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Abstract
The external bandwidth of germanium waveguide photodetectors (PDs) decreases with the device length due to the load and parasitic effects even if the internal one is less affected. Shortening PDs raises the external bandwidth but lowers the responsivity, introducing a trade-off between the two figures of merits. Here, we present a scheme of waveguide PDs based on total internal reflections of corner reflectors. The reflector can be easily fabricated with the standard photolithography at the end of PDs to efficiently reflect optical power back to germanium for additional absorption, allowing for further size reduction. The structure may render the optimization of PDs more flexible.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thanks to the growth of high-quality germanium (Ge) on silicon (Si) substrates [1,2], Ge photodetectors (PDs) fabricated in the waveguide (WG) scheme now play an essential part in Si photonic integrated circuits at the receiver end [3–5]. Over the past decades, Ge PDs with a bandwidth of several-ten gigahertz (GHz) have been demonstrated [6–17] and incorporated into commercial optical-communication modules based on Si photonics [18]. Under complementary-metal-oxide-semiconductor compatible fabrications preferred by most manufacturers, how to further push the bandwidth of Ge WG PDs to the hundred-GHz range while maintaining a proper responsivity becomes an inevitable issue as the amount of data keeps on growing.
Both the responsivity and bandwidth of Ge WG PDs should be high so that a significant signal-to-noise ratio and data transmission rate can be supported. The internal bandwidth of WG PDs is limited by the carrier transport transverse to the WG, which includes the transit process across reverse biased photon-absorptive media (transit response) and drift counterpart through conductive regions [resistance-capacitance ($RC$) response]. Both processes limit the internal bandwidth of Ge WG PDs. In the limit of lateral transport, the transit and $RC$ responses are both influenced by the thickness of Ge, and the latter is additionally affected by the size of WG cross section. While the internal bandwidth seems less affected by the WG length, the external bandwidth does depend on it. The external bandwidth is typically lower than the internal one and decreases with the increasing WG length due to the joint effect of load impedance and capacitance that scales with the WG length [6,12,14,17]. On the other hand, the longer Ge WG brings about the more thorough photon absorption and increases the responsivity. A trade-off between the bandwidth and responsivity is hence introduced due to the WG length even if the $RC$ time seems to be length-independent. The optimizations of these two figures of merits (FOMs) need to be compromised through the proper choice of WG lengths.
A strategy to increase the margins of these two FOMs is to redirect optical signals back to the Ge WG for the second-pass absorption. In this way, the increment of responsivity can be traded for the enhancement of bandwidth. Such a principle has been demonstrated by fabricating distributed Bragg reflectors (DBRs) in the form of etched grating structure behind Ge WG PDs [14,17]. In views of the tolerance, reproducibility, and yield of lithography, it is reasonable to adopt a reflective photonic structure which not only provides a decently high reflectivity but yet is as simple for fabrications as possible (more error tolerance).
In this work, we theoretically investigate the improvement on the responsivity and bandwidth of Ge WG PDs through the introduction of corner reflectors (CRs) at the end of WG. The CR is effectively a high-index prism in the shape of isosceles right triangle and was first examined as a feedback mirror for semiconductor lasers [19]. The working principle of CRs is based on the total internal reflection (TIR), which is also a candidate scheme for mode redirection in Si photonics [20,21]. Through the two successive TIRs of incoming waves incident onto the two short sides of CRs at an incident angle of about $\pi /4$, the optical power is sent back to the Ge WG for additional absorption. While the reflectivity of CRs may not be as high as that of the more elaborate photonic structures, it is sufficient for the strategy here. In fact, the true advantage of CRs lies in their easy fabrication through the standard ultraviolet (UV) photolithography. The linewidth issue is not critical for CRs.
The rest of the paper is organized as follows. In section 2., we will introduce the structures of Ge WG PDs with the CR. The DC and AC characteristics are presented in sections 3. Finally, a conclusion will be given in section 4.
2. Device structure
Figure 1(a) shows the schematic of the Ge WG PD in our study. The PD WG is extended along the $x$ axis and ended with a CR whose top view appears as an isosceles right triangle with a corner angle of $\pi /2$. The device is surrounded by silicon dioxide (SiO$_{2}$). The bulk Ge is responsible for photon absorption, and the corresponding Shockley-Read-Hall lifetime is set to 1.5 ns [22]. The height $h$ and width $w$ of Ge region are set to 300 nm and 3 $\mu$m, respectively, and its (effective) length $L$ (measured from the front of Ge region to the middle of CR) would be varied. The upper portion of Ge is diffusively n-doped with a peak doping density of $10^{18}$ cm$^{-3}$ while the lower part is also diffusively p-doped (unintentionally) due to the p-type Si with a doping density of $10^{19}$ cm$^{-3}$. The thickness of intrinsic Ge between the two doped regions is estimated as $200$ nm. Together with the 70 nm thick p-type Si layer (etched down from the 220 nm Si layer of a silicon-on-insulator substrate) beneath Ge, the Si-Ge heterostructure forms a p-i-n diode. Aluminum (Al) is adopted as the contact metal of the PD. A 600 nm wide stripe contact atop the n-type Ge and two counterparts on the 70 nm heavily p-doped Si layer, which are symmetrically located at a distance $d=2.5~\mu$m away from the centerline of PD, are responsible for outputting the photocurrent. The cross section of Ge is mostly aligned to that of the tapered Si, which is gradually expanded from a 450 nm $\times$ 220 nm Si WG in the butt-coupling scheme. The overlap area between the Si taper and Ge WG is 150 nm in height, as shown by the inset in Fig. 1(a). The input Si WG only supports one transverse-electric-like (TE-like) mode at wavelength $\lambda =1.55~\mu$m.
Fig. 1. (a) The schematic of the Ge WG PD. The Ge WG is ended with a CR. The inset shows the side view of Si tapered WG and Ge WG. (b) The trajectory of an incident ray parallel to the WG axis. The ray is redirected back to Ge by the two TIRs at the CR. (c) and (d) show the field patterns $|\mathbf {E}(\mathbf {r})|$ excited by the TE-like mode of Si WG along the mid horizontal plane of Ge region in the PDs with and without the CR, respectively. The interference near the CR is a feature of two TIRs. The longer intensity tail outside the PD in (c) indicates the more power leakage.
The schematic in Fig. 1(b) indicates how the optical field is reflected by the CR in views of ray optics. The incident field can be regarded as an optical beam traveling along the WG. A ray in the beam propagates along the WG and hits one short side of the CR at an incident angle of about $\pi /4$. The TIR that occurs at the corresponding Ge-SiO$_{2}$ interface redirects the ray toward the other short side at a similar incident angle. The second TIR then sends the ray back to the Ge WG for another round of absorption. The aforementioned picture would be valid if both the WG width $w$ and lateral extent of incoming field are much larger than the effective wavelength $\lambda /n_{\mathrm {eff}}$ in Ge, where $n_{\mathrm {eff}}$ is the effective index of the fundamental TE-like mode. The effective index $n_{\mathrm {eff}}$ of this mode is around 3.81 at $\lambda =1.55~\mu$m, corresponding to an effective wavelength of 0.407 $\mu$m. The WG with $w=3~\mu$m supports multiple propagating modes at $\lambda =1.55~\mu$m, and the second TE-like mode is present at $w>300$ nm. Although a width $w$ of 3 $\mu$m is only about a few effective wavelength wide, the picture of ray optics remains valid. Still, the incident field may contain components other than the fundamental TE-like mode of Ge WG, indicating that some of the incoming and reflected rays actually propagate with a noticeable off-axis angle with respect to the WG axis ($x$ axis).
With the TE-like mode of Si WG at 1.55 $\mu$m as the input, we numerically excite the optical fields $\mathbf {E}(\mathbf {r})$ around the PDs with and without the CR. The effective length $L$ of the Ge WG is set to $3~\mu$m. The field magnitudes $|\mathbf {E}(\mathbf {r})|$ along the mid horizontal ($xy$) plane of Ge region corresponding to the PDs with the CR and without the CR are shown in Fig. 1(c) and 1(d), respectively. An spotted interference pattern due to the incoming and redirected fields can be observed near the CR. On the other hand, the longer tail outside the PD without the CR indicates that a significant portion of optical power may leak out of it. For both PDs, the distribution of the interference does not appear as the simple standing-wave along the WG. This indicates the presence of the higher-order modes in those WGs. However, as long as the majority of optical power carried by these modes are absorbed by Ge, their presence would not affect the performance of PD much.
3. DC and AC characteristics
In this section, we present the comparison between Ge PDs with and without the CR through simulations. We note that our results are not meant for precise predictions of device parameters. The simulations cannot catch every detail of realistic devices such as the parasitic effect of circuits. Still, they are able to reflect the relative improvement of performances under specific changes of device scheme. Albeit various non-ideal effects present in realistic devices, an estimation of improvements based on our simulations is worthy of reference.
The enhanced reflection due to CRs raises the optical power absorbed by Ge in the PD. Accordingly, the external quantum efficiency and responsivity of the device are increased. Figures 2(a) and 2(b) show the fractions of absorbed power $F$ and responsivities, respectively, for two Ge WG PDs with and without the CR as a function of the WG length $L$. The faction $F$ is the ratio between the power $P_{\mathrm {abs,Ge}}$ absorbed by Ge and input power $P_{\mathrm {in}}$ into the Si WG:
Fig. 2. (a) The fractions of absorbed power $F$ as a function the WG length $L$ for the Ge PDs with (blue) and without (red) the CR. The CR enhances the absorption noticeably, especially at the shorter $L$. (b) The corresponding responsivities versus $L$. The responsivity is enhanced by the CR accordingly.
The optical power absorbed by the PD with the CR is noticeably higher than that of the PD without the CR at the shorter length side due to the more reflected power back into the Ge WG. At the longer WG length, the fractions of absorbed power and responsivities of the two PDs are similar since for both devices, the guided modes in the incoming optical fields would be absorbed completely before they reach the end of WG. At the same level of the responsivity, the required length $L$ of PDs with the CR is usually one half to two thirds of that corresponding to PDs without the CR, indicating that the former device may potentially exhibit an external bandwidth less limited by the inherent capacitance and load impedance.
We further note that the sharpness of the corners in the reflector is not necessarily critical to the enhancement of absorption and responsivity. Even if we replace the right-angled corner of the reflector with a $\pi /2$ sector whose curvature radius $r$ is not too large and do similarly to the other two $3\pi /4$ counterparts [see the inset in Fig. 3(a)], reflectivity of the PD can be still maintained at a decent level close to 90 %. In Fig. 3(a), we show the power reflectivity as a function of the curvature radius $r$ of the three corners. Here, the fundamental TE-like mode of the 3 $\mu$m wide WG composed of Si and Ge is utilized as the incident field, and the absorption coefficient of Ge is set to zero in order to isolate the effect of reflections. The power reflectivity is defined as the ratio between the reflected power of all the guided modes and incident power into the 3 $\mu$m wide WG. The power reflectivity changes little as $r$ is smaller than 200 nm. It decreases appreciably at the larger $r$ as the CR begins to behave like a submicron lens which focuses the beam outside the device. Since a characteristic size of 200 nm can be achievable through the standard UV photolithography, the resolution of fabrication would not be a limiting factor to the performance of Ge PDs with the CR.
Fig. 3. (a) The power reflectivity of the CR versus the curvature radius $r$ of the three corners. Below a radius of $200$ nm, the reflectivity changes little. The inset shows the schematic of a CR with a finite curvature radius $r$. (b) The power reflectivity of the CR versus its opening angle $\theta _{\mathrm {CR}}$. The reflectivity drops less prominently at the increasing side of $\theta _{\mathrm {CR}}$. The inset indicates the angle $\theta _{\mathrm {CR}}$. (c) The power reflectivities of the CRs with $r=0$ and $200$ nm versus the sidewall angle $\theta _{\mathrm {sw}}$. In both cases, the reflectivities are only reduced slightly as the angle decreases to 70$^{\circ }$. Around $\theta _{sw} \approx 90^{\circ }$, the difference between the two reflectivities is most prominent. The inset show the schematic of sidewall with angle $\theta _{\mathrm {sw}}$.
The ideal opening angle $\theta _{\mathrm {CR}}$ of the CR should be close to 90$^{\circ }$ [see inset of Fig. 3(b)]. However, the angle may deviate from the optimized one in realistic devices. Figure 3(b) shows the power reflectivity as a function of $\theta _{\mathrm {CR}}$. The reflectivity peaks near $\theta _{\mathrm {CR}} \approx 90^{\circ }$ and decreases toward both sides of the optimized angle. It drops more rapidly toward the decreasing side of $\theta _{\mathrm {CR}}$. If the deviation of $\theta _{\mathrm {CR}}$ from 90$^{\circ }$ is inevitable, an opening angle larger than 90$^{\circ }$ would be preferred.
The verticalness of the Si/Ge sidewalls is another factor that might lower the fraction of the power reflected back to the WG PD. In Fig. 3(c), we show the power reflectivities at curvature radii $r=0$ and $200$ nm, respectively, as a function of the sidewall angle $\theta _{\mathrm {sw}}$ [see the inset in Fig. 3(c)]. A perfectly vertical sidewall should have a $\theta _{\mathrm {sw}}$ of 90$^{\circ }$, and typical $\theta _{\mathrm {sw}}$ ranges from 70$^{\circ }$ to 90$^{\circ }$, depending on the details of fabrications. In our calculations, the slanted sidewalls are also present at the CR. As shown in Fig. 3(c), when the sidewall angle $\theta _{\mathrm {sw}}$ decreases to 70$^{\circ }$, the power reflectivity at $r=0$ nm is reduced slightly to 84 %. The reflectivity at $r=200$ nm is generally smaller than the counterpart at $r=0$ nm, but the former is only lower than later by 2 to 3% at $\theta _{\mathrm {sw}} \approx 90^{\circ }$, at which the decrement is most prominent.
The surface roughness on CRs can introduce additional scattering and reduce the reflectivity. We may make a quick estimation on the decrement using Tien’s method [23], which is valid for surface roughness with a long correlation length. In the CR, a ray experiences two TIRs and gets scattered twice. In the regime of Rayleigh scattering, each scattering event introduces a fractional power loss $f$ as follows [23]:
where $\sigma$ is the standard deviation of surface roughness; and the factor $\pi /4$ in the cosine represents the incident angle of TIRs. The overall power loss could then be estimated as $2f$. Substituting $n_{\mathrm {eff}}=3.81$ at $\lambda =1.55~\mu$m and set $\sigma =2$ nm [24], we obtain $2f\approx 0.38~\%$, which should be acceptable in views of other losses introduced during fabrications.The aforementioned discussions indicate that the functionality of CR is robust against the imperfections of fabrications and increases the margins of responsivity and external bandwidth.
Since the Ge PD with the CR redirects more power back to the device for absorption, the photocurrent is also more easily saturated by the high input power due to the screening effect brought by the optically generated carriers. This effect lowers the electric field which separates electrons and holes. Hence, the linearity of the device (photocurrent versus input power) would be potentially less ideal for the PD with the CR in the high-power regime. Figure 4 shows the responsivities of the Ge PD with and without CR as a function of the input power at $L=3~\mu$m. While the responsivity of the PD with the CR is larger at small input power, it also drops more with the input power. On the other hand, the significant decrement of the responsivity only occurs at a power level around 4 mW, which is higher than typical input power received by Ge PDs after the optical wave experiences various losses in a Si photonic chip. Therefore, even if the nonlinearity is more prominent for Ge PDs with the CR, it may be still tolerated.
Fig. 4. The responsivities versus input power for Ge PDs with and without the CR. Although the PD with the CR has the higher responsivity at low input power, it also decreases more prominently in the high-power regime.
Figure 5(a) shows the real and imaginary parts of simulated small-signal frequency response $H(\omega )$ (in linear scale) of the Ge PD with the CR versus the modulation frequency $\omega /2\pi$ of optical signals at $\lambda =1.55~\mu$m ($\omega$ is the angular frequency of modulation). The response is normalized so that its DC magnitude is unity. The WG length $L$ is set to $3~\mu$m, and the bias voltage $V$ is set to $-1$ V. In the reverse-bias regime, the electrons and holes in Ge are drifted by the electric field at the corresponding saturated velocities $v_{\mathrm {e}}=6 \times 10^{6}$ cm/s and $v_{\mathrm {h}}=5.4 \times 10^{6}$ cm/s, respectively (default values of Lumerical). To grasp the features of simulated response, we utilize the following expression [25] and least-square method to fit them:
Fig. 5. (a) The comparison between the simulated frequency responses $H(\omega )$ based on the effective 2D method and corresponding fittings for the Ge PD with the CR at $L=3$ $\mu$m and $V=-1$ V. The red and blue solid (dashed) curves are the real and imaginary parts of simulated results (fittings), respectively. (b) The magnitude $|H(\omega )|$ from simulations (dots). The corresponding fitting is shown in solid line. (c) The internal bandwidth versus $V$.
In Fig. 5(c), we present the internal bandwidth of the Ge PD with the CR at $L=3~\mu$m versus the bias voltage $V$ using the effective 2D method. In the revere-bias regime considered here, the internal bandwidth of the device can all exceed 61 GHz. We note that the internal bandwidth first increases but then decreases gradually with the bias voltage $V$. The voltage $V$ prolongs the depletion region of n-type Ge and therefore increases the effective height $h_{\mathrm {eff}}$ of Ge region. The effect of an increasing $h_{\mathrm {eff}}$ is two-fold. First, it can lower the inherent capacitance $C$ of the p-i-n diode and reduces the $RC$ time. On the other hand, it also increases the transit times $\tau _{\mathrm {t,e}}$ and $\tau _{\mathrm {t,h}}$ of carriers when they drift through the Ge region. As a result, the internal bandwidth is maximized at an optimal effective height $h_{\mathrm {eff}}$. We suggest that the initial increment followed by the slight decrement of the internal bandwidth with $V$ in Fig. 5(c) just reflects the crossing across this optimal effective height.
We further assume that the inherent capacitor of p-i-n diode in Ge PDs can be effectively modeled as a parallel-plate one with a capacitance $C=\epsilon _{\mathrm {Ge,s}}(A/h_{\mathrm {eff}})$, where $\epsilon _{\mathrm {Ge,s}}$ and $A~(\propto L)$ are the static permittivity and horizontal area (on the $xy$ plane) of the Ge region, respectively. In this way, we estimate $R \approx 371.17~\Omega$ and $C \approx 5.69$ fF for the Ge PD with the CR at $L=3~\mu$m and $V=-1$ V. These pieces of information can help us evaluate the influence of load impedance on the external bandwidths of the Ge PDs.
As a PD outputs the photocurrent to the external system, its external bandwidth would be lowered in general. Figure 6(a) shows the equivalent circuit of a generic PD. In the limit of large shunt resistance $R_{\mathrm {SH}}$, the photocurrent mainly flows to the outside system which is effectively modeled as an external load with impedance $Z_{\mathrm {L}}$. As a result, the $RC$ time constant in the response $H_{RC}(\omega )$ is replaced with $(R+Z_{\mathrm {L}})C$. Assuming that $Z_{\mathrm {L}}$ is mainly resistive, for example, 50 or 100 $\Omega$, which are typically adopted for most microwave and radio-frequency circuit systems, the overall charging/discharging time constant is prolonged, and hence the external bandwidth of PDs as seen by the outside system is reduced. The increment $Z_{\mathrm {L}}C$ of this time constant is less prominent if the capacitance $C$ is smaller. At the same level of responsivity, since the Ge WG PD with the CR is shorter than that without the CR, the external bandwidth of the former is less affected by $Z_{\mathrm {L}}$ than the latter is. Figure 6(b) shows the external bandiwdths versus $Z_{\mathrm {L}}$ in the resistive regime for the Ge WG PDs with the CR at $L=3~\mu$m and the counterpart without the CR at $L=6~\mu$m. The bias voltage is set to $V=-1$ V, and the two WG lengths are chosen such that they have similar responsitivities of about 1.03 A/W. Due to the smaller WG length $L$ (lower $C$), the external bandwidth of the Ge WG PD with the CR drops much less rapidly with $Z_{\mathrm {L}}$ than that of the device without the CR does. This indicates that CRs can help shorten the length of Ge WG PDs under the constraint of responsivity and prevent the reduction of external bandwidths.
Fig. 6. (a) The equivalent circuit model of a generic PD. A resistive load impedance $Z_{\mathrm {L}}$ prolongs the $RC$ time constant to $(R+Z_{\mathrm {L}})C$. (b) The external bandwidths of the PDs with ($L=3~\mu$m) and without the CR ($L=6~\mu$m) versus resistive $Z_{L}$ at $V=-1$ V. Due to the shorter $L$, the external bandwidth of the former is less affected by $Z_{\mathrm {L}}$.
In [14], the DBR at the end of a 10 $\mu$m long Ge PD could enhance the responsivity from 0.85 to 0.9 A/W at $\lambda =1.55~\mu$m (5.9 % improvement). In [17], the optimized DBR can increase the responsivity of a 5 $\mu$m long device from 0.6 to 0.72 A/W (20 % improvement). Also, at the same level of responsivity, the bandwidth of the 5 $\mu$m long device with DBR is 31.7 GHz, which is higher than that of the 10 $\mu$m long device without DBR (24.8 GHz). While a direct comparison between the numbers from our simulations and existing data in literature is inappropriate, a comparison on performance improvements may be taken for reference. Our simulated device with the CR at a device length of 3 $\mu$m could potentially exhibit a 25.6 % improvement on responsivity at $\lambda =1.55~\mu$m (0.82 to 1.03 A/W). At an external load impedance of 100 $\Omega$, the external bandwidth of the 3 $\mu$m long device with the CR can also be potentially higher than that of the 6 $\mu$m long device without the CR by 17%. These numbers show that the improvements brought by the CR may be comparable to those due to the more elaborate DBR. Yet, the structure of our proposed device is much simpler, and the fabrication would be less demanding.
4. Conclusion
In conclusion, we have investigated the effect of CRs on the responsivity and bandwidth of Ge WG PDs in Si photonic integrated circuits. With CRs at the end of Ge WGs, the optical power can be redirected back to the Ge region for the second-pass absorption. In this way, the responsivity is raised, which allows for the further reduction of the WG length and therefore the enhancement of external bandwidth. While other photonic structures may provide the higher reflection than CRs do, CRs are more easily incorporated into Ge WG PDs with the standard UV photolithography. They do not require a high-resolution fabrication process. Integrating CRs with Ge WG PDs can be a cost-effective method to increase the responsivity and external bandwidth.
Funding
Academia Sinica (AS-iMATE-109-41, RCAS Research Grant); Ministry of Science and Technology, Taiwan (MOST 105-2221-E-001-021-MY3, MOST 109-2224-E-992-001, MOST 108-2221-E-001-018-MY3).
Acknowledgments
The authors would like to thank Kuo-Fang Chung and Dr. Po-Han Fu for the helpful assistance and discussion.
Disclosures
The authors declare no conflicts of interest.
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