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We propose and analyze theoretically electro-absorption modulators with uniaxially tensile strained Ge/Si0.19Ge0.81 multiple quantum wells (MQWs). The effects of uniaxial strain on band structures including Γ-valley and L-valley are discussed. The simulation results indicate that the absorption contrast of TE mode is improved by 3.1 dB while the TM mode absorption is reduced by two-thirds under 1.6% uniaxial tensile strain. Zero-biased electro-absorption modulators covering 1380-1550 nm wavelength can be achieved by introducing 0.18%-1.6% uniaxial tensile strain. Taking into account the TE-polarized mode excited usually in integrated waveguides, the proposed scheme provides a promising approach to design highly efficient Ge/SiGe MQWs electro-absorption modulators for on-chip optical transmission and cross-connect applications.
© 2017 Optical Society of America
1. Introduction
Since the energy consumption and band width limit have restricted the performance of large-scale integration circuit, on-chip optical interconnect has been recognized as a potential solution. Silicon photonics is the most promising platform for integration of photonic and electronic components, however, is limited by the lack of efficient active devices compatible with traditional complementary metal-oxide-semiconductor (CMOS) technology. Ge and SiGe materials draw much attention as the corresponding wavelength of direct bandgap of Ge lies within the communication C-band, while the indirect L-valley band edge is only about 140 meV lower than the direct Γ-valley band edge. Several CMOS compatible schemes have been developed to engineer the band structure of Ge related materials, including Ge/SiGe quantum wells [1–3], suspended Ge microstructures [4–7], and GeSn alloys [8-9]. Ge/SiGe quantum wells electro-absorption modulators have shown advantages in aspects of small foot print and low energy consumption [10]. Because of strong quantum confinement, the direct bandgap absorption edge energy of Ge/SiGe quantum well is larger than that of bulk Ge. The 10 nm Ge well in [1] has an operating wavelength of ~1420 nm under zero-bias. Although the absorption edge can be shifted by applying different bias voltages, the degeneration of absorption contrast ratio is observed by the experiment [3]. High bias voltage is also not preferable for large-scale integration. The operating wavelengths of the Ge/SiGe MQWs modulators are thus limited.
Suspended microstructures have been used to enhance the tensile strains in bulk Ge. Different strains can be achieved by trimming the shape and size of microstructures. Up to now, 5.7% uniaxial tensile strain [6] and 2.3% biaxial tensile strain [7] have been realized in germanium membranes. Tensile strain in germanium decreases the bandgap of Ge material. As the Γ-valley deformation potential is much larger than that of the indirect L-valley, the difference between the direct bandgap and indirect bandgap is decreased or even removed. Recent work on Ge/SiGe MQWs microdisks achieved 0.51% biaxial tensile strain and revealed the feasibility of tuning the strain in Ge/SiGe quantum wells [11].
We propose a novel design of Ge/SiGe MQWs zero-biased electro-absorption modulators covering 1380-1550 nm wavelength. By introducing 0.18%-1.6% uniaxial tensile strain, those modulators can be fabricated on the 10 /12 nm Ge/Si0.19Ge0.81 MQWs chip. The simulation results show that uniaxial tensile strained Ge/SiGe quantum well has an enhanced TE mode absorption contrast and suppressed TM mode absorption for waveguide configuration. The proposed electro-absorption modulator with higher TE mode absorption contrast is benefit to the applications of on-chip optical communications.
2. Description of theoretical model
2.1 Modeling of strains in uniaxially strained Ge/SiGe MQWs
The structure we design and simulate is presented in Fig. 1(a). It consists of 10 × 10 /12 nm Ge/Si0.19Ge0.81 quantum wells surrounded by 40 nm intrinsic Si0.15Ge0.85 spacers, grown on 300 nm Si0.15Ge0.85 buffer layer. The whole structure is capped by a n-type Si0.15Ge0.85 layer as the top contact. The Ge content of buffer layer, spacers and cap layer is designed to compensate the strain. The Ge quantum well width and barrier Ge content are chosen as strong quantum confined Stark effect is verified by the experiment [1]. Previous works [1, 14] used large barrier widths in order to eliminate the influence of coupling effect of neighboring wells. As the electrons and holes are confined in the wells, large barrier width reduces the absorption contrast for waveguide configuration [12]. In our design, we set the barrier width to be 12 nm as it is wide enough to avoid coupling effect of neighboring wells. A strain compensated design requires that the lattice constant of the strained MQWs equals to that of the thermally expanded buffer layer, so that the MQWs region does not receive stress from the buffer layer during growth. Strain accumulation during the successive growth of quantum wells and barriers is avoided by taking the strain compensated design and the crystal quality is ensured in this way.
Fig. 1 (a) Epitaxy design of the MQWs structure. (b) Schematic of the electro-absorption modulator with suspended microbridge structure. (c) Cross-section map of the strain tensor element εxx distribution. W1 = 165 μm, W2 = 2 μm, L1 = 300 μm, L2 = 20 μm. (d) εxx distribution in the x-y plane with a distance of 150 nm away from the bottom of the buffer layer. (e) Field distribution of the TE field Ey for the quasi-TE fundamental mode at λ = 1550 nm. w = 600 nm, d = 512 nm.
The overall view of the device is shown by Fig. 1(b). A taper-shaped microbridge design is adopted as it moderates the strain concentration effect at the corner and reduces the optical mode mismatch loss [13]. There is a taper region at each side of the bridge. The total length of the device is 300 μm, with a taper width of 165μm. A COMSOL 3-D finite element method (3-D FEM) simulation is performed to evaluate the strain distribution of the bridge, as shown by Figs. 1(c) and 1(d). An initial biaxial strain of 0.18% is set in the SiGe buffer layer and the upper MQWs layer is simplified as an unstrained SiGe layer. The strain distribution on the narrow bridge is almost uniform with a uniaxial strain value of ~1.6%. The length of the narrow bridge is 20 μm which serves as an effective modulation region. The field distribution of the ridge waveguide is shown by Fig. 1(e). Optical confinement factor for the MQWs region of the quasi-TE fundamental mode is 0.44.
As the lattice constant of Ge is 4% larger than that of Si, direct epitaxy of Ge or Ge rich alloy on Si suffers from dislocations and surface roughness. The commonly used method is to grow a buffer layer on Si substrate followed by a high temperature anneal process [1, 14, 15]. Because of the thermal expansion coefficient difference between Si and Ge, there is 0.1%-0.3% residual tensile strain in the buffer layer when the wafer cools down. Germanium-on-insulator (GOI) material is fabricated by taking the heteroepitaxy and layer transfer techniques [16], which removes the Ge-Si interface and keeps the tensile strain. Here, we call the Ge rich buffer layer on insulator as GOI material for brevity. The subsequent quantum wells are grown at low temperature of 400-450 °C. During the low temperature growth of MQWs, the lattice aligns to the thermally expanded buffer layer and each layer is biaxially strained.
The fabrication of the suspended microbridge needs several steps of dry etching and vapor HF under-etching of the silicon dioxide [4]. After the suspended microbridge structure is formed, strain along the bridge is enhanced and turned to be uniaxial. The strain distribution on the bridge can be simulated by the finite element method which requires a lot of computational resource especially for multiple quantum wells structures. By assuming that the strain on the bridge is uniform and no bending effect exists, idealized strain models can be used for analyses and simulations. The schematic of the suspended microbridge structure and coordinates are shown by Fig. 1(b). For the [001] oriented crystal, when the microbridge is along the x axis ([100] direction) strain of the ith layer along the x direction is
where is the unstrained lattice constant of the buffer layer, is the unstrained lattice constant of the ith layer, is the uniaxial strain of the bridge. We describe the uniaxial strain as the strain value of the buffer layer along the bridge which is larger than the biaxial strain induced by thermal expansion. In determining the strain tensor elements in other directions, we combine the biaxial strain model in [17] and the uniaxial strain model in [18]. Strains in y and z directions go through a gradual relaxation with the increase of uniaxial strain along x direction, with where and are elastic stiffness constants, is the residual biaxial strain of the buffer layer induced by thermal expansion. This hybrid model applies to both the MQWs material and the single film material. When applied to the Ge microbridge in [19], the proposed strain model has −0.23% for. The X-ray scanning data has a maximum of −0.25% with the maximum uniaxial strain of 0.48%.Parameters of Si and Ge used in the strain calculation are listed in Table 1. The unstrained lattice constant of Si1-xGex is determined by [20]
while the elastic constants are linearly extrapolated for Si1-xGex alloys. The residual biaxial strain induced by thermal expansion in the buffer is assumed to be 0.18%.
Table 1. The parameters of Si and Ge used in the proposed strain model [2].
2.2 Modeling of band structures and absorption spectra
The band structure near Γ point is simulated by an 8 band k∙p model described in detail in [18]. The direct bandgap absorption coefficient is given by
where is the electron charge, is the refractive index of the QW, is the light speed in vacuum, is the free-space permittivity, is the free electron mass, is the round frequency, is the width of the QW, is the in-plane wave vector, is the momentum operator and the momentum matrix elements are calculated between conduction bands and valence bands, denotes the polarization. and are the energy dispersions of the first Brillouin zone, is the Lorentzian linewidth defined by Full-Width-at-Half-Maximum (FWHM). The Fermi functions, and quasi-Fermi levels, are detailed in [18]. is the Boltzman constant and is the temperature.Since Si and Ge are indirect bandgap materials, both L-valleys and Δ-valleys should be included to have a better simulation for the band structures of Ge/SiGe quantum well [21]. To provide a usable confinement for carriers, the barrier Ge content of Ge/SiGe quantum well is usually lower than 0.95 [1, 3, 14]. The commonly used strongly confined structure has a type-II alignment for Δ conduction band. Electrons of Δ conduction band are confined in the barriers while valence band holes are confined in the wells. The L-valley conduction subbands are simulated by the single band k∙p model [22]. The material parameters are listed in Table 2. For Si1-xGex alloys, the Dresslhaus-Kip-Kittel (DKK) parameters are decided by using the scheme in [23] and other parameters are linearly extrapolated. The indirect absorption coefficient is estimated utilizing the one-phonon model described in [24, 25]. The temperature is set to be 300 K with a broadening of = 3 meV.
Table 2. The parameters at 300 K for Si and Ge used in the calculation of band structures.
3. Results and discussions
The lowest confined electronic subband states of Γ-valley (Γe1, Γe2, Γe3), L-valley (Le1, Le2) and confined hole subband states (heavy hole HH1, HH2, HH3 and light hole LH1, LH2) are shown in Fig. 2. Under the situation of 0.18% biaxially strained buffer layer, the band offset of the light hole, heavy hole, and electron at the Γ point is calculated to be 76, 152, and 405 meV, respectively. Near the direct bandgap, the Γe1-HH1 and Γe1-LH1 transitions are allowed while the Γe1- HH2 and Γe1-LH2 transitions are forbidden by symmetry. Fig. 3. presents the energies of both direct band gaps (Γe1-HH1, Γe1-LH1) and indirect band gaps (Le1~LH1, Le1~HH1) with the change of uniaxial strain. The p-i-n junction works under reverse bias and the built-in electric field is 19.3 kV/cm. The band offset of L-valley conduction band is only about 50 meV and barely supports the type-I alignment. Experimental results have verified the absence of indirect bandgap transition peaks both under room temperature [1] and ultralow temperature [26] for 10 and 12 nm Ge wells, respectively. As the uniaxial tensile strain of the bridge increases, both direct and indirect bandgaps decrease. However the direct bandgap decreases more rapidly than the indirect one due to the larger deformation potential of Γ-valley. The uniaxial strain effectively tailors the direct bandgap and 1.6% uniaxial strain can shift the Γe1-HH1 transition to 0.8 eV which corresponds to the communication 1550 nm wavelength. Unlike bulk germanium material [27], the proposed Ge/Si0.19Ge0.81 quantum well has a separation energy of 0.05 eV between HH1 and LH1 when the uniaxial strain equals to 0.18%.
Fig. 2 Band edges (black lines) and squared wave functions (colored lines) of 10 /12 nm Ge/Si0.19Ge0.81 MQWs on 0.18% biaxially strained buffer: (a) conduction band, (b) valence band.
Figure 4 shows the energy dispersions along [100] and [110] directions. The electron dispersions are close to parabolic and the subbands have nearly constant effective masses. The valence subbands are labeled as light hole and heavy hole subbands according to their characters. Band mixing effects become significant when. The interaction between the light hole, heavy hole, spin-orbit splitting band and conduction band results in the nonparabolicities in the valence subbands [2]. It is interesting that the nonparabolicities degenerate under uniaxial tensile strain, especially for LH1 due to the separation of subbands. In Fig. 5, we show the normalized momentum matrix elements of TE and TM polarizations. For the waveguide configuration, y polarization corresponds to the TE mode and z polarization corresponds to the TM mode. Although the Γe1- HH2 transition is forbidden by selection rules, there are still transition peaks along [100] direction for both TE and TM polarizations due to the admixture effect of LH1 and HH1 into HH2. The absorption edge is dominated by the Γe1- HH1 transition near the zone center. The Γe1- HH1 transition is significantly suppressed for TM polarization and enhanced for TE polarization under 1.6% uniaxial strain. However, the situation is the opposite for Γe1- LH1 transition with slightly increased TM transition and decreased TE transition. The difference between heavy hole and light hole subbands is caused by the uniaxial strain in the Ge/SiGe quantum well, in a similar way with the III-V quantum well [28].
Fig. 4 Energy dispersions of Γ-valley electron subbands and valence hole subbands on 0.18% biaxially strained buffer (dash lines) and 1.6% unixially strained bridge (solid lines).
Fig. 5 Normalized squared momentum matrix elements for Γe1-HH1, Γe1-LH1 and Γe1-HH2 transitions on 0.18% biaxially strained buffer (dash lines) and 1.6% unixially strained bridge (solid lines): (a) TM (z polarization), (b) TE (y polarization).
Figures 6(a) and 6(b) show the TM and TE absorption coefficients under different voltages, simulated by using the 8 band k∙p model. Due to the lower barrier Ge content, the Γe1- HH1 exciton peak of the proposed structure on 0.18% biaxially strained buffer locates at 1380 nm, 30 nm shorter than [1]. Both TM and TE absorption edges are moved to the 1550 nm wavelength under 1.6% uniaxial strain, as expected by Fig. 3. The exciton peaks are caused by the intersubband transitions between conduction subbands and valence subbands. The Γe1- LH1 and Γe1- HH1 exciton peaks are marked by the colored arrows. The absorption coefficient of Γe1- HH1 exciton peak is calculated to be 6500 cm−1 on 0.18% biaxially strained buffer. It should be mentioned that all wells and barriers are included in the QW width in Eq. (5). With 1.6% tensile uniaxial strain applied, the TM absorption coefficient is decreased to 1850 cm−1 while the TE absorption coefficient is increased to 11200 cm−1. As the quantum confinement of the proposed well structure is quite strong and blocks the spatial separation of electrons and holes, the Stark shift is small without bias. Voltage swing of 1-2 V is needed to get a maximum absorption contrast for 1550 nm wavelength. The TE absorption contrast is improved by 1.2 dB for 0V/1V operation and 3.1 dB for 0V/2V operation under 1.6% uniaxial tensile strain. The voltage swing can be reduced by cutting the numbers of wells at the expense of waveguide optical confinement factor of the MQWs region [14]. The distance between p-type layer and n-type layer is reduced with a smaller number of wells. In other words, the thickness of the MQWs region is reduced. At the same time, the total thickness of the device is hard to reduce since buffer layer, cap layer and spacer layers are all essential. So the optical confinement factor of the MQWs region will be affected if we cut the number of wells. Because the electro-absorption modulation happens in the MQWs region, smaller optical confinement factor of the MQWs region means larger device length and insertion loss for a certain extinction ratio. Another feature of the uniaxially strained Ge quantum well material is that the Stark shift has little change with the uniaxial strain. Thus the modulators operating on different wavelengths can be driven by the same digital logic circuit. As the suitability at using Ge/SiGe MQWs modulator in DWDM optical interconnects has already been studied by [29], the application of uniaxial strain can provide a new method to achieve this goal. Because TE mode waveguide is often preferred in waveguide integration, the improvement of TE absorption contrast makes the device appropriate for waveguide integrated highly efficient modulation.
Fig. 6 The absorption spectra under different voltages on 0.18% biaxially strained buffer (dash lines) and 1.6% unixially strained bridge (solid lines): (a) TM (z polarization), (b) TE (y polarization).
Figure 7 shows the absorption coefficients of Γe1- LH1 and Γe1- HH1 excitons under different uniaxial strains. For TM polarization, the difference between Γe1- LH1 and Γe1- HH1 exciton peaks is increased as the Γe1- HH1 transition is greatly suppressed by the uniaxial strain. For TE polarization, the Γe1- LH1 exciton peak is smoothed out because of the reduction of Γe1- LH1 transition near the Brillouin zone center. It should be noted that the absorption coefficient at the location of Γe1- LH1 transition still keeps growing with the uniaxial strain. This is because the reduction of Γe1- LH1 transition is neutralized by the broadening effects of adjacent Γe1- HH1 and Γe2- HH2 transitions. To model the scattering relaxation effect, we include the Lorentzian function in the calculation of absorption coefficient in Eq. (5). Each k-pint contributes to the absorption in adjacent transition energy zone through the Lorentzian function. Since the Γe1- HH1 and Γe2- HH2 transitions are quite strong and widely distributed in the first Brillouin zone, they have significant lifting effects for the absorption coefficient at the location of Γe1- LH1 transition through scattering relaxation.
Fig. 7 The absorption coefficients at the locations of Γe1- LH1 and Γe1- HH1 exciton peaks under different uniaxial strains: (a) TM (z polarization), (b) TE (y polarization).
The total length of the modulator device is 300 μm and the effective modulation area length for 1550 nm wavelength is 20 μm. Under 0 V/2 V operation, the TE absorption coefficients of the bridge for 1550 nm are 1077 and 6934 cm−1 respectively. As for the taper regions, an averaged strain value of 0.3% is chosen and the corresponding TE absorption coefficients for 1550 nm wavelength are 63 and 65 cm−1 respectively. For the quasi-TE fundamental mode in Fig. 1(e), we get an extinction ratio of 17.8 dB under 0 V/2 V operation and the corresponding insertion loss is 9.5 dB. The insertion loss mainly comes from the material absorption loss of the waveguide. The taper regions have a total length of 280 μm but have little contribution to extinction ratio. The coupling loss of the bridge is negligible in compare with the material absorption loss. The total loss could be decreased as the 20 μm long bridge only introduces 4.2 dB loss.
Although the Ge-on-Si material has already achieved 2.5% uniaxial strain [30], the use of GOI structure allows for larger uniaxial strain and better device isolation. We also predict that biaxial strain which shifts the absorption edge to 1550 nm is much smaller than 1.6%. The biaxially strained Ge/SiGe MQWs material has the same absorption coefficient for TE and TM polarizations. As the commonly used micro-cross structure intended to induce biaxial strain is not suitable for waveguide integration, the biaxial strain required can be achieved by a SiN stressor layer [11, 31].
3. Conclusion
We analyze the absorption properties of uniaxially tensile strained 10 /12 nm Ge/Si0.19Ge0.81 multiple quantum wells. The band structure and quantum confined Stark effect are modeled by an 8-band k∙p model. The simulation results show that zero-biased electro-absorption modulators covering 1380-1550 nm wavelength can be fabricated on the same chip by tailoring the suspended microstructures. The proposed design provides a new method to design waveguide integrated Ge/SiGe MQWs electro-absorption modulators for on-chip optical communication applications. We also show that the uniaxially tensile strained Ge/SiGe quantum well has an enhanced TE absorption contrast and suppressed TM absorption for waveguide configuration. Under 1.6% uniaxial strain, the TE absorption contrast is improved by 3.1 dB for 0V/2V voltage swing while the TM absorption coefficient decreases by two-thirds. The improvement in TE absorption makes the device suitable for waveguide integrated highly efficient modulation.
Funding
National Natural Science Foundation of China (NSFC) (Grant No. 61435004).
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