Leviathan: /* Technology-Aware Thickness Model */

2026-05-26T01:02:59Z

Technology-Aware Thickness Model

← Older revision		Revision as of 01:02, 26 May 2026
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	=== Technology-Aware Thickness Model ===		=== 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):''' <math display="inline">3.0\ \mu\text{m}</math> * '''SKY130 (met5/Metal6):''' <math display="inline">1.26\ \mu\text{m}</math> * '''Generic Fallback:''' <math display="inline">1.0\ \mu\text{m}</math>		Since the sheet resistance and thickness of the top metal layer are process-dependent, the generator uses estimated top metal thicknesses:

	Using a safe, conservative maximum current density for Aluminum (<math display="inline">J_{max} = 2.0\text{ mA/}\mu\text{m}^2</math> at <math display="inline">125^\circ\text{C}</math>), the required width is computed as: <math display="block">W_{EM} = \frac{I_{steady\_state}}{J_{max} \cdot H}</math>		'''SG13G2 (TM2/Metal7):''' <math display="inline">3.0\ \mu\text{m}</math>

			'''SKY130 (met5/Metal6):''' <math display="inline">1.26\ \mu\text{m}</math>

			'''Generic Fallback:''' <math display="inline">1.0\ \mu\text{m}</math>

			Using a safe, conservative maximum current density for Aluminum

			<math display="inline">J_{max} = 2.0\text{ mA/}\mu\text{m}^2</math> at <math display="inline">125^\circ\text{C}</math>


			the required width is computed as: <math display="block">W_{EM} = \frac{I_{steady\_state}}{J_{max} \cdot H}</math>

Leviathan: Created page with " 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). ---- == 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 $150\text{ n..."$

2026-05-26T01:01:03Z

Created page with " 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). ---- == 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 (<math display="inline">t \approx 100\text{ ns}</math> to <math display="inline">150\text{ n..."

New page

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).
----

== 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 (<math display="inline">t \approx 100\text{ ns}</math> to <math display="inline">150\text{ ns}</math>). 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 (<math display="inline">\Delta T</math>): <math display="block">E_{Joule} = E_{Thermal}</math> <math display="block">I_{peak}^2 \cdot R \cdot t = m \cdot c_p \cdot \Delta T</math>

=== Parameter Substitution ===
The resistance of the wire is: <math display="block">R = R_{sheet} \cdot \frac{L}{W}</math>

The mass of the wire is: <math display="block">m = \text{density} \times \text{Volume} = d \cdot W \cdot H \cdot L</math>

Since thickness <math display="inline">H</math> and sheet resistance <math display="inline">R_{sheet}</math> are related to material resistivity <math display="inline">\rho</math> by <math display="inline">H = \frac{\rho}{R_{sheet}}</math>, we can express mass as: <math display="block">m = d \cdot W \cdot \frac{\rho}{R_{sheet}} \cdot L</math>

Substituting <math display="inline">R</math> and <math display="inline">m</math> back into the energy balance equation: <math display="block">I_{peak}^2 \left( R_{sheet} \frac{L}{W} \right) t = \left( d \cdot W \cdot \frac{\rho}{R_{sheet}} \cdot L \right) c_p \cdot \Delta T</math>

Notice that the wire length <math display="inline">L</math> cancels out from both sides: <math display="block">I_{peak}^2 R_{sheet} \frac{t}{W} = d \cdot W \cdot \frac{\rho}{R_{sheet}} \cdot c_p \cdot \Delta T</math>

Solving for <math display="inline">W^2</math>: <math display="block">W^2 = \frac{I_{peak}^2 \cdot R_{sheet}^2 \cdot t}{d \cdot \rho \cdot c_p \cdot \Delta T}</math>

Taking the square root gives the required width <math display="inline">W_{ESD}</math>: <math display="block">W_{ESD} = I_{peak} \cdot R_{sheet} \sqrt{\frac{t}{d \cdot \rho \cdot c_p \cdot \Delta T}}</math>

=== Material Constants (Aluminum) ===

* '''Density (<math display="inline">d</math>):''' <math display="inline">2700\text{ kg/m}^3</math>
* '''Resistivity (<math display="inline">\rho</math>):''' <math display="inline">2.65 \times 10^{-8}\ \Omega\cdot\text{m}</math>
* '''Specific Heat (<math display="inline">c_p</math>):''' <math display="inline">900\text{ J/(kg}\cdot\text{K)}</math>
* '''Safe Temp Rise (<math display="inline">\Delta T</math>):''' <math display="inline">300\text{ K}</math> (limits peak temperature to <math display="inline">\approx 600\text{ K}</math>, well below the Aluminum melting point of <math display="inline">933\text{ K}</math>)
* '''HBM Peak Current (<math display="inline">I_{peak}</math>):''' <math display="inline">1.33\text{ A}</math> (corresponding to a <math display="inline">2\text{kV}</math> HBM ESD event: <math display="inline">V_{ESD} / R_{HBM} = 2000\text{ V} / 1500\ \Omega</math>)
* '''Pulse Duration (<math display="inline">t</math>):''' <math display="inline">150\text{ ns}</math>

----

== 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 <math display="inline">J_{max}</math> of the metal: <math display="block">W \ge \frac{I_{steady\_state}}{J_{max} \cdot H}</math>

=== 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):''' <math display="inline">3.0\ \mu\text{m}</math> * '''SKY130 (met5/Metal6):''' <math display="inline">1.26\ \mu\text{m}</math> * '''Generic Fallback:''' <math display="inline">1.0\ \mu\text{m}</math>

Using a safe, conservative maximum current density for Aluminum (<math display="inline">J_{max} = 2.0\text{ mA/}\mu\text{m}^2</math> at <math display="inline">125^\circ\text{C}</math>), the required width is computed as: <math display="block">W_{EM} = \frac{I_{steady\_state}}{J_{max} \cdot H}</math>

Physics-Based Wire Sizing for I/O Pad Cells - Revision history

Leviathan: /* Technology-Aware Thickness Model */