Physics-Based Wire Sizing for I/O Pad Cells

This document details the physical equations, parameters, and assumptions used by the LibrePDK layout generator to size wires in pad cells (such as the ESD rails and main power rails).

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1. Transient ESD Wire Sizing (Human Body Model)

During an ESD event (e.g., a Human Body Model pulse), a high current passes through the ESD protection circuit in a very short duration ($t\approx 100\text{ ns}$ to $150\text{ ns}$). Because this event is extremely brief, we assume adiabatic heating—all heat generated by Joule heating is stored in the metal wire itself, and no heat is dissipated to the surrounding oxide/dielectric.

Energy Balance Equation

The electrical energy dissipated in the wire must not exceed the thermal capacity of the wire corresponding to a safe maximum temperature rise ($\mathrm{\Delta }T$): $$E_{Joule}=E_{Thermal}$$ $$I_{peak}^2\cdot R\cdot t=m\cdot c_p\cdot \mathrm{\Delta }T$$

Parameter Substitution

The resistance of the wire is: $$R=R_{sheet}\cdot \frac{L}{W}$$

The mass of the wire is: $$m=\text{density}\times \text{Volume}=d\cdot W\cdot H\cdot L$$

Since thickness $H$ and sheet resistance $R_{sheet}$ are related to material resistivity $\rho$ by $H=\frac{\rho }{R_{sheet}}$, we can express mass as: $$m=d\cdot W\cdot \frac{\rho }{R_{sheet}}\cdot L$$

Substituting $R$ and $m$ back into the energy balance equation: $$I_{peak}^2\left(R_{sheet}\frac{L}{W}\right)t=(d\cdot W\cdot \frac{\rho }{R_{sheet}}\cdot L)c_p\cdot \mathrm{\Delta }T$$

Notice that the wire length $L$ cancels out from both sides: $$I_{peak}^2{R}_{sheet}\frac{t}{W}=d\cdot W\cdot \frac{\rho }{R_{sheet}}\cdot c_p\cdot \mathrm{\Delta }T$$

Solving for $W^2$: $$W^2=\frac{I_{peak}^2\cdot R_{sheet}^2\cdot t}{d\cdot \rho \cdot c_p\cdot \mathrm{\Delta }T}$$

Taking the square root gives the required width $W_{ESD}$: $$W_{ESD}=I_{peak}\cdot R_{sheet}\sqrt{\frac{t}{d\cdot \rho \cdot c_p\cdot \mathrm{\Delta }T}}$$

Material Constants (Aluminum)

• Density ($d$): $2700{\text{ kg/m}}^3$
• Resistivity ($\rho$): $2.65\times 10^{-8}\mathrm{\Omega }\cdot \text{m}$
• Specific Heat ($c_p$): $900\text{ J/(kg}\cdot \text{K)}$
• Safe Temp Rise ($\mathrm{\Delta }T$): $300\text{ K}$ (limits peak temperature to $\approx 600\text{ K}$, well below the Aluminum melting point of $933\text{ K}$)
• HBM Peak Current ($I_{peak}$): $1.33\text{ A}$ (corresponding to a $2\text{kV}$ HBM ESD event: $V_{ESD}/R_{HBM}=2000\text{ V}/1500\mathrm{\Omega }$)
• Pulse Duration ($t$): $150\text{ ns}$

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2. Power Rail Sizing (Electromigration Limit)

For continuous, steady-state operating currents, the sizing constraint is governed by electromigration (EM), where momentum transfer from electrons physically moves metal atoms over time, leading to voids or shorts.

To prevent electromigration, the current density must not exceed the maximum allowable threshold $J_{max}$ of the metal: $$W\ge \frac{I_{steady\mathrm{\_}state}}{J_{max}\cdot H}$$

Technology-Aware Thickness Model

Since the sheet resistance and thickness of the top metal layer are process-dependent, the generator uses estimated top metal thicknesses: * SG13G2 (TM2/Metal7): $3.0\mu \text{m}$ * SKY130 (met5/Metal6): $1.26\mu \text{m}$ * Generic Fallback: $1.0\mu \text{m}$

Using a safe, conservative maximum current density for Aluminum ($J_{max}=2.0\text{ mA/}\mu {\text{m}}^2$ at $125^{\circ }\text{C}$), the required width is computed as: $$W_{EM}=\frac{I_{steady\mathrm{\_}state}}{J_{max}\cdot H}$$