Tianshu Hou
Abstract
The electromigration-induced reliability issues (EM) in very large scale integration (VLSI) circuits have attracted continuous attention due to technology scaling. Traditional EM methods lead to inaccurate results incompatible with the advanced technology nodes. In this article, we propose a learning-based model by enforcing physical constraints of EM kinetics to solve the EM reliability problem. The method aims at solving stress-based partial differential equations (PDEs) to obtain the hydrostatic stress evolution on interconnect trees during the void nucleation phase, considering varying atom diffusivity on each segment, which is one of the EM random characteristics. The approach proposes a crafted neural network-based framework customized for the EM phenomenon and provides mesh-free solutions benefiting from the employment of automatic differentiation (AD). Experimental results obtained by the proposed model are compared with solutions obtained by competing methods, showing satisfactory accuracy and computational savings.
- [1] . 2016. TensorFlow: A system for large-scale machine learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI’16). 265–283.Google Scholar
- [2] . 2004. Design Tool and Methodologies for Interconnect Reliability Analysis in Integrated Circuits. Ph.D. Dissertation. Massachusetts Institute of Technology.Google ScholarDigital Library
- [3] . 2015. Automatic differentiation in machine learning: A survey. (2015).
arxiv:1502.05767 Google Scholar - [4] . 1969. Electromigration–A brief survey and some recent results. IEEE Trans. Electron. Devices 16, 4 (1969), 338–347.Google ScholarCross Ref
- [5] . 1976. Electromigration in thin aluminum films on titanium nitride. J. Appl. Phys. 47, 4 (1976), 1203–1208.Google ScholarCross Ref
- [6] . 2018. Power grid electromigration checking using physics-based models. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. 37, 7 (2018), 1317–1330.Google ScholarCross Ref
- [7] . 2016. Analytical modeling and characterization of electromigration effects for multibranch interconnect trees. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. 35, 11 (2016), 1811–1824.Google ScholarDigital Library
- [8] . 2020. Fast analytic electromigration analysis for general multisegment interconnect wires. IEEE Trans. Very Large Scale Integ. Syst. 28, 2 (2020), 421–432.Google ScholarCross Ref
- [9] . 2021. A fast semi-analytic approach for combined electromigration and thermomigration analysis for general multisegment interconnects. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. 40, 2 (2021), 350–363.Google ScholarDigital Library
- [10] . 1981. Numerical differentiation of analytic functions. ACM Trans. Math. Softw. 7, 4 (1981), 512–526.Google ScholarDigital Library
- [11] . 2003. Computer Algebra: Historical Development, Characterization, and Prospects. Springer-Verlag, Berlin, 1–9.Google Scholar
- [12] . 2000. The effects of the mechanical properties of the confinement material on electromigration in metallic interconnects. J. Mater. Res. 15, 8 (2000), 1797–1802.Google ScholarCross Ref
- [13] . 2013. Electromigration early failure void nucleation and growth phenomena in Cu and Cu(Mn) interconnects. In Proceedings of the IEEE International Reliability Physics Symposium. 2C.1.1–2C.1.6.Google ScholarCross Ref
- [14] . 2016. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 770–778.Google ScholarCross Ref
- [15] . 1989. Multilayer feedforward networks are universal approximators. Neural Netw. 2, 5 (1989), 359–366.Google ScholarCross Ref
- [16] . 2022. A space-time neural network for analysis of stress evolution under DC current stressing. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. (2022), 1–14.
DOI: Google ScholarCross Ref - [17] . 2016. Physics-based electromigration models and full-chip assessment for power grid networks. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. 35, 11 (2016), 1848–1861.Google ScholarDigital Library
- [18] . 2014. Physics-based electromigration assessment for power grid networks. In Proceedings of the 51st Annual Design Automation Conference. 1–6.Google ScholarDigital Library
- [19] . 2021. EMGraph: Fast learning-based electromigration analysis for multi-segment interconnect using graph convolution networks. In Proceedings of the 58th ACM/IEEE Design Automation Conference (DAC). 919–924.Google ScholarDigital Library
- [20] . 2021. Data-driven electrostatics analysis based on physics-constrained deep learning. In Proceedings of the Design, Automation Test in Europe Conference Exhibition (DATE). 1382–1387.Google ScholarCross Ref
- [21] . 1993. Stress evolution due to electromigration in confined metal lines. J. Appl. Phys. 73, 8 (1993), 3790–3799.Google ScholarCross Ref
- [22] . 1998. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9, 5 (1998), 987–1000.Google ScholarDigital Library
- [23] . 2020. Fourier neural operator for parametric partial differential equations. CoRR abs/2010.08895 (2020).
arXiv:2010.08895 Google Scholar - [24] . 2008. New models for interconnect failure in advanced IC technology. In Proceedings of the 15th International Symposium on the Physical and Failure Analysis of Integrated Circuits. 1–7.Google ScholarCross Ref
- [25] . 2019. PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs. (2019).
arxiv:1909.10145 Google Scholar - [26] . 2008. Power grid analysis benchmarks. In Proceedings of the Asia and South Pacific Design Automation Conference. 376–381.Google ScholarCross Ref
- [27] . 1998. In Reliability and Failure of Electronic Materials and Devices. Elsevier.Google Scholar
- [28] . 2017. Automatic differentiation in Pytorch. In Proceedings of the NIPS Workshop.Google Scholar
- [29] . 2017. DeepXplore: Automated Whitebox Testing of Deep Learning Systems. (2017).
arxiv:1705.06640 Google Scholar - [30] . 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 (2019), 686–707.Google ScholarCross Ref
- [31] . 2018. Deep Neural Networks Motivated by Partial Differential Equations. (2018).
arxiv:1804.04272 Google Scholar - [32] . 2003. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 192, 7 (2003), 941–954.Google ScholarCross Ref
- [33] . 2016. Postvoiding stress evolution in confined metal lines. IEEE Trans. Device Mater. Reliab. 16, 1 (2016), 50–60.Google ScholarCross Ref
- [34] . 2020. EMSpice: Physics-based electromigration check using coupled electronic and stress simulation. IEEE Trans. Device Mater. Reliab. 20, 2 (2020), 376–389.Google ScholarCross Ref
- [35] . 2021. Fast physics-based electromigration analysis for full-chip networks by efficient eigenfunction-based solution. IEEE Trans. Comput.-Aid. Des. Integ. Circ. Syst. 40, 3 (2021), 507–520.Google ScholarCross Ref
- [36] . 2021. Predicting the dynamic process and model parameters of the vector optical solitons in birefringent fibers via the modified PINN. Chaos, Solitons Fract. 152 (2021), 111393.Google ScholarCross Ref
- [37] . 2020. Weak adversarial networks for high-dimensional partial differential equations. J. Comput. Phys. 411 (2020), 109409.Google ScholarCross Ref
Index Terms
- A Deep Learning Framework for Solving Stress-based Partial Differential Equations in Electromigration Analysis
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