## 3 Device Measurement Results  
Figure 3a,b shows the measured bias-dependent dark current, photocurrent, and operation gain of the demonstrated APDs, subject to different optical pumping powers at optical wavelengths of 1.31 \(\upmu\)m and 1.55 \(\upmu\)m for devices A and B. As can be seen, the measured V\({}_{\text{br}}\) and V\({}_{\text{pt}}\) are around \(-\)19.5, \(-\)43 V and \(-\)7.2, \(-\)24.2 V for devices A and B. The smaller V\({}_{\text{pt}}\) and V\({}_{\text{br}}\) of Device A is attributed to the thinner absorber layer and thicker M-layer than that of Device B. Moreover, under 0.9 V\({}_{\text{br}}\) operation, Device B has a higher responsivity than that of Device A which is due to the thicker absorption layer (2 versus 0.4 \(\upmu\)m). In addition, when the reverse bias is over V\({}_{\text{pt}}\), the photocurrent of Device B ramps up more quickly than that of Device A. This can be attributed to the precise control of charge layer doping in Device B which allows the E-field in its dual M-layers to be high enough to trigger the cascade avalanche process and generate a significant amount of multiplication gain. We assume a zero coupling loss and single-pass of injected light into the absorption layer of our device. The theoretical maximum unit gain responsivity of Device A (B) with a 0.4 (2.0) \(\upmu\)m-thick In\({}_{0.53}\)Ga\({}_{0.47}\)As absorption layer, will be around 0.4 (1.0) A/W at the 1.31(1.55) \(\upmu\)m wavelength. Here, the photo-absorption constant used for the In\({}_{0.53}\)Ga\({}_{0.47}\)As layer at wavelengths 1.31(1.55) \(\upmu\)m are around 1.2 (0.8) \(\upmu\)m\({}^{-1}\)[15; 16]. The gain versus bias voltages under different optical pumping power (1 to 500 \(\upmu\)W) are also provided in the  
Figure 2: (a) electric field profiles at V\({}_{\text{pt}}\) and V\({}_{\text{br}}\) along the vertical direction. Top-view of the fabricated (b) Device A and (c) Device B.  
figures for reference. As can be seen, there is a significant reduction in all the measured operation gain when the reverse bias voltage is over V\({}_{\text{br}}\) due to the tremendous increase in the dark current, which occupies most of the measured total current (i.e., summation of photocurrent and dark current). In addition, we can clearly see that the maximum operation gain gradually decreases with increasing pumping power. This phenomenon can be explained by the space charge screening (SCS) effect induced by the photo-generation of holes in the undoped In\({}_{0.53}\)Ga\({}_{0.47}\)As absorption layer, which reduces the net E-field and multiplication gain in the M-layer [11]. Figure 4a,b shows the 3-dB O-E bandwidth response, under a low optical pumping power of 50 (1) uW and operation gain at the 1.31(1.55) uW wavelength of Devices A and B, respectively. In both cases, under such a low optical pumping power, we can clearly see a gradual decrease in the measured O-E bandwidth with an increase of the reverse bias voltage (operation gain). The maximum 3-dB O-E bandwidths of Devices A and B are around 18 and 1.25 GHz, respectively. The degradation in the O-E bandwidth with the increase of operation gain is very common in APDs and can be attributed to the increase of the avalanche induced delay time with the gain. Figure 5a,b shows the bias dependent O-E frequency responses of Devices A and B measured under a high input optical power (1 mW and 240 uW), respectively. As can be seen, the 3-dB O-E bandwidths for devices A and B are pinned at 16 and 1.3 GHz, respectively, regardless of the changes in the reverse bias voltages. These bandwidth values are close to the maximum O-E bandwidths of the device measured under low power excitation, as shown in Figure 4a,b. Such invariant high-speed performance can be attributed to the significant decrease in the value of the operation gain, becoming much less sensitive to the reverse bias voltage under high-power operation, as shown in Figure 3a,b.