RF sheet resistance in multilayer media
Updated February 13,
here to go to our page on skin depth
here to go to our page on transmission line losses
with this discussion we offer a download
that calculates equivalent RF sheet resistance for metals that
have up to three different layers. If you are a thin film vendor
and would like to sponsor this outstanding page and beat your competitors
to the punch, contact
By the way, we use
the metric system for this discussion, and in the download. Why? because there are no
popular units of bulk resistivity in the English system. Regarding
skin depths and thin film thicknesses, if you want to convert from
microns to micro-inches exactly, divide microns by 0.0254. There
are approximately 40 microinches in one micron, that's close enough
when you are dealing with plating.
Here is a clickable index for
this web page:
the skin depth equation for multiple layers
use our free download to calculate RF sheet resistance
useful examples of RF sheet resistance calculations
Solving the skin depth equation
for multiple layers
Quite often we use a stack of
multiple metals when constructing a thin film or a MMIC. To reduce
processing cost, resistor material is often placed underneath the
primary conductor metal(s), such as gold. How do you analyze this?
Look at the image below...
Here are equations for sheet conductivity, starting with one metal:
Now two metals:
Now three metals:
See the pattern? See how we "integrate" the conductivity across the X-axis where that axis is in skin depths, not microns?
Below is the normal current density versus skin depth curve, which you should have memorized. Someone spelled "number" wrong....
Now check out what happens in a two metal system, at three frequencies:
Come back in a few days, we'll annotate this a lot better!
use our free download to calculate RF sheet resistance over frequency
More to come!
Below are some examples of data
generated by the Microwaves101 RF sheet resistance calculator download.
Let's look at the effect of 50
micro-inches of nickel plating. As you can see from the plot below,
from 1 GHz on up you have over five skin depths. So you have achieved
the "maximum RF sheet conductivity" of the metal at any
Below is what that means in terms
of RF sheet resistance. A "good" value for RF sheet resistance
is perhaps 0.03 ohms per square at X-band. Here we have about 1
ohm per square at X-band. If your transmission lines have any length
at all, you have some serious attenuation in a fifty-ohm system.
Note that if you over plate the nickel with gold, you don't fix the
Let's look at the RF sheet resistance
of two thickness of tantalum nitride (TaN) resistors. In the first
case, the Tan film to achieve a DC sheet resistance of 50 ohms per
square is about is 2 microinches, or 0.05 microns. This sheet resistance
is a popular value with thin film vendors. This thickness of Tan
is less than 1% of a skin depth at X-band, so the RF sheet resistance
is very nearly equal to the DC value. The plot below shows how the
RF skin depth varies over frequency; the error is only about 1%
all the way up at W-band, less at lower frequencies. In this case there is almost nothing to
concern yourself with.
Below we have simulated a thicker
film of TaN, this time approximately 20 micro-inches, which provides
a DC sheet resistance of 5 ohms per square. In this case the film
is about 6% of a skin depth a X-band, and about 20% at W-band. Below
the RF sheet resistance is plotted over frequency. Now we see a
3% error at X-band and a 10% error at W-band. Something to think
about next time you are trying to trying to design an accurate attenuator
at 100 GHz!
In the case of a
thin film that has a resistor layer below the gold, does the resistor
metal have an appreciable effect on the transmission line attenuation?
Here you want the resistor material to be a fraction of a skin depth
(a DC value of 50 ohms/square is good), while the gold to be at
least three skin depths. Then you won't be able to tell the difference
between a gold plated over Tan, or pure gold!