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

Problem Overview

An ASIC design team optimizes the layout of a semiconductor die containing an active logic core of $48\text{ mm}^2$ bounded by $2\text{ mm}$ side margins and $1.5\text{ mm}$ top/bottom margins. We formulate total silicon area $A = W \cdot H$, reduce variables, and verify the global minimum using the Second Derivative Test.

Step-by-Step Mathematical Solution

Step 1: Formulate Overall Dimensions and Objective Function

The governing process follows the Applied Optimization Modeling Framework: (Defined in Applied Optimization Modeling Framework.)

Let $x$ be the width of the active logic core (in mm) and $y$ be the height of the active logic core (in mm).

1. Total Overall Die Dimensions:
   • Total Width:
     $$W = x + 2(2) = x + 4\text{ mm}$$
   • Total Height:
     $$H = y + 2(1.5) = y + 3\text{ mm}$$
2. Objective Function: Minimize Total Silicon Die Area:
   $$A(x, y) = W \cdot H = (x + 4)(y + 3)$$
3. Core Area Constraint:
   $$x \cdot y = 48 \implies y = \frac{48}{x}$$

Step 2: Variable Reduction (Substitution)

Substitute $y = \frac{48}{x}$ into the area function:

Step 3: Establish the Context-Driven Domain

(Governed by Context-Driven Domain in Geometric Optimization.)

• Active core dimensions must be strictly positive: $x > 0$.
• Feasible physical domain is the open interval $x \in (0, \infty)$.

Step 4: Differentiate and Find Critical Numbers

Differentiate $A(x)$ with respect to $x$ using the Power Rule:

Step 5: Verify Global Minimum via Second Derivative Test

Compute the second derivative:

Step 6: Compute Final Core and Die Dimensions

1. Active Core Height ($y$):
   $$y = \frac{48}{8} = 6\text{ mm}$$
2. Overall Die Dimensions ($W \times H$):
   • Total Width: $W = 8 + 4 = 12\text{ mm}$
   • Total Height: $H = 6 + 3 = 9\text{ mm}$
3. Minimum Total Silicon Die Area:
   $$A_{\min} = 12 \times 9 = 108\text{ mm}^2$$

VLSI Engineering Insight: The active core aspect ratio is $\frac{x}{y} = \frac{8}{6} = \frac{4}{3}$, while the overall die aspect ratio is $\frac{W}{H} = \frac{12}{9} = \frac{4}{3}$. The optimal layout aligns the core aspect ratio precisely with the margin ratio $\frac{\Delta W}{\Delta H} = \frac{4}{3}$, ensuring balanced silicon utilization.