Relaxation of Plasmon-Induced Hot Carriers - ACS Photonics (ACS

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Relaxation of Plasmon-Induced Hot Carriers Jun Liu, Hui Zhang, Stephan Link, and Peter Nordlander ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00881 • Publication Date (Web): 08 Dec 2017 Downloaded from http://pubs.acs.org on December 12, 2017

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Relaxation of Plasmon-Induced Hot Carriers Jun G. Liu,1,2 Hui Zhang,1,3 Stephan Link,1,3,4 and Peter Nordlander1,2,3* 1 2 3

Laboratory for Nanophotonics

Department of Physics and Astronomy, MS 61

Department of Electrical and Computer Engineering, MS 366 4

Department of Chemistry, MS 60

Rice University, Houston, Texas 77005, United States *Corresponding author: [email protected]

Plasmon-induced hot carrier generation has attracted much recent attention due to its promising potential in photocatalysis and other light harvesting applications. Here we develop a theoretical model for hot carrier relaxation in metallic nanoparticles modeled using a fully quantum mechanical jellium model. Following pulsed illumination, non-radiative plasmon decay results in a highly nonthermal distribution of hot electrons and holes. Using coupled master equations, we calculate the time-dependent evolution of this carrier distribution in the presence of electron-electron, electron-photon, and electron-phonon scattering. Electron-electron relaxation is shown to be the dominant scattering mechanism and results in efficient carrier multiplication where the energy of the initial hot electron-hole pair is transferred to other multiple electron-hole pair excitations of lower energies. During this relaxation, a small but finite fraction of electrons scatter into luminescent states where they can recombine radiatively with holes by emission of photons. The energy of the emitted photons is found to follow the energies of the electrons and thus redshifts monotonically during the relaxation process. When the energies of the electrons approach the Fermi level, electron-phonon interaction becomes important and results in heating of the nanoparticle. We generalize the model to continuous-wave excitation and show how nonlinear effects become important when the illumination intensity increases. When the temporal spacing between incident photons is shorter than the relaxation time of the hot carriers, we predict that the photoluminescence will blueshift with increasing illumination power. Finally we discuss the effect of the photonic density of states (Purcell factor) on the luminescence spectra.

Keywords: Plasmon, hot carriers, photoluminescence, electron-electron scattering, Purcell effect

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Localized surface plasmon resonances (LSPRs), the collective and coherent oscillations of conduction electrons in metallic nanoparticles (NPs) have attracted considerable recent interest due to their significant light harvesting capabilities1–4 and ability to induce large local near-fields in hot-spots outside the NP surface.5–8 The LSPR energies depend on the geometry of the NP,9–11 and can readily be tuned to desired energies such as excitation energies of nearby molecules.12,13 In such a situation, the LSPR can serve as a nanoscale lens and drastically enhance the rate of electronic or vibrational transitions in nearby atoms and molecules.14 Another attractive property of LSPRs is their ability to generate hot electrons (HE) and hot holes (HH) through nonradiative plasmon decay.15 Due to the large cross section of plasmon excitations, which can exceed the geometrical cross section of the NP by several orders of magnitudes, LSPR excitation is a much more efficient process for hot carrier (HC) generation than direct excitation of metal specific interband transitions.16 Such HCs can be utilized in a variety of light harvesting applications such as photodetection,17–20 or photocatalysis.21–28 In photodetection, a plasmonic antenna is placed near an acceptor such as a semiconductor.29 Following plasmon excitation HEs can be injected into the conduction band of the acceptor and detected as a photocurrent with a spectral responsivity determined by the plasmonic antenna.17 In photocatalysis, HCs can transfer into electronic states of a nearby molecule and induce bond breaking or desorption.22,30 HCs are also believed to play a significant role in the photoluminescence (PL) of plasmonic NPs.31 While numerous experimental studies have explored applications of plasmon induced HCs, much less is known about the fundamental mechanisms for HC generation and relaxation.32–34 A recent study17 showed that the distribution of HCs created by plasmon decay can be very different than HCs generated by direct excitation. In both cases, the energy of the initial electron-hole pair is equal to the photon energy but for plasmon decay, the electrons tend to have higher energies than for direct excitation which produces relatively low energy electrons but more energetic holes.19 Another difference is that plasmon induced HCs are created by the plasmonic near-field24 which contains higher multipolar components while direct excitation results in dipolar transitions which in principle can decay radiatively without scattering. While the instantaneous generation of HCs from plasmon 2


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decay has been studied recently,16,35,36 much less is known about the relaxation and lifetimes of such carriers. Since the ultimate efficiencies of HC induced processes are expected to strongly depend on HC lifetimes, the mechanisms by which HCs relax are clearly a critical issue to resolve. These relaxation dynamics are less important in photodetection applications where the HCs typically transfer quickly directly into an acceptor with a continuum of empty states and thus are unlikely to revert back to the NP even in the case of no bias or Schottky barrier. It is especially important to understand HC relaxation for catalytic reactions on the surface of the NPs since charge transfer and changes to bonding leading to chemical transformations typically occur on the femtosecond to picosecond timescales. To maximize quantum efficiencies for catalytic reactions HC lifetimes need to be comparable and ideally even exceed the time for a chemical transformation to occur. The dynamics of the HCs is also important for the understanding of the PL in plasmonic NPs. Since HCs are in general produced from dark transitions induced by the plasmonic near-field, PL can only appear after the electrons and/or holes have undergone electron-electron scattering and end up in states that can decay into the far-field through dipolar transitions. Several studies have made significant progress elucidating the role of

electron-electron and

electron-phonon scattering during HC relaxation.37,38 Brown et al. used a first principle atomistic approach and clarified the role of NP surface and geometry.33 Saavedra et al. and Govorov et al. investigated the effects within the independent-electron picture.39,40 All of these methods can predict the lifetime of individual HC but require exact matrix element calculations which involve a 4-fold wavefunction integration. The numerical complexity of such calculations is very severe thus limiting such approaches to only very large nanoparticles, where a bulk formalism can be used, or to very small nanoparticles where the plasmon resonances are not fully formed.41 In a previous publication, we introduced a fully quantum mechanical jellium model for the modeling of HC generation from plasmon decay.16 The jellium model has a long and successful history in plasmonics and has in many cases been shown to provide quantitative agreement with experimental results.42 The simplifications introduced in the jellium model allow the approach to be extended to large realistically sized nanoparticles where the optical resonance can be characterized as plasmonic. Here we extend our model to study the relaxation dynamics of such carriers. Specifically, we assume a pulsed excitation of the plasmon and study how the HC distribution generated from plasmon 3


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decay relaxes toward equilibrium. The relaxation mechanisms include electron-electron scattering and electron-photon scattering (radiative decay through emission of a photon) and electron-phonon scattering (nonradiative decay through local heat generation). The electron-electron and electron-photon scattering matrix elements are calculated numerically using our quantum mechanical model but the electron-phonon scattering is described using empirical parameters. Because of its complexity, our quantum mechanical model is limited to systems containing less than 1000 electrons. To extend its applicability to more realistically sized NPs, we introduce a parameterization of the electron-electron scattering process, in which the rates are assumed to be a function of the energy difference between scattering states. Using this parameterization, the theory can be applied to larger more realistic systems provided the electronic energy bands are given. Using this model we employ coupled master equations to calculate the instantaneous energy distribution of plasmon-induced HCs as a function of time following a pulsed plasmon excitation. As some of the HCs scatter into luminescent states, they can recombine radiatively and photons are emitted by spontaneous emission. As the energies of the HCs decrease, the instantaneous PL redshifts. The lifetimes of the HCs can be extracted directly from the time-dependence of the carrier distributions. Finally we generalize the results to continuous wave excitation and show that when the temporal spacing between incident photons becomes smaller than the HC relaxation time, the PL blueshifts with increasing light intensity, due to a multi-photon effect.

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Figure 1. Plasmon-induced HC generation in a silver jellium NP and decay mechanisms. (a) Schematic representation of the system under study. We consider a jellium nanosphere of diameter D. We model the conduction electrons of this structure as free electrons of a density corresponding to Ag in a finite spherical potential well of depth V0 and radius D/2. (b) HC distribution immediately after plasmon decay in a Ag jellium sphere with D=6 nm. The number of HEs (red) and HHs (blue) per unit volume are plotted as a function of energy relative to the Fermi level. The incident photon energy is 3.5 eV (resonant excitation). The illumination power is 1 mWμm-2. (c) Illustration of the decay channels for HC relaxation: electron-electron; electron-photon; and electron-phonon scattering.

Hot carrier relaxation: Exact matrix elements Our initial system is a D=6 nm diameter jellium nanosphere with parameters as appropriate for

Ag: work function 4.5 eV; a dielectric background screening from the ions of =4.18; and an

electron density corresponding to a Wigner Seitz radius of 3 Bohr corresponding to a bulk plasmon frequency =

,

where is the electron charge, the electron mass and the electron

density. For a Ag nanosphere, the energy of the plasmon resonance is lower than the interband transition threshold and the plasmon is well described by a screened jellium model.16 The plasmon is resonantly excited by a laser pulse as illustrated in Fig 1a. As discussed in our previous work,16 the

independent electron approximation can be used to model plasmon decay. The eigenenergies and eigenstates |⟩ are obtained analytically from the radial Schrödinger equation.

HC generation following plasmon decay is calculated using the formalism introduced in the previous publication.16 Briefly, the incident light is assumed to be dipolar with a spatial

dependence = − , where is the amplitude of the electric field, the radial position, and

the polar angle. The rate of electron-hole pair generation (excitation from state |⟩ to |!⟩) is

calculated as:

1 4 . "→$ = '⟨'(V+ + )'!⟩' . (1) ( − $ + ). + ℏ. & 1. &

In this expression V+ is the plasmon-induced electric field: V+ (, ) = 4(5)7. 4(5)16

NP and 5< 5(57=)

4(5)16 9 8 /(8 . ) 4(5)7.

inside the

outside the NP calculated using the Drude model: () = −

with parameters appropriate for Ag: a corrected damping accounting for the finite size effect

@ > = > + ℏA/. with > = 0.06 and vE = 1.4 × 10G cm/s is the Fermi velocity.43 The incident

?

light frequency is and the lifetime & refers to the natural linewidth of the excitation and is chosen

so long that the results for the initial carrier generation after plasmon decay do not change with further 5


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increases of & (& = 1 H). It is important to note that this quantity does not refer to the lifetime of the HCs which will be calculated explicitly below by considering electron-electron scattering. The initial HC distribution after illumination with a light pulse of duration Δt (10 K) is:

L (0) = K( ) − M "→$ K( ) N1 − KO$ PQ ΔR + M "$→ KO$ PO1 − K( )PΔR , (2) $

$

where L is the probability for occupation of the -th state and K( ) is the Fermi-Dirac distribution. Since we will focus here on the initial relaxation (< 2 ps) where the carriers still remain excited, we assume a zero background temperature. The incident photon energy is assumed resonant with the Ag plasmon energy of 3.5 eV. Fig. 1b shows the HC distribution immediately after pulsed excitation calculated using Eq. (2), which reveals similar features as what has been obtained using different theoretical approaches.36,44 After excitation, the HCs start to relax through electron-electron, electron-photon and

electron-phonon scattering as depicted in Fig. 1c. The time-dependent occupations L (R) of the electronic states are obtained from the coupled master equations: TL (R) = − M "$→UV L L$ (1 − LU )(1 − LV ) + M "UV→$ LU LV (1 − L )O1 − L$ P TR $UV

$UV

− M "→$ L O1 − L$ P + M "$→ L$ (1 − L ) $

=

$

=

− M "→$ L O1 − L$ P + M "$→ L$ (1 − L ). (3) $

+

$

+

The right hand side of Eq. (3) is the sum of all transitions in and out of state i. The first two terms

are the electron-electron scattering contributions which due to the size of the scattering rate "$→UV is

the dominant scattering mechanism during the relaxation process.

The third and fourth terms

represent radiative recombination, i.e. PL. The electron-photon rate "→$ is around six orders of =

magnitude smaller than the electron-electron scattering rate. The last two terms are the contributions from electron-phonon coupling "→$ which occurs over a slower picosecond timescale.45–47 While no +

conclusive experimental measurements of the timescales for HC relaxation has been reported so far, it is generally believed that electron-electron interactions occur on a timescale of 200 ~ 500 K45,48–51 and that electron-phonon coupling to equilibrate electron and lattice temperatures takes a few 6


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picoseconds45,52–54 in gold and silver. Thus, the energetic HCs will be quickly thermalized through electron-electron scattering during the first picosecond after which electron-phonon scattering becomes the dominant relaxation channel. In this work, we focus on the dynamics of the energetic

HCs and parameterize the electron-phonon interaction using the empirical relaxation time 1/Γ + ≈

1 H,45,54 which is much longer than individual electron-electron scattering events on the order of

10 K .33 Since the radiative decay rate is around 6 orders of magnitude smaller than the electron-electron interaction rate, the relaxation dynamics during the first picoseconds is essentially independent of electron-photon and electron phonon interactions and determined by electron-electron scattering. The electron-electron scattering rate is calculated from the electronic wavefunctions of the NP and can be expressed as:

1 4 . Γ$→UV = 'M$→UV ' (4) ( + $ − U − V ). + ℏ. & 1. &

where |, !⟩ and |], ^⟩ are the initial and final states, respectively. M$→UV = . |⟨, !||], ^⟩ − ⟨, !||^, ]⟩| is the transition matrix element, and N_, _ ′ Q =

′' `ab@ '_`_′

′' '_1_′

6

with ]cE ≈ 1 dℎ (Ag)

is the Yukawa potential resulting from Thomas-Fermi screening of the Coulomb potential. The matrix element M$→UV is an ℝg integral and very time-consuming to implement which limits the exact

approach to systems less than a thousand electrons. For most first principle calculations for bulk systems, electron-electron interaction is given by the imaginary part of the quasiparticle self-energy using Bloch states.33 However, in a finite nanoparticle a real-space approach is more suitable. The effect of screening is a reduction of the magnitude of the matrix element. The screening of the Coulomb interaction plays a very important role for the matrix element. For an unscreened Coulomb interaction, the matrix elements would typically be two orders of magnitude larger resulting in much shorter HC lifetime distributions. In the supplementary information we show the effects of different approaches to screening on the HC distribution relaxation. Using the bare

Coulomb potential but restricting the spatial integrations to |h − hi | < 1/]cE decreases the lifetimes by 20%. The screening due to d-electrons reduce the matrix element by a multiplicative factor

1/(k − 1) where k − 1 is the polarizability of the lattice. For Ag, where k = 5 this effect is substantial and increases the lifetimes by a factor of 3.3. However since the d-electrons are not 7


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explicitly accounted for in a jellium model, we will neglect their contribution to screening in the present study. The PL is believed to include many complex processes. A recent study attributes emission to electronic Raman scattering.55 While we are not ruling out this mechanism, in the present paper we show that PL can also be generated from Purcell enhanced radiative recombination of plasmon induced HCs. The HCs formed by plasmon decay are generated by the inhomogeneous internal electric field of the nanoparticle. This internal field is a superposition of multipolar components. While the bright dipolar HCs can directly decay radiatively, the dark HCs need to interact and scatter into bright states before radiative recombination becomes possible. This electron-electron scattering is the dominant HC relaxation mechanism and results in redshifted emission. The instantaneous PL spectrum at each time step is calculated as: lmn (, R) =

M "$→ (ω)L (R)(1 − L$ (R)) , (5)

pq 1pr s5

=

where "$→ is the spontaneous emission rate given by:56 =

"$→ = =

√9 v|⟨||!⟩|. (6) 3uℏ 9

where is the speed of light, and v = w

9 . w .7p

is the Purcell factor.57 As will be demonstrated below,

the Purcell factor plays a very important role for the shape of the PL spectrum. When an electronic transition coincides with a peak in the photonic density of states (PDOS), the emission can be strongly enhanced. In Fig. 2 we show the instantaneous HC distribution and luminescence following a pulsed laser excitation of the plasmon in a small nanosphere calculated using numerical integration of the electron-electron scattering matrix element. For this system, plasmon decay predominantly results in cold holes and HEs in two states located around 1 and 2 eV above the Fermi level. Within the first picosecond, the electrons in the 2 eV state begin to scatter into the 1 eV state and to some extent, into a state located around 1.5 eV. This is shown in Fig. 2 as a reduction in the population of the 2 eV state and an increased population of the 1 eV state. The population of the 1.5 eV state remains relatively low. As the electrons scatter into these lower energy states, the PL spectrum changes. The initial PL 8


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spectrum is dominated by two peaks: a 1.5 eV photon and a 3 eV photon corresponding to transitions of the 1 and 2 eV HEs into holes below the Fermi level. As the 2 eV HEs scatter into the 1 eV state the intensity of the 3 eV photon becomes smaller while the intensity of the 1.5 eV PL peak increases.

Figure 2. Instantaneous HE (red), HH (blue) distribution, and PL for a Ag jellium sphere of D=1.5 nm (100 electrons) illuminated for 10 fs with light intensity of 1 mW/μ. . The colored solid lines are the results using the numerically calculated matrix elements Eq. (4) and the dotted lines are for the parameterization Eq. (7). The inset shows the carrier distribution immediately after plasmon decay.

The small system described in Fig. 2 exhibits significant quantum size effects and has a very granular electronic structure which would be absent in the much larger NPs typically used in experiments. The extension of our approach to realistically sized nanoparticles poses a significant theoretical challenge. The initial dynamics (< 2 ps) is determined by electron-electron scattering M$→UV in Eq. (6) which is a six dimensional spatial integral and very time-consuming to compute. In

addition, the number of matrix elements scales as the fourth power of the number of electrons x . To develop a more general approach suitable for larger NPs as well as nanostructures of different shapes, we introduce a statistical parameterization procedure for the electron-electron interaction matrix elements.

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Figure 3. Parameterization of electron-electron interaction. (a) Scattering matrix elements as a function of α: exact matrix elements calculated numerically (green dots); averaged over an interval of 80 meV (purple dots); parameterization Eq. (7) for the system used in Fig. 1 (red line). (b) Matrix element parameterization coefficients as a function of x . The dots are obtained from the exact matrix elements and the solid lines are the fits to Eq. (8). The diameters of the spheres are 1.5, 1.85, 2.12, 2.34, 2.52, 2.68, and 2.82 nm.

Hot carrier relaxation: extrapolation to large systems Our basic assumption is that the parameterized electron-electron scattering matrix elements only depend on the energies of the electronic states. Theoretically the matrix elements depend on both energy levels and other spatial quantum numbers such as ^, for spherical particles. However, as

will be demonstrated below by explicit comparison, the overall HC relaxation rates calculated using the exact (non parametrized) and the parametrized matrix elements agree remarkably well. Only with such parametrized matrix elements it becomes possible to model HC relaxation in large relevant nanoparticles. We emphasize that our parametrization only refers to the HC distribution, not to the scattering of individual HCs. For a jellium sphere, the quantum numbers labeling the states are (, ^, ) with m referring to

the azimuthal orientation of the wavefunction. Thus the degeneracy of a state of energy ,V is 2m+1. We average all matrix elements y$→UV with the same state energies (i.e., , $ , U , V ), so that the 10


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dependence is removed.

After carefully examining the calculated averaged transition matrix

elements between the different states, we find that the majority of the matrix elements can be expressed as a function of a parameter z = { Nw'pq 1pr '1|pa 1p| |wQ , which can also be expressed as

z = 'O , $ P − (U , V )' = '}~O , $ P − }~(U , V )' using energy conservation. This ansatz is physically reasonable because the scattering depends on the energy difference between initial

and final states, but not on the absolute value of the individual energy levels. The argument z of the

Gaussian is carefully chosen to satisfy three criteria, (i) z will be expressed by the four energies for the four states (from , ! to ], ^); (ii) z should keep the symmetry of the matrix element, i. e. 'y$→UV ' =

'y$→UV ' = 'y$→VU ' = 'yUV→$ '; (iii) z will be able to show the amount of energy difference during the electron-electron interaction, typically the difference between two initial states and two final states. Additionally, the matrix elements are also proportional to the density of states of the four energy states.

Fig. 3a shows the numerically calculated scattering matrix elements as a function of α (green dots). There are several different matrix elements with the same α. Most of the matrix elements are very small but for a given α, the matrix elements range from small to a maximum. The area filled by the green dots is bell shaped and suggests that a Gaussian fit of the matrix elements may be appropriate. To see this more clearly, we uniformly divide the α range into small intervals (80 meV) and average the numerically calculated matrix elements within each interval. These averages are the blue points in Fig. 3a and clearly suggest a Gaussian line shape. The averaged matrix elements are then parameterized as: 1N 'M$→UV ' (z) = }T T$ TU TV e .

1 Q , (7)

with }, , being three fitting parameters. Here T is the electronic density of states (EDOS) for the

states = , !, ], ^. For a small system, we use single electron states, i.e.

with more continuous energy levels, T would be the appropriate EDOS.

T = 1. For larger systems

The dotted lines in Fig. 2 show the relaxation dynamics for the small x = 100 electron system

calculated using Eq. (7) with: } = 0.0364 . ; = 1.16 ; and = 0.59 . The colored lines represent the calculations using the exact transition matrix elements. As can clearly be seen, the agreement is almost perfect suggesting that our parameterization of the electron-electron scattering matrix elements captures the essential physics. In order to extend our parametrization process to larger 11


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systems, we now investigate how the three parameters a, b, and c depend on the size of the NP. To do this, we follow the procedure discussed above and perform exact calculations of the electron-electron scattering matrix for nanospheres with x = 200 − 700 electrons and diameters ranging from 1.5 to 2.82 nm. The calculated parameters a, b, and c are the dots in Fig. 3b. Their x dependence can be well described by the following expressions:

3.576 . x 0.4068x + 199.6 (x ) = eV, (8) x + 106.6 c(x ) = 0.645 eV

}(x ) =

The fits are shown as lines in Fig. 3b and reproduce the matrix element parameterization very well. We now assume that our fits can be extrapolated to more realistic NPs. Figure 4 shows the evolution of the HC distribution and PL following pulsed illumination of a D=6 nm Ag jellium sphere containing 6750 electrons. Compared with the smaller system discussed in Fig. 2, the HC distribution immediately following plasmon decay exhibits more pronounced lower energy peaks close to Fermi level. The reason for these peaks is that the plasmon decay transition matrix elements are relatively large for states close to the Fermi level and the larger size of the NP results in a more homogeneous electronic structure with a higher density of states in this energy range. The left column shows snapshots of the HC distributions from 100 fs to 1 ps (the HC distribution immediately after plasmon decay is shown in Fig. 1b). Similar to the smaller system in Fig. 2, the HCs gradually relax to lower energies. Importantly, these results clearly demonstrate carrier multiplication: During the relaxation process, the initially created HE loses energy by exciting an another electron with an energy closer to the Fermi level thereby transferring some of its energy to a new less energetic electron-hole pair that subsequently also will relax through electron-electron scattering. After around 1 ps, the carrier distribution can be characterized as two distinct peaks above and below the Fermi level. However, we note that the carriers still are relatively hot with average energies of a few 100 meV and that the total energy deposited by plasmon excitation has not yet been dissipated to the lattice. While the timescale for an individual electron-electron scattering event is very fast (a few fs), the electrons that emerge on a much longer, hundreds of femtoseconds timescale from many scattering events are still hot. The lifetime of the HC distribution is therefore significantly 12


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longer than the lifetime of a distinct HC state. The carriers have to undergo hundreds of scattering events to reach a thermal equilibrium distribution according to Fermi-Dirac statistics. Based on a metric that 99% hot carriers have reach an energy within 1 eV of the Fermi level during the evolution of the carrier distributions in Fig. 4, we deduce a qualitative HC distribution lifetime of around 1 ps for the 6 nm NP. This lifetime is shorter than the 2 ps lifetime inferred from the evolution of the carrier distribution for the 1.5 nm NP in Fig. 2. The HC decay rates in larger NPs are expected to be larger than those in smaller systems with a granular density of states and reduced phase space for the scattered electrons. Several previous theoretical studies have discussed the thermalization time. For individual HCs, Brown et al. reported the relaxation time is around 10 fs, which is much smaller than the thermalization of the entire energy states.33 Saavedra et al. also reported a collective relaxation time of about 1~3 ps with the help of master equations.39 Several experiments have probed the thermalization time due to electron-electron scattering.32,45,48–51,58 For example, Sun et al. reported 500 fs for a gold film,49 and Fann et al. reported a relaxation time ranging from 730 fs to 1 ps.58 Voisin et al. reported a value of 250 fs for Ag nanospheres with diameter of 6 nm.51 Our calculations match these measurements quite well, further supporting the validity of our parameterization of the electron-electron interactions in Eqs. (7) and (8).

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Figure 4. Instantaneous HC distribution (left) and PL (right) at 100, 250, 500, and 1000 fs after pulsed excitation of the plasmon in a D=6 nm Ag sphere. The organization of the figure is as in Fig. 2. The incident photon energy is 3.5 eV (resonant excitation). The illumination power is 1 mWμm-2 and the pulse duration is 10 fs.

The PL is shown in the right column of Fig. 4 and is dominated by two peaks: one from around the plasmon energy at 3.5 eV and one from lower energy carriers around 0.5 eV. The 3.5 eV feature corresponds to transitions of HEs to cold holes just below the Fermi level. The 0.5 eV feature corresponds to transitions from cold electrons and holes that artificially accumulate around the Fermi level because of our rough approximation for electron-phonon scattering. As time passes, the HEs keep interacting with electrons of lower energies, leading to the relaxation of HCs and an overall redshift of the PL. Eventually only the lower-energy peak remains.

Hot carrier distribution relaxation The lifetime of HCs plays a crucial role in many applications. As discussed above, the timescale for electron-electron scattering is very short (a few fs) but the electrons that emerge from scattering remain hot. Thus several hundreds of collisions are needed to thermalize the hot electrons. In order to quantify the hot electron relaxation dynamics, we group the electronic states based on the HE 14


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distribution peaks above the Fermi level into nonoverlapping ~0.5 eV energy intervals beginning at 0.4 eV above the Fermi energy. These energy parameters are chosen to maximize the overlap with the most prominent peaks in the instantaneous HC distributions. Fig. 5a shows the HC distribution immediately following plasmon decay for the same system as in Fig. 4. The colored regions indicate the energy intervals that are monitored. As the carrier distribution is propagated forward in time, the number of electrons within each interval is recorded. Fig. 5b illustrates how the number of hot electrons in each interval evolves in time. The hottest electrons (dark blue interval) exhibit a monotonous decay as a function of time. The number of electrons in the slightly lower energy interval (purple) initially

Figure 5. HCE relaxation. (a) HC distribution immediately after plasmon excitation. The energy intervals are 0.4-0.9 eV (green), 0.9-1.6 eV (yellow), 1.6-2.3 eV (purple), and 2.3-3.0 eV (dark blue). (b) Number of HEs per unit volume in the different energy intervals defined in (a) as a function of time. The system is the same as that in Fig. 4.

increases due to the influx of “blue” electrons but eventually exhibits a monotonous decay. The number of electrons in the next lower energy interval (yellow) exhibits a more complicated behavior with two different timescales. The initial fast decrease is due to the relatively large number of such electrons formed directly by plasmon decay. At around 0.5 ps, the decay rate of these electrons decreases because of the influx of electrons scattering out of the higher energy states. As can be 15


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expected, the number of electrons in the lowest energy interval (green) also exhibits multiple timescale dynamics since they can be populated by scattering from many higher energy states. The instantaneous population increases to a maximum around 0.5 ps and then undergoes a relatively slow relaxation. Even at 2 ps, there is a significant fraction of HCs in this energy interval (0.4-0.9 eV). Although not as energetic as the “blue” electrons, their energy is still well above thermal energy.

We

note that our present model neglects electron-phonon interactions which become the dominant scattering mechanism when the energies of the HEs start to overlap with phonon modes, i.e. after 2 ps. One approach for quantifying the lifetimes in the present approach is to determine the time when the number of HEs in each energy interval is reduced to 1/e of its initial value. This procedure results in lifetimes of 0.8, 2.0, and 1.5 ps for the “blue”, “purple”, and “yellow” electrons respectively. Since almost no “green” electrons are generated by plasmon decay, our procedure for lifetime extraction cannot be applied here. If instead we define the lifetime as the time for the population to reach 1/e of its maximum value, we get a lifetime of 2.5 ps. Since the timescale for individual electron-electron scattering events is of the order of a fs, the results in Fig. 5b clearly show that multiple events are involved in the HC relaxation. Fig. 5b also suggests that the lifetimes for HCs decrease with increasing energy. While this behavior is expected from Fermi liquid theory, this assumption cannot be justified for structures where the plasmon energy is similar to the work function of the metal. For such systems the HEs are delocalized with wavefunctions that extends away from the NP with very little overlap with the remaining conduction electrons. Such a situation applies for intermediate sized aluminum NPs where the combination of large bulk plasmon energy and a low work function allows for the placement of the plasmon energy just below the vacuum level. The discussions above focused on pulsed excitation to clearly illustrate the dynamics of the HC relaxation. We now turn to continuous wave (CW) excitation. For such illumination, the HC distribution will be power dependent. In contrast to fermionic excitations where the Pauli principle restricts the population of quantum states to a single occupation, plasmons are bosonic excitations which allow for multiple occupations of the same mode. During CW illumination, the occupation number of the plasmon will reach a steady state value where the damping rate is equal to the excitation rate. If the time between successive plasmon excitation events is shorter than the timescale for HC relaxation, the steady state HC distribution will be an average of the carrier distributions 16


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remaining from previous plasmon excitation events as shown in Fig. 4. As the illumination power increases, the time between excitation events decreases and the average carrier distribution will be skewed to higher energies. While this effect clearly is important for light harvesting applications such as photodetection and photocatalysis it is also important for understanding the power dependence of PL.

Figure 6. Absorption (top), PDOS (middle) and PL (bottom panel) from a 6 nm Ag sphere (red) and a 10x70 nm nanorod (blue). The absorption above 2.5 eV is multiplied by 10. The yellow dashed lines indicate the dipolar and octupolar nanorod resonances, while the black line represents the dark mode. The green line denotes the dipole resonance of the sphere as well as the transverse mode of the nanorod. The PL calculation was for an incident electric field perpendicular to the nanorod with a photon energy 3.6 eV. The insets are the charge plots for the different nanorod modes. For Ag we used Palik’s dielectric data.59

Photoluminescence Before discussing the power dependence of the PL, it is appropriate to discuss the effect of the

Purcell factor v in Eq. (6). This factor is the enhancement of the photonic density of states (PDOS)

for photons and plays a significant role in shaping the emission spectrum. For a low loss plasmonic material like Ag, the Purcell factor can be large. In general, both bright and dark plasmonic modes can

contribute strongly to the Purcell factor since v represents the emission caused by a local dipole. For small spherical NPs the plasmon resonances are determined by material properties and the PL 17


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spectrum is not tunable. However, metallic nanorods are highly tunable by simply changing their aspect ratio. In Fig. 6, we show the normalized absorption, PDOS and the steady state PL spectrum for the same Ag nanosphere as in Fig. 4 and a 10x70 nm Ag nanorod under CW illumination with 3.6 eV photons. The electronic states of the nanorod are calculated the same way as for the nanosphere. For the present small Ag NPs, only the bright dipolar modes (green line) are visible in the absorption spectrum. Due to the very small intrinsic damping of Ag at 3.6 eV, the dipolar Purcell factor is very

large v=25. The PDOS are surface averaged values calculated using the boundary element method (MNPBEM package).60–62 All modes in the system can contribute to the PDOS and the Purcell factor. The vertical dashed lines represent the bright dipolar ( v = 60, yellow line) and octupolar

(v = 7.5, yellow line) and dark quadrupolar (v=13, black line) nanorod modes. The nanorod PL

was calculated for transverse excitation polarization. The highest energy resonance corresponds to the transverse dipolar nanorod mode which has a similar energy as the sphere dipole. The PL spectra clearly look very different from the absorption spectra because of their dependence on the PDOS. For instance, dark modes such as the quadrupole could in principle show up in PL spectra. PL can occur when HEs scatter into bright electronic states, i.e. states which can relax radiatively through dipolar transitions as described by Eq. (6). Thus if successive plasmon excitations occur before the HCs have enough time to relax, one would expect a blueshift of the steady state HC distribution and hence a blueshift of the PL spectra. Since the temporal spacing between the incident photons is inversely proportional to the incident power, this blueshift will be power dependent. When incident power is reduced and the time interval between consecutive plasmon excitations becomes longer than the lifetime of the HCs, the excitation-decay events would be independent of each other resulting in PL spectra with fixed features but with an intensity that decreases linearly with incident power. Both of these predictions are seen experimentally in the PL spectra of gold nanorods.63 This discussion shows that CW excitation in principle could be used to probe the ultrafast electronic relaxation dynamics. However, it is of course not as precise as ultrafast measurements since the temporal spacing between consecutive photons always fluctuates.64 For a laser intensity of

0.1~1mWμm1., a NP with radius of 10 nm, the temporal photon spacing can be as long as 0.5~5 ps, which is of the same order as the time scales for HC relaxation in Fig. 5. We assume that the temporal spacing between consecutive plasmon excitations is inversely proportional to the incident light 18


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intensity I according to ∆R = , where ∆R is the temporal photon spacing, β =

ℏ5

is a

proportionality constant that depends on the photon energy (ℏ) and the absorption cross section of the NP.

The detailed modeling of the PL process would require the inclusion of the band structure of the metal. Since a realistic band structure cannot be directly incorporated within a jellium model, we here focus on the average HC distribution energy,

T (R, ) 〉 , (9) () = 〈 T (R, )

where 〈 〉 refer to the steady state time average for an incident power of . Since the PL emission spectrum reflects the HC distribution, the steady state average PL emission energy will reflect this quantity. The power dependence of () can be understood using simple and universal arguments. Figure 7 illustrates the mechanism. For low incident light intensity, the temporal spacing between consecutive photons (and plasmon excitations) is large compared to the HC lifetime and PL occurs as an independent and uncorrelated process. As the incident light intensity increases, the HCs generated by plasmon decay do not have sufficient time to completely relax to the initial conditions before the next plasmon excitation creates more HCs. Continued excitation increases both the total and the average energy in the HC distribution. The qualitative aspects of this phenomenon can be illustrated

by a simple three-level system |0⟩ and |1⟩ and |2⟩ with arbitrary degeneracies. These three levels

represent the ground state and two distinct HE continua. Once a photon excites a plasmon, it quickly

decays and creates one HE in state |2⟩. The electron in |2⟩ then undergoes electron-electron scattering and carrier multiplication and decays on a timescale of &. to a finite number .6 of less

hot electrons in |1⟩ and then from |1⟩ to |0⟩ on a timescale of &6 . Although this model is not an

attempt to simulate HC relaxation, the results from Fig. 4 show that indeed two such HE continua are

prominent during the relaxation: a continuum around . ≈3.5 eV and a continuum around 6 ≈1.5 eV with lifetimes &6 ≫ &. . As discussed above, we assume that plasmon excitation and decay events

occur at equally spaced time intervals R =

19


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¡ ¢£ (¤) from Eq. (11) Figure 7. Schematic illustrating a mechanism for blueshift of the HC distribution. The red curve is calculated using ¥ =1.5eV, ¦ = §. ¨©ª, «¥ = ¨¬¬®, «¦ = ¥¬¬®, and ¯ = ¨ × ¥¬§ ©ª°±1¦ . The left inset (yellow) shows the average PL when consecutive excitations occur with temporal spacings exceeding the HC relaxation time. The right inset (purple) shows the case when the plasmon excitations occur faster than the HC relaxation resulting in a blueshifted PL.

In steady state, the populations of |1⟩ and |2⟩ just before the next photon arrives can be approximated as a geometric series:

β exp ½− ¾ µ & 1 · 1 { 1 . ³ L. () = ¶ ¸ + ¸ + ¸ + … º = β ³ 1 − exp ½− & ¾ . β ´ exp À− Á · { &6 ³ L () = ¶ 1¸{ + 1¸{ + 1¸{ + ⋯ º = . (10) 6 .6 .6 ³ β 1 − exp À− Á ² &6

The steady state time average of the energy of the HC distribution is then = (L6 6 + L. . )/ (L6 + L. ) and can be expressed in terms of Â() = L. ()/L6 () as: () =

½6 + Â(). ¾ . (11) ½1 + Â()¾

For . Ã 6 , the average energy of the HC distribution () increases monotonically with light

intensity . The red curve in Fig. 7 shows the blueshift of () calculated using semi-realistic

̅ () will follow the average energy parameters. The average energy integrated PL emission energy lmn of the HC distribution () and exhibit a blueshift with increasing light intensity . This

qualitative description of the PL blueshift with increasing intensity does not include the strong effect that the Purcell effect can have on the shape of the PL spectrum. As was demonstrated above for the Ag nanorods, the PDOS emission enhancement can be very large and confined to specific narrow 20


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wavelength intervals. These emission resonances will then dominate the PL spectra making the blueshift smaller and more difficult to observe experimentally. For sufficiently large incident power, an additional important effect not captured by the simple model discussed above can contribute to a larger a blueshift of the PL. Plasmon decay results in an electron-hole pair of energy equal to the plasmon quantum. However, depending on the energy of the initial electron, the absolute energies of the electrons and holes can vary. An electron at the Fermi level can be excited resulting in a cold hole (CH) just below the Fermi level and a HE. An electron well below the Fermi level can be excited to just above the Fermi level resulting in a HH and a cold electron (CE). Thus if the HE thermalization occurs slower than the temporal spacing of the incident photons, PL could be generated by radiative transitions between HEs and HHs created by consecutive photons. The maximum energy of such upconverted emission is two times the incident photon energy. Since the lifetimes of HE and HH are likely to be different (we expect a HH to have shorter lifetime than a HE because of its localization inside the NP), the intensity threshold for this process will be determined by the lifetime of the HH. We now turn to our fully quantum mechanical modeling. Figure 8 shows the calculated PL for CW illumination of our 6 nm jellium Ag nanosphere using different photon energies. The two upper panels show relatively weak PL emission resulting from direct intraband excitations and the lower panel shows the strongly enhanced PL resulting from plasmon excitation. Both the intensity and shape of the PL spectra clearly depend strongly on incident photon energy. The top panel shows the PL spectrum for incident photon energy of 2.5 eV and is dominated by two emission resonances at 2.5 and 3.5 eV. The 2.5 eV feature is reemission through the radiative recombination of the HCs before the onset of significant electron-electron scattering. The PDOS enhancement at this energy is negligible resulting in a Purcell factor close to unity. The intensity of this resonance increases linearly with incident power. The second resonance at 3.5 eV is an upconverted signal caused by transitions from HEs to HHs generated by consecutive plasmon excitation events. This emission feature is

strongly enhanced by the Purcell factor v=25 at 3.5 eV and depends much more strongly on intensity since it represents a multiphoton event. However, these results demonstrate that coherent two-photon absorption is not necessary for PL upconversion, but can also be achieved by CW illumination. The fact that experimentally PL upconversion following multi-photon excitation has 21


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been mostly observed for plasmonic nanostructures when using ultrafast pulsed excitation could, based on our results, be due to the much higher excitation rate and not because of a coherent multi-photon electronic excitation.65,66

Figure 8. Power dependence of PL under CW excitation for different incident energies: 2.5 eV (a); 3.0 eV (b); and 3.5 eV (c). The incident powers are: 1 (yellow); 2 (green); 4 (blue); and 10 ÅÆ1. (purple). The dashed vertical lines show the average emission energies for the same incident powers. The NP considered here is the 6 nm Ag jellium sphere. The insets show the emission intensity at specific photon energies: (2.5 eV in (a) and 3.0 eV in (b) (dark blue), 3.5 eV (red)) as a function of incident power.

The middle panel for incident photon energy of 3.0 eV shows more features. The two dominant emission resonances at 3.0 and 3.5 eV have the same origin as for 2.5 eV excitation. The 3.0 eV peak represents HCs that decay radiatively directly after being created without having undergone electron-electron scattering. The 3.5 eV peak with its superlinear power dependence is an upconverted event involving transitions from HEs to HHs generated by different plasmon excitation events. The 22


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weaker features at 2.2 and 2.7 eV represent emission from intermediate luminescent states that are populated transiently through electron-electron scattering during the HC relaxation process. The

̅ () for 2.5 and 3.0 eV excitations exhibits clear blueshifts with average PL emission energies lmn ̅ () blueshifts is increasing incident power as discussed above. The dominant contribution to the lmn from the onset of upconversion. The mechanism illustrated in Fig. 7 also contributes but plays a more minor role for Ag spheres because of the large Purcell factor at 3.5 eV. The PL for resonant excitation using 3.5 eV photons (lower panel) looks qualitatively different. Most notably, the PL is around two orders of magnitude larger than for non-resonant excitation. The PL spectra are dominated by the very strong Purcell factor enhanced emission at 3.5 eV. A lower energy continuum between 2 and 3 eV is also present and originates from HCs that have lost some of their initial excitation energy on their way to the ground state. In contrast to direct intraband excitation (2.5 and 3.0 eV), The PL spectrum does show a significant blueshift with increased excitation intensity. The reason is the strong energy dependence of the Purcell factor which is maximal at the

plasmon resonance and close to 1 for off-resonance excitations. Thus the PL spectrum for 3.5 eV excitation is completely dominated by the Purcell effect and blueshifted emission is obscured by the strong signal at 3.5 eV. Experimentally a blueshift of a few meV has been observed for the longitudinal mode of gold nanorods using excitation powers on the order of mW/µm2.63 Conclusions Plasmon-induced HC generation has stimulated considerable interest across the broad area of nanoscience because of important applications in photodetection and photocatalysis. Once HCs have been generated by plasmon decay, they relax through electron-electron scattering, radiative recombination and eventually through electron-phonon coupling. In this paper, we developed a fully quantum mechanical model for the HC relaxation through electron-electron scattering and radiative recombination (luminescence). Electron-electron scattering is found to be the dominant relaxation channel. Individual scattering events occur on a timescale of a few femtoseconds. However, the emerging electrons remain hot and the full thermalization through electron-electron scattering requires thousands of scattering events. The overall HC relaxation times are found to be around a picosecond depending on the energy of the initial HEs. We also analyzed in detail radiative recombination of HCs. For pulsed plasmon excitation, the instantaneous PL is found to redshift as the HCs relax toward the 23


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Fermi level. For CW illumination, the steady state HC distribution and PL blueshift with increasing incident power. When the temporal spacing between consecutive plasmon excitation events becomes shorter than the HC lifetimes, upconverted light emission arising from the radiative recombination of HEs with HHs generated from a previous photon is predicted. Finally, the shape of the PL spectrum is found to be strongly sensitive to the photonic density of states through the Purcell factor. Our model can straightforwardly be extended to other metals like gold and aluminum where inter-band transitions play a more important role than in silver. Our predictions that the power dependence of the PL reflects the lifetimes of the HC may then be used to quantify the role of interband transitions on HC lifetimes. Our model and results provide a solid theoretical framework for the description of plasmon-induced HC decay processes and pave the way for the design and optimization of HC based devices and photocatalysts.

Supporting Information A figure showing the effect of screening on the timescales for hot carrier relaxation.

Conflict of interest The authors declare no conflict of interest.

Author Contribution P.N. and S.L conceived the project. J.G.L. performed the calculations. All authors contributed to the writing and editing of the manuscript.

Acknowledgements The authors acknowledge extensive and stimulating discussions with Dr. Alejandro Manjavacas, Dr. Wei-Shun Chang, and Yi-Yu Cai. This work was supported by the Air Force Office of Scientific Research (MURI FA9550-15-1-0022) and the Robert A. Welch Foundation under grants C-1222 (P.N) and C-1664 (S.L.).

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For Table of Contents Use Only. Relaxation of plasmon-induced hot carriers by J.G. Liu, H. Zhang, S. Link, and P. Nordlander. The graphic shows the instantaneous hot hole (blue) and electron (red) distributions 200 and 1500 femtoseconds after plasmon decay. Plasmon induced hot carrier generation is a promising mechanism for sustainable catalysis.

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Electron-electron

200 fs

Number of hot carriers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

ACS Photonics

EF

Electron-photon

1.5 ps

EF photon ~~~

~~~

Electron-phonon

EF ACS Paragon Plus Environment

Energy (eV)

phonon~~~

~~~

Relaxation of Plasmon-Induced Hot Carriers - ACS Photonics (ACS

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