A Study of Sheet Resistance and Thickness Correlation

Whether it’s a thick-film or thin-film application, shielding effectiveness is best calculated via sheet resistivity. To better understand the complexities of the measurement process, we will compare conductive paint, a popular thick-film shielding, to vacuum vapor-deposited aluminum (Al) film. Our conclusions may surprise you since even though the sheet resistance (R) characterizes the shielding effectiveness, it tells only part of the story.

Sheet resistivity or simply sheet resistance is given in ohms per square. This concept was developed in the thin-film circuitry industry to design traces with the same resistance by dividing each trace into squares.

Resistance of a thin-film resistor of length (L), width (W), and thickness (t) is given by R = rL/tW, where r is the bulk resistivity given in ohm·meter or ohm·centimeter. For a square resistor, L = W and R = r/t, a value that depends only on r and t.

Interestingly, R can be measured by placing the leads of an ohmmeter closely together on the thin film to be characterized and recording R directly, provided the measurement is done a reasonable distance from any discontinuity. To prove this, a thin film is formed on a flat substrate on which a long, thin strip is masked so that it is electrically isolated from the rest of the thin film. The resistance of this strip then is measured and its value divided by the number of squares of the strip, where the number of squares simply is the length of the strip divided by the width.

This value is compared to the value measured on the rest of the thin film by directly placing the leads closely together on the film. Since most ohmmeters have a lower-end resolution of tenths of ohms, the strip resistance measurements might yield a two to three significant figure value, while the direct measurement usually is a single-digit value.

For example, a strip with 200 squares gave a reading of 89.2 W while the direct reading read 0.4 W or 0.5 W depending on the contact resistance of the ohmmeter leads. The value of the strip measurement is 89.2/200 = 0.446. Since R depends on r and t only, process control requires that r and t are well controlled. Thickness is determined by the mass of Al charge.

In vacuum vapor-deposited Al for background pressures of less than

10^-4Torr, r is a constant. Even though it is not the value of bulk aluminum, it is reliably constant and only several times that found in the bulk. When the Al film is exposed to the air, a thin oxide layer that inhibits further oxidation is formed.

For conductive paints, r is controlled by the manufacturer and the end user. r rises with time, and the shelf life must be strictly observed. r also varies with agitation of the pot. Care must be taken to agitate constantly and uniformly and assure steady usage so the paint in the line to the gun does not stagnate. Generally, the sprayer must apply a thickness in excess of average requirements because of variations in r beyond his control.

Vacuum Vapor-Deposited Al Films

Experimental Procedure

Four substrates for each thickness of vapor-deposited Al film were placed in a vacuum chamber and corresponding masses of Al charge were vaporized. For set A, 250 mg of Al were vaporized. For set B, 500 mg were vaporized, ending with 1,800 mg for the sixth set, F.

As a result, six films, roughly from 1,000 Å to 10,000 Å, were produced in this manner. Only the 10,000-Å film gave reliable results in a thickness measurement. The thickness of the other films had to be deduced from the Al charge, given that 1,800 mg of Al had yielded a 10,000-Å film thickness.

Experimental Results

Figure 1 shows R vs t, the film thickness, along with a theoretical plot R = r/t where the value of r was calculated from the 10,000-Å film with R for that film having been measured along with t. As can be seen, r is virtually constant, and no systematic deviation of the data is apparent.

Conductive Paint

Experimental Procedure

Four samples, each for six different thicknesses, were processed as follows: The first set of samples was sprayed until opacity was reached. The number of spray passes was noted and then referred to as x. Then the next five sets of samples denoted as B, C, D, E, and F were given 3x, 4x, 6x, 8x, and 10x spray-gun passes. Two more samples were given 15x spray-gun passes and denoted as G.

The sheet resistance was measured by cutting the samples with a razor blade so an electrically isolated narrow strip of paint of length L and width W was created. Again, the resistance of this strip was measured and its value divided by the number of squares in the strip, given by L/W.

The paint thickness was measured as the difference between the total substrate plus coating thickness and the substrate thickness without coating.

Experimental Results

Figure 2 shows that the bulk resistivity in a conductive paint coating is not a constant until a threshold thickness is reached. The region of dramatic change in bulk resistivity also is the region in which the thickness is not proportional to the number of spray-gun passes. Since the thicknesses were measured with a micrometer, the peak height of the paint particles above the substrate gave the thickness result. A microscopic scratch technique gave roughly the same results.

Figure 3 shows R vs coating thickness along with calculated values of R for paint where the bulk resistivity remained constant. The sharp rise of the bulk resistivity in thin coats may be a function of the paint drying before the conductive particles have settled because the films are somewhat porous.

Conclusion

In vapor-deposited Al films, a simple correlation between R and t exists: R = r/t. In conductive paints, r varies with t when t is smaller than some threshold that must be established for each paint. For example, when R for a vapor-deposited film is known, its thickness can be inferred. However, a value of R in a conductive paint coat does not relate to its thickness directly and can be found only if r(t) is known.

In terms of quality control, since R gives the variable most directly related to RFI shielding, it must be measured. To generate a coating with one-half the measured R value, for example, would not necessarily mean twice the paint thickness as is the case in vapor-deposited Al films.

In addition, multiple passes of the gun do not guarantee thickness proportional to the number of passes, except beyond a threshold that must be experimentally established. This is particularly important in terms of cost calculations.

To find the threshold, the point at which the sheet resistance is inversely proportional to the coating thickness must be found. This threshold also coincides with the point at which the coating thickness is proportional to the number of passes of the spray gun and appears to be the point at which the paint coat reaches its minimum porosity or maximum density.

About the Author

Hilarion Braun, Ph.D., is president of Summit Coating Technologies. His professional career spans two decades, beginning as a scientist at Mead Digital Systems, followed by several years at both Eastman Kodak and at Soltex Digital Printing. Dr. Braun earned his doctorate and M.S. from the University of Vermont and a B.S. in physics from Central Connecticut State College. Summit Coating Technologies, 25 N. 43^rd Ave., Phoenix, AZ 85009, (602) 455-9365.

December 1999