Open Access
Add to BibSonomy
Abstract
The photothermal properties of graphene plasmonic waveguides (GPWs) are numerically investigated, while most of existing studies focus on their optical properties. A three-dimensional (3D) coupled optical-thermal model based on finite element method (FEM) is presented. The graphene sheet is treated as an graphene equivalent impedance surface. Transient thermal responses and peak temperature of the GPWs are captured using time-domain FEM (TDFEM). The effectiveness of the proposed method is validated by two examples of hybrid GPWs. Numerical results present the main factors that influence the photothermal properties of the GPWs, including the conductivity of graphene, and the wavelength and power density of incident light. The findings unveil that the temperature increase is an underlying factor influencing the maximum integration density of GPWs in optical interconnect.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Due to the diffraction limit of light, it is difficult to fabricate and integrate optical devices at nanoscale. Surface plasmon polaritons (SPPs) are light induced resonances of free electrons at the metal/dielectric interfaces, which enable the optical devices to break through the diffraction limit. SPP is one of the most promising approaches to manipulating light at nanoscale, which has attracted extensive attention [1–3]. Plasmonic waveguides (PWs) based on SPPs have been widely adopted in optical interconnects, and exhibit promising application in manufacturing highly integrated photonic chips [4–7]. Graphene is a two-dimensional (2D) crystal comprised of a one-atom thick layer of carbon atoms arranged in a honeycomb lattice. Graphene has an easily tunable surface conductivity, which supports SPPs from terahertz to mid-IR [8–13]. Compared with conventional PWs based on noble metals, graphene PWs (GPWs) exhibit many remarkable characteristics, such as relatively lower loss, longer propagation distance, and deeper sub-wavelength. Moreover, the optical properties of GPWs can be constantly adjusted by changing external voltage or chemical doping [14–22]. Hence, the GPWs have become one of the most promising candidates for optical interconnects at mid-infrared waveband.
When light propagates in a GPW, the inevitable absorption of light by graphene is converted to heat. As a positive effect, the photothermal effects of metals and graphene have been exploited in many applications [23–31]. However, as a negative effect, the photothermal effect can seriously affect the integration and performance of optical devices [32]. At present, most of the research work focuses on the optical properties of GPWs, while the photothermal properties of GPWs is less studied and still need to be further explored. In addition to the optical feature size, the thermal feature size is another factor that restricts the maximum integration density for GPWs in optical interconnect, which should be minimized in practical design. Therefore, a thorough characterization of the photothermal properties of GPWs is essential for the design and optimization of GPWs. Numerical modeling method provides an efficient way to predict and analyze the photothermal properties of GPWs, and plays a vital role in the photothermal study of GPWs [33–40].
In this article, the photothermal effects of GPWs are modeled and characterized using an efficient numerical framework based on finite element method (FEM). While most of numerical researches devote to the optical properties of GPWs, we establish a coupled optical-thermal model to investigate the photothermal properties of GPWs. The graphene sheet is treated as an equivalent impedance surface. Compared with conventional voluminal modeling of graphene sheet, this treatment significantly reduces the number of unknowns and improve the computational efficiency. Based on the heating generated from light absorption, the transient thermal responses and the peak temperature of the GPWs are captured using time-domain FEM (TDFEM). The photothermal effects of two hybrid GPWs are investigated. Numerical results present the main factors that influence the photothermal properties of the GPWs, including the conductivity of graphene, and the wavelength and power density of incident light. These findings reveal the physical properties of GPWs from the photothermal perspective rather than conventional optical perspective, and indicate that the thermal feature size is an underlying factor influencing the maximum integration density of GPWs, which provides useful insights for the design and adoption of GPWs in optical interconnect.
2. Method and formulations
2.1 Equivalent impedance surface
The 2D nature of the graphene materials implies an inherent anisotropy in vectorial 3D formulations [41,42]. Assuming a graphene sheet normal to the z-axis, this type of anisotropy is modeled by a surface conductivity tensor of
Fig. 1. The surface conductivity $\sigma$ of graphene varying with (a) chemical potential (T = 293.15 K and $f = 30\textrm{ THz}$), and (b) temperature (${\mu _c}$ = 500 meV and $f = 30\textrm{ THz}$).
Fig. 2. The surface conductivity $\sigma$ of graphene varying with temperature for (a), (b) real part and imaginary part of $\sigma$ (${\mu _c}$ = 500 meV, $\mu \textrm{ = 2}0,000\textrm{ c}{\textrm{m}^\textrm{2}}/(\textrm{V} \cdot \textrm{s})$ and $f = 30\textrm{ }\textrm{THz}$). (c), (d) real part and imaginary part of $\sigma$ (${\mu _c}$ = 600 meV, $\mu \textrm{ = 2},000\textrm{ c}{\textrm{m}^\textrm{2}}/(\textrm{V} \cdot \textrm{s})$ and $f = 40\textrm{ }\textrm{THz}$).
For three-dimensional (3D) numerical analysis, to model a graphene sheet, one can approximate the graphene layer by a thin slab with a small but finite thickness. The main drawback of this method is that a realistic model of graphene will leads to a very high aspect ratio of its length and thickness. The numerical computation of photothermal responses in such a structure will therefore require a very high mesh density and subsequent high computational costs. Since the skin-depth is much larger than the thickness of the graphene sheet, it can be modeled as an equivalent impedance surface to treat this problem with low computational costs [45,46]. As an impedance thin sheet, the tangential electric field ${\textbf E}_t^{}$ is proportional to the electric current density ${\textbf J}$ at its surface as ${\textbf E}_t^{} = {Z_s}{\textbf J}$. Here, ${Z_s} = 1/\sigma$ is the equivalent surface impedance of the graphene. Based on this, the boundary condition cross the graphene sheet should satisfy
2.2 Optical analysis
The optical properties of GPWs can be characterized by the theory of electromagnetic wave. The behavior of surface plasmon wave in a GPW is governed by Maxwell’s equation [47]. Assuming time-harmonic electric field, the electric field distribution in a GPW can be calculated by using the 3D vector wave equation
where ${\textbf E}$ denotes the electric field, ${k_0}$ is the free-space wavenumber, $\mu _r^{}$ is the relative permeability, and ${\varepsilon _r}$ is the relative permittivity. Since the graphene sheet is modeled as an equivalent impedance surface, we impose an impedance boundary condition at this surface to generate a complete boundary value problem. Derived from Eq. (4), the impedance boundary condition at the graphene surface can be described asIn 3D FEM analysis, we use tetrahedral elements to discretize the whole computational domain [48]. In each element, we expand ${\textbf E}$ using vector edge basis function ${\textbf N}$ as
where $\varphi$ is the unknown coefficient of electric field to be solved, and m denotes the number of edge basis functions in an element. Using the Galerkin’s method, a spatially discrete matrix equation can be derived from Eqs. (5) and (6) as where2.3 Thermal evolution
Based on the optical analysis, the heat source ${Q}$ in the GPWs is obtained by computing the electromagnetic power density
where ${Q}$ represents the heat power density, ${\sigma _r}$ is the real part of the surface conductivity of graphene, and ${\textbf E}$ denotes the obtained electric field in the optical analysis.Then, the transient temperature evolution can be obtained by solving the 3-D heat conduction equation
where T is the unknown temperature as a function of time and space, $\rho$ is the mass density of material, c refers to the specific heat capacity, $\kappa$ represents the thermal conductivity.To describe the heat exchange between the GPW and the air, a convection boundary condition is employed on the GPW-air interface as
where h denotes the convective heat transfer coefficient, and ${T_0}$ represents the surrounding temperature.For the solution of Eq. (11), an initial condition should be introduced, which can be specified as
The time-domain FEM (TDFEM) is implemented to solve Eq. (11) so as to compute the transient temperature response of the GPWs. For the mesh discretization, the thermal domain shares the same set of elements with the electromagnetic domain. The temperature T in each element is expanded by scalar nodal basis functions N as where u is the unknown coefficient of temperature to be solved, and s denotes the number of nodal basis functions in an element. Combining with Eqs. (10) and (12), the spatially discrete form of Eq. (11) can be obtained with the Galerkin’s scheme as whereFor an unconditionally stable time-domain solution, the backward difference scheme is employed for the temporal discretization of Eq. (11). By dividing the total time t into p time steps with an identical size of $\Delta t$, we have
where the superscript n denotes the number of time step. By combining Eqs. (15) and (19), we can get the temperature update equation The coefficient of temperature u can be obtained by solving Eq. (20). Then, by using Eq. (14), we can get the transient distribution of temperature in a GPW at any time step. The computational costs for the whole photothermal computation are mainly dependent on the number of unknown after mesh discretization. Parallel computation can be employed to improve the efficiency of photothermal analysis.3. Results and discussion
Two types of hybrid GPWs that can be used in optical interconnect are considered for the numerical modeling of photothermal effects, which are a rectangular hybrid GPW and a Y-type hybrid GPW. They consist of silicon (Si) cap and substrate, two layers of silicon dioxide (SiO2), and a graphene layer between two SiO2 layers. Table 1 shows the material parameters of Si and SiO2 for the numerical computations. The permittivities are from Refs. [49] and [50], and the thermal parameters are from COMSOL material library. The proposed optical and thermal co-analysis method is applied for the photothermal characterization. All the computations are performed on a workstation with parallel computation environment.
Table 1. Material parameters of Si and SiO2 for the photothermal analysis.
3.1 Rectangular hybrid GPW
A rectangular hybrid GPW is first considered for the photothermal analysis. Figure 3 presents the configuration and dimensions of this GPW [50]. The surrounding temperature is set to be ${T_0} = 293.15\textrm{ }\textrm{K}$, and the convective heat transfer coefficient is set to be $h = 20\textrm{ W}/({\textrm{m}^2} \cdot \textrm{K})$. The TM-polarized incident light is induced through the GPW along z direction, which has awavelength of 10 µm and a power density of 1 µ$\textrm{W}/$µm2. The chemical potential of the graphene layer is first set to be ${\mu _c} = \textrm{5}00\textrm{ meV}$. According to Eqs. (2) and (3), we can get a surface conductivity of the graphene of $\sigma = 1.6824 \times {10^{ - 6}} - j3.0741 \times {10^{ - 4}}$. Tetrahedral elements are adopted for the mesh discretization of computational domain in the FEM analysis. Since the graphene layer is treated as an impedance surface, the voluminal discretization of graphene is avoided, which greatly reduces the computational costs.
Fig. 3. Configuration of the rectangular hybrid GPW. (a) 3D view. (b) Side view, front view, and dimensions, where w = 150 nm, h = 20 nm, t1 = 60.5 nm, t2 = 80 nm, and L = 1000 nm.
First, the light propagation in the GPW is simulated by the proposed method. The frequency of the light is 30 THz. The electric field distributions in the GPW are presented in both x-z plane and x-y plane. Figure 4(a) shows the y-component of electric field (${E_y}$) in x-z plane, which reveals the behavior of the SPP wave along the direction of propagation. As a comparison, the results of COMSOL software are provided in Fig. 4(b). For the COMSOL simulation, the graphene layer is treated as a thin slab with a thickness of 0.34 nm, whose permittivity is derived from the formula of surface conductivity. Furthermore, the calculated results of ${E_y}$ in x-y plane are provided together with the COMSOL results in Figs. 4(c) and 4(d). Good agreement can be observed, which demonstrates the accuracy of the proposed method.
Fig. 4. The electric field distributions in the rectangular hybrid GPW at 30 THz in x-z plane and x-y plane. (a), (c) Calculated results based on equivalent impedance surface model of graphene. (b), (d) COMSOL results based on volume model of graphene.
Then, we conduct the thermal analysis based on the obtained electric field distribution. The transient temperature responses are calculated by using the proposed TDFEM. 30 ms total time and 300 time steps are adopted for the time-domain evolution. Figure 5 presents the maximum temperature of the GPW varying with time. It can be seen that the maximum temperature rises gradually until it reaches a steady state.
Fig. 5. The maximum temperatures in the rectangular hybrid GPW varying with time for the incident lights with different frequencies.
Besides, the transient temperature responses are also calculated for different frequency of incident light. As is also shown in Fig. 5, with the frequency of incident light increasing, the maximum temperature decreases. It should be noted that this conclusion is not universal, which will be demonstrated in the next example, since the temperature evolution is dependent on structures and material parameters. The steady-state distributions of temperature increase in the GPW at 30THz are determined in both x-z plane and x-y plane, as shown in Fig. 6. From these results, we can see that the frequency of incident light obviously affects the temperature increase. The reason is that the frequency affects not only the electric field distribution, but also the resistive loss of graphene that produces heating.
Fig. 6. The steady-state distributions of temperature increase in the rectangular hybrid GPW at 30 THz in (a) x-z plane, (b) x-y plane.
Moreover, we change the chemical potential of graphene to investigate the maximum temperature increase. As can be seen from Fig. 7, different chemical potentials lead to different maximum temperature increases. For some chemical potentials, the temperature rise is obvious due to the drastic enhancement of electric field. More specifically, different chemical potentials leads to different conductivities of graphene. For some conductivities, the enhancement of electric field can be much larger than those for the others, which can be regarded as resonance points. The value of the resonance point is closely related to the structure and material of the GPW, and the frequency of light. Generally, GPWs with different structures, materials and operating frequencies exhibit different resonance points. Hence, in practical design, when we change the chemical potential to obtain different optical properties of graphene, the consequent photothermal effect should be taken seriously. We are also required to take the frequency of light into account to avoid excessive temperature increase.
Fig. 7. The maximum temperature increase of the rectangular hybrid GPW varying with the chemical potential of graphene at 30 THz.
Finally, we test the influence of the electron mobility of graphene $\mu$ on the temperature response. Two cases including $\mu =$2,000 and 10,000 $\textrm{c}{\textrm{m}^\textrm{2}}/(\textrm{V} \cdot \textrm{s})$ are analyzed at 30 THz incident light and 500 meV chemical potential. Figures 8(a)–8(c) shows the transient temperature responses and the electric field distributions in the GPW in x-z plane. For comparison, the results of $\mu =$20,000 $\textrm{c}{\textrm{m}^\textrm{2}}/(\textrm{V} \cdot \textrm{s})$ are also plotted in Fig. 8(a). It can be seen that, lower mobility means faster attenuation of electric field along the direction of propagation from Figs. 8(b) and 8(c), and higher temperature increase from Fig. 8(a). The reason is that lower mobility leads to larger propagation loss. Hence, the adoption of graphene with high mobility can reduce the temperature increase.
Fig. 8. The temperatures and electric field distributions in the rectangular hybrid GPW for different electron mobilities of graphene. (a) Transient temperature responses. (b), (c) Electric field distributions in x-z plane for the mobility $\mu =$2,000 and $\mu =$10,000 $\textrm{c}{\textrm{m}^\textrm{2}}/(\textrm{V} \cdot \textrm{s})$.
3.2 Y-type hybrid GPW
The second example analyzes the photothermal effect of a Y-type hybrid GPW. Figure 9 presents the configuration and dimensions of this GPW. The dimensions of three ports, the surrounding temperature and the incident light information are the same as the last example. The chemical potential of the graphene layer is first set to be ${\mu _c} = \textrm{6}00\textrm{ meV}$, and we can obtain a surface conductivity of $\sigma = \textrm{1}\textrm{.6744} \times {10^{ - 6}} - j\textrm{3}\textrm{.7067} \times {10^{ - 4}}$ for graphene. Tetrahedral elements are adopted for the mesh discretization of computational domain in the FEM analysis. The behavior of the SPP wave in the Y-type GPW is first simulated by the proposed method. The frequency of the light is 30 THz. The electric field distributions are presented in both x-z plane and x-y plane. Figures 10(a) and 10(b) shows the calculated ${E_y}$ in x-z plane and x-y plane.
Fig. 9. Configuration of the Y-type hybrid GPW. (a) 3D view. (b) Vertical view and dimensions, where w = 150 nm, L1 = 400 nm, and L2 = 300 nm. The thicknesses of the Si and SiO2 layers are the same as those in the rectangular hybrid GPW example.
Fig. 10. The calculated results of electric field distributions in the Y-type hybrid GPW at 30 THz in (a) x-z plane, (b) x-y plane.
Then, the transient temperature responses are calculated by using the TDFEM. 30 ms total time and 300 time steps are adopted for the time-domain evolution. Figure 11 presents the maximum temperature of the Y-type GPW varying with time. It can be seen that the maximum temperature rises gradually until it reaches a steady state.
Fig. 11. The maximum temperatures of the Y-type hybrid GPW varying with time for the incident lights with different frequencies.
Besides, incident lights with different frequencies are employed to examine the transient temperature responses. Figure 11 also shows the maximum temperature varies with the frequency of incident light increasing. It can be seen that, different from the last example, the maximum temperature does not monotonically decrease with the frequency of incident light increasing. This is because the frequency of incident light relates to the conductivity of graphene, and the SPP-based field enhancement is structure-dependent and material parameter-dependent. The steady-state distributions of temperature increase in the Y-type GPW are also drawn in both x-z plane and x-y plane in Fig. 12.
Fig. 12. The steady-state distributions of temperature increase in the Y-type hybrid GPW at 30 THz in (a) x-z plane, (b) x-y plane.
Moreover, the maximum temperature increase varying with the power density of incident light is investigated. Figure 13 shows the maximum temperature increases as a function of light power density in the Y-type GPW. The rate of this maximum temperature increase is 10.33 $\textrm{K} \cdot$µm2/µW. Hence, the power density of incident light also affects the photothermal effect, which should be determined carefully in practical design and adoption of GPWs.
Fig. 13. The maximum temperature increase of the Y-type hybrid GPW varying with the power density of incident light.
Finally, we study the influence of different buffer materials on the temperature responses. The comparison of transient temperature responses of the Y-type waveguides at 30 THz with different buffer layer materials of Al2O3 and SiO2 are presented in Fig. 14. The permittivity of Al2O3 are from Ref. [49] as ${\varepsilon _r} = 8$, and its thermal parameters are from COMSOL material library as the mass density $\rho = \textrm{3965 kg}/{\textrm{m}^3}$, specific heat capacity $c = \textrm{730 J}/(\textrm{kg} \cdot \textrm{K})$ and thermal conductivity $\kappa = \textrm{35 W}/(\textrm{m} \cdot \textrm{K})$. It can be seen from Fig. 14 that the GPW with Al2O3 buffer layer obtains lower temperature increase compared with that with SiO2 buffer layer. This is mainly due to the much higher thermal conductivity of Al2O3. In practical design, one often needs to test various materials and take into account both the optical and thermal performances. Based on this, the photothermal modeling is useful for guiding the design of GPWs.
Fig. 14. The maximum temperatures of the Y-type hybrid GPW varying with time at 30 THz for different buffer layer materials of Al2O3 and SiO2.
4. Conclusion
A 3D optical and thermal co-analysis of the GPWs is studied in this article. Numerical framework based the FEM are introduced for the photothermal modeling and characterization of GPWs. Transient temperature responses are obtained using the TDFEM. Performance of the proposed method are validated by two examples of hybrid GPWs. The findings presents the main factors that affect the photothermal properties of GPWs, such as the frequency and power density of incident light, and the conductivity of graphene. These findings reveal the physical properties of GPWs from the photothermal perspective rather than conventional optical perspective, and provide insights into the underlying factors influencing the adoption of GPWs for optical interconnect. According to the results, the thermal feature sizes of GPWs can exceed their optical counterparts due to large temperature increase. In these cases, the maximum integration density for GPWs will be limited by their thermal feature sizes even though the GPWs can confine the light beyond the diffraction limit. To reduce the heating, one needs to adjust the parameters of graphene and incident light to reduce resistive loss of graphene which fundamentally forms the heat sources. Although the examples focus on the hybrid Si-SiO2-graphene GPWs, this photothermal modeling method can be extrapolated to other types of GPWs. Besides, the model can be further ameliorated by set periodic heat flux condition in the horizontal sides of the GPW, so that the waveguide is no longer treated as a standalone object, but as an array with a lattice constant, which can be more valid for describing the practical boundary.
Funding
National Natural Science Foundation of China (61601240); Open Research Program Foundation of State Key Laboratory of Millimeter Waves (K201806).
References
1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef]
2. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
3. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]
4. 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]
5. 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]
6. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef]
7. C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17(13), 10757–10766 (2009). [CrossRef]
8. A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009). [CrossRef]
9. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef]
10. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]
11. T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014). [CrossRef]
12. G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557(7706), 530–533 (2018). [CrossRef]
13. S. Cakmakyapan, P. K. Lu, A. Navabi, and M. Jarrahi, “Gold-patched graphene nano-stripes for high-responsivity and ultrafast photodetection from the visible to infrared regime,” Light: Sci. Appl. 7(1), 20 (2018). [CrossRef]
14. D. Ansell, I. P. Radko, Z. Han, F. J. Rodriguez, S. I. Bozhevolnyi, and A. N. Grigorenko, “Hybrid graphene plasmonic waveguide modulators,” Nat. Commun. 6(1), 8846 (2015). [CrossRef]
15. J. T. Kim and S. Y. Choi, “Graphene-based plasmonic waveguides for photonic integrated circuits,” Opt. Express 19(24), 24557–24562 (2011). [CrossRef]
16. L. F. Ye, K. H. Sui, Y. H. Liu, M. Zhang, and Q. H. Liu, “Graphene-based hybrid plasmonic waveguide for highly efficient broadband mid-infrared propagation and modulation,” Opt. Express 26(12), 15935–15947 (2018). [CrossRef]
17. X. T. 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]
18. J. Li, J. Tao, Z. H. Chen, and X. G. Huang, “All-optical controlling based on nonlinear graphene plasmonic waveguides,” Opt. Express 24(19), 22169–22176 (2016). [CrossRef]
19. X. He, T. Ning, R. Li, L. Pei, J. Zheng, and J. Li, “Dynamical manipulation of Cosine-Gauss beams in a graphene plasmonic waveguide,” Opt. Express 25(12), 13923–13932 (2017). [CrossRef]
20. M. Miscuglio, D. Spirito, R. P. Zaccaria, and R. Krahne, “Shape Approaches for Enhancing Plasmon Propagation in Graphene,” ACS Photonics 3(11), 2170–2175 (2016). [CrossRef]
21. Q. Zhang, X. Li, M. M. Hossain, Y. Xue, J. Zhang, J. Song, J. Liu, M. D. Turner, S. Fan, Q. Bao, and M. Gu, “Graphene surface plasmons at the near-infrared optical regime,” Sci. Rep. 4(1), 6559 (2015). [CrossRef]
22. Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, and D. N. Basov, “Edge and Surface Plasmons in Graphene Nanoribbons,” Nano Lett. 15(12), 8271–8276 (2015). [CrossRef]
23. X. Huang, P. K. Jain, I. H. El-Sayed, and M. A. El-Sayed, “Plasmonic photothermal therapy (PPTT) using gold nanoparticles,” Laser Med. Sci. 23(3), 217–228 (2008). [CrossRef]
24. Y. Liu, P. Bhattarai, Z. Dai, and X. Chen, “Photothermal therapy and photoacoustic imaging via nanotheranostics in fighting cancer,” Chem. Soc. Rev. 48(7), 2053–2108 (2019). [CrossRef]
25. X. Huang, P. K. Jain, I. H. El-Sayed, and M. A. El-Sayed, “Photothermal engineering of graphene plasmons,” Phys. Rev. Lett. 121(5), 057404 (2018). [CrossRef]
26. P. Matteini, F. Tatini, and L. Cavigli, “Graphene as a photothermal switch for controlled drug release,” Nanoscale 6(14), 7947–7953 (2014). [CrossRef]
27. Q. Wei, H. Ni, X. Jin, and J. Yuan, “Graphene oxide wrapped gold nanorods for enhanced photo-thermal tability,” RSC Adv. 5(68), 54971–54977 (2015). [CrossRef]
28. K. A. Knapper, E. H. Horak, C. H. Vollbrecht, K. D. Heylman, and R. H. Goldsmith, “Phase-sensitive photothermal imaging of ultrahigh-Q polyoxide toroidal microresonators,” Appl. Phys. Lett. 113(23), 231105 (2018). [CrossRef]
29. M. Zhu, G. Baffou, N. Meyerbröker, and J. Polleux, “Micropatterning thermoplasmonic gold nanoarrays to manipulate cell adhesion,” ACS Nano 6(8), 7227–7233 (2012). [CrossRef]
30. D. S. Lim, M. H. Shin, H. S. Oh, and Y. J. Kim, “Opto-thermal analysis of novel heat assisted magnetic recording media based on surface plasmon enhancement,” IEEE Trans. Magn. 45(10), 3844–3847 (2009). [CrossRef]
31. X. Chen, Y. Chen, M. Yan, and M. Qiu, “Nanosecond photothermal effects in plasmonic nanostructures,” ACS Nano 6(3), 2550–2557 (2012). [CrossRef]
32. Q. Li, W. Zhang, H. Zhao, and M. Qiu, “Two-dimensional analysis photothermal properties in nanoscale plasmonic waveguides for optical interconnect,” J. Lightwave Technol. 31(24), 4051–4056 (2013). [CrossRef]
33. W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. express 13(13), 4818–4827 (2005). [CrossRef]
34. X. Y. Zhang, A. Hu, J. Z. Wen, T. Zhang, X. J. Xue, Y. Zhou, and W. W. Duley, “Numerical analysis of deep sub-wavelength integrated plasmonic devices based on Semiconductor-Insulator-Metal strip waveguides,” Opt. express 18(18), 18945–18959 (2010). [CrossRef]
35. G. Baffou, R. Quidant, and F. J. G. de Abajo, “Nanoscale control of optical heating in complex plasmonic Systems,” ACS Nano 4(2), 709–716 (2010). [CrossRef]
36. G. Baffou, R. Quidant, and C. Girard, “Thermoplasmonics modeling: A Green's function approach,” Phys. Rev. B 82(16), 165424 (2010). [CrossRef]
37. Y. Du, Y. Zhao, Y. Qu, C. H. Chen, C. M. Chen, C. H. Chuange, and Y. Zhu, “Enhanced light-matter interaction of graphene-gold nanoparticle hybrid films for high-performance SERS detection,” J. Mater. Chem. 2(23), 4683–4691 (2014). [CrossRef]
38. J. Wang, Y. Chen, X. Chen, J. Hao, M. Yan, and M. Qiu, “Photothermal reshaping of gold nanoparticles in a plasmonic absorber,” Opt. express 19(15), 14726–14734 (2011). [CrossRef]
39. A. Hatef, S. Fortin-Deschênes, E. Boulais, F. Lesage, and M. Meunier, “Photothermal response of hollow gold nanoshell to laser irradiation continuous wave short and ultrashort pulse,” Int. J. Heat Mass Transfer 89, 866–871 (2015). [CrossRef]
40. B. Gallinet, J. Butet, and O. J. F. Martin, “Numerical methods for nanophotonics standard problems and future challenges,” Laser Photonics Rev. 9(6), 577–603 (2015). [CrossRef]
41. B. Wang, S. Blaize, J. Seok, S. Kim, H. Yang, and R. Salas-Montiel, “Plasmonic-Based Subwavelength Graphene-on-hBN Modulator on Silicon Photonics,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–6 (2019). [CrossRef]
42. D. Chatzidimitriou, A. Pitilakis, and E. E. Kriezis, “Rigorous calculation of nonlinear parameters in graphene-comprising waveguides,” J. Appl. Phys. 118(2), 023105 (2015). [CrossRef]
43. G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Comput. Phys. 103(6), 064302 (2008). [CrossRef]
44. J. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, “Intrinsic and extrinsic performance limits of graphene devices on SiO2,” Nat. Nanotechnol. 3(4), 206–209 (2008). [CrossRef]
45. P. Li, L. J. Jiang, and H. Bagcı, “A Resistive Boundary Condition Enhanced DGTD Scheme for the Transient Analysis of Graphene,” IEEE Trans. Antennas Propag. 63(7), 3065–3076 (2015). [CrossRef]
46. I. Llatser, C. Kremers, A. Cabellos-Aparicio, J. M. Jornet, E. Alarcon, and D. N. Chigrin, “Graphene-based nano-patch antenna for terahertz radiation,” Photonic. Nanostruct. 10(4), 353–358 (2012). [CrossRef]
47. W. C. Chew, Waves and fields in inhomogeneous media. IEEE Press, 1995.
48. J. M. Jin, The Finite Element Method in Electromagnetics, II ed., John Wiley & Sons, Inc., 2002.
49. M. Hajati and Y. Hajati, “High-performance and low-loss plasmon waveguiding in graphene-coated nanowire with substrate,” J. Opt. Soc. Am. B 33(12), 2560–2565 (2016). [CrossRef]
50. W. Du, K. Li, D. Wu, K. Jiao, L. Jiao, L. Liu, F. Xia, W. Kong, L. Dong, and M. Yun, “Electrically controllable directional coupler based on tunable hybrid graphene nanoplasmonic waveguide,” Opt. Commun. 430, 450–455 (2019). [CrossRef]
