Amplified Generation of Hot Electrons and Quantum Surface Effects in

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Amplified Generation of Hot Electrons and Quantum Surface Absorption of Light in Nanoparticle Dimers with Plasmonic Hot Spots Lucas Vazquez Besteiro, and Alexander O. Govorov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b05968 • Publication Date (Web): 05 Aug 2016 Downloaded from http://pubs.acs.org on August 7, 2016

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The Journal of Physical Chemistry

1

Amplified Generation of Hot Electrons and Quantum Surface Effects in Nanoparticle Dimers with Plasmonic Hot Spots

Lucas V. Besteiro* and Alexander O. Govorov* Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, United States

Abstract

Plasmonic excitations in optically driven nanocrystals are composed of excited single-particle electron-hole pairs in the Fermi sea. In large nanostructures, most of the excited plasmonic electrons have relatively small excitation energies due to the conservation of linear moment. However, small optically driven nanocrystal may have large numbers of hot electrons with large energies. In this study, we develop the concept of hot electron generation further by considering the effect of a plasmonic hot spot. Plasmonic hot spots are areas in a nanostructure with highly inhomogeneous and enhanced electric fields. In our model of a nanoparticle dimer, the hot spot region appears near the gap between the nanoparticles. We then apply the quantum formalism based on the density matrix to describe this system. We show that the electromagnetic enhancement and the non-conservation of linear momentum in the hot spot of the nanoparticle dimer lead to strongly increased rates of generation of energetic (hot) electrons. The rates of hot electron generation grow faster than the absorption cross section and the electromagnetic

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2 enhancement factor with the decrease of the gap between the nanoparticles. This happens due to the breaking of the linear momentum conservation of electrons in the hot spot regions. We also show that hot electron generation effect leads to the quantum mechanism of surface-induced absorption in nanocrystals that is an intrinsic property of any confined plasmonic system. The results obtained in this study can be useful for understanding and designing plasmonic photodetectors and hybrid materials for efficient photocatalysis.


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3 1. INTRODUCTION

Hot electron generation is a very active research topic within the study of plasmonic phenomena in nanostructures. This topic concerns two types of applications, plasmonic systems for enhanced photochemistry1-9

and

photodetectors

and

solar

cells

based

on

plasmon-enhanced

photocurrents.10-19 Recently, an interesting question has been posed: Whether plasmonic hot spots in metal nanocrystals can be used to amplify the rates of generation of energetic electrons.20 Plasmonic hot spots are small spaces in metal nanocrystal assemblies where the electromagnetic field of incident light becomes strongly enhanced due to the geometry of the system. Obviously, photochemistry and photocurrents can be strongly affected by such spots. Two mechanisms can contribute to the generation of energetic electrons by the hot spot regions:21 (1) The plasmon enhancement effect and (2) the non-conservation of linear momentum of electrons in a confined system. The first mechanism is classical, where the second one is a quantum phenomenon. The effect of hot spots on the photochemistry has been proposed in ref 19 and several experimental and theoretical studies have recently confirmed the importance of hot spots for phenomena related to generation of energetic (hot) electrons.21-23 Then, certainly, the effect of hot-spots in the generation of energetic carriers, which can be used in photocatalysis, is an interesting and promising direction of study. Here we study theoretically the quantum structure of the plasmonic excitations in single nanoparticles (NPs) and in nanoparticle dimers with hot spots. The NP dimer structure is an excellent model system to study the hot spot effects.24 In this study, we apply a hybrid approach, adapted from the one we previously developed to describe plasmonic kinetics of metal nanostructures.21,25-27 First, we describe a classical electrodynamic dielectric model, in which we



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4 compute the electromagnetic fields using numerical finite-elements methods (COMSOL). For the case of a single NP, we can use the analytical solution. In the second step, we utilize the quantum formalism of the density matrix and the approach of two relaxation times. Using this hybrid approach based on the self-consistent field approximation, we are able to treat the quantum effect of generation of hot electrons in a complex plasmonic system such as the NP dimer with a plasmonic hot spot. As a result, we obtain the energy distributions of hot electrons and related energy dissipation rates. The rates of generation of energetic electrons significantly increase in the structures with intense hot spots. Two physical effects contribute to this behavior: The classical effect of plasmonic enhancement and the quantum mechanism that generates highenergy electrons in the hot spot region. The latter happens due to the breakdown of momentum conservation of electrons in the presence of the non-uniform electric fields of the hot spot. The concept of plasmonic hot spots became very popular since the discovery of Surface Enhanced Raman Scattering.28,29 In the SERS phenomenon, Raman signals of small dye molecules attached to metal nanocrystals are greatly amplified due to the plasmonic field enhancement effect. Another related effect, developed recently in refs 30 and 31, is the plasmoninduced circular dichroism of biomolecules by which enhanced optical signals of chiral molecules/UV-emitters appear in the visible wavelength interval. The effect of generation of hot plasmonic electrons due to the hot spots, which we discuss in this paper, is one more phenomenon related to the concept of hot spots. This phenomenon, however, is significantly different from SERS and plasmon-enhanced optical activity since it relies more on the intrinsic quantum properties of the excited Fermi gas in metal nanocrystals. Regarding theoretical methods, the problem of hot plasmonic electrons has recently been treated with a variety of related methods, involving perturbative approaches for the injection


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5 currents, Fermi’s golden rule, non-equilibrium Green functions, quantum kinetic approaches, etc.13,32-35 Our approach is based on the perturbative solution for the density matrix with two relaxation times.21,25-27 The convenience of our approach resides in that it provides (1) a realistic description of electronic and plasmonic kinetics in real metals and (2) the ability to directly calculate the energy distributions of hot plasmonic electrons generated in typical metal nanocrystals. Importantly, in contrast to the field theories based on the quantization of plasmons as quasi-particles,36,37 we look at the properties of the many-body wave function using the approach of the one-electron density matrix. We should also mention the phothermal applications of plasmonic nanocrystals in chemistry.38-40

The concept of hot electrons and the photothermal effect in plasmonic

nanocrystals are related, but have an important difference: These two phenomena involve two different time scales. Optically excited hot electrons live for very short times and represent a quantum effect. The effect of photoheating has much longer lifetimes due to its dependence on classical heat diffusion processes. However, both phenomena concern optical energy being stored in a metal nanocrystal.



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6

2. FORMALISM

2.1 ELECTROMAGNETIC MODELS AND THE GEOMETRIES

The electrodynamic part of the problem of hot electrons relies on the standard approach based on the local dielectric constants of the materials (Figures 1a,b). Our models include one or two metal nanoparticles (NPs). We denote the dielectric constants of the matrix and the metal as ε0 and ε Au (ω) , respectively. In our calculations, we will use ε0 = 2 , which corresponds to that of a polymer matrix. For the optical dielectric constant of gold, we will use the data from ref 41. Besides, we denote the NP radius and the gap between the NPs as R0 and ∆ , respectively. The dimers consist of two Au NPs with identical radii. In our models of small nanocrystals, we assume that the external electric field is uniform and monochromatic,

Eexternal = 2 ⋅ E0 cos (ω ⋅ t ) = E0 ⋅ e−iωt + E0 ⋅ e+iωt .

Here E0 is the field amplitude. Using this input, we compute numerically the near-field, which has a complex spatial structure in the NP-NP geometry, using the COMSOL software. For these fields we adopt the notation Eω . Then, the physical electric field can be found via the following standard equation:


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7

Eoptical = Eω ⋅ e−iωt + Eω* ⋅ e+iωt .

(1)

Figure 1b shows the numerical maps for the amplitude of the local field Eω , in units of E0 = E 0 . For small NP-NP gaps, we observe the formation of electromagnetic hot spots in the

central region, as expected.24 We now introduce a couple of optical parameters that are convenient for the understanding of our systems. First, we define the local enhancement factor of the optical electric field,

Plocal (r) =

Eω ⋅ Eω* . E02

(2)

In the NP dimers, we observe the formation of plasmonic hot spots near the NP-NP gap region where the local enhancement factor is stronger (Figure 1b). The plasmonic hot spots in the dimers appear due to the interaction between NPs.24 Of course, for large gaps the hot spot effect vanishes, as we can see in Figure 1b, and the remaining local enhancement factor inside the NPs is then caused by dynamic charge screening, occurring in the isolated NPs due to their different material properties with relation with the matrix, as well as their individual geometry. The following integral describes the overall strength of the electromagnetic field enhancement in our systems

Penh (ω ) =

1 2 ⋅ VNP

Eω ⋅ Eω* ∫ E02 dV , NPs


(3)


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8 where the integral is taken over the volumes of the NPs. The rate of absorption of light energy by the NPs is given by

Qabs =

∫

= Im(ε Au )

dV j ⋅ E

NPs

time

ω 2π

∫

dV Eω ⋅ Eω* ,

(4)

NPs

where jω is the dynamic current in the NPs. Correspondingly, the absorption cross section is given by

σ abs (ω ) =

Q abs , I0

I0 =

c0 ε0 2 ⋅ E0 . (5) 2π

In Figure 2, we show the numerical results for the absorption cross section and the integrated enhancement factor of the NP dimers with different gaps. For small gaps, we observe the typical blue shift of the plasmon resonance, which comes from the NP-NP coupling.24 For the case of a single NP, the equations for the absorption rate and the integrated enhancement factor are simple,42

σ abs , NP =

2

4π R03 3ε 0 Im(ε Au ), ω⋅ 3 ε Au + 2ε 0 ε0

1 c0

2

3ε 0 Pabs , NP (ω ) = . ε Au + 2ε 0

For real metals, the above absorption rate can be split into two parts in the following way:

σ abs , NP = σ NP, intraband + σ NP, interband . The above terms describe the intraband and interband processes, respectively (see Supporting Information and also ref 26 for more details). Both


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9 quantities Pabs , NP and σ abs , NP will be used in the following sections for interpretation of the results. We now turn to the quantum microscopic part of the paper.

2.2 KINETIC MASTER EQUATION FOR THE DENSITY MATRIX WITHIN THE PERTURBATION THEORY

Before considering specific systems, let us briefly describe the theoretical formalism, for which a detailed derivation can be found in our previous work.21,25,26 We will then show the results of applying this formalism to calculate the generation of energetic carriers in metal NP dimers with plasmonic hot spots. We model the electronic system through the one-particle density matrix ρnm (t ) , which contains the information about the statistical energy distributions of excited plasmonic carriers. Within the many-body theory of electron gas,43,44 we can reduce the many-body wave function of an electronic system to the one-particle density matrix:

ρ nm (t ) = Ψ (t ) cˆm+ cˆn Ψ (t ) ,

(6)

where Ψ (t ) is the time-dependent many-body function of a nanocrystal’s electrons in the presence of the continuous wave (CW) illumination; cˆn and cˆm+ are the second quantization



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10 operators for the corresponding single-particle states. In order to compute the non-equilibrium density matrix, we now write down the kinetic equation of motion

h

∂ρˆ = i  ρˆ , Hˆ sp  − Γˆ ρˆ , ∂t

(7)

where ρˆ is the operator of the density matrix that generates the elements ρ nm = n ρˆ m ; Hˆ sp is the single-particle Hamiltonian within the random-phase approximation (RPA)42 and Γˆ is the relaxation operator,

 ρˆ mn − f F (ε n )  , m = n h τε   m Γˆ ρˆ n =    h ρˆ mn , m≠n   τ p 

(8)

where τ p and τ ε are the characteristic relaxation times for momentum and energy, respectively;

f F (ε ) is the equilibrium Fermi distribution function at room temperature. In our model, illustrated in Figure 3a, the momentum relaxation time is short and leads to the broadening of all resonances; this is the coherence time of the plasmon excitation. The second time describes energy relaxation and it is typically longer. The table in Figure 3 provides the chosen relaxation times, which correspond to gold.45 The Hamiltonian in eq 7 includes the one-electron energy and the optically induced selfconsistent potential,

h2 2 Hˆ sp (r, t ) = − ∇ + U 0 (r) + Vôptical (r, t ) , (9) 2m0


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11 where U0 (r) is the confining potential of a nanocrystal.

The time-dependent perturbation,

Vôptical (r , t ) , originates from the external CW excitation:

Vôptical (r, t ) = Va (r )e − iωt + Vb (r )eiωt , Va = eϕω (r ), Vb = eϕω* (r ),

(10)

where ϕω (r) is the complex amplitude of the electric potential induced inside a nanocrystal and e = − e is the electron charge. We assume that our nanocrystals are small and the near-field

quasi-static approximation is applicable. The total dynamic potential in eq 10 has the following structure:

ϕω (r) = ϕexternal ,ω (r) + ϕinduced ,ω (r)  , ϕexternal ,ω (r) = −E0 ⋅ r, where the total dynamic potential is split into two parts. The potential ϕexternal ,ω is created by the external driving field defined in the previous section. The induced potential ϕinduced ,ω comes from the dynamic surface charges on the NPs. To find the total dynamic potential for the dimer, we should solve Maxwell’s equations numerically. For the case of a single sphere, the total potential inside the NP has a simple solution, given previously. In the next step, we apply the perturbation theory to solve eq 7. The perturbative solution for the diagonal matrix elements reads:21,26

ρ nn = Ψ (t ) cˆn+ cˆn Ψ (t ) ≈ f F (ε n ) + δρ nn(0) , δρ nn(0) =

 f f ( − ) V ∑ γ ε n ' n ' n  nn ', a 2

2

γp 2 ( hω − ε n + ε n ' ) + γ p2

+ Vnn ',b


2

 , 2 ( hω + ε n − ε n ' ) + γ p2 

γp

(11)


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12 where f n = f F (ε n ) is the value of the Fermi distribution at energy ε n . The transition matrix elements in eq 11 are defined as

Vnn′, a = n eϕω (r ) n′ .

Here n

(12)

and ε n are the single-particle states and their energies, correspondingly. The two

important broadening parameters involved in eq 11 are given by γ p = h / τ p and γ ε = h / τ ε , respectively. We have previously derived eq 11 in ref 26 as part of the Kinetic DFT formalism that incorporates the two relaxation times illustrated in Figure 3. We continue by computing the population of excited states in the optically driven NPs. This population should be calculated as the sum:

δ n(ε ) = 2∑δρnn ⋅ P(ε − ε n ) .

(13)

n

In eq 13, the coefficient 2 originates from the spin states and the function P(ε −εn ) is a pulse function given by

P (ε ) =

1

δε

P (ε ) = 0,

, ε < δε / 2,

ε > δε / 2,

where δ ε is the broadening that should be taken as a small value in the numerical computations. We note that the perturbative approach,21,26 which gives eq. 11, is very productive since it reduces a very complex many-body problem to a large set of numerical one-electron coefficients

δρnn(0) . Then, we use these elements to create the non-equilibrium distribution function of


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13 plasmonic electrons, which provides us with the picture of the coherent population of electronic states in the localized plasmon waves in NCs.

The optical matrix elements (eq 12) use the total potential ϕω (r) . However, it is typical for numerical electrodynamic packages to solve for the electric field instead. Therefore, it is convenient to modify eq 12 to the following form:

ϕnn ', a

r h2 = n ϕω (r ) n ' = dV ⋅ ψ E ⋅ ∇ ψ n ' , (14) n ω m (ε n − ε n ' ) ∫

(

)

r where Eω = −∇ϕω is the complex amplitude of the total electric field inside a nanocrystal. We

defined this magnitude in the previous section. Another important element of the calculations for plasmonic nanocrystals is the averaging over the nanocrystal size.21,26 The averaging is crucial since the non-equilibrium energy distributions of the NPs typically exhibit very strong quantum oscillations as a function of energy. Physically, this averaging describes an ensemble of NPs with a dispersion of sizes. All results presented in this paper have been averaged by using the following procedure:

δ n (ε ) =

1

πδR

+∞

∫ dR ⋅e

 R − R0  −   δR 

2

⋅ δ n (ε , R ) ,

(15)

−∞

where δ R is the parameter describing a dispersion of NP sizes and R0 is the average size of a NP. When considering the dual sphere configuration, we apply the same averaging approach (15) also to the inter-particle gap ∆ . To do this, we express the gap ∆ as a fraction of R and then perform the averaging.



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14

3. RESULTS

3.1. HOT ELECTRONS IN A SINGLE PLASMONIC SPHERE

We start with the case of a single NP. In the equilibrium, the electrons in Au NPs form a Fermi gas with the Fermi energy of EF = 5.5 eV (Figure 3a); the corresponding Fermi velocity is then equal to vF = 1.38 ⋅108 cm / s . Under illumination, the mobile electrons of a NP screen the external dynamic field. The resulting dynamic potential inside a single NP has the form

ϕω (r ) = γ sphere (ω ) ⋅ ϕexternal ,ω (r ), γ sphere (ω ) =

3ε 0 . ε Au (ω ) + 2ε 0

For the single-electron states of a spherical well, we have

 α nr l r   ⋅ Ylm (θ , φ ) , R  

ψ n ( r , θ , φ ) = Rl  εn =

h 2α n2r l 2m0 R 2

,

where Rl ( r ) are the Spherical Bessel functions, being α nr l its nr th zero, and Ylm (θ , φ ) are the spherical harmonics. The state n is now described by the three quantum numbers, nr , l, m .


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15 These wave functions and their energies are involved in the optical matrix elements given by eq 14. We now compute the energy distribution of electrons in the non-equilibrium steady states of spherical NPs using eqs 11 and 13. Figure 3b shows the typical graphs for the energydistribution function δ n (ε ) . Values for δ n (ε ) are positive for ε > E F and negative for ε < EF . Physically, we create the occupied electronic states (excited electrons) above the Fermi level and the empty states (holes) below the Fermi level. The function δ n (ε ) has a very characteristic shape. We observe the pronounced peak-dip structure near the Fermi level when ε ≈ E F . This peak-dip structure describes the excited states of electrons and holes with small excitation energies. In large nanocrystals, we typically use the Drude model to describe such low-energy excitations and the related Joule heating effect. Then, we see two plateaus in the regions EF − hω < ε < EF and E F < ε < E F + hω . These structures correspond to the hot plasmonic

electrons generated in a NP due to the non-conservation of momentum. This is a quantum effect and originates from the quantum transitions described by the optical matrix elements (eq 12). The plateaus include electrons with high energies that can be used for photocurrents and photochemistry. From Figure 3b, we see that the energy distribution of excited carriers is very sensitive to the choice of the relaxation rates. Therefore, a realistic description of a plasmonic nanocrystal requires the right choice of the relaxation parameters. Figures 4, 5 and 6 provide a summary of the properties of a single plasmonic NP. To compute the distributions, we used the following parameters for the size averaging: δ R = 1 nm , 1 nm , and 0.5 nm for the NP sizes R0 = 3 nm , 2 nm , and 1 nm , respectively. The above

choice of the size broadenings satisfies R 0 ≥ 2δ R .



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16 We now look at the very characteristic properties of the function δ n (ε ) . In Figure 4, we observe that the intensity of the peak-dip structure grows rapidly with the NP size, approximately as δ nmax ∝ R05 for large NP radii. This is a bulk effect, which comes from the quantum transitions with low excitation energies in the bulk of the NP. The width of the maximum of the function

δ n (ε ) is about hk NP vF , where kNP = π / 2R0 is the characteristic momentum transfer due to the size quantization in a NP. For our NP sizes, we observe that the maximum width is still in the quantum regime, i.e. hk NP vF >> k BT , where kBT is the room-temperature thermal energy.

The

plateaus in Figure 4 behave very differently, since they originate from the quantum effect of dynamic scattering of electrons at the surfaces of the NP. Due to the surface, the electrons in the NP do not conserve their linear momenta and, in the presence of the external driving field, these electrons can absorb the full photon quantum hω . For the internal photoelectric effect in solids, this mechanism of absorption has been described in the seminal paper in ref 46. Correspondingly, we observe in the numerical calculations that the generation of electrons with high energies is proportional to the surface area of the NP: δ nplateaus ∝ R02 (Figure 5a). To better understand the physics behind the hot electron distributions, we now introduce some characteristic parameters. In particular, we define the rates for the total numbers of excited electrons (holes) and for the high-energy electrons:

Rateelectrons , tot = Rateholes , tot = Ratelow−energy electrons =

1

τε

Ratehigh−energy electrons =

1

τε

1

τε

EF + 0.5 eV

∫

∞

EF

∫ dε ⋅δ n (ε ) = − τ ε ∫ dε ⋅δ n (ε ),

EF

dε ⋅δ n (ε ),

EF ∞

∫

1

d ε ⋅δ n (ε ).

EF + 0.5 eV


0

(16)

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17 Figure 5b shows that these rates are comparable for small NP radii ( R0 ≤ 2 nm ) and then the total rate dominates: Rateelectrons,tot >> Ratehigh−energy _ electrons for R0 > 2 nm . This is an expected behavior since the total rate comes from both bulk and surface effects, where the rate of generation highenergy electrons is only a surface effect. From the fitting in Figure 5b, we observe the expected dependences: Rateelectrons ,tot ∝ R04 and Ratehigh − energy electrons ∝ R02 . We now look at the rate of energy dissipation in a single NP. In our picture, external illumination first creates excited states of electrons and, in the second step, the electrons transfer energy to the lattice via the process of phonon emission. The rate of energy dissipation due to the relaxation of carriers is given by another integral

Qtot =

1

τε

∞

∫ dε ⋅ε ⋅ δ n (ε ) = 0

1

τε

∞

∫ dε ⋅ (ε − E ) ⋅ δ n (ε ) . F

(17)

0

We distinguish again between the “bulk” and “surface” effects in the rate of dissipation. For this, we split the integral in eq 17 into two parts:

Qtot = QLow− Energy + QHigh− Energy , QLow− Energy = QHigh− Energy =

1

τε 1

τε

EF + 0.5eV

∫

dε ⋅(ε − EF ) ⋅ δ n (ε ),

EF −0.5eV

∫

dε ⋅(ε − EF ) ⋅ δ n (ε ).

EF −0.5eV >0


(18)


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18 Figures 6a shows the calculated behaviors of the above parameters as a function of the NP radius. We again observe the characteristic bulk and surface effects

QLow − Energy ∝ R03 , QHigh − Energy ∝ R02 . As expected, the low-energy term is due to the optical absorption in the bulk of the NP. Simultaneously, the high-energy term originates in the quantum transitions near the surfaces, where the electrons do not conserve their momenta due to scattering at the boundary. For large sizes, R0 → ∞ , the low-energy term starts to dominate and the total dissipation behaves as

Qtot ∝ R03 . The behavior Qtot ∝ R03 indicates that the classical regime of dissipation should occur for large NPs. It is now time to compare our quantum calculations with the classical Drude model, which should be applied for NPs with large radii. The Drude model should describe the dissipation of the low-energy electrons, whereas the generation of high-energy electrons is a quantum effect and is beyond the Drude approach. Indeed, our model perfectly reproduces these qualitative expectations. In Figure 6a, we now compare the quantum calculation and the classical intraband absorption and we observe excellent agreement. Indeed, these two absorption rates are very similar:

2

QLow− Energy ≈ Qabs, intraband = I 0 ⋅ σ abs, intraband

2R3 3ε 0 = ω ⋅ 0 E02 Im(ε Au , intraband ) . (19) 3 ε Au + 2ε 0

The classical calculation for the rate of dissipation of low-energy Drude electrons (see eq 19 above) was performed using the empirical dielectric constant of gold and the standard treatment


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19 of the interband and intraband processes (see Supporting Information). In particular, we used in eq 19 the formula for the intraband absorption σ abs, intraband , which is given in Supporting Information. In Figure 6a, we still see significant quantum effects for NPs with R0 ~ 10nm since

Qtot > QLow−Energy ≈ Qabs, intraband and the difference between Qtot and QLow− Energy is noticeable. Now we will introduce a quantum parameter that describes the role of quantum effects in the absorption. The absorption rate QLow− Energy ≈ Qabs, intraband has a classical nature and we can describe it by the classical Drude model. The electrons with small excitation energies experience a so-called “frictional force” that is the feature of the famous Drude model. The purely quantum effect in the absorption, as described in our formalism, is the generation of hot electrons with large excess energies ∆ε ~ hω . This generation is given by the dissipation rate QHigh−Energy , and this effect is a quantum surface effect. Then, we introduce the related quantum parameter: QP =

QHigh − Energy Qtot

,

where the absorption rates were defined by eqs 18. Figure 6b shows the typical behavior of the quantum parameter as a function the NP radius. This parameter decreases with the NP radius, since the surface effects become less important for large volumes. We also observe that our NPs, with sizes chosen in the range 1 nm ≤ R0 ≤ 12 nm , are in the quantum regime, in where quantum surface effects make an important contribution to the total rate of absorption. For large NP radii, we expect QP ∝ 1 / R0 , since the QHigh − Energy is a surface effect and Qtot becomes mostly a bulk effect for R0 → ∞ .



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20

3.2 NANOPARTICLE DIMER WITH A PLASMONIC HOT SPOT

The most interesting case is a physical system with a plasmonic hot spot, i.e. with a spatial region where incident electromagnetic fields become amplified and concentrated. Another property of a hot spot region is a very strong inhomogeneity of plasmonic fields. We show such case in Figure 1b. In our model, the electrons are confined in two spherical wells. Then, these electrons experience the action of the plasmonic field created by external illumination. The formalism is similar to the case of a single NP, but there is an important difference. In the NP dimer case, the plasmonic fields become non-uniform and strongly amplified at some positions due to the hot spot effect. In our formalism, the transition matrix elements (eq 14) directly depend on these non-uniform and amplified plasmonic fields. In the sum responsible for the hot electron distribution (eqs 11 and 13), we now include thousands of matrix elements that describe the corresponding quantum transitions induced by the non-uniform plasmonic fields. Figures 7 to 11 show the results for the NP dimers. Figure 7 reveals the typical energy distributions of excited electrons in the plasmonic resonance, including both the high-energy plateaus and the low-energy peak-dip (Drude) spectral structures. Then, we plot in Figure 8 the integrated generation rates of low- and high-energy electrons defined by eqs 16. Overall we see that the optical generation of hot carriers becomes amplified for small NP-NP gaps. In principle, such behavior is expected since the electric fields are amplified in general in the NP dimers (Figure 2b), but now we look into the details of the amplification process. The process of generation of carriers is, of course, resonant and the corresponding rates exhibit the plasmon resonance of the dimer (Figure 9). We observe this resonant behavior for both low- and high-


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21 energy electrons. Mathematically, the plasmon resonance in the carrier generation rates appears via the matrix elements of the electric field (eq 14). These matrix elements involve singleelectron wave functions that exhibit rapid oscillations in space and, therefore, the carrier generation rates are, of course, not proportional to the integrated enhancement factor Penh given by eq 3. At this point, it is interesting to compare the generation rates and the enhancement factor

Penh for different NP-NP gaps keeping the NP radius fixed. Then, we observe an interesting behavior: The rate of generation of high energy electrons increases faster than the enhancement factor when we decrease the NP-NP gap. We can see this in Figure 10a, where we plot the carrier rate and the enhancement factor at the plasmonic peak for different gaps and radii. Among other details, we observe that these parameters approach their asymptotic values (those of the isolated sphere, shown as horizontal lines) for large NP-NP gaps. We also can compare the rates and the absorption cross sections at the plasmonic wavelength (Figure 10b) and we again observe the same feature. The rate increases faster than the absorption when the NP-NP gap becomes small. We will now interpret and explain this difference in the behaviors of the optical parameters ( Penh and σabs ) and the rate ( Ratehigh−energy electrons ). The process of generation of high-energy electrons comes from the quantum transitions between pairs of states separated by the photon energy, i.e. from the transitions ε n → ε m = ε n + hω . Such process really requires a large change in the electron momentum. The violations of the momentum conservation in NPs can originate from the quantum confinement or from the strong and non-uniform fields in the hot spots. In the case of the dimers with small gaps, we create the second mechanism, the plasmonic hot spots. Therefore, we can expect a stronger amplification of the generation rates for small gaps, since in tightly packed NP dimers the effect of the hot spots becomes much stronger.



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22 It is also interesting to look at the quantum parameter for the dimers. Figure S1 in Supporting Information gives the dissipation rates for the high-energy and low-energy carriers defined by eqs 18. From the data in that figure, we plot the quantum parameter for the dimers with different NP radii (Figure 11). We again see that the role of the quantum effects is reduced with increasing the NP radius. When all sizes of the system increase, we expect that the quantum parameter should decrease. Behavior that we indeed observe in Figure 11. Eventually, for

R0 → ∞ , the quantum parameter QP will approach zero. In our simulations for the dimers computing time becomes a limitation, as the problems involve a very large number of rapidlyoscillating single-particle states and their matrix elements.21 Therefore, we have data for a limited interval of R0 in Figure 11 we have data for a limited interval of R0 .

4. CONCLUSIONS We investigated theoretically the generation process of hot electrons in single NPs and in NP dimers with a plasmonic hot spot. In a single NP, we describe the characteristic properties of the excited plasmonic steady states, such as the formation of optically excited populations with lowenergy and high-energy electrons. We also show that the generation of high-energy electrons is equivalent to the quantum effect of absorption at the surface of the NP. As predicted, we found that the quantum effects due to surface scattering of electrons are only important for small NPs. In the more interesting case of the NP dimer with a hot spot, we observe the increase of hot carrier generation rates. Interestingly, the rates of hot electron generation typically grow faster than the enhancement factor and the absorption cross section. We then show that the amplification of hot electron generation has two origins: (1) Global field enhancement in the NP dimer and (2) non-conservation of linear momentum of an electron due to the non-uniform fields


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23 of the hot spot region. Overall, we have found that the hot spot effect in plasmonic nanostructures looks promising for applications in photochemistry where hot plasmonic electrons can be utilized to promote various chemical reactions.

Supporting Information: Details regarding the calculation of optical absorption for real metals. Additional data for the dissipation rates in the dimer system.

ACKNOWLEDGEMENTS This work was supported by Volkswagen Foundation (Germany) and by the Army Office of Research (MURI Grant W911NF-12-1-0407).

Corresponding Authors *Contact details: [email protected], (740) 597-2661; [email protected], (740) 593-9430.



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24

FIGURES


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25

Figure 1. (a) Geometry and relevant parameters for the dimer system. (b) Hot spot intensity, represented through the square root of local field enhancement inside the Au-NPs for different inter-particle distances, as obtained by classical electrodynamic calculation. (c) Cartoon of an energy diagram depicting optical excitation of carriers inside the metal and their transfer to neighboring materials.



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26

Figure 2. Classical responses of the dimer under CW illumination. Results for NPs with R0 = 3 nm . (a) Absorption cross section per NP in the dimer, compared with that of an isolated

spherical NP. (b) Local enhancement factor, calculated as described in the equation in the left, for each sphere of the dimer, compared with a single NP.


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27

Figure 3. (a) Schematic of the relaxation processes involved in our kinetic model, along with the numerical values of relaxation parameters employed in this work. (b) Typical excited-carrier distributions obtained with our model. These carrier distributions depend strongly on the choice of the relaxation parameters.



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28

Figure 4. Excited carrier distributions for isolated NPs of a range of sizes, obtained with our kinetic model. The subfigures show the same data at different scales to make evident specific features. (a) There is a peak-dip structure immediately around the Fermi energy. This structure reveals the excited holes and electrons with low energy. (b) At higher energies, we encounter that the excited electron population exhibits a plateau in the region ε < EF + hω . The excited hole population shows a similar but inverted plateau for ε > EF − hω . Insets: Model of a spherical NP and the energy diagram for the excited energetic (hot) electrons.


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29

Figure 5. Energetics of the single sphere under CW illumination for various sizes. (a) Height of the population distribution of hot electrons at the plateau region. We show that its value is proportional to R02 . (b) Comparison between the total rate of generation of excited electrons and the excitation rate of electrons with large energies ε ≥ EF + 0.5 eV . They grow with the size of the NP as R04 and R02 , respectively. Inset: Same plot at small values of R0 .



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30

Figure 6. (a) Energy dissipation rates. The contribution of the high-energy electrons (depends on the surface area) dominates over the one by low energy electrons (depends on the volume) when

R0 < 8 nm . We also see that the energy absorbed by the low-energy electrons is well described by the Drude model. (b) Quantum parameter of a NP as a function of the radius. When the size of the NP is small, the quantum effect, which is due to the surface-induced generation of carriers with high energies, becomes dominant. For larger radii, the quantum parameter decreases and the NP gradually develops towards the classical regime of dissipation.


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31

Figure 7. Excited carrier population distributions in the dimer. Here we show the populations per NP. Each plot includes the results of systems with different inter-particle gap for comparison, along with the results of the isolated sphere. The plots are grouped by energy of the incident illumination (increasing from left to right) and by the radii of the NPs in the dimer (decreasing from top to bottom).



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32

Figure 8. Electron excitation rates within each NP in the Au-NP dimer as a function of the interparticle

gap,

comparing

those

of

low

(

ΕF < ε < ΕF + 0.5 eV

)

and

high

( ΕF + 0.5 eV < ε < ΕF + hω ) energy electrons. The plots are grouped by NP radius (decreasing from left to right) and photon energy (increasing from top to bottom).


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33

Figure 9. Electron excitation rates within each NP in the Au-NP dimer as a function of the photon energy. The upper panel presents the high-energy electrons and the lower one the lowenergy excitations. Results are grouped by NP radius, decreasing from left to right. The excitation rate of electrons is strongly impacted by the plasmonic enhancement of the fields inside the particles, exhibiting the same resonant response observed in the classical electrodynamic calculation (see Figure 2).



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34

Figure 10. Comparison of the excitation rates of high-energy electrons in the dimer alongside the enhancement factor (a) and the peak optical absorption in the NP (b), showing the values for the isolated spheres as the asymptotes for large NP-NP gaps. This figure gives the data at the plasmonic peak wavelengths.


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35

Figure 11. Quantum parameter for the dimers as a function of the NP radius. We again see that the quantum effects become less important when increasing the size of a system.



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