EM_analytical

A matlab tool that analytically calculates the transient Electromigration (EM) stress at discrete spatial points in multi-segment lines of power grids. This method exploits the specific form of the discrete stress coefficient matrix and constructs a closed-form with almost linear complexity without the need of time discretization. Then, this closed-form equation can be used at any given time in transient stress analysis by executing the EM_analytical_stress_calculation.m script.

If you use any part of this project, please cite:

Olympia Axelou, Nestor Evmorfopoulos, George Floros, George Stamoulis, Sachin Sapatnekar. 2022. A Novel Semi-Analytical Approach for Fast Electromigration Stress Analysis in Multi-Segment Interconnects. In Proceedings of 2022 IEEE/ACM International Conference On Computer Aided Design (ICCAD). ACM, San Diego, CA, USA.

Guideline to use EM_analytical

Matlab files hierarchy

• Matlab script files:
  • EM_analytical_wrapper.m - The wrapper of our tool. It includes the input parsing, the EM analytical functions calling and plotting of the results.
  • EM_analytical_formulation.m - This function corresponds to the one-time-cost Algorithm 1 of our paper. It forms the time-invariant matrices of our closed-form solution.
  • EM_analytical_stress_calculation.m - This function corresponds to Algorithm 2 of our paper. It is called for each different time where we want to calculate the transient stress, σ(t) and provides the final solution.
  • (auxiliary) read_inputs.m - Auxiliary function that parses the input files (line geometry and current density files) and returns the lengths and the current densities of each segment of the line.
  • (auxiliary) matrix_formulation.m - Auxiliary function that forms the system matrices A and B.
• Folder dct/ includes third-party software containing the Discrete Cosine Transform Type II (DCT-II) and the Inverse Discrete Cosine Transform Type II (IDCT-II).
• Folder benchmarks/ contains the input files for our tool.

Inputs

The input files in order to execute the tool as standalone are:

• The segment lengths. An indicative line of this file would be:

  <segment name> <node1> <node2> <length (m)>
  

• The current densities. For example:

  <segment name>, curden = <current density (A/m^2)>
  

How to run

This tool was developed using MATLAB R2021a.

Tool as standalone

In order to use the tool as standalone, and calculate the EM stress at a specific time, one has to:

• Set the time t = 6.38e8; (20y) and the discretization point dx = 1e-6; (m)
• Define the line and current densities input files

  benchmark = '5-segment';
  tree_file = strcat('benchmarks/', benchmark, '/tree_line.txt');
  current_densities_file = strcat('benchmarks/', benchmark, '/tree_curden.txt');
  

• Set the physical parameters of the line (in our example we have used Cu DD)

Integration into other projects

Inputs:

• System matrix B
• Current densities u
• Total discretization points nx_total

Steps:

1. Execute function EM_analytical_formulation.m in order to find the time-invariant parts of the closed-form equation:
   • The eigenvalues of matrix A (lambdas)
   • A temporary matrix right_side_matrix
2. Execute function EM_analytical_stress_calculation.m at each t to find stress across the line at the corresponding time.

In case of transient analysis, one can execute Step 2 for multiple points in time. Our tool supports parallel execution.

Example (5-segment-line)

In this repository, we provide an example of 5-segment line as shown here:

Performing stress analysis with our tool at t=6.38e8, i.e. 20years, and with discretization step dx=1e-6 gives as the following results: