## Abstract

The effect of nonlocal optical response is studied for a novel silicon hybrid plasmonic waveguide (HPW). Finite element method is used to implement the hydrodynamic model and the propagation mode is analyzed for a hybrid plasmonic waveguide of arbitrary cross section. The waveguide has an inverted metal nano-rib over a silicon-on-insulator (SOI) structure. An extremely small mode area of~10^{−6} *λ*^{2} is achieved together with several microns long propagation distance at the telecom wavelength of 1.55*μ*m. The figure of merit (FoM) is also improved in the same time, compared to the pervious hybrid plasmonic waveguide. We demonstrate the validity of our method by comparing our simulating results with some analytical results for a metal cylindrical waveguide and a metal slab waveguide in a wide wavelength range. For the HPW, we find that the nonlocal effects can give less loss and better confinement. In particular, we explore the influence of the radius of the rib’s tip on the loss and the confinement. We show that the nonlocal effects give some new fundamental limitation on the confinement, leaving the mode area finite even for geometries with infinitely sharp tips.

©2013 Optical Society of America

## 1. Introduction

In order to realize electronic photonic integrated circuits, we need to develop optical waveguides capable of guiding light with deep subwavelength confinement. It is well known that surface plasmon polariton (SPP) waveguides can provide subwavelength confinement in the scale of 100nm or less, by coupling the electromagnetic waves to electron oscillations at the metallic surface. In the past years, people have demonstrated several novel structures, such as metal slot waveguides [1, 2], channel SPP (CPP) waveguides [3, 4], and wedge plasmon polariton (WPP) waveguides [5–8]. However, they lead to high propagation loss. Recently, several hybrid plasmonic waveguides have been proposed to achieve both subwavelength mode confinement and relatively low loss [9–13], which makes them attractive as potential candidates for the realization of nano-photonic circuits. As enhancing light confinement while maintaining the transmission loss is a goal for most of the hybrid waveguide designs, the trade-off between the mode confinement and loss still exists. Currently, the miniaturization trend in the design of plasmonic devices is reaching the limit where the local solutions of macroscopic Maxwell’s equation can no longer accurately describe their electromagnetic properties due to a longitude mode near the metal surface. To overcome this, a sophisticated nonlocal material model is required, such as the hydrodynamic model of the electron gas as discussed by Boardman [14]. Recently, a number of nonlocal solutions of Maxwell’s equations in two dimensions (2D) or 2.5D or 3D (Cartesian coordinates or cylindrical coordinates) have been proposed based on numerical approaches such as FDM [15], FDTD [16] or FEM [17–22] for frequency domain stationary application or transient propagation application. An insightful transformation-optics method [23] has also been proposed to study the optical properties of plasmonic nanostructures and has revealed the nonlocal effect to the reduced field enhancement for e.g. metallic dimer of touching metal nanowires. An experiment has also shown that the intrinsic nonlocality of the metal limits the field enhancement [22]. In the analysis of propagation mode for some plasmonic waveguide, early formulation treatments have been given for a metal slab waveguide [24, 25], a metal cylindrical waveguide [26] or aggregates thereof [27]. For a plasmonic waveguide of large size with apex feature, the analytical method in [27] is invalid. Although there has been an enormous advance in the theoretical analysis of nonlocal effects in plasmonic structures, to the best of our knowledge, there is no report on a propagation mode analysis of nonlocal effect in a plasmonic waveguide of arbitrary cross-section.

In this letter, we introduce a novel highly confining hybrid plasmonic waveguide with an inverse triangular metal rib separated from a high-index material by a low-index material. The mode area of the hybrid mode can be reduced to about 10^{−6}*λ*^{2}, while still maintaining propagation distances exceeding several microns at telecom wavelengths. We also investigate how nonlocal effects alter the confinement of the hybrid plasmonic waveguide. However, we don’t consider the quantum effect at the metal surface [28, 29]. This is because the effect of electron tunneling between the metal and the dielectric is weaker than that between the metal nanoparticles. First, we present a FEM nonlocal solution of Maxwell’s equation using the weak formulation. Then we validate the simulation method by comparing the simulation results with the analytical results for a metal cylindrical waveguide and a metal slab waveguide. Moreover, we compare the results simulated by the FEM nonlocal method and the traditional FEM Drude method. At last, we investigate how the nonlocal effects change the sensitivity of the electromagnetic field to the radius of the tip. We demonstrate that the nonlocal effects drastically change the nanoscale confinement performance of plasmonic tips when the radius of the tip is comparable to the scale of the longitudinal mode penetration depth.

## 2. Theoretical formalism

We firstly describe the basic formalism of our nonlocal FEM model. The Maxwell inhomogeneous wave equation at frequency *ω* can be described by the following equation:

**J**(inside a metal) induced by an electric field

**E**can be represented as

*ε*

_{0}is the vacuum permittivity, and

*γ*and

*ω*are the damping coefficient and the plasma frequency, respectively, which also appear in the conventional Drude formula,

_{p}*ε*(

*ω*)

*=*1

*-*(

*ω*

_{p}^{2}/(

*ω*)). The nonlocal parameter

^{2}+ iγω*β*, which measures approximately the speed of sound in Fermi-degenerate plasma of conduction electrons, is proportional to the Fermi velocity

*v*

_{F}of the metal. It is closely related to the penetration depth

*δ*of the longitudinal mode excited in the metal structure, due to the nonlocal permittivity. In this paper, no interband transition is considered, nor the dependence of

_{L}*β*on the frequency due to transition absorption.

In the bulk metal, the dispersion relation of the longitude mode can be described by the following equation:

Thus, we can write the longitudinal mode penetration depth*δ*in the bulk metal

_{L}*ω*is higher or

*β*is larger,

*δ*becomes larger. Thus, the nonlocal effect becomes important at high frequencies. The penetration depth is in the order of 0.1nm in noble metals, such as gold or silver [20]. Therefore, the theoretical investigation of plasmonic phenomena in this sub-nanometer structure requires the hydrodynamic model.

_{L}We follow the approach outlined in Ref [17]. to solve the above equations and use a highly adaptive mesh through a commercial FEM package, Comsol Multiphysics. We take advantage of the fact that **J** and **E** have the same propagation constant. We can write Eq. (2) into a weak form resulting in an integral expression [30]

_{$\tilde{J}$}denotes the vector-valued test function of

**J**. Note that one may obtain a more simplied (but less accurate) integral expression as [21] $\underset{\Omega}{\int}\left(-{\beta}^{2}\left(\nabla \xb7J\right)\left(\nabla \xb7\tilde{J}\right)+\omega \left(\omega +i\gamma \right)J\xb7\tilde{J}-i\omega {\omega}_{p}^{2}{\epsilon}_{0}E\xb7\tilde{J}\right)}d\Omega =0.$ The additional boundary condition is the continuity of the normal component of

**E**, and the continuity of the normal component of

**J**at the surface due to the finite values of charge and current densities [31].

## 3. Validation for benchmark problems

For a validation of the present numerical approach, we simulate a metal cylindrical waveguide and a metal slab waveguide, where the cylindrical symmetry allows for an analytical solution in terms of modified Bessel and Hankel functions [24, 25], and the slab waveguide allows for an analytical solution in terms of exponential functions [26], both for Drude and nonlocal responses. Figure 1
summarizes the results for our benchmark problems, and illustrates the strong dependence of the nonlocal effects on the subwavelength size of the nanostructures. Figure 1 shows the effective index *n _{eff}*, the propagation distance

*L*and the mode profile for the metal cylindrical waveguide and the metal slab waveguide. The propagation distance is given by

_{prop}*L*= 1/(2Im(

_{prop}*n*)

_{eff}*k*

_{0}), where

*k*is the wave number in vacuum. We take parameters for silver, namely, the plasma frequency

_{0}*ηω*= 8.59eV, Drude damping coefficient

_{p}*ηγ*= 0.075eV, and nonlocal parameter

*β =*0.0036

*c*[20] in Fig. 1. However, the interband transition is not considered here, nor the dependence of

_{0}*β*on the frequency due to transition absorption.

Figures 1(a1)-(a4) compare the numerical results for nanowires of radius *R* = 2nm with the exact analytical solution, both for local (*β* = 0) and nonlocal response (*β* = 0.0036*c _{0}*). For these tiny nanowires, the nonlocal effect can be considerable. To give two examples, the relative difference of

*L*between the nonlocal and local responses in the figure can be up to 4.6%. The relative difference of

_{prop}*n*between the nonlocal and local responses in the figure can be up to 2.5%. The mode profile in the metal is different due to the fact that the nonlocal parameter

_{eff}*β*will increase the longitudinal mode penetration depth

*δ*. Effectively, the electric field penetration into the metal is increased at the same time.

_{L}The analytical and numerical curves overlap almost completely for the Drude model, and likewise for the nonlocal model. Hence only two of the four curves are visible. The relative errors of the numerically computed *L _{prop}* and

*n*for the nonlocal response are always less than 0.2% and 0.4%, respectively, while for the local response the relative errors are always less than 10

_{eff}^{−5}.

Figures 1(b1)-(b4) show the corresponding four curves as Fig. 1(a1)-(a4), but now for a metal slab waveguide with thickness *d* = 2nm. As one can see, the maximum relative differences in *L _{prop}* and

*n*mode profile between the nonlocal and local models are 1.8% and 0.9%, respectively. This indicates that the infinity along one dimension of the slab waveguide (which has only one-dimensional confinement) reduces the difference between the local and nonlocal models as compared to the cylinder waveguide, which has a two-dimensional confinement. The maximum relative errors in the numerically computed

_{eff}*L*and

_{prop}*n*for the nonlocal response in Fig. 1(b) are 0.16% and 0.07%, respectively, while for the local response it is always smaller than 10

_{eff}^{−5}.

In summary, our numerical implementation for the metal cylindrical waveguide and the metal slab waveguide is accurate for the distribution of the electromagnetic field in a wide range of wavelength and length scales.

## 4. Novel silicon hybrid plasmonic waveguide

Having addressed the metal cylindrical waveguide and the metal slab waveguide, we now consider a silicon-based hybrid plasmonic waveguide with an inverted metal nano-rib, which can reduce the mode area to the order of 10^{−6}*λ*^{2}, while the propagation distance is still several microns simultaneously at telecom wavelengths. The cross section of the present silicon hybrid plasmonic waveguide is shown in Fig. 2(a)
. It consists of a SOI rib with an inverted metal nano-rib on the top. There is a low-index material (e.g., air) between the SOI rib and the metal structure. This novel hybrid plasmonic waveguide combines the advantage of the silicon hybrid plasmonic waveguide structure (which has low propagation loss and strong field confinement in the vertical direction) and the advantage of the WPP waveguide (low propagation loss and strong field confinement in the horizontal direction). When the metal rib height is zero (i.e., a metal slab) in Fig. 2(c), or the SOI rib is absent in Fig. 2(d), the fundamental mode field is not well confined in any of the two directions and the mode area is about 10^{−4}*λ*^{2}. However, when there is an inverted metal nano-rib, the field distribution of the guide mode is significantly modified and the field is strongly confined at the tip of the metal rib (see Fig. 2(b)) without much decrease in the propagation distance.

Figure 3
displays the behavior of HPW modes when *w*_{co} = 22nm, *h*_{metal} = 10nm, *h*_{rib} = 10nm, *h*_{Si} = 50nm, *θ* = 10°, *g* = 0.5nm and *R* = 0.5nm. The frequency-dependent permittivity of the metal, SiO_{2} and Si are the same as those in Ref [32]. We do not take interband transition into consideration here, nor the dependence of *β* on the frequency due to transition absorption. Panel (a) shows the dispersion relation of the fundamental mode using the Drude method (line) or the nonlocal method (line). The mode has no cutoff wavelength. Panel (b) shows the propagation distance *L _{prop}* and the mode area

*A*

_{m}_{.}The mode area

*A*is defined as the ratio of the total mode energy to the peak energy density [9]

_{m}*W*is the time averaged electromagnetic energy density. In the Drude model, the average energy density in the metal can be expressed as follows [33]

*β*= 0 in the Drude model, Eq. (9) can be converted to Eq. (7) [33]. On the right side, the second term represents the dissipation caused by the Joule heating, with which we can have a deeper insight of the relative energy loss.

*p*is the ratio of the total dissipation energy to the total mode energy. When

*λ*= 1.55

*μ*m, we have

*p*= 2.17% in the nonlocal model and

*p*= 2.18% in the Drude model for this HPW. Here we can see that a relatively lower loss is obtained in the nonlocal model. This agrees with the simulation result that the propagation distance is longer in the nonlocal model than that in the Drude model. An amazing point is that in the meantime the mode area is smaller in the nonlocal model than that in the Drude model because the continuity of the normal component of

**E**in the nonlocal model enables a peak energy density located in the metal side at the interface between the metal and dielectric. When the propagation distance is 2.5

*μ*m, the mode area is 4.4 × 10

^{−6}

*λ*

^{2}in the traditional Drude model at 1.55

*μ*m. The mode area is 2.8 × 10

^{−6}

*λ*

^{2}and the propagation distance is 2.6

*μ*m in the hydrodynamic model at 1.55

*μ*m. To the best of our knowledge, this is the best result that has ever been reported about the mode area (note that the energy density distribution given in Ref [35]. is not correct because they used the Drude model at the sharp corner and did not use the correct definition of Eq. (7) for the energy density).

Yet we cannot say that the trade-off between the mode confinement and the propagation length has been broken through since not that much improvement lies here. From the Panels 3(a), (b), we can also see, the propagation distance is decreased as *λ* decreases, while the ratio of the mode area to the *λ ^{2}* is increased as

*λ*decreases. Panels 3(c), (d) show the electromagnetic energy density distribution at a wavelength of 1.55

*μ*m for both the nonlocal model and the Drude model. The difference between the energy density distributions of the nonlocal model and the Drude model is associated with the boundary condition and the changes in the induced-charge distribution. In the Drude model, the charge is strictly a surface charge, while in the nonlocal model the charge density is finite and the longitudinal mode penetration depth

*δ*tends to spatially smear out the charge distribution. Effectively, this smearing increases the electric field penetration into the metal, which causes the energy density strong in the thin layer near the metal surface.

_{L}We further explore the influence of the gap on the confinement and loss of this hybrid plasmonic waveguide. We sweep the gap from 0.5nm to 7nm at a fixed wavelength of 1.55*μ*m. In Figs. 4(a)
and 4(b), when the gap height decreases from 7nm to 3nm, the mode area keeps decreasing while the propagation distance does not change notably. When the gap height is reduced further, the mode area decreases to the order of 10^{−6}*λ*^{2}. The results of the nonlocal hydrodynamic model and the traditional Drude model have the same tendency. However, the difference between the mode areas simulated by the two models increases as the gap increases.

We also use a dimensionless figure of merit (FoM) to evaluate the confinement and propagation distance. FoM is defined as the ratio of the propagation distance (*L _{prop}*) to the effective mode radius (

*R*) [36]

_{m}*R*is defined as the radius of the effective mode area (

_{m}*A*). The FoM for our novel hybrid plasmonic waveguide can be as high as 2700 (or 4000) in the Drude (or hydrodynamic) model. While the FoMs of previously studied wedge plasmonic waveguide, cylinder plasmonic waveguide, channel plasmonic waveguide and the hybrid plasmonic waveguide in [37] are just around 1500 at the smallest mode area of 2.5 × 10

_{m}^{−3}

*λ*

^{2}at the telecom wavelength of 1.55

*μ*m in the Drude model. Obviously, our novel HWP can reduce the mode area and in the mean time improve the FoM.

The finite value of the charge density and the longitude plasmonic mode also allow resolving the field in the proximity of very sharp corners and tips, which is similar to the finite SER enhancement factor for an infinitely sharp tip, as discussed in Ref [19]. In Fig. 5(a)
, we reduce *R* from 1nm down to 0.005nm and see the difference between the nonlocal model and the Drude model. As expected, the difference between the two models increase as the tip radius *R* approaches the length scale of the longitudinal mode penetration depth *δ _{L}* (~0.1nm). In the Drude model, when

*R*is approaching zero, the mode area monotonously decreases. When

*r*is 0.005nm, the mode area approaches 10

^{−8}λ and a sharp peak appears in the energy density distribution as shown in Fig. 5(b). However, there is a fundamental saturation of the mode area (

*A*= 3 × 10

_{m}^{−7}

*λ*at

^{2}*r*= 0.005nm) in the nonlocal model, as the nonlocal effect relaxes the sharpness of the tip and the energy density distribution is smeared out as shown in Fig. 5(b).

## 5. Conclusions

In conclusion, we use finite element method for implementing the hydrodynamic model to analyze the propagation mode of a plasmonic waveguide of arbitrary cross section. To the best of our knowledge, this is the first time that the nonlocal effect is numerically implemented for the mode analysis of a plasmonic waveguide. We have proposed a SOI hybrid plasmonic waveguide with an inverted metal nano-rib, which enables nanoscale light confinement. Our theoretical investigation has shown that the mode area could be of 10^{−6}*λ*^{2} size in a wide wavelength range from 1.3*μ*m to 2*μ*m, and the propagation distance still remains several microns. We have studied theoretically how the nonlocal mode affects this new hybrid plasmonic waveguide. We have found that the nonlocal effects can reduce the loss and improve the confinement. We have shown that when the radius of the rib’s tip is approaching zero, the mode area gets saturated in the nonlocal model, instead of monotonous decrease in the Drude model.

## Acknowledgment

We thank Peipeng Xu and Yingchen Wu for valuable discussions. This work is partially supported by the National Natural Science Foundation of China (Nos. 60990322 and 61178062), the National High Technology Research and Development Program (863) of China (No. 2012AA012201) and Swedish VR (No. 621-2011-4620).

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