# What does mΩ/sq/25 μm mean?

I came across this specification on Saralon.com where they mentioned for one of their products

"Stretchable silver ink for stretchable electrical connections. Specially designed for bending applications such as TPU or bendable plastics with a good sheet resistance (30 mΩ/sq/25 μm)."

So, does this mean the Sheet resistance is 30 mΩ/sq for thickness(sheet) 25 μm?

• Your conclusion is (probably) good. Converting it to correct units is valid, but may make it harder to understand what is achieved in 'the real world'. ie For a layer 25 µM thick you get 30 milli-ohm per square. (As you concluded). For 10 µM it will be 75 µM/square Oct 3 at 22:43

What does mΩ/sq/25 μm mean?

Surface resistance is measured in resistance units per square. This is because the resistance between two opposite sides of a square of some approximately 2 dimensional material is (approximately) a constant, regardless of the size of the square! The larger the size of the square, the more distance between the opposite sides, but also the wider the path for current to flow. The two effects cancel each other.

This explains the unit $$\m\Omega/sq\$$.

Now, the resistance of a 3 dimensional approximation of a two dimensional square will also depend on the thickness of the material. In this case ink. The thicker the material, the lower the resistance. So, for this ink, we have a resistance per square that depends on the thickness of the ink layer. What you should have (thanks Hearth) is

$$m\Omega/sq \cdot \mu m$$

or just

$$m\Omega\cdot\mu m$$

The unit $$\m\Omega/sq/\mu m\$$ used by the vendor is in error.

• It should be mΩ·μm, not mΩ/μm. Oct 2 at 14:09
• Yes, you are right. Oct 2 at 14:11
• Understood. Really well laid out. So, Resistivity of the ink = Surface Resistance x Thickness i.e., 30 mΩ/sq x 25 μm = 0.75 μΩ.m. Is this right? Oct 2 at 14:11
• @Swarup Yes, that gives you the bulk resistivity of the ink. To find the resistivity of an actual square, take this bulk resistivity and divide by actual thickness of a layer of ink. Oct 2 at 14:29

It's wrong.

What they're trying to say is that the sheet resistivity of the material is 30 mΩ/⬜ for a sheet thickness of 25 μm. However, that wouldn't be expressed the way they wrote it--it should be 30·25 mΩ·μm/⬜, or a bulk resistivity of 750 nΩ·m, or 75·10⁻⁸ Ω·m*. This seems reasonable for silver ink, as it's a little worse than most metals (between about 1·10⁻⁸ and 20·10⁻⁸ Ω·m), but substantially better than semiconductors (graphite, for instance, is very roughly 20000·10⁻⁸ Ω·m).

You can divide this number by the sheet thickness to get the sheet resistivity.

Footnotes:
*The reason I use 10⁻⁸ Ω·m, rather than μΩ·m or nΩ·m, as a unit here is because this is what metal resistivities are usually quoted in in the literature. It works out to small numbers greater than 1 for most metals, and makes it easy to compare without having to move decimal places around. I think nΩ·m would have been a better unit to use overall, being a standard SI prefix, but the decision to use 10⁻⁸ Ω·m was likely made well over a century ago, before SI prefixes went smaller than μ.

⁻⁸ No, this is an exponent, not a footnote. What are you doing down here?

• what does the ⬜ symbol mean here? Oct 3 at 6:05
• @ilkkachu probably "per area" (square) without specific units like "per m2". It might be due to it not being important if only measuring the resistance of the thickness from top to bottom? It's just my guess.. The new question here would probably be "what does /sq mean?" Or sq is implicit for sqm, but that's error prone as it can be "sq ft" too.
– Raf
Oct 3 at 7:27
• @Raf It's neither "per square meter," nor "per square foot," nor per any unit of area, but rather just "per square." If you have a rectangular trace whose length is equal to its width, then that trace is "1 square," so given a sheet resistance of 30 mΩ per square, the resistance of that trace is 30 mΩ. If you have a rectangular trace whose length is 10 times its width, then that trace is "10 squares," so at 30 mΩ per square, its resistance is 300 mΩ. The aspect ratio (length divided by width) is important, but once you have the aspect ratio, the actual length, width, and area are irrelevant. Oct 3 at 11:25
• @ilkkachu It means square. Read Cassie's comment for details; Raf is wrong. It's a dimensionless unit, hence why I dropped it partway through. Oct 3 at 12:27
• @Hearth any reason you're not using □ (U+25A1)? The square you're using is quite large and has an off-white infill in my browser (Chrome on Win 10).
– BrtH
Oct 3 at 14:07