We present detailed investigations of the local strain distribution and the induced second-order optical nonlinearity within strained silicon waveguides cladded with a Si3N4 strain layer. Micro-Raman Spectroscopy mappings and electro-optic characterization of waveguides with varying width wWG show that strain gradients in the waveguide core and the effective second-order susceptibility χ(2)yyz increase with reduced wWG. For 300 nm wide waveguides a mean effective χ(2)yyz of 190 pm/V is achieved, which is the highest value reported for silicon so far. To gain more insight into the origin of the extraordinary large optical second-order nonlinearity of strained silicon waveguides numerical simulations of edge induced strain gradients in these structures are presented and discussed.
© 2013 Optical Society of America
Recently, the induction of second-order nonlinear optical effects in silicon nanophotonic structures has gained considerable interest for a wide field of applications [1–14]. Due to the inherent absence of a second-order susceptibility in silicon, resulting from its centro-symmetric crystal structure, different approaches of functionalizing silicon waveguides have been theoretically and experimentally investigated. For example filling silicon slot waveguides with nonlinear optic polymers for high-speed electro-optic modulators [2–5] or modifying the silicon surface with HBr chemistry for terahertz emitters  are efficient methods to create useable second-order nonlinear optical effects in silicon structures. Another prominent way is to induce asymmetric mechanical strain inside silicon waveguides by means of a silicon nitride (Si3N4) strain layer. This concept was first presented by Jacobsen et al  and Fage-Pedersen et al  in 2006. Thereafter several groups gave attention to this approach [9–13]. Recently Avrutsky and Soref  presented a theoretical study on phase-matched second harmonic generation (SHG) in strained silicon waveguides with a Si3N4 strain layer. Furthermore, Cazzanelli et al  realized SHG in Si3N4 cladded multi-mode waveguides, however, without demonstrating phase matching. Also Bianco et al  investigated the stress and strain distributions in these waveguides using micro-Raman mappings conducted on the waveguide facet.
In our previous work we presented for the first time a fully integrated electro-optic modulator based on a linear electro-optic (Pockels) effect in silicon rib waveguides coated with a Si3N4 strain layer . With this method an effective χ(2)yyz of 122 pm/V for the strained silicon waveguide was demonstrated. Furthermore, we used spatially resolved micro-Raman and terahertz difference frequency generation experiments to identify the vertical waveguide edges as regions of pronounced strain asymmetry and dominant second-order optical activity. Through this we could confirm the close relation between strain gradients and the electro-optic effect in silicon.
To make the concept of the strained silicon waveguide competitive with the established LiNbO3 modulator technology further improvement of the structure is necessary. In particular the reduction of Vπ L ≈110 Vcm (55 Vcm using electrodes in push-pull configuration ) to a value that can be supplied by state-of-the-art high-speed drivers at reasonable length is essential. To accomplish this, two approaches are viable: First, the efficiency of the field coupling could be improved by optimizing the electrode design so that for a given modulation voltage the induced electric field inside the waveguide is increased. Second, the effective χ(2)yyz of the waveguide could be enhanced by optimizing the strain profiles within the structure. For this a fundamental understanding of the strain generation is critical.
In this work we investigate the influence of the waveguide geometry, especially the waveguide width wWG, on the strain distribution and the effective χ(2)yyz in strained silicon rib waveguides. Spatially resolved Micro-Raman Spectroscopy (MRS) mappings conducted across waveguides with different wWG reveal large strain gradients in the vicinity of the waveguide sidewalls. Electro-optic characterization of Mach-Zehnder interferometer (MZI) based modulators with varying wWG show a strong impact of wWG on the effective χ(2)yyz. In complementary numerical simulations the conditions for edge induced strain gradients are investigated.
2. Experiments and results
For the experiments presented in this work MZI modulators based on strained silicon rib waveguides with varying width wWG, ranging from 300 nm to 2 μm, were fabricated. In Fig. 1 a schematic cross-section (a) and top view (b) image of the investigated device is shown. On these devices MRS mappings as well as electro-optic characterizations were conducted. The fabrication process and the experimental methodologies used in this work are, unless otherwise stated, identical to previous experiments and are described in detail in .
2.1 Spatially resolved Raman mappings
The first experimental investigation focuses on the analysis of the local strain distribution inside strained silicon waveguides with different wWG as the asymmetric distortion of the silicon lattice is responsible for the generation of the χ(2) in silicon. To get information about the strain distribution MRS line scans were conducted across different waveguides in y-direction. At each measuring point the exact spectral position and the full width at half maximum (FWHM) of the silicon Raman peak close to ωc = 520.5 cm−1 is analyzed. Deviations in the spectral position of the silicon Raman peak Δω reveal the type of predominant strain in the probed volume while an increase of the FWHM is interpreted as inhomogeneity, i.e. gradient in the strain distribution . Before and after each line scan the spectrum of a bulk silicon reference sample is measured and only the relative Raman shift Δω is considered. Thereby any influence of the calibration of the Raman Spectrometer is eliminated. The spatial resolution of the setup is given by the diameter of the gaussian-shaped laser spot which is approximately 800 nm (1/e). Together with a step size of 100 nm this results in a spatial overlap of the probed material volume for adjacent measuring points. Figure 2 displays MRS line scans measured on 2 μm and 500 nm wide waveguides including the relative Raman shift (black lines) and the FWHM of the silicon peak (blue line) on strained waveguides (solid line) and equivalent unstrained waveguides without Si3N4 layer (dotted line). The measurements on the reference waveguides show typical values for unstrained silicon (Δω ≈0, FWHM ≈3 cm−1) at the waveguide core and slight compressive strain in the slab region that originates from the Si/SiO2 interface. In case of the strained waveguides the Raman shift close to the waveguide sidewalls considerably changes its sign. Consequently the type of the predominant strain changes from compressive (positive Δω) to tensile (negative Δω) back to compressive. As can be seen from the FWHM this asymmetric strain profile is reflected in a prominent broadening of the Raman peak at the edges of the waveguides. Hence, the highest second-order nonlinearity is expected to be located in these regions.
For the optimization of the strained waveguide structure regarding an electro-optic modulator application the overlap of areas with high χ(2)yyz with the modal field distribution inside the waveguide is crucial. Therefore the strain distribution at the center of the waveguide (y = 0), where the highest mode field amplitudes are located, is of particular interest. In the 2 μm wide strained waveguide the core region features a maximum Δω, but at the same time a minimum of the Raman peak width. This indicates a strong but rather homogeneous strain profile and therefore points to a small effective χ(2)yyz. On the contrary the 500 nm wide strained waveguide exhibits a significant broadening of the Raman peak in the core region. Thus, in this waveguide a much higher effective χ(2)yyz can be expected. The asymmetry in the measurements conducted on the 500 nm wide waveguides is caused by the limited resolution of the micro-Raman system. To experimentally verify the influence of wWG on the nonlinear optical properties of strained silicon waveguides electro-optic measurements were conducted in the next step.
2.2 Electro-optic characterization
The electro-optic characterization is performed by measuring the spectral transfer functions (STF) of different MZI modulator devices in TE polarization at the C-band wavelength window under varied modulation voltages . From the STF the induced phase shift and thus the change of the effective waveguide index Δneff is extracted. Taking into account the specific confinement factor for high-index-contrast waveguides  the index change ΔnSi of the silicon part of the waveguide structure is calculated. The electric modulation field EMod inside the silicon waveguide is determined using numerical simulation. Finally the effective second order susceptibility of the silicon part of the waveguide structure χ(2)yyz = nSi·ΔnSi/EMod is calculated. In our evaluation of the χ(2) the measured phase shift is fully attributed to the strain induced Pockels effect inside the silicon waveguides and as in  parasitic carrier-based effects are neglected. Even though no detectable current (I < 1 nA) is observed during the measurements and the number of residual charges due to a background doping of 1·1015/cm3 is low possible carrier-based contributions to the observed phase shift are subject to further investigations.
The investigated modulators are identical except for the waveguide width wWG below the electrodes. The center electrode has a width of 70 µm while the side electrodes are 150 µm wide. Electrodes are spaced by 50 µm and have a length of 1.73 cm or 0.76 cm. The insertion loss of the modulators ranges from 43 dB to 38 dB.
Figure 3 shows the extracted χ(2)yyz values plotted versus wWG. A significant enhancement of the effective χ(2)yyz for reduced waveguide widths can clearly be identified. In the smallest waveguides (wWG = 300 nm) a mean effective χ(2)yyz of 190 pm/V was achieved, which represents a new record for a silicon structure. In contrast to that the broadest waveguides (wWG = 500 nm) exhibit a mean effective χ(2)yyz of only 75 pm/V. The χ(2)yyz values presented here are in good agreement with our previous experiments and confirm the conclusions stated in the preceding paragraph. For comparison the largest χ(2) value of LiNbO3 is 360 pm/V . Apart from the material properties of the silicon waveguides the Vπ L (without push-pull electrodes) of the modulator devices were calculated. The mean values shown in the inset of Fig. 3 range from 89 Vcm for modulators with wWG = 300 nm to 161 Vcm for modulators with wWG = 500 nm. Note that although the confinement factor is reduced in smaller waveguides, i.e. less light is guided in the silicon part of the waveguide structure an overall improvement in Vπ L could be achieved.
2.3 Simulation of strain profiles
In addition to the mapping of Raman and electro-optic responses numerical process simulations were performed to obtain information on the strain distribution within the strained silicon waveguides. The strain profiles after processing are simulated using the Sentaurus Process tool by Synopsys. The quasi-static equations of force equilibrium are solved to calculate the stress and strain profile within the structure. The simulated volume has been chosen large enough so that the impact of the Dirichlet boundary conditions on the calculated stress can be neglected. Si3N4 is treated as a purely viscous material while silicon is assumed to be elastic. As the viscosity of Si3N4 depends on the temperature the thermal history is important for the calculation of the stress and strain profiles. It is also crucial for a realistic simulation to take into account the temperature dependence of the expansion coefficients for both Si3N4 and silicon. Here we use values reported in the literature .
The actual growth process of the Si3N4 layer is not simulated. Instead a 2D scale model (Fig. 4(b)) of the waveguide cross section is created based on scanning electron microscope (SEM) images of the real structure (Fig. 4(a)). With this model the temporal evolution of the temperature within the waveguide structure during the rapid thermal annealing step is calculated. In particular the quench from maximum temperature (1050° C) to room temperature is taken into account. Another relevant factor for the simulation is the initial stress that is induced during the Si3N4 deposition before the annealing step. From earlier Raman experiments we know that the Si3N4 covered silicon waveguides features tensile stress directly after deposition. To consider this we have introduced a complementary compressive stress in the Si3N4 layer with a magnitude of 1 GPa. The simulation result is a 2D strain profile of the waveguide in the y-z-plane.
For this study we focus on the strain component εyy because the gradient δεyy/δz generates a polarization in the silicon crystal that is aligned in z-direction  parallel to the electric modulation field. Therefore we assume this gradient along the z-axis is the main origin of the χ(2)yyz in the strained silicon waveguides. Furthermore the strain component εyy is also probed in the MRS measurements. Figure 4(a) shows a cross-section SEM image of the investigated structure: a silicon rib waveguide cladded with a 350 nm thick Si3N4 strain layer. Due to the fact that a low temperature plasma-enhanced chemical vapor deposition process is used to deposit the Si3N4, layer growth takes place on all faces of the waveguide simultaneously. The resulting Si3N4 layer is therefore not planar but features pronounced gaps located close to the waveguide edges. These gaps strongly influence the formation of the strain profile within the silicon waveguide as they interrupt the Si3N4 layer and lead to a local strain relaxation. Thus this feature is also implemented in the simulation model shown in Fig. 4(b). Figure 5 displays multiple simulated strain profiles. In this graphic presentation blue colors indicate compressive strain while red colors indicate tensile strain in the crystal lattice ranging from εyy = −1⋅10−3 to + 1⋅10−2. To validate the simulations several parameter variations were carried out.
First the influence of the gap in the Si3N4 layer on the strain distribution is examined. As it is not possible to clearly identify the exact depth of the gap from the SEM image we have conducted strain simulations on identical silicon waveguides with different Si3N4 gaps. Three exemplary results of this investigation are shown in Figs. 5(a)-5(c). In case of a closed Si3N4 layer without a gap the strain distribution inside the silicon waveguide is nearly homogenous. This changes dramatically if a gap is included. Depending on the depth of the gap pronounced strain gradients with strong tensile and compressive components are induced in the vicinity of the vertical waveguide edges. The strain gradient is maximal if the gap reaches down to the silicon slab.
In a second investigation the waveguide width wWG was varied. Figures 5(d) and 5(e) show exemplary strain simulations of a 2 μm and a 400 nm wide waveguide with the gap reaching down to the silicon slab. The most significant difference between these structures is that the strain gradient δεyy/δz in the waveguide center (y = 0) is much stronger in the narrow waveguide. A numerical analysis of this characteristic is displayed in Fig. 6. Here the strain component εyy is plotted along the z-axis (at y = 0) for different wWG ranging from 300 nm to 2 μm. It can be seen that εyy exhibits the steepest slope and therefore the strongest gradient |δεyy/δz| for the 300 nm wide waveguide. With increasing width |δεyy/δz| gradually decreases. At wWG = 2 μm still a gradient can be detected but it is comparatively small. This correlation is in good agreement with the Raman experiments and explains the observed increased Raman line broadening in the waveguide center of small waveguide structures. Furthermore, the simulations are consistent with the results of the electro-optic experiments. There a larger strain gradient in the center of smaller waveguides induces a higher electro-optic effect in the respective modulator devices.
The presented results demonstrate that increasing the optical non-linearity of strained silicon waveguides is possible by optimizing the geometry of the waveguides. Similar implications can be derived from the work of Cazzanelli and Bianco [12,13], who have measured spatially resolved strain profiles on the cross-section of large multi-mode waveguides and correlated them with χ(2) values obtained from SHG experiments. Interestingly, they found that the effective χ(2) is higher in larger waveguides. This discrepancy is probably caused by fundamental differences in the structures. The most obvious distinctive feature is that their Si3N4/SiNx straining layer is located exclusively on top of the waveguide. Therefore the strain originates solely from the upper interface of the waveguide. In our structures we found that the largest strain gradients are located at the vertical waveguide sidewalls that are in our case also covered with Si3N4. For the optimization of the effective χ(2) in our waveguides a smaller wWG is required to increase the relevant strain gradient |δεyy/δz| at the waveguide center (y = 0). On the contrary the strain profiles shown in [12,13] do not show distinct maxima at the waveguide sidewalls. Obviously this can lead to a completely different waveguide width dependency of the strain gradients and consequently the effective χ(2).
An overall improvement regarding the modulation efficiency of strained silicon rib waveguides was achieved in this work by reducing wWG. As one way to further improve the applicability of strained silicon waveguides for an efficient modulator application more sophisticated changes in the waveguide structure like the use of asymmetric waveguide geometries are suggested.
In summary we have shown that the waveguide geometry in particular the width of the waveguide wWG has a strong influence on the induced strain asymmetry and the resulting effective χ(2)yyz in strained silicon waveguides. We have demonstrated a device with a new record of χ(2)yyz in strained silicon and have proven that by reducing wWG the strain gradient and therefore the effective χ(2)yyz can be significantly enhanced. In complementary process simulations we reproduced the strain distribution within the Si3N4 covered waveguides. In particular gaps in the Si3N4 layer allowing a local strain relaxation are identified as source for the large strain gradients close to the vertical waveguide edges. With specific waveguide structures exploiting these findings in a favorable way may open the road to challenge the well established LiNbO3 modulators.
This work was partially funded by the Federal Ministry of Education and Research (BMBF) as part of the MISTRAL project (contract# 01BL0800).
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