## 3 Device Measurement Results
Figure 8: Conceptual cross-sectional view of (**a**) Device C (**b**) BER values versus optical modulation amplitude (OMA) of our receiver modules at the 1.31 \(\upmu\)m optical wavelength and 25.78 Gbit/s data rate. BER curves of two commercially available ROSAs [4; 9] for reference. Thus, to further improve the bandwidth, we fabricated a back-illuminated device (Device A\({}^{\prime}\)) with a smaller active diameter of 18 versus 25 \(\upmu\)m, share the same epi structure as that of Device A. The back-illuminated device offers the advantage of less coupling loss as compared to the top-illuminated device in which the top metal layer blocks light from entering the device [5]. Figure 10a,c show top-views of the fabricated device before and after flip-chip bonding, respectively. Compared with our previous design [21], the geometric size and layout of the flip-chip bonding co-planar waveguide (CPW) pads has further been optimized to not only minimize any ripples on the measured O-E frequency responses, but also to enhance the reliability of the flip-chip bonding process due to the increase in bonding area. We use gold-tin as the material for the bumps on the AlN substrate, because it has good thermal conductivity and low dielectric loss, for flip-chip bonding process. Figure 10d shows the measured bias-dependent dark current, photocurrent, and operation gain of the demonstrated APDs, subject to different optical pumping powers at an optical wavelength of 1.31 \(\upmu\)m. As can be seen, the measured breakdown voltage and punch through voltage are the same as for the top-illuminated counterpart. Figure 11a,b represents the bias dependent O-E frequency response of the back side illuminated Device A\({}^{\prime}\) measured under an optical pumping power of 20 \(\upmu\)W at the 1.3 \(\upmu\)m wavelength. As can be seen, the measured 3 dB bandwidth of Device A\({}^{\prime}\) is 24.5 GHz under low gain (Mg = 3) and 18.6 GHz under 0.9 \(\mathrm{V_{br}}\) (Mg = 8.2). This is comparatively higher than those of the top illuminated device measured under the same operation gain condition. Here, we choose \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Physical Meaning** & **A** \\ \hline Cj & Junction Capacitance (ff) & 155 \\ Rj & Junction Resistance (k\(\Omega\)) & \(>\)600 \\ Rc & Contact Resistance (\(\Omega\)) & 8 \\ Cp & Parastic Capacitance (ff) & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the circuit element. Figure 9: (**a**) equivalent-circuit-model. VCCS: Voltage controlled current source. Device A (**b**) measured (blue line) and fitted (red line) S22 parameters from near dc to 40 GHz under a fixed dc bias \(-\)13V and (**c**) measured O-E, extracted RC-limited, transient time, and fitted O-E frequency responses. the responsivity (\(\sim\)0.51 A/W) at the punch-through point as our unit-gain responsivity. The corresponding GBP value under 0.9 V\({}_{\text{br}}\) operation is around 153 GHz. Figure 12a,b show the bias dependent O-E frequency responses of Device A\({}^{\prime}\) measured under a high input optical power (1 mW). As can be seen, Device A\({}^{\prime}\) behaves same as Device A. The maximum 3-dB O-E bandwidths are pinned at around 25 GHz, regardless of the changes in the reverse bias voltage. It is worth mentioning that under both low and high gain, our back side illuminated Device A\({}^{\prime}\) has 6-8 GHz higher bandwidth compared to the top illuminated Device A. Figure 13a,b shows the dc output photocurrent versus the input optical power for Devices A\({}^{\prime}\) and B, respectively. As can be seen, the dc saturation current of Device A\({}^{\prime}\) is higher than that of Device B (5 versus 3 mA) at near V\({}_{\text{br}}\) bias, due to the thinner depletion layer thickness. The output saturation power of our demonstrated APD is tested using a two-laser heterodyne-beating setup at the 1550 nm wavelength. Figure 14a,b show the measured photo-generated radio frequency (RF) power versus output photocurrent obtained under different reverse biases.