Landau Damping and Limit to Field Confinement and Enhancement in

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Landau Damping and Limit to Field Confinement and Enhancement in Plasmonic Dimers Jacob B. Khurgin, Wei-Yi Tsai, Din Ping Tsai, and Greg Sun ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00860 • Publication Date (Web): 10 Oct 2017 Downloaded from http://pubs.acs.org on October 10, 2017

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Landau Damping and Limit to Field Confinement and Enhancement in Plasmonic Dimers Jacob Khurgin1, Wei-Yi Tsai2, Din Ping Tsai2,3, and Greg Sun5 1

Department of Electrical and Computer Engineering, John Hopkins University, Baltimore, Maryland 21218. 2

3

5

Department of Physics, National Taiwan University, Taipei 10617, Taiwan.

Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan.

Department of Engineering, University of Massachusetts Boston, Boston, Massachusetts 02125.

Abstract Plasmonic dimers and other similarly-shaped plasmonic nanoantennas are capable of achieving large field enhancements inside a narrow gap where surface plasmon polaritons (SPPs) are excited. As the electric field concentration increases, two primary non-local effects emerge: an increase in energy dissipation and an expansion of the region in SPP mode (diffusion). While phenomenological theories of nonlocality exist, fundamentally nonlocality is very well-described by Landau damping, i.e. direct excitation of electron-hole pairs in the metal by the highly confined electric field of SPPs. This work verifies and extends our original, simple, self-consistent model by (1) calculating the effect of Landau damping on the field enhancement, effective volume, and linewidth of the SPP mode in the plasmonic dimer, and (2) demonstrating with extensive numerical simulations that major changes of SPP properties occur in the dimers with gaps as large as 1-2 nm, where they cannot be caused by the electron tunneling. Landau damping presents the most practically-relevant limit to the achievable degree of plasmonic enhancement.

Keywords: Landau damping, surface plasmon, plasmonic enhancement

1

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Introduction The most remarkable feature of plasmonic structures in the optical wavelength range is their ability to concentrate the optical field into the surface plasmon polariton (SPP) modes with volumes that can be orders of magnitude less than (λ/2n)3 where λ is the wavelength in vacuum and n is the refractive index. If the SPP mode can be coupled to the propagating electro-magnetic wave, the peak field inside the mode can be enhanced by orders of magnitude relative to the incident wave. In such structures, called (plasmonic) nanoantennas, all processes, both linear and especially nonlinear, are enhanced [1-3]. In particular, the Purcell effect amplifies the rate of spontaneous emission of radiation by as much as three orders of magnitude. A typical nanoantenna incorporates a dielectric gap between two metallic structures of various geometry; the concentrated electric field is confined within this gap. One example of a nanoantenna is a plasmonic dimer (Fig. 1a) consisting of two metal spheres of radius a separated by a dielectric gap d into which the object of the plasmonic enhancement (an atom, a molecule or a quantum dot) is placed [4-6]. Because the shape of this dimer can be both (1) simply modeled mathematically and (2) fabricated relatively easily, it has been studied extensively. Most of the conclusions obtained for this dimer are easily applicable to many other classes of nanoantennas (e.g., half-wave, Yagi, patch) which almost always incorporate two metal fixtures separated by a gap [7,8].

Nanoantennas constructed with plasmonic dimers have been successfully employed to observe surface-enhanced

Raman

spectroscopy

(SERS)

[9,10]

as

well

as

two-photon

excitation

photoluminescence [11]. Techniques that place active molecules at the hotspots of plasmonic dimers for sensing applications have also been explored [12]. Naturally, reduction of the gap size causes decrease in the SPP volume which is expected to be accompanied by the commensurate increase in the energy 2


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density of the SPP field and Purcell factor. However, once the gap size decreases to a few nanometers and less, both the field enhancement and Purcell factor cease to increase while the linewidth of the resonance eventually broadens to the degree where the SPP resonances are no longer discernible [1315]. Hence the size of the gap always needs to be optimized in order to attain maximum enhancement for a given optical process.

In this paper we show convincingly that all the observed changes occurring as the gap shrinks to a few nanometer size can be traced to Landau damping, i.e. direct absorption of SPP’s caused by large wave vectors in their spatial spectrum. Expanding our previous basic analysis [16] of Landau damping and electric field confinement limits: we develop self-consistent method that shows how the SPP loss increases dramatically causing not only large broadening and slight spectral shift of the SPP resonances, but also expansion (or “diffusion”) of the mode inside the metal, previously analyzed only using phenomenological treatment. Finally, we show that in the real space all absorption takes place within a few nanometers from the sharp surface thus confirming that Landau damping can be interpreted as surface-collision assisted damping [17].

Existing treatments of nonlocality A number of theories have been proposed to explain the cause for the saturation and ultimately decline of the SPP field enhancement in the plasmonic gap [14,18-22]. For extremely narrow gaps, two quantum phenomena that are ultimately responsible for limiting the maximally achievable field confinement in the gap are known to take place. One is the tunneling of electrons between the two constituent particles of the dimer, each acting as an electrode. This effect is expected to become 3


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substantial when the tails of two wavefunctions associated with the two particles begin to overlap in the gap, which happens when the gap size becomes comparable to a few times of the decay length of the electron wavefunction in the gap dielectric. The decay length can be estimated through its relation with the metal work function Φ as Ld ~ h / 2mΦ

1 Å for Φ 4-5 eV, which puts the onset of tunneling at

the gap size of about a fraction of a nanometer. The other quantum effect is finite extent of the electron wavefunction. Since the maximum Fourier component of electron wavefunction cannot exceed the Fermi wave vector k F , any change in electron density must occur over a spatial range that is greater than the Thomas-Fermi screening length λTF

k F−1 . This spread (or spill-over), on the order of a few

tenths of a nanometer for all noble metals, can be interpreted by the semi-classical non-locality theory [14,15] in which longitudinal dielectric constant of the metal possesses spatial dispersion can be written as

ε (ω, k ) = ε b (ω ) −

ω 2p , ω 2 + iωγ b − β 2k 2

(1)

where ε b (ω) is the complex dielectric constant of bound electrons, ω p is the plasma frequency, γ b is the scattering rate in the bulk metal, β 2 = 3 / 5v F2 , and vF is Fermi velocity. According to Eq. (1) the nonlocality changes primarily in the real part of the permittivity and one can obtain the equation for the current density [14,15]

β 2∇(∇⋅ J ) + (ω 2 + iωγ b )J = iω p2ε 0ε bω E

(2)

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According to [16] this equation can be re-written for the induced electron density n(r ) = −iω −1∇⋅ J / e as eβ 2∇2 n(r ) + e(ω2 + iωγ b )n(r ) = ωp2ε 0ε b∇⋅ E .

(3)

The first term in Eq. (3) is known as the “quantum pressure” or “convection term”. It indicates that the gradient of electron density near the surface can no longer be described by the δ-function and its spatial extent is on the order of β / ω p ≈ vF / ω p ≈ λTF , i.e. just a few Å as predicted above. However, when one looks at the experimental data, the spectra of the dimers start deviating from the predictions based on local theory by exhibiting (relatively small) blue shift and very significant broadening when the size of the gap is still a few nanometers. The broadening is always accompanied by the quenching of the electric fields as has been demonstrated experimentally using impressive range of linear and nonlinear measurements, such as photoluminescence [23], SERS [24], Four-wave mixing [25] and third harmonic generation [26] (An excellent review of the experimental work is given in [19]). Since neither tunneling nor spill-over can cause the field quenching and resonance broadening occurring at the nanometer scale, one should look for the different mechanisms. One way to address the problem, is to consider extremely time consuming first principle calculations [27], or slightly less complicated density functional studies [28], yet the issue can also be addressed as in [15,29] by adding imaginary part to the convection β 2 → β 2 − iωD in Eq. (3), where the newly-introduced diffusion constant is

D ≈ vF2 γ b−1 . As a result Eq. (3) takes form  β 2 ∇2 + ω 2 + iω(γ b − vF2 γ b−1∇2 )  n( r ) = iω p2ε bω∇ ⋅ E .

(4)

The diffusion term introduces additional loss on the scale of γ dif = vF2 γ b−1d eff−2 where d eff is the spatial extent of the field and if one compares this term with the bulk energy dissipation term γ b in Eq. (4) one can see

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that once d eff < vF / γ b , the diffusive broadening becomes dominant. This result agrees well with the phenomenological theory of Kreibig [17]. According to this theory the origin of loss can be traced to electrons colliding with the metal surface [30] and the frequency of collisions is γ s ≈ vF / d , where d is the characteristic dimension of the nanoparticle. This result of damping being inversely proportional to d can also be interpreted through the ratio of the surface area (proportional to d 2 ) to the volume of the nanoparticle (proportional to d 3 ). Phenomenologically, the origin of surface collision damping rate γ s can be explicated by considering direct intraband transition between two free electrons with energies E1 and E2 and wave vectors k1 and

k2 , respectively, as shown in Fig. 1b. This transition is prohibited because of the momentum mismatch, and a collision with a third particle (say a phonon or lattice defect) is required to provide a recoil necessary to match the momenta. Collision with metal surface can also provide the recoil and consequently a photon (or, more properly, a SPP) with energy hω = E2 − E1 ends up being absorbed. With the electric field of the SPP confined within the distance d eff from the surface, it is that effective distance that shall be used in the aforementioned Kreibig’s expression for the surface damping rate γ s [17]. For instance, consider the example of a SPP that propagates on the metal/dielectric interface and whose electric field inside the metal (Fig. 2a) is given as

E ( x, z ) ~ E0 exp( − x / d eff ) exp(ik z z ).

(5)

As shown in [16,31,32] the surface collision damping of the propagating SPP is γ s = 3 4 vF / d eff . Of course, this abnormal increase of loss near the surface of the metals has long been known as anomalous skin effect [33].

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A rather simple quantum mechanical interpretation of surface damping could be given using uncertainty principle. Indeed, if one considers an electron undergoing collisions with the mean interval τ b between those collisions, the energy of this electron can only be determined within ∆Eb = hγ b where γ b = 1/ τ b is the bulk damping constant. By the same rationale, if the electric field is confined on the scale of d eff then the energies of free electrons interacting with the field is defined only within the interval

∆Es = hγ s ≈ hvF / deff = h / τ , where τ = deff / vF is the time of flight. The bulk and surface damping mechanisms then add up according to Matthiesen rule and the effective dielectric constant of the SPP mode can be found as

ω p2 ε eff (ω , deff ) = ε b (ω ) − 2 . ω + iω (γ b + γ s )

(6)

This expression appears to be technically correct, yet it is flawed and flawed deeply. According to Eq. (6) ε eff of a given SPP mode is determined only by its size and not by its shape. Therefore, whether a sharp boundary (surface) is present or the mode shape is smooth, the damping determined by the finite time-of-flight τ = d eff / vF is always present. Nonlocality as a manifestation of Landau damping To understand why this interpretation is inaccurate we go back to Fig. 1b and linearize the dispersion near Fermi level to obtain the frequency of the direct transition between two states E1 and E2 = E1 + hω as ω ≈ vF ⋅ (k2 − k1) , where the wavevector mismatch ∆k = k2 − k1 ≥ k0 and the offset vector of Landau damping k0 = ω / vF [34-36]. Therefore, once the magnitude of the wavevector k of longitudinal electromagnetic wave E (r ) ~ E0 exp(ik ⋅ r ) exceeds k0 , direct absorption by the free electrons becomes

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allowed. This process is of course nothing but the Landau damping and k0 = ω / vF is its characteristic wavevector. At first glance, it seems that Landau damping becomes a problem only when the SPP mode dimensions become comparable k0−1 = vF / ω ≈ 0.5 nm for the wavelength of λ = 800 nm , but this conclusion is erroneous. The power spectrum of the electromagnetic wave insider the metal, E(k )

2

contains wide range of spatial frequencies and the components with any spatial frequency k ≥ k0 all gets absorbed. To further elucidate this fact we now proceed by following the derivation first given in [31] starting with the expression for the dielectric constant of the metal, derived by Lindhard [37] ε (ω , k ) = ε b +

3ω 2p k 2 vF2

 ω + kvF  ω ln 1 −   2 kvF ω − kvF 

.

(7)

Introducing normalized (to the onset of Landau Damping) wavevector q = k / k0 we then obtain the expressions for the real and imaginary parts of ε . For the real part ε r (ω, q) = ε r (ω,0) + ∆ε r (ω, q) , where

ε r (ω ,0) = ε b − ω p2 / ω 2 is a long wavelength Drude dielectric constant and ω p2 ω p2  3 3 1+ q ∆ε r = 2 g r (q ) = 2 1 + 2 − 3 ln  1− q ω ω  q 2q

   

(8)

is plotted in Fig. 3a. An important feature of ∆ε r is the change of sign near q = 1 , therefore, as shown below, the change in the real part of permittivity is relatively small and only leads to a relatively minor shift of the SPP spectrum. The imaginary part of the dielectric constant, on the other hand, remains zero for small wavevectors but for the larger wavevectors k > ω / vF (q > 1) Landau damping gives rise to the imaginary part of dielectric constant, 3 ωp , 2 ω 2 q3 2

ε i (ω, k > ω / vF ) = 3πω p2ω / 2k 3vF3 = π

(9)

8


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as plotted in Fig. 3b. For each mode E (r ) one can find a spatial Fourier transform | F (k ) | and a power density spectrum | F (k )|2 of the electric field inside the metal. According to Eq. (9) all the longitudinal field components with then k x > ω / vF get Landau damped and contribute to the imaginary part of the 2

2

permittivity. For each value of k the power density of the longitudinal field is F (k ) = F (k ) ⋅ k / k 2 . One can then evaluate the effective dielectric constant ε eff (ω ) by computing the overlap of F ( k ) with 2

ε (ω, k ) as seen in Fig. 3a and 3b for real and imaginary parts of ε , the latter being

ε eff ,i =

3πω 2p 2ω 2

∞

∫ q >1

∞

2

∫0 | F (q) |

q −3 F (q) d 3q

2

d 3q .

(10)

Using Drude formula ε eff ,i = ω p2γ s / ω 3 and then the surface collision damping can be introduced as γs =

∞

2 3πω q −3 F ( q ) d 3 q ∫ 2 q >1

∞

∫0 | F (q ) |

2

d 3q .

(11)

Considering the above-mentioned example of the propagating SPP shown in Fig. 2a and with electric field described by the Eq. (5) one can find the power spectrum in normalized units 2

F ( q) =

2vF

1

π d eff ω q 2 + (1 / k 0 d eff )2

≈

2vF

π d eff ω q 2

(12)

where the last step is performed under a reasonable assumption deff >> k0−1 ≈ 0.5 nm . Using Eq. (12) in Eq. (11) one immediately obtains

γ s = 3 4 vF / d eff

(13)

which is in full agreement with the Kreibig interpretation [17] of the surface scattering rate according to which the photon (SPP) absorption takes place when the electron collide with the surface.

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The situation could not be more different in the absence of sharp boundary. Consider the optical field that is confined inside the metal with a smooth Gaussian mode, E ( x , z ) ~ E0 exp( − x 2 / 2d eff2 ) exp(ik z z )

(14)

whose normalized longitudinal spectrum is obviously also Gaussian 2

F (q) =

2 2 ω d exp  − ( k 0 d eff q )  .   π 1/2 v f eff

(15)

Substitution of Eq. (15) into Eq. (11) yields γs ≈

3π 1/2 v f exp  − k0 d eff 2 d eff 

(

)

2

 .

(16)

For deff >> k0−1 ≈ 0.5 nm the damping is orders of magnitude lower than in the presence of sharp 2

interface (13). Note that “Kreibig-like” result γ s ~ v f / d eff is the consequence of F (k ) decays as k −2 for large wavevectors and k −2 is the power spectrum of the Heaviside step function describing the sharp metal surface. Therefore, the phenomenological model introduced by Kreibig [17] in which the photon (SPP) absorption occurs at the surface of the metal is a correct physical interpretation, while invoking the uncertainty principle would be erroneous. Note that since the electron density at the surface cannot be described by the step function due to the aforementioned electron spill out, technically speaking the upper limit of integration in Eqs. (10) and (11) should be qmax = 2kF / k0 rather than infinity, which in principle should mitigate Landau damping. However, for the parabolic band the value of the maximum wave-vector is about qmax ~ 4EF / hω , i.e., about 10 for visible light and the value of integral in Eq. (10) barely changes.

10


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We shall return to the question of where exactly the SPP absorption takes place in the penultimate section of this work, but for now we turn our attention to the real part of the dielectric constant and calculate the change in it as ∆ε eff ,r (ω ) =

ω 2p ∞ 2 g r ( q ) F (q ) d 3 q 2 ∫ ω q >1

∞

∫0 | F(q) |

2

d 3q

(17)

Since the function gr (q) introduced in Eq. (8) and plotted in Fig. 3a changes sign at q = 1 the ensuing cancellation makes the integral in Eq. (17) small and overall change in real permittivity very close to what one would expect from phenomenological approach and simply take the difference between the permittivity with and without surface damping  ω p2  ω p2 ω p2 ∆ε eff , r (ω ) = Re  2 − 2 ≈ γ (2 γ + γ ) .  s b s ω4  ω + iωγ b ω + iω (γ b + γ s ) 

(18)

Thus even when the surface damping dominates, the relative change of the real part of the permittivity even is only about (γ s / ω ) , i.e., a fraction of a percent. 2

Landau damping and the shape of the SPP modes The key conclusion, made in our previous work [31] and elaborated in [16] is that the surface collisions not only cause additional damping of SPPs, but also limit the extent to which the electric field can be concentrated in plasmonic structures. In [31], using the propagating SPPs (Fig.2a) as an example, we have developed a self-consistent method according to which we first calculated the field distribution E (1) ( r ) in the absence of surface collision damping, i.e. assuming γ = γ b , then evaluated the power 2

spectrum F||( 0) ( q ) and used Eq. (10) to evaluate the surface collision damping γ s(1) and hence total damping γ (1) = γ b + γ s(1) , where the superscript indicates the number of iteration. With the new value of damping the next iteration we could now obtain new field distribution E (2) (r ) and repeat the process until it converged after a few iterations. The main result of [16,31] can be summarized as follows: once 11


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the penetration depth d eff decreases to about 10-15 nm the surface collision damping becomes the dominant damping mechanism. The electric field gets “expelled” from the metal and the maximum attainable wavevector of the SPP cannot exceed the wavevector of a plane wave in the dielectric by more than a factor of 4 or 5, no matter how low is the loss in the bulk metal. Another key result important to the main subject of this work, dimers had been obtained in [38] where we have considered spherical nanoparticles capable of supporting a large number of SPP modes with mode index l = 1, 2,L with resonant frequencies ωl = ω p /[1 + (1 + 1/ l )ε d ]1/2 ,

ranging from the

ω1 = ω p / (1 + 2ε d )1/2 for the lowest order radiating dipole mode ( l = 1 ) to ωsp = ω p / (1 + ε d )1/2 for the highest order modes ( l → ∞ ) that behave as SPPs propagating along the nanoparticle surface. The higher order modes tend to have smaller effective volume Veff ,l ≈ 4π a 3 / (l + 1)2 ε d [39] and therefore the maximum amplitude of the field, occurring near the metal surface increases with l + 1 while the metal penetration depth increases as deff = a / (l + 1) where a is the radius of the nanospheres. According to Kreibig’s phenomenological theory surface collision damping rate of the l-th mode should then be γ s ,l ≈ A(l + 1)vF / (2a ) where A is a constant of the order of unity. Precise numerical modeling performed

in [39] indeed confirms that surface collision damping increases with the mode order as roughly γ s ,l ≈ lvF / 2 a . This damping becomes eventually so strong that higher order modes are no longer

resolvable in spectral measurements [40]. To conclude this section, we note that besides [16,31] the connection between Landau damping and SPP spectra broadening has been made in recent works [28] and [41] where both spill-over and Landau damping have been related to the Feibelman parameters [42].

Landau damping in dimers We can now apply the knowledge gained by considering simple SPP modes in [31] and [38] to the dimer of Fig. 1a. Consider an intuitive picture of how surface collision damping places the limit on 12


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the degree to which the field can be squeezed using the example of the propagating SPP (Fig. 2a) considered in [31]. For the overdamped case, i.e. when coupling rate between the incoming wave and the SPP mode is much lower than the SPP damping, the higher is the damping rate the less energy gets coupled into the mode. One can then state that the light tends to be couple into the modes with higher Q factors, and, if the shape of the mode can be somehow changed to reduce damping (while, of course, still satisfying Maxwell equations) then the shape with the highest Q will describe the actually excited mode. This argument is somewhat similar to the one used to justify the variational method in quantum mechanics. Therefore, as the surface collision damping of propagating SPP increases near the surface plasmon resonance, the mode re-arranges itself and spreads out so that the damping is reduced. Note, however, that in the so-called gap SPPs [43-45] shown in Fig. 2b, the boundary conditions do not allow the mode to spread out as the gap decreases. Therefore, the increased damping can only reduce propagation length and ability to couple the energy in and out of gap SPP mode, but cannot change the shape of the mode. The situation is quite different in the dimer shown in Fig. 1a where the optical field can indeed be “squeezed” or “pushed” out of the gap. As the gap gets smaller the shape of the mode shape is expected to change leading to the reduction in confinement. Let us consider a simple coupled mode model [46,47] to get an intuitive picture of how the spreading of the mode occurs. The SPP mode inside the gap can be represented as a coherent superposition of the individual modes of two spheres as Edim (r ) =

∞

∑ ∑C

k ,l

Ek ,l (r ) . The mode confinement increases with l, hence the more terms are present

k =1,2 l =1

in the expansion the stronger is the confinement. The incoming light first couples directly into the dipole

l = 1 mode of the each sphere, and from there into the l > 1 modes of the other sphere. The contribution of the l-th mode can be shown to be γ Ck ,l ~ κ1l / (ωl − ω − jγ l )

(19)

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where the inter-mode coupling coefficient κ1l increases with the gap decrease. Therefore as gap decreases progressively higher and higher order modes get mixed into the dimer mode Edim (r ) whose effective volume gets smaller. This process of mixing higher l modes eventually gets truncated because the surface collision damping of higher modes γ s ,l ≈ lvF / 2a in the denominator of Eq. (19) ultimately becomes so strong that these modes can no longer effectively couple into the mixture and the size of the mode can no longer shrink. This limitation becomes especially severe if the radius a is small. To test these intuitive considerations we solve the problem self-consistently using essentially the same iterative method as used for the propagating SPP in [31]. First we solve the Maxwell equations numerically based on the finite element method (FEM) method using the COMSOL Multiphysics Modeling Software and assuming that no surface damping is in the Drude expression for the dielectric (1) permittivity in Eq. (6) i.e. γ s(0) = 0. Once the first iteration field of the mode, E dim ( r ) , and its effective

(1) volume Veff is calculated we can find the first iteration of the surface damping rate γ s(1) . All one needs

(1) to do is to compute spatial Fourier transform of E dim ( r ) and then use Eq. (11). With new damping rate

(2) γ s(1) in Eq. (6) the next iteration yields new shape of the mode E dim ( r ) that now is wider and has

(2) (1) effective volume Veff > Veff . Wider mode has smaller surface damping rate γ s(2) < γ s(1) which can now

be used as input for the next iteration. As demonstrated in Fig. 4a and 4b the process converges after a small number of iterations and yields actual γ s , field shape E dim ( r ) and also provides the magnitude of the field enhancement. Figure 4a shows the results for the silver dimer of radius a = 2.5 nm and gap size of d = 0.5 nm with its dipole resonance at λ= 663 nm. After a few rather wild oscillations surface damping rate converges at a value of γ s = 1.9 ×1015 s −1 . This value is about two orders of magnitude larger than bulk 14


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silver damping rate γ b = 3.2 × 1013 s −1 . Note that the value of γ b (ω ) are obtained from the analytical fit in [48] in which one part of the damping is associated with conduction electrons and included in Drude expression as γ b and the other part is associated with interband absorption and included in ε b (ω ) . At the same time, according to Fig. 4b the effective volume of the mode also increases as the −3 mode gets “pushed” out of the gap and eventually settles at Veff = 4.7 nm . Using a simple

phenomenological Kreibig theory described above and taking into account of the scattering by the two surfaces associated with the two nanospheres in the dimer, one would expect to find

γ s ~ vF / (Veff / 2)1/3 ~ 1.1×1015 s−1 , close enough to the numerically computed result. It confirms that results of Fig. 4a are indeed self-consistent. Results for gold dimer of radius a = 2.5 nm and gap

d = 0.5 nm with the dipole resonance at λ= 742 nm are shown in Figs. 4c and 4d for its damping rate and effective mode volume, respectively. Once again, the surface damping rate converges to

γ s = 1.6 ×1015 s −1 , also about two orders of magnitude greater than the bulk gold damping rate, and Veff reaches ~8.7 nm-3. Figure 5 shows the influence of surface scattering on the shape and magnitude of the optical field in the SPP mode of the same silver and gold dimers of radius a = 2.5 nm and gap d = 0.5 nm at the resonance wavelengths of λ = 663nm and λ = 742 nm, respectively. In the absence of surface collisions the enhancement factor for silver reaches 800 (Fig. 5a) but once the surface damping is taken into account the enhancement ends up reduced by almost two orders of magnitude (Fig. 5b) – which is more or less the ratio of γ s / γ b . A similar field enhancement reduction occurs in gold dimer as shown in Fig. 5c and 5d. Even more prominent are the changes in the absorption spectrum of the silver or gold dimer as shown in Fig. 6. Without surface collisions one can see a number of narrow resonances for both silver 15


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and gold dimers with radius of 2.5 nm and gaps of 0.5 nm and 1 nm, consistent with the prior work on coupled nanoparticles [46,47] in which the strong coupling of the two nano-spheres in the dimer causes the shift of the dipole mode ( l = 1) and excitation of the higher order modes ( l > 1). As seen in Fig. 6a and 6c, the smaller gap of 0.5 nm has indeed led to a red-shift of the dipole mode and stronger excitation of higher order modes relative to that of 1 nm gap in Fig. 6b and 6d for silver and gold, respectively. However, once the surface damping is introduced, only a single broad spectral feature remains, pointing to the fact that the plasmonic enhancement has reached its limit. In Fig. 7a one can see how the total damping including surface collision and bulk damping (solid lines) increases as the gap of the silver dimer narrows, while the bulk damping (dashed line) remains unchanged. Figure 7b shows its effective volume gets smaller as the gap narrows. For larger gaps, the nanospheres in a dimer are essentially uncoupled, and Veff is basically the same no matter whether surface collisions are taken into account or not. But at smaller gaps, Veff decreases very sharply (dashed lines) when the surface effect is disregarded, once γ s is incorporated the decrease (solid lines) is not that significant. As expected, the mode gets “squeezed out” of the gap. A similar behavior can be observed for the gold dimer in Fig. 7c and 7d. Let us now examine the maximum achievable optical field enhancement in the gap of either silver or gold dimer and reveal the impact of surface collisions as the gap shrinks. One can see in Fig. 8a that for a silver dimer of any radius the maximum field enhancement in the gap increases rapidly with the decrease of the gap size when only bulk damping is accounted for (dashed curves), although the degree of enhancement varies for dimers of different sizes because of their different dipole moments and effective mode volumes that collectively determine the strength of SPP field in the gap as explained in [46,47]. It should be pointed out that the maximum enhancement is always calculated at the resonance of the lowest order (dipole) mode at each dimer radius and gap (inset of Fig. 8a, only a = 5 nm is shown 16


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here). However, once the surface collision damping is taken into account, the increase of enhancement is quite modest (solid curves) as the gap in the dimer shrinks for a range of nanosphere radius. The largest increase is actually for a silver dimer with larger nanospheres ( a = 50 nm) of greater effective mode volume, suggesting that greater enhancement is achieved when the SPP field spreads out more inside of the metal rather than being tightly bunched near the metal boundary – the origin of strong surface damping. A similar behavior is seen in the gold dimer (Fig. 8b). It should be noted that all enhancements for both silver and gold dimers were calculated at their resonant frequency of the lowest order (dipole) mode which is below the onset of interband absorption from d to s band in the metal [48]. We can also look at the maximum achievable field enhancement in the gap of a silver (Fig. 9a) or gold (Fig. 9b) bow-tie antenna, where once again the impact of surface collisions is dramatic. Enhancement at larger gaps ( d > 5 nm) remains almost an order of magnitude smaller than what is expected to be achievable when only bulk damping is considered, for smaller gaps, the promise of significant higher enhancement (dashed curve) simply disappears once the surface damping is taken into account.

Landau damping in real space Before concluding remarks, we also would like to address the question of where exactly the absorption of electro-magnetic energy by metal nanoparticles takes place in real space, even though up to now we have developed our theory entirely in the Fourier domain. When the SPPs get absorbed, the electrons get excited into the wide energy range from the Fermi level EF to EF + hω and the holes correspondingly in the range from EF − hω to EF . These states have wavevectors in the range from kF to kF + ω / vF ( kF − ω / vF for the holes). Due to the boundary condition at the surface all the wavefunctions are in phase and form the wave packet with FWHM of roughly ∆L ~ π vF / 2ω = 1 4 λ (vF / c) 17


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or about 1.2 nm for λ = 600 nm. The total energy absorbed is obviously equal whether it is calculated in real space or Fourier space according to Parceval’s theorem. Performing inverse Fourier transform on the imaginary part of polarization in Fourier space, Pim ( k1 ) ~ ε im ( k )F ( k ) results in the expression for the absorbed power in real space Wabs ( r ) ~ F(r )

∫

ε im ( k ) F ( k )sin( k ⋅ r )dk

(20)

k > k0

which is plotted in Fig. 10 for the case of a dimer with 5-nm radius and a 2-nm gap. (The sine in Eq. (20) is the consequence of boundary conditions on the surface). As expected the absorption takes place within a thin layer ∆L ~ π / 2k0 near the surface, in agreement with both Kreibig’s theory and the prior works [14,15] referred above. At the same time it is important to note that while the wave packets for electrons and holes are excited very close to the surface they can only be “detected” when they experience collision, i.e., within a mean free path from the surface, or if they get emitted outside (important for surface chemistry [49]) – any attempt to detect them closer to the surface would change the mean free path. Therefore, from practical point of view, all detectable effects, such as electron heating, can only be detected within the mean free path from the surface, i.e. they are not nearly as localized as Fig.10 would suggest.

Conclusions In this paper we have investigated the non-local effects in metal dimers by applying selfconsistent theory in which the nonlocality is represented as manifestation of Landau damping that occurs

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as the extent of electro-magnetic mode inside the metal decreases to roughly ten nanometers or less. While the results are essentially similar to those obtained using phenomenological surface collision damping theory of Kreibig [17] as well as hydrodynamic model including diffusion of Mortensen et el [14,15], our model provides a clear physical insight while being computationally simple. We have also cleared a number of misconceptions and have shown that the electrons do get excited at the very surface of nanoparticles and the presence of relatively abrupt interface is necessary for Landau damping. The results show that when the size of the gap in the dimer decreases below 10 nanometers the Landau damping becomes the dominant loss mechanism and the self-consistent solution for the dimer mode spreads out of the gap, putting a severe limit on the maximum amount of field enhancement attainable in the dimer. The “clamping” of the field enhancement in dimers occurs at gap sizes much larger than those at which either electron tunneling or wavefunction spill-over can take place in consistence with the experimental data [23-26,50] and the most recent numerical analysis in [51] Since our results can be easily applied to other types of optical nanoantennas with gap, the main implication is that for gap dimensions of less than a few nanometers the loss associated with bulk metal becomes irrelevant as Landau damping alone determines the maximum attainable degree of plasmonic enhancement of optical processes.

AUTHOR INFORMATION Corresponding Author *Email: [email protected] Author Contributions The results and manuscript were produced through contributions of all authors.

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Notes The authors declare no competing financial interests.

Acknowledgments J. B. Khurgin acknowledges support from NSF (1507749) and ARO (W911NF-15-1-0629). W.-Y. Tsai and D. P. Tsai acknowledge support from Ministry of Science and Technology, Taiwan (Grant No. MOST-105-2745-M-002-005-ASP), Academia Sinica (Grant No. AS-103-TP-A06), National Center for Theoretical Sciences, NEMS Research Center of National Taiwan University, and National Center for High-Performance Computing, Taiwan. G. Sun acknowledges support from AFOSR (FA9550-17-10354) and AOARD (A2386-17-1-4100).

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Figure 1. (a) Illustration of the SPP field concentration in plasmonic dimer and (b) origin of Landau Damping – direct transition between two states with different wave-vectors k1 and k2.

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Figure 2 Propagating SPP (a) at the metal-dielectric interface and (b) in a dielectric gap between metal.

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Figure 3 Spatial dispersions of (a) real and (b) imaginary parts of metal dielectric constant and their overlaps with spatial power spectrum of the electric field in the metal.

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Figure 4. Self-consistent iterative calculation of (a) the damping rate (total damping γ s + γ b in blue and bulk damping γ b in red) and (b) effective mode volume Veff of the silver dimer with radius of 2.5 nm and gap of 0.5 nm at the dipole resonance λ= 663 nm, and corresponding results for a gold dimer of same dimensions in (c) and (d) at the dipole resonance λ= 742 nm.

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Figure 5. SPP Field distribution in the silver dimer ( a = 2.5 nm and d = 0.5 nm) taking into account (a) only bulk damping and (b) both bulk and surface damping at the dipole resonance λ=663 nm, and corresponding results for the gold dimer of equal dimensions in (c) and (d) at the dipole resonance λ=742 nm.

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Figure 6. Absorption spectra of silver (a,b) and gold (c,d) dimers of a = 2.5 nm for two different gaps (

d = 0.5 nm and 1 nm) with and without surface collision damping.

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Figure 7 (a) Total damping γ s + γ b (solid lines) in the silver dimer as a function of gap width for three different radii compared with bulk damping γ b (dashed line). (b) Effective mode volume ( Veff ) as a function of gap width with (solid lines) and without (dashed lines) surface collisions. Corresponding results for the gold dimer in (c) and (d).

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Figure 8. Maximum enhancement as a function of gap width in (a) silver and (b) gold dimers of four different radii with (solid lines) and without (dashed lines) surface collision damping.

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Figure 9. Field enhancement as a function of gap width in (a) silver and (b) gold bow-tie antennas with (solid lines) and without (dashed lines) surface collision damping. Insets: field distributions for 2nm-gap silver and gold bow-ties.

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Figure 10. Power absorption in a metal nanoparticle dimer at λ = 650 nm.

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