Simulation of On Chip Interconnects with Three-Dimensional ...



Simulation of On Chip Interconnects with Three-Dimensional FDTD Alternating-Direction-Implicit (ADI) Method

X. Shao1 ,2, Neil Goldsman2, O. Ramahi2,3,4, P. N. Guzdar5

1National Space Science Data Center, Goddard Space Flight Center, NASA, Greenbelt, MD 20171.

2Electrical and Computer Engineering Department, 3Mechanical Engineering Department, 4CALCE electronic Products and Systems Center, 5Institute of Plasma research, University of Maryland, College Park, MD 20742.

Introduction:

Simulation of modern on chip interconnects or Metal-Insulator-Silicon Substrate (MIS) structure poses following challenges: 1. Need to resolve structure much smaller than the wavelength, e.g., the metal skin depth, SiO2 layer, both of which are in the micro meter range at the frequency of interest. 2. Being able to simulate broadband digital signal propagation. 3. Being able to simulate the induced substrate current loss. While conventional explicit Finite-Difference-Time-Domain (FDTD) method for solving Maxwell’s equation can meet the latter two challenges. It is limited by the first challenge since the simulation time step needs to satisfy the Courant’s condition ([pic]). With the smallest grid size on the order of 0.1 um, in order to resolve the skin depth, the time step needs to be < 4×10-16 sec. But, we are only interested in the pico-second signal propagation. [1, 2] introduced Alternating-Direction-Implicit (ADI) method to solve the Maxwell’s Equation by breaking the Courant’s limit. The ADI method is well-suited for studying signal propagation along the MIS structure since the simulation step is not limited by the finest grid size. In this paper, we developed a three-dimensional multi-grid FDTD-ADI code to study the digital signal propagation along on chip interconnects.

Simulation Method:

We briefly summarize the ADI method. The Maxwell’s equation is discretized on Yee’s [3] grid and two equations are in the form of Equ. 1 and 2. At the first step, the first half of the left side in Equ. 1 and the first half of the left side in Equ. 2 are treated as implicit. We plug Equ. 2 ([pic]) back to Equ. 1 and obtain Equ. 3. Equ. 3 can be easily solved with a tri-diagonal matrix solver. This procedure is conducted for all three components of the electric field. In the next step, we treat the other half of the Equ. 1 and 2 as implicit and obtain equation similar to Equ. 3. These two steps are alternated thereafter. The detailed algorithm of ADI method can be found in [1, 2].

[pic] (1)

[pic] (2)

[pic] (3)

Simulation Results:

To test the performance of our code in resolving the metal skin depth, we performed a 2D simulation of EM wave propagation under a metal strip. The configuration of the simulation is shown in Figure 1a. The metal strip conductivity= 3.9×107 S/m. The domain bottom is bounded with Perfect Electric Conductor (PEC) .The smallest grid size of 0.1 um is placed inside the metal. The metal is of thickness= 1.8 um. The grid along Z direction is of uniform size=150 um. The Courant condition requires ∆t ................
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