Integrated Circuit Die Silicon Area & Routing Margins (Electrical & Computer Engineering)

In semiconductor physics and VLSI (Very Large Scale Integration) microchip design, silicon manufacturing cost scales quadratically with total die area. A microchip die consists of an active central logic core (transistors, logic gates, and cache memory) surrounded by peripheral silicon margins allocated for input/output (I/O) wire-bond pads, electrostatic discharge (ESD) protection circuits, power supply distribution rings, and wafer dicing scribe lines.

A computer engineering team at UNLV is laying out a custom ASIC (Application-Specific Integrated Circuit) processor. Architectural timing simulations dictate that the central active logic core must have an area of exactly $48\text{ mm}^2$.

       +---------------------------------------------+
       |           Top Margin (1.5 mm)               |
       +-------+-----------------------------+-------+
       |  Left |                             | Right |
       | Margin|      Active Logic Core      | Margin|
       | (2 mm)|        Area = 48 mm^2       | (2 mm)|  H
       |       |       (Width x, Height y)   |       |
       +-------+-----------------------------+-------+
       |          Bottom Margin (1.5 mm)             |
       +---------------------------------------------+
                              W

Design rule constraints require the following peripheral margins around the core:

• A $2\text{ mm}$ margin on both the left and right sides for high-speed differential serial I/O transceivers and wire-bond pads.
• A $1.5\text{ mm}$ margin on both the top and bottom edges for ground/power distribution buses and test probe pads.

1. Express the overall die width $W$ and overall die height $H$ in terms of the core dimensions $x$ (core width) and $y$ (core height).
2. Formulate the total silicon die area function $A(x, y)$ and apply the logic core area constraint to reduce the model to a single-variable function $A(x)$.
3. Establish the Context-Driven Domain for core width $x$.
4. Determine the core dimensions ($x$ and $y$) and overall die dimensions ($W$ and $H$) that minimize total silicon die area.
5. Use the Second Derivative Test to verify that your critical point yields an absolute minimum die area, and compute the minimum total die area.