer, Berlin, 1995).
1703.05874.extracted_bib
{mortensen-pn13 N. A. Mortensen, %Nonlocal formalism for nanoplasmonics: Phenomenological and semi-classical considerations, Photonic. Nanostruct. 11, 303 (2013).
1703.05874.extracted_bib
{mortensen-nc14 N. A. Mortensen, S. Raza, M. Wubs, T. Sondergaard, and S. I. Bozhevolnyi, %A generalized non-local optical response theory for plasmonic nanostructures, Nat. Commun. 5, 3809 (2014).
1703.05874.extracted_bib
{landau L. D. Landau and E. M. Lifshitz, {Electrodynamics of Continuous Media (Elsevier, Amsterdam, 2004).
1703.05874.extracted_bib
{sm T. V. Shahbazyan, %Landau damping of surface plasmons in metal nanostructures. Phys. Rev. B 94, 235431 (2016).
1703.05874.extracted_bib
{balian-ap70 R. Balian and C. Bloch, %Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surface, Ann. Phys. 60, 401 (1970).
1703.05874.extracted_bib
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