Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 141−145
pubs.acs.org/JPCL
Origin of Plasmon Lineshape and Enhanced Hot Electron Generation in Metal Nanoparticles Xinyuan You,† S. Ramakrishna,‡ and Tamar Seideman*,‡ †
Graduate Program in Applied Physics and ‡Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: Plasmon-generated hot carriers are currently being studied intensively for their role in enhancing the efficiency of photovoltaic and photocatalytic processes. Theoretical studies of the hot electrons subsystem have generated insight, but we show that a unified quantum-mechanical treatment of the plasmon and hot electrons reveals new physical phenomena. Instead of a unidirectional energy transfer process in Landau damping, back energy transfer is predicted in small metal nanoparticles (MNPs) within a model-Hamiltonian approach. As a result, the single Lorentzian plasmonic line shape is modulated by a multipeak structure, whose individual line width provides a direct way to probe the electronic dephasing. More importantly, the hot electron generation can be enhanced greatly by matching the incident energy to the peaks of the modulated line shape.
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single-particle excitations to be modeled within a consistent quantum-mechanical framework, where the interplay between them is explicit. The theory is then applied to the plasmoninduced hot electron generation in small MNPs. Based on the coupled dynamics, we can not only deepen the understanding of the line shape, but also propose a novel mechanism to greatly enhance the hot electron generation. We consider only the conduction electrons of the metal in our model, which is a very good approximation for silver, where the energy difference between the d-band electrons and the Fermi level is larger than the plasmon energy.13 Treating the ionic background of the nanoparticle as a jellium sphere of radius R, the system Hamiltonian is written as Ä 2 ÉÑ N Å ÅÅÅ pi ÑÑÑ 1 N e2 H = ∑ ÅÅÅÅ + U (ri)ÑÑÑÑ + ∑ Å 2me ÑÑ 2 i ≠ j |ri − rj| i=1 Å (1) ÅÇ ÑÖ
lasmons are collective oscillations of the conduction electrons in a MNP.1,2 Due to their ability to focus and enhance electromagnetic fields, surface plasmons have inviting photovoltaic3−7 and photocatalytic8−11 applications. It has been shown that in small MNPs the plasmon decays mainly through the excitation of electron−hole pairs (Landau damping)12−17 that results in the generation of hot electrons (hot as they have high temperature relative to their environment). Following the generation of hot carriers is complex electronic dynamics, with electron−electron thermalization (∼100 fs) and electron−phonon relaxation (∼1 ps).12,18−20 When the MNP is in close contact with a semiconductor, hot electrons having sufficient energy to overcome the Schottky barrier tunnel into the semiconductor and thus improve the efficiency of the solar cell.21,22 Recent calculation also demonstrates a new mechanism through which plasmons can delocalize directly into electrodes, leading to instantaneous electron injection.23 The theory of plasmon-induced hot electron generation is currently under intense study. A typical method to study the generation of hot electrons is Fermi’s golden rule, in which plasmons are described either classically, as oscillating electromagnetic fields, 24−28 or as quantized plasmon modes.29 However, the plasmon damping is described merely via a fixed parameter in the dielectric function, which neglects the possible interplay between the hot electron and plasmon. Although these studies provide important insight into the hot electron generation, a complete understanding of the coupled dynamics is still missing, which can be crucial to improve the efficiency of hot electron generation. In this Letter, we go beyond previous theories by treating the plasmon damping as a quantum scattering process. This allows the plasmon and © XXXX American Chemical Society
where U(r) is the single-particle confining potential.30 The total Hamiltonian is solvable only for small clusters, containing a few atoms. In practice, it is useful to separate the total Hamiltonian into center-of-mass motion and relative motion as, H = Hc.m. + Hrel + Hc, where Hc.m. = ℏωpb+b describes the center-of-mass motion, b+ and b are the ladder operators for the plasmon within the harmonic approximation, and ωp is the collective oscillation (plasmon) frequency. The relative degrees of motion are treated using a mean-field approximation Hrel = ∑αϵαC+αCα, where C+α and Cα are the creation and annihilation operators for the single-particle state |α⟩, and ϵα is the Received: November 25, 2017 Accepted: December 19, 2017 Published: December 19, 2017 141
DOI: 10.1021/acs.jpclett.7b03126 J. Phys. Chem. Lett. 2018, 9, 141−145
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The Journal of Physical Chemistry Letters
well-defined. However, an effective temperature T* and chemical potential μ* can be obtained via the conservation of energy and particle number.35 As nonradiative Landau damping dominates over radiative decay in small MNPs, there is no additional Lindblad term for plasmon decay in our model. The absorption spectrum is given as, σabs = ω/(ϵ0c)Im[α(ω)], where the polarizability tensor α(ω) is calculated as the ratio of the Fourier transform of the total dipole to that of the electric field. Figure 1 shows the plasmon and hot electron dynamics. A Gaussian laser pulse of duration 20 fs (centered at t = 0 fs),
corresponding energy. The energy and dipole matrix element of single-particle states are obtained by solving the threedimensional Schrödinger equation in a spherical finite step potential with radius R and a height taken to be the sum of the Fermi energy ϵF and the work function W (see Supporting Information for details). Exchange-correlation effects alter the results only in a minor way compared to the free-electron energy.24 Hc = Λ(b++b)∑αβdαβC+αCβ is the coupling between the center-of-mass motion (plasmon mode) and the relative motion (single-particle excitation), where dαβ is the dipole matrix element between two single-particle states, and Λ = ℏmeωp3/2N is the coupling constant. By separating the many-body Hilbert space into low- and high-energy singleparticle excitations, one obtains the plasmon as arising from a coherent superposition of a large number of states with low energy. Thus, only the single-particle states that are in the high-energy sector can be excited during plasmon damping, and the summation in the coupling term should only include states whose energy differences are larger than a cutoff energy.31 When the nanoparticle is irradiated by a laser field, a light−matter interaction term Hd = −E(t)[ds(b++b) + ∑αβdαβC+αCβ] is added, where the electric field denoted by E(t) excites both the plasmon and the single-particle excitations. ds is the transition dipole moment of the plasmon.32 The evolution of the system is governed by the quantum Liouville equation in density matrix formalism, ∂tρ̂ = [H,ρ̂]/iℏ+L(ρ̂). A numerically efficient method is to propagate ρα,α′ and ρs,s′ via two coupled differential equations (see Supporting Information for details), where α and s denote the single-particle state and the plasmonic state, respectively. The price to be paid is the loss of global coherence between the two subsystems.33 However, in the specific case we are interested, this is a valid approximation, since the decoherence rate is much larger than the coupling strength between them. Note that, although the global coherence is gradually lost, the coupling still exists, and the coherence in each subsystem is maintained. The relevant dephasing and relaxation process is described using the Lindblad formalism,6,7,34 with the superoperator L given through
Figure 1. Plasmon and hot electron dynamics. Evolution of plasmonic energy (a), and electronic energy (b) for various relaxation and dephasing conditions. Dotted lines: γe−p = 1.3 meV, γp = 20 meV; Solid lines: γe−p = 1.3 meV, γp = 2 meV; Dashed lines: γe−p = 13 meV, γp = 20 meV. The electron distribution (c) and the electron distribution relative to the initial distribution (d) at different time steps for γe−p = 1.3 meV and γp = 20 meV.
central photon energy 3.65 eV (same as the plasmon energy ℏωp24), and maximum electric field 5 × 108 V/m interacts with a 1 nm silver nanoparticle. The dotted curves in Figure 1a,b correspond to the relaxation and dephasing rates as γe−p = 1.3 meV (0.5 ps), γe−e = 6.5 meV (0.1 ps) and γp = 20 meV (33 fs), which is typical for a silver nanoparticle.26 The plasmon starts to gain energy at the onset of the laser pulse, and decays after the laser disappears at a specific rate, which depends strongly on its interaction with the electronic subsystem (explained later). The related electronic total energy also increases with a slight temporal lag behind the plasmon excitation, since the hot carriers are generated mainly by the plasmon rather than by the external laser field. After the plasmon completely decays, the hot carriers relax at a much slower rate, determined by the electron−phonon interaction, which transfers energy to the atomic lattice. The evolution of the hot electron energy distribution is shown for four representative moments in Figure 1c (The multiple occupation values for single energy reflects the degeneracy of the spherical symmetry). The black curve displays a room temperature (300 K) Fermi−Dirac distribution prior to laser excitation at t = −50 fs. At t = 50 fs (red curve), when the plasmon has already been excited and starts to decay, the electronic subsystem is in a highly nonequilibrium state, showing an increase of electronic population above the Fermi level and a decrease below it. The energy distribution becomes close to a Fermi− Dirac distribution again at t = 170 fs (blue curve). While the high-energy excitations are reduced, the distribution takes a
Lα = α′ = −γe − e[ρα , α − fFD (ϵα , μ*, T *)] − γe − p[ρα , α − fFD (ϵα , μ0 , T0)] Lα ≠ α′ = −γpρα , α′ − γe − eρα , α′{1 − [fFD (ϵα , μ*, T *) + fFD (ϵ′α , μ*, T *)] /2} − γe − pρα , α′{1 − [fFD (ϵα , μ0 , T0) + fFD (ϵ′α , μ0 , T0)] /2}
(2)
where f FD denotes the Fermi−Dirac distribution, with ϵ, μ, and T being the energy, chemical potential, and temperature, respectively. The diagonal part describes the electron−electron thermalization and the electron−phonon relaxation, with corresponding rates γe−e and γe−p. The first term in the offdiagonal part describes the pure dephasing of electronic coherence with rate γp, which only affects the off-diagonal density matrix element. The other two terms are the dephasing rates associated with the diagonal relaxation and thermalization. μ0 and T0 are the initial chemical potential and temperature when equilibrating with the lattice. During the excitation of hot electrons, the system is in a nonequilibrium state, where the temperature and chemical potential are not 142
DOI: 10.1021/acs.jpclett.7b03126 J. Phys. Chem. Lett. 2018, 9, 141−145
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The Journal of Physical Chemistry Letters moderate slope (compared with t = 50 fs) near the Fermi energy, which is due to the energy conservation property of the electron−electron thermalization. Finally, at t = 730 fs (green curve) or longer time, the slope of the distribution becomes steep again (lower temperature) as a result of electron− phonon relaxation. The energy distribution difference with respect to the initial time (t = −50 fs) is shown in Figure 1d. Note that now the degenerate populations are summed over for each energy level. Next, the dependence of the coupled dynamics on the relaxation and dephasing rates is investigated and found to be of crucial importance. The solid curves in Figure 1a,b correspond to a slow pure dephasing γp = 2 meV (0.3 ps). Compared with the faster dephasing rate (dotted curve), in which the plasmon monotonically decreases, we find oscillations in both the evolution of plasmon energy and the electronic energy in the slow dephasing case. The opposite phases of these two oscillations indicate that energy is transferred back and forth between the plasmon and the hot carriers, which supports a recent observation on a Ag55 nanocluster that used real-time time-dependent density functional theory.36 This behavior emerges from the dipole− dipole interaction term in Hc, with (b+ + b) and ∑αβdαβC+αCβ proportional to the dipole of the plasmon and single-particle excitation, respectively. Thus, a slow dephasing rate results in a long lifetime for coherent dynamics, while a fast dephasing rate eliminates the possibility of back energy transfer. The dashed curves in Figure 1a,b exhibit the case when the electron− phonon interaction is increased. The electron energy evolution is significantly affected, while the plasmon relaxation is only marginally impacted. We argue here that it is important to treat the plasmon in a unified picture with the hot electron. Previous theories account for plasmonic effect via prespecified dielectric function, which makes the dynamics of the plasmon independent of the actual interaction of hot electrons.25,27 By contrast, in our approach, the plasmon decays with a rate determined by the energy transfer between plasmon and hot carriers (Figure 1a). In addition, as we are considering the small size nanoparticles (R ≤ 10 nm), all the energy absorbed by the plasmon should transfer to the hot carriers.37 However, the hot carrier generation rate determined by Fermi’s golden rule alone does not satisfy this condition. And the conservation of energy need to be additionally imposed by rescaling the generation rate with an appropriate factor.24 In our unified approach, the conservation of energy between plasmon and hot carriers is guaranteed automatically. This makes our approach more appropriate for calculating transient dynamics, where both the plasmon and hot carrier energy evolve rapidly in time. We then show the effect of the coupled dynamics on the optical response, and explain the results within our unified picture. The absorption spectra of MNPs under various conditions are shown in Figure 2. From the discussion above, coherence-induced back energy transfer affects the plasmon dynamics drastically. Since the coherence is dominated by the electronic pure dephasing, the absorption spectrum is expected to display such a dependence as well. In both insets in Figure 2, the pure dephasing rate is reduced from γp = 20 meV (red curve) to γp = 10 meV (black curve). In the R = 3 nm case, the absorption spectrum is modulated by multiple peaks, whereas in the R = 2 nm case the peaks become more prominent. Fitting of the individual peaks with Lorentzian functions shows that the width of these peaks is twice the pure dephasing rate,
Figure 2. Absorption spectrum of silver nanoparticle under various conditions. The main axis shows the absorption spectrum for nanoparticles of different radius: 4 nm (dotted line), 3 nm (dashed line) and 2 nm (solid line). The left (right) inset shows absorption spectrum for a 3 nm (2 nm) nanoparticle with different relaxation and dephasing rates. Black, red, blue lines correspond to γp = 10 meV, 20 meV, 200 meV, with the same γe−p = 1 meV.
and indicates that these peaks originate from single-particle excitations (see Supporting Information for details). Furthermore, it provides an experimental way to determine the electronic dephasing from measurement of the plasmonic absorption spectrum. As discussed in the context of Figure 1, a small pure dephasing rate enables back energy transfer, hence these peaks provide direct evidence of the coupled dynamics between the hot electron and the plasmon mode. For a large dephasing rate, γp = 200 meV (blue curves), the absorption spectrum becomes a perfect Lorentzian, because the individual peaks broaden and overlap with adjacent peaks. The critical dephasing rate that characterizes the transition from a multipeak to a single-Lorentzian feature correlates with the energy spacing or size of the particle (smaller particle has larger energy spacing). Therefore, a 2 nm MNP shows more prominent peaks compared with a 3 nm MNP with the same pure dephasing rate. On the other hand, the line width of the overall line shape is not sensitive to electronic dephasing. Instead, it is strongly affected by the size of the particle, as shown in Figure 2. The appropriate quantity to characterize the overall line width is the coupling strength Λ between the center-of-mass motion and the relative motion, as indicated in the literature. 30 From the dependence of Λ on N (Λ = ℏmeωp3/2N ), a smaller MNP with less number of electrons N would have a larger coupling strength, and hence faster plasmon decoherence and larger plasmon line width. The unified treatment of plasmon and single-particle excitation provides insight into the origin of the plasmon line shape, and attempts to answer whether the plasmon line width is mainly determined by electronic dephasing,26,38 or by particle-hole excitations.30,39 Our results indicate that the overall plasmon line width is dominated by energy transfer from the center-of-mass to the relative motion, while the line width of individual peak is determined by electronic dephasing, which becomes insignificant for a larger nanoparticle (but still in the range R ≤ 10 nm). Experimentally, the multipeak structure is rarely observed,40,41 mostly because the electronic pure dephasing is large compared with the energy spacing, and broadens the signal of the single-particle excitation. One direct outcome of the coupled nature of the plasmon and single-particle excitation is the impact of electronic dephasing on hot electron generation. In the above discussion, 143
DOI: 10.1021/acs.jpclett.7b03126 J. Phys. Chem. Lett. 2018, 9, 141−145
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The Journal of Physical Chemistry Letters
typical Lorentzian to a multipeak line shape. The line width of the additional peaks provides an experimental way to measure the electronic dephasing in small MNPs. Finally, we have shown that the hot electron generation can be enhanced substantially by matching the incident wavelength to the emerging peaks due to electronic coherence.
we have already seen the electronic dephasing has a significant influence on the absorption spectrum, especially for small nanoparticles. Due to conservation of energy, the energy injected into the hot carriers should be controlled by the electronic dephasing as well. Figure 3a shows the absorption
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b03126. Method to calculate single-particle states; explicit form of master equations; derivation of Lindblad equations; relation between line widths of individual peaks and electronic pure dephasing rates (PDF)
Figure 3. Hot electron generation dependence on dephasing rates. (a) Absorption spectrum for a 2 nm silver nanoparticle with different pure dephasing rates. Black (red) arrow points to the absorption spectrum at photon energy ℏω = 3.805 eV for γp = 20 meV (200 meV). Blue arrow points to the absorption spectrum at photon energy ℏω = 3.784 eV. (b) Energy distribution of hot electrons at steady state. The two solid lines correspond to excitation energy ℏω = 3.784 eV. The black (red) dashed line corresponds to the excitation energy ℏω = 3.805 eV for γp = 20 meV (200 meV). The inset is a zoom-in near the positive peak.
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spectrum for a 2 nm MNP with two different electronic pure dephasing rates γp = 20 meV (black line) and γp = 200 meV (red line). The multipeak structure obtained with a slow dephasing rate, leads to a 50% increase in the absorption spectrum for specific incident photon energy (e.g., 3.805 eV pointed by black and red arrows). The corresponding steady state hot electron population is shown in Figure 3b (red and black dash lines). It is clear that there are more hot electrons for the pure dephasing rate whose absorption spectrum is larger at the given energy. As a comparison, we study the incident energy where the two spectra have the same absorption cross-section (e.g., 3.784 eV, pointed by blue arrow) in Figure 3a. In this case, the two hot electron populations are pretty close, as shown by two solid lines in Figure 3b. Thus, the enhancement of hot electron generation has a wavelength dependence determined by electronic dephasing rate. This result relies on the fact that the absorption spectrum is affected by the electronic dephasing, which can only be revealed via a quantum description of combined plasmon−hot electron dynamics. Another possible effect of electronic dephasing on hot carrier generation is broadening of the density of states, which leads to a hot carrier distribution closer to the Fermi energy.24 In the case of a 2 nm MNP, the electronic dephasing (20 meV) has already saturated the density of states to a fully homogeneous distribution. Therefore, further increase of the electronic dephasing does not change the homogeneity of the density of states, or the hot carrier distribution. In summary, we have calculated plasmon absorption spectrum and plasmon-induced hot electron generation within a unified, quantum-mechanical picture. The coupled plasmon− hot carrier dynamics indicates that the plasmon decay and hot electron generation are correlated. Electronic coherence leads to the possibility of back energy transfer, which is controlled by the electronic dephasing. Moreover, this back energy transfer channel modulates the plasmon absorption spectrum from a
ACKNOWLEDGMENTS The authors thank the Atomic, Molecular, and Optics Sciences (AMOS) of the Department of Energy (Award Number DEFG02-04ER15612/0013) and the National Science Foundation (Award Number CHE-1465201) for support.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Xinyuan You: 0000-0002-9178-9419 Notes
The authors declare no competing financial interest.
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