## Abstract

We investigate the effect of the electron wave function producing permittivity (epsilon) near zero in sub-nanometer gaps and at surfaces. The field enhancement is calculated for gaps and nanoparticles, as well as the absorption from nanoparticles. Our modified quantum corrected model shows reduced absorption for nanoparticles due to “cloaking” of the epsilon near zero region, which has lower loss than the bulk region. We demonstrate that a modified quantum corrected model finite-difference time-domain simulation of metal slits with sub-nanometer gaps are in good agreement with the analytic expression for the quantum corrected plasmonic resonance wavelength as a function of gap size coming from *Re{ε}* = 0.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The ultimate limits of plasmonic enhancement have been considered in terms of non-local effects and quantum tunneling. The non-local effects caused by the Thomas-Fermi pressure term have been theorized to reduce the plasmonic shift in small gaps (and also the local field enhancement) [1]; however, two experimental works have found that local models are sufficient in this regime [2,3]. Time-dependent density functional theory simulations have shown that the electron spill-out with screening actually produces an effectively smaller gap, and so the plasmon shift is opposite to that predicted by the non-local model [4].

Separate from the non-local effects, tunneling has been postulated to reduce the gap plasmon enhancement for very small gaps by shunting out the gap. Linear spectra [5], third harmonic generation [6] and surface-enhanced Raman spectroscopy (SERS) [7] have shown an abrupt transition at the onset of tunneling. A quantum corrected model (QCM) was proposed to describe this effect [8,9]. Initially, the QCM modified the scattering rate in the Drude model, however, this was revised to modify the plasma frequency [6,10]. The physical basis for this revised model is that the carrier density is reduced away from the metal as the electron wave function decays.

A recent work has shown that including this model in gap plasmons results in increased absorption [11], which bares similarity to experimental reports [12–14]. This comes as a result of the localization of the field in the epsilon near zero (ENZ) region of gaps. ENZ has been studied in detail for field enhancement properties, as well as “supercoupling” [15] and large nonlinear responses [16–19]. Previously, it has been shown in experiment that ENZ materials can enhance the local field and thereby increase SERS significantly [20,21].

In this work, we consider the impact of the quantum plasmonic ENZ in metal nanostructures. We treat field enhancement and the resultant absorption from nanoparticles that arise due to ENZ regions within the QCM model. *Surprisingly, our calculations show that the absorption density decreases due to “cloaking” in the ENZ region.* These results are similar to “ENZ cloaking”, which has been reported previously [22–25]. On the other hand, the enhanced losses were observed in small metal nanoparticles and have been attributed to “surface scattering” [26,27]. We find an analytic expression for the quantum corrected plasmonic resonance wavelength as a function of gap size, which agrees with the maximum field enhancement in a QCM finite-difference time-domain simulation. These results are applicable for finding regimes of enhanced Raman from small gap structures—the optimal sizes for different wavelengths are produced in this work.

## 2. Quantum Corrected Model Revisited

For modeling a sub-nanometer metal gap filled with a self-assembled monolayer (SAM) and for nanoparticles, we used the QCM, which considers an effective conductive material to mimic the electron tunneling in the gap region [8,9] or the electron spill-out in the region around the nanoparticle. Using an effective Drude model, the permittivity in the QCM region is:

where*ε*is the background permittivity,

_{∞}*ω*and

_{g}*γ*are plasma frequency and damping parameter (collision frequency), of this effective material. γ

_{g}_{g}can be estimated as equal to the damping frequency of the metal,

*γ*.

_{p}In the past, we have considered that *ω _{g}* can be modified to be [6,10]:

*ω*is the plasmon frequency of the metal and

_{p}*T*(

*x*) is the electron tunneling probability at distance

*x*from the center of the gap. Here, instead of

*T*(

*x*), we use the probability of finding electron, which is the square of the normalized electron wave function, ${\left|\psi (x)\right|}^{2}$. As a result, the revised formula for the permittivity inside the gap region may be written as a following:

*k*which is given as:where

*m**is the effective mass of electrons in the metal and

_{e}*φ*is the barrier height. In vacuum, the barrier height is equal to the work function [28]. When the gap is filled with a dielectric like SAM, we use the barrier height between the metal and the dielectric inside the gap instead of using the work function of the metal [29,30].

_{B}## 3. Nanoparticle Calculations

Silver nanoparticles were calculated using a quasi-static model in MATLAB. The 0.5 nm region outside of the nanoparticle consisted of 10^{6} layers of Drude model materials. Drude parameters for silver (*ω _{p, Ag}* = 8.9eV/

*ћ*= 1.35215 × 10

^{16}rad/s and

*γ*= 1/(17 fs) = 5.88235 × 10

_{p, Ag}^{13}rad/s) were taken from Ref [31]. We chose the background permittivity of

*ε*= 1. The work function of silver, which is the barrier height here, is 4.5 eV [32]. The background permittivity was assumed to terminate at the edge of the silver region—the rationale for this is that the deeply bound electrons will spill out much less, and so their influence is negligible.

_{∞}For electron spill-out around a nanoparticle, the approximate dependence of the wave function is given by:

where*r*is the radius of the nanoparticle. This gives an ENZ region for the nanoparticles where

_{0}*Re{ε}*= 0. As a result, the field will be concentrated in this region, which is expected to have significant impact on the scattering results. A quasi-static model is used to solve for the field around the nanoparticle, matching the constant and dipole fields in each region with sub-picometer steps to ensure convergence.

Figure 1 shows the localization of the electric field around the region where *Re{ε}* = 0. The field is normalized to the incident plane wave value showing an intensity enhancement of 20 million for silver in a very narrow region. The imaginary part of the permittivity at this position is reduced from the value inside the metal due to the reduced electron density in the spill-out region. Therefore, the overall losses actually go down within this model.

Figure 2 shows the absorption density given by:

*V*is the volume of nanoparticle,

*E*is the electric field,

*r, θ,*and

*ϕ*are spherical coordinates. The values are normalized to the absorption density of the classical model in the quasi-static regime for the same incident field. It is a surprising result that the absorption density actually decreases due to cloaking from the ENZ region. This is contrary to the recent report on QCM absorption in grooves [11]. In that work, the absorption in the taper is enhanced by the quantum corrected formulation; however, for the spherical geometry considered here, the absorption is reduced. Further investigation is required to explore this effect since we have assumed that the scattering is the same as the bulk value. In other works, it has been shown that surface scattering values can be higher than the bulk [26,27].

It is also interesting to note that the ENZ region gives a very thin slice of extremely enhanced field. This may create larger interactions with molecules confined at the surface, for example, to increase their Raman cross section. Studies on Raman of self-assembled monolayer (SAM) have typically followed the expected quasi-static near-field dependence; however, one work showed an elevated signal with respect to theory for the molecular vibration closest to the surface [33]. This warrants further experimental investigation to determine if the field localization (and gradient of field) is playing a role in this regime.

## 4. Gap Calculations

We simulated the metal sub-nanometer gap in 2D using the finite-difference time-domain (FDTD) numerical software, Lumerical Solutions Inc., release 2017b, version 8.18.1365. A single Drude model material was used to model two gold rectangles (70 nm × 80 nm) with the background permittivity of 1 and using the plasma and collision frequencies of gold (*ω _{p, Au}* = 1.3713 × 10

^{16}rad/s and

*γ*= 4.05 × 10

_{p, Au}^{13}rad/s) from Ref [34]. and the effective mass of electrons in gold (

*m**) from Refs [35,36]. Different gap sizes in the range of 0.2 to 1 nm were simulated. In the classical electromagnetic model (CEM), the SAM was assumed to have a refractive index of 1.5.

_{e, Au}≃ 1.1 m_{e}In quantum corrected model (QCM), the gap region was modeled with layers of Drude model materials, each with thickness equal to the size of the mesh and permittivity calculated from Eqs. (3) with the background permittivity of *ε _{∞}* = 2.25. The barrier height between the metal and SAM inside the gap was considered as

*φ*= 1.5 eV [29,30]. We chose uniform mesh type with staircase mesh refinement. The mesh size was set to 0.00625 nm in

_{B}*x*direction and 0.2 nm in

*y*direction. The FDTD simulation region had the length of 24 nm in

*x*direction (from −12 nm to 12 nm, with

*x*= 0 in the center of gap) and the length of 25 nm in

*y*direction (from 5 nm distance from gold surface to 20 nm inside the gold and gap regions). However, our monitor geometry was 2 nm × 13 nm. We used a y-polarized plane wave source with wavelength range from 500 to 1200 nm, located 4 nm away from the gold surface. The monitor collected the data for 100 wavelength points. Furthermore, we set time stability factor to 0.99 and auto shutoff to 1 × 10

^{−5}.

A 1-D approximation to the wave function in the gap is given as:

where*x*is the distance from the center of the gap and

*d*is the gap size [Fig. 3]. Equation (7) was plugged into Eq. (3) in this simulation.

Figure 4 shows the simulation results of electric field intensity for different gaps in QCM and CEM conditions. Figure 5 depicts spatial distribution of electric field intensity in the gap of *d* = 0.5 nm for QCM FDTD simulation as an example.

## 5. Discussion

As can be seen in Fig. 4 and 5, QCM results shows enhancement of electric field in the center of gap, which is 20 times larger than CEM results. We attribute this field enhancement to the tunneling on ENZ in the gap region. The enhancement of field in gaps and its application such as surface-enhanced Raman spectroscopy (SERS) [20,21], surface-enhanced infrared absorption (SEIRA) [37,38], terahertz structures [39–41] and optical gap antenna [42,43] were realized experimentally by using either SAM [2,6,10], Al_{2}O_{3} deposited by atomic layer deposition (ALD) [39–41,44–47], or graphene [39] as the spacer in the nanometer gap.

The effect of the tunneling on ENZ in gaps leads to field enhancement. For finding an analytic expression for the quantum corrected plasmonic resonance wavelength as a function of gap size, Eqs. (3) and (7) can be used. Considering *Re(ε) = 0* at the center of the gap (*x = 0*) and knowing *γ _{p}^{2} << ω^{2}*, the resonance frequency

*ω*and resonance wavelength

_{r}*λ*as a function of gap size

_{r}*d*are given by:

*λ*), predicted by Eq. (8b) and QCM simulation, versus gap size

_{r}*d*. As can be seen in Fig. 6, the simple expression of Eq. (8b) is in good agreement with the FDTD QCM simulation, which confirms this basic physical interpretation.

## 6. Conclusions

We used a modified version of QCM to investigate the quantum plasmonic epsilon near zero phenomena for nanoparticles and slits. Our calculations show that the electron spill-out will yield both cloaking and field enhancement due to the real part of the permittivity going through zero in the QCM. We found an analytic expression for the quantum corrected plasmonic resonance wavelength as a function of gap size using *Re{ε}* = 0. This analytic expression and QCM FDTD simulations for slits agree, and the results may be tested with future experimental works using field enhancements in slits from ALD or SAMs as probed by harmonic generation or Raman scattering.

## Funding

Natural Sciences and Engineering Research Council of Canada Discovery Grant.

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