pvd modeling
PVD modeling is the calculation of where sputtered or evaporated atoms come to rest, and at the roughly 5 mTorr pressure a physical-vapor-deposition chamber runs, the mean free path is tens of centimeters — longer than the throw distance — so atoms cross the chamber in straight lines and the whole problem collapses to geometry: what fraction of the source can a given point on the wafer still see? A point on open field sees the entire source and coats at the nominal rate; a point at the bottom of a contact via sees only the sliver of source framed by the mouth, and that sliver is what every PVD model, from a one-line analytic estimate to a full Monte-Carlo transport code, is really computing.
**The quantity PVD modeling actually solves for is the arrival-angle distribution, not the deposition rate.** A sputter target emits with a near-cosine angular law — flux per unit solid angle falls off as $\cos\theta$ from the surface normal — so a flat wafer facing the target integrates that law over the full hemisphere and coats uniformly. Drop a feature into the surface and each interior point now integrates the same law over only the solid angle its walls leave unshadowed. For a cylindrical via of depth $d$ and width $w$ the mouth seen from the bottom centre subtends a half-angle $\theta$ with $\tan\theta = w/2d = \tfrac{1}{2\,\mathrm{AR}}$, and the cosine-weighted fraction that gets through is $\sin^2\theta$. That one expression is the backbone of every first-order PVD deck.
**Bottom coverage collapses as one over aspect ratio squared, and no amount of target power changes it.** Evaluating $\sin^2(\arctan[1/2\,\mathrm{AR}])$ gives 20% at aspect ratio 1, 5.9% at 2, 2.7% at 3, and just 0.25% at aspect ratio 10 — a factor-of-80 loss across a span of features a modern interconnect stack crosses routinely. Turning the magnetron up scales every one of those numbers by the same multiplier, so the ratio between field and bottom is invariant to power; it is fixed by geometry alone. This is why unaided PVD cannot fill, or even reliably line, a high-aspect-ratio hole, and why the real engineering is about reshaping the arrival-angle distribution rather than raising the flux.
**A collimator buys directionality by throwing most of the metal on the floor.** Inserting a honeycomb baffle of cell aspect ratio $\mathrm{AR_c}$ between target and wafer removes every atom whose trajectory tilts more than $\arctan(1/\mathrm{AR_c})$ off vertical, so the flux that survives is forward-directed and reaches deeper — a collimator of $\mathrm{AR_c}=2$ lifts the bottom-to-field ratio about 5×. But the same truncation passes only $\sin^2(\arctan[1/\mathrm{AR_c}])$ of the source: 50% at $\mathrm{AR_c}=1$, 30.8% at 1.5, 20% at 2, and 10% at 3. The discarded metal coats the collimator itself, which then flakes and drives particles, so the SEMATECH-era collimated Ti/TiN process traded throughput and particle budget for one modest reshaping of the angular distribution.
**Long-throw geometry narrows the same cone and pays in the same currency.** Moving the target far from the wafer — Novellus and Lam ran throw distances near 250-300 mm against a 200 mm wafer — lets only the near-normal atoms reach the substrate while the off-axis ones diverge onto the shields. The surviving cone narrows to a half-angle of about $\arctan(R/L)$ while the rate falls as $\dfrac{1}{1+(L/R)^2}$: at a throw of three target radii the arrival half-angle tightens to 18° but the rate drops to 10% of the close-coupled value. Long throw and collimation are the same idea built in vacuum versus in hardware, and both hit the same wall — the cone only narrows by discarding the atoms that were not already aimed where you wanted them.
**Ionizing the metal flux is the only fix that steers atoms instead of discarding them.** In ionized PVD — Applied Materials' Endura ionized-metal-plasma (IMP) source and its self-ionized-plasma (SIP) mode are the production examples — a secondary RF coil or very high target power ionizes a large fraction of the sputtered metal, and the wafer sheath then accelerates those ions straight down regardless of the angle they left the target. A modeled 85% ionized fraction holds bottom coverage near 85% all the way to aspect ratio 5, where bare PVD is already under 1%; only once the feature mouth narrows below the ion angular spread does it fall, to 57% at aspect ratio 7 and 28% at 10. Ionization energy, sheath voltage and gas rarefaction now enter the model, so an IPVD deck couples a plasma calculation to the transport calculation — but the reward is a directed flux instead of a decimated one.
**Wafer bias turns the substrate into a second, downward-pointing sputter source.** Once the metal arrives as ions, a bias on the wafer sets their landing energy, and above roughly 100-200 eV they resputter atoms already deposited on the via bottom. Modeling that resputtering is what lets a barrier or seed be redistributed onto the lower sidewalls: material knocked off the bottom corner redeposits on the walls, so net sidewall coverage rises even while bottom coverage is held deliberately flat. Push the bias too hard and the resputter yield exceeds the arrival rate at the bottom corner, the corner clears down to the underlying dielectric, and the model predicts the faceting and corner-clipping a real Ta/TaN barrier shows in cross-section.
**The ceiling on PVD fill is the overhang at the top, not the starvation at the bottom.** The upper corner of a feature sees more than a hemisphere — it collects flux from the field and from the opposite wall — so it deposits faster than any other point and builds a lip that leans over the opening. Every surface-evolving transport model, a level-set or string front driven by the local arrival integral as in SIMBAD or SPEEDIE, shows that lip closing the mouth before the bottom fills and sealing a keyhole void. This bread-loafing is why PVD copper fill gave way to electroplating and PVD barriers are yielding to ALD: past an aspect ratio near 2-3 the overhang wins, and the honest output of the model is a void, not a fill.
| Method | Arrival half-angle | Relative rate | Bottom/field @ AR 3 | Where it is used |
|---|---|---|---|---|
| Conventional magnetron | ~60° | 100% | 2.7% | field metal, thick films |
| Long-throw | ~18° | 10% | 27% | 200 mm liners |
| Collimated (AR_c 2) | ~27° | 20% | 13.5% | Ti/TiN glue and barrier |
| Ionized PVD (IMP/SIP) | ~5° | 60% | 85% | Ta/TaN barrier, Cu seed |
```flowchart
Target emission (cosine law) -> Gas-phase transport (ballistic, mfp >> chamber)
-> Arrival-angle distribution at feature mouth
-> Local solid-angle shadowing + ion steering / resputter (if IPVD)
-> Surface evolution (level-set / Monte-Carlo) -> Predicted profile: coverage or void
```
Read PVD modeling through a *transport-geometry* lens rather than a *chemistry* lens: unlike CVD or ALD, where the answer is set by reaction rates and precursor coverage, a PVD profile is set almost entirely by which atoms can travel in a straight line from source to surface without being intercepted. Collimation, long throw, ionization and resputter are not four unrelated tricks but four operations on one object — the arrival-angle distribution — and every hard problem in the field, from step coverage to overhang to sidewall symmetry, is a different question about the same distribution. Get that distribution right in the model and the deposited profile follows; get it wrong and no amount of chemistry or power will rescue the fill.