Article Cite This: ACS Photonics 2018, 5, 2584−2595
Relaxation of Plasmon-Induced Hot Carriers Jun G. Liu,†,‡ Hui Zhang,†,§ Stephan Link,†,§,∥ and Peter Nordlander*,†,‡,§ †
Laboratory for Nanophotonics, ‡Department of Physics and Astronomy, MS 61, §Department of Electrical and Computer Engineering, MS 366, ∥Department of Chemistry, MS 60, Rice University, Houston, Texas 77005, United States
ACS Photonics 2018.5:2584-2595. Downloaded from pubs.acs.org by WASHINGTON UNIV on 08/09/18. For personal use only.
S Supporting Information *
ABSTRACT: 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 using a fully quantum mechanical jellium model. Following pulsed illumination, nonradiative plasmon decay results in a highly nonthermal distribution of hot electrons and holes. Using coupled master equations, we calculate the timedependent 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 dominant 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
L
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 that produces relatively low energy electrons but more energetic holes.19 Another difference is that plasmon-induced HCs are created by the plasmonic nearfield,24 which contains higher multipolar components, while direct excitation results in dipolar transitions that, in principle, can decay radiatively without scattering. While the instantaneous generation of HCs from plasmon 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 HCinduced 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
ocalized 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 NP9−11 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 magnitude, LSPR excitation is a much more efficient process for hot carrier (HC) generation than direct excitation of specific interband transitions in metal.16 Such HCs can be utilized in a variety of light harvesting applications such as photodetection17−20 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 © 2017 American Chemical Society
Special Issue: Recent Developments and Applications of Plasmonics Received: August 5, 2017 Published: December 8, 2017 2584
DOI: 10.1021/acsphotonics.7b00881 ACS Photonics 2018, 5, 2584−2595
Article
ACS Photonics
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 multiphoton effect.
important in photodetection applications where the HCs typically transfer quickly directly to 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 time scales. 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 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 wave function 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 NPs 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 decay relaxes toward equilibrium. The relaxation mechanisms include electron−electron scattering and electronphoton 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 parametrization 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 parametrization, 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 plasmoninduced 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
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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 ε∞ = 4.18; and an electron density corresponding to a Wigner Seitz radius of 3 Bohr corresponding to a bulk plasmon frequency ωB =
4πe 2n m
,
where e is the electron charge, m is the electron mass, and n is 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 Figure 1a. As discussed in our previous
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 mW μm−2. (c) Illustration of the decay channels for HC relaxation: electron−electron, electron− photon, and electron−phonon scattering.
work,16 the independent electron approximation can be used to model plasmon decay. The eigenenergies εi and eigenstates |i⟩ 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 Vext = −E0r cos θ, where E0 is the amplitude of the electric field, r is the radial position, and θ is the polar angle. 2585
DOI: 10.1021/acsphotonics.7b00881 ACS Photonics 2018, 5, 2584−2595
Article
ACS Photonics
HC relaxation have been reported so far, it is generally believed that electron−electron interactions occur on a time scale of 200−500 fs45,48−51 and that electron−phonon coupling to equilibrate electron and lattice temperatures takes a few 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 parametrize the electron−phonon interaction using the empirical relaxation time 1/Γep ≈ 1 ps,45,54 which is much longer than individual electron−electron scattering events on the order of 10 fs.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 wave functions of the NP and can be expressed as
The rate of electron−hole pair generation (excitation from state |i⟩ to |j⟩) is calculated as Γ iex→ j =
4 1 |⟨i|e(Vp + Vext)|j⟩|2 τ (εi − εj + ω)2 + ℏ2τ −2 (1)
In this expression Vp is the plasmon-induced electric field: ϵ(ω) − 1 Vp(r , ω) = ϵ(ω) + 2 E0r cos θ i n s i d e t h e N P a n d ϵ(ω) − 1 3 D /(8r 2)E0 ϵ(ω) + 2
cos θ outside the NP calculated using the
Drude model: ε(ω) = ε∞ −
ωB2 ω(ω + iγ )
with parameters appro-
priate for Ag: a corrected damping accounting for the finite size ν effect γ = γ0 + ℏD F/ 2 with γ0 = 0.06 eV and νF = 1.4 × 108 cm/ s is the Fermi velocity.43 The incident light frequency is ω and the lifetime τ refers to the natural line width of the excitation and is chosen so long that the results for the initial carrier generation after plasmon decay do not change with further increases of τ (τ = 1 ps). 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 fs) is Pi(0) = f (εi) −
∑ Γiex→ jf (εi)(1 − f (εj))Δt + j
Γ ijee→ kl =
e−kTF|r − r ′|
j
element, and V (r, r′) = |r − r ′| with kTF ≈ 1 Bohr (Ag) is the Yukawa potential resulting from Thomas-Fermi screening of the Coulomb potential. The matrix element Mij→kl is an 6 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 2 orders of magnitude larger resulting in much shorter HC lifetime distributions. In the Supporting 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 |r − r′| < 1/k TF decreases the lifetimes by 20%. The screening due to d-electrons reduces the matrix element by a multiplicative factor 1/(ϵ∞ − 1), where ϵ∞ − 1 is the polarizability of the lattice. For Ag, where ϵ∞ = 5 this effect is substantial and increases the lifetimes by a factor of 3.3. However, since the d-electrons are not 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
where Pi is the probability for occupation of the ith state and f(εi) is the Fermi−Dirac distribution. Since we will focus here on the initial relaxation (
