In this work the impedance of separate-absorption-charge-multiplication Ge/Si avalanche photodiodes (APD) is characterized over a large range of bias voltage. An equivalent circuit with an inductive element is presented for modeling the Ge/Si APD. All the parameters for the elements included in the equivalent circuit are extracted by fitting the measured S22 with the genetic algorithm optimization. Due to a resonance in the avalanche region, the frequency response of the APD has a peak enhancement when the bias voltage is relatively high, which is observed in the measurement and agrees with the theoretical calculation shown in this paper.
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In fiber-optic communication systems, highly sensitive photodetectors are desirable to have long reach in ultra long-haul networks. Additionally, in access networks (like GPON-FTTH systems), low-cost highly-sensitive photodetection is essential for short distances (several tens of kilometers) because the optical power is split multiple times before reaching the user terminals. Hence, avalanche photodetectors (APDs) with their internal gain are a natural choice for such high sensitivity applications.
For APDs, the gain-bandwidth product (GBP) is one of the most important figures of merit. For a traditional InP-based APD receiver, the GBP is usually about 100 GHz due to the large k value (~0.4-0.5) . In contrast, silicon has a low k-value (<0.1), which makes it one of the most promising candidates for APDs that have both high gain and high bandwidth simultaneously. In order to make Si-APDs available in the infrared regime, a material with a high absorption coefficient in the infrared, like InGaAs  or Ge [3-5] is used for the absorption layer along with the Si multiplication layer. Ge is attractive since it is possible to develop an APD based on a complementary metal-oxide-semiconductor (CMOS) compatible process. Although these APDs tend to have higher dark current because of threading dislocations due to the lattice mismatch between Ge and Si , it is possible to minimize the impact of dislocations with careful processing and device design. In Ref , Kang et al. reported CMOS-compatible Ge/Si APDs with a GBP as high as 340 GHz by using a separate-absorption-charge-multiplication (SACM) structure. Zaoui et al. [6, 7] examined higher voltage operation of these diodes where space charge current is dominant and resonant effects become apparent. In this regime, extremely high gain bandwidth products (840 GHz) were observed.
In this paper, we examine the origin of these resonant effects in the region where space charge effects are important and give a characterization (the frequency response and the impedance) of a normal-incidence SACM Ge/Si APD. The circuit model presented includes carrier transit-time effects and the effect of parasitics. S parameters are widely used for the characterization and modeling of high-frequency and high-speed devices  (e.g., the APDs considered here). From the measured S22, one can extract the circuit component values for the various elements (R, C, L) in the equivalent circuit. In this paper, we use the measured S22 parameters to extract all the component values at different bias voltages simultaneously. Based on the extracted circuit model, we calculate the device frequency response, which is in good agreement with measurements.
2. The device structures
Figure 1(a) shows the cross section of the present normal-incidence illuminated SACM Ge/Si APD, which is the same as that in Ref . The structure consists of a Si multiplication layer, a Si charge layer, and a Ge absorption layer. The thicknesses and the doping concentrations for all layers are shown in Fig. 1(b). The charge layer is very thin and has the correct doping so that one obtains sufficient gain via a high electric field in the Si multiplication layer, but the electric field in the absorber is low enough that avalanche gain in the absorption layer is too low to affect the performance of the APD. A silicon-nitride film was deposited and serves as an anti-reflection coating in the 1310nm window to improve the quantum efficiency.
3. Results and discussion
Here we measured three APDs with different diameters, D = 150, 80, and 50 μm. We used a temperature controller and set the stage temperature to T = 25°C. Figures 2(a) -2(c) respectively show the frequency responses measured by a lightwave component analyzer for each SACM APD (with D = 150, 80, and 50μm) at different bias voltages. Here we only show results at bias voltages close to the avalanche breakdown point Vb. The input optical power is –14dBm and the wavelength is 1310nm. Devices with larger diameters have larger junction capacitances, which will limit the bandwidth. This can be seen clearly in Figs. 2(a)-2(c). Here the APD with D = 50μm has the largest bandwidth. We note that the bandwidth is also strongly dependent on the bias voltage. When the bias voltage is relatively low (e.g., –23V), very little impact ionization occurs and consequently the frequency response is similar to that of a PIN photodetector. When the bias voltage increases, more impact ionization occurs and the DC gain increases (as can be seen in the responses in the low-frequency range shown in Fig. 2). When the bias voltage increases further (e.g., |V|>24.6V for D = 80μm), the response at low frequency decreases while there is an enhancement in the high frequency range. The enhancement becomes greater as the bias voltage increases. This behavior is observed in all three APDs with different diameters. For example, for the APD with D = 80μm, the maximal peak enhancement is as large as 4dB when it operates at Vbias = –26.6V. Such behavior can be used to enhance the bandwidth of the APD.
Figure 3 gives a comparison between the 3dB-bandwidths for the three SACM APDs with different diameters as the bias voltage is varied. One sees that the bandwidths at low voltages for D = 50, 80 and 150μm are about 7.1, 4.4, and 1.3 GHz, respectively. The bandwidth is inversely proportional to the area of the APD. At high voltages, the bandwidth increases significantly due to the peak enhancement (see Figs. 2(a)-2(c)) as discussed above. For example, for the case of D = 50μm, the bandwidth is as high as 16GHz at a high voltage (e.g., V = –27.6V). A similar phenomenon has been reported for in Si-SiGe-based APDs (at 830 nm) , Si APDs , and InGaAs/InAlAs APDs . On the other hand, when the APD operates at a high bias voltage, the dark current (shown in Fig. 4 ) will be relatively high and the gain will become small. This will limit the operation at high bias voltage.
In order to understand the peak enhancement shown in Figs. 2(a)-2(c), we examined the APD impedance by measuring the microwave reflection parameter S22 as the bias voltage is varied by using an Agilent E8364A network analyzer. The frequency range is from 45MHz to 30GHz. The results for D = 150μm, 80μm, and 50μm are shown in Figs. 5(a) -5(c), respectively. From this figure, one sees that S22 changes with bias voltage in the same way for all three devices. Since the shapes of the curves for different diameters are similar, here we focus our discussions on the case of D = 80μm. For the APD with D = 80μm, when the bias voltage is low (e.g., V = –24.6V), the entire curve is below the line Г i = 0. This corresponds to the expected frequency response similar to that of a PIN photodetector (which corresponds to a resistor and capacitor in parallel, representing the diode capacitance and diode resistance). For a higher bias voltage, one has Г i>0 in a certain frequency range. The phenomenon becomes stronger when the bias voltage increases further. Figure 6 (a) and 6(b) show the real part Zr and the imaginary part Zi of the impedance for the APD with D = 80μm, respectively. Both parts are strongly voltage-dependent. At relatively high voltage (e.g., V>25.4V), one sees that the real part Zr of the APD impedance has a peak at a certain frequency f r. The imaginary part Zi of the APD impedance has a transition from a positive to a negative value at almost the same position f r. This is what usually occurs when there is a resonance. Since the field distribution in the presented Si/Ge APD is similar to that in an impact ionization avalanche transit-time (IMPATT) diode structure, we make an analysis similar to that shown in Ref . In the avalanche region, the impact ionization avalanche will introduce a delay between the AC current and the electric field (i.e., the AC voltage). With a small signal model, this delay due to the impact ionization avalanche in the Si multiplication layer is equivalent to an inductance. Therefore, an equivalent circuit model with an LC-circuit for the avalanche region will be presented below.
Figure 7 shows the equivalent circuit used for the SACM APD. The part in the dashed area in Fig. 7 represents the equivalent impedance of contact pads and metal interconnects. For the avalanche region, we include the LC-circuit (L A and C A) [12, 13] and the resistances (R A and R l) which are lossy elements in the avalanche region due to the finite reverse saturation current and field-dependent velocity . R l represents the leakage of the diode and RA is the series resistance of the inductor. The small signal model analysis shows that the inductance should be inversely proportional to the current density J 0 , which will be verified below. The resistance R d connected to the LC circuit is for the resistance in the drift region.
The carrier transit-time effect is also included in this model by using an additional RC circuit (the left part in Fig. 7) , which is coupled to the right part through a current-controlled current source; i.e., the source current I in = gI t (where g is a constant related to the gain).
All the parameters for the elements included in the equivalent circuit were extracted by fitting the measured S22 with a genetic-algorithm (GA) optimization. In order to obtain more reasonable fitting parameters, here we take the measured S22 parameters at a series of reverse bias voltages (e.g., Vbias = –26.6, –26.4, –26.2 and –26.0V) and extract all the parameters at each bias voltage. At different bias voltages, all the parasitic impedances (R s, C p, L p, R p) should be the same while the other parameters will change as the gain changes.
Figures 8(a) -8(d) show the measured (dotted curves) and fitted (solid curves) S22 parameters for the APD with D = 80μm at different bias voltages Vbias = – 26.6, – 26.4, – 26.2, and – 26V, respectively, when the input optical power is P = –14dBm. The fitted results for the R s, C p, L p, and R p are: R s = 16.76Ω, C p = 0.193pF, L p = 0.082nH, and R p = 6.65Ω, independent of the bias voltage. The fitted parameters for all other bias-dependent elements are shown in Table 1 . For the avalanche region, the capacitance C A changes very slightly while the inductance decreases as the bias voltage increases (which is due to the variation of current density as theoretically predicted). The measured currents are I = 9.66, 8.06, 6.65, and 5.36 mA for the case of the bias voltage Vbias = –26.6, –26.4, –26.2, and –26V, respectively. Correspondingly, the products L A × I are 29.74, 29.78, 29.33, and 29.18 (nH·mA), respectively. One sees that this product is almost constant as the bias voltage varies. This indicates that the inductance L A is almost inversely proportional to the current density, which is similar to the theoretically predicted relationship for an IMPATT diode in Ref .
We also measured S22 for the case of P = –20dBm. The parameters R s, C p, L p, and R p are the same as in the case of P = –14dBm since they are independent of the optical power. The other fitting parameters are shown in Table 2 . The values are slightly different from the case of P = –14dBm. The products L A × I are 29.12, 29.3, 29.2, and 29.1 for the case of Vbias = –26.6, –26.4, –26.2, and –26V, respectively. This verifies that the products (L A × I) are almost constant as the bias voltage varies.
From the equivalent circuit with the extracted parameters, we calculate APD frequency responses for Vbias = –26.6, –26.4, –26.2 and –26.0V. Here we only show the results for the case of P = –14dBm in Figs. 9(a) -9(d) since the results for P = –20dBm are similar. The responsivity for a gain of one is about 0.55A/W in the calculation here. The fitting parameters are shown in Table 3 . The transit-time changes slightly as the bias voltage varies. The fitted gain is about 14.8, which is close to the measured DC gain (about 15). From Fig. 9(a)-9(d), one sees that the simulated curve (dashed) and measured data (circled) agree well with each other, especially the peak-enhancement at the high frequency. Such a peak enhancement increases the bandwidth, which is similar to results for other types of APDs reported in Ref.s [11–13].
In summary, we have characterized a SACM Ge/Si APD and also given an equivalent circuit. A genetic algorithm has been used to extract the equivalent circuit model based on the measured S22 parameters. In this equivalent circuit, one of the key elements is the inductance, which is from a delay between the AC current and the electric field. This delay is introduced by the impact ionization avalanche in the avalanche region. It has been shown that this inductance is inversely proportional to the injected current, which is in agreement with the theoretical prediction. With these fitted parameters, we have also calculated the frequency responses, which agree well with the measured one. Due to the LC resonance, the present Ge/Si APD shows a peak enhancement at high frequency range when the bias voltage is high. Consequently a large bandwidth is achieved at high bias voltage.
This work was sponsored by the Defense Advanced Research Projects Agency (DARPA) under contract number HR0011-06-3-0009. The authors thank H. Kroemer, J. C. Campbell, M. Rodwell, M. Piels, A. Ramaswamy, and A. Pauchard for useful discussions.
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