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The influence of BaTiO3 ferroelectric domain orientations for high efficiency electro-optic modulation has been thoroughly analyzed. The Mach-Zehnder modulator structure is based on a CMOS compatible silicon/BaTiO3/silicon slot waveguide that supports both TE and TM polarizations whereas the Pockels effect is exploited by the application of a horizontal electric field with lateral electrodes placed on top of the BaTiO3 layer. The influence of the waveguide parameters has been optimized for each configuration and the lowest Vπ voltage combined with low losses has been determined. A VπL as low as 0.27 V·cm has been obtained for a-axis oriented BaTiO3 and TE polarization by rotating the waveguide structure to an optimum angle.
© 2015 Optical Society of America
1. Introduction
Nanophotonic integrated circuits (ICs) have the potential to replicate the microelectronic revolution of the past decades. The integration of complex optical functionalities with high performance will lead to a huge development in the field of nanophotonics for a broad range of applications such as information and communication technologies, biophotonics and sensing. While there are several existing technologies to design and build networks and systems, the introduction of a viable photonic ICs technology capable of creating a broad range of optical functions out of a single fabrication process is the fundamental breakthrough required to reduce costs dramatically.
Silicon photonics is currently the leading technology for the implementation of low-cost photonic integrated devices. The main argument in favour of this technology is its compatibility with mature silicon IC manufacturing and the availability of high quality silicon-on-insulator (SOI) wafers, an ideal platform for creating planar waveguide circuits [1]. Nevertheless, despite its huge potential there are fundamental constraints arising from material properties, which limit the complete development of commercial devices. One of them is the lack of linear electro-optic (EO) coefficient due to the crystallographic nature of silicon. Several approaches have been pursued to overcome such limitation. The most effective mechanism for changing the refractive index in silicon at a fast rate is the plasma dispersion effect, which consists in varying the free-carrier concentration in doped silicon, but it suffers from an inherent trade-off between low losses and low driving voltage [2].
EO modulation via the Pockels effect would allow achieving high modulation efficiency without penalizing the insertion losses. Pockels effect in silicon has been demonstrated by breaking the crystal symmetry [3]. Thereby, promising results has been achieved by exploiting the strain induced by silicon nitride (SiN) on top of narrow silicon waveguides [4]. However, efforts are still required to get practical devices. The integration of materials compatible with silicon CMOS photonics has therefore become a promising way to achieve EO modulation via Pockels effect. Polymers with a high second-order nonlinearity coefficient have been used as cladding [5] or in silicon based slot waveguides [6]. However, high temperature processes are usually required which makes more difficult the integration with standard CMOS steps. More recently, ferroelectric oxides have been attracting an increasing interest due to their high EO coefficients. The most known ferroelectric oxide is lithium niobate, LiNbO3, which is currently used in commercial EO modulators. Although different attempts have been investigated, the integration of high-quality films on silicon has only been achieved via complex and costly layer-bonding approaches [7, 8].
Barium titanate, BaTiO3, is also a well-known ferroelectric oxide, which exhibits a Pockels coefficient that is several times higher than LiNbO3 [9]. Integrated photonic devices based on BaTiO3 have been largely investigated on magnesium oxide (MgO) substrates [10–14]. Modulation with VπL over 0.5Vcm and bandwidths above 40GHz has been achieved. However, the recent demonstration of high quality BaTiO3 thin layers grown on SrTiO3-templated silicon [15] and SOI substrates [16,17] has opened a path towards the development of hybrid silicon BaTiO3 modulators with high performance [18,19].
In this work, the optimum BaTiO3 ferroelectric domain orientation to enhance EO modulation in a silicon CMOS compatible waveguide structure has been investigated via simulations. Furthermore, the expressions to calculate the Vπ voltage have also been derived for both BaTiO3 orientations and light polarizations. Thereby, all possible cases have been analyzed and compared in order to design the optimum modulation structure.
2. Modulator and design of the optical waveguide structure
The modulator device, shown in Fig. 1(a), is based on a Mach-Zehnder interferometer in which the active arm is formed by the waveguide structure shown in Fig. 1(b). The active length has been fixed to 2mm. The waveguide structure is fabricated as follows. The BaTiO3 layer is firstly grown on top of a SOI wafer. Optical confinement is then obtained by depositing a layer of amorphous silicon and etching this layer to form the waveguide. Electrodes are placed at both sides of the waveguide. The main advantage of this waveguide structure is that the BaTiO3 layer does not need to be etched. Furthermore, electrodes can be placed directly on top of the BaTiO3 layer that will allow achieving a stronger electric field in the active region and thus a more efficient EO performance. The waveguide structure is covered by a SiO2 cladding.
Fig. 1 (a) Schematic of the Mach-Zehnder modulator and (b) waveguide structure with electrodes. W is the waveguide width, G is the waveguide-to-electrode separation and taSi, tBTO and tSi are the amorphous, BaTiO3 and crystalline silicon thicknesses.
The thickness of the silicon layer in the SOI wafer has been optimized to tSi = 100nm for enhancing the optical confinement in the BaTiO3 layer when compared to the more standard value of 220nm [19]. A stronger optical confinement will not only allow a higher EO efficiency but also designing more compact structures and therefore improve the device footprint. The BaTiO3 thickness, tBTO = 50nm, and the rest of the parameters, amorphous silicon thickness and waveguide width, have been selected to achieve single-mode operation with negligible losses and high optical confinement for both TE and TM operation. The design has been carried out with a commercial finite-element based mode solver [20]. It should be noticed that the waveguide structure can suffer from propagation losses if coupling to slab modes in the silicon layer below the BaTiO3 occurs. Undesired coupling is obtained when the effective index of the waveguide structure decreases below the effective index of the slab modes. Figure 2(a) shows the effective index as a function of the waveguide width for both TE and TM polarization. It can be seen that the effective index of the fundamental TM mode is lower than for TE polarization because the mode is more confined in the BaTiO3 layer due to the slot effect [21]. High confinement is desired to enhance the EO performance. However, the thickness of the amorphous silicon layer needs to be increased to avoid propagation losses due to coupling to slab modes. Therefore, an amorphous silicon thickness of taSi = 220nm has been initially chosen to have a waveguide structure with negligible propagation losses for both polarizations.
Fig. 2 (a) Effective index and (b) upper bound of the EO overlap integral as a function of the waveguide width taking into account tBTO = 50nm and taSi = 220nm. Propagation losses due to electrodes are also shown for c) TE and (d) TM and different waveguide-to-electrode separations. The horizontal dashed line is for 1dB/cm propagation losses.
Single-mode transmission is also achieved for waveguide widths between 400nm and 700nm, as it can be seen in Fig. 2(a). The waveguide becomes multimode for widths above 700nm for TE polarization and 1100nm for TM polarization. Light confinement of the fundamental mode has been analyzed by simulating the effective index change due to a uniform change of the BaTiO3 refractive index. In such a way, an upper bound of the EO overlap integral between the electric field and optical mode can be estimated. Figure 2(b) shows the simulated results as well as the TE and TM mode profiles for a waveguide width of 600nm. It can be seen that in agreement with the effective index results, the upper bound of the EO overlap integral is higher for the TM fundamental mode. A value of about 30% is achieved, which in addition is kept almost constant for the different waveguide widths. On the other hand, a maximum value of about15% is achieved for TE fundamental mode. In this case, the mode is pushed into the amorphous silicon and is therefore less confined in the BaTiO3 layer when the waveguide width increases. Therefore, the effective index increases while the upper bound of the EO integral decreases for wider waveguides.
The propagation losses induced by the electrodes as a function of the waveguide width is shown in Fig. 2(c) for TE polarization and Fig. 2(d) for TM polarizations. CMOS compatible Aluminum electrodes (nAl = 1.5137 + j15.234) have been considered. It can be seen that, despite the higher confinement factor, TM polarization presents higher losses due to the stronger interaction between the horizontal metallic contacts and the vertically oriented electric field component of the optical mode. Furthermore higher losses are experienced by both polarizations when the waveguide width decreases because the optical mode is expanded laterally. A waveguide width of W = 600nm and a waveguide-to-electrode separation of G = 1μm have been initially selected in order to have negligible losses for both polarizations due to the electrodes.
3. Analysis and design optimization of the electro-optic performance
BaTiO3 is a ferroelectric material that has a tetragonal crystal structure at room temperature, which is shown in Fig. 3(a), being the spontaneous polarization parallel to the crystallographic c-axis. The EO effect can be modelled by using the index ellipsoid, which is sketched in Fig. 3(b). BaTiO3 has a negative uniaxial anisotropic performance and therefore the ordinary index (no = 2.444) is higher than the extraordinary index (ne = 2.383). The index ellipsoid in the presence of an electric field (Ex, Ey, Ez) is given by
where rij are the linear (or Pockels) EO coefficients. The considered values are r13 = 8pm/V, r33 = 28pm/V and r51 = 800pm/V [9]. It should be noticed that such values have been measured in bulk BaTiO3 crystals. The coordinate system with axes x, y, and z of the index ellipsoid is aligned along the crystallographic axes, a1, a2 and c, respectively, of the BaTiO3 crystal structure. Therefore, the EO performance will depend on how the BaTiO3 is grown to fabricate the waveguide structure. The tetragonal form of the crystal structure implies that the material can be grown with two different orientations depending on the process conditions [22]. The so-called a-axis or c-axis orientations will depend on if the optical axis (z-axis) is in-plane or out-of-plane in the BaTiO3 layer. A single ferroelectric domain has been assumed.Fig. 3 (a) BaTiO3 crystal structure and (b) index ellipsoid. The crystallographic axes, a1, a2 and c, are aligned along the coordinate system with axes x, y, and z, respectively.
3.1 Electro-optic performance for a-axis oriented BaTiO3
Figure 4 shows the possible waveguide orientations with respect to the coordinate system defined by a-axis oriented BaTiO3. The x-axis is perpendicular to the horizontal plane but the same performance would be achieved by considering the y-axis perpendicular. Two extreme cases can be initially distinguished: on one hand when the electric field is parallel to the optical axis, Fig. 4(a), and on the other hand when the electric field is perpendicular to the optical axis, Fig. 4(b). In the former, it can be obtained from Eq. (1) that modulation is determined by the r33 coefficient for TE polarization and by the r13 coefficient for TM. However, in the latter there will not be modulation neither for TE nor for TM polarizations.
Fig. 4 Waveguide structure for a-axis oriented BaTiO3. The optical axis (z-axis) is in-plane in the BaTiO3 layer and can be (a) parallel or (b) perpendicular to the applied electric field. (c) The waveguide can also be rotated by an angle of ϕ in the yz plane to enhance the EO performance.
The EO performance can be enhanced by rotating the waveguide in the zy plane by a certain angle, ϕ, as depicted in Fig. 4(c), to achieve an EO coefficient that will be a linear combination of r13, r33 and r51. To analyze the EO performance, it is more convenient to define the coordinate system along the applied electric field and the propagation of light, z’ and y’ axes respectively in Fig. 4(c). The applied electric field is mainly in the horizontal direction so it can be assumed that Ex = 0. The index ellipsoid in the new coordinate system is obtained by using the following transformation
So, by operating in Eq. (1), the refractive index and EO coefficient can be derived for TE polarization and for TM polarizationThe refractive index for TE polarization will change from the ordinary to the extraordinary value when the rotation angle changes from 0° to 90°. On the other hand, the EO coefficient as a function of the rotation angle is shown in Fig. 5(a). It can be seen that the maximum value is achieved for an angle of 55°. This angle is somewhat higher than the 45° value that has been previously claimed in a similar waveguide structure [17, 18]. It should be noticed that this optimum angle is independent of the waveguide structure as it has been analytically derived from the index ellipsoid of BaTiO3. By looking at Fig. 5(a), it is also confirmed that there will not be any index modulation for a rotation angle of 90° and, from Eq. (6), the same will occur for TM polarization. The maximum EO coefficient for TM polarization will be given at 0°. However, the value is much lower than for TE polarization as only r13 is involved.
Fig. 5 (a) EO coefficient for TE polarization. (b) EO overlap and (c) Vπ voltage as a function of the rotation angle for both polarizations.
Figure 5(b) shows the EO overlap integral as a function of the rotation angle for both TE and TM polarizations, which has been calculated by
where Ee(ϕ) and Eo(ϕ) are the simulated amplitudes of the electric and optical fields, respectively, at each rotation angle. It should be noticed that the permittivity of BaTiO3 will also depend on the rotation angle so that we have considered that εz’ = 2200·sin(ϕ) + 56·cos(ϕ) and εx = 2200 [9]. The EO overlap integral will be almost constant with the rotation angle and with values, ΓTE≈11% and ΓTM≈29%, very close to the maximum ones estimated in Fig. 2(b). The EO overlap integral decreases at small rotation angles due to the low permittivity of the BaTiO3 at such angles. The refractive index and EO coefficients described in Eq. (3)-(6) have been used to simulate the change in the effective index of the fundamental mode and extract the Vπ voltage of the modulator as a function of the rotation angle. Results are shown in Fig. 5(c) for both TM and TE polarization.The Vπ voltage can also be analytically calculated as
where λ is the wavelength, S is the separation between the electrodes, L is the active length and ΓTE and ΓTM are the EO overlap integral for TE and TM polarizations. In our case, λ = 1550nm, S = 2.6μm, L = 2mm, ΓTE = 11% and ΓTM = 29%. The Vπ voltages calculated by Eq. (8) and (9) have also been plotted in Fig. 5(c). It can be seen that there is a very good agreement between simulation and analytic results except for smaller rotation angles where the EO overlap integral decreases and therefore the assumption of a constant value is no longer valid. The optimum EO performance is obtained for TE polarization and a rotation angle of 55°, in agreement with Fig. 5(a), for which a Vπ voltage as low as 2V is achieved. For TM polarization, although the overlap integral is almost two times higher due to the stronger optical confinement, a drastically higher Vπ voltage of 64V is obtained at the optimum angle around 10°.The influence of the amorphous silicon thickness, waveguide width and waveguide-to-electrode separation on the EO performance has also been analyzed via simulations for TE polarization and the optimum rotation angle. Figure 6(a) shows the Vπ voltage as a function of the amorphous silicon thickness, taSi, for different waveguide widths. It can be seen that a Vπ voltage as low as 1.35V (VπL = 0.27 V·cm) can be achieved by using thinner amorphous silicon layers. However, there is a minimum thickness that cannot be exceeded to avoid losses due to coupling to slab modes. Furthermore, the minimum thickness, which has been chosen to ensure propagation losses below 1dB/cm, will depend on the waveguide width. Wider waveguides, which have a higher effective index, allow using thinner amorphous silicon layers. However, it can be seen in Fig. 6(a) that the minimum achievable Vπ voltage is almost the same independently of the waveguide width. Wider waveguide would be preferred to minimize additional propagation losses due to sidewall roughness.
Fig. 6 Vπ voltage for TE polarization and ϕ = 55° as a function of (a) the amorphous silicon thickness and G = 1μm and (b) the waveguide-to-electrode separation and taSi = 220nm. Results are shown for different waveguide widths.
Figure 6(b) shows the Vπ voltage as a function of the waveguide-to-electrode separation, G, for different waveguide widths and taking into account an amorphous silicon thickness of 220nm. Electrodes can be placed closer to wider waveguides, as it was shown in Fig. 2(c), and hence the Vπ voltage can be decreased. However, there is a lower bound in the minimum waveguide-to-electrode separation in this case to avoid losses due to the interaction of the optical mode with the metallic contacts. The interesting point is that there will also be a minimum achievable Vπ voltage independently of the waveguide width. Therefore, the optimum EO performance can be achieved independently of the waveguide width by optimizing the amorphous silicon thickness and waveguide-to-electrode separation.
3.2 Electro-optic performance for c-axis oriented BaTiO3
The definition of the waveguide structure is shown in Fig. 7(a) for c-axis oriented BaTiO3. In this case, the optical axis (z-axis) is perpendicular to the horizontal plane. However, with the application of an electric field, the index ellipsoid is rotated around the y-axis and therefore the principal axes in the transversal plane are no longer aligned with the waveguide structure, as depicted in Fig. 7(b). The angle between the rotated x’z’ and the original xz coordinates can be derived from Eq. (1) as
by assuming that the applied electric field is mainly in the x-axis, i.e. Ez = Ey = 0. On the other hand, unlike what happens for a-axis oriented BaTiO3, the rotation of the waveguide in the xy plane will not affect to the EO performance.Fig. 7 Waveguide structure for c-axis oriented BaTiO3. (a) The optical axis (z-axis) is out-of-plane in the BaTiO3 layer. (b) However, there is a rotation in the xz plane (θ-angle) when a voltage is applied in the electrodes.
Figure 8(a) shows the induced rotation angle of the BaTiO3 principal axes in the transversal plane as a function of the applied voltage and for different waveguide-to-electrode separations. The induced rotation will affect the optical signal by rotating the input polarization. This undesired effect can be neglected if the Vπ voltage is small enough to minimize the induced rotation. Furthermore, smaller Vπ voltages will be required if the waveguide-to-electrode separation is decreased, as it can be seen in Fig. 8(a). The Vπ voltage for TE and TM polarizations can be derived from Eq. (1) as
when low θ-angle values (θ<10°) are considered. The square root causes a non-linear phase shift variation with the applied voltage, as it is shown in Fig. 8(b), and therefore a non-periodic intensity response of the Mach-Zehnder modulator [10,23,24]. It should also be noticed that the phase shift will be positive (ΔnTM>0) for TM polarization but negative (ΔnTE<0) for TE polarization.Fig. 8 (a) Induced rotation angle of the BaTiO3 principal axes as a function of the applied voltage and for different waveguide-to-electrode separations. (b) Simulated and analytic phase shift for both polarizations as a function of the applied voltage for G = 1μm.
The absolute value of the simulated phase shift has been compared in Fig. 8(b) with the calculated analytically by assuming a constant electric field, Ex = V/S, and the maximum achievable overlap integral plotted in Fig. 2(b). In this case, BaTiO3 permittivity is εx = 2200 and εz = 56. It can be seen that there is a good agreement between both results thus indicating an efficient EO performance. The Vπ voltage is smaller for TM, Vπ,TM = 4.5V, than for TE, Vπ,TE = 7.2V, due to the higher optical mode confinement in the BaTiO3 layer. However, the same waveguide-to-electrode separation (G = 1μm) as well as amorphous silicon thickness (taSi = 220nm) have been considered for both polarizations.
From Eq. (11) and (12), the Vπ voltage of TE and TM polarizations can be related as
by assuming that the ordinary and extraordinary indices have a comparable value (no≈ne). This expression indicates that the lower overlap integral for TE (ΓTM>ΓTE) could be compensated by using a smaller electrode separation (STE<STM) thus yielding to lower Vπ voltages.Figure 9 compares the Vπ voltage for (a) TE and (b) TM polarizations by changing the amorphous silicon thickness and waveguide-to-electrode separation. It can be seen that a larger improvement can be achieved for TE polarization by optimizing these parameters. Furthermore, the impact of minimizing the waveguide-to-electrode separation is stronger than decreasing the amorphous silicon thickness, in agreement with Eq. (13). Therefore, the lowest value, Vπ,TE = 4.75V, has been achieved by reducing the waveguide-to-electrode separation to 550nm and using a slightly lower amorphous silicon layer thickness of 210nm. The limit for each configuration has also been determined for having propagation losses below 1dB/cm. Nevertheless, it should be noticed that the polarization rotation effect will be more accentuated for smaller waveguide-to-electrode separations, as it was shown in Fig. 8(a). However, this effect will be almost negligible due to the low Vπ voltages achieved.
Fig. 9 Vπ voltage for (a) TE and (b) TM polarizations as a function of the amorphous silicon thickness taken into account different waveguide-to-electrode separations and a waveguide width of 600nm.
In the case of TM polarization, which is shown in Fig. 9(b), the amorphous silicon thickness cannot be decreased below 220nm to avoid losses. However, the waveguide-to-electrode separation can be decreased to 900nm to improve the Vπ voltage. Thereby, a Vπ voltage as low as 4.25V can be achieved which overcomes the lowest value obtained with TE polarization. On the other hand, the influence of the waveguide width on the Vπ voltage has also been analyzed. As for a-axis oriented BaTiO3, the Vπ voltages cannot be significantly improved by using a different waveguide width.
4. Conclusion
The optimum BaTiO3 ferroelectric domain orientation to enhance the EO modulation in a silicon CMOS compatible waveguide structure has been thoroughly analyzed. The best EO performance has been achieved for a-axis oriented BaTiO3 and TE polarization. In this case, the Vπ voltage can be significantly improved by rotating the waveguide structure with respect to the principal axes of BaTiO3. Hence, a Vπ voltage as low as 1.35V has been obtained for a modulation length of 2mm, i.e. VπL = 0.27 V·cm, at the optimum rotation angle of 55°.
On the other hand, c-axis oriented BaTiO3 allows achieving an optimum EO performance for both TE and TM polarizations though the phase shift variation with the applied voltage is no longer linear as it happens for a-axis oriented BaTiO3. A Vπ voltage of 4.25V has been achieved for TM polarization, which is slightly smaller than the value of 4.75V obtained for TE polarization. Such values are small enough to neglect the undesired effect of polarization rotation that can be induced in c-axis oriented BaTiO3.
The optical confinement could be improved by using thicker BaTiO3 layers, which would increase the EO overlap integral. However, it should be noticed that even by assuming an ideal overlap integral of 100%, a Vπ voltage of 2.29V for TM polarization and 1.57V for TE polarization would be achieved, both still higher than the obtained for a-axis oriented BaTiO3.
On the other hand, it should be noticed that a multi-domain structure is usually formed during the fabrication of thin-film BaTiO3 layers [25]. The influence of a multi-domain structure on the electro-optic performance could be analyzed by comparing experimental results with the ones obtained in this work. However, some poling approach would be required in a real device to orientate the ferroelectric domains in a single direction, i.e. to obtain a single-domain structure. High DC bias voltages above the coercive field have been used for trying to orientate the ferroelectric domains [15, 17].
Acknowledgment
This work was supported by the European Commission under project FP7-ICT-2013-11-619456 SITOGA. Financial support from TEC2012-38540 LEOMIS is also acknowledged. We also acknowledge Stefan Abel and Sébastien Cueff for their helpful comments.
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