## Surface Roughness

At a microscopic scale, no surface appears perfectly smooth. All materials exhibit surface irregularities and bumps. One measure of surface roughness is the rms (root-mean-square) height h of the surface bumps.

At low frequencies, the depth of penetration of current (the skin depth) exceeds h. Low-frequency currents, therefore, submarine below the surface bumps, unaffected by surface roughness.

High-frequency currents, on the other hand, remain tightly bound to the surface at all times. At frequencies so high that the skin depth δ shrinks to less than the rms bump height h, the current follows the contours of the surface, over hill and down dale, as it flows along the conductor. At these high frequencies, the apparent resistance of the material increases to a value representative of the additional distance over which the current must flow to traverse the contours of the surface.

### How severe is the surface-roughness effect?

If the average inclination of the conducting surface is 60° (as if patterned with an infinite array of equilateral ridges perpendicular to the direction of current flow), the ultimate increase in surface resistance is 100 That's a pretty severe effect. An infinite array of equilateral 3-D pyramids induces a less severe effect, because, as anyone who hikes in the wilderness can tell you, you don't have to traverse every peak to make your way through the mountains.

### At what frequency does surface roughness take hold?

The onset frequency for an rms surface roughness of h=1 micron is 4.37 GHz, decreasing with the square of h, like this:

At the onset frequency, the effective resistance of the conductor has progressed 60% of the way from the low-frequency asymptote (no effect) to the high-frequency asymptote (full effect).

##### How rough is FR-4?

The roughness of the copper layers used in pc-board materials is often rated in terms of the average peak-to-valley height. Typical roughness treatments suitable for use with FR-4 dielectrics produce heights of 6 to 18 microns. Even if you choose the low end of this roughness range, an average peak-to-valley height of 6 microns corresponds to an rms height of about one-fourth that value and an onset frequency of slightly more than 1 GHz. At less than 1 GHz, you don't notice the roughness effect; at greater than 1 GHz, you do.

### What can you do to make things better?

The exact dependence of skin-effect resistance on surface roughness defies analysis, as it depends not only on the height of the bumps but also on their horizontal extent, spacing, and exact shape. Long, sinuous undulations cause few problems, whereas short, choppy, steep-faced bumps significantly increase the path length of the current. Talk with your pc-board vendor about the variety of available surface treatments that are compatible with your dielectric material.

Vendors of pc-board materials refer to the toothing profiles of their cores when speaking about surface roughness. Toothing profiles are purposefully etched into the copper to facilitate adhesion between layers.

You may have little choice about the surface treatment used on the inside surfaces of a core layer; the core manufacturers make them pretty rough. You can control the surfaces on the outside of the core that your pc-board fabricator processes. Stack your board so these heavy current-carrying surfaces are on the bottom (reference-plane-facing side) of your highest-frequency traces. On that side, the current density is the highest, and surface roughness matters the most.

PC-board fabricators have access to many surface treatments. The RTF (reverse-treat foil) process makes a surface that looks like the Himalayas. Sulfuric-peroxide treatments add a dense forest of bushy trees to the landscape. The black-oxide treatment produces square-looking crystalline shapes.

One of the most aggressive treatments, from the standpoint of good surface adhesion, is the double-treat process (Figure 1). It grows long, dendritic fingers that stick straight out from the copper surface but leaves the underlying surface fairly smooth. Of all the choices, I like this one best. My theory is that current on a double-treated surface will remain mostly bound to the smooth underlying surface, flowing like a river around the dendritic columns.