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Direct Parameter-Extraction Method for Laser Diode Rate-Equation Model Jianjun Gao, Xiuping Li, Jens Flucke, and Georg Boeck, Senior Member, IEEE
Abstract—A new direct extraction method to determine the small-signal and rate-equation model parameters for laser diode is presented in this paper. This method differs from previous ones by extracting the whole model parameters without global numerical optimization techniques. The main advantage of this method is that a unique and physically meaningful set of extrinsic and intrinsic parameters are extracted by using a set of closed-form expressions based on the input reflection coefficients and modulation responses taken from on-wafer measurement. Simulated and measured results for the input reflection coefficients and modulation responses exhibit good agreement over a wide range of bias points. Index Terms—Laser diode (LD), parameter extraction, rateequation model.
wire inductance, and chip parasitic elements can be extracted from on-wafer-based input reflection coefficient measurements. After de-embedding the effects of parasitic elements, the intrinsic small-signal model parameters can be obtained from the small-signal modulation response. The rate-equation model parameters are extracted from the intrinsic small-signal model parameters at multiple bias points to verify the validity of this approach. This paper is organized as follows. Section II gives the small-signal-equivalent circuit model of the LD and the basic formalism used in the extraction procedure. Section III then discusses the basic procedure for extracting the small-signal element parameters. Section IV gives the extraction results and discussion. The conclusions are discussed in Section V.
I. INTRODUCTION
S
EMICONDUCTOR lasers are the important light sources for high-bit-rate lightwave communication system because of their large intrinsic modulation bandwidth. Accurate extraction of the small-signal- and large-signal-equivalent circuits for laser diodes (LDs) is extremely important for optimizing the device performance. At microwave frequencies, the feeding transmission lines, the source and cavity impedance, and parasitic effects due to the chip and package geometry will have significant influence on the measured results. The parasitic elements and rate-equation model parameters can be obtained from reflection coefficient, modulation response, and the relative noise intensity measurements by using numerical optimization techniques [1]–[8]. However, the accuracy of the numerical optimization methods that minimize the difference between measured and modeled data can vary depending upon the optimization method and starting values, while the analytical methods allow us to extract the equivalent circuit model parameters in a straightforward manner. In order to overcome these difficulties, a full analytical and accurate method for extracting the small-signal and large-signal model parameters is proposed in this paper. The extraction procedure allows direct and fast calculation of a unique physically meaningful set of parasitic elements by using a set of closed-form expressions based on the input reflection coefficients and modulation responses on wafer measurement. First, the parasitic elements including pad capacitance, series
Manuscript received November 5, 2003; revised March 22, 2004. J. Gao, J. Flucke, and G. Boeck are with the Institute of High-Frequency and Semiconductor System Technologies, Technische Universität Berlin, Berlin 10587, Germany (e-mail:
[email protected]). X. Li is with the Department of Telecommunication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China. Digital Object Identifier 10.1109/JLT.2004.829211
II. LASER DIODE RATE-EQUATION MODEL Many characteristics of the LD behavior can be modeled by a set of coupled rate equations, which describe the relation between the carrier density and photon density and are written as (1) (2) where is the photo density averaged over the nominal modal volume, is the electron density averaged over the volume of the active region, is the optical confinement factor, is the with being the gain slope coefficient given by is the electron density at which the net group velocity, and gain is zero. is the spontaneous recombination lifetime, is the photon lifetime, is the fraction of spontaneous emission coupled into the lasing mode, is the volume of the active re), is the gion multiplied by the electronic charge ( current injected into the active region, and is nonlinear gain compression factor. The intrinsic small-signal-equivalent circuit model of LD can be obtained from the steady-state analysis by using the direct transform of rate equations and Fourier transform of the linear rate equations (Fig. 1). In this model, charge storage in the active layer is modeled by the diffusion capacitance , and the small-signal photo storage is proportional to the is modeled by the inductance , and small-signal light output intensity. The relationship between the linear equivalent circuit elements and rate-equation model parameters can be written as [2], [3]
0733-8724/04$20.00 © 2004 IEEE
(3)
GAO et al.: DIRECT PARAMETER-EXTRACTION METHOD FOR LD RATE-EQUATION MODEL
Fig. 1.
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Small-signal model of LD.
(4) (5) (6) (7) is the threshold current of the LD. where A typical parasitic network model (Fig. 1) includes a pad carepresenting the wirebond, pacitance , a series inductance representing the contact capacitance, a shunting capacitance a series resistance representing the contact resistance and Bragg mirror stacks, and the space-charge capacitance of the . is the source impedance, heterojunction is modeled by and represents the inductance of the cable between the source and the microstrip feeding line of the test fixture. To calculate the normalized small-signal modulation response of the laser, both the photon response and the effect of the parasitic elements have to be taken into account. The total can be expressed as small-signal normalized response (8)
Fig. 2. LD test structure.
can be obtained from the -parameter measurement of the cable. The values of the other parasitic elements (including , , , and ) can be obtained from the measurements of microwave reflection coefficient under zero bias and above-threshold bias conditions. Fig. 2 shows the microstrip test fixture with the mounted laser. ), the Under above-threshold bias condition ( space-charge capacitance of heterojunction has a low impedance and low-junction dynamic resistances shorting it. The input impedance of LD is very small and can be modeled by using a short circuit. Therefore, the equivalent circuit becomes becomes much simpler; the -parameter
is the small-signal current transformation function where of the parasitic network (9)
(11) can be extracted directly from low-frequency measurements
where = ; = ; = ; = ; = . The small-signal current transformation function of the intrinsic network can be determined from the rate-equation description of the dynamics (10) where =
.
(12) and
can be expressed as (13) (14)
After de-embedding the effects of the parasitic elements ( , , and ), the contact capacitance and the zero bias spacecan be determined from the -paramcharge capacitance eter of zero bias condition
III. PARAMETER-EXTRACTION PROCEDURE A. Extraction of Extrinsic Elements is The internal resistance of the modulation source usually known and is typically 50 , and the cable inductance
(15) (16)
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where and represent the one-port -parameter under above-threshold bias and zero bias conditions, respectively. B. Intrinsic-Elements Extraction and Determination of the and After obtaining the intrinsic modulation responses, and can be determined as follows: (17)
R
versus frequency under above-threshold bias condition
L
versus frequency under above-threshold bias condition
Fig. 3. Extracted I ). (I
(18) represents the small-signal frequency response, and where is the peak response at the resonant frequency . With (4) and (6) substituted in (18), we have (19) and can be obtained from the plot of versus . The interception gives the value , and from the slope can be extracted. Therefore, all intrinsic elements can be determined from (6), (17), and (18).
Fig. 4. Extracted (I I ).
C. Extraction of Other Rate-Equation Parameters There are eight model parameters in the rate equation ( , , , , , , , and ), and all are bias independent. The parameter can be calculated from the known device dimensions, and can be calculated with a three-dimensional-equivalent refractive-index method [9]. Then, , , , and can be determined in sequence, as described hereafter. 1) Gain Slope Coefficient : Since the space-charge capacis smaller when compared with , the relaxation ositance cillation resonance frequency can be expressed approximately as
The spontaneous emission factor lower bias condition as follows:
can be determined at a
(25) This method also can be considered as an initial guess of a subsequent optimization procedure leading to the final model parameters. IV. RESULTS AND DISCUSSION
(20) i.e., (21) 2) Zero-Gain Electron Density : can be determined directly from threshold current measurement (22) 3) Gain Compression Factor : It is found that is almost a constant under a high-current-bias condition and can be expressed as (23) i.e., (24)
To illustrate the previously described method, we present the extracted model parameters for several quantum-well (QW) LDs. The active region consists of six 7-nm QWs in an InGaAsP separate confinement herostructure (SCH) region, op4.4 mA (ambient temperature). erating at 1310 nm with The measurements were carried out with the laser mounted in the microstrip test fixture and the light output detected using a high-speed photodiode. A network analyzer was used to determine the transfer function of the laser–photodiode combination. The -parameter measurements for model extraction and verification were made on wafer up to 40 GHz using Cascade Microtech’s Air-Coplanar Probes ACP50-GSG-100, with all instruments under IC CAP software control. at different bias currents The extracted contact resistance in the low-frequency range is shown in Fig. 3. The magnitude variation of is very small (less than 8% from mean value) and can be neglected in a first-order approximation. Therefore, can be considered to be independent of bias. Figs. 4 and 5 show and . and can be the extracted parasitic elements extracted with the de-embedding procedure as shown in Figs. 6
GAO et al.: DIRECT PARAMETER-EXTRACTION METHOD FOR LD RATE-EQUATION MODEL
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TABLE I PARASITIC ELEMENTS
Fig. 5. Extracted I ). (I
C
versus frequency under above-threshold bias condition
Fig. 8. Extracted condition. Fig. 6.
R =R
versus frequency under three different biased
Extracted C versus frequency under zero bias condition.
Fig. 7. Extracted C
versus frequency under zero bias condition.
and 7. Rather constant values can be observed. The extracted extrinsic elements are summarized in Table I. versus frequency at three difThe extracted ratio of ferent bias conditions is shown in Fig. 8; rather constant values are observed over a wide frequency range. Fig. 9 shows the plot versus . The extracted spontaneous recombination of lifetime and photon lifetime are 0.84 ns and 6.0 ps, respectively. It can be found that the measured data at low-bias-conis very useful to dition region with low resonant frequency and . improve the extraction accuracy of the By combining (6), (17) , and (18) , all the intrinsic elements can be obtained. The extracted intrinsic small-signal parameters are summarized in Table II, and the corresponding rate-equation parameters calculated from (20) (25) are summarized in Table III. It is noted that we have assumed a linear relationship between gain and carrier density in the active region, and a sublinear relationship caused by the size quantization effect in QW is neglected (typically a logarithmic formula has been chosen
Fig. 9. Plot of 1=R
C
versus f .
TABLE II EXTRACTED INTRINSIC ELEMENTS
based on phenomenological considerations). The larger value of the spontaneous emission coupling coefficient is required to simulate the large damping.
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TABLE III RATE-EQUATION PARAMETERS
Fig. 11.
Comparison of modeled and measured modulation response.
Fig. 12. Comparison of modeled and measured resonant frequency versus bias current.
V. CONCLUSION An accurate method for extracting LD parasitic intrinsic as well as rate-equation model elements has been presented. The extraction procedure allows direct and fast calculation of a unique physically meaningful set of parasitic elements. The excellent agreement between measured and modeled data at different bias points has proven the accuracy of the proposed extraction method. Fig. 10. Comparison of modeled and measured S -parameter for the LD. Bias: (a) I 80 mA and (b) I 0 mA.
=
=
Fig. 10 compares the measured and modeled -parameters for the LD in the frequency range of 1–40 GHz under two dif80 mA and 0 mA). Good ferent bias conditions ( agreement over the whole frequency range is obtained. Measured and modeled small-signal frequency response for the QW laser is shown in Fig. 11 at four different bias currents above thethreshold, and a comparison between modeled and measured resonance frequency versus bias current is given in Fig. 12. Good agreements are obtained between simulated and measured modulation response data over a wide range of bias points. The discrepancy between measured and modeled small-signal frequency response may be duo to the assumption of the a linear relationship between gain and carrier density in the active region; a sublinear relationship caused by the size quantization effect in QW is neglected.
ACKNOWLEDGMENT The authors would like to thank Dr. A. Huelsmann and Mr. F. Fidorra of Merge Optics company for valuable discussion and for proving the laser diode. REFERENCES [1] C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv, “Noise equivalent circuit of a semiconductor laser diode,” IEEE J. Quantum Electron., vol. QE-18, pp. 333–337, Mar. 1982. [2] R. S. Tucker and I. P. Kaminow, “High-frequency characteristics of directly modulated InGaAsP ridge waveguide and buried heterostructure lasers,” J. Lightwave Technol., vol. LT-2, pp. 385–393, Aug. 1984. [3] R. S. Tucker and D. J. Pop, “Microwave circuit models of semiconductor injection lasers,” IEEE Trans. Microwave Theory Tech., vol. 31, pp. 289–294, Mar. 1983. [4] L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol., vol. 14, pp. 839–850, May 1996. [5] J. C. Cartledge and R. C. Srinivasan, “Extraction of DFB laser rate equation parameters for system simulation purposes,” J. Lightwave Technol., vol. 15, pp. 852–860, May 1997.
GAO et al.: DIRECT PARAMETER-EXTRACTION METHOD FOR LD RATE-EQUATION MODEL
[6] M. Bruensteiner and G. C. Papen, “Extraction of VCSEL rate-equation parameters for low-bias system simulation,” IEEE J. Select. Topics Quqntum Electron., vol. 5, pp. 487–494, May/June 1999. [7] M. L. Majewski and D. Novak, “Method for characterization of intrinsic and extrinsic components of semiconductor laser diode circuit model,” IEEE Microwave Guided Wave Lett., vol. 1, pp. 246–248, Sept. 1991. [8] J. Lee, S. Nam, S. H. Lee, and J. Jeong, “A complete small-signal equivalent circuit model of cooled butterfly-type 2.5 Gbps DFB laser modules and its application to improve high frequency characteristics,” IEEE Trans. Adv. Packag., vol. 25, pp. 543–548, Nov. 2002. [9] J. Shimizu et al., “Optical-confinement-factor dependencies of the K factor, differential gain, and nonlinear gain coefficient for 1.55 m InGaAs/InGaAsP MQW and strained-MQW lasers,” IEEE Photon. Technol. Lett., vol. 3, pp. 773–776, Sept. 1991.
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Xiuping Li received the B.S. degree from Shandong University, Shandong, China, in 1996 and the Ph.D. degree from the Beijing Institute of Technology, Beijing, China, in 2001. She worked on multilayer microstrip antenna design and analysis from 1996 to 2001. From 2001 to 2003, she was a Research Fellow with the Positioning and Wireless Technology Center, Nanyang Technological University, Singapore, where she was involved in the research and development of radio frequency identification (RFID) system. In the microelectromechanical systems group, her work included micromachined filter design, fabrication, and coaxial and on-wafer measurement. In 2003, she was a Research Professor with Yonsei University, Seoul, South Korea. Since 2004, she has been an Associate Professor with the Telecommunication Engineering School at Beijing University of Posts and Telecommunications, Beijing, China. Her current research interests include radio-frequency and microwave devices for communications, microwave filters and antennas, and coaxial and on-wafer measurement.
Jens Flucke, photograph and biography not available at the time of publication. Jianjun Gao was born in Hebei Province, China, in 1968. He received the B.Eng. and Ph.D. degrees from the Tsinghua University, Tsinghua, China, in 1991 and 1999, respectively, and the M.Eng. degree from Hebei Semiconductor Research Institute, China, in 1994. He was a Postdoctoral Research Fellow from 1999 to 2001 with the Microelectronics R&D Center, Chinese Academy of Sciences, Beijing, China, developing a PHEMT optical modulator driver. In 2001, he joined the School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore, as a Research Fellow in semiconductor device modeling and on wafer measurement. In 2003, he joined the Institute of High-Frequency and Semiconductor System Technologies, Berlin University of Technology, Berlin, Germany, as a Research Associate working on InP heterojunction bipolar transitor (HBT) modeling and circuit design for high-speed optical communication.
Georg Boeck (M’93–SM’00) was born in Wertingen, Germany, in 1951. He received the Dipl.Ing. degree in electrical engineering and the Ph.D. degree from the Berlin University of Technology, Berlin, Germany, in 1977 and 1984, respectively. In 1984, he joined the Siemens Research Laboratories, Munich, Germany, where his research areas were fiber optics and GaAs electronics. From 1988 to 1991, he was a Full Professor of electronic devices and circuits at the Fachhochschule Regensburg, Germany. Since 1991, he has been a Full Professor of microwave engineering at the Berlin University of Technology. His main areas of research are characterization, modeling, and design of microwave semiconductor devices, monolithic integrated circuits (MICs), and microwave monolithic integrated circuits (MMICs) up to the millimeter-wave range.
