## Abstract

A vertical slot LiNbO_{3} waveguide with an Ag nanowire and 3L MoS_{2} embedded in the low-refractive index slot region is proposed for the purpose of improving light confinement. We find that the proposed waveguide has a novel dielectric based plasmonic mode, where local light field is enhanced by the Ag nanowire. The mode exhibits an extremely large figure of merit (FoM) of 6.5×10^{6}, one order of magnitude larger than that the largest FoM of any plasmonic waveguide reported in the literature to date. The waveguide also has an extremely long propagation length of 84 cm in the visible wavelength at 680 nm. Furthermore, the waveguide has a low sub-micro bending loss and can be directly connected to all-dielectric waveguides with an extremely low coupling loss. The proposed vertical slot LiNbO_{3} waveguide is a promising candidate for the realization of ultrahigh integration density tunable circuits in the visible spectral range.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Despite rapid progress in the development of photonic integrated circuits, the lateral dimension of a dielectric waveguide is still typically of the order of the wavelength (i.e. μms) [1]. Even though a dielectric waveguide can have a long propagation length, the field confinement in the dielectric waveguide is weak compared to that of a surface plasmon (SP) waveguide. SP waveguides are not bounded by the diffraction limit and thus enable nano-scale optical waveguiding and light confinement. SPs are thus a promising candidate for photonic integrated circuits (PICs) with an ultrahigh integration density [2,3]. The bending loss, direction couplers and the coupler structures between plasmonic waveguides and dielectric waveguides have been studied in [4–9]. A limitation of conventional nano-plasmonic waveguides is that the propagation distance is usually limited to the order of a few micrometers due to large intrinsic loss in the metal. To overcome this challenge, many different kinds of long-range surface-plasmon-polariton waveguides and hybrid plasmonic waveguides have been proposed [2–17]. Small footprint, high performance lasers, modulators and detectors based on such hybrid plasmonic waveguides have been proposed [18–26]. Hybrid plasmonic waveguides made by embedding a metal in the middle of a vertical or horizontal slot waveguide [27–34] can further extend the propagation length and maintain subwavelength confinement. Low loss and high confinement ring resonators, passive waveguides, and modulators have been proposed based on this kind of hybrid plasmonic waveguides [34–36]. The largest figure of merit (FoM) that has been reported to date is 139027 [29].

Hybrid plasmonic waveguides integrated with 2D materials such as graphene, WS_{2}, WSe_{2}, MoSe_{2} and MoS_{2} have been studied [13,17,19,21,22,27]. Graphene in a hybrid plasmonic waveguide is used to support SPP modes [13,17]. Single layer WS_{2}, WSe_{2}, and MoSe_{2} based on high refractive index dielectric materials can also enhance field confinement in a hybrid plasmonic waveguide [27]. Single layer MoS_{2} can, not only enhance field confinement in the waveguide, but also be used as a gain or tunable material [18,21,22,25]. However, in the visible range, single layer MoS_{2} will also increase unwanted loss, and consequently shorten the waveguide propagation length. By utilizing multiple MoS_{2} layers, the complex refractive index *N* will change [37,38]. A 3-layer (3L) MoS_{2} has zero optical loss from 680 nm to 710 nm [38], and is a good candidate to enhance mode confinement, while maintaining long propagation length, in a plasmonic waveguide. To the best of our knowledge, there is to date no report on plasmonic waveguide with 3-layer (3L) MoS_{2}.

It has been shown that the FoM of a hybrid plasmonic waveguide increases as the width of metal nanowire or metal decreases [27,29,32,33,36], however, there is no study on the FoM when the width of the metal nanowire is smaller than 5 nm. It would thus be of interest to study the FoM and mode characteristics of a hybrid plasmonic waveguide as the metal structure becomes smaller. In order to study the light confinement in a metal sub-nanometer structure, the nonlocal dynamic in the metal described by the hydrodynamical model becomes increasingly important [15,39].

In this paper, we propose a LiNbO_{3} based vertical slot waveguide with an Ag nanowire and 3L MoS_{2} embedded in the low-refractive index slot region that enhances light confinement and enables an ultra-long propagation length. Using COMSOL Multiphysics software [40] with a hydrodynamic model [15,39], the guiding properties of the proposed waveguide, namely, the effective index, the propagation length, the mode area, the FoM, the bending loss, the coupling length, and the insertion loss were studied. The wavelength of operation is 680 nm, which is in the transparent band of the 3L MoS_{2}. The proposed waveguide has a dielectric based plasmonic mode, rather than a hybrid plasmonic mode or a long-range surface-plasmon-polariton mode. The achievable propagation length is 84 cm with strong confinement, and the normalized mode area is 0.11. An FoM of 6.5×10^{6}, which is more than one order of magnitude larger than the best result that has been reported previously [29], is achieved. This kind of waveguide is then suitable for plasmonic circuits with high performance requirements. Thanks to the large electro-optical coefficient of LiNbO_{3}, the effective refractive index of the waveguide is also tunable. Our study thus paves a way for future tunable subwavelength integrated optoelectronics.

## 2. Waveguide structure and analysis

A cross-sectional view of the proposed vertical slot LiNbO_{3} waveguide with an Ag nanowire and 3L MoS_{2} embedded is shown in Fig. 1. The Ag nanowire is placed over the 3L MoS_{2}, that has a thickness of 2.1 nm. The radius of the Ag nanowire is denoted by *r* and the gap between the nanowire and the 3L MoS_{2} is denoted by *s*. The vertical slot waveguide over the thick SiO_{2} substrate, is made of X-cut lithium niobate (LiNbO_{3}), which has low insertion loss in the 680 nm ∼ 710 nm range [41,42]. Compared to other low loss materials such as SiO_{2}, Si_{3}N_{4}, TiO_{2} and AlN, LiNbO_{3} has a large electro-optical coefficient and a strong optical nonlinearity, which is of importance e.g. to realize high-performance tunable optical devices [41]. The bottom LiNbO_{3} and top LiNbO_{3} have the same height, denoted by *h _{LN}*. Silica is chosen to form the low-index gap in the vertical slot waveguide. The Ag nanowire and the 3L MoS

_{2}are located in the silica and have the same distance, denoted by

*h*(see Fig. 1(b)), away from the nearest LiNbO

_{SiO2}_{3}surface. The refractive indices of SiO

_{2}and 3L MoS

_{2}used are

*n*= 1.456,

_{SiO2}*n*= 5.38 [38] at 680 nm. The ordinary (

_{3L MoS2}*n*) and extraordinary (

_{o}*n*) refractive indices of X-cut LiNbO

_{e}_{3}are

*n*= 2.275 and

_{o}*n*= 2.192 at 680 nm [41]. The absorption indices of LiNbO

_{e}_{3}and SiO

_{2}are 7×10

^{−8}[42] and 1×10

^{−7}[43], respectively. The pertinent Ag parameters are the plasmonic frequency $\hbar $

*ω*= 8.59 eV, the damping coefficient $\hbar $

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

*B*= 0.0036

*c*[15], where $\hbar $ is the reduced Planck constant. In our simulation, by using perfectly matched layer boundary conditions [40], the radiative losses can be taken into consideration by solving the complex propagation constant of the waveguide.

_{0}We use the propagation length, the normalized mode area, and the figure of merit to evaluate the performance of the proposed waveguide. The propagation length is a measure of the waveguide transmission loss, which is defined as the length at which the initial normalized electromagnetic energy of the supported mode has decreased by *1/e* [29], and is calculated as

*β*is the propagation constant and

*n*the effective refractive index of a guided mode.

_{eff}The mode area is a measure of energy confinement and is defined as the ratio of the total electromagnetic energy to the peak energy density of the guided mode, [29] i.e.:

where*W*is the time averaged electromagnetic energy density. For dispersive and lossy materials,

*W*can be expressed as [15]:

In the hydrodynamic model the time averaged electromagnetic energy density in the metal can be expressed as (the assumptions used in this computational model is the same as those in Refs. [15] and [39], i.e., the electron gas at the metal surface has only the kinetic energy and is similar to the hydrodynamic flow which causes a hydrodynamic polarization current; More detailed assumptions about the nonlinear effect and magnetic induction effect in the hydrodynamic model can be found in Ref. [44]):

**J**is the current inside the metal induced by the electric field

**E**,

*ω*is the plasmonic frequency, and

_{p}*B*is the nonlocal parameter [15].

We use the normalized mode area, defined as *A _{m}/A_{0}*, to quantitatively evaluate the mode confinement.

*A*is the diffraction-limited mode area in free space, where

_{0}= λ_{0}^{2}/4*λ*is the operation wavelength.

_{0}The figure of merit is defined as the ratio of the propagation distance to the effective mode area radius as shown in Eq. (5) [29]:

The mode properties of the proposed waveguide as functions of the Ag nanowire radius are shown in Fig. 2. Here, we choose *w* = 0.08 μm, *h _{LN}* = 0.6 μm,

*h*= 2 nm, and

_{SiO2}*s*= 0.2 nm. In Fig. 2(a), both the plasmonic mode and the dielectric based plasmonic mode exist in the proposed waveguide. We find that there is a large difference between

*n*of the plasmonic mode and

_{eff}*n*of the dielectric based plasmonic mode when

_{eff}*r*is below 2.5 nm. The effective index

*n*of the dielectric based plasmonic mode is almost the same as

_{eff}*n*of the proposed waveguide without the Ag nanowire. The proposed waveguide does not support the hybrid plasmonic modes [10,12]. The difference between

_{eff}*n*of the dielectric based plasmonic mode and

_{eff}*n*of the plasmonic mode becomes smaller, as the radius of the Ag nanowire increases. When the radius of the Ag nanowire is larger than 20 nm, the coupling between the dielectric based plasmonic mode and the plasmonic mode is stronger and these two modes will turn in to hybrid plasmonic modes [10,12]. The dielectric based plasmonic mode does not belong to the long-range surface-plasmon-polariton, which exists in finite width thin metal film in an infinite dielectric [3]. Below, we focus the study of this new dielectric based plasmonic mode.

_{eff}From Figs. 2(b-d), we see that the Ag nanowire influences the propagation length, the light field distribution and the FoM of the dielectric based plasmonic mode. The electric field distribution of the dielectric based plasmonic mode and the plasmonic mode of the proposed waveguide are shown in Figs. 2(e,f). In Fig. 2(e), we find that the electric field is locally enhanced at the surface of the Ag nanowire. In Fig. 2(f), we find that the light in the plasmonic mode is confined more strongly than the case for the dielectric based plasmonic mode as seen in Fig. 2(e). From Figs. 2(b-d), we find that the dielectric based plasmonic mode has a longer propagation length, a larger mode area and a larger FoM as compared to the plasmonic mode. When the *r* is reduced from 2.5 nm to 0.1 nm, the propagation length and the normalized mode area of the dielectric based plasmonic mode increase and tend to that of the waveguide without metal. When the *r* is 0.2 nm, the FoM of the dielectric based plasmonic mode reaches its maximum of 6.5×10^{6}, which is more than an order of magnitude larger than the largest such previously reported [29]. In addition, from Fig. 2(d), we find that for the dielectric mode in the proposed waveguide without metal, the FoM is 3.87×10^{6}, somewhat smaller than the maximum for the proposed dielectric based plasmonic mode. Thus, as compared to the dielectric mode, the dielectric based plasmonic mode in the proposed waveguide can simultaneously enhance the mode confinement and increase the FoM. For the plasmonic mode, the propagation length and the normalized mode area are reduced when *r* is reduced from 2.5 nm to 0.15 nm. The plasmonic mode cannot be found in the hydrodynamic model when the radius is below 0.15 nm. When *r* is 0.15 nm, the FoM of the plasmonic mode is 46, and the mode area can be as small as 5.9×10^{−7}.

In Table 1 we compare our results with previous works. We can see that the dielectric based plasmonic mode in the proposed waveguide has the best performance, indeed much better than any previously reported results, in terms of FoM. However, the fabrication complexity of the proposed waveguide is high.

We study the mode properties at varying number of MoS_{2} layers, the results of which are shown in Fig. 3. It can be seen that the proposed waveguide with 3 MoS_{2} layers has the smallest mode area, i.e., the 3L MoS_{2} can enhance the mode confinement most significantly. The propagation length for 1L MoS_{2} or 2L MoS_{2} is one order of magnitude smaller than for the proposed waveguide with 3L MoS_{2} and thus the FoM of the proposed waveguide with 3L MoS_{2} is the largest, as shown in Fig. 3(c). It should be noted that if we were to remove the 3L MoS_{2} and reduce the SiO_{2} gap by the equivalent thickness, the FoM would be 4.1×10^{6}, smaller than that of the proposed waveguide with 3L MoS_{2}.

We further analyze the effects that changes in geometry have on the mode characteristics, i.e. the propagation length, the normalized mode area and the FoM, by varying the gap between the Ag nanowire and the 3L MoS_{2}, *s*; the waveguide width, *w*; the thickness of the SiO_{2}, *h _{SiO2}*; and the height of LiNbO

_{3},

*h*. In Fig. 4 we see that, by reducing

_{LN}*s*,

*w*,

*h*and

_{SiO2}*h*, the normalized mode can be reduced and the FoM can increase slightly. However, if

_{LN}*w*and

*h*, are further reduced the dielectric based plasmonic mode cannot be supported in the proposed waveguide. By comparing Figs. 2–4, we find that the geometry parameters,

_{LN}*s*,

*w*,

*h*, and

_{SiO2}*h*have a limited effect on the propagation length, the normalized mode area, and the FoM as compared to that of the radius of the Ag nanowire or the number of the MoS

_{LN}_{2}layers, i.e., the geometry parameter that the mode characteristics are most sensitive to is the size of the Ag nanowire.

The proposed waveguide can be fabricated by conventional planar fabrication technology in the thin-film LiNbO_{3} platform [42] by the following process sequence: (1) a thin SiO_{2} film is coated on top of LiNbO_{3}; (2) the 3L MoS_{2}, which can be fabricated by mechanical exfoliation [37], is placed over the thin SiO_{2} film; (3) a very thin SiO_{2} layer is deposited on the top the 3L MoS_{2}; (4) Alignment markers are patterned on the samples; (5) A 0.4 nm silver nanowire can be fabricated by electro-/photochemical redox reaction [45]; (6) A silver nanowire is aligned and placed on the top by combining fluidic alignment with surface-patterning techniques [46]; (7) Alignment markers are patterned based on the silver nanowires’ position; (8) a thin SiO_{2} film is deposited on the top again; (9) another thin-film LiNbO_{3} is flipped and bonded on the top; (10) the bonded LiNbO_{3}substrate is removed; (11) the waveguide patterns are aligned based on the predefined alignment markers and then the waveguides are fabricated using electron beam lithography and the inductively coupled reactive ion etching.

The actual fabrication for the Ag nanowire with *r* = 0.2 nm may introduce some imperfection. Therefore, we study the effect of the misalignment δx between the actual Ag nanowire and the designed center of the proposed waveguide in the horizon direction on the FoM and propagation length, as shown in Fig. 5. Here we can see that the imperfection effect is less than 0.1% fluctuation for the propagation length, normalized mode area and FoM, which means that it is insensitive to the misalignment δx within a variation range of 3 nm.

When introducing some random changes in the radius of the Ag nanowire along the waveguide, we need to consider the insertion loss caused by the mode mismatch. The insertion loss caused by the radius variation of the nanowire is shown in Fig. 6 (calculated by the overlapping integral between modes), from which we can see that the insertion loss is below 0.001 dB when the radius changes from 0.1nm to 0.3nm. Furthermore, Fig. 2(b-d) shows that when the radius changes by 10% from 0.2 nm (i.e., from 0.18 nm to 0.22 nm), the imperfection effect is about 7.6% variation for the propagation length, 14.9% variation for the normalized mode area, and 3.7% variation for FoM. Thus, the mode properties remain acceptable to some fabrication imperfection of the radius variation.

As is well-known, by applying an external electric field, the ordinary, *n _{o}*, and the extraordinary,

*n*, indices of X-cut LiNbO

_{e}_{3}can be changed and we can utilize the change of the effective index of refraction,

*n*, of the dielectric based plasmonic waveguide to design tunable devices. Here we discuss the effect of the variation of

_{eff}*n*and

_{o}*n*of LiNbO

_{e}_{3}on

*n*of the proposed waveguide. From Figs. 7(a,b), we find that, in the proposed waveguide,

_{eff}*n*is more sensitive to a variation of

_{eff}*n*in the X-cut LiNbO

_{o}_{3}. As

*n*can be controlled by applying a vertical electric field it is thus more suitable to design tunable devices by utilizing an applied vertical voltage to change the refractive index of the LiNbO

_{o}_{3}. For example, the resonance wavelengths of some ring resonators or 1D photonic crystal nanocavities made by the proposed waveguide can be tuned by an externally applied electric field.

In order to improve integration density of photonic circuits, a short coupling length, a low bending loss, and a low insertion loss are required. Figures 8(a-c) show the bending loss, the coupling length, and the insertion loss of the proposed waveguide with structural parameters being: *w *= 0.08 μm, *h _{LN}* = 0.6 um,

*h*= 2 nm,

_{SiO2}*s*= 0.2 nm, and

*r*= 0.2 nm. The bending loss, shown in Fig. 8(a), is given by the transmission loss through a 90-degree arc. The 90-degree bending loss increases almost exponentially as the bending radius decreases, which is similar to a dielectric waveguide. When the bending radius is above 0.4 μm, the bending loss is below 1 dB. The coupling length is given by

*L*Re[

_{c}= π/*β*], where

_{o}– β_{e}*β*and

_{o}*β*are the propagation constants for the odd and even modes. The coupling length, shown in Fig. 8(b), increases almost exponentially as the distance between the two waveguides increases, which is also similar to a dielectric waveguide. When the distance D between two of the proposed waveguides is 0.2 μm, the coupling length 3.3 μm.

_{e}In order to ensure compatibility with conventional all-dielectric waveguides, the end-butt insertion loss between a dielectric waveguide and the proposed waveguide is studied as shown in Fig. 8(c). The insertion loss is about 0.0001 dB and is insensitive to the waveguide width variation and thus it is clear that our proposed waveguide can easily be integrated with conventional dielectric waveguide circuits and hence enable high density integrated circuits.

## 3. Conclusion

A new dielectric based plasmonic mode has been discovered in a vertical slot LiNbO_{3} waveguide with an embedded Ag nanowire and 3L MoS_{2} in the low-refractive index slot region. Compared to hybrid plasmonic waveguides, the dielectric based plasmonic mode in our proposed waveguide has a low propagation loss and the FoM is an order of magnitude larger. Our calculations have furthermore shown that the proposed waveguide can also be utilized for tunable devices due to the large electro-optic coefficient of LiNbO_{3}. The proposed waveguide is easy to connect to all-dielectric waveguides through an end-butt coupling structure with an extremely low insertion loss, which makes it convenient to integrate with all-dielectric waveguide circuits. With these excellent properties, the proposed waveguide shows great promise for realizing compact high-performance and tunable devices, such as wavelength tunable nano-lasers and ultra-compact tunable photonic circuits, for high integration density optoelectronics.

## Funding

National Key Research and Development Program of China (2018YFB2200200, 2017YFA0205700); Fundamental Research Funds for the Central Universities (2019FZA5002); National Natural Science Foundation of China (91833303); National Natural Science Foundation of China (61774131).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **L. Thylén, S. He, L. Wosinski, and D. Dai, “The Moore’s law for photonic integrated circuits,” J. Zhejiang Univ., Sci., A **7**(12), 1961–1967 (2006). [CrossRef]

**2. **L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**(17), 6645–6650 (2005). [CrossRef]

**3. **P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmon-polariton waveguides,” J. Appl. Phys. **98**(4), 043109 (2005). [CrossRef]

**4. **W. Wang, Q. Yang, F. Fan, H. Xu, and Z. L. Wang, “Light Propagation in Curved Silver Nanowire Plasmonic Waveguides,” Nano Lett. **11**(4), 1603–1608 (2011). [CrossRef]

**5. **D. J. Dikken, M. Spasenovic, E. Verhagen, D. Oosten, and L. Kuipers, “Characterization of bending losses for curved plasmonic nanowire waveguides,” Opt. Express **18**(15), 16112–16119 (2010). [CrossRef]

**6. **D. Zeng, L. Zhang, Q. Xiong, and J. Ma, “Directional coupler based on an elliptic cylindrical nanowire hybrid plasmonic waveguide,” Appl. Opt. **57**(16), 4701–4706 (2018). [CrossRef]

**7. **M. Z. Alam, J. Niklas Caspers, J. S. Aitchison, and M. Mojahedi, “Compact low loss and broadband hybrid plasmonic directional coupler,” Opt. Express **21**(13), 16029–16034 (2013). [CrossRef]

**8. **S. H. Badri and M. M. Gilarlue, “Coupling Between Silicon waveguide and Metal-Dielectric-Metal Plasmonic Waveguide with Lens-Funnel Structure,” Plasmonics **15**(3), 821–827 (2020). [CrossRef]

**9. **S. H. Badri and M. M. Gilarlue, “Coupling a silicon-on-insulator waveguide to a metal-dielectric-metal plasmonic waveguide through a vertical and lateral taper-funnel structure,” Appl. Phys. B **126**(5), 94 (2020). [CrossRef]

**10. **D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express **17**(19), 16646–16653 (2009). [CrossRef]

**11. **L. Ding, J. Qin, K. Xu, and L. Wang, “Long range hybrid tube-wedge plasmonic waveguide with extreme light confinement and good fabrication error tolerance,” Opt. Express **24**(4), 3432–3440 (2016). [CrossRef]

**12. **R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics **2**(8), 496–500 (2008). [CrossRef]

**13. **X. Zhou, T. Zhang, L. Chen, W. Hong, and X. Li, “A Graphene-Based Hybrid Plasmonic Waveguide With Ultra-Deep Subwavelength Confinement,” J. Lightwave Technol. **32**(21), 4199–4203 (2014). [CrossRef]

**14. **K. Zheng, X. Zheng, Q. Dai, and Z. Song, “Hybrid rib-slot-rib plasmonic waveguide with deep-subwavelength mode confinement and long propagation length,” AIP Adv. **6**(8), 085012 (2016). [CrossRef]

**15. **Q. Huang, F. Bao, and S. He, “Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement,” Opt. Express **21**(2), 1430–1439 (2013). [CrossRef]

**16. **V. A. Zenin, S. Choudhury, S. Saha, V. M. Shalaev, A. Boltasseva, and S. I. Bozhevolnyi, “Hybrid plasmonic waveguides formed by metal coating of dielectric ridges,” Opt. Express **25**(11), 12295–12302 (2017). [CrossRef]

**17. **S. He, X. Zhang, and Y. He, “Graphene nano-ribbon waveguides of record-small mode area and ultra-high effective refractive indices for future VLSI,” Opt. Express **21**(25), 30664–30673 (2013). [CrossRef]

**18. **X. Meng, R. R. Grote, W. Jin, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, “Rigorous theoretical analysis of a surface-plasmon nanolaser with monolayer MoS2 gain medium,” Opt. Lett. **41**(11), 2636–2639 (2016). [CrossRef]

**19. **L. Xu, F. Li, S. Liu, F. Yao, and Y. Liu, “Low Threshold Plasmonic Nanolaser Based on Graphene,” Appl. Sci. **8**(11), 2186 (2018). [CrossRef]

**20. **M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photonics Rev. **8**(3), 394–408 (2014). [CrossRef]

**21. **L. Xu, F. Li, L. Wei, J. Zhou, and S. Liu, “Design of Surface Plasmon Nanolaser Based on MoS2,” Appl. Sci. **8**(11), 2110 (2018). [CrossRef]

**22. **X. He, F. Liu, F. Lin, G. Xiao, and W. Shi, “Tunable MoS2 modified hybrid surface plasmon waveguides,” Nanotechnology **30**(12), 125201 (2019). [CrossRef]

**23. **Y. J. Lu, J. Kim, H. Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C. Y. Wang, M. Y. Lu, B. H. Li, X. Qiu, W. H. Chang, L. J. Chen, G. Shvets, C. K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science **337**(6093), 450–453 (2012). [CrossRef]

**24. **R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**(7264), 629–632 (2009). [CrossRef]

**25. **I. Suárez, A. Ferrando, J. Marques-Hueso, A. Díez, R. Abargues, P. J. Rodríguez-Cantó, and J. P. Martínez-Pastor, “Propagation length enhancement of surface plasmon polaritons in gold nano-/micro-waveguides by the interference with photonic modes in the surrounding active dielectrics,” Nanophotonics **6**(5), 1109–1120 (2017). [CrossRef]

**26. **J. Guo, J. Li, C. Liu, Y. Yin, W. Wang, Z. Ni, Z. Fu, H. Yu, Y. Xu, Y. Shi, Y. Ma, S. Gao, L. Tong, and D. Dai, “High-performance silicon-graphene hybrid plasmonic waveguide photodetectors beyond 1.55 μm,” Light: Sci. Appl. **9**(1), 29 (2020). [CrossRef]

**27. **K. Zheng, Y. Yuan, J. He, G. Gu, F. Zhang, Y. Chen, J. Song, and J. Qu, “Ultra-high light confinement and ultra-long propagation distance design for integratable optical chips based on plasmonic technology,” Nanoscale **11**(10), 4601–4613 (2019). [CrossRef]

**28. **Y. Bian and Q. Gong, “Low-loss light transport at the subwavelength scale in silicon nano-slot based symmetric hybrid plasmonic waveguiding schemes,” Opt. Express **21**(20), 23907–23920 (2013). [CrossRef]

**29. **W. Wei, X. Zhang, and X. Ren, “Asymmetric hybrid plasmonic waveguides with centimeter-scale propagation length under subwavelength confinement for photonic components,” Nanoscale Res. Lett. **9**(1), 599 (2014). [CrossRef]

**30. **W. Ma and A. S. Helmy, “Asymmetric long-range hybrid-plasmonic modes in asymmetric nanometer-scale structures,” J. Opt. Soc. Am. B **31**(7), 1723 (2014). [CrossRef]

**31. **Y. Binfeng, H. Guohua, J. Yang, and C. Yiping, “Characteristics analysis of a hybrid surface palsmonic waveguide with nanometric confinement and high optical intensity,” J. Opt. Soc. Am. B **26**(10), 1924–1929 (2009). [CrossRef]

**32. **C. Y. Jeong, M. Kim, and S. Kim, “Circular hybrid plasmonic waveguide with ultra-long propagation distance,” Opt. Express **21**(14), 17404–17412 (2013). [CrossRef]

**33. **C. C. Huang, “Ultra-long-range symmetric plasmonic waveguide for high-density and compact photonic devices,” Opt. Express **21**(24), 29544–29557 (2013). [CrossRef]

**34. **Y. Su, P. Chang, C. Lin, and A. S. Helmy, “Record Purcell factors in ultracompact hybrid plasmonic ring resonators,” Sci. Adv. **5**(8), eaav1790 (2019). [CrossRef]

**35. **A. O. Zaki, K. Kirah, and M. A. Swillam, “Hybrid plasmonic electro-optical modulator,” Appl. Phys. A **122**(4), 473 (2016). [CrossRef]

**36. **J. T. Kim, J. J. Ju, S. Park, M. Kim, S. K. Park, and S. Shin, “Hybrid plasmonic waveguide for low-loss lightwave guiding,” Opt. Express **18**(3), 2808–2813 (2010). [CrossRef]

**37. **O. Salehzadeh, M. Djavid, N. H. Tran, I. Shih, and Z. Mi, “Optically Pumped Two-Dimensional MoS2 Lasers Operating at Room-Temperature,” Nano Lett. **15**(8), 5302–5306 (2015). [CrossRef]

**38. **C. Hsu, R. Frisenda, R. Schmidt, A. Arora, S. M. Vasconcellos, R. Bratschitsch, H. S. J. der Zant, and A. Castellanos-Gomez, “Thickness-Dependent Refractive Index of 1L, 2L, and 3L MoS2, MoSe2, WS2, and WSe2,” Adv. Opt. Mater. **7**(13), 1900239 (2019). [CrossRef]

**39. **G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A.-P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics **2**(3), 161–166 (2013). [CrossRef]

**40. **COMSOL, COMSOL Multiphysics, https://www.comsol.com/, accessed: May. 2020.

**41. **D. E. Zelmon, D. L. Small, and D. H. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide–doped lithium niobate,” J. Opt. Soc. Am. B **14**(12), 3319–3322 (1997). [CrossRef]

**42. **B. Desiatov, A. Shams-Ansari, M. Zhang, C. Wang, and M. Lončar, “Ultra-low-loss integrated visible photonics using thin-film lithium niobate,” Optica **6**(3), 380 (2019). [CrossRef]

**43. **R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. **46**(33), 8118–8133 (2007). [CrossRef]

**44. **K. R. Hiremath, L. Zschiedrich, and F. Schmidt, “Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nédélec finite elements,” J. Comput. Phys. **231**(17), 5890–5896 (2012). [CrossRef]

**45. **B. H. Hong, S. C. Bae, C. W. Lee, S. Jeong, and K. S. Kim, “Ultrathin Single-Crystalline Silver Nanowire Arrays Formed in an Ambient Solution Phase,” Science **294**(5541), 348–351 (2001). [CrossRef]

**46. **Y. Huang, X. Duan, Q. Wei, and C. M. Lieber, “Directed Assembly of One-Dimensional Nanostructures into Functional Networks,” Science **291**(5504), 630–633 (2001). [CrossRef]