## Abstract

We present a design of monolithically integrated GeSi electro-absorption modulators and photodetectors for electronic-photonic integrated circuits on a silicon-on-insulator (SOI) platform. The GeSi electro-absorption modulator is based on the Franz-Keldysh effect, and the GeSi composition is chosen for optimal performance around 1550 nm. The designed modulator device is butt-coupled to Si(core)/SiO_{2}(cladding) high index contrast waveguides, and has a predicted 3 dB bandwidth of >50 GHz and an extinction ratio of 10 dB. The same device structure can also be used for a waveguide-coupled photodetector with a predicted responsivity of > 1 A/W and a 3 dB bandwidth of > 35 GHz. Use of the same GeSi composition and device structure allows efficient monolithic process integration of the modulators and the photodetectors on an SOI platform.

©2007 Optical Society of America

## 1. Introduction

The combined integration of electronic and photonic circuits has become an increasingly promising technology for high functionality extension of traditional technology shrink [1,2]. Among the key components of Si-based photonic technology are high performance photonic modulators and photodetectors. In recent years, significant progress has been made in Si modulators based on free carrier plasma dispersion, and the bandwidth has reached a few GHz [3,4]. High performance, waveguide-integrated Ge photodetectors on Si have also been demonstrated recently. A responsivity of ∼1.0 A/W and a 3 dB bandwidth greater than 4 GHz have been achieved [5].

Electro-absorption (EA) modulators are desirable for electronic-photonic integration due to their high speed and relatively low power consumption. Recently, we have demonstrated an enhanced Franz-Keldysh (FK) effect in tensile strained, epitaxial Ge-on-Si. [6]. The absorption contrast Δ*α*/*α*∼3.0 at 1647 nm, where *α* is the absorption coefficient at an electric field of 14 kV/cm and Δ*α* is the absolute change in the absorption coefficient due to the FK effect when the electric field increases to 70 kV/cm. Such an absorption contrast is comparable to the EA effect in Ge multiple quantum wells [7]. The optimal operation wavelength of our tensile strained Ge material is around 1647 nm due to the tensile strain induced direct band gap shrinkage [8,9]. To shift the optimal wavelength to 1550 nm, an effective way is to add a small amount of Si into Ge in order to increase its band gap.

In this paper we present a design of monolithically integrated, high performance Ge_{1-x}Si_{x} EA modulators and photodetectors on a silicon-on-insulator (SOI) platform. Since the FK effect takes place in sub-ps time scale [10], the speed of the EA modulator based on the FK effect is only limited by the RC delay and can be designed to achieve a bandwidth of >50 GHz. With adequate design of butt-coupling to high index contrast Si(core)/SiO_{2}(cladding) waveguides, a high extinction ratio of 10 dB can be achieved. The same device structure can also be used as a waveguide-coupled photodetector with a predicted responsivity of >1.0 A/W and a bandwidth of >35 GHz. Therefore, a monolithic integration of modulators, waveguides, and photodetectors with CMOS electronics can be achieved with our design.

## 2. Material design of the GeSi EA modulator

The EA property of the Ge_{1-x}Si_{x} material is modeled using the generalized FK theory [11]. We only consider the FK effect of the direct band gap and neglect that of the indirect gap because the former is three orders of magnitude stronger than the latter [12]. This model agrees well with the experimental results of epitaxial Ge-on-Si [6]. The input material parameters required in this model are the direct band gap (*E*
_{g}
^{Γ}), the effective mass of electrons (*m _{e}*) and holes (

*m*), the optical transition matrix element (

_{h}*E*), and the real part of the refractive index (

_{P}*n*).

_{r}The band gap is affected by the composition and the strain of Ge_{1-x}Si_{x}. The direct band gap of unstrained Ge_{1-x}Si_{x} is *E*
_{g}
^{Γ}(Ge_{1-x}Si_{x})=(0.8+3.26*x*)*eV* [13,14]. For optimal operation around 1550 nm we need a direct band gap slightly larger than 0.80 eV, therefore the Si composition is <3% in a candidate composition for EA modulation at 1550 nm. Tensile strain can enhance the EA effect in Ge [6], and it is preferred for modulator devices. About 0.2% thermally induced tensile strain is introduced to our epitaxial Ge-on-Si [8,9]. When *x*<0.03 the thermal expansion coefficient of Ge_{1-x}Si_{x} is very similar to Ge [14], so the tensile strain in epitaxial Ge_{1-x}Si_{x}-on-Si is also around 0.2% for *x*<0.03. Since the required length of the Ge_{1-x}Si_{x} EA modulators (tens of μm) is much larger than its cross-sectional dimensions (<1 μm), we have found that the strain in the transverse direction is almost fully relaxed and the structure is mainly strained along the longitudinal direction. Details about the experimental results of strain analysis will be reported elsewhere. Assuming the Ge_{1-x}Si_{x} modulator is oriented along the [110] direction on the Si wafer, the most common crystallographic direction of patterned rectangular features on Si due to the alignment of lithography, the band gaps from the maxima of light hole, heavy hole and split-off bands to the minima of the Γ valley *E*
_{g}
^{Γ}(*lh*), *E*
_{g}
^{Γ}(*hh*) and *E*
_{g}
^{Γ}(*so*) are calculated by the deformation potential theory for the case of uniaxial strain along the [110] direction [15]. The deformation potential, elastic constants and split-off energy are linearly interpolated between Ge and Si [9,13,14]. With the approach described above, the band gap of the Ge_{1-x}Si_{x} material can be obtained given the composition and the strain.

The effective mass of electrons and holes of Ge_{1-x}Si_{x} is almost the same as Ge for *x*<0.03, and the difference is actually within the experimental error [13]. Therefore, we simply use the electron and hole effective mass of Ge in our simulation for Ge_{1-x}Si_{x} with *x*<0.03, i.e., electron effective mass *m _{e}*=0.038

*m*(note that this is the electron effective mass at Γ valley corresponding to the direct band gap), light hole effective mass

_{0}*m*=0.043

_{lh}*m*and heavy hole effective mass

_{0}*m*=033

_{hh}*m*, where

_{0}*m*is the mass of a free electron. From the k∙p theory, the optical transition matrix element is given by [16]

_{0}The value of *E _{P}* can be easily calculated for Ge

_{1-x}Si

_{x}since we have already determined

*m*, and we can calculate the band gaps

_{e}*E*

_{g}

^{Γ}(

*lh*),

*E*

_{g}

^{Γ}(

*hh*) and

*E*

_{g}

^{Γ}(

*so*) with the methods described in the previous paragraph. Finally, the real part of the refractive index of Ge

_{1-x}Si

_{x}at 1550 nm is

*n*(Ge

_{r}_{1-x}Si

_{x}) =4.10–0.64

*x*[13,14].

With the material parameters described above, we are able to calculate the absorption coefficient of uniaxially strained Ge_{1-x}Si_{x} (*x*<0.03) in the presence of an electric field. In particular, we are most interested in the absorption contrast of Ge_{1-x}Si_{x} (Δ*α*/*α*) at 1550 nm. The breakdown field of intrinsic Ge is 125 kV/cm [17] and Ge_{1-x}Si_{x} with *x*<0.03 should have a similar breakdown behavior, so we adopt a maximum applied electric field of 100 kV/cm in our Ge_{1-x}Si_{x} EA modulator design. Figure 1(a) plots Δ*α*/*α* at 1550 nm as a function of Si content, assuming an applied electric field of 100 kV/cm at optical off-state and 10 kV/cm built-in electric field at optical on-state (0V bias) in the intrinsic Ge_{1-x}Si_{x} layer of a *p-i-n* diode structure. The optimal composition is around *x*=0.75%, giving an absorption contrast (Δ*α*/*α*) of ∼3.0. Figure 1(b) shows the absorption coefficient of Ge_{0.9925}Si_{0.0075} at 1550 nm as a function of electric field. At 10 kV/cm and 100 kV/cm the absorption coefficients are 158 and 633/cm, respectively. Due to the relatively high absorption coefficient of Ge_{0.9925}Si_{0.0075} at optical on-state (158/cm), the length of the EA modulator should be less than 70 μm for an insertion loss of less than 5 dB. For photodetectors, the high absorption state at a high electric field conveniently corresponds to the condition under reverse bias, resulting in an enhanced quantum efficiency together with a higher bandwidth. In the next section we will focus on the device design of monolithically integrated Ge_{0.9925}Si_{0.0075} EA modulators and photodetectors.

## 3. Device design of Ge_{0.9925}Si_{0.0075} EA modulators and photodetectors

The structure of a Ge_{0.9925}Si_{0.0075} EA modulator and a photodetector monolithically integrated on an SOI platform is schematically shown in Fig. 2. High index contrast Si(core)/SiO_{2}(cladding) waveguides are butt-coupled to the modulator and the photodetector. The Si waveguide is 500 nm wide and 200 nm high for single mode operation at 1550 nm. Currently, a fiber-to-waveguide coupling loss of <1 dB [18] and a propagation loss of ∼0.35 dB/cm in the Si waveguide [19] can be achieved. Since this paper focuses on the modulator and detector design, we do not include the Si waveguide loss into the design parameters. The GeSi EA modulator and the photodetector are of the same vertical Si/Ge_{0.9925}Si_{0.0075}/Si *p-i-n* diode structure with a doping level of 2× 10^{19}/cm^{3} in *n*
^{+} and *p*
^{+} Si, and their heights (H) and widths (W) can be designed for the optimal device performance. The only difference in the dimensions of the GeSi EA modulator and the photodetector is that the latter is longer than the former (L_{2}>L_{1}) to increase the absorption. Use of the same GeSi composition and device structure allows efficient monolithic process integration of the modulators and the photodetectors on an SOI platform.

For electronic-photonic integration the TE mode in Si waveguides is preferred because a small bending radius of ∼1 μm can be achieved with low loss. Therefore, we will focus our discussions on the coupling of TE modes in these devices. The modes of the Si waveguide and the GeSi EA modulator are calculated with *Apollo Photonics Solution Suite 2.3* using the complex refractive indexes of the materials and the full vector method in the finite domain analysis. Both the mode profile and the complex effective indexes are obtained from the mode solver. Two factors should be considered in calculating the insertion loss and the extinction ratio of the GeSi EA modulator: the modal overlap Ω in the *x-y* plane (see Fig. 2) between the Si waveguide and the GeSi EA modulator, and the transmittance *t* through the modulator in the *z* direction. The modal overlap (Ω) is obtained by calculating the overlap integral between the TE modes of the Si waveguide and the GeSi EA modulator. The transmittance (*t*) in the *z* direction is calculated by the propagation matrix element method, using the complex effective indexes of the GeSi EA modulator and the Si waveguides. The transmittance (*t*) thus calculated takes into account both the impedance mismatch in the *z* direction and the absorption in the GeSi EA modulator. The insertion loss and extinction ratio are given by

where Ω(0), *t*(0), Ω(V) and *t*(V) are the modal overlap and transmittance at 0 bias and a reverse bias of *V*, respectively. The square terms of Ω in the equations take into account the coupling loss into and out of the GeSi EA modulator due to modal mismatch in the *x-y* plane. As the electric field has little effect on the real part of the refractive index of Ge_{0.9925}Si_{0.0075} (|Δ*n _{r}*|≤10

^{-3}), the mode profile in the GeSi EA modulator is not affected by the electric field. Therefore Ω(0) ≈ Ω(V) to obtain the approximation for the extinction ratio in Eq. (2b).

Figure 3(a) plots the extinction ratio over insertion loss of 50 μm-long GeSi EA modulators with different widths (W) and thicknesses (H) of the GeSi active layers. Since we want a high extinction ratio and a low insertion loss, the value on the vertical axis in Fig. 3(a) should be maximized with the optimal design. In this calculation, we specify that the reverse bias applied at the optical off-state of the modulator should not exceed 3.3 V to be compatible with 180 nm CMOS technology. Nor should the electric field exceed 100 kV/cm due to the limit of material breakdown. The optimal GeSi thickness is determined to be ∼400 nm. Below this thickness the optical confinement in the GeSi active layer decreases significantly and the extent of optical modulation is reduced. Above this thickness the modal overlap with the Si waveguide (Ω) and the electric field at 3.3 V reverse bias both decrease, leading to a higher coupling loss and a lower extinction ratio. The optimal width is ∼700 nm. However, for H=400 nm the values of extinction ratio over insertion loss are almost identical for W=600–800 nm. To reduce the capacitance, a narrower device is preferred. Therefore, we choose a dimension of H=400 nm and W=600 nm. Figure 3(b) plots the bandwidth, extinction ratio and insertion loss as a function of device length for a Ge_{0.9925}Si_{0.0075} EA modulator with W=600 nm and H=400 nm. We assume 50 × 50 μm^{2} metal contact pads with a vertical distance of 4 μm from the substrate, so the pad capacitance is 21.6 fF. For a 50 μm-long GeSi modulator we predict a high extinction ratio of 10 dB and a 3 dB bandwidth of >50 GHz. The corresponding insertion loss is 5 dB, including 1 dB coupling loss budget and 4 dB material absorption loss. The coupling loss can be decreased to 0 with optimal coupling design. Depending on the applications, the extinction ratio can be traded off for lower material loss by decreasing the device length. The size of the metal contact pad can be decreased to further increase the RC-limited bandwidth. With 180 nm CMOS fabrication technology the misalignment between the waveguide and the modulator can be easily controlled below 50 nm, and our simulation results show that the insertion loss increases by <0.5 dB and the extinction ratio is not affected in that case. Therefore, the design can be well implemented with 180 nm CMOS technology.

The same *p-i-n* diode structure can also be used as a waveguide-integrated photodetector. The responsivity *R* at a reverse bias of *V* is given by

where *r* is the reflectance at the input port of the photodetector, and *α*
_{eff}(*V*) is the effective absorption coefficient of Ge_{0.9925}Si_{0.0075} in the detector at a reverse bias of *V*. Equation (3) is adapted from the responsivity of a normally incident photodetector [20] by considering the modal overlap and the effective absorption coefficient in the case of waveguide coupling. Note that *α*
_{eff}(*V*) is different from the material absorption coefficient of Ge_{0.9925}Si_{0.0075} since not all the light is confined in GeSi. To obtain *α*
_{eff}(*V*) we set the imaginary indexes of the *n*
^{+} and the *p*
^{+} Si to 0 and calculate the imaginary effective index *n*
_{i,eff}(*V*) of the detector structure. The mode profile remains the same in this way, and *n*
_{i,eff}(*V*) is only due to the absorption of Ge_{0.9925}Si_{0.0075}. So we have *α*
_{eff}(*V*) =4π*n*
_{i,eff}(*V*)/*λ*. The 3 dB frequency of a photodetector is determined by the carrier transit time and the RC delay. Figure 4 plots the responsivity and 3 dB frequency of waveguide-coupled Ge_{0.9925}Si_{0.0075} photodetectors as a function of device length. A high responsivity of ∼1.1 A/W and a 3 dB bandwidth of >35 GHz can be achieved with an 80 μm-long device. By shortening the device we can also trade off lower responsivity for higher speed, depending on the requirement of applications. High bandwidth driver circuits for the modulator and trans-impedance amplifiers (TIA) for the photodetector can be achieved with SiGe bipolar CMOS (BICMOS) technology and integrated with the modulator and the detector devices [21].

## 4. Conclusions

We have presented a design of monolithically integrated GeSi EA modulators and photodetectors on a SOI platform. The GeSi EA modulator is optimized for operation at 1550 nm, and has a predicted 3 dB bandwidth of >50 GHz and an extinction ratio of 10 dB. The photodetector has a predicted responsivity of ∼1.1 A/W and a 3 dB bandwidth of >35 GHz. Both devices utilize the same material composition and device structure, so they can be monolithically integrated on-chip.

Early portion of this work was partially supported by Pirelli Lab, S.p.A. This research was sponsored under the Defense Advanced Research Projects Agency’s (DARPA) EPIC program supervised by Dr. Jagdeep Shah in the Microsystems Technology Office (MTO) under Contract No. HR0011-05-C-0027. The authors would like to thank Dr. John Yasaitis for helpful discussions.

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