Tunable Lifetime and Nonlinearity in Two Dimensional Materials Plasmonic-Photonic Absorber

We investigate a framework of local field, quality factor and lifetime for tunable graphene nanoribbon plasmonic-photonic absorbers and study the second order and third order nonlinear optical response of surface plasmons. The energy exchange of plasmonic-photonic absorber occurs in two main ways: one way is the decay process of intrinsic loss for each resonant mode and another is the decay process of energy loss between graphene surface plasmon (GSP) mode and the external light field. The quality factor and lifetime of the plasmonic-photonic absorber can be obtained with using the coupled mode theory (CMT) and finite difference time domain (FDTD) method, which are effectively tunable with changing Fermi energy, carrier mobility and superstrate refractive index. The evolutions of total energy and lifetime of GSP are also shown, which are helpful for the study of micro processes in a two-dimensional material plasmonic-photonic absorber. The strongly localized fundamental field induces a desired increase of second harmonic (SH) wave and third harmonic (TH) wave. The manipulation of the quality factor and lifetime of the GSP makes graphene an excellent platform for tunable two-dimensional material plasmonic-photonic devices to realize the active control of the photoelectric/photothermal energy conversion process and higher harmonic generation.


Field Enhancement and Photon Flux Density of GSP
We concentrated on the properties and behavior of collections of photons, which is investigated by the nature of the GN surface plasmon wave. The absorptions with different width and lattice period using the FDTD simulation are shown here in Figure 1b The anisotropic dielectric tensor is given by ε gra = (ε 11 , ε 22 , ε 33 ). The two components of dielectric tensor in xy-plane are set with ε 11 = ε 22 = ε 0 (1 + iσ gra/ (ε 0 ω∆)). The component with ε 33 = ε 0 was along the z direction. Here, the electric charge is e, and the reduced Planck's constant ish. The vacuum permittivity with ε 0 , Fermi velocity with ν f , Fermi energy with E f , carrier mobility with µ and carrier relaxation time with τ were used. The parameters about graphene nanoribbon were used here with E f = 0.64eV, µ = 1 m 2 /(V·s), ν f = 10 6 m/s and τ = (µE f )/(eν f 2 ). When the incident light S 1,in was coupled into the nanostructured GN grating, the characteristics of the GSP was analyzed using theoretical CMT. The field a m (m = 1, 2, 3) of GSP modes has equation da m /dt = −iω m a m , where the resonance frequency is ω m . The evolution for a m can be described with the theoretical CMT model of the nanostructured GN grating [39,40]: 1/Q m = 1/Q im + 1/Q wm .
where S 1,in , S 2,in , S 1,out , and S 2,out represent the amplitude for incoming and outgoing waves, respectively. |a represents the amplitude of resonant GSP modes and |K is the coupling coefficient between GSP modes and light field. k m stands for the coupling between each GSP mode and external light field. The Ω matrix represents resonant frequencies, the Γ w matrix represents external loss rate, the Γ i matrix represents the intrinsic loss rate, and the M matrix represents coupling coefficients, respectively. The m and n are set to 1, 2 and 3. If n = m, ω mn , γ wmn , γ imn , Q wmn and Q imn are all equal to zero; if m = n, we have the relations ω mn = ω m , γ imn = γ im , γ wmn = γ wm , Q wmn = Q wm , Q imn = Q im , µ mn = 0, γ wm = ω m /(2Q wm ) = 1/(2τ wm ), Q wm = ω m τ wm , γ im = ω m /(2Q im ) =1/(2τ im ) and Q im = ω m τ im . The decay rate γ im represents the intrinsic loss process for m-th GSP mode, and Q im is the quality factor for the corresponding decay process of intrinsic loss. The decay rates γ wm represent the energy coupling loss process between the m-th GSP mode and external light field; and Q wm is the quality factor for this energy coupling loss process. The coupling coefficient is µ mn , which represents the coupling between three resonant GSP, which has the relation µ mn = µ nm . The τ wm and τ im are the lifetime. The relations between the quality factor and lifetime for corresponding processes at the m-th GSP mode have the equations Q wm = ω m τ wm and Q im = ω m τ im . For the GN plasmonic-photonic absorber structure, the total quality factor Q m , the total lifetime τ m and ω m have the relations in Equations (7) and (8).
With the initial condition S 2, in = 0, the reflection function, transmission function and absorption A(ω) with using the CMT method are obtained with: where γ m = −(ω m -ω)i − γ wm − γ im . The χ 1 = iµ 12 , χ 2 = iµ 13 , χ 3 = iµ 23 . G 0 , G 1 , G 2 and G 3 are the function of γ m , χ 1 , χ 2 and χ 3 , We can compare the absorption spectra A(ω) obtained by the CMT theory in Equation (11) with the absorption simulated by FDTD method. With using Q wm = ω m /(2γ wm ) and Q im = ω m /(2γ im ), we can get values such as ω m , Q wm , Q im , τ wm and τ im , respectively. Moreover, the total quality factor Q m and total lifetime τ m for the m-th GSP mode can be calculated.
It is represented that the distributions of the electric field component E z , electric field component E x , and photon flux density Φ at λ 1 = 3.9 µm, λ 2 = 4.35 µm and λ 3 = 4.91 µm are obtained with FDTD simulation in Figure 1d-l, respectively. The distributions of the electric field component E z at λ 1 , λ 2 , and λ 3 are shown in Figure 1d-f, respectively. The distributions of the electric field component E x at λ 1 , λ 2 , and λ 3 are shown in Figure 1g-i, respectively. The distributions of the photon flux density Φ at λ 1 , λ 2 , and λ 3 are shown in Figure 1j-l, respectively. The left and right edges corresponding to the polarization of the incident light have strong local photon flux density Φ at λ 1 = 3.9 µm and λ 2 = 4.35 µm, whose magnitude decays very fast outside the graphene nanoribbons edge in Figure 1j-k. The Φ at λ 2 = 4.35 µm is also seen to be partly localized inside the graphene nanoribbons region due to the short-range interaction in Figure 1k. The Φ with λ 3 = 4.91 µm is seen to be located at the four corners of the graphene nanoribbon, which is a corner effect as shown in Figure 1l. The photon flux density Φ at λ 3 = 4.91 µm has reduced to almost zero inside the center region of graphene nanoribbons.
The amplitude of electric field |E(λ m )| for the m-th GSP mode had a function of r 0 , Fermi energy E f , carrier mobility µ and refractive index n 1 , which can be calculated as |E(λ m , r 0 , E f , µ, n 1 )| with FDTD simulation: where r 0 is a position in the grapheme region. We concentrated on the property and behavior of collections of photons, which is determined by the nature of graphene surface plasmon wave. Monochromatic light of a frequency ω m (m = 1, 2, 3) and intensity I (w/m 2 ) carries a mean photon flux density Φ. The distribution of photon flux density Φ for the m-th GSP mode (m = 1, 2, 3) can be calculated as: For tunable graphene nanoribbons plasmonic-photonic absorber, we can get the tunable ability of electric field |E(λ m )| and photon flux density Φ using Fermi energy E f , carrier mobility µ and refractive index n 1 .
To get more insight into the field localization and corresponding photonic localization, electric field and photon flux density with different modulated parameters E f are studied with FDTD simulation in Figure 2a (10 20 ). For plasmonic-photonic absorber, we can get the tunable ability of electric field amplitude |E(λ m )| and photon flux density Φ using Fermi energy E f , which can be adjusted with applied voltage bias or doping concentration.
shown in Figure 1l. The photon flux density Φ at λ3 = 4.91 μm has reduced to almost zero inside the center region of graphene nanoribbons. The amplitude of electric field |E(λm)| for the m-th GSP mode had a function of r0, Fermi energy Ef, carrier mobility μ and refractive index n1, which can be calculated as |E(λm, r0, Ef, μ, n1)| with FDTD simulation: |E(λm, r0, Ef, μ, n1)| = sqrt(Ex 2 + Ey 2 + Ez 2 ) (12) where r0 is a position in the grapheme region. We concentrated on the property and behavior of collections of photons, which is determined by the nature of graphene surface plasmon wave. Monochromatic light of a frequency m (m = 1, 2, 3) and intensity I (w/m 2 ) carries a mean photon flux density Φ. The distribution of photon flux density Φ for the mth GSP mode (m = 1, 2, 3) can be calculated as: For tunable graphene nanoribbons plasmonic-photonic absorber, we can get the tunable ability of electric field |E(λm)| and photon flux density Φ using Fermi energy Ef, carrier mobility μ and refractive index n1.
To get more insight into the field localization and corresponding photonic localization, electric field and photon flux density with different modulated parameters Ef are studied with FDTD simulation in Figure 2a (10 20 ). For plasmonicphotonic absorber, we can get the tunable ability of electric field amplitude |E(λm)| and photon flux density Φ using Fermi energy Ef, which can be adjusted with applied voltage bias or doping concentration. If we set Fermi level Ef = 0.64 eV and n1 = 1, we can study similar field localization and corresponding photonic localization with different carrier mobility μ. The amplitude of electric field |E(λm)| and photon flux density Φ(λm) at λ1, λ2 and λ3 inside the graphene region with various carrier mobility μ are depicted in Figure 2b,e. The values of |E(λm)|, µ, (f) n 1 at resonant wavelengths λ 1 , λ 2 and λ 3 inside the graphene region with FDTD simulation, respectively.
If we set Fermi level E f = 0.64 eV and n 1 = 1, we can study similar field localization and corresponding photonic localization with different carrier mobility µ. The amplitude of electric field |E(λ m )| and photon flux density Φ(λ m ) at λ 1 , λ 2 and λ 3 inside the graphene region with various carrier mobility µ are depicted in Figure 2b (10 20 ). The electric field |E(λ m )| and photon flux density Φ can be adjusted with the type and quantity of impurities, and working temperature. The corresponding sensors probe the impurity and analyzing temperature [41].

Tuning Quality Factor and Lifetime of GSP
The field localization and corresponding photonic localization can be adjusted with applied voltage bias, doping concentration, impurities, working temperature and refractive index n 1 of superstrate. The quality (Q) factor of the system may vary greatly with different applications and requirements. The system, with an emphasis on damping, only needs a low-quality factor. The Q factor of atomic clock, accelerators, laser or other optical resonators, which need strong resonance or frequency stability, is high. And their Q factor can reach 10 11 or even higher. Trapped light within the radiation continuum has been experimentally measured, and the ability to tune the maximal radiative Q from infinite to finite is an unique property that may be exploited [42]. High Q-factor indicates that the energy loss rate of the oscillator is slow, and the vibration lasts for a long time. It is necessary that the quality factor and lifetime of GSP in plasmonic-photonic devices can undergo tuning.
The energy exchange of plasmonic-photonic absorber has mainly two ways: one way is the decay process of intrinsic loss for each resonant mode and another is the decay process of coupling loss between the GSP mode and the external light field. Q im and τ im are the quality factor and lifetime of the first decay process. Q wm and τ wm are the quality factor and lifetime of the second decay process. For the plasmonic-photonic absorber, the total quality factor Q m and total lifetime τ m can be obtained with the relations in Equations (7) and (8).
The lower mobility in the graphene nanoribbon corresponds to higher loss with fixed Fermi energy E f = 0.64 eV and n 1 = 1. The evolution of optical absorption spectra for different carrier mobility µ is investigated with FDTD simulation as shown in Figure 4a. The resonant modes with λ 1 = 3.9 µm, λ 2 = 4.35 µm and λ 3 = 4.91 µm keep unchanged with different carrier mobility µ. The absorption intensity of the three resonance modes in GSP system possesses the exponential decay with the decrease of carrier mobility µ. The lower mobility corresponds to a higher loss in the GN grating. The fitting parameters Q wm , Q im , γ im , γ wm , τ im and τ wm can be resented by comparing absorption A(ω) with absorption obtained by FDTD simulation. Then we can get Q im , Q wm ,τ wm , τ im , Q m and τ m (m = 1, 2, 3) with various carrier mobility µ from µ = 0.1 to µ = 1 m 2 /(V·s). For graphene plasmonic-photonic structure at m-th GSP mode (m = 1, 2,3), the total quality factor Q m and total lifetime τ m are the function of carrier mobility µ obtained using CMT method in Figure 4b. In Figure 4b, the Q m increase with the increase of µ. The total lifetime τ m increase as the increasing of carrier mobility µ in Figure 4b. With the CMT method, the quality factors Q w2 , Q w3 and lifetimes τ w2 , τ w3 with different carrier mobility µ is shown in Figure 4c. With the CMT method, the quality factors Q i2 , Q i3 and lifetimes τ i2 , τ i3 with different carrier mobility µ is represented in Figure 4d. All the fitting values of parameters for the quality factors are obtained as follows: Q w1 = 7.7 × 10 9 exp(−12µ 1/5 ), Q w2 = 3.2 × 10 3 − 8.9 × 10 3 µ + 1.5 × 10 4 µ 2 − 7.5 × 10 3 µ 3 , Q w3 = 2.2 × 10 3 − 6.1 × 10 3 µ + 1.1 × 10 4 µ 2 − 5.2 × 10 3 µ 3 , Q i1 = 2.3 × 10 −6 + 172µ+ 191µ 2 , Q i2 = 3.4 × 10 −7 + 107µ + 119µ 2 and Q i3 = −1.5 × 10 −7 + 94µ+ 104µ 2 . The theoretical descriptions and data fitting of decay Nanomaterials 2022, 12, 416 9 of 16 rates or lifetimes will make it useful in to apply the methods for the change of carrier mobility µ in future modulated graphene devices. According to the different applications and requirements, the total quality factor Q m and lifetime τ m can be modulated by carrier mobility by changing the type and quantity of impurities and working temperature. and data fitting of decay rates or lifetimes will make it useful in to apply the methods for the change of carrier mobility μ in future modulated graphene devices. According to the different applications and requirements, the total quality factor Qm and lifetime τm can be modulated by carrier mobility by changing the type and quantity of impurities and working temperature. Here, we fix the values of Ef = 0.64 eV and μ = 1 m 2 /(V·s). The evolution of simulated optical absorption spectra for different refractive index n1 is investigated with the FDTD method, as shown in Figure 5a. The fitting values of Qwm, Qim, τwm, τim, γwm and γim can be represented after comparing absorption A(ω) in Equation (11) with that obtained by FDTD simulation in Figure 5a. The three resonance GSP modes have the red-shift with increasing of refractive index n1 in Figure 5a. This quasi-linear response characteristic between the n1 and resonant wavelength is especially valuable for the sensing application of graphene. For the graphene plasmonic-photonic structure at the m-th GSP mode (m = 1, 2, 3), the total quality factor Qm and total lifetime τm are shown as the function of the refractive index n1 of superstrate using CMT method in Figure 5b. In Figure 5b, the total quality factors Qm decreased as the refractive index n1 increased from 1 to 1.8, while the total lifetime τm remained nearly unchanging with the various refractive index n1. With the CMT method, the quality factors Qw2, Qw3 and lifetimes τw2, τw3 with different refractive index n1 is shown in Figure 5c. With the CMT method, the quality factors Qi2, Qi3 and lifetimes τi2 and τi3 with different refractive index n1 are shown in Figure 5d. All the values of quality factors are represented as follows: Qw1 = − 22,375 n1 + 1.1726 × 10 5 , Qw2 = − 784 n1 + 2902, Qw3 = −696 n1 + 2526, Qi1 = 10 n1 + 38, Qi2 = −5 n1 + 17, Qi3 = 4 n1 + 15. The theoretical descriptions and data fitting of quality factors and lifetimes with different n1 of superstrate, such as aqueous solution, will make it useful to apply the methods for future 2D materials modulation and sensing devices. Here, we fix the values of E f = 0.64 eV and µ = 1 m 2 /(V·s). The evolution of simulated optical absorption spectra for different refractive index n 1 is investigated with the FDTD method, as shown in Figure 5a. The fitting values of Q wm , Q im , τ wm , τ im , γ wm and γ im can be represented after comparing absorption A(ω) in Equation (11) with that obtained by FDTD simulation in Figure 5a. The three resonance GSP modes have the red-shift with increasing of refractive index n 1 in Figure 5a. This quasi-linear response characteristic between the n 1 and resonant wavelength is especially valuable for the sensing application of graphene. For the graphene plasmonic-photonic structure at the m-th GSP mode (m = 1, 2, 3), the total quality factor Q m and total lifetime τ m are shown as the function of the refractive index n 1 of superstrate using CMT method in Figure 5b. In Figure 5b, the total quality factors Q m decreased as the refractive index n 1 increased from 1 to 1.8, while the total lifetime τ m remained nearly unchanging with the various refractive index n 1 . With the CMT method, the quality factors Q w2 , Q w3 and lifetimes τ w2 , τ w3 with different refractive index n 1 is shown in Figure 5c. With the CMT method, the quality factors Q i2 , Q i3 and lifetimes τ i2 and τ i3 with different refractive index n 1 are shown in Figure 5d. All the values of quality factors are represented as follows: Q w1 = − 22,375 n 1 + 1.1726 × 10 5 , Q w2 = − 784 n 1 + 2902, Q w3 = −696 n 1 + 2526, Q i1 = 10 n 1 + 38, Q i2 = −5 n 1 + 17, Q i3 = 4 n 1 + 15. The theoretical descriptions and data fitting of quality factors and lifetimes with different n 1 of superstrate, such as aqueous solution, will make it useful to apply the methods for future 2D materials modulation and sensing devices.
How to describe the Figure of merit (FOM) of a sensor? It can be related to the resonance wavelength shifts at certain refractive index n 1 . The sensitivity can be defined as [38,39]. Resonant wavelength variation ∆λ can be changed by the refractive index change ∆n of superstrate environment.
The carrier dynamic of a saturable structure plays an important role to determine how a short pulse can be produced in large-scale vertical bilayer junctions [23]. Let us assume that the incident field is ultrashort pulse here [43]. The short pulse with two different central frequencies ω 2 and ω 3 can be written as E(r,t) = E(r)exp(−(t − t 0 ) 2 /(t d 2 )) (e −iω2t + e −iω3t ), where delay of time is t 0 and t d is 200 fs. How to describe the Figure of merit (FOM) of a sensor? It can be related to the resonance wavelength shifts at certain refractive index n1. The sensitivity can be defined as [38,39]. Resonant wavelength variation Δλ can be changed by the refractive index change Δn of superstrate environment.
The carrier dynamic of a saturable structure plays an important role to determine how a short pulse can be produced in large-scale vertical bilayer junctions [23]. Let us assume that the incident field is ultrashort pulse here [43]. The short pulse with two different central frequencies ω2 and ω3 can be written as E(r,t) = E(r)exp(−(t − t0) 2 /(td 2 )) (e −iω2t + e −iω3t ), where delay of time is t0 and td is 200 fs.
Oscillations and plasmon energy shift are studied in gold nanorods [44]. Experiment of the carrier dynamics in perovskite was observed [45,46]. The transient optical responses and dynamic evolution of the carrier in the 2D materials junction samples can be characterized using femtosecond differential transmission spectroscopy [19]. For thorough investigation of total energy W and lifetime τm for m-th resonant GSP mode, the transient dynamic evolution of optical responses can be investigated here using the FDTD code. The evolution of total energy W and total lifetime τm can be investigated with using the pulse excitation. The relation between the lifetime τm of the m-th resonant mode is the equations τm = W/P. The W is the total energy stored inside the computational region while P is the power radiation out from the grapheme region. ωm represents frequency of the resonant GSP modes. The manipulation of the quality factor and lifetime of the GSP makes graphene an excellent application platform of the photoelectric/photothermal energy conversion process and higher harmonic generation.
First, we investigate the total energy W for m-th resonant GSP mode, which is stored inside the computational region. For the fixed value with μ = 1 m 2 /(V·s) and n1 = 1, the transient dynamic evolution of total energy at wavelengths λ2 and λ3 various with three different Fermi energy Ef from 0.8 eV to 0.4 eV obtained with FDTD simulation are plotted in Figure 6a-c, respectively. The total energy for Ef = 0.40 eV has a more rapidly decay than that of Ef = 0.60 eV and Ef = 0.80 eV. Moreover, the total energy of the mode λ2 has a more rapidly exponential decay than that of λ3. For plasmonic-photonic absorber, we can study the time evolution of total energy and its photon flux density by changing applied voltage bias. For the fixed value of Ef = 0.6 eV and n1 = 1, the time evolution of total energy at λ2 and λ3 with three different carrier mobility μ = 0.1 eV, 0.6 eV and 1 m 2 /(V·s) are represented with FDTD simulation in Figure 6d-f, respectively. The total energy has a more rapid decay when carrier mobility μ decreased from μ = 1 m 2 /(V·s) to μ = 0.1 m 2 /(V·s). The time evolution of total energy with different carrier mobility is helpful to probe the impurity and analyzing temperature. Oscillations and plasmon energy shift are studied in gold nanorods [44]. Experiment of the carrier dynamics in perovskite was observed [45,46]. The transient optical responses and dynamic evolution of the carrier in the 2D materials junction samples can be characterized using femtosecond differential transmission spectroscopy [19]. For thorough investigation of total energy W and lifetime τ m for m-th resonant GSP mode, the transient dynamic evolution of optical responses can be investigated here using the FDTD code. The evolution of total energy W and total lifetime τ m can be investigated with using the pulse excitation. The relation between the lifetime τ m of the m-th resonant mode is the equations τ m = W/P. The W is the total energy stored inside the computational region while P is the power radiation out from the grapheme region. ω m represents frequency of the resonant GSP modes. The manipulation of the quality factor and lifetime of the GSP makes graphene an excellent application platform of the photoelectric/photothermal energy conversion process and higher harmonic generation.
First, we investigate the total energy W for m-th resonant GSP mode, which is stored inside the computational region. For the fixed value with µ = 1 m 2 /(V·s) and n 1 = 1, the transient dynamic evolution of total energy at wavelengths λ 2 and λ 3 various with three different Fermi energy E f from 0.8 eV to 0.4 eV obtained with FDTD simulation are plotted in Figure 6a-c, respectively. The total energy for E f = 0.40 eV has a more rapidly decay than that of E f = 0.60 eV and E f = 0.80 eV. Moreover, the total energy of the mode λ 2 has a more rapidly exponential decay than that of λ 3 . For plasmonic-photonic absorber, we can study the time evolution of total energy and its photon flux density by changing applied voltage bias. For the fixed value of E f = 0.6 eV and n 1 = 1, the time evolution of total energy at λ 2 and λ 3 with three different carrier mobility µ = 0.1 eV, 0.6 eV and 1 m 2 /(V·s) are represented with FDTD simulation in Figure 6d-f, respectively. The total energy has a more rapid decay when carrier mobility µ decreased from µ = 1 m 2 /(V·s) to µ = 0.1 m 2 /(V·s). The time evolution of total energy with different carrier mobility is helpful to probe the impurity and analyzing temperature. Nanomaterials 2022, 11, x FOR PEER REVIEW 11 of 16 For the case μ = 1 m 2 /(V·s) and n1 = 1, the dynamics evolution of lifetimes at wavelengths λ2 and λ3 various with three different values of Ef = 0.4 eV, 0.6 eV and 0.8 eV are represented with FDTD simulation in Figure 6g-i, respectively. The lifetimes can almost reach 1.4 ps for the cases Ef = 0.6 eV and 0.80 eV. The lifetime of the mode λ2 has a more rapid exponential decay than that of λ3. For trapped light within the radiation continuum, the ability to tune the maximal radiative Q from infinite to finite is a unique property that may be exploited [42].
For the ultrafast dynamics of excited carriers in graphene, the time, energy, and momentum-resolved statistical distribution of hot electrons in quasi-free-standing graphene was directly measured after a photoexcitation process, which plays a central role for many electronic and optoelectronic applications [31]. The photoinduced carrier multiplication and carrier density were obtained from the electronic temperature. To study the dynamic evolution of the SP-induced hot carrier in the heterostructure of gold-graphene [44], the time-resolved differential reflection measurements were performed. After photoexcitation, the strongly out-of-equilibrium photocarriers rapidly thermalize distribution. The dominant mechanism for SP induced hot electron generation in the graphene originates from the near-field enhancement of direct photoexcitation in the graphene.

Second-Order and Third-Order Nonlinearity of GSP
The high-efficiency second and third harmonic generation effects have been experimentally investigated in monolayer graphene-based transistors and exfoliated BP [21,22].
Ef=0.6eV Ef=0.8eV For the case µ = 1 m 2 /(V·s) and n 1 = 1, the dynamics evolution of lifetimes at wavelengths λ 2 and λ 3 various with three different values of E f = 0.4 eV, 0.6 eV and 0.8 eV are represented with FDTD simulation in Figure 6g-i, respectively. The lifetimes can almost reach 1.4 ps for the cases E f = 0.6 eV and 0.80 eV. The lifetime of the mode λ 2 has a more rapid exponential decay than that of λ 3 . For trapped light within the radiation continuum, the ability to tune the maximal radiative Q from infinite to finite is a unique property that may be exploited [42].
For the ultrafast dynamics of excited carriers in graphene, the time, energy, and momentum-resolved statistical distribution of hot electrons in quasi-free-standing graphene was directly measured after a photoexcitation process, which plays a central role for many electronic and optoelectronic applications [31]. The photoinduced carrier multiplication and carrier density were obtained from the electronic temperature. To study the dynamic evolution of the SP-induced hot carrier in the heterostructure of gold-graphene [44], the time-resolved differential reflection measurements were performed. After photoexcitation, the strongly out-of-equilibrium photocarriers rapidly thermalize distribution. The dominant mechanism for SP induced hot electron generation in the graphene originates from the near-field enhancement of direct photoexcitation in the graphene.

Second-Order and Third-Order Nonlinearity of GSP
The high-efficiency second and third harmonic generation effects have been experimentally investigated in monolayer graphene-based transistors and exfoliated BP [21,22].
It was found that the local SP can also enhance the second and third harmonic generation (THG). The strongly localized fundamental field induces a desired increase of second harmonic wave and third harmonic wave. Here, we investigated the TH wave and SH wave including SHG as well as the SFG and DFG signals, whose results re calculated by the FDTD simulation in Figures 7 and 8. with frequencies ω1 and ω2: E(r,t) = E2e + E3e . Here, ki (I = 2, 3) is the corre-sponding wave vector. Using the express of E(r,t), we can obtain the polarization of second-order nonlinear P (2) (r,t): (2) :[E2E2e 2i(k2·r−ω2t) + E3E3e 2i(k3·r−ω3t) + 2E2E3e i[(k2+k3)·r−(ω2+ω3)t] + 2E2E3*e i[(k2−k3)·r−(ω2−ω3)t] + cc]+2ε0χ (2) : (E2E2* + E3E3*) (14) The terms in the Equation (15) have the second harmonic generation (SHG) signal at frequencies 2ω2 and 2ω3, difference frequency generation (DFG) signal with (ω3 − ω2) and sum frequency generation (SFG) signal with (ω2 + ω3), respectively. The value of χ (2) can be obtained from Equation (15). The Equation (15) represents the nonlinear optical processes, including SHG, SFG, and DFG. When the fundamental wave (FW) light with two frequencies is incident upon the GN grating, SHW will be excited. It is noted that observation of optical second harmonic generation from suspended single-layer and bi-layer grapheme was experimentally reported [21]. Let us now assume that the incident electromagnetic field is the superposition of two monochromatic plane waves. These incident electric fields with two frequencies ω2 and ω3 can be written as: E(r,t) = E0(r)(e −iω2t + e −iω3t ). For incident wave with λ2 = 4.35 μm and λ3 = 4.91 μm, the Fourier spectrum of Ex propagating away from structure is shown in Figure   Figure 7.   Figure 7b-e, respectively. The distribution of SH photo flux density Φ at wavelength λ4 is seen to be mainly localized inside the graphene nano ribbons center region due to the short-range interaction in Figure 7b. The distribution o SH wave photon flux density Φ at wavelength λ6 is seen to be mainly localized along th short edge region in Figure 7d. The distribution of SH wave photon flux density Φ a wavelength λ7 is seen to be mainly localized along the four-edge region in Figure 7e. Log plots of the SH enhancement factor with three different Fermi energy Ef = 0.4 eV, 0.60 eV and 0.80 eV are obtained in Figure 7f. Log plot of the SH enhancement facto with three different position z = 0, z = 10 nm and z = 100 nm away from the graphene laye with Fermi energy Ef = 0.64 eV are plotted in Figure 7g, respectively.
The third order nonlinear optical property in nonlinear 2D material graphene pla monic-photonic absorber is investigated here. The Kerr effect of the third-order nonlinea polarization of graphene is expressed as following: . (15 In this equation, third order susceptibility χ (3) of graphene can be obtained in nonlin ear 2D material. The graphene lattice with D6h space group is centrosymmetric. A dire implication of this property is that second-order nonlinearity is forbidden. However, non linearity for TH wave is allowed and particularly strong in graphene. The quadratic opt cal nonlinearity of graphene can be described with using the nonlinear optical conductiv ity tensor σ3. The current density of third order nonlinear is j 3nl (r,t) = σ3E(r,t)|E(r,t)| 2 . Th nonlinear conductivity has the form [47]: σ3(ω) = ie 2 Ef/πћ 2 (ω + iτ 1 )] + 3ie 2 (eνf) 2 (1 + α)/(32πћ 2 Ef/ω 3 ). (16 Here, the imaginary of σ3 is negative, which describes the self-focusing type nonlin ear response in graphene. Both linear conductivity σgra in Equation (1) and nonlinear con ductivity σ3 in Equation (17) nonlinear conductivity are highly dependent on fermi en ergy, which could provide a way to get an electrically controlled optical biostability. Afte considering nonlinear effect, total conductivity has the form: Ef=0.4eV Ef=0.6eV The incident electromagnetic field is composed of two monochromatic plane waves with frequencies ω 1 and ω 2 : E(r,t) = E 2 e i(k2·r−ω2t) + E 3 e i(k3·r−ω3t) . Here, k i (I = 2, 3) is the corresponding wave vector. Using the express of E(r,t), we can obtain the polarization of second-order nonlinear P (2) (r,t): The terms in the Equation (15) have the second harmonic generation (SHG) signal at frequencies 2ω 2 and 2ω 3 , difference frequency generation (DFG) signal with (ω 3 − ω 2 ) and sum frequency generation (SFG) signal with (ω 2 + ω 3 ), respectively. The value of χ (2) can be obtained from Equation (15). The Equation (15) represents the nonlinear optical processes, including SHG, SFG, and DFG. When the fundamental wave (FW) light with two frequencies is incident upon the GN grating, SHW will be excited. It is noted that observation of optical second harmonic generation from suspended single-layer and bilayer grapheme was experimentally reported [21].
Let us now assume that the incident electromagnetic field is the superposition of two monochromatic plane waves. These incident electric fields with two frequencies ω 2 and ω 3 can be written as: E(r,t) = E 0 (r)(e −iω2t + e −iω3t ). For incident wave with λ 2 = 4.35 µm and λ 3 = 4.91 µm, the Fourier spectrum of E x propagating away from structure is shown in Figure 7a. There are four GSP modes for second-order nonlinear spectrum with four resonant wavelengths λ 4 = 2.17 µm, λ 5 = 2.31 µm, λ 6 = 2.45 µm, and λ 7 = 38.87 µm in Figure 7a. The SHG modes with λ 4 = 2.17 µm and λ 6 = 2.45 µm are resulted from the FW with wavelengths λ 2 = 4.35 µm and λ 3 = 4.91 µm due to the SHG effect, respectively. The SFG field has the wavelength λ 5 = 2.31 µm. The resonant wavelength is λ 7 = 38.87 µm for DFG field. The distributions of SH photon flux density Φ for λ 4 = 2.17 µm, λ 5 = 2.31 µm, λ 6 = 2.45 µm, and λ 7 = 38.87 µm are represented in Figure 7b-e, respectively. The distribution of SH photon flux density Φ at wavelength λ 4 is seen to be mainly localized inside the graphene nanoribbons center region due to the short-range interaction in Figure 7b. The distribution of SH wave photon flux density Φ at wavelength λ 6 is seen to be mainly localized along the short edge region in Figure 7d. The distribution of SH wave photon flux density Φ at wavelength λ 7 is seen to be mainly localized along the four-edge region in Figure 7e.
Log plots of the SH enhancement factor with three different Fermi energy E f = 0.40 eV, 0.60 eV and 0.80 eV are obtained in Figure 7f. Log plot of the SH enhancement factor with three different position z = 0, z = 10 nm and z = 100 nm away from the graphene layer with Fermi energy E f = 0.64 eV are plotted in Figure 7g, respectively.
The third order nonlinear optical property in nonlinear 2D material graphene plasmonicphotonic absorber is investigated here. The Kerr effect of the third-order nonlinear polarization of graphene is expressed as following: In this equation, third order susceptibility χ (3) of graphene can be obtained in nonlinear 2D material. The graphene lattice with D 6h space group is centrosymmetric. A direct implication of this property is that second-order nonlinearity is forbidden. However, nonlinearity for TH wave is allowed and particularly strong in graphene. The quadratic optical nonlinearity of graphene can be described with using the nonlinear optical conductivity tensor σ 3 . The current density of third order nonlinear is j 3nl (r,t) = σ 3 E(r,t)|E(r,t)| 2 . The nonlinear conductivity has the form [47]: σ 3 (ω) = ie 2 E f /πh 2 (ω + iτ 1 )] + 3ie 2 (eν f ) 2 (1 + α)/(32πh 2 E f /ω 3 ).
Here, the imaginary of σ 3 is negative, which describes the self-focusing type nonlinear response in graphene. Both linear conductivity σ gra in Equation (1) and nonlinear conductivity σ 3 in Equation (17) nonlinear conductivity are highly dependent on fermi energy, which could provide a way to get an electrically controlled optical biostability. After considering nonlinear effect, total conductivity has the form: σ = σ gra + σ 3 |E(r,t)| 2 .
When the FW wave with one frequency is incident upon the GN grating, third-order nonlinearity will be excited. Log plot of the TH enhancement factor with three different Fermi energy E f = 0.40 eV, 0.60 eV and 0.80 eV are shown for mode λ 2 in Figure 8a. TH enhancement factor is about 10 −4 . The THG mode with wavelength λ 2 (blue-dotted line) was excited by the FW wave with wavelength λ 2 (blue-dotted line). Illuminated with an x-polarized plane wave at the fundamental frequency λ 2 , the polarization state of the SH emission for amplitude of electric field (|E|) at λ 2 from an array of two graphene nanoribbons is shown in Figure 8b. The TH signal is a function of the angle (not the polarization of the incident field); the θ = 0 corresponds to the x axis. The log plot of the TH enhancement factor at TH mode λ 8 with three different position z = 0, z = 10 nm and z = 100 nm away from the graphene layer is shown in Figure 8c. Recently, quantum confinement-induced enhanced third-order nonlinearity and carrier lifetime modulation in two-dimensional tin sulfide were observed with Z-scan measurements and fs-resolved transient absorption spectroscopy [48].

Conclusions
We investigate a framework of the local field enhancement, photon flux density, quality factor and lifetime for tunable graphene plasmonic-photonic structure and study the second order and third order nonlinear optical response of grapheme surface plasmons. The quality factor and lifetime for each process of intrinsic loss or coupling loss have been studied. We have investigated the modulated plasmonic-photonic absorber in two graphene nanoribbons grating using Fermi energy, carrier mobility and refractive index. The theoretical descriptions and data fitting will make it useful to apply the methods for future 2D material plasmonic-photonic structures, modulation, and devices application. The modulated 2D plasmonic-photonic absorber results from the enhanced local field. The strongly-localized fundamental field induces a desired increase of TH wave and SH wave, including SHG, as well as the SFG and DFG signals. The proposed configuration and results could provide the guidance for designing quality factor and lifetime modulated 2D material plasmonic-photonic devices and the active control of the photoelectric/photothermal energy conversion process such as solar energy conversion, nanoantenna, higher harmonic generation and high-sensitivity sensing.