Landau Damping and Limit to Field Confinement and Enhancement in

Article Cite This: ACS Photonics 2017, 4, 2871-2880

pubs.acs.org/journal/apchd5

Landau Damping and Limit to Field Confinement and Enhancement in Plasmonic Dimers Jacob Khurgin,† Wei-Yi Tsai,‡ Din Ping Tsai,‡,§ and Greg Sun*,∥ †

Department of Electrical and Computer Engineering, John Hopkins University, Baltimore, Maryland 21218, United States 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, United States ‡

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 nonlocal 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 welldescribed 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 line width 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

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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 3 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 (Figure 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 © 2017 American Chemical Society

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 a decrease in the SPP volume, which is expected to be accompanied by the commensurate increase in the energy 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 line width of the resonance eventually broadens to the degree where the SPP resonances are no longer discernible.13−15 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. Received: August 1, 2017 Published: October 10, 2017 2871

DOI: 10.1021/acsphotonics.7b00860 ACS Photonics 2017, 4, 2871−2880

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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.

Expanding our previous basic analysis16 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

β 2∇(∇·J ) + (ω 2 + iωγb)J = iωp2ε0εbωE

According to ref 16, this equation can be rewritten for the induced electron density n(r) = −iω−1∇·J/e as eβ 2∇2 n(r ) + e(ω 2 + iωγb)n(r ) = ωp2ε0εb∇·E

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 substantial when the tails of two wave functions 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 wave function in the gap dielectric. The decay length can be estimated through its relation with the metal work function Φ as Ld ∼ ℏ/ 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 wave function. Since the maximum Fourier component of electron wave function cannot exceed the Fermi wave vector kF, any change in electron density must occur over a spatial range that is greater than the ThomasFermi screening length λ TF ∼ kF−1. This spread (or spillover), on the order of a few tenths of a nanometer for all noble metals, can be interpreted by the semiclassical nonlocality theory14,15 in which longitudinal dielectric constant of the metal possesses spatial dispersion can be written as

[β 2∇2 + ω 2 + iω(γb − vF2γb−1∇2 )]n(r ) = iωp2εbω∇·E (4)

The diffusion term introduces additional loss on the scale of −2 γdif = vF2γb−1deff , where deff is the spatial extent of the field, and if one compares this term with the bulk energy dissipation term γb in eq 4, one can see that once deff < vF/γb, 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 surface30 and the frequency of collisions is γs ≈ 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 d2) to the volume of the nanoparticle (proportional to d3).

ωp2 ω 2 + iωγb − β 2k 2

(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/ωp ≈ vF/ωp ≈ λTF, that is, 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 fourwave mixing,25 and third harmonic generation26 (an excellent review of the experimental work is given in ref 19). Since neither tunneling nor spillover 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 timeconsuming first principle calculations,27 or slightly less complicated density functional studies,28 yet the issue can also be addressed as in refs 15 and 29 by adding an imaginary part to the convection β2 → β2 − iωD in eq 3, where the newly introduced diffusion constant is D ≈ vF2γb−1. As a result, eq 3 takes the form

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ε(ω , k) = εb(ω) −

(2)

(1)

where εb(ω) is the complex dielectric constant of bound electrons, ωp is the plasma frequency, γb is the scattering rate in the bulk metal, β2 = 3/5vF2, and vF is the 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 density14,15 2872


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NONLOCALITY AS A MANIFESTATION OF LANDAU DAMPING To understand why this interpretation is inaccurate, we go back to Figure 1b and linearize the dispersion near Fermi level to obtain the frequency of the direct transition between two states E1 and E2 = E1 + ℏω, as ω ≈ vF·(k2 − k1), where the wavevector mismatch Δk = |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 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 a 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 ref 31, starting with the expression for the dielectric constant of the metal, derived by Lindhard37

Phenomenologically, the origin of surface collision damping rate γs 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 Figure 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 ℏω = E2 − E1 ends up being absorbed. With the electric field of the SPP confined within the distance deff from the surface, it is that effective distance that shall be used in the aforementioned Kreibig’s expression for the surface damping rate γs.17 For instance, consider the example of a SPP that propagates on the metal/dielectric interface and whose electric field inside the metal (Figure 2a) is given as E(x , z) ∼ E0 exp(−x /deff )exp(ikzz)

Article

(5)

ε(ω , k) = εb +

(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, εr(ω,q) = εr(ω,0) + Δεr(ω,q), where εr (ω , 0) = εb − ωp2 /ω 2 is a long wavelength Drude dielectric constant and

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

Δεr =

is plotted change of change in only leads

As shown in refs 16, 31, and 32, the surface collision damping of the propagating SPP is γs = 3 4 vF/deff . Of course, this abnormal increase of loss near the surface of the metals has long been known as anomalous skin effect.33 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 τb between those collisions, the energy of this electron can only be determined within ΔEb = ℏγb, where γb = 1/τb is the bulk damping constant. By the same rationale, if the electric field is confined on the scale of deff then the energies of free electrons interacting with the field is defined only within the interval ΔEs = ℏγs ≈ ℏvF/deff = ℏ/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 εeff (ω , deff ) = εb(ω) −

3ωp2 ⎡ ω + kvF ⎤ ω 1 ln − ⎥ ⎢ 2kvF ω − kvF ⎦ k 2vF2 ⎣

ωp2 ω

g (q) = 2 r

ωp2 ⎛ 1+q ⎞ 3 3 ⎟⎟ ⎜1 + 2 − 3 ln 2⎜ 1−q ⎠ ω ⎝ q 2q

(8)

in Figure 3a. An important feature of Δεr is the sign near q = 1; therefore, as shown below, the the real part of permittivity is relatively small and to a relatively minor shift of the SPP spectrum.

ωp2 ω 2 + iω(γb + γs)

(6)

This expression appears to be technically correct, yet it is flawed and flawed deeply. According to eq 6, εeff 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 = deff/vF is always present.

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. 2873


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spectrum of the Heaviside step function describing the sharp metal surface. Therefore, the phenomenological model introduced by Kreibig17 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/ℏω, that is, about 10 for visible light and the value of integral in eq 10 barely changes. 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

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, εi(ω , |k| > ω/vF) = 3πωp2ω/2k3vF3 =

2 3 ωp π 2 3 2 ωq

(9)

as plotted in Figure 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 kx > ω/vF get Landau damped and contribute to the imaginary part of the permittivity. For each value of k the power density of the longitudinal field is |F∥(k)|2 = |F(k)·k|2/k2. One can then evaluate the effective dielectric constant εeff(ω) by computing the overlap of |F∥(k)|2 with ε(ω,k), as seen in Figure 3a,b for real and imaginary parts of ε, the latter being εeff, i =

3πωp2 2ω

2

∞

∞

∫q> 1 q−3 |F (q)|2 d3q/∫

|F (q)|2 d3q

0

Δεeff, r (ω) =

(10)

γs =

3πω 2

∫

q−3 |F (q)|2 d3q /

q>1

∫0

∞

|F (q)|2 d3q

2vF 2vF 1 ≈ 2 2 πdeff ω q + (1/k 0deff ) πdeff ωq2

∞

∫q> 1 gr (q)|F (q)|2 d3q/∫

∞

|F (q)|2 d3q

0

Since the function gr(q), introduced in eq 8 and plotted in Figure 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 a phenomenological approach and simply take the difference between the permittivity with and without surface damping

(11)

Considering the above-mentioned example of the propagating SPP shown in Figure 2a and with electric field described by eq 5, one can find the power spectrum in normalized units |F(q)|2 =

ω

2

(17)

Using Drude formula εeff, i = ωp2γs/ω3 and then the surface collision damping can be introduced as ∞

ωp2

⎡ ⎤ ωp2 ωp2 ω2 ⎥ ≈ γ (2γ + γ ) p Δεeff, r (ω) = Re⎢ 2 − 2 s b s ⎢⎣ ω + iωγb ω4 ω + iω(γb + γs) ⎥⎦

(12)

(18)

where the last step is performed under a reasonable assumption deff > >k 0−1 ≈ 0.5 nm . Using eq 12 in eq 11 one immediately obtains

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

γs = 3 4 vF/deff

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LANDAU DAMPING AND THE SHAPE OF THE SPP MODES The key conclusion, made in our previous work31 and elaborated in ref 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 (Figure 2a) as an example, we have developed a self-consistent method according to which we first calculated the field distribution E(r) in the absence of surface collision damping, i.e. assuming γ = γb, then evaluated the power spectrum |F||(0)(q)|2 and used eq 10 to evaluate the surface collision damping γs(1) and, hence, total

(13)

which is in full agreement with the Kreibig interpretation17 of the surface scattering rate according to which the photon (SPP) absorption takes place when the electron collide with the surface. 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, 2 E(x , z) ∼ E0 exp( −x 2/2deff ) exp(ikzz)

(14)

whose normalized longitudinal spectrum is obviously also Gaussian 2 ω |F(q)| = 1/2 deff exp[−(k 0deff q)2 ] π vf

damping γ (1) = γb + γs(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 refs 16 and 31 can be summarized as follows: once the penetration depth deff 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 the loss is in the bulk metal.

2

(15)

Substitution of eq 15 into eq 11 yields γs ≈

3π 1/2 vf exp[−(k 0deff )2 ] 2 deff

(16)

For deff ≫ k 0−1 ≈ 0.5 nm the damping is orders of magnitude lower than in the presence of sharp interface.13 Note that “Kreibig-like” result in γs ∼ vf/deff is the consequence of |F(k)|2 decays as k−2 for large wavevectors and k−2 is the power 2874


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Figure 4. Self-consistent iterative calculation of (a) the damping rate (total damping γs + γb in blue and bulk damping γb 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 the same dimensions in (c) and (d) at the dipole resonance λ = 742 nm.

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 rearranges itself and spreads out so that the damping is reduced. Note, however, that in the so-called gap SPPs43−45 shown in Figure 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 Figure 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 model46,47 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 ) = ∑k = 1,2 ∑l = 1 Ck , lEk , l(r ). The mode confinement increases with l; hence, the more terms are present in the expansion, the stronger is the confinement. The incoming light first couples directly into the dipole l = 1 mode of each sphere and, from there, into the l > 1 modes of the other sphere. The contribution of the lth mode can be shown to be

Another key result that is important to the main subject of this work, dimers had been obtained in ref 38, where we have considered spherical nanoparticles capable of supporting a large number of SPP modes with mode index l = 1, 2, ..., with resonant frequencies ωl = ωp/[1 + (1 + 1/l)εd]1/2, ranging from the ω1 = ωp/(1 + 2εd)1/2 for the lowest order radiating dipole mode (l = 1) to ωsp = ωp/(1 + εd)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 ≈ 4πa3/(l + 1)2εd39 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 lth mode should then be γs,l ≈ A(l + 1)vF/(2a), where A is a constant of the order of unity. Precise numerical modeling performed in ref 39 indeed confirms that surface collision damping increases with the mode order as roughly γs,l ≈ lvF/2a. 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 refs 16 and 31, the connection between Landau damping and SPP spectra broadening has been made in recent works, refs 28 and 41, where both spillover and Landau damping have been related to the Feibelman parameters.42

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LANDAU DAMPING IN DIMERS We can now apply the knowledge gained by considering simple SPP modes in refs 31 and 38 to the dimer of Figure 1a. Consider an intuitive picture of how surface collision damping places the limit on the degree to which the field can be squeezed using the example of the propagating SPP (Figure 2a) considered in ref 31. For the overdamped case, that is, when the 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

γCk , l ∼ κ1l /(ωl − ω − jγl)

(19)

where the intermode 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 γs,l ≈ lvF/2a in the denominator of eq 19 ultimately becomes so strong that 2875


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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.

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 and 1 nm) with and without surface collision damping.

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 ref 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 permittivity in eq 6, that (1) (r ), is, γs(0) = 0. Once the first iteration field of the mode, Edim

iteration of the surface damping rate γs(1). All one needs to do is (1) (r ) and then use to compute spatial Fourier transform of Edim

eq 11. With new damping rate γs(1) in eq 6 the next iteration (2) (r ) that now is wider and yields a new shape of the mode Edim (2) (1) > V eff . Wider mode has smaller has an effective volume V eff

surface damping rate γs(2) < γs(1), which can now be used as input for the next iteration. As demonstrated in Figure 4a,b, the process converges after a small number of iterations and yields actual γs, field shape Edim(r) and also provides the magnitude of the field enhancement.

(1) is calculated; we can find the first and its effective volume V eff

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Figure 7. (a) Total damping γs + γb (solid lines) in the silver dimer as a function of gap width for three different radii compared with bulk damping γb (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).

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.

radius a = 2.5 nm and gap d = 0.5 nm with the dipole resonance at λ = 742 nm are shown in Figure 4c,d for its damping rate and effective mode volume, respectively. Once again, the surface damping rate converges to γs = 1.6 × 1015 s−1, also about 2 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 = 663 and 742 nm, respectively. In the absence of surface collisions the enhancement factor for silver reaches 800 (Figure 5a), but once the surface damping is taken into account the enhancement ends up reduced by almost 2 orders of magnitude (Figure 5b), which is more or less the ratio of γs/γb. A similar field enhancement reduction occurs in gold dimer, as shown in Figure 5c,d. Even more prominent are the changes in the absorption spectrum of the silver or gold dimer, as shown in Figure 6. Without surface collisions one can see a number of narrow

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 γs = 1.9 × 1015 s−1. This value is about 2 orders of magnitude larger than bulk silver damping rate γb = 3.2 × 1013 s−1. Note that the value of γb(ω) are obtained from the analytical fit in ref 48 in which one part of the damping is associated with conduction electrons and included in Drude expression as γb, and the other part is associated with interband absorption and included in εb(ω). At the same time, according to Figure 4b, the effective volume of the mode also increases as the mode gets “pushed” out of the gap and eventually settles at Veff = 4.7 nm−3. 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 γs ∼ vF/(Veff/2)1/3 ∼ 1.1 × 1015 s−1, close enough to the numerically computed result. It confirms that results of Figure 4a are indeed self-consistent. Results for gold dimer of 2877


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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 2 nm gap silver and gold bow-ties.

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 (Figure 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 (Figure 9a) or gold (Figure 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.

resonances for both silver and gold dimers with radius of 2.5 nm and gaps of 0.5 and 1 nm, consistent with the prior work on coupled nanoparticles46,47 in which the strong coupling of the two nanospheres in the dimer causes the shift of the dipole mode (l = 1) and excitation of the higher order modes (l > 1). As seen in Figure 6a and c, 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 Figure 6b and d 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 Figure 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 γs 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 Figure 7c,d. 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 Figure 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 refs 46 and 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 Figure 8a, only a = 5 nm is shown 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

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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 + ℏω and the holes correspondingly in the range from EF − ℏω 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 wave functions are in phase and form the wave packet with fwhm of roughly ΔL ∼ πvF/2ω = 1 4 λ(vF/c) 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, P(k1) ∼ εim(k)F(k) results in the expression for the absorbed power in real space Wabs(r ) ∼ F (r ) 2878

∫k>k

εim(k)F(k)sin(k·r )dk 0

(20)


ACS Photonics

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

attainable degree of plasmonic enhancement of optical processes.

which is plotted in Figure 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 ΔL ∼ π/2k0 near the surface, in agreement with both Kreibig’s theory and the prior works14,15 referred to 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, that is, within a mean free path from the surface, or if they get emitted outside (important for surface chemistry49), 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 Figure 10 would suggest.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Din Ping Tsai: 0000-0002-0883-9906 Greg Sun: 0000-0002-8001-5020 Author Contributions

The results and manuscript were produced through contributions of all authors. Notes

The authors declare no competing financial interest.

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CONCLUSIONS In this paper we have investigated the nonlocal effects in metal dimers by applying self-consistent theory in which the nonlocality is represented as manifestation of Landau damping that occurs 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 Kreibig17 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 nm 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 wave function spillover can take place in consistence with the experimental data23−26,50 and the most recent numerical analysis in51 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

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ACKNOWLEDGMENTS

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REFERENCES

J.B.K. acknowledges support from NSF (1507749) and ARO (W911NF-15-1-0629). W.-Y.T. and D.P.T. acknowledge support from the Ministry of Science and Technology, Taiwan (Grant No. MOST-106-2745-M-002-003-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.S. acknowledges support from AFOSR (FA9550-17-1-0354) and AOARD (A2386-17-1-4100).

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