In this article, we proposed and numerically studied the surface plasmon polariton whispering gallery mode properties of the graphene coated InGaAs nanowire cavity. The quality factor and the mode area were investigated as a function of the chemical potential, the cavity radius and the wavelength. A high cavity quality factor of 235 is predicted for a 5 nm radius cavity, accompanied by a mode area as small as, when the chemical potential is 1.2 eV. The proposed structure offers a potential solution to high density integration of the nanophotonic devices with an ultra-compact footprint.
© 2014 Optical Society of America
Whispering gallery mode (WGM) in the dielectric disk cavity or ring cavity has been studied over a wide range of applications [1–3]. However, the quality factor (Q factor) of the traditional dielectric microcavity decreases drastically when the cavity radius is scaling down due to the increasing of the radiation loss . On the other hand, surface plasmon polaritons (SPPs) attracted intensive interest in the field of nanophotonic devices [4–7], since lightwave is concentrated to deep subwavelength scales far below the diffraction limit. But the Q factors of plasmonic devices were typically low in both for visible and near-infrared wavelengths due to the ohmic loss [8, 9]. Very recently, many plasmonic WGM structures have been experimentally and theoretically studied [10–13]. Among the reported structures, Min et al. demonstrated experimentally a high-Q plasmonic whispering gallery microdisk fabricated by coating the surface of a high-Q silica microresonator with a thin layer of silver . Later on, Xiao et al. proposed a kind of metal-coated microtoroid supporting high-Q plasmonic whispering gallery mode . Q factor of this structure exceeds 1000 in the near infrared at room temperature, which is close to the theoretical metal-loss-limited Q factors. Nevertheless, the modal volumes of those structures are still in square micrometer scale. Furthermore, most of the devices work in cryogenic condition , which limits their practical applications. Compared with dielectric materials, III-V compound semiconductors offer the desirable advantage of high optical gain for the window of telecommunication due to the direct energy bandgap nature. Active nanophotonic devices are possible as long as the III-V semiconductors are employed as the cavity materials.
Recently, graphene, which is composed of a single carbon atom layer, attracts intensive investigation [15–17] due to its excellent optical properties [18–20]. In a span of appropriate frequency, a single layer of graphene behaves as a very thin ‘metal’ layer, which is capable of supporting transverse magnetic (TM) SPPs along it [21, 22]. Graphene supported SPP waves have the unique properties compared with those supported by noble metals, such as relatively low damping loss and tight confinement on the interface between graphene and the free space. In addition, the properties of the SPP on graphene are controlled by the surface conductivity, which is further determined by the chemical potential, and the chemical potential can be further tuned by the doping or gated voltages. Such tunability has also been experimentally demonstrated and used to control SPPs and explore the dynamics between matter and light in graphene [23–25]. Carrier concentration as high as has been achieved, which corresponds to a chemical potential higher than 1 eV, when the temperature is below 250K [26, 27]. In order to take full advantages of the whispering gallery modes and the high confinement of SPP waves, we propose the optical structure of semiconductor nanowires coated by graphene monolayer instead of noble metals in this work. We also assume that the temperature of the devices is already below 250K to ensure the high chemical potential of the graphene monolayer is achievable. In the cavity with a radius of 5 nm, an effective mode area of (is the free space wavelength) and a Q factor of as high as 235 are achieved in the near infrared spectrum. Additionally, we investigate the influence of the chemical potential of graphene on the nanocavity quality factor for different azimuthal mode number of the plasmonic whispering gallery mode. The Q factor increases with the increasing chemical potential, when the chemical potential of graphene is over.
2. Plasmonic whispering gallery mode of nanowire cavity for different azimuthal number
The system in interest is composed of an infinite long InGaAs nanowire coated by graphene with the radius of r. The properties of the SPP WGMs are numerically studied using the software COMSOL multi-Physics RF module, version 4.3b. The length of the nanowire is infinite in the direction perpendicular to the plane of the WGM. So we only consider 2 dimensional pictures of the proposed structure, and the schematic is as shown in Fig. 1(a). Here, and. It should be pointed out that the graphene is one-atom layer and the “thickness” loses the meaning. So we treat it as a zero thickness layer with surface conductivity [16, 17], which, according to Kubo formula, is determined by the chemical potentials of graphene. Thus, the boundary conditions along the circumference is governed by the Ohm’s law, i.e., where J is the surface current density, E is the electric component of the electromagnetic (EM) field, andis the surface conductivity of graphene. We first investigate the electric field distribution of SPP WGM in the proposed structure. In an InGaAs nanowire cavity with r = 5 nm and chemical potential of graphene μc = 0.9 eV, plasmonic WGM with an azimuthal number (m) of 9 can be excited at a resonant wavelength of 1550 nm in free space. The electric field (Ex ) profiles are shown in Figs. 1(b) and (c). It is obvious in Fig. 1(b) that the electric field is tightly localized on the surface of the graphene-coated cavity. Also, the Ex field profiles of various azimuthal numbers from 2 to 9 are revealed in Fig. 2.The electric field becomes tighter and tighter along the circumference of the nanocavity as the azimuthal mode number increases. The Ex component is anti-symmetric inside and outside the InGaAs resonator due to the TM field nature, while the azimuthal number is different in InGaAs resonator from that in air region owing to the different refractive index and the non-zero current density in the graphene layer. According to the eigenfrequency analysis by COMSOL, we get eigenvalue ,which has an imaginary partrepresenting the eigen frequency of the resonator, and a real part δ representing the damping. The Q factor is defined as28]Fig. 3(a).Figure 3(b) plots the Q factor and the effective area as a function of the azimuthal number m. The Q-factor increases with the increasing the azimuthal number before it reaches the maximum 192.1 at m = 8, and then falls down slightly when m further increases. However, the effective area reduces monotonously with the increasing of the azimuthal number. This trend means that with the increasing of the azimuthal number, the confinement of the EM field becomes tighter and would enhance the EM field intensity at the interface. In general, the loss of plasmonic resonator is composed of radiation loss and metallic absorption loss. With the increasing of the resonance wavelength, the metallic absorption loss decreases, leading to the Q factor going up. While the resonant wavelength increases to a certain degree, radiation loss begins to increase gradually and even exceeds the reduction of absorption loss, resulting in the falling of Q factor . In our proposed graphene-coated InGaAs nanowire structure, the effective mode area is typically smaller than with the resonance wavelength from 1370 nm to 2620 nm, much smaller than the conventional optical whispering-gallery mode with the same wavelength.
As a comparison, we simulate the InGaAs nanowire cavity coated by Au and Ag thin films with a thickness of 2 nm. If the resonance wavelength is kept at 1.55 µm, it is found that the quality factor of the Au-coated nanocavity is around 0.33 with an azimuthal mode number of 2, and a radius of 54.5 nm. The Q factor of the Ag-coated InGaAs nanocavity is 0.14 with the same azimuthal number. For the graphene coated InGaAs nanowire cavity, the azimuthal number m is 109 with the same radius in which the chemical potential is 0.9 eV. This is in stark contrast to the cavity coated with the conventional metal films. Therefore, graphene is much more effective than Au and Ag in achieving a high Q factor for the nanowire resonators.
3. The quality factor of the plasmonic whispering gallery mode cavity with different radius and refractive index of the nanowire and chemical potential of the graphene
We also study the Q factor and the azimuthal number with the change of the radius, when the chemical potential is 0.9 eV. The resonance wavelengths are fixed to 1500 nm (black), 1550 nm (blue) and 1620 nm (purple), respectively. One can see from Fig. 4 that the quality factor is stable at around 192 with all the three resonance wavelengths as the radius increases from 5 nm to 35 nm, while the azimuthal mode number increases linearly. The absorption loss of the SPP mode decreases with the decreasing of the radius due to the reduction of the propagation distance along the circumference. This effect would result in increasing of the quality factor. On the other hand, the confinement capability of the InGaAs cavities without coating degrades as the radius decreases, which would induce a decreasing quality factor. When these two effects balance each other, the total loss during one round trip along the circumference becomes stable, which finally results in stable quality factors of the various wavelengths and radius. In a SPP WGM cavity, the azimuthal number is roughly satisfied by Fig. 4. Interestingly, as for the Q factors of the nanowire cavities, there is no obvious influence from the refractive index of the nanowire material ranging from 3.05 to 3.85. This may suggest that the characteristics of the nanowire cavities be mainly determined by the radius and chemical potential in this regime of refractive index.
Finally, we study the quality factors and the azimuthal mode numbers of the nanowire cavities with the radius r = 5 nm versus the chemical potential at the resonant wavelength λ = 1.56 µm (blue), 1.61 µm (black), which are shown in Fig. 5(a).It is obvious that the Q factor increases with the increasing of the chemical potential while the azimuthal mode number decreases. When the chemical potential reduces to 0.5 eV from higher level, the azimuthal mode number rises sharply and the quality factor is less than 20. As is well known, the propagation loss of graphene supported SPPs is determined by the real part of the surface conductivity of graphene, and finally determined by the value comparison between the chemical potential and half of the energy of the SPP . When the chemical potential is much higher than , the surface conductivity is intraband electron-photon scattering dominated, which results in low loss of SPP mode on graphene. Thus the Q factor of the WGM is high. When the doping level is lowered, the chemical potential is lower and there are more states available for interband electron transitions. As shown in Fig. 5(b), the real part of surface conductivity is therefore gradually dominated by interband electron transitions and the loss increases, which would lead to lower Q factor or even finally disappearance of supported SPP WGM . Indeed, no meaningful mode was found when the chemical potential is lower than half of the photon energy. One should make the chemical potential much higher than half energy of the SPP modes to avoid the high loss, which would increase the quality factor of the nanocavity, as shown in Fig. 5(a). A quality factor as high as 235 is achieved when the chemical potential is 1.2 eV, accompanied by a mode area as small as. In order to understand Fig. 5(a), the effective index of SPP wave as a function of the chemical potential of flattened graphene is plotted in Fig. 6.For a certain frequency, as the chemical potential approaches to, which is around 0.4 eV in our case, the effective index increases drastically, which means the wavelength of the SPP decreases accordingly. Eventually the azimuthal number increases. Albeit the relationship between the effective refractive index and the chemical potential is not exactly the same as that in our WGM case, we can still believe the trend is adoptable to understand Fig. 5(a).
In this paper, we proposed the structure of the graphene-coated InGaAs nanowire cavities and numerically studied the surface-plasmonic whispering-gallery properties. The proposed nanocavities exhibit high Q factors and extremely small mode areas, which are superior to those InGaAs nanowire cavities coated by noble metals. The proposed surface-plasmonic whispering-gallery nanocavity could be one of the key components for a wide range of nanophotonic devices such as single photon source, broadband cavity quantum electrodynamics (QED), subwavelength single photon transistor, and low threshold light sources. In addition, from the theoretical aspect, the hybrid modes exhibit the characteristics of both the whispering gallery mode and surface plasmon mode. This result shows a potential scheme to further shrink the size of nanocavities and improve the device performance simultaneously.
The authors are grateful to the support by the National Science and Technology Major Project under Grant No 2011ZX02708, the National 863 project under Grant No 2012AA012203, the Opened Fund of the State Key Laboratory on Integrated Optoelectronics under grant No. IOSKL2012KF12, and the Opened Fund of the Key Laboratory on Semiconductor Materials under grant No. KLSMS-1201.
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