electromigration simulation
**Electromigration (EM) simulation** predicts the **lifetime and failure risk** of metal interconnects under current stress — modeling how sustained current flow causes metal atoms to migrate along the conductor, eventually forming voids (open circuits) or hillocks (short circuits).
**What Electromigration Is**
- When current flows through a metal conductor, the "electron wind" exerts a force on metal atoms, causing them to diffuse in the direction of electron flow.
- Over time, this migration creates:
- **Voids**: Metal atoms leave, creating gaps that increase resistance and eventually cause open-circuit failure.
- **Hillocks/Extrusions**: Metal atoms accumulate, forming protrusions that can short to adjacent lines.
- EM is a **wear-out** failure mechanism — it develops gradually over the operating lifetime of the chip.
**The Physics of Electromigration**
- **Driving Force**: The electron wind force $F = Z^* e \rho j$, where $Z^*$ is the effective charge, $\rho$ is resistivity, and $j$ is current density.
- **Diffusion Paths**: Atoms migrate along the path of least resistance — grain boundaries, interfaces (metal/barrier, metal/cap), and surfaces.
- **Black's Equation**: The empirical lifetime model:
$$t_{50} = A \cdot j^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right)$$
Where $t_{50}$ is median time to failure, $j$ is current density, $n$ ≈ 1–2, and $E_a$ is activation energy.
- **Blech Effect**: Below a critical length-current-density product ($jL_{crit}$), mechanical back-stress prevents void growth — short lines are **immortal** to EM.
**What EM Simulation Predicts**
- **Current Density Distribution**: Map $j$ across all metal segments — identify hot spots where current density exceeds the EM design rule limit.
- **Void Nucleation Sites**: Predict where voids will form based on flux divergence — typically at via connections, width transitions, and grain boundary triple junctions.
- **Time to Failure**: Estimate the lifetime of each interconnect segment under operating conditions.
- **Temperature Effects**: Higher temperature exponentially accelerates EM — thermal simulation feeds temperature data to EM analysis.
**EM Design Rules**
- **Maximum Current Density**: $j_{max}$ limits for each metal layer and via — typically **1–2 MA/cm²** for DC and higher for AC (due to self-heating recovery).
- **AC Enhancement**: Bidirectional current flow partially reverses EM damage — AC currents can tolerate higher $j$ than DC.
- **Width and Length Dependence**: Wider lines and shorter segments are more EM-resistant.
**Simulation Workflow**
1. **Extract** the power grid and signal net current distributions from circuit simulation.
2. **Map** current densities onto the physical layout.
3. **Check** all segments against EM design rules.
4. **Fix** violations by widening metal, adding vias, or redistributing current.
Electromigration simulation is **critical for reliability** — at advanced nodes with shrinking metal dimensions and increasing current densities, EM is one of the primary lifetime-limiting failure mechanisms.