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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.

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Figures

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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.

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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).

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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.

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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

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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.

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Fig. 6

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

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Fig. 7

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

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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.

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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.

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