0
Research Papers: Heat Transfer in Manufacturing

Transient Heat Conduction in On-Chip Interconnects Using Proper Orthogonal Decomposition Method

[+] Author and Article Information
Banafsheh Barabadi

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Satish Kumar

G. W. Woodruff School of Mechanical
Engineering,
Georgia Institute of Technology,
801 Ferst Drive,
Atlanta, GA 30306

Yogendra K. Joshi

G. W. Woodruff School of Mechanical
Engineering,
Georgia Institute of Technology,
801 Ferst Drive,
Atlanta, GA 30306
e-mail: yogendra.joshi@me.gatech.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 1, 2013; final manuscript received January 25, 2017; published online March 21, 2017. Assoc. Editor: Leslie Phinney.

J. Heat Transfer 139(7), 072101 (Mar 21, 2017) (10 pages) Paper No: HT-13-1223; doi: 10.1115/1.4035889 History: Received May 01, 2013; Revised January 25, 2017

A major challenge in maintaining quality and reliability in today's microelectronics chips comes from the ever increasing levels of integration in the device fabrication, as well as from the high current densities. Transient Joule heating in the on-chip interconnect metal lines with characteristic sizes of tens of nanometer, can lead to thermomechanical fatigue and failure due to the thermal expansion coefficient mismatch between different materials. Full-field simulations of nearly a billion interconnects in a modern microprocessor are infeasible due to the grid size requirements. To prevent premature device failures, a rapid predictive capability for the thermal response of on-chip interconnects is essential. This work develops a two-dimensional (2D) transient heat conduction framework to analyze inhomogeneous domains, using a reduced-order modeling approach based on proper orthogonal decomposition (POD) and Galerkin projection. POD modes are generated by using a representative step function as the heat source. The model rapidly predicted the transient thermal behavior of the system for several cases, without generating any new observations, and using just a few POD modes.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Lloyd, J. R. , and Thompson, C. V. , 1993, “ Materials Reliability in Microelectronics,” MRS Bull., 18(12), pp. 16–18. [CrossRef]
Phan, T. , Dilhaire, S. , Quintard, V. , Lewis, D. , and Claeys, W. , 1997, “ Thermomechanical Study of AlCu Based Interconnect Under Pulsed Thermoelectric Excitation,” J. Appl. Phys., 81(3), p. 1157. [CrossRef]
Bilotti, A. A. , 1974, “ Static Temperature Distribution in IC Chips With Isothermal Heat Sources,” IEEE Trans. Electron Devices, 21(3), pp. 217–226. [CrossRef]
Shen, Y. L. , 1999, “ Analysis of Joule Heating in Multilevel Interconnects,” J. Vac. Sci. Technol., B: Microelectron. Nanometer Struct., 17(5), pp. 2115–2121. [CrossRef]
Teng, C. C. , Cheng, Y. K. , Rosenbaum, E. , and Kang, S. M. , 2002, “ Item: A Temperature-Dependent Electromigration Reliability Diagnosis Tool,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 16(8), pp. 882–893. [CrossRef]
Chen, D. , Li, E. , Rosenbaum, E. , and Kang, S. M. , 2002, “ Interconnect Thermal Modeling for Accurate Simulation of Circuit Timing and Reliability,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 19(2), pp. 197–205. [CrossRef]
Stan, M. R. , Skadron, K. , Barcella, M. , Wei, H. , Sankaranarayanan, K. , and Velusamy, S. , 2003, “ Hotspot: A Dynamic Compact Thermal Model at the Processor-Architecture Level,” Microelectron. J., 34(12), pp. 1153–1165. [CrossRef]
Gurrum, S. P. , Joshi, Y. K. , King, W. P. , Ramakrishna, K. , and Gall, M. , 2008, “ A Compact Approach to On-Chip Interconnect Heat Conduction Modeling Using the Finite Element Method,” ASME J. Electron. Packag., 130(3), p. 031001. [CrossRef]
Celo, D. , Ming, G. X. , Gunupudi, P. K. , Khazaka, R. , Walkey, D. J. , Smy, T. , and Nakhla, M. S. , 2005, “ Hierarchical Thermal Analysis of Large IC Modules,” IEEE Trans. Compon. Packag. Technol., 28(2), pp. 207–217. [CrossRef]
Christopoulos, C. , 2002, “ The Transmission-Line Modeling Method: TLM,” IEEE Antennas Propag. Mag., 39(1), p. 90. [CrossRef]
De Cogan, D. , O'connor, W. J. , and Pulko, S. , 2005, Transmission Line Matrix (TLM) in Computational Mechanics, CRC Press, Boca Raton, FL.
Barabadi, B. , Joshi, Y. K. , Kumar, S. , and Refai-Ahmed, G. , 2010, “ Thermal Characterization of Planar Interconnect Architectures Under Transient Currents,” ASME Paper No. IMECE2009-11996.
Barabadi, B. , Joshi, Y. K. , Kumar, S. , and Refai-Ahmed, G. , 2010, “ Thermal Characterization of Planar Interconnect Architectures Under Different Rapid Transient Currents Using the Transmission Line Matrix and Finite Element Methods,” 12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), Las Vegas, NV, June 2–5, pp. 1–8.
Ait-Sadi, R. , and Naylor, P. , 1993, “ An Investigation of the Different TLM Configurations Used in the Modelling of Diffusion Problems,” Int. J. Numer. Modell.: Electron. Networks, Devices Fields, 6(4), pp. 253–268. [CrossRef]
Smy, T. , Walkey, D. , and Dew, S. , 2001, “ Transient 3d Heat Flow Analysis for Integrated Circuit Devices Using the Transmission Line Matrix Method on a Quad Tree Mesh,” Solid-State Electron., 45(7), pp. 1137–1148. [CrossRef]
Antoulas, A. , Sorensen, D. , and Gugercin, S. , 2001, “ A Survey of Model Reduction Methods for Large-Scale Systems,” Contemp. Math., 280(2001), pp. 193–220.
Glover, K. , 1984, “ All Optimal Hankel-Norm Approximations of Linear Multivariable Systems and Their L,∞-Error Bounds†,” Int. J. Control, 39(6), pp. 1115–1193. [CrossRef]
Kokotovic, P. V. , 1976, “ Singular Perturbations and Order Reduction in Control Theory—An Overview,” Automatica, 12(2), pp. 123–132. [CrossRef]
Pearson, K. , 1901, “ LIII on Lines and Planes of Closest Fit to Systems of Points in Space,” London, Edinburgh, Dublin Philos. Mag. J. Sci., 2(11), pp. 559–572. [CrossRef]
Feldmann, P. , and Freund, R. W. , 1995, “ Efficient Linear Circuit Analysis by Padé Approximation Via the Lanczos Process,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14(5), pp. 639–649. [CrossRef]
Grimme, E. J. , 1997, “ Krylov Projection Methods for Model Reduction,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Champaign, IL.
Jaimoukha, I. M. , and Kasenally, E. M. , 1997, “ Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations,” SIAM J. Matrix Anal. Appl., 18(3), pp. 633–652. [CrossRef]
Tan, B. T. , 2003, “ Proper Orthogonal Decomposition Extensions and Their Applications in Steady Aerodynamics,” Ph.D. thesis, Ho Chi Minh City University of Technology, Ho Chi Minh, Vietnam.
Ahlman, D. , Jackson, J. , Kurdila, A. , and Shyy, W. , 2002, “ Proper Orthogonal Decomposition for Time-Dependent Lid-Driven Cavity Flows,” Numer. Heat Transfer, Part B, 42(4), pp. 285–306. [CrossRef]
Berkooz, G. , Holmes, P. , and Lumley, J. L. , 1993, “ The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annu. Rev. Fluid Mech., 25(1), pp. 539–575. [CrossRef]
Berkooz, G. , Holmes, P. , and Lumley, J. , 1996, “ Turbulence, Coherent Structures, Dynamical Systems and Symmetry,” Cambridge Monographs on Mechanics, Cambridge University Press, New York, pp. 1200–1208.
Cusumano, J. P. , Sharkady, M. T. , and Kimble, B. W. , 1994, “ Experimental Measurements of Dimensionality and Spatial Coherence in the Dynamics of a Flexible-Beam Impact Oscillator,” Philos. Trans. R. Soc., A, 347(1683), pp. 421–438. [CrossRef]
Feeny, B. F. , and Kappagantu, R. , 1998, “ On the Physical Interpretation of Proper Orthogonal Modes in Vibrations,” J. Sound Vib., 211(4), pp. 607–616. [CrossRef]
Atwell, J. A. , and King, B. B. , 2001, “ Proper Orthogonal Decomposition for Reduced Basis Feedback Controllers for Parabolic Equations,” Math. Comput. Modell., 33(1–3), pp. 1–19. [CrossRef]
Liang, Y. , Lin, W. , Lee, H. , Lim, S. , Lee, K. , and Sun, H. , 2002, “ Proper Orthogonal Decomposition and Its Applications—Part II: Model Reduction for MEMS Dynamical Analysis,” J. Sound Vib., 256(3), pp. 515–532. [CrossRef]
Codecasa, L. , D'amore, D. , and Maffezzoni, P. , 2003, “ An Arnoldi Based Thermal Network Reduction Method for Electro-Thermal Analysis,” IEEE Trans. Compon. Packag. Technol., 26(1), pp. 186–192. [CrossRef]
Bialecki, R. , Kassab, A. , and Fic, A. , 2005, “ Proper Orthogonal Decomposition and Modal Analysis for Acceleration of Transient FEM Thermal Analysis,” Int. J. Numer. Methods Eng., 62(6), pp. 774–797. [CrossRef]
Bialecki, R. , Kassab, A. , and Fic, A. , 2003, “ Reduction of the Dimensionality of Transient FEM Solutions Using Proper Orthogonal Decomposition,” AIAA Paper No. 2003-4207.
Fic, A. , Bialecki, R. A. , and Kassab, A. J. , 2005, “ Solving Transient Non-Linear Heat Conduction Problems by Proper Orthogonal Decomposition and FEM,” Numer. Heat Transfer, Part B, 48(2), pp. 103–124. [CrossRef]
Bleris, L. G. , and Kothare, M. V. , 2005, “ Reduced Order Distributed Boundary Control of Thermal Transients in Microsystems,” IEEE Trans. Control Syst. Technol., 13(6), pp. 853–867. [CrossRef]
Raghupathy, A. P. , Ghia, U. , Ghia, K. , and Maltz, W. , 2010, “ Boundary-Condition-Independent Reduced-Order Modeling of Heat Transfer in Complex Objects by POD-Galerkin Methodology: 1d Case Study,” ASME J. Heat Transfer, 132(6), p. 064502. [CrossRef]
Jolliffe, I. T. , and Myilibrary , 2005, Principal Component Analysis, Springer, New York.
Grossman, R. L. , and Kamath, C. , 2001, Data Mining for Scientific and Engineering Applications, Kluwer Academic Publishers, Norwell, MA.
Kirby, M. , and Sirovich, L. , 2002, “ Application of the Karhunen-Loeve Procedure for the Characterization of Human Faces,” IEEE Trans. Pattern Anal. Mach. Intell., 12(1), pp. 103–108. [CrossRef]
Eriksson, P. , Jiménez, C. , Bühler, S. , and Murtagh, D. , 2002, “ A Hotelling Transformation Approach for Rapid Inversion of Atmospheric Spectra,” J. Quant. Spectrosc. Radiat. Transfer, 73(6), pp. 529–543. [CrossRef]
Liang, Y. , Lee, H. , Lim, S. , Lin, W. , Lee, K. , and Wu, C. , 2002, “ Proper Orthogonal Decomposition and Its Applications—Part I: Theory,” J. Sound Vib., 252(3), pp. 527–544. [CrossRef]
Holmes, P. , Lumley, J. L. , and Berkooz, G. , 1998, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, New York.
Chatterjee, A. , 2000, “ An Introduction to the Proper Orthogonal Decomposition,” Curr. Sci., 78(7), pp. 808–817.
Rolander, N. W. , 2005, “ An Approach for the Robust Design of Data Center Server Cabinets,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA.
Bizon, K. , Continillo, G. , Russo, L. , and Smula, J. , 2008, “ On POD Reduced Models of Tubular Reactor With Periodic Regimes,” Comput. Chem. Eng., 32(6), pp. 1305–1315. [CrossRef]
Graham, M. D. , and Kevrekidis, I. G. , 1996, “ Alternative Approaches to the Karhunen-Loeve Decomposition for Model Reduction and Data Analysis,” Comput. Chem. Eng., 20(5), pp. 495–506. [CrossRef]
Rowley, C. W. , Colonius, T. , and Murray, R. M. , 2001, “ Dynamical Models for Control of Cavity Oscillations,” AIAA Paper No. 2001-2126.
Ding, P. , Wu, X. H. , He, Y. L. , and Tao, W. Q. , 2008, “ A Fast and Efficient Method for Predicting Fluid Flow and Heat Transfer Problems,” ASME J. Heat Transfer, 130(3), p. 032502. [CrossRef]
Ly, H. V. , and Tran, H. T. , 2001, “ Modeling and Control of Physical Processes Using Proper Orthogonal Decomposition,” Math. Comput. Modell., 33(1), pp. 223–236. [CrossRef]
Rambo, J. , and Joshi, Y. , 2007, “ Reduced-Order Modeling of Turbulent Forced Convection With Parametric Conditions,” Int. J. Heat Mass Transfer, 50(3–4), pp. 539–551. [CrossRef]
Rambo, J. D. , 2006, “ Reduced-Order Modeling of Multiscale Turbulent Convection: Application to Data Center Thermal Management,” Ph.D. thesis, Geogia Institute of Technology, Atlanta, GA.
Ostrowski, Z. , Białecki, R. , and Kassab, A. , 2008, “ Solving Inverse Heat Conduction Problems Using Trained POD-RBF Network Inverse Method,” Inverse Probl. Sci. Eng., 16(1), pp. 39–54. [CrossRef]
Ostrowski, Z. , Białecki, R. A. , and Kassab, A. J. , 2005, “ Estimation of Constant Thermal Conductivity by Use of Proper Orthogonal Decomposition,” Comput. Mech., 37(1), pp. 52–59. [CrossRef]
Temam, R. , 1997, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York.
Zhang, Z. M. , 2007, Nano/Microscale Heat Transfer, McGraw-Hill Professional, New York.
Barabadi, B. , Joshi, Y. , and Kumar, S. , 2011, “ Prediction of Transient Thermal Behavior of Planar Interconnect Architecture Using Proper Orthogonal Decomposition Method,” ASME Paper No. IPACK2011-52133.

Figures

Grahic Jump Location
Fig. 1

Schematic of the computational domain with a cross-sectional area of 1.44 μm × 720 nm. It consists of a set of 360 nm × 360 nm interconnects that are evenly spaced and embedded in the dielectric. The mesh used in the POD and FE models is shown. In this study: Hint = Hde = 360 nm and P = 4Hint = 1.44 μm.

Grahic Jump Location
Fig. 2

Different types of heat sources used in this study. Case 1: only step function (solid line) and case 2: sinusoidal and step function (dashed line).

Grahic Jump Location
Fig. 3

Eigenvalues or energy percentage in log form versus number of the POD modes. In order to build a reliable reduced order model, the number of basis functions used for the projection was chosen such that the cumulative correlation energy of the modes are greater or equal to 99.99%. The first two modes capture over 98% of the energy.

Grahic Jump Location
Fig. 4

Comparison of temporal dependence of temperature rise in the left-most node of the top edge (node 1 in the interconnect), x = 0 using 17, 9, 5, and 2 observations for case 1

Grahic Jump Location
Fig. 5

First five POD modes or basis functions plotted in 2D contours. The POD modes are normalized by the total sum of the modes chosen for each study.

Grahic Jump Location
Fig. 6

First five b-coefficients versus time using the Galerkin projection technique for case1

Grahic Jump Location
Fig. 7

Spatial variation of temperature rise after 20 μs for FE (top) and POD (bottom) models using five basis functions for case 1

Grahic Jump Location
Fig. 8

Spatial variation of temperature rise after 18 μs in x direction for the upper edge of the structure (a), y direction for the left edge of the structure (c), and along the diagonal (e). Spatial variation of temperature after 19.2 μs in x direction for the upper edge of the structure (b), y direction for the left edge of the structure (d), and along the diagonal (f). The FE results are plotted in solid lines, and the POD results using five basis functions are plotted in circular markers. The results are for case 2.

Grahic Jump Location
Fig. 9

Comparison of temporal dependence of temperature rise at nodes 1–5 along the diagonal for case 2. The FE results are plotted in circular markers, and the POD results are shown by dashed lines.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In