Calculating Ohms-cm, Ohms-per-square, or Sample Thickness When Two of the Three are Known
& What Constitutes a Thin Film as Opposed to a Bulk Material?
The term ohms-cm (“ohms centimeter”) refers to the measurement of the “volume” resistivity (also known as “bulk” resistivity) of a semiconductive material. The value in ohms-cm is the inherent resistance of a given material regardless of the shape or size.
Many materials that are thick or relatively large such as silicon ingots (as opposed to a thin film or layer), can be measured using a four point probe to determine the volume resistivity. Sheet resistance is expressed as “ohms-per-square” and it is used when measuring a layer or thin film of a semi-conductive material.
The determination of what constitutes a thin film is based upon the relationship between one of the four point probe tip spacings and the thickness of the layer. The sheet resistance of a given material will change depending on the thickness of the layer. The following briefly explains how to calculate sheet resistance, volume resistivity, and thin film thickness if only two of these three properties are known.
Sheet resistance (ohms-per-square) multiplied times the thickness of the material in centimeters, equals the volume resistivity (ohms-cm).
Answers to questions, by John Clark, C. Eng, M.I.Mech.E., F.B.H.I., Founder of Jandel Engineering Ltd.
Q.How thick can a sample be and have it still be measured as a thin film, express in ohms-per-square? In other words, at what point is a sample so thick that it is no longer valid to measure it as a thin film?
A. When the thickness exceeds 5/8 (62.5%) of the spacing between two needles – after which sheet resistance needs more than 1% correction. So, 0.625mm (625 microns) for a probe head with 1mm needle spacing.
Q. If I am measuring a thick material to determine volume resistivity expressed in ohms-cm, how thick must a sample be so that it can be considered a semi-infinite volume for which I do not need to apply a correction factor?
A. 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%
Q. I have heard that the calculations for sheets resistance still apply at sample thicknesses of up to 40% of the tip spacing between two pins, however, this information is saying that it is okay up to 62.5%, which means wafers up to 625 microns thick can be measured using sheet resistance calculations. Don’t most companies use volume resistance measurements when measuring bare silicon wafers, most of which are about 550 microns thick?
A. It is a question of what you consider to be okay. From the graph at https://four-point-probes.com/page16.pdf we can see that at t/s = 0.625 the correction is 0.9898 – effectively 0.99 and within 1%. Less than 40% of the tip spacing and the measurements need no correction. I think most companies measure volume resistance of their wafers, but not by using a volume resistance equation – this is why it is necessary to know the thickness of the wafers – if they were using the volume resistance equation they would not need to know the wafer thickness.
If one has a unit that assumes wafer thickness of 550 microns it can measure sheet resistance and multiply its result by 0.055 to give volume resistance. From the graph at https://four-point-probes.com/page14.pdf it would appear that if you measure bulk on a 550 micron wafer with a 1.591mm probe head then t/s = 0.34 and a correction of 0.25 would need to be applied.
The Calculations
A customer mentioned that his tantalum film was supplied to him with a sheet resistance value of 8.0389 ohms-per-square, a film thickness of 2500 angstroms, and a volume resistivity of 201.94 micro-ohms-cm. Normally you would not know all three of these, and so you could use a four point probe to determine the thickness if the volume resistivity was supplied, or you could determine the volume resistivity if the thickness was supplied. If you didn’t have a four point probe, but knew the film thickness and the volume resistivity, you could calculate the sheet resistance of the sample. The relationship between these values is as follows:
Calculating Volume Resistivity from Sheet Resistance and Thickness:
The thickness of the layer in centimeters times the sheet resistance value expressed as ohms-per-square is equal to the volume resistivity in ohms-cm. For the above mentioned tantalum, it works out to: 0.000025 (thickness in cm) x 8.0389 (ohms-per-square value) = 0.0002009725 which equals 200.9725 micro- ohms-cm (which deviates less than 0.5% from the 201.94 micro-ohms-cm value supplied).
Calculating Thickness from Volume Resistivity and Sheet Resistance:
To calculate the thickness of the layer using the supplied volume resistivity value and the (measured) sheet resistance value, you would divide the bulk resistivity by the sheet resistance value. So, again for the above mentioned tantalum sample, the 0.00020194 (201.94 micro-ohms-cm) / 8.0389 (ohms-per-square) = 2.5120352287004440906094117354364e-5 which is 0.00002512035228700 centimeters, or 2512.0352287 angstroms (which matches the expected 2500 angstroms).
Calculating Sheet Resistance from Film Thickness and Bulk Resistivity:
The bulk resistivity divided by the thickness of the layer in centimeters equals the sheet resistance. So, for an aluminum layer that is 200 microns thick (or 0.02cm thick), since the bulk resistivity of aluminum is 0.0000027 ohms-cm 0.0000027 / 0.02cm =0.000135 ohms-per-square or 1.35 x 10^-4. This assumes that the aluminum film is pure, since the bulk resistivity value was taken from the periodic table of the elements.
These web pages may be helpful in making these calculations:
http://onlineconversion.com/length_all.htm (opens new window)
http://www.csgnetwork.com/sntodeccalc.html (opens new window)
More questions with answers by John Clark, C. Eng, M.I.Mech.E., F.B.H.I., Founder of Jandel Engineering Ltd.
Q. “What is the range of bulk resistivity (ohms-cm) that the Jandel RM2 Test Unit can measure in Ohms-cm?”
A. This crops up regularly, and it is hard to answer – let me give you an example to show the problem. [Update: Please note that the RM2 Test Unit was replaced years ago by Jandel’s newer versions of the Test Unit. The current version of the Test Unit and all versions subsequent to the RM2, the Jandel RM Series Test Units, include PC software that simplifies the task of calculating bulk resistivity for wafers and bulk materials such as ingots. Some of the newer versions also auto-range to determine the optimum choice of input current].
The normal range of sheet resistance which the (now discontinued) Jandel RM2 Test Unit could measure was between 1 and 10^7 ohms per square. The volume resistivity would be numerically equal to the sheet resistance if the specimen was 1 cm thick and made from the same material from which the sheet resistance figure was derived. It is difficult to define the limits of volume resistivity that the RM2 Test Unit could measure – for example we could not measure the volume resistivity of a block of platinum 1 cm thick because it is too highly conducting for the RM2 Test Unit to obtain a reading. If it was a platinum film 200 Angstroms thick then we could measure the sheet resistance easily and it would be approx 100 ohms per square.
Let’s consider a specific wafer sample:
Let us assume that the wafer is 0.5mm thick and its resistivity is 0.005 Ohms-cm. By pressing the ohm/sq button, we can set the RM3-AR to deliver 4.5324 mA so the mV displayed is numerically equal to the sheet resistance in ohms/square. In this situation we can say that:
bulk resistivity = sheet resistance x thickness in cm.
i.e., 0.005 = | |
= 0.10mV |
This would be the displayed reading.
Of course, if it was a thin film, the thickness would be much less than 0.5mm and the sheet resistance correspondingly more so it would be possible to calculate the bulk resistivity more accurately. It is the eternal problem of low resistivity materials, where a supplementary voltmeter able to read a microvolt or less, is desirable. Such a voltmeter would be useful for the 0.005 Ohm-cm material -essential if it was thicker than 0.5mm.
If we were to assume that we were talking about the bulk resistivity of silicon wafers, then, using the formula rest=2 x pi x s x V/I we calculate that with a probe tip spacing of 1.00mm, V=2V, I=10 nanoamperes the MAXIMUM value of resistivity would be about 10^8 ohm.cm. Using V=10 x 10^-6V and I=10×10^-3 amperes the MINIMUM value would be about 6 x 10^-4 ohm.cm.
There are limitations on these values – in practice it may not be possible to drive the minimum current in the high resistivity material owing to contact resistance, and equally for the low resistivity material we are only looking at a single digit at the end of the voltage display. So, for the now discontinued RM2 Test Unit we quoted something a little less ambitious say 10^-2 up to 10^6 ohms.cm. However, for the RM3 Series Test Unit (the current version) we quote the bulk resistivity measurement range from 10^-3 up to 10^6 ohms-cm. System accuracy is within 0.3%
The equation for calculating Ohms-cm without converting from sheet resistance is:
2 x s x Pi(π) x V/I
Where s is the spacing between each of the four point probe tips in cm. If one uses a probe head with tip spacing of 1.591mm (62.6 mils), since 1.591mm is 1/(2 x pi)cm, it cancels out to V/I.
Q. I understand that the RM3 Test Unit can read out directly in ohms-per-square for use when measuring thin films (sheet resistance), but how do I measure thick materials that are measured in ohms-cm (volume resistivity)? What current level do I select for a particular material?
A. [Update: The RM3 Test Unit has been replaced with the RM Series Test Unit which includes PC software which allows volume resistivity of both thick materials as well as thin films to be calculated and saved in CSV format. The new RM Series Test Units can read out directly on the screen in ohm-cm (volume resistivity) if one inputs the sample thickness or if one indicates that a bulk material is being measured, in addition to reading out in either sheet resistance expressed in ohms-per-square or in millivolts. The RM Series Test Units also have a button which can be pressed to auto-range to determine the best choice of input current for the material being measured. When calculating the volume resistivity of thin films, the thickness of the layer must be known. If one prefers to calculate volume resistivity without using the RM Series software, or if using an measurement electronics which will not read out in ohms-cm, there is a relatively simple way: Toggle from the ohms-per-square mode so that the electronics reads out in millivolts and follow these instructions:]
When making an ohm-cm measurement, ideally you will want to use a current which will simplify the math.
The formula is 2 x pi x s x V/I where s is the spacing between each of the needles in cm. If you use a probe with tip spacing of 1.591mm (~same as 62.6 mils), it makes the math easier since 0.1591 is 1/(2 x pi). Therefore we would have:
Resistivity = V/I
This means that if 1mA current is used, then the measured voltage value (in millivolts) = the resistivity of the sample in ohms-cm.
If you want to measure on the ‘High’ range it may be that the voltage will be too high to measure. In this case you could try 100uA and the mV result would need multiplying by 10. If the voltage value is quite low (maybe 9mV or so) you could increase the current to 10mA and then the mV result could be divided by 10 to give the resistivity (higher currents can sometimes offer more stable results). [More on the subject of measuring volume resistivity without using software can be seen here: https://four-point-probes.com/hm21-srm-hand-held-meter-with-srm-probe-head/ . The same procedure described in relation to the HM21 Hand Held Meter applies to the RM Series Test Units as well, however, later versions of the RM Series Test Unit will read-out on the display in ohms-cm if one enters the film thickness or if one indicates that a volume resistivity material is being measured. When compared to the HM21, the RM Series Test Units have a greater measurement range on both ends of the scale and it they auto-range, whereas the HM21 does not auto-range. Both the RM Series Test Unit and the HM21 include software that can be used to calculate volume resistivity.]
Some information about selecting the best choice of input current when using four point probe electronics that do not auto-range can be found here: https://four-point-probes.com/reversing_current.pdf
Q. What if I want to measure volume resistivity using a probe with 1mm tip spacing instead of 1.59mm spacing? Can I set the input current so that the millivolt value is still equivalent to the volume resistivity in ohms-cm?
A. This can be done, however, the mathematics in this instance would be carried out by adjustment of the current rather than adjustment of the spacing. Therefore:
R(b) = 2 x pi x s x V/I
R(b) = 0.62832 x V/I
We multiply by the spacing (1/0.62832 giving 0.1591mm) but we are dividing by the current, so the current should be 628.32uA (0.62832mA)
This current can be increased / reduced by a factor of 10 where merited by the sample. (see the following link for information about selecting the best choice of input current when using four point probe electronics that do not auto-range: https://four-point-probes.com/reversing_current.pdf
Such measurements need to be done with a programmable current source such as the RM Series Test Unit which allows such an input current to be used.
Q. The RM Series Test Unit has a stated resistivity limit of 10^-3 to 10^6ohms-cm. So, for metals with 10^-6 ohm-cm, such as silver, copper, gold how can those be measured?”
A. When Jandel assigns an ohms-cm limit for one of their four point probe measurement electronics, the value in ohm-cm is with reference to actual bulk samples. If you want to measuring a layer of silver, copper, or gold which has a volume (or bulk) resistivity lower than 10^-6 ohms-cm, that may or may not be possible to measure, depending on the thickness of the layer. For example, working out the sheet resistance of a copper layer:
Resistivity of copper is about 1.68 x 10^-6 ohm.cm For a layer R(sheet) x t(cm) = Resistivity
For thickness of 16 microns
t=16 x 10^-4
R(sheet) = 1.05 x 10^-3 = 1.05 mohm/square
So, for a copper layer that is 16 microns or thinner, the RM Series can be used. If you were to try and measure a thick piece of copper, say 1mm thick, then it could not be measured with the four point probe technique. A four point probe can be used to calculate sheet resistance of very conductive materials, and if one knows the thickness of the layer, then the bulk resistivity can be calculated. Or, if one knows the bulk resistivity of a material, one can measure a thin film to calculate the thickness of the layer.
Four-Point-Probes is a division of Bridge Technology. To request further information please call Bridge Technology at (480) 219-9007 or send e-mail to Joshua Bridge at: sales@bridgetec.com