IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 3, MAY/JUNE 2011 531 Thermal Analysis: VCSELs on an SiOB C. C. Chen, Y. C. Chen, Chin-Ta Chen, Hsu-Liang Hsiao, Chia-Chi Chang, Y. T. Cheng, Senior Member, IEEE, and Mount-Learn Wu Abstract—This paper introduces a method combining a general electrothermal network π-model in system level and the associ- ated mathematical technique, Green’s theorem, in terms of the adopted materials and system geometries to build up an equiva- lent electrothermal circuit model (EETCM) for efficient thermal analysis and behavior prediction in a thermal system. Heat conduc- tion and convection equations in integral forms are derived using the theorem and successfully applied for the thermal analysis of a 3-D optical stack, vertical-cavity surface-emitting lasers (VCSELs) on a silicon optical bench. The complex stack structure in con- ventional simulators can be greatly simplified using the method by well-predicting probable heat flow paths, and the simplifica- tion can eventually achieve the goal of CPU time saving without having complicated mesh designing or scaling. By comparing the data from the measurement, the finite-element simulation, and the method calculation shows that an excellent temperature match within ±∼0.5 ◦ C and 90% CPU time saving can be realized. Index Terms—3-D optical stacks, Green’s theorem, conduction, convection, equivalent electrothermal circuit model (EETCM), general electrothermal network π-model, heat transfer equation, silicon optical bench (SiOB), thermal management, vertical-cavity surface-emitting laser (VCSEL). I. INTRODUCTION H IGH SPEED, low power consumption, robust perfor- mance, and small form factor are four primary design issues in the development of the next generation microelec- tronic systems [1], [2]. However, undesired nonuniformly ther- mal effects due to drastic increase of the power dissipation within intensively operating chipsets have become significant design problems in the system development [1]–[4] and even restrained the design flexibility in terms of associated configu- rations of device packaging system and maximum power per- formance of integrated system circuits [1]. Several strategies and techniques concerned with the predictions of the undesired thermal effects on the microsystem performance and reliabil- Manuscript received July 8, 2010; revised August 31, 2010; accepted October 4, 2010. Date of publication December 16, 2010; date of current version June 8, 2011. This work was supported by the Optical Sciences Center, National Central University, Taiwan, under Project 99-EC-17-A-07-S1–001. C. C. Chen, Y. C. Chen, and Y. T. Cheng are with the Microsys- tems Integration Laboratory, Department of Electronics Engineering and the Institute of Electronics, National Chiao Tung University, Hsinchu, Tai- wan 300 (e-mail: gettgod.ee92g@nctu.edu.tw; ycchen.ee97g@g2.nctu.edu.tw; ytcheng@mail.nctu.edu.tw). C.-T. Chen, H.-L. Hsiao, C.-C. Chang, and M.-L. Wu are with the Department of Optics and Photonics, Institute of Optical Sci- ences, National Central University, Jhong-Li, Taiwan 320 (e-mail: cs13579@hotmail.com; s972406006@dop.ncu.edu.tw; 972406010@cc.ncu. edu.tw; mlwu@dop.ncu.edu.tw). 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.2010.2090037 Fig. 1. Scheme of the general electrothermal network π -model for thermal management [8]. By analogy with the common electric circuit π -model, three main blocks, heating sources, propagated resistances, and common base resis- tances, are adopted to present the thermal sources, thermal flow paths, and the common base, respectively. ity have been presented to conquer the circumstance [1]–[8]. So far, the establishment of equivalent electrothermal circuit model (EETCM) is the most efficient thermal analysis scheme that can be easily implemented in computer-aided design (CAD) pro- grams for optimal system-IC designs to avoid undesired system functionality degradation and device failure resulted by excess thermal accumulation. In comparison with the other analytical methods for the prediction of nonuniformly thermal effects, such as numerical solutions based on Laplace’s equation [1], finite- element analysis (FEA), or boundary element method (BEM) for computer simulators [5]–[7], the EETCM can also effectively prevent data unwieldiness and time consuming in the thermal analysis resulted by the complicated boundary conditions (BCS) associated with a system configuration. The complexity of BCS will cause a large amount of meshing work in CAD simula- tion, which would affect the accuracy of outcomes and raise the analysis time and cost. Therefore, for developing an efficient method for the ther- mal analysis of a microelectronic system with corresponding behavior predictions, this paper will illustrate a previously de- veloped general electrothermal network π-model in the system level [8], as shown in Fig. 1. By employing the concept of the π-model in electronic transistors, a thermal system can be de- composed into three main subcomponents, which are heating sources, propagated resistances, and common base resistances, respectively. From the model, the thermal interconnects from one component to the others or to external boundaries can be definitely clarified, and the external boundaries connected to the main components can be decomposed into several thermal paths to simplify the BCS. Thence, by introducing Green’s theorem for resolving the boundary problems in the Poisson’s equation of power density and heat flux gradient, new heat conduction and convection equations in integral forms can be derived to clearly specify the thermal propagations within the system, and the thermal interactions between the system and its surrounding environment. Therefore, the thermal fields within the system and on the external surfaces under the prescribed thermal-field- BCS can be defined, as well as the temperature distribution of 1077-260X/$26.00 © 2010 IEEE