Why a flat model of a via misses its ends

What a via is and why its coupling matters

A via is a tiny vertical metal connection. In a modern package, thousands of them run through a glass or silicon carrier, called an interposer, to join one chip to another. The wider family of such connections is described in the overview of through-silicon vias.

Two vias that sit close together act on each other electrically. A signal on one pushes a little charge onto its neighbour. Engineers call this coupling, and it decides whether a signal arrives clean or arrives with noise on it.

Design teams cannot build a package to find out. They estimate the coupling in software first, so they can choose spacing and margins before anything is made. This job is called parasitic extraction.

What a flat model leaves out

A common fast approach models a via as a very long straight rod and look only at a slice across it. That slice is flat, which is why it is called a two-dimensional model. It is quick to solve, which makes it attractive for parasitic extraction, and it works well in the middle of a long rod.

A real via is not infinitely long. It has a top and a bottom. At each end the electric field spreads out into the space around the pad and the surrounding material. A flat slice cannot see that spreading, because the slice has no ends.

Why now: chip makers are moving to packages that hold several chiplets, and Intel has announced glass substrates for such packages, planned for the latter part of this decade. Thin glass carriers mean short vias, where the ends matter most.

So a flat model is not wrong everywhere. It is blind in one specific place. The question for a design team is how much that blindness costs for the geometry they actually plan to build.

• If the vias are long and thin, the ends are a small part of the whole.
• If the vias are short and wide, as in a thin glass carrier, the ends are a larger part.
• If the surroundings are crowded with pads and metal planes, the field near the ends changes again.

What a three-dimensional solver adds

A three-dimensional solver models the whole shape, ends included. It cuts the surface of the conductors into small pieces and solves for how charge spreads over them. This family of methods is usually called the boundary element method, and a public example is FastCap, distributed by the MIT Computational Prototyping Group.

The price is time. A three-dimensional solve costs far more than a flat slice, so it is used to check, not to run on every candidate layout. A sensible workflow uses the fast flat model to search and the slower solver to confirm.

ChipletOS built its own three-dimensional solver for this purpose. A solver written by the same team that uses it needs checking before anyone should rely on it, and that is the part the published record describes.

How the published record checks the solver

The published record for this result lists the checks the solver passed before it was used. They are of two kinds.

The first kind compares the solver with shapes whose exact answers are known from textbooks, such as a sphere, two spheres, and a short coaxial cable. The record says the solver was validated against those cases, and gives the size of each gap.

The second kind compares it with a solver the team did not write. The record reports a cross-check against FastCap, with a gap of 0.80% on the main diagonal terms and 1.04% on the largest off-diagonal term. An excerpt of the record’s own text follows.

The size of the end effect depends on the geometry, so this post does not state one. What matters here is how the solver was checked.

Why the check order matters

A solver is only useful if its mistakes are smaller than the thing it measures. That sounds obvious, but it shapes how the checking has to be done.

The order goes like this:

• Start with cases where the answer is not in doubt. A single sphere has an exact textbook capacitance, so a solver that cannot reproduce it has a bug.
• Add two spheres and a short coaxial cable, which bring in more geometry and still have known answers.
• Only then compare with another solver, on shapes that have no closed-form answer.

This order separates two kinds of error. A gap against a textbook shape is an error in the solver. A gap against another solver on a hard shape may be an error in either, and it tells you how much the two methods disagree in practice.

Then there is the question of what to do with a disagreement. A responsible report states the gap and leaves the reader to judge, which is what the published record does.

A plain reading of the numbers

A gap of about one part in a hundred against an outside solver is small for engineering work, but only for the cases tried. The record does not say that every geometry will behave the same way, and neither does this post.