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We propose a novel plasmonic photodetector with high responsivity, utilizing nano-scale active regions. This design can be applied to diverse materials (group III-V or IV materials) and different operation wavelengths covering the O-U bands. The periodic structure utilizing Surface Plasmon Polariton Bloch Waves (SPP-BWs) has low optical power loss. FDTD simulation shows an absorptance of 74.4% which means a responsivity of about 0.74 A/W at 1550 nm. The low capacitance brings low noise, reduced power consumption, and a high electrical bandwidth which is estimated to be 140 GHz. Among the plasmonic PDs with inherent high speeds but low responsivities, our design makes the obvious progress on improving the absorptance.
© 2016 Optical Society of America
1. Introduction
Plasmonics, which utilize materials supported surface plasmon modes to substantially reduce optical wavelengths compared to free space, has drawn much attention from researchers [1]. Despite the parasitic absorption associated with plasmonics, certain devices can benefit from the resulting increase in intensity, such as photodetectors (PDs). The plasmonic concentrators and resonators can couple light into deep sub-wavelength volume below the diffraction limit. They makes use of optical antennas to capture energy from free space [2]. The enhancement and confinement of light on the nano-scale active region can provide a plasmonic PD with good electrical performance [3].
Periodic metallic structures give the momentum compensation for converting the incoming light to the Surface Plasmon Polariton Bloch Waves (SPP-BWs) [4–6].The bullseye grating surrounds a single small active region and converts the large area illumination into the Surface Plasmons (SPs), as a consequence the aperture where the active region locates gets intense electromagnetic fields [7, 8]. The efficiency of dots-in-a-well (DWELL) detector increases when the interaction length between the incoming light and the active material is extended with plasmonic crystals [9–11]. Another kind of nanopillar PD which benefits from the SPP-BWs has good performances and ingenious fabrication procedure [12].
An alternative plasmonenhanced scheme is the localized surface plasmon resonance (LSPR) [13] which takes advantage of the nano-scale metallic structures. Metal− semiconductor −metal (MSM) PDs consisting of deepsubwavelength volumes and resonant antennas show integration advantage and high polarization sensitivity [14, 15]. Arrays of metal nanodipole antennas with active volumes located in gaps are reported to have good performances [16]. Besides, the extraordinary optical absorption (EOA) is found in the resonant plasmonic nanostructures [17]. A nanopillar-based plasmonically-enhanced photodetectors (NP-PEPDs) utilizing both SPP-BWs and LSPRs have been reported [18].
The most important criterions of PD are responsivity, 3dB modulation bandwidth, and dark current. To remove the trade-off between responsivity and bandwidth of the traditional PD, the waveguide photodetectors are researched with good performances, e.g., the device in [19] has a responsivity of 0.74 A/W at 1550 nm and a 3-dB bandwidth of 67 GHz. The tiny active regions of plasmonic PDs may lead to high speed and low dark current, but it also brings large optical power loss. The application of the plasmonic PDs needs obvious responsivity improvement.
In this work, our novel plasmonic PD design focuses on the responsivity improvement. The periodic structure with short-cycle and large active regions has a large absorptance. Since the area sizes are kept in nano-scale, the degraded 3dB bandwidth decided by the capacitance remains over 140 GHz. Then we get a plasmonic PD with high responsivity as well as high speed. In Section 2, we discuss the evaluation criterions for the light concentration PDs and propose the design principle. Section 3 contains the parametric study with physical mechanism discussions and a final optimized example device under 1550 nm operation. In Section 4, we demonstrate that the design is feasible by presenting the device-to-fiber coupling analysis, electrical bandwidth calculation, and fabrication procedure.
2. Discussion and operation principle
Normalized absorption cross section (NACS) has been widely used for light-concentrating PDs, which is given by
where ω is the angular optical frequency, imag(ε) is the imaginary part of the dielectric permittivity of absorption material, Isource is the intensity of incident light, is the electric field intensity, vabs the volume of the active region, and Aabs the cross-section area of the active region. NACS measures the ratio between the absorbed power in the active region and the incident power on the active area. It is obviously that the absorptance η iswhere Asource is the area of source and RA is the ratio between the areas of active region and source. The optical field distribution in the active region reflects the light concentration performance, and the Average Electric field intensity (AE) in the active region normalized by the source amplitude Esource is given byAE evaluates the light concentration effects with the influence from absorption property of material removed. Large NACS (and AE) is good for the plasmonic PDs. However, that may leads to low η, e.g., the PDs with bullseye metallic grating can get a large NACS which benefits from a large Asource, but relatively low η since it is hard to reduce the power loss on so large illumination area. The periodic active region designs utilizing the SPP-BWs can have the larger RA and η, since the active regions in different units absorb the in-plane resonance waves, the optical scattering loss can be much reduced with a proper design. A high absorptance has been predicted by achieving critical coupling in the analytical study [9].From the point of view of application, the plasmonic PDs with nano-scale active regions have inherent advantages in electrical bandwidths and dark currents. In this work, we argue that the key points are improving the responsivity (to reach the commercial p-i-n PDs), coupling to the commercial fibers, and fabrication feasibility.
We propose and investigate a periodic plasmonic photodetector with nano-scale active regions and high responsivity. As shown in Fig. 1, the top metal layer with height hm is etched by periodic circles arranged into a two-dimensional square array. P denotes the period and wetch denotes the radius of the etched circles. In one unit cell, the active region placed in the center has horizontal sizes wx and wy. hd denotes the height of active region in the dielectric layer. The dielectric layer sandwiched between the top metal layer and the substrate acts as an electrical isolation layer and supports the SPP-BWs. hsub denotes the height of substrate ridge, which is embedded into the dielectric layer.
Fig. 1 Schematic of the proposed device: (a) overview; (b) top view; (c) structure of a unit cell; (d) cross-section in x-z plane.
The periodic metallic structures provide the necessary momentum conservation for the coupling from incident light to the SPP-BWs on the interface between the metal and the dielectric. The enhanced near field [6] which benefits from the resonance waves leads to light concentration in the active regions. Besides, the interactive length between the incident light and the active region increases as Fig. 1(a) shows. Assuming the active regions as perturbation, the face-centered structure array can be approximated as a square hole array, its absorption resonance properties can be predicted from the SPP-BWs analysis [10, 18]. However, this approximation is not precise enough, and the actual resonance wavelength is longer than the analytic prediction. In other words, we can get a short period for the certain operation wavelength. A short period results in a relatively large RA (RA = wx × wy/P2), which makes high absorptance η possible. In addition, the bandstructure calculations are used for precise analysis.
With optimization of the geometric parameters, we can get high NACS, AE, and η (the key criterion). The metal material options are Au, Al, and Ag. The dielectric material can be Si3N4 or SiO2. Both InGaAs (substrate: InP) and Ge can be applied to this design, and we choose the former one in presenting this work. In the process of material choosing and parametric study, our investigation concentrates on how to improve the responsivity.
3. Simulation results and parametric study
The finite difference time domain (FDTD) method [20] is used for all simulations. Optical properties of Au and Ag come from Johnson and Christy [21], Palik’s material data [22] are used for Al and InP. InGaAs is from [23], and Si3N4 from [24]. The boundary conditions are periodic boundary conditions in the x- and y- directions and Perfectly Matched Layer (PML) boundary conditions in the z-direction .The incident illumination is a x-polarized plane wave source in the negative Z direction above the top metal layer. A convergence analysis is presented in the appendix.
3.1 Materials choosing
For the two-dimensional square array, the well-known momentum-matching condition for coupling the incident radiation to the SPP-BW modes is
where λ is the resonance wavelength, εm and εd are respectively the dielectric constants of the metal and the dielectric material in contact with the metal, and i, j are the scattering orders of the array. The most widely used plasmonic metals are Au [10], Ag [9], and Al [25]. For the SPP(0, ± 1) and SPP( ± 1,0) modes, the periods are calculated for different wavelengths and dielectric refractive indices using different metals as shown in Figs. 2(a)-2(c).Fig. 2 Period P versus wavelength for different dielectric refractive indices when metal is (a) Ag, (b) Al, (c) Au. The wavelength dependent refractive indices of Si3N4 and SiO2 are plotted in (a). (d) Absorptance η versus wavelength with parameters: hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wetch = 0.27 μm, wx = wy = 0.21 μm. Materials are InGaAs, Si3N4, and InP for the active region, dielectric layer, and substrate, respectively.
The metals have similar dielectric constants with large negative real parts and small positive imaginary parts, so they have similar performance in the plasmonic devices. What matters is the ohmic loss of metal which is decided by the imaginary part of the metal permittivity [26]. As Fig. 2(d) shows, Ag which has the lowest ohmic loss performs best in the design, so it is chosen as the metal material.
In Fig. 2(a), the wavelength dependent refractive indices of SiO2 and Si3N4 are plotted. To achieve 1550 nm operation, the Si3N4 and SiO2 devices have P of 0.7 μm and 1 μm, respectively. The short-period Si3N4 device has the larger RA which fits the high absorptance demand. Besides, the higher order SPP-BW modes which need longer periods are not applicable for the same reason. Therefore, the selected materials in this presentation are InGaAs, Si3N4, InP, and Ag for the active region, dielectric layer, substrate, and metal layer, respectively.
3.2. Preliminary parametric optimizations
There are several important parameters which can be classified into two types, one has much influence on the resonance wavelength and one has little. The simulations with the coarse mesh dimensions (not listed here) suggest that the hsub, hd, and wetch belong to the latter. The preliminary parametric optimizations focus on these parameters.
3.2.1 hsub and hd
The vertical light confinement is important during the coupling from the incident light to the in-plane SPP -BWs. Here we study the vertical structure of dielectric layer. The devices with varied hsub and hd are simulated with the results presented in Fig. 3.
Fig. 3 (a) Absorptance η versus hd and hsub at 1550 nm. (b) Absorptance η versus wavelength with hd = 0.2 μm, hsub = 0.7 μm. The other parameters: hm = 0.22 μm, p = 0.68 μm, wetch = 0.27 μm, wx = wy = 0.21 μm.
Figure 3(a) suggests the optimized thickness of dielectric layer (hsub + hd) to be 0.8~0.9 μm for high η. The actual SP modes are supported by the Ag-Si3N4-InP (Si3N4: 1.99, InP: 3.16 at 1550 nm) structure. When the Si3N4 thickness is suitable, the slot Surface Plasmon (SP) mode has the similar effective index and mode distribution with the SP mode on the Ag-Si3N4 interface in the simplified assumption. For too thin dielectric layer, the vertical slot SP modes concentrate power in the Si3N4-InP surface rather than near the active region. When the dielectric layer is too thick, the high order vertical slot SP modes have high electric fields at both ends of the Si3N4 layer, and these high order mode excitations degrade the device performance, too. Figure 4 shows all these three cases. A lager hd means a higher absorptance but a lower AE, then we choose hd = 0.2 μm and hsub = 0.7 μm with overall consideration.
Fig. 4 Schematic diagram of the coupling between the incident light and the slot Surface Plasmon modes when the dielectric layer is (a) too thin, (b) suitable, (c) too thick, the orange lines denote roughly the slot Surface Plasmon mode intensity distributions.
3.2.2 Wetch
The simulation results for varied wetch are presented in Fig. 5.
Fig. 5 Absorptance η versus wavelength and wetch, the other parameters: hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wx = wy = 0.21 μm, P = 0.68 μm.
From the analytical study [27] of enhanced optical transmission through periodically nano-structured metal film, we know that the periodic metallic structure can excite the SPP-BWs, i.e., the primary design with zero wetch works as well. The red shift of resonance peak with increase of wetch has been analyzed in the small hole cases [27]. For the large hole cases, each hole acts like a new point source and the larger holes couple light more strongly [28]. We set wetch to be 0.27 μm for the latter study. In addition, the large wetch is good for decreasing the stray capacitance. Optical field distributions of different wavelengths are presented in Fig. 6.
Fig. 6 (normalized to Esource) in x-z plane in the center of one cell at wavelength (a) 1400 nm, (b) 1550 nm, (c) 1700 nm, in x-y plane in the center of metal layer at wavelength (d) 1400 nm, (e) 1550 nm, (f) 1700 nm, Parameters: hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wx = wy = 0.21 μm, P = 0.68 μm, wetch = 0.27 μm.
We note from Fig. 6 that the strong coupling from illumination to the SPP-BWs occurs at the resonance wavelength (1550 nm), thus the optical field in the active region gets enhanced. In the meantime, most power is reflected at the non-resonant wavelengths. At shorter wavelength like 1400 nm, the active region acts as a waveguide core, and the fundamental x-polarized mode is excited and turns leaky when propagating in the negative z-direction since the core size is in the sub-wavelength scale [29]. The etched holes are hardly involved in this process.
3.3 Design for particular operation wavelength
The influence of hm on resonance wavelength is studied in Section 3.3.1. In Section 3.3.2, we analyze the light concentration mechanism during the parameters wx&y and P optimizations. Section 3.3.3 shows the final optimized device under 1550 nm operation. In practice, the overall optimizations of these parameters enable designs for particular operation wavelengths covering the whole O-U bands.
3.3.1 Thickness of metal (hm)
For the free-standing metal films with sub-wavelength holes, the effects of metal thickness on the extraordinary optical transmission (EOT) have been researched experimentally [30] and theoretically [31]. While it is reported that an increasing metal layer thickness causes blue shift of the resonance peak, red shift of the absorption peak with increasing hm is found in this design, as shown in Fig. 7.
Fig. 7 Absorptance η versus wavelength with parameters: hd = 0.3 μm, hsub = 0.6 μm, wetch = 0.27 μm, wx = wy = 0.21 μm, P = 0.68 μm.
The varied metal thickness leads to the contrary resonance wavelength shifts for the EOT and the light absorption enhancement. We argue that the part of active region in metal layer plays an important role, and the majority of absorption occurs there (see Fig. 6). The metal layer should be thick enough for high performance.
3.3.2 Light concentration mechanism
To shed light on the light concentration mechanism, the optical antenna effect (i.e. LSPR) of a single unit cell is studied. In LSPR simulations, the structure includes a single active region and four surrounding etched holes, PML boundary conditions are applied to all directions and the source is changed to the x-polarized total field scattered field (TFSF) [18]. The results of periodic simulations and LSPR simulations are compared in Fig. 8. In the LSPR simulations, P defines the locations of etched holes. From Fig. 8, one can clearly see that the LSPR effect is not the main contribution of the enhanced absorption. The much larger NACS values in periodic simulations confirm that the SPP-BWs contribute to the absorption resonance mainly.
Fig. 8 Normalized absorption cross section (NACS) values from LSPR simulations and periodic simulations versus wavelength and P with device parameters: hm = 0.22 μm, hd = 0.2μm, hsub = 0.7μm, wetch = 0.27μm, (a) wx = wy = 0.15 μm (b) wx = wy = 0.22 μm.
The resonance wavelengths of LSPR effect are decided by the sizes of active regions, while the locations of etched holes (P) only have influence on the intensities and bandwidths of NACS spectrums. In periodic simulations, the resonance wavelengths are sensitive to both the active region horizontal sizes and P, and Fig. 9 gives more absorptance results for varied wx&y and P.
Fig. 9 Absorptance versus wavelength and P with device parameters: hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wetch = 0.27 μm,wx = wy = (a) 0.07 μm,(b) 0.1 μm, (c) 0.15 μm, (d) 0.2 μm, (e) 0.21 μm, (f) 0.22 μm.
From Figs. 9(a)-9(f), the RA increases from about 1% to about 10%, and the absorptance increases from about 3% to 70%. To explain the influence of wx&y and P on resonance wavelengths, the equivalent medium theory (EMT) [32] can be used . As shown in Fig. 10, the dielectric layer in each period is approximated by an equivalent material with equivalent refractive index ncenter sandwiched by the dielectric material, and this sandwiched structure is approximated by a final equivalent material with equivalent refractive index nequ.
The fill factor f is defined as f = wx / P. Since the SPP waves are TM modes, the equivalent refractive indices are written as [32]
By replacing εd with nequ2 in Eq. (4), the resonance wavelengths (λEMT) are calculated and pointed in Fig. 9. Since the effect of hm is not taken into consideration and the active regions in metal layer are ignored, the EMT estimation has a limited accuracy. Besides, the square array assumption is inapposite unless f is far below 1. So we can find mismatches between the resonance wavelengths from EMT and FDTD.Bandstructure calculations are more precise for resonance analysis of this complex structure. Several excitation dipoles are placed randomly and the Bloch boundary conditions are applied in the xz and yz-direction. Several randomly distributed time monitors record the time decay of the electric fields, and then the total electric field is computed by the Fast Fourier Transform (FFT). The bandstructure (see Fig. 11) maps consist of the results for different x-direction momentums kinplane.
Fig. 11 FDTD Bandstructure for varied wx&y = (a) 0 μm, (b) 0.07 μm, (c) 0.12 μm, (d) 0.17 μm, (e) 0.22 μm. Other parameters: hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wetch = 0.27 μm, P = 0.63 μm.
The SPP-BW modes denoted in the white lines (Fig. 11(a)) are calculated analytically:
The y- direction propagating SPP(0, ± 1) modes are degenerate for different x-direction momentums kinplane. Unlike the analytic prediction, the SPP( ± 1,0) are non-degenerate at gamma point according to the numerical calculation, and the resonance wavelength of SPP(−1,0) without active region (wx&y = 0) is longer than the analytic result at gamma point. With wx&y increasing, the large active regions make both the square array and hexagonal array approximations improper, thus the analytic explanations are less accurate as mentioned before, and we can find that the active region changes the bandstructure significantly. The SPP(−1,0) band breaks in two bands when wx&y increases from 0.07 to 0.12 μm as the red circle in Fig. 11(c) shows. When wx&y equals 0.22 μm, the lower band in gamma point corresponds to wavelength of about 1550 nm (Fig. 11(e)), and it is close to the absorption peak wavelength 1560 nm. In the meantime, the strong resonance near 1300 nm (about 230 THz) doesn’t contribute to the absorption in active region. The light concentration resonance can be attributed to the SPP-BWs of the face-centered structure, which is, however, different from the SPP( ± 1,0) mode of the square array.3.3.3 Optimized device under 1550 nm operation and discussion
The parameters of the optimized device are hm = 0.22 μm, hd = 0.2 μm, hsub = 0.7 μm, wetch = 0.27 μm, wx = wy = 0.22 μm, and p = 0.63 μm. A bulk p-i-n PD with the same absorptance needs an absorption layer thickness of 1.4 μm, thus the height and area of this device are respectively 31.4% and 12.2% of the reference PD. The optical field distributions in x-z plane and x-y plane are respectively presented in Figs. 12(a) and 12(b). The responsivity Resp is obtained by
where q is the unit charge, h is Plank’s constant, ν is the optical frequency, and ηi is the internal quantum efficiency of InGaAs, which is taken to be 0.8 [33].Fig. 12 (normalized to Esource) at 1550 nm (a) in x-z plane in the center of one cell ;(b) in x-y plane in the center of metal layer.(c) Absorptance η and responsivity on the left and right vertical axes, respectively.
As shown in Fig. 12(c), the center wavelength is 1560 nm, and the absorption spectrum bandwidth is 60 nm with over 60% absorptance. At 1550 nm, the responsivity is 0.74 A/W. The resonance wavelength of the square array ignoring the active region is 1300 nm from Eq. (4), with EMT correction, it increases to 1465 nm. The face-centered structure has a much longer resonance wavelength than the square array assumption. Both the high refractive index dielectric material and the face-centered structure enable a short period for particular operation wavelength. In this optimized device, a large RA of 12.2% provides possibility for high absorptance.
We list the performance comparisons between reported typical plasmonic PDs and this work in Table 1. The most remarkable advantage of our design is the high responsivity. We argue that there are three key points in the absorptance improvement of a plasmonic PD. Firstly, the low loss metal like Ag can be adopted, then a large RA (more than 10%) should be achieved by material choosing and structure design, and then a high-efficiency resonant structure is needed. The active regions in our design are not as small as those in the nanodipole optical antenna design, but high-speed operation and low dark current can also be realized, besides, the active region with hundreds of nanometers rather than only a few tens of nanometers has better fabrication tolerance.
Table 1. Parameter and performance comparisons between typical plasmonic PDs.
4. Feasibility analysis
In this part, we discuss three important issues to analyze the device operation feasibility. Firstly, the device-to-fiber coupling is analyzed.
In practice, the periods in both x and y directions should be enough to ensure accuracies of the infinite periodic simulations. In [28],the 15 × 15 sub-wavelength hole array on metal film shows a saturation transmission spectrum in the EOT research. Here we analyze the device with 20 × 20 tiled array.
As Fig. 13(b) shows, plane wave source in substrate illuminates in the + z direction. The near field above the metal layer is collected. The far-field at 70 μm above the device is calculated by Fast Fourier Transform (FFT) on the total near field of the whole array apodized by a Gaussian function. The |Ex| component in Fig. 13(c) is highly consistent with the mode field of commercial SMF-28 fiber [36] with mode field diameter of 10.4 μm. Thus the device-to-fiber coupling could be efficient due to time-reversal symmetry [9].
Fig. 13 (a) Overall view of the far-field calculation in both Cartesian and polar coordinates, (b) detailed structure of the part in the circle of overall view diagram, (c) far-field |Ex| component distribution in polar coordinate with λ = 1550 nm, N = 20, L = 70 μm, inset: enlarged view of far-field and mode field of commercial SMF-28 fiber.
To estimate the resistor–capacitor (RC) time constant, we calculate device’s effective capacitance C. Stray capacitances are ignored, and the abrupt junction barrier capacitance in one cell is
where q is unit charge, and XD is the depletion width. The voltage VD is set to 3V, and the doping concentrations in p and n regions NA and ND are both set to 1 × 1018 cm−3, then we have XD = 81.6 nm and 400 × CT = 21.1 fF. Assuming load resistance as 50 Ω, we have RC limited bandwidthThe transit time limited bandwidth is [37]The electron saturation drift velocity in InGaAs Vsat is 7 × 106 cm/s [38]. Then we haveWith help of micro-lens, photo-sensitive area decreases, e.g. device with 10 × 10 array has a smaller effective capacitance, a lager fRC = 603 GHz, and a lager f3dB = 320 GHz.
At last, we present the fabrication procedure in Fig. 14. The first step is growing epitaxial InGaAs on InP using Metal-organic Chemical Vapor Deposition. The nano-scale active region and InP ridge can be defined utilizing electron beam lithography (EBL) and Inductively Coupled Plasma (ICP).Then the Si3N4 layer is deposited on the wafer using Plasma Enhanced Chemical Vapor Deposition. Then EBL and ICP are used again to remove the Si3N4 covering the InGaAs. With help of a third time EBL, the lift-off process and electron beam evaporation are applied to deposit Ag film on the Si3N4 layer with etched holes. The Si3N4 layer and Ag film may have non-ideal sunken edges near the active regions after etching Si3N4. Besides, the three times EBL may be the main challenge in fabrication. Selective area nanopillars growth [12] is an alternative to this scheme. This design’s fabrication is harder than commercial PDs, but it can be realized with mature fabrication technologies.
5. Conclusion
We investigate the evaluation criterions of plasmonic photodetectors and get motivated to improve the absorptance of PDs with nano-scale active regions, which usually have high speed but serious optical power loss. In this study, we propose a periodic plasmonic PD with enhanced light concentration in the nano-scale active regions utilizing the SPP-BWs. The face-centered structure enables a short period which makes for high absorptance. The optimized device has an absorptance of 74% and a responsivity of 0.74 A/W under 1550 nm operation from FDTD simulation. Its electrical bandwidth is predicted to be 140 GHz for 20 × 20 array. The far-field analysis confirms the high-efficiency fiber-to-device coupling. We can adjust the device parameters for particular operation wavelength covering the O-U bands. The bandwidth of responsivity spectrum is wide enough for good fabrication tolerance. The design can be applied to both group III-V and IV materials. The nano-scale active regions bring low noise, small capacitance, and reduced power consumption. We have proposed the fabrication procedure performed by mature technologies to show the design can be realized, and the experimental fabrication is under way.
Appendix
As Fig. 15 shows, the simulation precision analysis is done by using varied mesh dimensions (dx × dy × dz) of the active region in metal layer.
Fig. 15 (a) Average electric field intensity (AE) and (b) absorptance η versus wavelength for the same structure with varied mesh dimensions. Device parameters: hm = 0.22 μm, hd = 0.2μm, hsub = 0.7 μm, wetch = 0.27μm, wx = wy = 0.1μm, P = 0.68 μm. Materials are Ag, InGaAs, Si3N4, and InP for the metal layer, active region, dielectric layer, and substrate, respectively.
To make balance the simulation precision with computing time, we set the mesh dimensions of active region in metal layer to be 1.5 x 1.5 x 2 nm, corresponding to the red curve. Since the red curve and pink curve (for 1.0 x 1.0 x 1.0 nm) almost overlap, we estimate that this reduces the absolute error for η to below 3%.
Funding
National Hi-Tech Research and Development Program of China (2008AA1Z207); Natural Science foundation of Hubei Province, China (2010CDB01606); Fundamental Research Funds for the Central Universities (HUST: 2016YXMS027); Huawei innovation Research Program (YJCB2010032NW, YB2012120133, YB2014010026, YB2016040002); Scientific Research Foundation for the Retuned Overseas Chinese Scholars.
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