1328 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 5, SEPTEMBER/OCTOBER 2010 Derivation of the Small Signal Response and Equivalent Circuit Model for a Separate Absorption and Multiplication Layer Avalanche Photodetector Daoxin Dai, Member, IEEE, Mark J. W. Rodwell, Fellow, IEEE, John E. Bowers, Fellow, IEEE, Yimin Kang, Member, IEEE, and Mike Morse, Member, IEEE Abstract—A small signal analysis for a separate-absorption- charge-multiplication (SACM) avalanche photodetector (APD) is presented for the general case when the electrons and the holes have different ionization coefficients and different velocities. The analytic expressions for the impedance and frequency response are given and a simplified equivalent circuit (including an inductance with a series resistance in parallel with a capacitance) for the APD is obtained. The calculation and experimental results show that the impedance of the APD operated at high bias voltages has a maximal value at a certain frequency due to the resonance of the LC-circuit, and this is the origin for a peak-enhancement of the frequency response. Index Terms—Avalanche photodetector (APD), Ge, Si. I. INTRODUCTION H IGHLY sensitive photodetectors are desirable for a wide range of applications. One important metric is the gain- bandwidth product (GBP), which is typically in the 80–200 GHz range for III–V avalanche photodetectors (APDs), as is expected for a k value (i.e., the ionization ratio) ∼0.4–0.5 [1]. Silicon has a low k-value (<0.1), which is desirable for a high GBP, and can be applied to communication applications at 1.3 and 1.55 μm, respectively, when combined with a material with a high absorption coefficient in the infrared, such as InGaAs [2] or Ge [3]–[5]. Kang et al. reported CMOS-compatible Ge/Si separate-absorption-charge-multiplication (SACM) APDs with a GBP as high as 340 GHz [3]. Zaoui et al. [6], [7] examined higher voltage operation of these diodes and observed a peak en- hancement of the frequency response and high gain bandwidth products (840 GHz). Peak enhancement has also been observed for other types of APDs [9]–[11] and is sometimes explained as due to space charge effects. In our previous paper [12], we noted that the impedance for the Ge/Si APD could be fit with an equiv- alent circuit containing an inductive element and the parameters Manuscript received September 1, 2009; revised November 7, 2009; accepted November 29, 2009. Date of publication February 5, 2010; date of current ver- sion October 6, 2010. This work was supported by the Defense Advanced Research Projects Agency under Contract HR0011-06-3-0009. D. Dai, M. J. W. Rodwell, and J. E. Bowers are with the Department of Electri- cal and Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106 USA (e-mail: dxdai@ece.ucsb.edu; rodwell@ece.ucsb.edu; bowers@ece.ucsb.edu). Y. Kang and M. Morse are with Intel Corporation, Santa Clara, CA 95054 USA (e-mail: yimin.kang@intel.com; mike.morse@intel.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTQE.2009.2038497 for the elements in the equivalent circuit were extracted by fit- ting the measured S 22 with a genetic algorithm optimization. It was shown that the inductive element plays an important role for the peak enhancement of the frequency response. Small signal modeling based on fundamental equations (e.g., the Poisson equation and the semiconductor transport equa- tions) is a fundamental and important approach to understand the RF behavior of an APD. Small signal analysis is helpful to make clear the physical mechanism and consequently has been used very widely, e.g., for an RF analysis for impact ionization avalanche transit-time (IMPATT) diodes [13], [14], and for p-n junctions in breakdown [15]. The previous small signal model was developed for IMPATT oscillators and usually with the as- sumption that the electrons and the holes have equal ionization rates and velocities [15]. In [16], Manasse et al.. considered the differences in hole and electron velocities and ionization rates and gave an improved dispersion relationship of a p-n junction avalanche diode, however, for the case with a single uniform depletion layer. In this paper, we present a small signal analysis to calcu- late the impedance characteristic and the frequency response of an SACM APD, which includes different parameters for the absorption layer and the multiplication layer. Specifically, we consider the general case that the electrons and the holes have different ionization coefficients and different velocities. It is the first derivation of the peak enhancement of the frequency response. Furthermore, we use this analysis to derive the equiv- alent circuit model for an SACM APD. From the small signal analysis, we obtain analytical expressions for the impedance and the short-circuit frequency response of the APD and compare to experimental results for Ge/Si APDs as an example. II. THEORY Fig. 1 shows the schematic configuration of an SACM APD. There is thin charge layer between the multiplication layer and the absorption layer, which is doped such that one obtains suffi- cient gain via a high electric field in the multiplication layer, but the electric field in the absorber is low enough that avalanche gain almost does not occur in the absorption layer (see Fig. 1). The charge layer is usually very thin (100 nm in [3]), so we simplify the APD structure into an avalanche layer and an ab- sorption layer, with uniform electrical fields in each. In the following analysis, we consider the avalanche region and the drift region separately. 1077-260X/$26.00 © 2010 IEEE