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A graphene-embedded tunable plasmonic nanodisk resonator operating at near-infrared wavelength range is proposed, in which a certain resonant mode among multiple whispering-gallery modes (WGMs) can be selected as a dominant mode by modulating the Fermi level of the graphene. Our theoretical investigation reveals that the dominant mode selection mechanism in the proposed resonator is governed by the figure-of-merit (FOM) of the one-dimensional (1D) waveguide of the resonator’s vertical structure, which is defined as a propagation length to mode size ratio. As the conductivity of the graphene changes with a gating voltage, the wavelength dependence of the FOM changes and a WGM closest to the maximum FOM wavelength is selected. Partial tuning of the selected dominant mode is incurred by the change of the effective index of the 1D waveguide. This novel mode selection mechanism of the proposed resonator can be adopted to realize an optically pumped tunable nanolaser with a wide wavelength tuning range.
© 2014 Optical Society of America
1. Introduction
Since the advent of the wavelength division multiplexing revolutionized optical communication systems, realization of an efficient tunable laser device has been one of the primary subjects in optical communication society. In addition, as the on-chip communication systems have been focused on the integration in recent years, compact-sized laser devices were also extensively investigated. However, realization of laser devices having both compactness and wide tunability is challenging [1]. In order to achieve tuning of the resonant wavelength, various approaches, such as gas condensation [2], temperature change [3], optical pump spot change [4, 5], carrier injection [6], micro electro mechanical system (MEMS) technology [7–9], external cavity [10], and mechanical stretching [11], have been reported previously. However, tuning range and speed of the resonant wavelength are fairly limited with these methods. Especially for the MEMS technology or the external cavity structure, which provide a large wavelength tuning range, its drawback is a rather large device volume. In this regard, more efficient tunable resonators with respect to the device size and the wavelength tuning range are demanded, and a graphene-based resonator can be a potential candidate for this purpose. The graphene is a one-atom thick material and its material property can be widely tuned by manipulating its Fermi level, which can be simply realized through electrical gating [12–15]. Graphene-based tunable photonic devices have been extensively investigated and successfully demonstrated in both theory and experiment in various applications including nano-antenna [16], optical imaging [17], metamaterials [18–22], and optical modulators [23–26]. The tuning can be further enhanced via surface plasmon effects of the graphene [27–33]. If the graphene of this great tunability is combined with a nano-resonator, an electrically tunable laser cavity of both the compact size and the wide tuning range can be realized. So far, to the best of our knowledge, the graphene-based tunable resonator has not been reported.
In this work, we propose a graphene-embedded tunable plasmonic resonator, which consists of a monolayer graphene embedded in a plasmonic resonator structure. It is shown that the monolayer graphene in the proposed resonator enables to significantly modify a resonant mode spectrum and selectively enhance an intensity of a particular resonant mode by properly adjusting the graphene Fermi level. One-dimensional (1D) analysis of the proposed resonator reveals that such resonant mode selection is incurred at the wavelength closest to the maximum figure-of-merit (FOM) rather than at the wavelength showing the lowest loss. The selected dominant mode shows a wide tuning range of ~750 nm in near-infrared wavelength via Femi level change from 0.48 eV to 0.85 eV. These novel characteristics of the proposed resonator are believed to be useful for the realization of widely tunable lasers through the applied gate voltage. In this work, for the theoretical study of the proposed resonator was carried out by using the three-dimensional (3D) finite-difference time-domain (FDTD) method (Lumerical FDTD Solutions 8.0), and the 1D waveguide analysis was carried out by solving Maxwell’s equations analytically.
2. Structure and resonant mode spectrum of a graphene-embedded plasmonic resonator
Figure 1(a) shows a nanodisk hybrid plasmonic resonator, which consists of a metal disk on top of a dielectric slab.In this structure, the coupling between the surface plasmon polariton (SPP) at the Ag-SiO2 interface and the slab waveguide mode of the SiO2-InGaAs-InP structure results in a hybrid plasmonic mode traveling along a x-y plane. This hybrid plasmonic mode confines most of field in the SiO2 layer, so that both the loss and the modal confinement are simultaneously improved compared to conventional plasmonic modes [34]. Due to the cylindrical symmetry of the resonator, the traveling hybrid plasmonic mode forms whispering-gallery-modes (WGMs) in the horizontal direction. Figure 1(b) shows the proposed resonator structure in which an additional monolayer graphene is embedded between the SiO2 and the InGaAs layers. Due to the strong field confinement in the SiO2 layer, the resonant modes in the proposed structure can be strongly affected by the graphene, and at the same time, the proposed structure facilitates an effective control of the carrier concentration (Fermi level) of the graphene by applying a gate voltage. If the InGaAs layer and the InP substrate are properly doped, a gate electrode for the graphene can be placed on the InP as depicted in Fig. 1(b). Since we focus on the resonant mode properties of the proposed resonator in this work, In0.2Ga0.8As of 1.24 eV bandgap (λg ~1 μm) is used for the high-index layer and the proposed structure is treated as a passive resonator in the telecommunication wavelength range. For a laser operation, In0.53Ga0.47As or InxGa1-xAsyP1-y, which has an optical gain in the telecommunication wavelength range and is lattice matched to InP, will be used as the high index layer. The dominant mode control concept proposed in this work, which will be discussed later, does not depend on the material of the high index layer as far as the hybrid plasmonic mode is supported.
Fig. 1 A schematic of the resonator (a) without and (b) with a monolayer graphene. SiO2 is placed between the metal and the graphene so that a capacitive effect is induced. A gate electrode for the graphene is placed on the InP substrate with properly doped InGaAs and InP assumed. The calculated resonant mode spectra of the resonator (c) without and (d) with the graphene layer. Insets show field (Ez) profiles of each resonant mode in the horizontal and the vertical directions. Azimuthal modal number N and quality factor Q are also denoted for each resonant peak. For the vertical field profile, the 1D field profile is also plotted. The graphene is assume to be neutral and geometrical parameters of tM, tL, tG, and tH are fixed to 200 nm, 5 nm, 1 nm, and 300 nm, respectively.
Thickness of the monolayer graphene is assumed to be 1 nm (tG = 1 nm) [14, 15, 17, 25, 26] and the rest of the geometrical parameters are set to be tM = 200 nm, tL = 5 nm, tH = 300 nm, and R = 400 nm throughout this work. The refractive indices of InP, In0.2Ga0.8As, and SiO2 are assumed to be nInP = 3.14, nInGaAs = 3.72, and nSiO2 = 1.45, respectively. Since the wavelength range of main interest in this work is enough below the bandgap of In0.2Ga0.8As (λg ~1 μm), the fixed index assumption (nInGaAs = 3.72) is valid enough. The Drude model is taken into account for the silver permittivity [35, 36], such that
where the background dielectric constant ε∞, the plasma frequency of silver ωp, and the collision frequency at room temperature γ are 3.1, 1.4x1016 s−1, and 3.1x1013 s−1, respectively. For the Fermi level dependent permittivity of the graphene, we used the general expression of the graphene conductivity referred in several previous works [14–17, 27], which is derived from the Kubo formula under the random phase approximation in the local limit, such thatwithkB is the Boltzmann constant, T is the temperature, ω is the operating angular frequency, EF is the Fermi level of graphene, e is the elementary charge, and τ is the carrier relaxation time defined as τ = μEF/evF [27–29, 31, 33], in which μ is the mobility whose value is fixed to 104 cm2/Vs [27–29, 31, 33] and vF is the Fermi velocity that is fixed to 2.5 × 106 m/s [37]. The first term of the graphene conductivity is attributed to intraband transitions that become dominant when the Fermi level of the graphene is increased higher than half of the incident photon energy, and the second term represents interband transitions that become dominant when the Fermi level is smaller than half of the incident photon energy. The Fermi level dependent permittivity of the graphene can be derived from the equation [15, 16, 21, 22, 26]where εo is the vacuum permittivity and tG is the thickness of the monolayer graphene. Since the equation of the graphene conductivity includes the integral form of the interband transition that cannot be analytically solved, the equation is needed to be numerically treated to obtain the conductivity values (Wolfram Mathematica 8.0 is used.). In the numerical derivation of the graphene conductivity with the given equation, however, numerical error becomes large as EF approaches to 0 eV. Therefore, the graphene conductivity of the neutral state is replaced with that of EF = 0.01 eV in this work [30].Figures 1(c) and 1(d) show the calculated resonant mode spectra of the resonators depicted in Figs. 1(a) and 1(b), respectively. The FDTD method was used to calculate the spectra. The perfectly matched layer was used for the calculation domain boundaries and the non-uniform mesh ranging from 0.2 nm to 20 nm was used. In the calculation, all the resonant modes present in the resonator were excited by a transverse magnetic (TM) polarized dipole source of broad bandwidth (a temporally short pulse) and let to evolve in time for long enough. Note that the hybrid plasmonic mode supports only TM polarized mode. The dipole source was located at the center of the InGaAs layer in the vertical direction and 10 nm away from the edge in the horizontal direction. Fields at several different positions were used to calculate the spectra by taking the discrete Fourier transforms. The obtained spectra were then normalized with the source spectra, so that the mode of a higher Q, which decays slowly, shows a relatively higher peak intensity in the spectra. Mode profile (Ez), azimuthal number (N), and Q of each resonant mode are represented in insets of Figs. 1(c) and 1(d). For the field profiles, the horizontal field profiles were obtained along the center of the SiO2 layer, and only the vertical field profile for the mode of N = 4 was plotted since all the resonant modes show very similar vertical field distributions. Due to the intrinsic loss of the graphene, the linewidths of the resonant peaks are broadened by the introduction of the graphene layer while the peak positions are almost unchanged. It should be noted that embedding the graphene significantly changes the relative peak intensity distribution of the resonant mode spectrum. For example, the intensity of N = 9 resonant mode having the largest intensity in Fig. 1(c) is greatly suppressed by introducing the graphene and that of N = 6 mode becomes the largest in Fig. 1(d). This implies that the resonant modes in the proposed resonator are highly affected by the graphene. Therefore, it is expected that the properties of the resonant modes can be dynamically varied with the change of material properties of the graphene.
3. Dominant mode selection in a graphene-embedded plasmonic resonator
Figure 2(a) shows a resonant mode spectrum of the proposed resonator for the increased graphene's Fermi level of EF = 0.6 eV. Quality factor Q and azimuthal number N are denoted for each resonant peak, and insets show horizontal and vertical field (Ez) profiles for the resonant mode of N = 4. The spectrum was obtained in the same way with the same source as the previous calculation, and only the permittivity of the graphene was changed. One can see that the peak intensity and Q of N = 4 resonant mode is remarkably enhanced and thus, becomes a dominant mode among the azimuthal modes for EF = 0.6 eV. In this calculation, only the passive characteristics of the resonator are considered without optical gain, and the relative peak intensities in the spectrum just reflect the relative magnitudes of Q of the resonant modes. However, if proper optical gain is provided for a laser operation with optical pumping, the dominant mode will be selected as a lasing mode obviously. So, it seems that the mode selection property of the proposed resonator can be used to tune the lasing wavelength by modulating the graphene's Fermi level. Since Q of the dominant mode of the proposed resonator is higher than or comparable to those of the previously reported plasmonic lasers [38, 40, 41] and the active layer is thicker than those plasmonic lasers, its lasing with optical pumping seems to be promising. In the nanopatch laser case [38], 200 nm thick InGaAsP active layer enabled a lasing with a lower Q of ~80. Although light extraction from the proposed resonator based laser would not be a problem as in the case of the nanopatch laser, more efficient light extraction from the WGM may be achieved by placing a waveguide or a metallic wire close to the resonator.
Fig. 2 Resonant mode spectrum of the proposed resonator for EF = 0.6 eV. The spectrum is plotted in both (a) a linear scale and (b) a logarithmic scale. The spectrum for the neutral graphene is also plotted in (b) for a reference. Insets show resonant mode profiles (Ez) of the selected dominant mode in the horizontal and the vertical directions. The horizontal mode profile is obtained along the center of the graphene layer, and a magnified-plot of the vertical mode profile shows the strong field confinement in both the SiO2 and the graphene layers.
Figure 2(b) shows the spectrum of Fig. 2(a) in a logarithmic scale, where the spectrum for the neutral graphene is also plotted for a reference. The suppressed resonant peaks are clearly seen in Fig. 2(b). This implies that the resonant modes of relatively small Q in Fig. 2(a) do not disappear but are suppressed down. From the enlarged vertical field profile in the inset, it is also noted that in the case of EF = 0.6 eV, the field confined in the graphene layer is enhanced in comparison to the neutral graphene case. This enhanced confinement in the graphene layer contributes to the increased Q of N = 4 resonant mode of the resonator compared to the neutral graphene.
The intensities of the resonant peaks are somewhat dependent on the locations of the field monitors in the FDTD calculations. However, it was verified that the selected dominant mode is not varied with respect to the resonant wavelength and Q, and thus, is almost independent of the locations of the field monitors even though the relative resonant peak intensities are slightly changed.
In order to understand the dominant mode selection mechanism in the proposed resonator, we investigated the influence of the graphene's Fermi level change on the resonant mode properties in detail. First, we investigated the effect of the graphene's dispersive characteristic. Figure 3(a) shows absolute values of graphene's refractive indices (|n|) for the neutral graphene and EF = 0.6 eV as a function of wavelength. For EF = 0.6 eV, |n| shows a minimum value at λ = 1.372 μm, while showing monotonic increase for the neutral case. Black and blue dashed lines indicate the wavelengths of the resonant modes in the proposed resonator with EF = 0.6 eV; the black one corresponds to the wavelength of the selected dominant mode (N = 4) and the blue ones correspond to the wavelengths of the adjacent modes (N = 3 and 5). In a previous research on a graphene-embedded plasmonic waveguide device, it was shown that |n| played an important role for the device performances [26]. However, as one can see in Fig. 3(a), it does not seem that |n| has any direct correlation with the dominant mode selection in our resonator.
Fig. 3 Property of the 1D waveguide structure of the proposed resonator. (a) Absolute values of graphene’s refractive indices for the neutral graphene and EF = 0.6 eV as a function of wavelength. (b) Schematic of the 1D waveguide structure. Electric field (Ez) distributions at (c) λ = 1.22 μm, (d) λ = 1.517 μm, and (e) λ = 1.85 μm in EF = 0.6 eV case. (f) Propagation length Lp and (g) FOMs Lp/Am as a function of wavelength for EF = 0.6 eV. Black dashed lines in (a), (f), and (g) indicate the wavelength of the selected dominant mode and blue dashed lines indicate the wavelength of the resonant modes of N = 5 (λ = 1.22 μm) and N = 3 (λ = 1.85 μm).
Then, we have analyzed the characteristics of the one-dimensional (1D) waveguide of the same vertical structure as the proposed resonator since the performance of the WGM is strongly dependent on the wave propagation characteristics in the horizontal direction [34]. Figure 3(b) depicts the 1D waveguide structure and its geometrical parameters are fixed to tm = 200 nm, tl = 5 nm, tg = 1 nm, and th = 300 nm. Figures 3(c)-3(e) show the calculated field (Ez) profiles at the resonant wavelengths of N = 3, 4, and 5 (corresponding to the modes at λ = 1.22 μm, 1.517 μm, and 1.85 μm) in EF = 0.6 eV case, respectively. The well-confined guided modes are found at all 3 resonant wavelengths as expected. The sign of the real part of the graphene's permittivity changes at λ = 1.372 μm, so that the signs of the fields in the graphene layer also change accordingly. Although the guided mode at the resonant wavelength of N = 4 mode shows rather higher field confined in the graphene layer, no distinct difference is found among the 3 field profiles at a glance. So, for more detailed analysis, propagation lengths (Lp = (2Im[β])−1) and figure-of-merits (FOMs) defined as Lp/Am [34], where Am is a mode size, for the 1D waveguide with EF = 0.6 eV have been calculated and plotted as a function of wavelength in Figs. 3(f) and 3(g), respectively. The mode size of the 1D waveguide is defined as the ratio of the total energy to the maximum energy density such that [34]
where W(ω, z) is the electromagnetic energy density,with εR being the real part of the permittivity [39]. Since a well-confined guided mode disappears at λ = 1.372 μm for the EF = 0.6 eV case, where |n| becomes close to zero, the propagation length (Lp) becomes zero. This is why there are dips in Figs. 3(f) and 3(g). In Fig. 3(f), one can see that the propagation length shows a higher value at λ = 1.85 μm (N = 3 resonant mode) than at λ = 1.517 μm (N = 4 resonant mode, the dominant mode). This implies that the loss by itself is not the major factor in the dominant mode selection mechanism of the resonator. Whereas, in Fig. 3(g), it is noted that the resonant wavelength of the selected dominant mode is very close to the maximum of the FOM and thus, the dominant mode experiences the highest FOM among all the resonant modes. Therefore, it seems that the FOM of the 1D waveguide structure is the major factor in the dominant mode selection in our resonator. This is in good agreement with the previous finding in [34] that Q of a resonant mode in a metal nanodisk based plasmonic resonator is closely related to the FOM of the 1D waveguide structure. Both the vertical modal confinement and the propagation loss affect Q of the resonator. As seen in Fig. 3(d), at the wavelength of the selected dominant mode (λ = 1.517 μm), the field is more concentrated in the graphene layer resulting in the stronger vertical confinement. At the same time, the loss is rather higher due to the graphene's loss resulting in the smaller propagation length than the case of λ = 1.85 μm as seen in Fig. 3(f). Somehow, at λ = 1.517 μm, the enhanced vertical confinement effect compensates for the increased loss effect and the optimum condition for maximizing Q of the resonator is obtained under EF = 0.6 eV. The FOM of the 1D waveguide includes both the modal confinement and the propagation length effects and that is why its maximum is closely related to the dominant mode selection of the resonator.To clearly verify the role of the FOM of the 1D waveguide in the dominant mode selection in the proposed resonator, for various values of graphene's Fermi levels, the resonant mode spectra in the resonator and the FOMs of the corresponding 1D waveguide have been calculated and plotted in Figs. 4(a) and 4(b). For a reference, the calculated propagation lengths are also plotted in Fig. 4(c). In Fig. 4(a), one can see that the mode of N = 4 is selected as a dominant mode for 0.58 eV ≤ EF ≤ 0.62 eV, and for 0.66 eV ≤ EF ≤ 0.72 eV, the mode of N = 5 becomes a dominant mode. Within a certain range of EF in which the resonant mode of the same azimuthal number is selected as a dominant mode, the resonant wavelength of the dominant mode shows a blue shift due to the increase of |n| with increasing EF. For a further change of EF above the range, the resonant mode of the next azimuthal number is selected as a dominant mode.
Fig. 4 Dominant mode wavelength tuning with various values of graphene’s Fermi levels ranging from EF = 0.58 eV to 0.72 eV. (a) Resonant mode spectra of the proposed resonator, (b) FOMs and (c) propagation lengths of the corresponding 1D waveguide as a function of wavelength for various EF. Insets of (a) show resonant mode profiles (Ez) for the modes of N = 4 and N = 5 at the wavelength near λ = 1.5 µm and 1.3 µm, respectively.
By comparing FOMs and propagation lengths between the resonant modes of N = 4 and 5 for each EF, one can find that the resonant mode with a higher FOM is always selected as a dominant mode. For example, in the case of EF = 0.66 eV (red curves in Figs. 4(a)-4(c)), the resonant mode of N = 5 is a dominant mode due to the higher FOM instead of the smaller Lp than the resonant mode of N = 4. Therefore it is clear that the dominant mode selection in the proposed resonator is governed by the FOM of the 1D waveguide structure rather than the loss alone. When EF ~0.64 eV, which is not shown here, it was found that both modes of N = 4 and 5 show peaks of comparable intensities and the dominant mode is not clearly determined since those two modes experience FOMs of very close values. In this case, if the resonator is pumped for a laser operation, it will show mode hopping behavior.
In the proposed resonator, there can be a forbidden wavelength range for the dominant mode as shown in Fig. 4(a). The embedded graphene does not produce a new resonant mode, but only selectively enhances Q of a particular resonant mode. So, if the wavelength interval between two adjacent azimuthal modes is large enough, the forbidden wavelength range is inevitable. However, if the wavelength interval between azimuthal modes is narrowed sufficiently by increasing the diameter of the resonator, almost continuous tuning of the dominant mode wavelength in between two adjacent azimuthal modes may be possible via the change of graphene's |n|.
In order to confirm the dominant mode selection in our passive resonant mode calculation, the dependence of the source center wavelength has also been investigated. Even in the case that the source center wavelength very close to a resonant mode of a smaller FOM was used, the calculated resonant mode spectrum showed the dominant mode selection with a higher FOM.
We further explored a dominant mode selection in a wide wavelength range over for a large EF change. Figures 5(a) and 5(b) show the calculated resonant mode spectra and the FOMs of the corresponding 1D waveguide for various EF. As EF changes from 0.48 eV to 0.85 eV, a mode close to the FOM maximum is selected as a dominant mode and the azimuthal mode number of the dominant mode varies from N = 3 to 6, resulting in a wide wavelength tuning ranging from 1.12 μm to 1.87 μm.
Fig. 5 Dominant mode wavelength tuning over a wide wavelength range with a large EF change. (a) Resonant mode spectra of the proposed resonator and (b) FOMs of the corresponding 1D waveguide for various EF ranging from EF = 0.48 eV to 0.85 eV. Insets of (a) show horizontal mode profiles (Ez) of each selected dominant peak.
In the proposed resonator, the Fermi level of the graphene layer is modulated by a gate voltage. The gate voltage required to obtain a certain value of EF can be estimated from the induced carrier density of the graphene layer (NG) and the capacitance between the metal disk and the graphene. The graphene’s carrier density is related to EF as NG = EF2/(πħ2vF2), where ħ is the reduced Plank constant and vF is the Fermi velocity. From the simple capacitive model, the gate voltage (VG) is given by VG = qNGtL/εoεSiO2. For example, EF = 0.85 eV requires VG = 3.64 V, which corresponds to the electric field of ~7.28 MV/cm in the oxide region. This is less than the SiO2 breakdown voltage of ~13 MV/cm [42].
The focus of this work is just to demonstrate the dominant mode selection mechanism through graphene's permittivity modulation. Therefore, in this work, resonator structure optimization to minimize the gate voltage for the dominant mode tuning is not performed. In this regard, if the resonator is properly designed, the required gate voltage can be further reduced and the tuning wavelength range can also be increased.
4. Conclusion
In this work, we proposed a graphene-embedded plasmonic resonator for a tunable laser and investigated its dominant resonant mode selection mechanism. It was shown that a particular mode out of WGMs supported in the resonator can be selected as a dominant mode with the highest Q by manipulating the graphene's Fermi level. It was also revealed that the dominant mode selection is determined by not only the wave propagation loss in the horizontal direction but also the vertical mode confinement. Therefore, the WGM which experiences the highest value of FOM defined as a propagation length to mode size ratio is selected as the dominant mode. A wide tuning range of a dominant mode wavelength from 1.12 μm to 1.87 μm was numerically demonstrated for EF changes from 0.48 eV to 0.85 eV in the proposed resonator. This novel dominant mode selection of the proposed resonator will lead to realization of nanolasers with an extremely wide wavelength tunability.
Acknowledgment
This work was supported by National Research Foundation of Korea Grant (NRF-2014-006720).
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