Kinetic Density Functional Theory for Plasmonic Nanostructures

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Kinetic Density Functional Theory for Plasmonic Nanostructures: Breaking of the Plasmon Peak in the Quantum Regime and Generation of Hot Electrons Alexander O Govorov, and Hui Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp512105m • Publication Date (Web): 04 Feb 2015 Downloaded from http://pubs.acs.org on February 10, 2015

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The Journal of Physical Chemistry

Kinetic Density Functional Theory for Plasmonic Nanostructures: Breaking of the Plasmon Peak in the Quantum Regime and Generation of Hot Electrons Alexander O. Govorov1* and Hui Zhang1,2*

1

2

Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701

Current address: Rice University, Physics and Astronomy, Houston, TX 77251 *Corresponding Author: [email protected]

Abstract We develop a quantum kinetic theory of the dynamic response of typical noble metals. Our approach is based on the Density Functional Theory (DFT) and incorporates new important elements as compared to the conventional time-dependent DFT formalism. The kinetic DFT is derived starting from the master equation of motion for the density matrix, which involves both momentum and energy relaxation processes. Therefore, the quantum system is described by two relaxation parameters, unlike the conventional time-dependent DFT incorporating only one relaxation parameter.

This allows us to describe both the absorption of light and the generation

of hot plasmonic electrons. Using our kinetic DFT theory, we also observe the transition from the multiple peaks in small size-quantized systems to the intensive plasmonic resonance in large classical systems. Unlike the standard picture of collisional broadening of the plasmon peak in

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small systems, we observe a very different scenario – the formation of multiple plasmonic-like peaks in small quantized systems. These peaks are the result of a hybridization of the collective plasmon mode and the single-particle transitions in a quantized electron gas. There are a few experimental observations that seem to correlate with such scenario of the plasmonic broadening in small systems. Our approach also incorporates the interband transitions, which are important for a qualitative description of gold and silver. Although this paper gives an application of our kinetic DFT only to the slab geometry, our theory can be applied to nanocrystals of arbitrary shape. The kinetic DFT formalism developed here can be employed to model and predict a variety of metal and hybrid nanostructures for applications in photocatalysis, sensors, photodetectors, metamaterials, etc.

Keywords: Hot plasmonic electrons, Density Functional Theory, plasmonic nanocrystal and nanoparticle, plasmon resonance, plasmonic catalysis, injection of charge carriers.


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1. INTRODUCTION

The Density Functional Theory (DFT) is at the heart of the modern physics of materials.1,2,3 It is probably the best known approach to predict and describe quantum states and quantum properties of molecules, 3D crystals, nanostructures, etc. Plasmonic nanocrystals made of metals are an important class of nanomaterials that exhibit strong plasmon resonances in optical and photocatalytic responses. These physical and chemical properties come from collective and single-particle excitations in metals and look very interesting for energy applications.4,5 One particular physical effect peculiar for the metal nanocrystals is optical generation of hot plasmonic carriers.6,7 This effect is utilized in the two types of experimental applications. The first one concerns photodetectors based on the Schottky junction8 and the second one involves surface photocatalysis.4,9

The obvious advantage of plasmonic injectors of hot electrons is in

large and resonant absorption cross sections of metal nanocrystals and nanoantennas. Over the last years, a large number of experimental investigations has been reported in this area. These studies involved both solid-state structures10,11,12,13,14,15,16,17,18,19,20,21,22,23 used for photocurrents and colloidal systems24,25,26,27,28,29,30,31,32,33,34,35,36 involved in photochemistry experiments. Here we develop a new computational formalism called a Kinetic DFT. The derivation of the Kinetic DFT is based of the equation of motion for the density matrix.37,38,39 Our approach incorporates two relaxation rates (for momentum and energy) and the interband transitions in a metal. Importantly, all parameters in our theory come from experimental data for a bulk metal and, therefore, we do not use any free fitting parameters. Our theory allows us to compute both the energy distribution of coherent energetic carriers and the optical spectra. As expected, the efficiency of generation of hot plasmonic electrons decreases rapidly with increasing 3 ACS Paragon Plus Environment


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nanostructure size. High efficiencies can be achieved only for nanostructures with small and ultra-small sizes or in nanostructures with electromagnetic hot spots.40 In thin metal films with a quantized spectrum, our theory shows the formation of multiple peaks, which can be understood as a hybridization of a collective plasmonic mode and single-particle transitions. Simultaneously, we do not observe any additional broadening of the plasmon peak due to collisions with the surface, as commonly accepted in the related literature. It is essential to compare this paper with our previous publications7,40,41. In our previous studies,7,40,41 we have used a semi-quantitative analytical approach to this problem by incorporating classical solutions for the electromagnetic field inside a slab. Our previous approach7,40,41 ignored an important effect of a finite dynamic screening length near the surfaces of a nanostructure and involved only one common relaxation time. In this paper, we use exact numerical solutions for the local electric field and local electron density in the framework of the DFT-based jellium formalism and, in addition, carefully construct the collision integrals by including both energy and momentum relaxation rates. Although we computed only the simplest slab geometry, we believe that our results can be qualitatively applied to more complex geometries. In one of our previous papers on this topic,40 we applied our semi-qualitative approach to the plasmonic spheres and cubes and found an effect of strongly-enhanced generation of hot carriers due to the presence of plasmonic hot spots. Extension of our approach to 3D nanostructures, such as spheres, cubes and prisms, is a next logical step. We now briefly review related theoretical studies. A theoretical paper by Schatz and Kraus42, that is a prominent publication in the field of plasmonic nanocrystals, treated plasmon resonances using a dielectric function of a quantized metal nanocrystal (NC) as a single parameter. The approach of the paper42 did not include the dynamic screening inside a NC and 4 ACS Paragon Plus Environment

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used the single-particle transition amplitudes. Such approach is valid for the case of localized electrons separated in space and assumes a mean-field approximation. The reality with the many-electron confined systems is, of course, much more complicated.

The problem of a

confined many-electron NC is nonlocal and cannot be solved analytically. In the analytical theory by Schatz and Kraus42 and the previous analytical theories using similar approximations,43 the plasmon peak acquires an additional collisional broadening, which is proportional to the Fermi velocity and inversely proportional to the NC size. Basically, this additional quasiclassical broadening is given by the frequency of collisions of the Fermi electrons with the walls of a NC. In the full version of the self-consistent theory (often regarded also as the Random Phase Approximation), the dielectric function of a confined NC is a matrix of infinite dimensionality applied to the matrix elements of the dynamic light-induced Hartree potential and, therefore, the analytical diagonalization is impossible. Our exact numerical DFT-based diagonalization of the self-consistent problem of a slab with carefully-defined relaxation rates shows a different picture of the plasmon broadening in small confined systems. In our picture, the plasmon peak becomes split into multiple peaks due to the hybridization of the collective plasmon mode and the single-particle quantum transitions. We should note however that, despite the strong simplifications used in the theories based on a dielectric function as a single parameter,42,43 such single-parameter treatment reproduces some important features of the quantum problem. For example, this approach has been used to explain the blue shift of the plasmon in the experiments on silver NCs.44 Time-dependent DFT (TD-DFT) combined with the jellium model has been applied to small metal NCs in the paper.45 Ref.45 aimed to mimic optical properties of real plasmonic nanocrystals, but utilized artificially-chosen parameters of a metal host including, for example, 5 ACS Paragon Plus Environment


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an arbitrary value of the energy broadening of the plasmon resonance. Further studies in the direction led to hybrid quantum-classical models used to describe plasmonic dimers with hot spots.46 The study of Ref.47 presented a direct evaluation of quantum temporal dynamics of electron density in small metal NCs within the jellium TD-DFT. In addition, the study of Ref.47 reported the formation of two types of excitations in small NCs, quantum core plasmons and classical surface-like plasmons. The paper of Ref. 48 reported the atomistic time-dependent DFT for the Ag561 cluster and related semi-empirical dielectric models of small nanoparticles. In additional, the study of Ref.48 discussed the role of interband transitions in small and large NCs. Generally, the role of interband transitions in plasmonic dynamics of noble nanocrystals can be very significant49,50 since these transitions can be very intensive in the visible and UV wavelength intervals (gold and silver). An experimental observation of hot plasmonic electrons can be done via photochemistry or photocurrents. Theoretically, the effect of photo-injection of electrons in a semiconductormetal Schottky-barrier device is described by the Fowler theory51, which is based on the assumption of isotropic and energy-independent generation of excited electrons.

Recent

theoretical papers reported further developments of the Fowler theory considering geometrical effects52,53 and plasmonic fields.54,55,56 Quantum theories for the internal and external photoelectric effects in bulk materials offer the equations for the photo-generated currents across an interface of two media.57,58 These equations were recently employed for an evaluation of the photo-injection efficiency in plasmonic nanostructures.59,60 A recent theoretical paper of Ref.61 treated a steady state of an optically-driven metal NC using an effective temperature approach and kinetic equations.

In small quantized systems, generation and energy distribution of hot


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plasmonic carriers can also be evaluated using the Fermi’s golden rule.62,63 Finally, there is a prediction that plasmonic electrons can lead to an optical up-conversion effect.64 In contrast to the several theoretical papers described above, we develop a DFT-based theory starting from the quantum kinetic equation that naturally incorporates the most relevant relaxation rates. In addition, our Kinetic DFT takes into account the interband transitions in a metal host and has no fitting parameters. The involvement of interband transitions is crucial for a realistic description of optical properties of gold and silver NCs. We organize the paper in the following way. Section 2 provides the derivations for the key equations of the Kinetic DFT. Section 2 is liked to Supporting Information that provides the details of derivations. The following sections 3 and 4 include applications of the Kinetic DFT to the gold and silver slabs with nanoscale sizes.

The applications concern the effect of generation of hot plasmonic

electrons and the quantum effects in the plasmonic resonances.

2. KINETIC DENSITY FUNCTIONAL THEORY

Metal nanocrystals (NCs) may have various shapes (Figure 1a). Incident light creates dissipative currents inside metal NCs and becomes efficiently absorbed (Figures 1b). Optical absorption spectra of noble metal NCs exhibit strong plasmonic resonances, which depend on a shape of NC (Figure 1b). These near-field plasmonic resonances in NCs are typically described by the classical theory based on the Maxwell’s equations (Figure 1b). When the size of a metal NC exceeds several nanometers, the classical theory based on the local dielectric function gives a very good approximation. However when we move to the smaller sizes in the range of a few



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nanometers, the quantum description becomes important because of the quantization of motion of electrons in a confined geometry.

Simultaneously, the electronic structure of the plasmon

resonance is a purely quantum problem that would tell us about the energy-distribution of excited electrons which form the localized plasmon mode (Figures 1c and d). For very small sizes and small numbers of atoms in a NC, the system becomes cluster-like and requires a quantum atomistic description.65 Here we will work in the intermediate regime of a few nanometers and use the so-called jellium model66 when the potential of ions is approximated as a uniform positively-changed background. In addition, we will assume the near-field regime of interaction between light and NCs. In other words, we will consider NCs with relatively small sizes R EF , where EF = 5.5eV is the Fermi energy of gold. The distribution

δ n(ε ) is positive for ε > EF indicating the presence of optically-excited energetic electrons. Simultaneously, δ n(ε ) is negative in the interval ε < EF where we see the generation of energetic holes. Both intervals have two characteristic features.7,40,41 (1) The first feature is the the characteristic spectral structures in Fig. 6 with low excitation energies. These structures for the excited electrons and holes have a triangular shape with a width

δε ≈ hvF qNC , where qNC = π / LNC is the linear momentum transfer from the nanocrystal to an electron in the process of optical excitation and vF = 1.4 ⋅108 cm / s is the Fermi velocity of gold.

These

triangular structures are “bulk” contributions and look very similar to the energy distribution of hot carriers in a bulk plasmon.7,40,41 (2) The second feature of the function δ n(ε ) in Figure 6 is the nearly flat regions (a positive plateau for electrons and a negative one for holes) with the widths of optical energy hω . These regions with highly-excited carriers come from the optical transitions with conservation of energy ε n → ε n + hω with non-conservation of linear momentum.

The effect of non-conservation of the linear momentum occurs due to the

confinement which results in scattering of electrons by the walls.

These flat parts of the

spectrum δ n(ε ) are a “surface” effect. Figures 7 and 8 show the results for various slab sizes and we can see clearly the trends. The integrated number of carriers in the peaks with low



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excitation energies (the triangular structures) is proportional to the volume of a slab V0 = LNC ⋅ L20 . Mathematically, Figure 8b shows the rate

Ratelow-energy electrons = Ratehigh-energy electrons =

1

EF + hvF q NC

∫

τε

.

∞

1

τε

δ n(ε )d ε ,

EF

∫

(12)

δ n(ε )d ε .

EF + 0.7 eV

As we can see, the above rate of generation of high-energy electrons is calculated for the interval

EF + 0.7eV < ε < EF + hω . We take here the excess energy of 0.7eV because it is a typical energy for the Schottky barrier and also for the excess energies of chemical reactions. For large sizes, the rate Ratelow-energy electrons ∝ V0 = LNC ⋅ A and, therefore, Ratelow-energy electrons ∝ LNC for a fixed surface area A . Figure 8b shows that. Simultaneously, again for large sizes, the amplitude of the triangular peaks scales with the size as δ nmax ∝ LNC 2 since the width of these peaks on the energy axis is given by δε ≈ hvF qNC ∝ 1/ LNC . This proves that we deal here with a “bulk” effect. Interestingly, the rate of generation of high-energy carriers in the flat high-energy regions ( EF + 0.7eV < ε < EF + hω )

has

a

very

different

behavior

for

large

sizes:

Ratehigh-energy electrons ( LNC ) ∝ L0NC = const for a fixed surface area of a slab (Figure 8b). In other

words, the rate Ratehigh-energy electrons is independent of the slab size for large sizes. This is a proof that it is a “surface” effect. It is also interesting to look at the shape of the important function Ratehigh-energy electrons , which describes the generation of energetic carriers (Figure 8b). For large

sizes, Ratehigh-energy electrons becomes saturated, as expected for any “surface” effect. However, for small sizes, it increases strongly and we attribute this behavior to the quantum effects. For ultra-


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small sizes (~3nm), the quantization is very strong and the surface effect plays the important role. These surface and quantization effects result in non-conservation of the linear momentum and lead to large quantum amplitudes for transitions that generate high-energy electrons in the whole allowed interval EF < ε < EF + ω . The same effect occurs for the high-energy holes. We can also estimate the efficiency of generation of energetic electrons with excitation energies ε > EF + 0.7eV . And we define the efficiency as

Eff ( LNC ) =

Ratehigh-energy electrons Rateabsorbed photons

,

where Ratehigh-energy electrons is the rate of generation of hot electrons with ε > EF + 0.7eV (see Eq. 12) and Rateabsorbed photons is the number of absorbed photons by the slab per second, which is calculated from the absorption. We now look at the efficiencies of hot electron generation with excitation at λ0 = 516nm and λ0 = 619nm (Figure 8c). We see a rapid decrease of efficiency for slabs with relatively large sizes. This rapid decrease of the efficiency was already predicted by our analytical semi-quantitative theories,7,40,41 but now we can calculate this effect rigorously by using the Kinetic DFT. In a simplified picture valid for large sizes, the efficiency

Eff ( LNC ) = surface property/bulk property ∝ 1/ LNC and, therefore, it is a decreasing function of the size. A relatively large number for the efficiency for LNC = 3nm in Figure 8b also comes from the quantum effects that enable the non-conservation of linear moment and, in this way, enhance strongly the generation of energetic electrons for small sizes (Figure 8b). To repeat ourselves, the physical reason is in the breaking of linear-momentum conservation for small sizes and in resulting large amplitudes for transitions creating high-energy excitations. This is the



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result coming from the quantum kinetic DFT. At the same time, this effect requires nanoplatelets and films of relatively small sizes, 3nm or so. Typically, the experimental sizes of colloidal Au nanocrystals of various shapes are about or larger than 5-10nm. Simultaneously, a fabrication of ultra-thin films, platelets and prisms with widths of 3nm or so is certainly possible. Considering ultra-thin films for hot-electron applications, such films do not need to be continuous or uniform. As expected, we see that the efficiency is low for Au nanocrystals with sizes about 10nm or larger. Two important effects contribute here. (1) First, the inter-band transitions dominate the absorption at the plasmon peak hω ≈ 2.5eV

(see Supporting

information) and, therefore, the rate of creation of intraband electron-hole pairs is relatively small. Simultaneously, the interband transitions create low-energy electrons in the sp-band and high-energy holes in the d-band.7 Here we count only high-energy electrons in the sp-band and, therefore, the classical limit for the efficiency is 17% at 516nm (2.4eV) and 58% at 619nm (2eV); this classical limit is estimated as the ratio between the total absorption and the intraband absorption shown in Figure S5. More discussions on this topic can be found in the recent review paper.7 (2) The second effect, that makes the efficiency smaller, is the character of the energy distribution of hot electrons for large sizes that comes from the non-conservation of linear momentum (Figures 6-8). For large sizes, the majority of hot electrons has small excitation energies and do not contribute to the number of energetic electrons Ratehigh-energy electrons . Therefore, for large sizes, we get Eff ∝ 1/ LNC , whereas for ultra-small sizes, the behavior is more complex because of the strong quantum effects. In this paper, we deal with the simplest plasmonic geometry that is the slab geometry. The plasmon peak in this geometry for gold is at about 500nm. At this wavelength, the interband transitions are very strong and, therefore, the efficiency of generation of sp-electrons is relatively 22 ACS Paragon Plus Environment

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small, as it was discussed above. At the same time, such Au nano-platelets, prisms and films can efficiently generate hot holes which also can be used in the photo-catalysis applications36. To have a large number of energetic sp-electrons, one can use nanorods and other geometries exhibiting plasmon resonances with λ plasmon > 600nm .7 Another possibility to get large numbers of intraband energetic carriers is to use nanostructures with electromagnetic hot spots as described in Ref.40. Finally, to conclude this Section, we show the effect of the relaxation rates on the generation of high-energy carriers (Figure 9). Our kinetic DFT incorporates two rates, γ ε and

γ p . Both rates are crucial for the quantitative description of generation of hot carriers. We can see this in Figure 9. When we remove the two-rate approximation and assume γ ε = γ p , the carrier distribution changes dramatically. From the equation (9) and the expression for δρ nn (Supporting Information), we can write

δ n plateau ∝

1

, δ nmax ∝

γε

γp . γε

These dependences have been derived earlier in Refs.7,40,41.

4. QUANTUM REGIME FOR THE PLASMONS IN A SILVER PLATELET:

BREAKING THE PLASMON PEAK

As promised above, we now discuss the evolution of the plasmon resonance for the Ag slabs of different sizes. We consider silver of high quality and assume a small Drude decay constant, 23 ACS Paragon Plus Environment


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which was taken from the fit to the bulk experimental data.68

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We start with the model case

when we intentionally remove the interband transitions from the picture and look what happens with the absorption spectra. Then, the matrix dielectric constant, ε matrix , metal , will be first set to 7. Figure 10 shows the results of the Kinetic DFT. For small sizes, we observe multiple plasmonic peaks and, for large sizes, the spectrum exhibits a single plasmonic peak. The result for large sizes is very close to the classical Drude spectrum (Figure 10 for the 10nm-slab). For small sizes, each peak in the set has a narrow width (about the Drude width). The scenario for the breaking of the plasmon peak in the strongly quantum regime is very different to the traditional picture of collisional broadening of the single plasmonic resonance.42,43

In the traditional

picture,42,43 the plasmon peak in a small system acquires an additional collisional broadening, which is ~ hvF / R , where R is the NC size.

Our result is very different: Instead of the

broadening effect, we observe the multi-peak structure and this observation can be easily understood in terms of the hybridization model (inset in Figure 10). In our scenario, the system has a strong collective resonance (hΩ p ) and a set of single-particle resonances in the quantized Fermi gas ( ∆ε i ) with smaller transition intensities. The plasmon mode interacts with the singleparticle resonances and the spectrum acquires a main resonance and several weaker side peaks. In

the

next

step,

we

include

the

interband

transitions

and

take

ε matrix , metal (ω ) = 7 + ε inteband, Ag (ω ) . For the interband dielectric function, we use the data from the experiment.68 To get the function ε inteband,Ag (ω ) , we have subtracted the Drude contribution from the full experimental dielectric function. Figure 11 shows the results. The broad interband structure at 320nm tends to mask the interesting multi-peak structure which comes from the quantum intraband single-particle transitions. Nevertheless, the quantum multi-peak structure 24 ACS Paragon Plus Environment

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can be seen in the absorption spectra (Figure 11). The role of the interband transitions is not only “masking” of the quantum multiple peaks. The interband transitions also create a strong additional broadening for the high-frequency quantum resonances and, therefore, we cannot see such resonances in the spectra on Figure 11 in the interval of λ0 < 320nm . The problem is that, in this interval, any intraband transition peak should acquire a strong broadening from the interband excitations. We also should note that small Ag nanoparticles and clusters indeed exhibit multi-peak structures in the plasmon-resonance region.65,73 An origin of the observed multi-peak structures of the plasmonic resonance needs more discussions and investigations. In principle, these multi-peak structures in the experiments65,73 can come from ligand-related transitions or they can be created via the quantum hybridization mechanism described in our Kinetic DFT. To conclude this Section, we like to comment more on the traditional picture of collisional broadening of the single plasmonic resonance.42,43 Ref. 42 derived the collisional broadening indirectly, from the high-frequency limit, while a redirect derivation would require to see the pole of the function 1/ ε eff (ω ) , where ε eff (ω ) is an effective dielectric constant of a nanocrystal. Within the analytical formalism of Refs.42,43, it is not easy to identify such poles. In our Kinetic DFT, we solve the problem numerically (since it not treatable analytically) and identify such plasmonic poles in the response function (absorption). And we see the multi-peak structure with one strong peak and a few smaller peaks. Another aspect of the present situation is that the approach of phenomenological collisional broadening was applied to many experiments and this approach was very useful for description. However, it can be that the plasmon peaks of small nanocrystals in the experimental data were broadened due to a size-dispersion effect or due to strong effects coming from a modified surface structure of nanocrystal. This question requires 25 ACS Paragon Plus Environment


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more studies including calculations of spherical and other nanocrystals using the Kinetic DFT. Our results here were obtained for the slab geometry. We found that collisions with the walls for small slab sizes are important and govern the plasmon peak structure and the non-equilibrium electron distribution. However, in 3D nanocrystals, such collisional effects can be even more important and the effect of hybridization can be expressed stronger. To clarify this issue, we need more modeling and calculations.

5. CONCLUSIONS

We have presented a new quantum kinetic formalism based on the DFT, random-phase approximation and kinetic master equation for the density matrix. Using our Kinetic DFT, we are able to offer a convenient description of the many-body wave function of the localized plasmon. In our theory, the populations of coherent hot electrons in a plasmonic wave depend on the two relaxation times. The momentum relaxation time governs the plasmon resonance broadening, while the energy-relaxation time is important to calculate the non-equilibrium population of hot plasmonic electrons. Our theory does not have any fitting parameters. All parameters and relaxation times are taken from the experimental data. The atomistic interband transitions play an important role in our calculations and are included via the background dielectric constant. Our Kinetic DFT suggests a new scenario of the breaking of the plasmon peak in the quantum systems. In our theory, the calculated spectra demonstrate a multi-peak structure and we explain this effect using a simple quantum hybridization model. The results


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obtained in this paper can be used for understanding and predicting a variety of plasmonic and photo-catalytic systems utilizing hot plasmon carriers.

Acknowledgements This work was supported by the Volkswagen Foundation. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract Number DEAC02-06CH11357 (Project: 37839).

Supporting Information Analytical derivations for the Kinetic DFT; a dielectric model of the metal slab; DFT calculations for the ground state; optical properties of Au and Ag nanocrystals based on the classical approach. This information is available free of charge via the Internet at http://pubs.acs.org.



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Figure 1: Optical and electronic properties of metal nanocrystals. a) Models of nanocrystals. b) Calculated classical absorptions of nanocrystals of various shapes. c) Energy diagram of an optically-driven metal nanocrystal coupled with a semiconductor or with an adsorbed molecule.

d) An example of energy distribution of excited carriers in a small nanocrystal driven by optical excitation.


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Figure 2: Description of the DFT model for the ground state of a metal NC. a) The NC potential and Fermi energy. b) The dielectric part of the model.



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Figure 3: Kinetic properties of a metal nanostructure involved in the DFT model. a) Dynamic dielectric model of a slab. The local dielectric constant includes the contributions from the bound charges and interband transitions in a metal. b) Schematic of relaxation processes in the electron plasma. Here we show both quasi-elastic relaxation of momentum and relaxation of energy.


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Figure 4: Absorption spectra of Au slabs calculated using the quantum Kinetic DFT and the classical theory. The surface area of slab is taken as A = L2NC . For the case of 10nm-slab, we also show the Drude-based classical absorption.



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Figure 5: Dynamic screening and dynamic Friedel oscillations in the plasmonic Au slabs with various widths. The plot shows the non-equilibrium change density in a slab induced by the external ac electric field.


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Figure 6: General structure of the distribution function for the coherent hot electrons and holes in the plasmonic excitation in a gold slab.



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Figure 7: Distribution functions of excited carriers for slabs with different sizes. We can see that the character of distribution of hot carriers changes dramatically with increasing size of a slab.


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Figure 8: a) Populations of hot electrons for different sizes of nano-slabs. Inset shows the interval with high-energy electrons, ε > EF . b) Rates of generation of low-energy and highenergy excitations as a function of the slab size. c) Calculated efficiency of generation of highenergy electrons with ε > EF + 0.7eV .



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Figure 9: Calculated populations of hot carriers in a slab for two sets of the kinetic parameters; the slab width is 5nm. We see that the populations are very sensitive to the choice of relaxation times.


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Figure 10: Absorptions of silver slabs with different widths, calculated without the interband transitions. The matrix dielectric constant is independent of the frequency and includes only the term due to bound-charges.

In other words, the interband transitions in the Ag matrix are

neglected. The lateral sizes are chosen as Llateral = LNC .



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Figure 11: Absorptions of silver slabs with different widths, calculated with the interband transitions.

The matrix dielectric constant is frequency-dependent and includes both the

interband term and the bound-charge term. The lateral sizes are chosen as Llateral = LNC .


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Figure for table of content


Kinetic Density Functional Theory for Plasmonic Nanostructures

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