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**I.**Introduction**II.**Present Status of Ge-Based Active Photonic Devices for Si Photonics**III.**Theoretical Formulas for Operation Frequency of pin PDs**IV.**Calculated Results and Discussions**V.**Summary**Appendix A.**3-dB Cutoff Frequency Due to Transit Delay and RC Delay**Appendix B.**3-dB Cutoff Frequency Due to Transit Delay Based on a Simplified Model**Appendix C.**3-dB Cutoff Frequency Due to Transit Delay Based on Continuity Equations

SECTION I

Si photonics is a technology that integrates photonic devices with electronic devices on an Si chip [1], [2], [3]. Applications include not only optical transceivers for relatively short-distance interconnects such as server-to-server and chip-to-chip interconnects but also on-chip optical interconnects such as intercore communications in the central processing units. Applications are not limited to the information technology but expanded to optical biological sensors [4], [5]. The photonic devices should be fabricated taking into account the compatibility with Si complementary metal–oxide–semiconductor (CMOS) fabrication process. The mass-productive CMOS process can lead to a low-cost fabrication of photonic–electronic integrated circuits.

In this paper, operation frequencies are discussed for near-infrared (NIR) photodiodes (PDs) using Ge epitaxial layers on Si, which are crucial for the photonic–electronic convergence on an Si chip. After describing the present status of Ge active photonic devices on Si, theoretical investigations on the operation frequency are presented for Ge pin PDs. According to the formula derived from the continuity equation, Ge pin PDs are found to operate with the 3-dB cutoff frequency as high as 80 GHz, which is limited by the slow diffusion current from the n and p layers. A new structure of p-Ge/i-Si/n-Si heterojunction is also examined to increase the operation frequency over 100 GHz.

SECTION II

The Si channel/rib optical waveguide is one of the most fundamental devices in Si photonics. Si is a superior material for NIR waveguides due to the transparency in the wavelength range of 1.3–1.6 $\mu\hbox{m}$ used in the optical fiber communications. The large difference of refractive index $n$ between the Si core $(n = 3.5)$ and the $\hbox{air/SiO}_{2}$ cladding $(n = 1.0/1.5)$ reveals a strong confinement of light in the Si core region. This property enables a small bending of waveguide with the radius less than 10 $\mu\hbox{m}$ as well as single-mode channel waveguides with the small width/height less than 0.5 $\mu\hbox{m}$. Such a narrow Si waveguide can be fabricated using a state-of-the-art lithography and an anisotropic dry etching of Si-on-insulator (SOI) wafers. The scattering losses due to the sidewall roughness [6] have been reduced as low as 3 dB/cm or below [7], and passive waveguide devices such as ring resonators and arrayed waveguide gratings have been realized. These passive devices satisfy the light propagation within a chip. However, the fabrication of active photonic devices such as light sources, optical modulators, and PDs remains as a challenging issue because of the small interaction between Si and the NIR light. Although Si pn-junction optical modulators with a Mach–Zehnder interferometer configuration [8], [9] have been developed utilizing the plasma-dispersion effect, where the refractive index is reduced by injecting free carriers, the size (more than 100 $\mu\hbox{m}$ in length) and the consumed power (at least, 5 pJ/bit) should be reduced for the integration with other devices. The use of ring resonators [10] can reduce the size and the power, while a severe control of resonance wavelength is necessary.

Ge is one of the group-IV semiconductors, having an indirect band structure similar to Si. From the viewpoint of photonic devices, Ge is often regarded as quasi-direct bandgap material [11], since the difference of conduction band energy between the indirect L valley and the direct $\Gamma$ valley is as small as 0.14 eV [11], [12]. It is important that the direct bandgap energy of 0.80 eV corresponds to 1.55 $\mu \hbox{m}$ in wavelength. As a result, Ge shows a large optical absorption below 1.55 $\mu\hbox{m}$, being suitable to PDs in the optical communication band. Although Ge possesses a large lattice mismatch of 4% with Si, high-quality Ge epitaxial layers can be grown using chemical vapor deposition (CVD) techniques with a low–high temperature two-step growth (typically, 350 and 600 °C), followed by a postgrowth annealing at a higher temperature of $\sim\!900\ ^{\circ}\hbox{C}$, as Luan *et al.* reported almost one decade ago [13]. The threading dislocation density (TDD) can be reduced below $10^{7}\ \hbox{cm}^{-2}$. After this breakthrough, NIR active photonic devices using Ge epitaxial layers on Si have been reported, such as PDs [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and electro-absorption optical modulators utilizing the Franz–Keldysh effect [28], [29]. Multiple quantum-well structures of Ge/SiGe heterojunction have been also applied to the fabrication of optical modulators using the quantum-confined Stark effect [30], as well as NIR PDs [31]. Ge-based light emitters have also attracted interests recently [32], [33], [34], [35], [36], [37], [38], [39], and an intense light emission around 1.55 $\mu\hbox{m}$ has been demonstrated from the forward-biased Ge pn junction [36], [37]. An optical gain [38] and a lasing [39] have been also reported for heavily doped n-Ge, where electrons fill the conduction band minima at the L valley, leading to an enhanced direct transition of electrons from the $\Gamma$ valley to the valence band, rather than to the L valley for the relaxation by the phonon emission. The details on the Ge growth and the device applications have been summarized in our previous paper [40].

Among the Ge-based photonic devices, Ge pin PDs on Si have been most widely investigated. So far, in addition to the free-space PDs, Ge pin PDs integrated with Si waveguides [41], [42], [43], [44], [45] have been reported for the photonic–electronic convergence. For the practical applications, PDs are required to possess 1) high quantum efficiency (high responsivity), 2) high sensitivity, and 3) high operation frequency.

The internal quantum efficiency has been reported to approach almost 100% [14], [15], [18], [19]. Large responsivities more than 0.1 A/W can be obtained in the most of communication bands from the O band (1.28–1.36 $\mu\hbox{m}$) to the L band (1.56–1.62 $\mu\hbox{m}$) as well as in the C band (1.53–1.56 $\mu\hbox{m}$) [15], [18], [19]. It is noted that the absorption edge due to the direct transition exceeds 1.6 $\mu\hbox{m}$, which is much larger than 1.55 $\mu\hbox{m}$ for the bulk Ge, as the authors reported [15], [18], [19] and several other groups also reported later [24], [27], [42], [46]. This red shift is derived from the narrowing of direct bandgap induced by the tensile strain of $\sim$0.2% [15], [18], [19]. Although the compressive strain in Ge due to the lattice mismatch is relaxed after the growth more than the critical thickness ($\sim$1 nm), such a tensile strain is accumulated during the cooling from the growth/annealing temperature to the room temperature due to the mismatch of thermal expansion coefficient [15], [18], [19]. To further enhance the responsivity, Ge/Si heterojunction avalanche PDs (APDs), which are composed of a Ge absorption layer and a Si multiplication layer, have been developed recently [47].

The second requirement is to realize the high sensitivity for the detection of weak optical signals. The dark leakage current is an important property to determine the sensitivity, although an input noise of transimpedance amplifier may be more important for the ultrahigh-speed applications, as suggested in a recent invited article in this journal [27]. The dark current in Ge pin PDs is typically 10 $\hbox{mA/cm}^{2}$ at the reverse bias of 1 V, when annealed Ge layers with TDD $\sim 10^{7}\ \hbox{cm}^{-2}$ are used [13], [14]. The minimum value reported is $\sim\!\!1\ \hbox{mA/cm}^{2}$ by optimizing the growth/annealing condition [24], [27]. Detailed discussions on the dark current can be seen in [27]. Although the dark current in Ge pin PDs on Si is larger than that for InGaAs PDs more than one order of magnitude, PDs are applicable to the detection of light with the intensity as small as, e.g., −20 dBm required for the receiver optical subassembly (ROSA) in the fiber-to-the-home applications. APDs [47] should help the increase of sensitivity. For integration with Si CMOS electronic devices, however, one of the critical issues is to eliminate high-temperature postgrowth annealing of Ge, which is against the fabrication of Ge devices at the back end of the Si CMOS process. The process temperature should be below $\sim\!450\ ^{\circ}\hbox{C}$, while the dark current is increased more than 100 $\hbox{mA/cm}^{2}$ for PDs using as-grown defective Ge layers $(\hbox{TDD} \sim 10^{9}\ \hbox{cm}^{-2})$ [13], [21], [27]. Further studies are necessary to prepare high-quality Ge without the postgrowth annealing.

The third requirement is to increase the operation frequency to convert high-speed optical signals to the electrical ones. Higher operation frequency is crucial for not only the optical interconnects but also other applications such as millimeter-wave generation [48] and high-resolution imaging with time-of-flight techniques. In recent years, several groups have reported the operation frequency beyond 30 GHz [20], [26], [42], [44], and the highest frequency reported is 49 GHz [26]. In order to further increase the operation frequency, the thickness of i layer in the pin structure should be reduced for the reduction of transit delay. However, the thickness reduction increases the electric field strength, leading to the increase of dark leakage current, as well as the breakdown of i layer. Furthermore, the slow diffusion current from the n and p contact layers injected to the i layer cannot be ignored for such PDs, as shown later. In the following chapters, the operation frequency for Ge pin PDs is discussed based on the theoretical formula. An approach for the frequency enhancement is also discussed, where a new structure of p-Ge/i-Si/n-Si is proposed.

SECTION III

The operation frequency of pin PDs is determined by two types of delay time: 1) carrier transit delay through the diode and 2) RC delay. The simplest equivalent circuit is shown in Fig. 1. Here, a PD generates a photocurrent $I_{PD}(\omega)$ with the angular frequency $\omega = 2\pi f$ ($f$: the frequency), which is supplied to a capacitor with the depletion layer capacitance $C_{PD} = \varepsilon A/W_{i}$ ($\varepsilon$ is the permittivity, $A$ the diode area, and $W_{i}$ the thickness of i layer) and a load resistor with the resistance $R_{load}$. The 3-dB cutoff frequency $f_{3\,dB}$ is given by TeX Source $$f_{3\,dB}\sim{1\over 2\pi}\sqrt{1\over \tau_{PD}^{2} + \tau_{RC}^{2}} = \sqrt{1\over 1/f_{PD}^{2} + 1/f_{RC}^{2}}\eqno{\hbox{(1)}}$$ where $\tau_{PD}$ is the transit delay, $\tau_{RC} = R_{load}C_{PD}$ the $RC$ delay, $f_{PD} = 1/2\pi\tau_{PD}$, and $f_{RC} = 1/2\pi \tau_{RC}$ [49]. The details to derive Eq. (1) are described in Appendix A.

In Ge-on-Si pin PDs, a pin structure is formed in a thin film of Ge grown on a Si substrate. Carriers are optically generated only in Ge, since there is no interband transition in the Si substrate for the wavelength more than $\sim\!\!1.1\ \mu\hbox{m}$. A schematic band diagram of a pin structure is shown in Fig. 2(a), together with optically generated carriers under a low-injection condition. There are three photocurrent components due to the carrier generation in Ge: 1) electrons and holes generated in the i-Ge layer, which are drifted by the electric field (drift current); 2) holes in the n-Ge layer; and 3) electrons in the p-Ge layer, which are diffused to the i-Ge layer (diffusion current). In order to analyze the transit-limited 3-dB cutoff frequency, a simplified model shown in Fig. 2(b) has been widely used, which is based on the assumption that a drift current flows due to the carrier injection at the one edge of i layer [50]. As in Appendix B, the 3-dB cutoff frequency $f_{PD}$ due to the transit delay is expressed as TeX Source $$f_{PD} \sim {0.443\over\tau^{i}}\eqno{\hbox{(2)}}$$ where $\tau^{i}$ is given by $\tau^{i} = W_{i}/\upsilon$ ($W_{i}$: the thickness of i layer, $\upsilon$: the saturation velocity of carriers under large reverse voltages). However, based on this formula, the operation frequency should be underestimated, since the transit delay through the i layer is overestimated due to the assumption that all the generated carriers need to transit throughout the i layer. Furthermore, the diffusion from the p and n layers is not included in Eq. (2). Particularly for high-speed pin PDs, an i-Ge layer with a reduced thickness $(<\ 1\ \mu\hbox{m})$ is used. In such a case, the thickness of i-Ge approaches to those for the n and p layers. The optical absorption should be comparable among these three layers, meaning that the diffusion current from the p and n layers significantly contributes to the photocurrent. Since the diffusion needs an additional transit length, the operation frequency obtained from Eq. (2) should be, in turn, overestimated. Although one may claim that the diffusion current is eliminated by increasing the doping concentration in the n and p layers due to the Auger recombination, the carrier diffusion length is estimated to be larger than the typical thickness for the p and n layers $(\sim\! 0.1\ \mu\hbox{m})$, even if the concentration is increased up to the highest limit of $\sim\!\! 10^{20}\ \hbox{cm}^{-3}$ [32]. From the fabrication point of view, the control of doping profile should also become difficult with increasing the doping concentration, while the thickness reduction of n and p layers for the reduction of diffusion current would increase the contact resistance.

The formulas, taking into account the diffusion current as well as the drift current with a more precise treatment, are obtained based on the continuity equations [50]. As in Fig. 2(a), the electric field $E$ in the i layer $(0 \leq x \leq W_{i})$ is assumed to be constant, while no field is assumed in the n $(-W_{n} \leq x \leq 0)$ and p $(W_{i} \leq x \leq W_{i} + W_{p})$ layers. For simplicity, photocarriers are generated uniform in depth $x$, but the generation rate oscillates with time $t$, i.e., TeX Source $$g(x, t) = g(t) = g_{0}(1 + Ae^{i\omega t}).\eqno{\hbox{(3)}}$$ Here, $\omega$ is the angular frequency and $A$ ($\leq$ 1) is the amplitude of oscillation. The spatially uniform generation should be valid for waveguide PDs, where a vertical pin structure is illuminated from the lateral direction. Even for the free-space configuration, Eq. (3) should give a good approximation if the penetration depth of light $1/\alpha$ ($\alpha$: the absorption coefficient) is larger than the thickness of pin structure. Although the details are described in Appendix C, the transit-limited 3-dB cutoff frequency $f_{PD} = \omega_{PD}/2\pi$ is obtained by solving $\Re_{PD}(\omega_{PD}) = 1/2$, where $\Re_{PD}(\omega)$ is given by TeX Source $$\Re_{PD}(\omega) = \left\vert{W_{i}\left[S_{n}^{i}(\omega) + S_{p}^{i}(\omega)\right] + W_{n}S_{p}^{n}(\omega) + W_{p}S_{n}^{p}(\omega) \over W_{i}\left[S_{n}^{i}(0) + S_{p}^{i}(0)\right] + W_{n}S_{p}^{n}(0) + W_{p}S_{n}^{p}(0)}\right\vert^{2}.\eqno{\hbox{(4)}}$$ Here TeX Source $$\eqalignno{S_{n}^{i}(\omega) =&\, 1/i\omega\tau_{n}^{i} - \left(1 - e^{-i\omega\tau_{n}^{i}}\right)\Big/\left(i\omega\tau_{n}^{i}\right)^{2}&\hbox{(5a)}\cr S_{p}^{i}(\omega) =&\, 1/i\omega\tau_{p}^{i} - \left(1 - e^{-i\omega\tau_{p}^{i}}\right)\Big/\left(i\omega\tau_{p}^{i} \right)^{2}&\hbox{(5b)}\cr S_{p}^{n}(\omega) =&\, {1\over W_{i} + W_{n}}\left[1 + {W_{i}\over W_{n}}\left(1 - e^{-i\omega\tau_{p}^{i}}\right)\Big/i\omega\tau_{p}^{i}\right] \cdot {L_{p}\over \beta_{p}} \cdot \tanh(\beta_{p}W_{n}/L_{p})&\hbox{(5c)}\cr S_{n}^{p}(\omega) =&\, {1\over W_{i} + W_{p}}\left[1 + {W_{i}\over W_{p}}\left(1 - e^{-i\omega\tau_{n}^{i}}\right)\Big/i\omega\tau_{n}^{i}\right] \cdot {L_{n}\over \beta_{n}} \cdot \tanh(\beta_{n}W_{p}/L_{n}).&\hbox{(5d)}}$$ In these equations, $\tau_{n}^{i}$ and $\tau_{p}^{i}$ are defined as $\tau_{n}^{i} = W_{i}/\upsilon_{n}$ and $\tau_{p}^{i} = W_{i}/\upsilon_{p}$ ($\upsilon_{n}$ and $\upsilon_{p}$ are the saturation velocities for electrons and holes), $L_{n} = \sqrt{D_{n}\tau_{n}}$ and $L_{p} = \sqrt{D_{p} \tau_{p}}$ are the diffusion lengths for electrons in the p layer and holes in the n layer, $\tau_{n}$ and $\tau_{p}$ are the recombination lifetimes for electrons and holes, and $\beta_{n}$ and $\beta_{p}$ are defined as $\beta_{n} = \sqrt{1 + i\omega \tau_{n}}$ and $\beta_{p} = \sqrt{1 + i\omega \tau_{p}}$, respectively.

SECTION IV

Fig. 3 shows a calculated result of 3-dB cutoff frequency due to the drift current as a function of i-Ge thickness (0.01–1 $\mu\hbox{m}$). As the first step, the diffusion current was ignored in the formula derived from the continuity equation, i.e., $W_{n} = W_{p} = 0$ in Eq. (4), in order to examine whether the 3-dB cutoff frequency from Eq. (2) for the drift component is underestimated, as mentioned above. The parameters used in the calculation are summarized in Table 1 [12], [51]. Basically, the values reported for bulk Ge were used, although the lattice strain in Ge on Si ($\sim$0.2% tensile strain in CVD-grown Ge [15], [18], [19]) may modify the transport property in addition to the optical absorption. The solid lines in Fig. 3 show the results without the $RC$ delay. It is found that the results based on the continuity equation (red line) shows 26% larger frequencies than those from Eq. (2) (black line), indicating that an underestimation occurs for Eq. (2), as anticipated. The effect of $RC$ delay was also included, as shown by the dashed lines for the diode areas of 10, 100, and 1000 $\mu\hbox{m}^{2}\ (R_{load} = 50\ \Omega)$. The frequencies are reduced with decreasing the i-Ge thickness due to the increase of $C_{PD}$. According to Fig. 3, the operation frequency more than 100 GHz can be realized when the i-Ge thickness and the diode area are reduced to $\sim\! 0.3\ \mu\hbox{m}$ and $\sim\!\! 10\ \mu\hbox{m}^{2}$, respectively.

Next, effects of the diffusion current and the breakdown of i layer were taken into account. The diffusion parameters of $\tau_{n} = \tau_{p} = 1\ \hbox{ns}$, $D_{n} = 20\ \hbox{cm}^{2}/\hbox{s}$, and $D_{p} = 10\ \hbox{cm}^{2}/\hbox{s}$ were used. These values are typical ones for heavily doped $(>\ 10^{18}\ \hbox{cm}^{-3})$n and p layers. The diffusion lengths are $L_{n} = 1.4\ \mu\hbox{m}$ for electrons in the p layer and $L_{p} = 1.0\ \mu\hbox{m}$ for holes in the n layer, which are larger than the thicknesses of p and n layers (0.1 $\mu\hbox{m}$ or below used in the calculation). Thus, the diffusion current flows from the n and p layers to the i layer. Note that, even if the lifetimes (or the diffusion coefficients) are reduced by two orders of magnitude, the diffusion length is still larger than 0.1 $\mu\hbox{m}$.

Fig. 4 shows the relationship between the 3-dB cutoff frequency and the i-Ge thickness. Even when the $RC$ delay (blue region in Fig. 4) is negligible due to the use of small area PDs, the diffusion current from the n and p layers (green region) is found to cause a reduction of 3-dB cutoff frequency from the result taking into account only the drift current through i-Ge (red line). This is ascribed to the larger transit delay for the diffusion current. In the case of 0.1-$\mu\hbox{m}$-thick n and p layers, corresponding to the blue line at the boundary between the diffusion-limited green region and the $RC$-limited blue region, the maximum frequency is found to be limited to $\sim$80 GHz. This frequency is obtained for the i-Ge thickness around 0.15 $\mu\hbox{m}$, while further reduction of i-Ge thickness decreases the frequency due to the reduced contribution of drift current in the i layer with the smaller transit delay. The maximum frequency of $\sim$80 GHz is much smaller than that in Fig. 3, which suggests that the operation frequency more than 100 GHz is realized by simply reducing the i-Ge thickness to $\sim\! 0.3\ \mu\hbox{m}$. In order to achieve 100-GHz operation, a precise control of n and p layer thicknesses and the doping concentrations should be required. However, it is not easy to form abrupt p-Ge/i-Ge and i-Ge/n-Ge interfaces, since the diffusion coefficients of dopant impurities are relatively large for Ge [12], [51].

Additionally, as shown by the red region in Fig. 4, the breakdown of i-Ge should be also important for thin i-Ge layers $(<\ 0.1\ \mu\hbox{m})$. Under the reverse bias of $V_{R} = 1\ \hbox{V}$, the avalanche breakdown of i-Ge occurs below the thickness of $\sim\! 0.05\ \mu \hbox{m}$, which was obtained from the sum of $V_{R}$ and built-in voltage ($\sim$0.6 V), divided by the breakdown field of $\sim$300 kV/cm for thin $(<\ 0.1\ \mu\hbox{m})$ Ge depletion layers [51], [52]. Even before the breakdown, the sensitivity of PDs would be degraded due to the increased dark leakage current under the high electric field, where the thermal generation of carriers is enhanced by the Frenkel–Poole emission [51], [53]. Such an increased dark current also suggests the difficulty of realizing the operation frequency over 100 GHz.

In order to increase the operation frequency, a new structure is examined, which is composed of p-Ge/i-Si/n-Si heterojunction, as shown in the right of Fig. 5. In this structure, i-Si and n-Si layers are used instead of i-Ge and n-Ge layers in the ordinary pin PDs. The band diagram for the p-Ge/i-Si/n-Si structure is shown in Fig. 6. According to Ref. [14], the valence band offset $\Delta E_{v}$ is 0.36 eV at the interface between Ge (relaxed) and Si, while the conduction band offset $\Delta E_{c}$ between the L point in Ge and the $\Delta$ point in Si is $\sim$0.1 eV. These values are in good agreement with the ones based on the alignment of hybrid orbital energy/charge neutrality level between Ge and Si $(\Delta E_{v} = 0.29\ \hbox{eV})$ [54]. The small conduction band offset allows the flow of electrons beyond the Ge/Si interface. It should be mentioned that a larger $\Delta E_{v}$ of 0.79 eV is obtained from the model-solid theory in [55]. A type-II band alignment is anticipated, where the conduction band bottom of Ge is located higher in energy than that of Si. Even in this case, the electron flow from Ge is not prevented. It should be also mentioned that 0.2% tensile strain in Ge grown on Si causes only a small influence on the band offsets [55], since the bandgap reduction is as small as $\sim$0.03 eV.

As in Fig. 6, although no optical absorption due to the interband transitions occurs in i-Si and n-Si (wavelength more than $\sim\!\! 1.1\ \mu \hbox{m}$), photocurrent flows due to the diffusion of electrons generated in p-Ge, followed by the collection with the i-Si layer having an electric field. A similar concept of separate absorption and carrier collection has been utilized in InGaAs/InP UTC PDs [48], [56] to obtain $f_{3\,dB}\ >\ 100\ \hbox{GHz}$, where the hole current with a low transport speed is prevented. In Ge PDs, the speed of hole flow, limited by the saturation velocity, is almost the same as that of electrons (see Table 1). Instead, the replacement from i-Ge to i-Si should effectively increase the 3-dB cutoff frequency due to 1) larger saturation velocity, 2) larger breakdown electric field strength for Si than that for Ge, which enables the use of thinner i layer, and 3) smaller dielectric constant (capacitance) to reduce the $RC$ delay. The use of wider gap Si is also effective to reduce the dark current, since the thermal generation of carriers through the midgap levels is suppressed. From the viewpoint of crystalline quality of i-Si, the structure shown in the bottom right ofFig. 5 is better rather than the top right, since the homoepitaxy of Si can be used to form i-Si on the n-Si substrate, although p-Si substrates have been widely used for Ge pin PDs on Si [14].

For p-Ge/i-Si/n-Si PDs, the 3-dB cutoff frequency is calculated, taking into account the diffusion current from the p-Ge layer to the i-Si layer. The 3-dB cutoff frequency $f_{PD} = \omega_{PD}/2\pi$ (without the $RC$ effect) is obtained by solving $\Re_{PD}(\omega_{PD}) = 1/2$, where $\Re_{PD}(\omega)$ is given by TeX Source $$\Re_{PD}(\omega) = \left\vert{S_{n}^{p}(\omega)\over S_{n}^{p}(0)} \right\vert^{2}.\eqno{\hbox{(6)}}$$ Here TeX Source $$S_{n}^{p}(\omega) = {1\over W_{i} + W_{p}}\left[1 + {W_{i}\over W_{p}}\left(1 - e^{-i\omega\tau_{n}^{i}}\right)/i\omega \tau_{n}^{i}\right] \cdot {L_{n}\over \beta_{n}} \cdot \tanh(\beta_{n}W_{p}/L_{n})\eqno{\hbox{(7)}}$$$\tau_{n}^{i}$ is defined as $\tau_{n}^{i} = W_{i}/\upsilon_{n}$, where $\upsilon_{n}$ is the saturation velocity for electrons in the i-Si carrier-collection layer.

Fig. 7 shows the 3-dB cutoff frequency as a function of i-Si thickness for p-Ge/i-Si/n-Si PDs. It is found that the red region, showing the breakdown, is reduced due to the larger breakdown field strength for i-Si ($\sim$900 kV/cm). In the case of a p-Ge thickness of 0.1 $\mu\hbox{m}$, corresponding to the boundary between the green region and the blue region in Fig. 7, the maximum 3-dB cutoff frequency of $\sim$90 GHz was obtained. This value is only slightly larger than that for the ordinary pin PDs ($\sim$80 GHz). However, the operation frequency more than 100 GHz can be obtained by slightly reducing the thickness of p-Ge layer from 0.1 $\mu\hbox{m}$, as shown by the green dashed lines in Fig. 7. It is important that the fabrication of p-Ge/i-Si/n-Si structure should be more robust than that for the ordinary pin structure in terms of the doping profile control, since the diffusion coefficients of dopant impurities are much smaller for Si [12], [51]. An abrupt doping profile should be realized in the heteroepitaxy/ion-implantation process around the p-Ge/i-Si interface, in comparison with n-Ge/i-Ge and p-Ge/i-Ge homointerfaces in the ordinary pin structure. Thus, it is concluded that the p-Ge/i-Si/n-Si structure is more promising for increasing the operation frequency over 100 GHz.

However, the optical absorption in p-Ge/i-Si/n-Si PDs is reduced in comparison with the ordinary Ge pin PDs, since the absorption is limited to the thin layer of p-Ge. In order to increase the optical absorption (quantum efficiency/responsivity), the waveguide PD configuration should be effective, because the optical absorption is dominated by the length of absorption layer rather than the thickness. Using the 2-D finite-difference time-domain (FDTD) method, the optical absorption in p-Ge/i-Si/n-Si PDs was calculated. Fig. 8 shows cross-sectional distributions of electric field for the TE-mode CW light propagating through a stack of Ge ($n = 4.0$ and absorption coefficient $\alpha = 4000\ \hbox{cm}^{-1}$ at 1.55 $\mu\hbox{m}$) and Si $(n = 3.5)$ films surrounded by the $\hbox{SiO}_{2}$ cladding $(n = 1.5)$. Here, the thickness of Si was fixed to be 0.2 $\mu\hbox{m}$, which is the typical height for Si channel waveguides fabricated from SOI wafers, while the Ge thickness was changed as a parameter (0, 0.1, 0.2, and 0.3 $\mu\hbox{m}$). It is found from Fig. 8 that the light intensity is significantly reduced during the propagation through the Ge/Si structures. Even for Ge with a thin layer of 0.1 $\mu\hbox{m}$ [see Fig. 8(b)], the intensity reduction can be clearly seen during the propagation in several micrometers. Quantitatively, an absorbance more than $\sim$80% is obtained for 10-$\mu\hbox{m}$-long waveguide PDs with the 0.1-$\mu\hbox{m}$-thick Ge layer, as shown in Fig. 9. The effective absorption coefficient $\alpha_{eff}$ is found to be 1700 $\hbox{cm}^{-1}$, corresponding to the optical penetration length $1/\alpha_{eff}$ of 5.9 $\mu\hbox{m}$. This length was found to be not sensitive to the width of waveguide PDs, according to the FDTD calculations. Thus, the waveguide PDs, having a width of less than $\sim\!\! 1\ \mu\hbox{m}$ and a length of several micrometers (the area below 10 $\mu \hbox{m}^{2}$), can show the 3-dB cutoff frequency above $\sim$100 GHz with keeping the high quantum efficiency/responsivity. As for the responsivity and dark current, further studies are necessary on the effect of defects at the p-Ge/i-Si interface due to the lattice mismatch. Applicability of p-Ge/i-Si/n-Si structures to high-speed APDs, where the i-Si layer acts as the multiplication layer in addition to the carrier-collection layer, will also be discussed elsewhere.

SECTION V

Operation frequency was discussed for NIR PDs using Ge layers on Si, which are crucial for the photonic–electronic convergence on an Si chip. Based on the formula derived from the continuity equation, Ge pin PDs were found to operate with the 3-dB cutoff frequency as high as 80 GHz, which is limited by the slow diffusion current from the n and p layers to the i layer. In order to increase the operation frequency, a new structure of p-Ge/i-Si/n-Si heterojunction was examined. In this structure, electrons generated in the p layer of Ge are collected by the i layer of wider gap Si, being similar to UTC PDs of InGaAs/InP in III–V systems. Higher operation frequencies were expected, reflecting the larger saturation velocity of carriers for i-Si. Fabrication of p-Ge/i-Si/n-Si structure should be more robust than that for the ordinary pin structure in terms of the doping profile control.

APPENDIX A

The operation frequency of PDs is described, taking into account 1) carrier transit delay and 2) $RC$ delay [49]. Based on the equivalent circuit in Fig. 1, the current supplied to the load resistor $I_{load}(\omega)$ is given by TeX Source $$I_{load}(\omega) = {1\over 1 + i\omega\tau_{RC}}I_{PD}(\omega) \eqno{\hbox{(A1)}}$$ where $\tau_{RC} = R_{load}C_{PD}$ is the $RC$ delay. The response of PD, i.e., $\Re(\omega)$, is defined as the output power normalized by that for $\omega = 0$, i.e., TeX Source $$\Re(\omega) = {R_{load}\left\vert I_{load}(\omega)\right\vert^{2}\over R_{load}\left\vert I_{load}(0)\right \vert^{2}} = \left\vert {I_{PD}(\omega)\over I_{PD}(0)}\right\vert^{2} \cdot {1\over 1 + \omega^{2}\tau_{RC}^{2}}. \eqno{\hbox{(A2)}}$$ When $\Re_{PD}(\omega)$ is defined as TeX Source $$\Re_{PD}(\omega) = \left\vert{I_{PD}(\omega)\over I_{PD}(0)}\right\vert^{2} = {1\over 1 + \omega^{2}\tau_{PD}^{2}}\eqno{\hbox{(A3)}}$$$\Re(\omega)$ is given by TeX Source $$\Re(\omega) = {1\over\left(1 + \omega^{2}\tau_{PD}^{2}\right) \left(1 + \omega^{2}\tau_{RC}^{2}\right)}.\eqno{\hbox{(A4)}}$$ The 3-dB cutoff frequency $f_{3\,dB} = \omega_{3\,dB}/2\pi$ is obtained by solving $\Re(\omega_{3\,dB}) = 1/2$ and approximately expressed as TeX Source $$f_{3\,dB} \sim {1\over 2\pi}\sqrt{1\over\tau_{PD}^{2} + \tau_{RC}^{2}} = \sqrt{1\over 1/f_{PD}^{2} + 1/f_{RC}^{2}}\eqno{\hbox{(A5)}}$$ where $f_{PD} = 1/2\pi \tau_{PD}$, and $f_{RC} = 1/2\pi\tau_{RC}$.

APPENDIX B

In the simplified model shown in Fig. 2(b), the photogenerated carriers are assumed to be injected at one edge of the i layer [50]. In this case, the current density due to the photocarriers $J(\omega)$ is given by the spatial average of the current density at the position $x$, $J(\omega, x) =$ $J_{0}e^{i\omega(t - x/\upsilon)}$, i.e., TeX Source $$J(\omega) = {1\over W_{i}}\int\limits_{0}^{W_{i}}J_{0}e^{i\omega(t - {x/\upsilon})}dx = J_{0}e^{i\omega t}{1 - e^{-i\omega\tau^{i}}\over i\omega\tau^{i}}\eqno{\hbox{(B1)}}$$ where $W_{i}$ is the thickness of i layer, $\upsilon$ is the saturation velocity of carriers in the case of large reverse bias, and $\tau^{i} = W_{i}/\upsilon$ is the transit delay. The response $\Re_{PD}(\omega)$ is given by TeX Source $$\Re_{PD}(\omega) = \left\vert{J(\omega)\over J(0)}\right\vert^{2} = {\sin^{2}(\omega\tau^{i}/2) \over (\omega\tau^{i}/2)^{2}}.\eqno{\hbox{(B2)}}$$ The 3-dB cutoff angular frequency $\omega_{PD}$ is obtained by solving $\Re_{PD}(\omega_{PD}) = 1/2$, leading to $\omega_{PD}\tau^{i}/2 \sim 1.392$. Thus, the 3-dB cutoff frequency $f_{PD}$ is obtained as TeX Source $$f_{PD} = {2 \times 1.392 \over 2\pi\tau^{i}} = {0.443 \over \tau^{i}}.\eqno{\hbox{(B3)}}$$

APPENDIX C

A more accurate formulation is firstly to solve 1-D continuity equations to determine electron and hole concentrations, i.e., for electrons TeX Source $${\partial n\over\partial t} = g_{n} - {n - n_{0}\over\tau_{n}} + n\mu_{n}{\partial E\over\partial x} + \mu_{n}E{\partial n\over\partial x} + D_{n}{\partial^{2}n\over \partial x^{2}}\eqno{\hbox{(C1)}}$$ and for holes TeX Source $${\partial p\over \partial t} = g_{p} - {p - p_{0}\over\tau_{p}} - p\mu_{p}{\partial E\over \partial x} - \mu_{p}E{\partial p\over \partial x} + D_{p}{\partial^{2}p\over \partial x^{2}}\eqno{\hbox{(C2)}}$$ where $n\ (p)$ is the electron (hole) concentration, $n_{0}\ (p_{0})$ the equilibrium electron (hole) concentration, $g_{n}\ (g_{p})$ the generation rate, $\tau_{n}\ (\tau_{p})$ the recombination lifetime of electrons (holes) in the p (n) layer, $\mu_{n}\ (\mu_{p})$ the mobility, $D_{n}\ (D_{p})$ the diffusion coefficient, and $E$ the electric field. The electron and hole current densities $J_{n}$ and $J_{p}$ can be determined using TeX Source $$\eqalignno{J_{n} =&\, qn \mu_{n}E + qD_{n}{\partial n \over \partial x}&\hbox{(C3)}\cr J_{p} =&\, qp\mu_{p}E - qD_{p}{\partial p \over \partial x}&\hbox{(C4)}}$$ where $q$ is the elemental charge.

Here, we assume the band diagram in Fig. 2(a) and the generation rate uniform in the space [50], i.e., TeX Source $$g(x, t) = g(t) = g_{0}(1 + Ae^{i\omega t}).\eqno{\hbox{(C5)}}$$ Note that formulas for the carrier generation exponentially decreased in depth, corresponding to the free-space PDs, can be obtained in the same manner shown below.

In the i layer, $\partial E/\partial x = 0$ and no recombination lead the continuity equations (C1) and (C2) to TeX Source $$\eqalignno{{\partial n\over\partial t} =&\, g_{n} + \upsilon_{n}{\partial n\over \partial x}&\hbox{(C6)}\cr{\partial p \over \partial t} =&\, g_{p} - \upsilon_{p}{\partial p\over\partial x}&\hbox{(C7)}}$$ where $\upsilon_{n}$ and $\upsilon_{p}$ are the drift velocities in the i layer for electrons and holes. The total current $J_{tot}^{i}$ is given using the electron and hole currents in the i layer $J_{n}^{i}$ and $J_{p}^{i}$ by TeX Source $$J_{tot}^{i} = {1\over W_{i}}\int \limits_{0}^{W_{i}}\left(J_{n}^{i} + J_{p}^{i} + \varepsilon{\partial E\over \partial t}\right)dx \sim {1\over W_{i}}\int\limits_{0}^{W_{i}}\left(J_{n}^{i} + J_{p}^{i}\right)dx\eqno{\hbox{(C8)}}$$ where the displacement current $\varepsilon \partial E/\partial t$ should be neglected under the low-injection condition. By solving Eqs. (C6) and (C7) under the boundary conditions $n \vert_{x = 0} \!\!= n\vert_{x = W_{i}} \!=\! 0$ and $p\vert_{x = 0} \!=\! p\vert_{x = W_{i}} \!=\! 0$, Eq. (C8) gives the ac current density with the angular frequency $\omega$, $[J_{n}^{i}(\omega) + J_{p}^{i}(\omega)] \cdot Ae^{i\omega t}$, which is expressed as TeX Source $$\eqalignno{J_{n}^{i}(\omega) =&\, qg_{0}W_{i} \cdot S_{n}^{i}(\omega)&\hbox{(C9a)}\cr S_{n}^{i}(\omega) =&\, 1\big/i\omega\tau_{n}^{i} - \left(1 - e^{-i\omega\tau_{n}^{i}}\right)\Big/\left(i\omega\tau_{n}^{i}\right)^{2}&\hbox{(C9b)}\cr J_{p}^{i}(\omega) =&\, qg_{0}W_{i} \cdot S_{p}^{i}(\omega)&\hbox{(C10a)}\cr S_{p}^{i}(\omega) =&\, 1\big/i\omega \tau_{p}^{i} - \left(1 - e^{-i\omega\tau_{p}^{i}}\right)\Big/\left(i\omega\tau_{p}^{i}\right)^{2}.& \hbox{(C10b)}}$$ Here, $\tau_{n}^{i} = W_{i}/\upsilon_{n}$ and $\tau_{p}^{i} = W_{i}/\upsilon_{p}$ are the transit delays for electrons and holes through the i layer.

On the other hand, in the n layer, the diffusion current flows due to the minority carriers of holes. The current can be obtained, starting from the continuity equation (C2), which is written as TeX Source $${\partial p \over \partial t} = g_{p} - {p - p_{0} \over\tau_{p}} + D_{p}{\partial^{2}p \over \partial x^{2}}\eqno{\hbox{(C11)}}$$ due to the absence of electric field. The total current from the n layer is obtained as TeX Source $$\eqalignno{J_{tot}^{n} =&\, {1\over W_{n} + W_{i}}\left[\int\limits_{-W_{n}}^{0}\left.J_{p}^{n}\right\vert_{x = 0}dx + \int\limits_{0}^{W_{i}}\left.J_{p}^{n}\right\vert_{x = 0}e^{-i\omega x/\upsilon_{p}}dx\right]\cr =&\, {W_{n}\over W_{n} + W_{i}}\left(1 + {W_{i}\over W_{n}}{1 - e^{-i\omega\tau_{n}^{i}} \over i\omega\tau_{n}^{i}} \right)\left.J_{p}^{n}\right\vert_{x = 0}.&\hbox{(C12)}}$$ Using the boundary conditions at the surface $s_{p}p\vert_{x = -W_{n}} = D_{p}(\partial p/\partial x)\vert_{x = -W_{n}}$ for the ac component, $s_{p}[p\vert_{x = -W_{n}} - p_{0}] = \partial p/\partial x\vert_{x = -W_{n}}$ for the dc component ($s_{p}$ is the surface recombination velocity for holes), and the boundary condition at the n/i interface $p\vert_{x = 0} = 0$, the ac current density with the angular frequency $\omega$ is expressed as $J_{p}^{n}(\omega) \cdot Ae^{i\omega t}$, where TeX Source $$\eqalignno{J_{p}^{n}(\omega) =&\, qg_{0}W_{n} \cdot S_{p}^{n}(\omega)&\hbox{(C13a)}\cr S_{p}^{n}(\omega) =&\, {1\over W_{i} + W_{n}}\left[1 + {W_{i}\over W_{n}} \left(1 - e^{- i\omega\tau_{p}^{i}}\right)/i\omega\tau_{p}^{i}\right] \cdot {L_{p}\over \beta_{p}}\cr&\cdot\ {- s_{p}\tau_{p} + s_{p}\tau_{p}\cosh (\beta_{p}W_{n}/L_{p}) + \beta_{p}L_{p}\sinh(\beta_{p}W_{n}/L_{p})\over \beta_{p}L_{p}\cosh(\beta_{p}W_{n}/L_{p}) + s_{p}\tau_{p}\sinh(\beta_{p}W_{n}/L_{p})}.&\hbox{(C13b)}}$$ Here, $L_{p} = \sqrt{D_{p}\tau_{p}}$ is the diffusion length for holes in the n layer, and $\beta_{p}$ is defined as $\beta_{p} = \sqrt{1 + i\omega\tau_{p}}$. When $s_{p}$ is small enough, the current density is expressed as TeX Source $$\eqalignno{J_{p}^{n}(\omega) =&\, qg_{0}W_{n} \cdot S_{p}^{n}(\omega)&\hbox{(C14a)}\cr S_{p}^{n}(\omega) =&\, {1\over W_{i} + W_{n}}\left[1 + {W_{i}\over W_{n}} \left(1 - e^{-i\omega\tau_{p}^{i}}\right)\Big/i\omega\tau_{p}^{i}\right] \cdot {L_{p}\over \beta_{p}} \cdot \tanh(\beta_{p}W_{n}/L_{p}).&\hbox{(C14b)}}$$ For the boundary condition at the n/i interface, it might be better to assume that the carriers are injected with the thermal velocity [56], i.e., $J_{p}\vert_{x = 0} = q\upsilon_{th - p}p\vert_{x = 0} = -qD_{p}(\partial p/\partial x)\vert_{x = 0}$, because they are similar to the source edge of MOS transistors. It should be mentioned that the change in the boundary condition to the use of thermal velocity slightly decreases the operation frequency.

Similar to the n layer, the current from the p layer to the i layer flows due to the electron diffusion, and the ac current density with the angular frequency $\omega$ is expressed as $J_{n}^{p}(\omega) \cdot Ae^{i\omega t}$, where TeX Source $$\eqalignno{J_{n}^{p}(\omega) =&\, qg_{0}W_{p} \cdot S_{n}^{p}(\omega)&\hbox{(C15a)}\cr S_{n}^{p}(\omega) =&\, {1 \over W_{i} + W_{p}}\left[1 + {W_{i} \over W_{p}}\left(1 - e^{-i\omega\tau_{n}^{i}}\right)\Big/i\omega\tau_{n}^{i}\right] \cdot {L_{n} \over \beta_{n}} \cdot \tanh(\beta_{n}W_{p}/L_{n}).&\hbox{(C15b)}}$$ Here, $L_{n} = \sqrt{D_{n}\tau_{n}}$ is the diffusion length for electrons in the p layer, and $\beta_{n}$ is defined as $\beta_{n} = \sqrt{1 + i\omega\tau_{n}}$.

The 3-dB cutoff frequency $f_{PD} = \omega_{PD}/2\pi$ is obtained by solving $\Re_{PD}(\omega_{PD}) = 1/2$, where $\Re_{PD}(\omega)$ is given by TeX Source $$\Re_{PD}(\omega) = \left\vert{J_{n}^{i}(\omega) + J_{p}^{i}(\omega) + J_{p}^{n}(\omega) + J_{n}^{p}(\omega) \over J_{n}^{i}(0) + J_{p}^{i}(0) + J_{p}^{n}(0) + J_{n}^{p}(0)} \right\vert^{2} = \left\vert {W_{i}\left[S_{n}^{i}(\omega) + S_{p}^{i}(\omega)\right] + W_{n}S_{p}^{n}(\omega) + W_{p}S_{n}^{p}(\omega)\over W_{i}\left[S_{n}^{i}(0) + S_{p}^{i}(0)\right] + W_{n}S_{p}^{n}(0) + W_{p}S_{n}^{p}(0)} \right\vert^{2}.\eqno{\hbox{(16)}}$$

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