## Abstract

We investigate the influence of the wavelength, within the 1.3*μ*m–1.63*μ*m range, on the second-order optical nonlinearity in silicon waveguides strained by a silicon nitride (Si_{3}N_{4}) overlayer. The effective second-order optical susceptibility
$\overline{{\chi}_{xxy}^{(2)}}$ evolutions have been determined for 3 different waveguide widths 385 nm, 435 nm and 465 nm and it showed higher values for longer wavelengths and narrower waveguides. For *w _{WG}* = 385 nm and

*λ*= 1630 nm, we demonstrated $\overline{{\chi}_{xxy}^{(2)}}$ as high as 336 ± 30 pm/V. An explanation based on the strain distribution within the waveguide and its overlap with optical mode is then given to justify the obtained results.

© 2014 Optical Society of America

## 1. Introduction

Silicon-based photonics has generated a strong interest in recent years, mainly for optical telecommunications and optical interconnects in microelectronic circuits. The main rationales of silicon photonics are the reduction of photonic system costs and the increase of the number of functionalities on the same integrated chip by combining photonics and electronics together [1].

Numerous optoelectronic building blocks including Laser sources, germanium photodetectors and optical modulators have been experimentally demonstrated for the development of silicon photonic circuits. Among them, optical silicon modulators are considered as one of the main components. However, bulk silicon possesses a centro-symmetry lattice. This symmetry inhibits the existence of nonzero elements in the second-order nonlinear susceptibility *χ*^{(2)} tensor and consequently disables linear electro-optic modulation of light based on Pockels effect [2]. Modulation based on this effect can possibly be driven at much lower power than those based on plasma dispersion or Kerr effects and its speed is not limited by charge mobility or charge recombination times [3].

Nevertheless, it has been demonstrated that the centro-symmetry of silicon can be broken by using a stress overlayer to induce strain in the crystal underneath [4–8] and since then strained silicon has gained considerable interest as a suitable material for optical nonlinear phenomena. Indeed second harmonic generation has been demonstrated in silicon strained by a Si_{3}N_{4} overlayer [7]. Furthermore, different research groups have successfully shown light modulation in strained silicon based on Pockels effect [4, 8–10], which makes it a promising material for high speed and low power optical modulation in silicon. Its potential for optical modulation and nonlinear optics, has motivated the pursuit of understanding the response of these effects to different silicon waveguide geometries and strain distributions [9, 11]. Indeed, Chmielak et al. [9] have recently studied the influence of the waveguide width on the nonlinear effects at a wavelength close to 1550 nm and were able to show that by decreasing the waveguide width, the efficiency of Pockels modulation could be highly improved and a maximum value of effective *χ*^{(2)} = 190 pm/V was reported. However no study has been made regarding how the wavelength influences the performance of Pockels based devices in strained silicon, in particular over the NIR range of the telecom windows from 1.3 *μ*m and 1.65 *μ*m. This knowledge is of particular importance to understand the response of a general strained silicon structure to different wavelengths in any telecom and datacom sectors.

In this paper, we investigate how different waveguide widths (*w _{WG}*) and different wavelengths (

*λ*) in the NIR range influence the Pockels electro-optic effect in Silicon-On-Insulator (SOI) strip waveguides strained by a Si

_{3}N

_{4}overlayer. As result, we present the measurements of the ${\chi}_{xxy}^{(2)}$ coefficient as a function of the wavelength in the range 1.3

*μ*m–1.63

*μ*m for three different

*w*. The results show a strong impact of both

_{WG}*λ*and

*w*on the final effective ${\chi}_{xxy}^{(2)}$ component and a justification based on the strain distribution inside the waveguide is proposed to justify the obtained results.

_{WG}## 2. Device and fabrication

#### 2.1. Stress study of the cladding layer

The silicon waveguides are surrounded by a cladding layer whose purpose is not only to induce strong stress, but also to completely isolate the optical mode from the metallic electrodes deposited on top. The mechanical stress is applied by a highly stressed-Si_{3}N_{4} layer in direct contact with the silicon waveguide and the good confinement of the mode can be supported by depositing SiO_{2} on top of the stress layer. However, since the maximization of stress is a determining factor, we analysed the effect of a shallow layer of SiO_{2} on the overall stress in the cladding layer.

The stress (*σ*) was obtained by measuring the change in the curvature radius of a Si wafer before and after the deposition of the materials using a commercial tool (FSM500TC, Frontier Semiconductor Measurements) based on the Stoney’s formula [12]. The results showed that a single 250nm-thick Si_{3}N_{4} layer presented *σ* ∼ 1200±25 MPa which dropped to *σ* ∼ 800±25 MPa when 250 nm of SiO_{2} was deposited on top. This shows that the SiO_{2} layer considerably reduces the overall stress in the silicon wafer, which can be attributed to some mechanical relaxation occurring in the interface between the two materials. Therefore, we used a single 700 nm layer of Si_{3}N_{4} (with *σ* ∼ 1000 ± 25 MPa) in order to achieve maximum stress and consequent stronger strain in silicon, while ensuring complete isolation of the optical mode from the covering metallic electrodes.

#### 2.2. Fabrication

Our device consisted of a set of asymmetric (Δ*L* = 50*μ*m) Mach-Zehnder Interferometers (MZI) with three different waveguide widths fabricated on a 260nm-thick, (100) oriented-top SOI substrate, with 2 *μ*m Buried Oxide (BOX), total length of 8 mm and two MZI arm lengths *L*, 3.3 mm and 5 mm. The layout was defined using e-beam lithography, the silicon waveguides were structured by Inductively Coupled Plasma (ICP) etching and the final measured waveguide widths after etching were 385nm, 435nm and 465nm, respectively. The dimensions of the waveguides were chosen so that they work as single TE-mode for all range of wavelengths in the analysed NIR spectrum (1.3*μ*m–1.63*μ*m)

A 700nm-thick layer of Si_{3}N_{4} was then deposited using Plasma Enhanced Chemical Vapor Deposition (PECVD). The conditions of this deposition were such that a high intrinsic stress was generated in the Si_{3}N_{4} layer, which is the source of strain in the silicon crystal. Finally, 10/150nm of Cr/Au electrodes were evaporated directly on this layer for the generation of the electric field required to induce the Pockels phase shift. Figure 1 shows a schematic cross section and top view of the waveguides in the MZI and the corresponding Scanning Electron Microscopy (SEM) picture. It can be seen that Si_{3}N_{4} uniformly covers the Si waveguide, which is oriented as **x̂** = [1̄10], **ŷ** = [100] and **ẑ** = [110]. The visible cracks in the Si_{3}N_{4} and Si structures were caused after cleaving and can be attributed to the high strain inside these materials.

## 3. Electro-optic characterization and results

For the electro-optics characterization, we applied a variable DC electric field **F** to the MZI branches, by changing the voltage *V* applied to electrodes from −30V to 30V in a push-pull configuration. The electrodes themselves were designed as shown in Fig. 1(a) so that the overall **F** electric field in the waveguide is oriented in the *y* direction (Fig. 1). The mean electric field inside the silicon core is similar (up to 1% difference) in the three waveguides and is given by *F _{y}/V* ≃ 17.5 × 10

^{3}m

^{−1}, calculated with a finite elements software.

We measured the shift Δ*λ _{r}* of the MZI resonance at wavelength

*λ*as a function of the applied voltage, as shown in Fig. 2. The good linearity of the graph

_{r}*λ*(

_{r}*V*) over positive and negative voltages, proves that the change in refractive index is due to Pockels effect. The slight change in transmission of the resonances when a voltage is applied, noticeable in Fig. 2, is attributed to a slight unbalance of the strained MZI coupler which is sensitive to any bias applied to it. The change of the effective index Δ

*n*of the propagating TE mode is given by Δ

_{eff}*n*(

_{eff}*λ*) = Δ

_{r}*λ*/(2

_{r}λ_{r}*L*×

*δλ*), where

_{r}*δλ*is the Free Spectral Range at

_{r}*λ*, can then be extracted for each of the available resonances

_{r}*λ*between 1300nm and 1630nm, allowing us to fully characterize the Pockels phenomenon.

_{r}The procedure described above was done for each of the studied *w _{WG}* and for

*L*= 3.3mm and

*L*= 5 mm and confirmed that Δ

*n*was independent of the device length

_{eff}*L*. The values Δ

*n*(

_{eff}*λ*), for each

*λ*presented in Fig. 3, correspond to the statistical mean of the values Δ

*n*(

_{eff}*λ*) measured in the different experiments, for all

_{r}*λ*within

_{r}*λ*± 10 nm. The error bars correspond to the statistical standard deviation of the data set used for each

*λ*.

#### 3.1. Second order nonlinear susceptibility

The change of the effective index of the guided mode, which is the quantity that we can extract directly from the experiments, is an overall effect which depends on the different contributions from the components of the *χ*^{(2)} tensor. It can be shown that for a general high-index contrast waveguide, oriented as that in Fig. 1, the change in effective index due to Pockels effect is given by [13]

*c*is the speed of light in vacuum,

*ε*

_{0}is the vacuum dielectric permittivity,

**E**and

**H**are the optical electric and magnetic fields, respectively, which propagate in the

*z*direction and

**F**is the DC electric field applied by the electrodes, which is oriented in the

**ŷ**direction, as referred in section 3. The integral of the numerator is done over the waveguide silicon core.

The most important component of the fundamental TE-mode oriented as in Fig. 1 is *E _{x}*, hence it is a good approximation to neglect the contributions of the other components in Eq. (1), which leaves
${\chi}_{xxy}^{(2)}$ as the only relevant nonzero component of the

*χ*

^{(2)}tensor [10]. The optical nonlinearity generated at the interface Si/SiN, where the centrosymmetry is broken, can be neglected as it is several orders of magnitude smaller than that induced by strain [14].

The *effective second order suceptibility*,
$\overline{{\chi}^{(2)}}$, is defined as the value *χ*^{(2)} would have if it was constant inside the waveguide. Equation (1) is then simplified to
$\mathrm{\Delta}{n}_{\mathit{eff}}=\overline{{\chi}_{xxy}^{(2)}}F\gamma $, where *γ* is a specific *mode confinement* for this particular situation, given by

The mode confinement *γ* can be easily computed for different *w _{WG}* and

*λ*using a finite element software. In our structures, its values range between 0,192 and 0,256, growing with

*w*and decreasing with

_{WG}*λ*. The previous equations allow us to extract the values of the $\overline{{\chi}_{xxy}^{(2)}}$ from Δ

*n*data plotted in Fig. 3(a), whose results are presented in Fig. 3(b). There, a clear trend can be detected: the effective second order nonlinear susceptibility in our strained silicon waveguides is higher for longer wavelengths and lower widths. Consequently, the highest value $\overline{{\chi}_{xxy}^{(2)}}=336\pm 30\hspace{0.17em}\text{pm}/\text{V}$ was achieved for

_{eff}*w*= 385 nm and

_{WG}*λ*= 1630 nm, corresponding to Δ

*n*= (3.48 ± 0.12) × 10

_{eff}^{−5}under 30V.

## 4. Discussion of the results

Despite the lack of general proof for this claim available in the literature yet, it has been widely accepted [7, 9, 15] that the second order nonlinear effects in strained silicon are caused by the variations of strain i.e. *strain gradients* inside the crystal. As a consequence, the resulting optical nonlinear effect is due to the interaction between the strain gradients and the electric field of the optical mode propagating in the waveguide.

Simulations have shown that the highest strain gradients are located closer to the edges of the waveguide [9] and hence the narrower the waveguide, the closer to its center these gradients get. This leads to higher overlap between the strain gradients and the optical mode, which is preferably located in the waveguide core (see Fig. 1(c)), resulting in higher
$\overline{{\chi}_{xxy}^{(2)}}$ values for narrower waveguides, as reported in Fig. 3. A more extensive study on the influence of the waveguide width on the effective *χ*^{(2)} for a fixed wavelength has been reported by Chmielak et al. [9], whose trend for *λ* ∼ 1550 nm is in good agreement with our results.

The same reasoning can be used to explain the wavelength dependence presented in Fig. 3. By increasing the wavelength, the TE fundamental mode spreads in the waveguide core, which leads to higher optical field intensity at the edges of the waveguide, where the strain gradients are mainly located. Consequently, the optical nonlinear effects induced by the strain should be stronger, which explains an higher effective *χ*^{(2)} for longer wavelengths.

## 5. Conclusion

To sum up, we analysed the wavelength dependence of the Pockels electro-optic modulation in silicon strained by a Si_{3}N_{4} stress overlayer in a wide range of the NIR spectra and for different *w _{WG}*. The presented results and analysis have clearly shown that the effective
${\chi}_{xxy}^{(2)}$ generated inside the waveguide increases with wavelength and decreases with waveguide width. Consequently, its highest value was measured for the narrowest waveguide (385 nm wide) and longest wavelength (1630 nm) as being
$\overline{{\chi}_{xxy}^{(2)}}=336\pm 30\hspace{0.17em}\text{pm}/\text{V}$, which is, for our knowledge, the highest value of
$\overline{{\chi}^{(2)}}$ reported in this type of structures.

These results, give us meaningful information to improve our understanding on the mechanisms by which the strain interacts with the optical electric field to induce second order non-linear effects. Moreover, the analysis of the effect of wavelength on the effective *χ*^{(2)} in silicon structures in the wide NIR range 1.3*μ*m – 1.63*μ*m opens a promising route for the implementation of strained silicon for Pockels modulation in any band of the telecom sector.

Finally, our strained silicon waveguides the *electro-optic coefficient*, given by
${r}_{ijk}={\chi}_{ijk}^{(2)}/{n}^{4}$ [16], present a maximum value of *r _{xxy}* ∼ 2.2 pm/V. This number is still around one order of magnitude smaller than that of LiNbO

_{3}, the preferable material for electro-optic modulation, reported as

*r*

_{33}∼ 33 pm/V, but comparable with GaAs or InP which have

*r*

_{41}∼ 1.5 pm/V [17]. This shows that further enhancement of the nonlinear effects and device optimization are required to get closer to the performances of LiNbO

_{3}-based devices and the improvements reported here prove to be a relevant step towards efficient optical modulation in strained silicon.

## Acknowledgments

The authors would like to acknowledge Frederic Boeuf from STMicroelectronics for financial support.

## References and links

**1. **J. M. Fedeli, L. D. Cioccio, D. Marris-Morini, L. Vivien, R. Orobtchouk, P. Rojo-Romeo, C. Seassal, and F. Mandorlo, “Development of silicon photonics devices using microelectronic tools for the integration on top of a cmos wafer,” Adv. Optical Technol. **2008**, 412518 (2008). [CrossRef]

**2. **J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photon. **4**, 535–544 (2010). [CrossRef]

**3. **G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photon. **4**, 518–526 (2010). [CrossRef]

**4. **R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature **441**, 199–202 (2006). [CrossRef] [PubMed]

**5. **C. Schriever, C. Bohley, and R. B. Wehrspohn, “Strain dependence of second-harmonic generation in silicon,” Opt. Lett. **35**, 273–275 (2010). [CrossRef] [PubMed]

**6. **C. Schriever, C. Bohley, J. Schilling, and R. B. Wehrspohn, “Strained silicon photonics,” Materials **5**, 889–908 (2012). [CrossRef]

**7. **M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, “Second-harmonic generation in silicon waveguides strained by silicon nitride,” Nat. Mater. **11**, 148–154 (2012). [CrossRef]

**8. **B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express **19**, 17212–17219 (2011). [CrossRef] [PubMed]

**9. **B. Chmielak, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Waldow, and H. Kurz, “Investigation of local strain distribution and linear electro-optic effect in strained silicon waveguides,” Opt. Express **21**, 25324–25332 (2013). [CrossRef] [PubMed]

**10. **M. W. Puckett, J. S. T. Smalley, M. Abashin, A. Grieco, and Y. Fainman, “Tensor of the second-order nonlinear susceptibility in asymmetrically strained silicon waveguides: analysis and experimental validation,” Opt. Lett. **39**, 1693–1696 (2014). [CrossRef] [PubMed]

**11. **F. Bianco, K. Fedus, F. Enrichi, R. Pierobon, M. Cazzanelli, M. Ghulinyan, G. Pucker, and L. Pavesi, “Two-dimensional micro-Raman mapping of stress and strain distributions in strained silicon waveguides,” Semicond. Sci. Technol. **27**, 085009 (2012). [CrossRef]

**12. **A. Tarraf, J. Daleiden, S. Irmer, D. Prasai, and H. Hillmer, “Stress investigation of PECVD dielectric layers for advanced optical MEMS,” J. Micromech. Microeng. **14**, 317–323 (2004). [CrossRef]

**13. **A. W. Snyder and J. D. Love, *Optical Waveguide Theory*, 1st ed. (Chapman and Hall, 1983), Chap. 31.

**14. **M. Falasconi, L. Andreani, A. Malvezzi, M. Patrini, V. Mulloni, and L. Pavesi, “Bulk and surface contributions to second-order susceptibility in crystalline and porous silicon by second-harmonic generation,” Surf. Sci. **481**, 105–112 (2001). [CrossRef]

**15. **S. V. Govorkov, V. I. Emel’yanov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, V. V. Yakovlev, and R. V. Khokhlov, “Inhomogeneous deformation of silicon surface layers probed by second-harmonic generation in reflection,” J. Opt. Soc. Am. B **6**, 1117–1124 (1989). [CrossRef]

**16. **M. Izdebski, W. Kucharczyk, and R. E. Raab, “On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities,” J. Opt. A: Pure Appl. Opt. **6**, 421–424 (2004). [CrossRef]

**17. **G. Li and P. Yu, “Optical intensity modulators for digital and analog applications,” J. Lightwave Technol. **21**, 2010–2030 (2003). [CrossRef]