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Si-waveguide-integrated lateral Ge p-i-n photodetectors using novel Si/SiGe buffer and two-step Ge-process are demonstrated for the first time. Comparative analysis between lateral Ge p-i-n and vertical p-Si/i-Ge/n-Ge p-i-n is made. Light is evanescently coupled from Si waveguide to the overlaying Ge- detector, achieving high responsivity of 1.16 A/W at 1550 nm with f3 dB bandwidth of 3.4 GHz for lateral Ge p-i-n detector at 5 V reverse bias. In contrast, vertical p-Si/i-Ge/n-Ge p-i-n has lower responsivity of 0.29 A/W but higher bandwidth of 5.5 GHz at -5 V bias. The higher responsivity of lateral p-i-n detectors is attributed to smaller optical mode overlap with highly doped Ge region as in vertical p-i-n configuration.