## Abstract

We propose and study a practical design of a Germanium photodetector implemented on a Silicon-on-insulator substrate to reach the critical coupling regime under vertical illumination at 1310 nm wavelength. With appropriate optimization procedures, a high efficiency bandwidth product larger than 50 GHz and a large 3dB spectral full width around 30 nm can be obtained given realistic material parameters and fabrication constraints. Our device is fully compatible to the state-of-art CMOS process technology, and may serve as a high performance, low cost solution for the optical receiver in Silicon photonics based optical interconnects.

© 2012 OSA

## 1. Introduction

Ge has recently become an attractive material in making near infrared photodetectors due to the advancement of direct epitaxy growth of high-quality Ge thin film on Si substrate. Its compatibility with CMOS process technology and low cost nature promise great potential in integrating with modern IC and in replacing conventional InGaAs photodetectors [1]. From the point of view of packaging, detectors with vertical illumination configuration can be easily handled by pick-and-place tools and is consequently a popular choice in making photoreceiver module. In the literature, vertically illuminated Ge on Si photodetectors with high responsivity [2,3] or high speed [4,5] at telecomm wavelengths have been demonstrated respectively, but the overall efficiency bandwidth product (EBP) is usually limited to ~5 GHz for the conventional designs without any additional optical coupling mechanism. One solution to the problem of low EBP is by incorporating Ge on a Silicon-on-insulator (SOI) waveguide [6–9] so that the photon absorption direction is no more parallel to the carrier drift direction. This way, the quantum efficiency and optical bandwidth can be individually optimized with little trade-off. For applications where off-chip coupling to fiber is needed, it is however necessary to interface the waveguide photodetectors with horizontal [10,11] or vertical [12,13] couplers, which inevitably degrade the overall external quantum efficiency by a few more dB. Moreover, the packaging scheme can be quite complex because, e.g., a facet polishing is needed for a horizontal spot-size-converter [10], and a non-90 degree emission will occur for a vertical grating coupler [12]. A very tight fiber alignment tolerance is another problem.

In early 90’s, it was known that the efficiency of a vertically illuminated photodetector can be increased without sacrificing its bandwidth by placing reflectors in adjunction with absorption region [14]. Due to the multiple reflections of incoming light by the reflectors, the absorption region length is effectively prolonged and such a technology is usually termed as resonant cavity enhanced (RCE) photodetector. See ref. 15 for a review on implementing RCE photodetectors on III-V semiconductor platform. In fact, unity quantum efficiency can be reached if the device is appropriately designed, which is a phenomenon tightly related to the so-called “critical coupling” that is routinely observed in different systems consisting of optical fiber or waveguide coupled to high *Q* microcavity [16,17]. In this paper, we apply the concept of critical coupling and design a practical Ge on SOI photodetector that is capable of improving the EBP by an order of magnitude compared to the conventional Ge on Si photodetectors [2–5]. We outline an optimization procedure and the calculations are furthered verified with numerical simulations. Note that although Si and Ge RCE photodetectors have been experimentally demonstrated [18,19] in recent years, they are far from the critical coupling regime to be discussed in this paper. Our work shall pave the way to realize the full potential of a vertically illuminated Ge photodetector.

## 2. Device design considerations and calculations

In Fig. 1(a) , we show the schematic plot of a Ge photodetector based on a commercially available SOI substrate with 250 nm thick crystalline Si (c-Si) and 3 μm thick buried oxide (BOX). Ge is assumed to be epitaxially grown on c-Si with high quality so that the defect induced leakage current arisen from Ge-Si interface lattice-mismatch can be minimized. Phosphorus and Boron are used for the n + and p + doped regions in Si and Ge, respectively. After Ge mesa etch, the Ge surface is passivated by a thin layer of amorphous Si (a-Si) [20] and NiSi contacts with Al electrodes are applied for backend process. The aperture accepting incoming light has a diameter equal to 14 μm, which is chosen to butt-couple a single-mode fiber with unity fill factor. The 5 μm and 1 μm design rules (from Ge mesa sidewall to n and p electrodes) as well as the 10 μm and 1 μm metal trace widths (for n and p electrodes) are limited by i-line lithography tool. In Fig. 1(b), we add a front side mirror by depositing Si and oxide to form a dielectric distributed-Bragg-reflector (DBR). A back side mirror is fabricated by first wet-etching the substrate Si to open up a V groove and then coat it with a thin Al layer.

The use of metal coating on BOX to form a back side mirror is based on three considerations. First, the BOX layer serves as a Si wet etch stop so that the cavity thickness can be precisely controlled. Second, we apply thin metal coating instead of thick DBR film on BOX to avoid stresses that can potential crack the wafer due to large V groove topography. Finally, a Si-oxide-Al layer offers ~98% reflectivity that is superior compared to a Si-Al layer with only ~89% reflectivity. The reflectivity of back side metallic mirror has a crucial impact on our device performance and will be discussed in detail in the following paragraphs.

To model our device, we construct an effective cavity structure shown in Fig. 2
. The BOX and Al layers are considered as a part of back side metallic mirror, and the DBR layers above a-Si are considered as a part of front side dielectric mirror. The reflection between Ge and a-Si or c-Si interface is neglected because it’s only ~0.8%, given Ge refractive index *n _{Ge}* = 4.2 and Si refractive index

*n*= 3.5. Such an approximation is valid in the critical coupling regime that we are interested in and will be confirmed numerically in Sec. 3. By applying the formalism of waveguide-ring resonator coupling [17], we can show that when the structure is vertically illuminated,

_{Si}*R*is the total reflectance of our structure.

*a*and

*b*are the incident and reflected fields.

*γ*

^{2}is the power decay ratio after a round trip, and

*r*

^{2}is the dielectric mirror reflectance when light is injected from air.

*θ*is the round trip phase shift. Note that analogy between Eq. (4) in ref. 17 and Eq. (1) here can be established if the waveguide input/output ports on chip are associated with the incident/reflected fields in free space. The condition of critical coupling occurs at

*r*and

_{M}*r*are the reflectivities of metallic and dielectric mirrors seen from the cavity, respectively.

_{D}*λ*

_{0}is the wavelength of interest.

*κ*is the imaginary part of Ge refractive index, and

_{Ge}*t*is the Ge layer thickness. When the condition in Eq. (2) is met, the light reflected by the dielectric mirror to air and the light transmitted from the cavity to air destructively interfere. All incoming light is therefore locked completely inside the cavity and dissipated by Ge and metal absorptions. In Fig. 3 , we plot the required

_{Ge}*t*to reach the critical coupling as a function of |

_{Ge}*r*|

_{M}^{2}and |

*r*|

_{D}^{2}given

*λ*

_{0}and

*κ*equal to 1310 nm and 0.08. It is shown that for a given

_{Ge}*t*, a critical coupling can happen as long as |

_{Ge}*r*|

_{M}^{2}> |

*r*|

_{D}^{2}. Note that the white region in Fig. 3 fails to satisfy this requirement.

For our Ge photodetector, only the optical power absorbed by Ge may contribute to the electrical current so the actual quantum efficiency at a critical coupling is calculated by

*η*is plotted as a function of |

*r*|

_{M}^{2}and |

*r*|

_{D}^{2}in Fig. 4(a) , and a near unity quantum efficiency can only be obtained when |

*r*|

_{M}^{2}is sufficiently close to 1. Next, we calculate the bandwidth of our Ge photodetector, which is controlled by the carrier transit time in the depletion region of p-i-n junction and the circuit delay time. The total optical bandwidth can be expressed bywhere the first and second terms in the square root correspond to carrier drift time and device resistance-capacitance delay time, respectively.

*v*is the hole (slower carrier) saturation velocity in Ge and is taken as 6x10

_{s}^{6}cm/s;

*R*is the series resistance and

*C*is the total capacitance. To accurately determine the bandwidth, we calculate

*R*using$R=\frac{\rho}{4\pi t}+\frac{\rho}{t}\frac{\delta}{2\pi {r}_{o}}\mathrm{coth}\frac{w}{\delta}$for a circular geometry [21], in which the first and second terms are caused by sheet resistance (thin doped regions in Ge and Si) and contact resistance (NiSi plus Al), respectively.

*ρ*and

*t*are the resistivity and thickness of the doping region.

*δ*is the transfer length of a contact and is equal to$\sqrt{{\rho}_{c}t/\rho}$where

*ρ*is the specific contact resistivity.

_{c}*r*is the distance between detector center and the middle of ring electrode trace.

_{o}*w*is the width of ring electrode trace. The following parameters [22] are used:

*ρ*= 9x10

^{−4}Ω·cm and 5x10

^{−4}Ω·cm for n + Si and p + Ge;

*t*= 150 nm;

*ρ*= 5x10

_{c}^{−8}Ω·cm

^{2}and 3.29x10

^{−6}Ω·cm

^{2}for n + NiSi-Si contact and p + NiSi-Ge contact. The final

*R*is equal to 9.85 Ω (Ge related) plus 4.92 Ω (Si related) ~15 Ω. In addition, $C={n}_{Ge}{}^{2}{\epsilon}_{0}A/{t}_{Ge}$is used to calculate the junction capacitance but neglect the parasitic capacitance associated with electrode pads.

*A*is the area of p-i-n junction. In Fig. 4(b),

*BW*at a critical coupling is plotted as a function of |

*r*|

_{M}^{2}and |

*r*|

_{D}^{2}. It can be seen that the maximum bandwidth can be obtained at

*t*~300 nm and its value is slightly larger than 50 GHz. We also plot the EBP at a critical coupling as a function of |

_{Ge}*r*|

_{M}^{2}and |

*r*|

_{D}^{2}in Fig. 4(c), and find that a > 50 GHz EBP can be readily reached if |

*r*|

_{M}^{2}is sufficiently close to 1.

Since the proposed Ge photodetector is based on reaching the critical coupling regime in a one-sided resonant cavity coupled to free-space, it is natural to ask whether the cavity linewidth is too small so the efficiency degrades significantly when there is an incident laser wavelength fluctuation or thermally induced wavelength drift. In Fig. 5(c)
, we plot Δ*λ*, the spectral full-width-half maximum (FHWM) of cavity resonance at a critical coupling, as a function of |*r _{M}*|

^{2}and |

*r*|

_{D}^{2}using the formula

*λ*<<

*λ*

_{0}.

*m*is the cavity order defined as the round rip phase shift at

*λ*

_{0}divided by 2π. Note that the a-Si thickness is adjusted for every given |

*r*|

_{M}^{2}and |

*r*|

_{D}^{2}so that

*λ*

_{0}is positioned at 1310 nm. In addition, a minimum a-Si thickness (see Fig. 5(b)) is chosen to have the lowest possible cavity order (see Fig. 5(a)), i.e., picking up the smallest possible cavity

*Q*factor, to maximize the spectral FWHM. It is shown that at |

*r*|

_{M}^{2}~98% and |

*r*|

_{D}^{2}~61%, a ~51 GHz EBP can be reach accompanied with a Δ

*λ*~51 nm (

*Q*~26). Such a broad spectral response suggests that our device is robust against the wavelength mismatch between incoming laser and photodetector.

## 3. Numerical simulations

Although the analysis discussed above has captured the main idea of an effective cavity structure shown in Fig. 2, it is necessary to perform numerical simulations to take into account the finer design details. In the following, we use a commercial finite-difference-time-domain (FDTD) simulation package [23] to investigate our critically coupled Ge photodetector. Here we target the point (|*r _{M}*|

^{2}, |

*r*|

_{D}^{2}) = (61.5%, 98.1%) on Fig. 4(c) with an EBP = 51.4 GHz. The corresponding

*t*is 304 nm. For the part of dielectric mirror, a three quarter-wave-plate structure, i.e., oxide-Si-oxide on top of a-Si, has a reflectance 65.6% and can be tweaked to 61.5% using DBR Si thickness ~93 nm and DBR oxide thickness ~226 nm. For the part of metallic mirror consisting of BOX and Al, we plot the simulated |

_{Ge}*r*|

_{M}^{2}as a function of

*t*in Fig. 6(a) . The maximum reflectance is 98.4% if the BOX thickness

_{BOX}*t*is equal to quarter wavelength in oxide, and drops to 89.1% if

_{BOX}*t*is equal to half wavelength in oxide. Eventually

_{BOX}*t*~2.922 μm (a number that is closest to 3 μm) is chosen to pick up 98.1% reflectance.

_{BOX}In Fig. 6(b), we first simulate *R* in the spectral domain with Ge and Al absorptions artificially turned off (blue curve). In this case the incoming laser is completely rejected by the cavity. When the Al absorption is turned on, a dip at 1.31 μm appears showing our design has a correct resonance wavelength (green curve). Finally, by turning both Ge and Al absorptions on, we observe *R* approaching zero at 1.31 μm, which is the direct evidence of reaching the critical coupling regime (red curve). Such an observation also justifies the previous assumption of neglecting the reflections at Ge-Si interfaces. Note that in the simulations the a-Si thickness *t _{a-Si}* is set to be ~125 nm; oxide refractive index

*n*= 1.447 is assumed.

_{Ox}As discussed previously, the actual quantum efficiency at a critcal coupling depends on the portion of optical power absorbed by Ge and is not unity when there is metal absorption. To simulate this quantity, we enclose the Ge region by Poynting vector moniters and calculate the the power difference between the output and input directions. This corresponds exactly to the actual quantum efficiency and is plotted in Fig. 7
(red-solid curve). At 1310 nm, *η* is ~95% and matches very nicely with analytical prediction with < 1% difference. On the other hand, Δ*λ* is ~30 nm (*Q* ~44) and is smaller than the analytical prediction. It can be understood as the thicknesses of dielectric and metal mirrors are neglected previously, which underestimates the cavity *Q* factor. Note that the above simulations are all done via 2D FDTD method with Bloch boundary condition in the horizontal direction. We have also performed a full-structure 3D FDTD simulation, and the resultant *η* is plotted in Fig. 7 (yellow-dashed curve) in which the peak value is reduced to ~90% at 1310 nm. This is due to a small portion of light diffracting horizontally out of the mirror coverage during cavity photon lifetime. In Fig. 7(a) and 7(b), we adjust *t _{a-Si}* and

*t*respectively to investigate the impact of layer stack uniformities. It is observed that the peak wavelength shift is quite sensitive to both

_{BOX}*t*and

_{a-Si}*t*variations (though the case of a-Si is stronger as expected), which suggests a precise control over layer stack thicknesses is crucial to having a high device yield over the whole wafer.

_{BOX}## 4. Summary

A critically coupled Ge photodetector fabricated on SOI substrate under vertical illumination is proposed and studied. The analytical calculations are verified with numerical simulations, and it is shown that a > 90% quantum efficiency and > 50 GHz optical bandwidth operation at 1310 nm wavelength is accessible. Consequently, the corresponding EBP is enhanced by an order of magnitude compared to conventional Ge-on-Si photodetectors. The spectral FWHM of an optimum design is ~30 nm, and further improvement on the spectral response to feature a “flat-top” shape may be obtained by designing a dielectric mirror with anomalous dispersion [24]. This can be an attractive solution not only as a robust, stand-alone photodetector but also for application of wavelength-division multiplexing (WDM) in vertical illumination configuration.

## Acknowledgments

The work is partially supported by Ministry of Education and National Tsing-Hua University (#100N7080E1) in Taiwan. T.T.W. and C.Y.C. thank Chih-Kuo Tseng for useful discussions on device fabrication.

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