Open Access
Add to BibSonomy
Abstract
The lack of a bulk second-order nonlinearity (χ(2)) in silicon nitride (Si3N4) keeps this low-loss, CMOS-compatible platform from key active functions such as Pockels electro-optic (EO) modulation and efficient second harmonic generation (SHG). We demonstrate a successful induction of χ(2) in Si3N4 through electrical poling with an externally-applied field to align the Si-N bonds. This alignment breaks the centrosymmetry of Si3N4, and enables the bulk χ(2). The sample is heated to over 500°C to facilitate the poling. The comparison between the EO responses of poled and non-poled Si3N4, measured using a Si3N4 micro-ring modulator, shows at least a 25X enhancement in the r33 EO component. The maximum χ(2) we obtain through poling is 0.30pm/V. We observe a remarkable improvement in the speed of the measured EO responses from 3 GHz to 15 GHz (3 dB bandwidth) after the poling, which confirms the χ(2) nature of the EO response induced by poling. This work paves the way for high-speed active functions on the Si3N4 platform.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Silicon nitride (Si3N4) is a high-performance platform for versatile on-chip photonic devices [1–4] because of its low propagation loss, broad transparency window (400-6700 nm [5]) and good compatibility with complementary metal-oxide semiconductor (CMOS) processing. However, Si3N4 lacks an intrinsic second-order nonlinearity (χ(2)) due to its centrosymmetric structure [6], which limits its applications from some critical active functions, such as electro-optic (EO) modulation and second harmonic generation (SHG).
Previous works have made great progress in realizing χ(2)-related phenomena on a silicon nitride platform. Pioneer studies report a SHG signal that originates from symmetry breaking at the interface of a multilayer stack of amorphous Si1-xNx:H films with various compositions [7]. An electro-optic response is also observed in a stacked Si3N4 multilayer [8]. Other works point out the existence of a bulk second-order nonlinearity in deposited Si3N4 films [9,10]; possible explanations include the presence of silicon nanocrystals [9] and a built-in static electric field [10] that leads to an electric field induced second harmonic generation (EFISHG). There are reports of SHG in Si-rich silicon nitride [11,12]. The use of high-quality micro-ring resonators [13] and nano gratings [14], in which the optical field at the pump frequency is amplified, further improves the efficiency of SHG in silicon nitride. Recently, an all-optical poling (AOP) method [15,16] has demonstrated record-breaking SHG [17] and sum-frequency conversion (SFC) [18] in Si3N4. The optical poling process in this approach automatically forms quasi-phase matching (QPM) gratings in the Si3N4 waveguide [15–20] and the QPM condition can be tuned by re-poling the device at the corresponding wavelength [19,20]. On the other hand, to realize Pockels EO modulation on the Si3N4 platform, other works have explored heterogeneous integration of other materials with a usable EO response, such as lithium niobate (LN) [21], barium titanate (BTO) [22], lead zirconate titanate (PZT) [23], zinc oxide and zinc sulfide [24], EO polymers [25], and two-dimensional (2D) materials [26–28], onto the Si3N4 platform. These approaches, however, complicate the fabrication process, introduce extra loss and/or are not CMOS-compatible. Alternatively, a very recent work by Zabelich et. al. [29] demonstrates an electro-optic response of tens of kHz on a pure Si3N4 platform induced by electrical poling at elevated temperatures.
In this paper, we propose and demonstrate an induction of a second-order nonlinearity for electro-optic modulation in silicon nitride through electrically poling the film and aligning the Si-N bonds. We show an enhancement of at least 25X of the EO response in a Si3N4 micro-ring resonator. The 3 dB bandwidth of the EO response after poling improves from 3 GHz to at least 15 GHz. Further analysis supports that this enhancement directly derives from the induction of χ(2) in Si3N4. Khurgin et al. [30] hypothesized that the Si-N bonds in Si3N4 possess a second-order hyperpolarizability comparable to the Ga-As bonds in gallium arsenide (GaAs), whose χ(2) is over 300pm/V [6,30]. Once the centrosymmetry in Si3N4 is broken, e.g., through electrical poling, a non-trivial bulk second-order nonlinearity will appear and thus enable the Pockels electro-optic effect. Electrical poling has been used to engineer material properties, such as to periodically pole various nonlinear crystals for QPM [6] and to induce a second order nonlinearity in polymers [31], silica glass [32], optical fibers [33], and very recently in Si3N4 [29].
2. Device design and fabrication
We pole silicon nitride by embedding a removable set of electrodes into the device. We deposit a pair of Molybdenum (Mo) electrodes sandwiching the Si3N4 ring (Fig. 1(a)). The gap between the electrodes and the ring waveguide is 300 nm (edge to edge). The Mo electrodes generate an average horizontal electric field of 4.23 × 103 V/cm per volt of bias applied in the Si3N4 waveguide (simulated with COMSOL Multiphysics). We apply 160 V across the Mo electrode pair and build a poling field of 0.676 MV/cm. We choose Mo as the material for the poling electrodes because it is compatible with the high temperatures reached during the poling process (See Section 3) and it can be removed (Fig. 1(b)) using xenon difluoride (XeF2) etch to eliminate the loss it introduces to the ring resonator. The XeF2 etch has a high selectivity to other materials used in the device and thus, it does not influence the poling results. We deposit another set of electrodes made of platinum (Pt) outside the Mo ones for characterizing the electro-optic response of the device after removal of the sacrificial Mo electrodes (Fig. 1(b)). To fabricate the device (Fig. 1(c)), we deposit 300 nm of stoichiometric Si3N4 (refractive index measured to be 1.99 at 1550 nm, which is in line with other reports [34]) using low pressure chemical vapor deposition (LPCVD) over 4 µm of thermally-grown SiO2 on a 4-inch Si wafer. We define the waveguide and the ring resonator using electron-beam lithography and etch the structure with inductively coupled plasma reactive-ion etching (ICP-RIE). We use an add-drop ring resonator as the drop port provides spectrum information with no background power for characterizing the performance of the ring. The waveguide and the ring have a width of 1 µm and the radius of the ring is 150 µm. To create the poling electrodes, we pattern them with deep ultraviolet (DUV) photolithography and sputter and lift-off 300 nm of Mo. 400 nm Pt is then deposited in the same way. Afterwards, we deposit a cladding layer of 2.3µm of SiO2 using plasma-enhanced chemical vapor deposition (PECVD) on top of the device for protection. We open vias using DUV photolithography and ICP-RIE to expose the Pt electrodes and for later removal of the Mo electrodes. Detailed dimensions of the device are shown in Fig. 1(a-b).
Fig. 1. Top view (top cladding hidden, not to scale) and typical cross section (not to scale) of the device (a) before and (b) after removal of Mo electrodes. A voltage is applied between the Mo and the Pt electrode pair for poling and high-speed modulation, respectively. (c) Fabrication process of the device.
3. Device poling
We pole the Si3N4 ring resonator by applying a strong electric field across it at high temperatures. High temperature makes Si-N bonds more susceptible to the applied poling field and therefore makes the poling process more efficient [32]. A source meter (Keithley 2470) generates the desired high voltage, and we apply it to the Mo electrodes through a pair of tungsten probes that is manipulated with micro-positioners. Before we enable the high voltage output, a continuous-wave (CW) CO2 laser beam is focused onto the device, as shown in Fig. 2(a), to pre-heat it to the desired temperature. We increase the laser power to approximately 10W and heat up the ring resonator to 550 ± 40°C, which we calculate from the resistance change of the Pt electrodes as temperature varies (See Supplementary Section 1). This value corresponds to the temperature at the middle of the heated ring resonator (Fig. 2(a-b)). At higher temperatures it becomes challenging to maintain steady contact between the tungsten probes and the electrodes. We also observe that the arcing threshold drops as we increase the temperature, compromising the maximum poling field we can apply. The highest voltage we apply at this temperature is 160 V, which generates a horizontal poling field of 0.676 MV/cm in the Si3N4 waveguide (Fig. 2(c)). We observe arcing between the exposed regions of the poling electrodes for a voltage of 180 V or higher. The corresponding horizontal poling field at this threshold is 0.760 MV/cm. The breakdown threshold of silicon nitride at room temperature is 3-8 MV/cm [35]. The lower arcing threshold in our experiment is a result of high temperature as well as that part of the Mo electrodes are exposed thus arcing can happen through the heated air. The collateral vertical field generated in the ring resonator is at least two orders of magnitude smaller than the horizontal field except at the corners, (Fig. 2(c)), so the poling of the Si3N4 resonator is mostly horizontal.
Fig. 2. (a) Schematic (top cladding hidden, not to scale) of poling the device at high temperature. (b) Microscope image of the device under room (left) and high (right) temperature. External illumination is turned off when taking the high-temperature image and the red color comes from thermal radiation of the chip. (c-d) Simulated (COMSOL Multiphysics) electric field profile in the waveguide (c) when applying 160 V bias to pole the device and (d) when applying a unitary volt for high-speed modulation. Note the Mo electrodes in (c) (dark blue regions where field is zero) are replaced with air in (d).
We enable the source meter output and pole the device for 5 minutes after it is heated up to the desired temperature. When the poling finishes, we turn off the heating laser immediately but maintain the poling field until the device dissipates its heat to the metal holder and cools down to room temperature (in approximately half a minute). The fast cooling while the poling field still exists helps to freeze the alignment of the poled Si-N bonds and minimizes their rollback to their original state. No significant damage or deformation is observed on the device during the poling process. The Mo electrodes used for poling are then removed using XeF2 dry etch to eliminate the optical loss it introduces to the ring resonator. The XeF2 etch is run at room temperature and has a high selectivity to Si3N4, SiO2, and Pt, so it does not undermine the already-built alignment of the Si-N bonds.
4. Measurement of the electro-optic coefficient of silicon nitride
To examine the EO coefficient of Si3N4 and the impact of the poling process, we apply a radio frequency signal across the ring resonator with the remaining Pt electrodes and analyze the corresponding RF component in the output optical signal. We couple light into the device from a tunable laser (1 mW) with a polarization-maintaining (PM) lensed fiber and collect the output optical power with a single mode (SM) lensed fiber (Fig. 3(a)). A fiber rotator at the input port enables launching TE or TM light into the device by rotating the PM fiber. The ring resonator is designed to support the fundamental modes only. The transmission spectrum, monitored by an optical power meter, shows a typical loaded quality factor (calculated using λres/FWHM, where λres is the wavelength of the resonance and FWHM is the full width at half maximum of the resonance) of approximately 5,000 for the TE0 mode (Fig. 3(b)) and 3,000 for the TM0 mode (Fig. 3(c)) after removal of the Mo electrodes. Corresponding coupling coefficients (same for the coupling of the bus waveguide and the drop port waveguide in our design) and propagation loss (loss from drop port excluded) are 25% and 50 dB/cm for TE0 mode and 57% and 90 dB/cm for TM0 mode, respectively, derived through Lorentzian fitting of the resonances [36]. The large propagation loss comes from residual Mo and the rough sidewall SiO2-air (Mo mostly removed) interface, which is only 300 nm away from the Si3N4 core, due to the isotropic nature of the XeF2 etch. The mode profiles calculated with the Finite Difference Method (FIMMWAVE, Photon Design) in Fig. 3(d-e) confirm considerable overlap between the mode and the interface. This loss can be reduced by optimizing the fabrication and the XeF2 etching process.
Fig. 3. (a) Schematic of the setup used to measure the EO response of the poled Si3N4 device. (b-c) Typical transmission spectra of the device working in the (b) TE0 and (c) TM0 modes. (d-e) Simulated (FIMMWAVE, Photon Design) electric field profile of the (b) TE0 and (c) TM0 modes after removal of the Mo electrodes.
We apply a modulation signal up to 15 GHz. The sinusoidal signal is generated using a signal generator and amplified to up to approximately 8 V (peak-to-peak value, see Supplementary Section 2) through a modulator driver. The modulation signal is delivered to the Pt electrodes with a pair of high-speed (40 GHz) probes and we terminate the transmission line with a 50Ω load to minimize signal reflection. We set the working wavelength of the laser at the slope of the resonance according to the transmission spectrum to optimize the EO response. The output of the photodetector is sent to a spectrum analyzer for characterizing the EO response of the device. Note that we amplify the output power to 10 mW, the input limit of the photodetector, using a C-band Erbium-Doped Fiber Amplifier (EDFA) to make the high-frequency components more distinguishable from the background noise of the spectrum analyzer.
The RF-frequency electrical power PSA measured at the spectrum analyzer allows us to quantitatively characterize the EO coefficient of the Si3N4 in our device. The corresponding refractive index change of nitride ΔnSiN can be determined through (detailed derivation is in Supplementary Section 3)
The radial-only asymmetry we introduce into the initially amorphous Si3N4 turns it into a rotationally symmetric structure with one axis of symmetry. Such a structure is analogous to the point group C∞ν with its axis of symmetry (Z direction) along the (radial) direction of the poling field, leading to a linear electro-optic tensor with only five non-zero terms:r13 = r23, r33, and r42 = r51 [6,38]. The electric field applied through the Pt electrode for the RF-speed test is also radial, thus the EO coefficients available to characterize in this device are r13 and r33, under the TM0 mode and the TE0 mode, respectively, using the equation [6]
where the amplitude of the applied RF modulation field Emod is proportional to the applied voltage Vmod (both half peak-to-peak value). The average horizontal field across the Si3N4 core is 4.83 × 102 V/cm per volt applied, according to simulations (Fig. 2(d)), while the collateral vertical field is at least one order of magnitude smaller in the Si3N4 core (except at the corners). We ignore the effect of the collateral vertical field in the calculations of the EO coefficient. We calibrate Vmod for all frequencies of interest (see Supplementary Section 2 and Fig.S2). The loss from the high-speed probe as a function of working frequency is provided by its manufacturer. We expect a relatively negligible change of refractive index of the Si3N4 from the poling and therefore treat it as isotropic in our calculations (using data from ellipsometry for different wavelengths).5. Results and discussion
Using the methods described in the previous section, we measure the electro-optic coefficients of poled silicon nitride at various modulation frequencies from 50 MHz to 15 GHz. For the r33 component measured with the TE0 mode (Fig. 4(a)), the poled device demonstrates an enhancement across the entire frequency range of testing, up to 39 ± 7fm/V (measured on the same day of poling at 250 MHz). This corresponds to a χ(2) of 0.30pm/V according to [39]
where εi is the permittivity of the material. The measured r13 component (Fig. 4(b)) is enhanced at high modulation speeds (> 5 GHz). Meanwhile, in both cases, the poled device shows an enhanced high-speed response with a 3 dB bandwidth of at least 15 GHz, while the non-poled device has a 3 dB bandwidth of around 3 GHz (we call them the ‘fast response’ and the ‘slow response’, respectively). As a result, the enhancement of the EO coefficients is most significant in the high-frequency regime (≥ 5 GHz, the blue markers in Fig. 4(a) and Fig. 4(b)); a > 25X enhancement is measured for r33 at 7 GHz. Note that the EO response of the non-processed device (green markers in Fig. 4(a) and Fig. 4(b)) is no longer distinguishable from the background noise of the spectrum analyzer (typically -130dBm at 1 GHz, increasing for higher frequencies) as we increase the modulation frequency to higher than 7 GHz and the markers only represent the results when using the noise level as the measurement. Hence, the lower error limits of these points are all set to zero and the upper error limits of the corresponding enhancements are set to infinity. (A detailed discussion upon sources of uncertainty in the error bar is provided in Supplementary Section 5) It is reasonable to infer that the actual responses (enhancements) at these frequencies follow the decaying (rising) trend line in the lower frequency regime. We track the induced 15 GHz EO response for over one week (red and pink curves in Fig. 4(a-b), Fig. 4(c)) and observe no significant decay, suggesting that the effect of the poling is long lasting, and very likely permanent.Fig. 4. Measured EO coefficients r33 ((a), (d)) and r13 ((b), (e)) of the poled device (dark red, red and pink curves in (a-b)), non-processed device (green curves), and devices processed only with high temperature (red curves in (d-e)) or strong electric field (blue curves in (d-e)). Dashed lines in (a-b, d-e) represent estimated trend lines of corresponding values at higher frequencies. From the beginning of these dashed lines, the response in the spectrum analyzer could not be distinguished from the background noise. Therefore, the lower error bars of these points go all the way to zero (EO coefficients) or infinity (enhancement). Performance of the poled device is tracked over one week by averaging the relative response variation at all frequencies (e).
The improvement of the response speed out of the poling under high temperature confirms that the process induces a second-order nonlinearity in the Si3N4 core through the alignment of Si-N bonds and breaking of the centrosymmetry, since no other mechanism can provide such a high-speed electro-optic response. We build a physical model to estimate the induced χ(2) in Si3N4 (See Supplementary Section 6). The fact that the characterized r13 value is smaller than the r33 value by a factor of approximately 3 matches well with the analytical result of the model.
We also study the EO coefficients of devices that are processed only with a strong poling electric field or a high temperature (Fig. 4(d-e)). No clear change appears after the field is solely applied, yet the heating process reduces the as-deposited EO response. The r33 component is suppressed by a factor of 5 while the r13 component is completely indistinguishable from the background noise.
The large bandwidth of the measured fast response indicates that the RC limit of our device is at least 15 GHz, and therefore the slow response must originate from a different mechanism of slower speed. Based on the characterization above, we infer that the slow response results from carrier effects in the Si3N4 waveguide due to its positive charging caused by defects [40,41]. The speed limit of the carrier effect electro-optic response is reported to be in the level of 1 GHz [42] because of the finite carrier lifetime, which agrees with the measured slow response speed. Further discussion is provided in Supplementary Section 7. Reported passivation of such defects in Si3N4 and related effects through annealing at 500°C [43] or higher temperatures [44] also agrees with our observations (Fig. 4(d-e), red curves) and supports our conclusions. The deterioration of the EO response after annealing, together with our model, explains the slight decrease of the r13 component in the sub-GHz regime after the poling process (Fig. 4(b)): the slow response is mostly removed in the high temperature while the newly built fast response is just not as large in the low-frequency regime. Recent work [29] on the EO response (up to tens of kHz) in Si3N4 devices electrically poled at an elevated temperature (260°C, not high enough for defect passivation [44]) likely shares the same carrier-based mechanism. The reported χ(2) (0.084pm/V) in this work matches the magnitude of the measured slow response in our work (10fm/V EO coefficient corresponds to about 0.08pm/V, according to Eq.(3)).
In conclusion, we demonstrate that a long-lasting second-order nonlinearity can be induced in amorphous silicon nitride by poling the material with a strong electric field at high temperature. The induced second-order nonlinearity gives rise to a non-trivial Pockels electro-optic effect in the silicon nitride. This work paves the way to achieve active functions on the Si3N4 platform. Currently, our result is mainly limited by not reaching a high enough temperature during the poling, which means the Si-N bonds do not have the optimium mobility to move under the poling field. By developing more efficient poling approaches to overcome this, we estimate that it is possible to induce a χ(2) of 15pm/V or more through poling the silicon nitride film (See Supplementary Section 6). With this EO response silicon nitride could become a platform that combines high-speed modulation with low loss, and CMOS compatibility to enable monolithic integration of photonics and CMOS electronics for photonic integrated circuits (PICs), LIDAR, data communications, and quantum optics.
Funding
National Science Foundation (ECCS 1941213).
Acknowledgments
This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant NNCI-2025233). This work was performed in part at the University of Rochester Integrated Nanosystems Center (URnano). This work was performed in part at the Semiconductor and Microsystems Fabrication Laboratory (SMFL) at Rochester Institute of Technology. The authors appreciate the staff at these facilities for their help on equipment during the preparation of the samples.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are available from the corresponding author by reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. D. J. Blumenthal, R. Heideman, D. Geuzebroek, A. Leinse, and C. Roeloffzen, “Silicon nitride in silicon photonics,” Proc. IEEE 106(12), 2209–2231 (2018). [CrossRef]
2. P. Muñoz, G. Micó, L. A. Bru, D. Pastor, D. Pérez, J. D. Doménech, J. Fernández, R. Baños, B. Gargallo, R. Alemany, A. M. Sánchez, J. M. Cirera, R. Mas, and C. Domínguez, “Silicon nitride photonic integration platforms for visible, near-infrared and mid-infrared applications,” Sensors 17(9), 2088 (2017). [CrossRef]
3. D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New CMOS-compatible platforms based on silicon nitride and Hydex for nonlinear optics,” Nat. Photonics 7(8), 597–607 (2013). [CrossRef]
4. W. D. Sacher, Y. Huang, G.-Q. Lo, and J. K. S. Poon, “Multilayer silicon nitride-on-silicon integrated photonic platforms and devices,” J. Lightwave Technol. 33(4), 901–910 (2015). [CrossRef]
5. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010). [CrossRef]
6. R. W. Boyd, Nonlinear Optics (Academic Press, 2020).
7. S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giorgis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81(25), 4706–4708 (2002). [CrossRef]
8. S. Miller, Y.-H. D. Lee, J. Cardenas, A. L. Gaeta, and M. Lipson, “Electro-optic effect in silicon nitride,” in CLEO: 2015 (OSA, 2015), p. SF1G.4.
9. T. Ning, H. Pietarinen, O. Hyvärinen, J. Simonen, G. Genty, and M. Kauranen, “Strong second-harmonic generation in silicon nitride films,” Appl. Phys. Lett. 100(16), 161902 (2012). [CrossRef]
10. M. W. Puckett, R. Sharma, H.-H. Lin, M. Yang, F. Vallini, and Y. Fainman, “Observation of second-harmonic generation in silicon nitride waveguides through bulk nonlinearities,” Opt. Express 24(15), 16923 (2016). [CrossRef]
11. E. Francesco Pecora, A. Capretti, G. Miano, and L. Dal Negro, “Generation of second harmonic radiation from sub-stoichiometric silicon nitride thin films,” Appl. Phys. Lett. 102(14), 141114 (2013). [CrossRef]
12. A. Kitao, K. Imakita, I. Kawamura, and M. Fujii, “An investigation into second harmonic generation by Si-rich SiNx thin films deposited by RF sputtering over a wide range of Si concentrations,” J. Phys. D: Appl. Phys. 47(21), 215101 (2014). [CrossRef]
13. J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19(12), 11415 (2011). [CrossRef]
14. T. Ning, H. Pietarinen, O. Hyvärinen, R. Kumar, T. Kaplas, M. Kauranen, and G. Genty, “Efficient second-harmonic generation in silicon nitride resonant waveguide gratings,” Opt. Lett. 37(20), 4269 (2012). [CrossRef]
15. A. Billat, D. Grassani, M. H. P. Pfeiffer, S. Kharitonov, T. J. Kippenberg, and C.-S. Brès, “Large second harmonic generation enhancement in Si3N4 waveguides by all-optically induced quasi-phase-matching,” Nat. Commun. 8(1), 1016 (2017). [CrossRef]
16. M. A. G. Porcel, J. Mak, C. Taballione, V. K. Schermerhorn, J. P. Epping, P. J. M. van der Slot, and K.-J. Boller, “Photo-induced second-order nonlinearity in stoichiometric silicon nitride waveguides,” Opt. Express 25(26), 33143 (2017). [CrossRef]
17. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “Efficient photoinduced second-harmonic generation in silicon nitride photonics,” Nat. Photonics 15(2), 131–136 (2021). [CrossRef]
18. D. Grassani, M. H. P. Pfeiffer, T. J. Kippenberg, and C.-S. Brès, “Second- and third-order nonlinear wavelength conversion in an all-optically poled Si3N4 waveguide,” Opt. Lett. 44(1), 106 (2019). [CrossRef]
19. E. Nitiss, O. Yakar, A. Stroganov, and C.-S. Brès, “Highly tunable second-harmonic generation in all-optically poled silicon nitride waveguides,” Opt. Lett. 45(7), 1958 (2020). [CrossRef]
20. E. Nitiss, J. Hu, A. Stroganov, and C.-S. Brès, “Optically reconfigurable quasi-phase-matching in silicon nitride microresonators,” Nat. Photonics 16(2), 134–141 (2022). [CrossRef]
21. S. Jin, L. Xu, H. Zhang, and Y. Li, “LiNbO3 thin-film modulators using silicon nitride surface ridge waveguides,” IEEE Photonics Technol. Lett. 28(7), 736–739 (2016). [CrossRef]
22. J. E. Ortmann, F. Eltes, D. Caimi, N. Meier, A. A. Demkov, L. Czornomaz, J. Fompeyrine, and S. Abel, “Ultra-low-power tuning in hybrid barium titanate–silicon nitride electro-optic devices on silicon,” ACS Photonics 6(11), 2677–2684 (2019). [CrossRef]
23. K. Alexander, J. P. George, J. Verbist, K. Neyts, B. Kuyken, D. Van Thourhout, and J. Beeckman, “Nanophotonic Pockels modulators on a silicon nitride platform,” Nat. Commun. 9(1), 3444 (2018). [CrossRef]
24. A. Hermans, M. Van Daele, J. Dendooven, S. Clemmen, C. Detavernier, and R. Baets, “Integrated silicon nitride electro-optic modulators with atomic layer deposited overlays,” Opt. Lett. 44(5), 1112 (2019). [CrossRef]
25. B. A. Block, T. R. Younkin, P. S. Davids, M. R. Reshotko, P. Chang, B. M. Polishak, S. Huang, J. Luo, and A. K. Y. Jen, “Electro-optic polymer cladding ring resonator modulators,” Opt. Express 16(22), 18326 (2008). [CrossRef]
26. C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015). [CrossRef]
27. I. Datta, S. H. Chae, G. R. Bhatt, M. A. Tadayon, B. Li, Y. Yu, C. Park, J. Park, L. Cao, D. N. Basov, J. Hone, and M. Lipson, “Low-loss composite photonic platform based on 2D semiconductor monolayers,” Nat. Photonics 14(4), 256–262 (2020). [CrossRef]
28. B. Wang, Y. Ji, L. Gu, L. Fang, X. Gan, and J. Zhao, “High-efficiency second-harmonic and sum-frequency generation in a silicon nitride microring integrated with few-layer GaSe,” ACS Photonics 9(5), 1671–1678 (2022). [CrossRef]
29. B. Zabelich, E. Nitiss, A. Stroganov, and C.-S. Brès, “Linear electro-optic effect in silicon nitride waveguides enabled by electric-field poling,” ACS Photonics 9(10), 3374–3383 (2022). [CrossRef]
30. J. B. Khurgin, T. H. Stievater, M. W. Pruessner, and W. S. Rabinovich, “On the origin of the second-order nonlinearity in strained Si–SiN structures,” J. Opt. Soc. Am. B 32(12), 2494 (2015). [CrossRef]
31. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994). [CrossRef]
32. R. A. Myers, N. Mukherjee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16(22), 1732 (1991). [CrossRef]
33. D. Z. Anderson, V. Mizrahi, and J. E. Sipe, “Model for second-harmonic generation in glass optical fibers based on asymmetric photoelectron emission from defect sites,” Opt. Lett. 16(11), 796 (1991). [CrossRef]
34. K. Luke, Y. Okawachi, M. R. E. Lamont, A. L. Gaeta, and M. Lipson, “Broadband mid-infrared frequency comb generation in a Si3N4 microresonator,” Opt. Lett. 40(21), 4823 (2015). [CrossRef]
35. C. M. S. Rauthan and J. K. Srivastava, “Electrical breakdown voltage characteristics of buried silicon nitride layers and their correlation to defects in the nitride layer,” Mater. Lett. 9(7-8), 252–258 (1990). [CrossRef]
36. J. Cardenas, M. Zhang, C. T. Phare, S. Y. Shah, C. B. Poitras, B. Guha, and M. Lipson, “High Q SiC microresonators,” Opt. Express 21(14), 16882 (2013). [CrossRef]
37. M. Bahadori, L. L. Goddard, and S. Gong, “Fundamental electro-optic limitations of thin-film lithium niobate microring modulators,” Opt. Express 28(9), 13731 (2020). [CrossRef]
38. P. N. Prasad and D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, 1991).
39. A. Yariv, Quantum Electronics (John Wiley & Sons, 1989).
40. S. Sharif Azadeh, F. Merget, M. P. Nezhad, and J. Witzens, “On the measurement of the Pockels effect in strained silicon,” Opt. Lett. 40(8), 1877 (2015). [CrossRef]
41. C. Schriever, F. Bianco, M. Cazzanelli, M. Ghulinyan, C. Eisenschmidt, J. de Boor, A. Schmid, J. Heitmann, L. Pavesi, and J. Schilling, “Second-order optical nonlinearity in silicon waveguides: inhomogeneous stress and interfaces,” Adv. Opt. Mater. 3(1), 129–136 (2015). [CrossRef]
42. M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “High-frequency electro-optic measurement of strained silicon racetrack resonators,” Opt. Lett. 40(22), 5287 (2015). [CrossRef]
43. I. Olivares, T. Angelova, and P. Sanchis, “On the influence of interface charging dynamics and stressing conditions in strained silicon devices,” Sci. Rep. 7(1), 7241 (2017). [CrossRef]
44. F. L. Martınez, I. Martil, G. Gonzalez-Dıaz, B. Selle, and I. Sieber, “Influence of rapid thermal annealing processes on the properties of SiNx:H films deposited by the electron cyclotron resonance method,” J. Non-Cryst. Solids 227-230, 523–527 (1998). [CrossRef]
