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# Sheet Resistance and the Calculation of Resistivity or Thickness Relative to Semiconductor Applications

Four point probe based instruments use a long established technique to measure the average resistance of a thin layer or sheet by passing current through the outside two points of the probe and measuring the voltage across the inside two points.

If the spacing between the probe points is constant, and the conducting film thickness is less than 40% of the spacing, and the edges of the film are more than 4 times the spacing distance from the measurement point, the average resistance of the film or the sheet resistance is given by:

Rs = 4.53 x V/I

The thickness of the film (in cm) and its resistivity (in ohm cm) are related to Rs by:

Rs = resistivity/thickness

Therefore one can calculate the resistivity if the thickness of a film is known, or one may calculate the thickness if the resistivity is known.

Glossary of Terms

Ohms-per-square: The unit of measurement when measuring the resistance of a thin film of a material using the four point probe technique. It is equal to the resistance between two electrodes on opposite sides of a theoretical square. The size of the square is unimportant.

Read more here at Wikipedia: http://en.wikipedia.org/wiki/Sheet_resistance

Ohms-Centimeter (Ohms-cm): The unit of measurement when measuring the bulk or volume resistivity of thick or homogeneous materials such as bare silicon wafers or silicon ingots, using the four point probe technique.

Q. Is sheet resistance an “inherent” property of a material, or is it a function of thickness?

RESISTIVITY is the inherent property of the material which gives it electrical resistance. It is sometimes called Specific Resistance. Sheet resistance is the resistance of a thin sheet of material which when multiplied by the thickness (in cm) gives the value of resistivity.

Q. How do I convert from ohms per square to ohms-centimeter?

The term Ohms-cm (Ohms centimeter) refers to the measurement of the “bulk” or “volume” resistivity of a semi-conductive material. Ohms-cm is used for measuring the conductivity of a three dimensional material such as a silicon ingot or a thick layer of a material. The term “Ohms-per-square” is used when measuring sheet resistance, i.e., the resistance value of a thin layer of a semi-conductive material.

To calculate Ohms-cm using a four point probe, one needs to know the thickness of the wafer (if it is a homogeneous wafer) or the thickness of the top layer that’s being measured, to be able to calculate Ohms-cm. The four point probe technique is used to measure one single layer or one single homogeneous material. If measuring a sample with two or more conducting layers, the result will be some type of meaningless average of all connected conductors.

As mentioned above, since the four point probe technique does not directly measure the thickness of thin films, if one knows two of the following three characteristics for a given sample, a four point probe can be used to determine the third characteristic: 1) the volume resistivity in ohms-cm, 2) the sheet resistance in ohms-per-square, 3) the sample thickness. More on this can be found here: http://www.fourpointprobes.com/understanding-volume-resistivity-measurements/

The equations for calculating bulk resistivity are different from those used to calculate sheet resistance, however, if one already knows the sheet resistance, bulk resistivity can be calculated by multiplying the sheet resistance in Ohms-per-square by the thickness of the material in centimeters.

Q. At what point do you stop multiplying the sheet resistance by the thickness in centimeters to arrive at Ohms-cm?

When the thickness exceeds 0.1 of the spacing between two needles – after which sheet resistance doesn’t apply. So, 0.1mm for a probe head with 1mm needle spacings. However, due to corrections, up to 0.3mm would be ok.

If the thickness is equal or greater than five times the probe spacing, the correction factor to be applied to the formula resistivity(rho) = 2 x pi x s x V/I is less than 0.1%. From the sheet resistivity point of view, the correction factor tables we have start at ratio thickness to probe spacing of 0.3, where the correction factor is unity, to a ratio of 2, where the correction factor is x0.6337.

I expect these tables can be extended up to a larger ratio, but clearly from a thickness of 2x spacing up to 5 x spacing is a bit of a no-mans land, but if one assumes that the situation is ‘bulk’ there are correction factors covering the ratio of thickness to spacing from 10 down to 0.4 where the correction factor is x0.288.

Read more here at Wikipedia: http://en.wikipedia.org/wiki/Resistivity

Four-Point-Probes is a division of Bridge Technology. To request further information please call Bridge Technology at (480) 988-2256 or send e-mail to Larry Bridge at: sales@bridgetec.com