This paper proposes and analyzes resonant Si-based electro-optical modulators and switches that use Ge-on-Si3N4 nanobeams (NBs) operating at 2 to 5 µm wavelengths. The wavelength of operation can be extended to 15 µm by mounting the Ge channel waveguides on a bulk Si chip. Electrons and holes are injected into the intrinsic Ge NB cavity center via thin P- and N- doped Ge wings on the NB (a lateral PIN diode at ~0.5 V forward bias). Simulations of the carrier-induced resonance-wavelength shift-and-damping in a 1 × 1 modulator show 6 dB of extinction at ~60 fJ/bit over the mid infrared. The NB’s active length is λ-scale. The cavity uses tapered-diameter air holes. Intensity modulation at ~1 Gb/s appears feasible. High-performance 2 × 2 switching is predicted by embedding one NB in each arm of a Mach-Zehnder device. The resonance of each identical NB is shifted by the same Δλ via carrier injection. Calculations show very low insertion loss and crosstalk in both the cross and bar states; however, the cross-to-bar energy, around 8 pJ/bit, is much higher than that in the 2 × 2 version that employs PN-junction carrier depletion.
© 2016 Optical Society of America
The germanium nanobeam (NB) is a Ge strip channel waveguide containing a one- dimensional photonic-crystal (1D PhC) lattice of air holes arrayed on the long axis to create a resonant mode—a strong transmission peak in the middle of the forbidden spectral band of the PhC. By itself, the NB is an inline electro-optical (EO) modulator having one input port and one output port (1 × 1). Modulation stems from an electrically controlled index perturbation in the cavity region. Beyond the 1 × 1, when identical Ge NBs are placed in a Ge-waveguided Mach-Zehnder interferometer (MZI), one NB in each arm of the MZI, the resulting dual nanobeam (DNB) device serves as a 2 × 2 EO switch (two inputs, two outputs) when the resonance of each NB is shifted identically along the wavelength scale by electrical means. A previous paper  shows specifically how this 2 × 2 DNB MZI switching is attained in silicon waveguides. The present paper deals with germanium as a practical 2 × 2 alternative. Here we propose that a germanium-waveguided DNB having the architecture of  will give effective switching especially in the 2 to 5 um mid-infrared.
This paper proposes and analyzes silicon-based Germanium-on-Nitride (GON) electro-optical modulators and switches (EOMs and EOSs) based upon resonant, crystalline GeNBs operating at 2 to 5 μm beyond the 1.55 μm telecommunications window. To place the present work in context, the prior art of waveguided NB EOMs consists of investigations on 1.55 μm silicon-on-insulator (SOI) devices. These were theoretical studies of PN-junction carrier depletion devices [1,2], experimental and theoretical studies of PIN-diode carrier injection devices [3,4], gas sensing devices , thermally tuned devices , and polymer-embedded electro-optic devices . All works indicate that NB modulation is a practical, high-performance approach. Advantageously, germanium offers transparency from 1.8 to 15 μm  while silicon experiences phonon-related absorption losses of 0.9 to 9 cm−1 at wavelengths from 8 to 18 μm. The thesis of this work is that germanium NBs have a role to play as a high-performance mid-infrared alternative to silicon NBs because Ge offers larger free-carrier effects and a wider spectrum of transmission. The substrate here is silicon coated with a thick film of Si3N4 which provides an optical cladding for the core, a lower cladding that is transmissive out to 7 μm [9,10] It should be noted that there is another approach that supplements GON, this is the Germanium-on-Silicon (GOS) technique where bulk silicon is the lower cladding and substrate. GOS is advantageous [11,12] because it has a wider transparency range than GON but with a drawback of lower refractive index contrast between the waveguide core and the lower cladding. As a result, the W × H cross section of the single- mode GOS channel will be larger than that of GON. Since the separation W between the P and N contact ribs will be significantly larger in the GOS PIN diode, this will increase the path resistance and will affect the diode’s switching speed. Because of the increased RC constant in the GOS device, we would expect the GOS NBs to switch at a somewhat slower speed than the GON NBs. Apart from this distinction, there are shared features. If we consider the optical insertion loss, optical crosstalk, and injection current in the GOS and GON DNB 2 × 2s, our simulations indicate that the GOS and GON devices will perform similarly in those aspects of switching. The simulation results presented in this paper show that the Ge NB can be switched between its off and on states with low energy and that the energy needed to keep the NB “on” is ultralow. Despite the fact that PN carrier depletion is faster than PIN carrier injection, such injection can be very fast; for example, the 2 Gb/s rate achieved by Shakoor et al . The limiting factor in that work was the carrier injection time and carrier removal time, the latter directly related to diffusion and surface recombination since there was no reverse bias to sweep out the carriers. While bare germanium has a lower surface recombination velocity than silicon, the surface recombination rate has been shown to increase through a reduction in waveguide size and through the introduction of additional defect states due to the device fabrication process [13–17]. This can cause the carrier lifetime in germanium waveguides to be comparable to their silicon counterparts . Therefore, we estimate that our device should be able to achieve > 1Gb/s PIN modulation. Also, the energies and speeds are expected to be competitive with those in Shakoor et al .
The devices in this paper are resonant structures in which the fundamental transmission mode of the nanobeam is designed to coincide with the wavelength of the laser selected for the on-chip system. Those devices differ from the non-resonant, broadband switches and modulators often seen in the integrated photonics art where several different-wavelength lasers are accommodated as chip-inputs within a wavelength band of several nanometers. Therefore resonant operation can be seen as a disadvantage; however, there is an advantage gained by the wavelength specificity - reduced energy per switching operation in the EO resonant devices compared to that in the EO broadband devices. So there is a tradeoff. Associated with the lower energy is the reduced footprint of the resonant devices relative to the non-resonant structures.
The Si-based GON platform is explored in this paper. As mentioned, GON and Germanium-on-Insulator (GOI) [19–22] offer tighter mode confinement than GOS due to their higher index contrast. The SiO2 in GOI has strong infrared absorption from 2.7 to 2.9 μm, and this oxide is then transparent from 3.0 to 3.8 μm, after which high absorption takes place beyond 3.8 μm . GOI waveguide experiments at 3.8 μm  suggest that GOI waveguides will indeed exhibit low loss over the 1.9 to 2.6 μm and 3.0 to 3.9 μm ranges. Compared to GOI, the GON waveguides have a wider spectral transmission range. The amorphous Si3N4 lower cladding of the Ge has a transparency range that depends upon the deposition method and conditions [9,10]. Specifically, the Si3N4 deposition techniques of RF magnetron sputtering and LPCVD (plus a high temperature anneal) give high mid infrared transmission with an absorption edge in the 5 to 7 μm range. For the best method, the transparency extends to 7 μm. That is why GON waveguides are viable to 7 μm. The GON can be fabricated readily by a bonding procedure in which a smart-cut crystal Ge wafer is bonded to a nitrided silicon wafer, after which lithographic etching is performed on the resulting Ge microlayer. The GON platform was explored recently as the basis of a Brillouin laser pumped at the 4 μm wavelength . That laser offered favorable acousto-optical overlap because of the microwave-acoustic properties of the nitride cladding. The GON is generally favorable for building a variety of low-loss on-chip high-index-contrast optical components.
In the Ge 2 × 2, the lateral PIN-diode regions are either in the fully injected state or the zero-injection state. Because the Ge is intrinsic, the uninjected state has the advantage of low-loss (high transmission) and high-Q resonance in both nanobeams since electro-refraction (ER) and electro-absorption (EA) are nominally zero. For the injected state, the ER shifts the NB transmission profile and the EA attenuates this profile. The strong electro-absorption in Ge is actually beneficial when the device is operated at λo, the resonance of the uninjected state. As discussed below, the metrics of insertion loss (IL) and crosstalk (CT) improve in proportion to the EA strength. The tradeoffs between the PN and PIN diode approaches are also discussed below.
The novelty of the present approach consists in: (1) the GON platform offering high index-contrast, strong optical mode confinement and low optical loss in on-chip components over the mid-infrared wavelength band, a band wider than that for GOI, (2) the lateral PIN-diode cavity actuation that gives high quality factor (Q) and low insertion loss in the non-injected state, (3) the use of resonance to provide low-energy switching and modulation in devices that have a very small footprint, and (4) the use of strong, free-carrier ER and EA in Ge to increase the 1 × 1 modulator contrast and to improve the 2 × 2 IL and CT in the injected state.
The organization of this paper as follows: We assign an air-hole array to the Ge NB 1D PhC resonator and design a zero-hole-defect resonator with tapered-diameter holes. By numerical modeling, an initial Q of about 3000 is found. Next, for assumed P and N doping of Ge side wings, the volume concentrations of electrons and holes ΔN and ΔP injected into the intrinsic waveguide’s mid-region as a function of forward bias voltage are determined by electrical software simulations. Free-carrier theory for Ge is applied to find the refractive index-perturbation Δn and Δk components within the NB PIN-diode region, after which the complex effective index changes of the NB waveguides are determined by modeling the index-distribution overlap with the fundamental TE00 mode. Next, an active modulation length of 0.2 to 5 wavelengths is determined, based on initial simulations showing that the cavity mode volume is of wavelength scale. The W × H cross-section dimensions of the single mode NB are known to scale with wavelength, allowing an optimized design at one wavelength (e.g. 2 μm) to be scaled easily to a different operational wavelength (5 μm). We then find the extinction ratio of the 1 × 1 device at two wavelengths. Next, the 2 × 2 design is presented, after which the spectral response of input-to-through and input-to-drop are simulated for both the fully injected and zero-injected states (cross and bar). Then the IL and CT are determined at λo. Finally, the off-to-on switching energy and the holding-on energy are determined.
2. Background discussion
The nanobeam is a strip channel waveguide containing a 1D photonic-crystal lattice of air holes, where the cavity can be determined by several different possible geometries, some of which include: (1) a few point-defects are placed in the middle of a uniform hole array, (2) a “zero defect” resonance is produced by continuously increasing the hole diameter from one end of the NB to the NB middle, and by then tapering down the diameter from the NB middle to the other NB end, (3) a Fabry Perot cavity is formed from two uniform mirror-hole-arrays plus a few tapered holes between mirrors. We have employed the quadratically tapered approach, but the expectation is that any of the resonator types would give equivalent performance. The NB offers a high value of peak transmission that is approximately independent of the wavelength used.
With its 1.96 index, the thick silicon nitride layer beneath the Ge provides a high index contrast with the Ge channel waveguide core. We assume that this Si3N4 layer was previously deposited upon a silicon substrate. The resulting proposed Ge/Si3N4/Si structure is related to the experimental sapphire-clad Ge/Al2O3/Si structure already attained by a wafer bonding technique . In a fashion similar to sapphire case, the GON wafer fabrication could proceed via a smart-cut wafer bonding process. The alternative method of Ge epitaxy upon nitrided silicon may have crystal-quality issues.
Our waveguided Ge devices are key components in silicon-based group-IV photonic integrated circuits (PICs), where group IV photonics is a more general form of silicon photonics that encompasses photonics constructed from group IV alloys. Supplementing the 1.55 μm telecom uses, there are important 2 to 5 μm applications that drive the need for group-IV PICs in this region. The principle applications are: (1) optical communications at ~2 μm [26–28] and (2) chemical, medical, biological and physical sensing over 2 to 5 μm [29–31]. Optical interconnects and communications in the 2 um band are enabled by a new generation of low-loss hollow-core photonic-bandgap fibers  and Ge 2 μm modulators come into play within a fiber-linked group-IV 2 μm transceiver chipset where the modulator impresses Gb/s information upon the cw laser emission. On-chip N × N EO switching will be valuable in future chips. Sensing applications are based upon the fundamental molecular “signatures” in the 2 - 5 μm range, and here the mid-infrared Ge modulator is a signal-processing element within the lab-on-a-chip sensor PIC, a chip that contains typically a spectrometer or refractometer. To attain these mid-infrared applications, knowledge of the Ge free-carrier physics  and related optical physics  is needed.
Experimental progress on Ge PIC-related structures has already been made . There are several examples: the first demonstration of low loss Ge MIR waveguiding , Ge planar concave gratings , low-loss Ge arrayed waveguide gratings , photonic microcavities , Ge thermo-optic phase shifters , as well as low-loss rib waveguides, grating couplers and MMIs .
Looking at the larger picture of these Si-based PIC structures, the Ge waveguided devices are part of the newly emerging on-chip infrared technology of SiGeSn active and passive waveguided heterostructures. Regarding electro-modulation, the complex free-carrier effects in bulk Ge have recently been investigated  and there it was found that Δn and Δk have a slightly irregular wavelength dependence but generally this ER and EA increase with increasing wavelength. Because Δn and Δk are “strong”  it is not necessary to attain a very high Q in the cavities. Even for Q in the range of 3000, the ER estimated here (see below) is sufficient to shift the resonance wavelength by one cavity linewidth as desired. Regarding the NB information bandwidth for the specific example of Q = 3000, we find that the bandwidth is 100 GHz at λ = 2 μm.
The GON and GOS NB platforms can be compared in terms of their Q’s. The maximum achievable Q in the GOS NB is actually lower than that of an identical GON NB because GOS has lower index contrast between its waveguide core and substrate. However, Q ~3000 is easily attained in GOS.
3. Design of the 1 × 1 Ge nanobeam electro-optical modulator
Our modulator design is composed of a germanium 1D nanobeam photonic crystal cavity integrated into a low-loss waveguide and follows the general methodology set forth in Quan et al . Starting at the center of the structure and using a zero-defect cavity design, the photonic crystal holes are quadratically tapered down in diameter, symmetrically in both directions, to the edges of the photonic crystal in order to create a linear change in the strength of the photonic-crystal mirrors. Such a design has been shown to provide high waveguide transmission as well as high cavity quality factor and small mode volume, the latter two being dependent on the total number of photonic crystal holes employed.
Two different germanium strip waveguide 1 × 1 devices, assumed to be sitting on top of an optically thick layer of Si3N4 were created with widths and heights chosen to provide TE00 single mode operation at wavelengths of 2 μm and 5 μm. The number of photonic crystal hole pairs was chosen to be 12, for a total of 24 holes, which gives a good tradeoff between Q and cavity lifetime. While a large Q makes modulation easy, it comes at the expense of increased cavity lifetime and, therefore, decreased device speed. Here are some specifics. At Q = 3000, the cavity lifetime is roughly 3 ps at the 2 μm wavelength and is 8 ps at the 5 μm wavelength. This corresponds to a theoretical maximum modulation frequency of 330 GHz at 2 μm and 125 GHz at 5 μm.
The choice of 12 hole pairs resulted in Q ~3000, deemed an appropriate value for efficient modulation and speed. Q’s were calculated by taking the transmission resonance center wavelength and dividing by the full linewidth at half maximum. The lattice spacing of the holes was then determined so that both devices would have a bandgap at the required operational wavelength. Hole sizes were calculated by using a fill factor of 0.2 at the center of the photonic crystal and by quadratically tapering down that fill factor to 0.1 at the edge of the photonic crystal. Relevant geometric parameters for each device are shown in Table 1.
In order to provide electrical contact to create the desired PIN structure, initially undoped germanium “wings” are placed on either side of the nanobeam, extending the entire length of the photonic crystal, Lw, and having a height equal to 20% of the nanobeam height. This is a very thin, lateral Ge “wing platform” of length Lw beneath the nanowire device as shown in Fig. 1. Localized P-type doping is then introduced into one wing and N-type doping into the other in order to create a modulation length Lm for the PIN junction, as shown in Fig. 1. The reason that Lw is always equal to the length of the photonic crystal is because large insertion losses were found when Lw was less than the photonic crystal length, hence Lm is always smaller than Lw. Donor and acceptor doping concentrations of 1*1018 were chosen for the doped regions in order to allow for efficient forward bias injection into the intrinsic germanium waveguide.
While fundamental TE mode operation was chosen for our device, slight design modifications to the nanobeam height and width could be implemented to achieve TM mode operation. High Q and wavelength-scale mode volume for TM and dual TE/TM photonic crystal nanobeams have been experimentally verified [42,43]. We have performed TMoo-mode simulations (not shown) which indicate that this mode also gives high transmission in the NB with 1 dB of IL. We also examined the Q values and the shift-and-reduction of the TMoo spectral resonance as carriers are injected and found that these TMoo responses were quite close to those determined for the TEoo mode.
In Fig. 1, the holes are fully etched through the Ge while the side wings are partially etched. This device could be achieved in a two-step lithography process using state-of-the-art electron beam lithography. In such e-beam fabrication, an overlay alignment accuracy of less than 10 nm is available between step 1 and step 2 which satisfies the requirements of the Fig. 1 device. In principle, a single-step process could use an atomic-force microscope hot-tip quasi-3D lithography tool.
4. Simulation of the 1 x 1 NB EOM
The optical and electrical behavior of the devices were determined using a commercial grade simulator based on the finite difference time domain method as well as a device simulator that self-consistently solves the Poisson and drift-diffusion equations . Optical simulations were performed first and were carried out by injecting a TE waveguide mode into one end of the NB device under consideration and by monitoring mode transmission through the device at the other end. After simulating the optical performance for the unmodulated/intrinsic devices, the performance for modulated devices of varying modulation length and varying injected carrier concentrations ΔP = ΔN = Conc were obtained. The carrier concentration values chosen and their respective changes in the real and imaginary parts of the Ge index of refraction are listed in Table 2, where Δn = Δne + Δnh and Δk = Δke + Δkh. These values were extracted from Nedeljkovic et al .
NB transmission spectra as a function of wavelength for various modulation lengths using concentrations of both 5*1017 and 1*1017 are shown in Figs. 2 and 3 along with a plot of the on/off extinction ratio as a function of modulation length for the two concentrations. Wavelengths are 2 um and 5 um in Figs. 2 and 3, respectively. Using the left two graphs, the extinction ratio values were calculated using the equation −10*log(Tm/To) where Tm is the modulated transmission value and To is the unmodulated transmission value. Taking 6 dB as a minimum required extinction ratio we see that a minimum injected carrier concentration of somewhere between 1 and 5 *1017 is required for the 2 μm device, while 1*1017 is sufficient for the 5 μm device. Required intensity-modulation lengths are wavelength-scale in size or sub-wavelength. As the modulation lengths begin to exceed the cavity mode length the extinction ratios begin to plateau. The photonic crystal cavity essentially increases the light matter interaction time, allowing for large changes to occur in such a small, subwavelength scale.
The extinction ratio in the 5 μm device (Fig. 3(c)) is much better than in the 2 μm device (Fig. 2(c)). The reason for this difference is that the extinction ratio is proportional to Δλs/Δλ, where Δλs is the wavelength red-shift of the transmission profile at a given Lm and Conc, and Δλ is the FWHM linewidth of the uninjected profile. We know that Δλs is proportional to the Δn in Table 2. Thus, if we examine the Δn/Δλ ratio in Figs. 2 and 3 for a particular pair (Lm, Conc) we find that this ratio gives quite accurately the comparative extinctions between Figs. 2(c) and 3(c).
5. Design and simulation of the Ge dual-nanobeam 2 × 2 EO switch
Figure 4 presents our proposed 2 × 2 MZI dual nanobeam design for the EO cross-bar switch. This follows the SOI layout in  except that the PN junctions of  have been replaced by PIN diodes as shown in the Fig. 4 electrical wiring diagram where both NBs are injected in the same way. A novel feature is the employment of germanium multimode-interference couplers to provide the desired 50/50 optical splits - a division also available in 3 dB Ge directional couplers. Because we have engineered very high NB transmission in the zero- injection state as per Figs. 2 and 3, the transfer matrix theory of the Fig. 4 uninjected switch shows that essentially all of this transmitted light exits the through port of the Fig. 4 device for light entering the input port. Considering in addition an independent signal entering the add port, we find that the add light exits the drop port; thus, we have a high quality cross state in Fig. 4 for zero injection. The bar state can be reached as described below.
To quantify switch performance, we put unit optical power into the input of Fig. 4, and simulated the input-to-through and input-to-drop power transmission, first at zero injection, and then at full injection, which is taken as ΔN = ΔP = 5*1017 cm−3. The resulting four spectral curves are plotted in Fig. 5 for the 2 μm device and in Fig. 6 for the 5 μm device. In all of these simulations, the active length for each NB (the length of P and N doped Ge wings) was fixed at 10 μm.
Now we take the wavelength of operation λ to be the same as the peak transmission wavelength λo of the zero-injection state. Having done that, we can determine the fractional transmission at λo for all four curves in Fig. 5 and in Fig. 6. Inspection of Figs. 5 and 6 for the zero-injection state (red curves) shows very strong transmission at input-to-drop and at add-to-through, meaning that an excellent cross state is achieved. Now we would like to illustrate a comparison of Ge PIN devices with Si PIN devices, considering the Si 2 × 2 as a reference device from which to point out differences with the Ge 2 × 2. This comparison is best done by considering the carrier-injected state in Figs. 5 and 6 (black curves) where, at λo, the drop output is high and the through output is low, as we desire for this bar state. The comparison with Si starts with a Si version of Fig. 4 having the same Lm and the same Conc as for Ge. Then we examine the induced change in the complex index of Si at λo = 2 μm, Lm = 10 μm, and ΔN = ΔP = 5 x 1017 cm−3. Using the free carrier theory of  for Si, we find that the bulk Si index changes by Δn + i Δk = 0.0025 + i 0.00017 compared to the change 0.0028 + i 0.00087 found in Table 2 for bulk Ge. This has two consequences: (1) the Δn’s are comparable for Si and Ge, hence the wavelength shift of the switch’s transmission curve will be essentially the same for Si and Ge since it is well known that the shift is proportional to Δn; (2) the EA of Ge is about five times larger than that of Si (which holds true also for 5 μm). This Δk property actually gives the advantage to Ge because in the bar state of Figs. 5 and 6, the larger the Δk, the higher is the drop output power (lower IL), and the lower is the through output power (lower CT). Therefore, for the bar state, we conclude that the Ge switch will always offer IL and CT metrics that are superior to those of the Si switch.
The fractions in the figures are then expressed in dB using P(dB) = −10 log (Pout/Pin), which then gives us the IL and CT for each of the two switch states. Our findings for the performance in Figs. 4 and 5 are summarized in Table 3. Generally, we predict very low cross-and-bar insertion loss, a performance notably better than in the PN case of . Also, the CT values are quite low and practical in Table 3. Because of the findings, the switches appear cascadable for N × N applications. Generally, a wide range of applications has been detailed  for such 2 × 2s.
The sources of IL in Table 3 are: (1) optical scattering loss at the left- and right-side cladding interfaces where the Ge core interfaces the 0.2H Ge rib platform and (2) free-carrier absorption loss from mode intensity that fringes out into the N and P doped regions. For (2), we estimate that the two rib contacts induce 0.1 dB of IL. In addition, in practice there will errors in fabrication such as wall roughness, hole distortion, and misalignment of N and P areas along the channel axis. Speaking generally, those errors can be minimized effectively by manufacturing the switch in a 45-nm CMOS foundry where the factory processing gives precise, excellent construction. During fabrication, some IL might arise from unwanted diffusion of dopants into the intrinsic core. That problem could be avoided by separating the N and P areas by a larger distance - for example, by a distance 1.2 W instead of W. However, that approach increases the electrical path resistance and the RC constant. Thus the reduced indiffusion will be traded off against a reduced switching speed, an increased drive voltage, and an increased switching energy (20% higher in the present example).
6. Electrical simulations and required switching energy
After the TEoo optical simulations were completed, electronic simulations were performed so that switching energies  could be determined. To address lifetime and recombination effects, the following material parameters were used: trap assisted (Shockley-Read-Hall) carrier lifetime for both electrons and holes of 0.1 ms, radiative electron-hole pair capture rate of 6.4*10−14 cm−3 s−1, and Auger carrier capture coefficients for both electrons and holes of 1.0*10−30 cm−6 s−1. Specific forward bias voltage values were used such that the intrinsic germanium waveguide would be injected with the same concentration of free carriers as used in the optical simulations. A current-voltage curve for both the 2 μm wavelength and 5 μm wavelength devices is shown in Fig. 7. For 5*1017 and 1*1017concentrations it was found that Vf = 0.6 V and Vf = 0.47 V were required. As seen in Fig. 8, the distribution of the injected carriers is highly uniform. The junction capacitance was then extracted from the electronic simulation results for each bias voltage value and modulation length. The off-to-on 1 × 1 switching energy could then be calculated using the following equation: . This 1 × 1 modulator is capable of digital and analog modulation, although the analog application is hampered by a nonlinear voltage-response characterstic. The principal application of our digital optical modulator would be to enable fast optical communications in the 2 μm wavelength region, a spectral band for optical interconnects that supplements the 1.55 μm telecom/datacom band . The Ge modulators would be integrated in a Si-based transceiver chip. The datacom link is formed by connecting two such transmit/receive chips with a low-loss 2 μm hollow core photonic bandgap fiber. Our digital emphasis here explains why our evaluation of energy consumption focuses on the energetics of a bit stream. Since a typical digital modulation scheme has a 50% probability of a bit flip , the average energy per bit is given by: . For the 2 × 2, Eb = Es.
Tables 4 and 5 list the required 1 × 1 switching energy for the various modulation lengths and injected carrier concentrations for the 2 μm and 5μm devices. The shaded regions of the tables correspond to extinction ratios that are greater than 6 dB. It was found that a minimum switching energy in the range of 63-240 fJ/bit is required, all with subwavelength modulation lengths. As a comparison, silicon-based SOI PN photonic-crystal nanobeam electro-optic depletion modulators operating at 1.55 μm can provide 6 dB of extinction with an energy cost of only 14 aj/bit. This is a big difference. The significant difference in the required switching energy (PIN vs PN) is a consequence of the additional capacitance associated with carrier injection in the Ge mid-infrared PIN device versus carrier depletion capacitance in the Si near-infrared PN device.
The 2 × 2 device is more demanding than the 1 × 1 device in the sense that the active length must be increased to several wavelengths and “high” injection must be used in order to produce a resonance shift of about one linewidth so as to attain the “true” bar state. The increased Lm and injection current in the 2 × 2 switch compared with that of the 1 × 1 causes an escalation of the switching energy. As a result, the cross-to-bar switching energy can reach into the domain of picojoules per bit. This is quantified in Tables 4 and 5 (doubling the listed value) where we take Lm = 10 μm and Vf = 0.6V for the 2 × 2, giving 5.6 and 8.0 pJ/bit at 2 and 5 μm, respectively.
The total energy required to operate either device consists of a DC holding energy in addition to the AC switching energy. The holding energy is the average of the energy required to maintain both the on and off states of the device and it can be calculated by taking the average DC power consumed in both states and multiplying that by the modulation speed of the device. Hence, the DC holding energy is the average energy required to maintain the device in a given state for a length of time equal to the transit time of one bit in a digital modulation scheme. Since the off state for our device corresponds to zero bias voltage, the holding energy here is only related to the on state where there is a non-zero bias voltage. Extracting the power consumption I*V from the electronic simulation and taking a modulation speed of 1Gb/s, we determine that the holding energy is only 2 or 3% of the off-to-on energy for all situations analyzed here. This implies that the total energy for the germanium PIN electro-optic modulator is dominated by the switching energy.
7. Discussion and conclusions
Our simulations of GON on-chip mid-infrared integrated devices show the feasibility of high transmission in the uninjected 1D PhC PIN Ge NB on resonance. The 1 × 1 modulation was found to be quite effective because the mid-infrared free-carrier effects are strong in Ge and because the ER and EA work in concert to shift and dampen the resonance. The 0.5 to 10 μm length P- and N- doped Ge wings are localized at the cavity center.
The 2 × 2 MZI architecture, embedding two identical nanobeams, is particularly effective for attaining very low IL and CT in both the cross and the bar states of the switch. This high performance is traceable to an “intrinsic” zero-injection state and to an effective-injection state. The Q ~3000 is low enough to allow information bandwidths of around 100 GHz. However, the technique of electron-and-hole injection exacts a price in the off-to-on switching energy of the nanobeams, a large penalty in energy with respect to the switching energy of PN-junction carrier-depletion nanobeams. The penalty arises from the injection current and PIN diode capacitance. Nevertheless, a few picojoules per bit is acceptable in many cases.
The switching and modulation techniques described here can be generalized for the wide context of Si-based group-IV mid-infrared photonics: (1) instead of Ge-on-nitride , the waveguide could just as well be SiGeSn on nitride or SiGeSn on silicon , (2) instead of the quadratic taper, the resonator could be two Fabry-Perot mirror hole-arrays separated by point defects, (3) instead of PIN injection, carriers could be depleted from a lateral PN-junction Ge NB cavity, and (4) instead of free-carrier ER and EA, an EA-dominant mechanism could be actuated by a reverse-biased PIN diode producing the Franz-Keldsyh effect  in the waveguide cavity.
In summary, we have designed and simulated GON PIN-diode dual-nanobeam switches and single-nanobeam modulators where electron-and-hole injection produces ER and EA in the resonator center. In the TE00-mode germanium waveguide, a photonic crystal lattice with a zero-hole-defect cavity and quadratic hole tapering was used to provide enhanced light-matter interaction and >90% transmission in the unmodulated state. Twelve air-hole pairs were chosen which lead to quality factors in the 103 range. The cavity lifetime associated with this Q allows for device operation at speeds exceeding 1Gb/s and, therefore, is not a limiting factor in device speed. External electrical contacts for bias voltage control were designed by locally doping an intrinsic germanium wing platform located on each side of the waveguide cavity center; the length of the doping region was equal to the modulation length and the wing height was 20% of the waveguide height. Using a forward bias of up to 600mV, extinction ratios and switching energies were determined for devices operating at 2 μm and 5μm. In the 1 × 1 intensity modulator, energy per bit values as low as 63 fJ were found to be sufficient for achieving 6 dB of extinction with subwavelength- scale modulation lengths. By replacing the underlying Si3N4 with Si, it is expected that such germanium based photonic crystal nanobeam electro-optic modulators could operate at wavelengths out to 15 μm with increased performance due to the increased change in the complex index of refraction with increasing wavelength.
A pair of identical nanobeams, one embedded in each arm of a mid-infrared Mach-Zehnder interferometer, was proposed as a technique for high-performance, low-energy switching. These NBs are turned on identically to go from the initial uninjected cross state to the injected bar state. Taking ΔN = ΔP = 5*1017 cm−3 at 0.6 V and Lm = 10 μm, we calculated 0.22 to 0.97 dB insertion loss and −14 to −27 dB crosstalk in both states. The cross-to-bar energy was ~8 pJ/bit. The conclusion drawn in this paper is that the PIN-injection 2 × 2 has IL-and-CT metrics generally superior to those of the PN-depletion 2 × 2 except for the big tradeoff in energy consumption. The analysis in  reveals that the 2 × 2 performance is rather sensitive to errors in the device dimensions and surfaces that arise during the fabrication process. Therefore, to minimize such fabrication errors, it is important to construct devices in a high-tech facility such as a 45-nm silicon foundry .
JH acknowledges support from the Air Force Office of Scientific Research (Program Manager Dr. Gernot Pomrenke) under contract number 15RYCOR159. RS appreciates sponsorship by the AFOSR on grant FA9550-14-1-0196 and by the UK EPSRC Project Migration.
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