Questions and Answers Regarding Resistivity, Resistance, Surface Resistivity, Sheet Resistance and Volume Resistivity

by John Clark, C. Eng, M.I.Mech.E., F.B.H.I., Managing Director of Jandel Engineering Ltd.

The following comments are based on elementary physics (before semiconductors!) and some years measuring semiconductors and thin films. They are my opinion only, but don’t get criticized in practice!

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Q. I would like to know why using a four point probe you don’t have trouble with contact resistances; using two current injection probes and two voltage probes, the voltage probes still probe the voltage through a Schottky barrier (for metal probes on an semiconductor), so they still ‘see’ the voltage drop through this barrier. So, could you tell me where I am wrong?

A. The very reason for “four point” probe measurements is to divorce the probes supplying the current from the probes measuring the voltage, so it is only necessary to consider the “voltage probes”. The device used to measure the voltage is provided with a very high input impedance (ASTM F84 recommends at least 10^6 x the resistivity of the specimen), thus the contact resistance is a small proportion of the resistance in the voltage measuring circuit. Compound semiconductors have relatively large contact resistance, and unless heavily doped are not measurable with a normal DC 4- point probe measuring system. The pressure of the 4-point probe needles invariably damages the crystal structure beneath the needles. We suppose that such damage promotes ohmic contact by largely eliminating the rectifying diodes you mentioned.

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Q. Is there a difference between sheet resistance and sheet resistivity? At least one author claims there is.

A. This is loose thinking. Sheet resistance is measured in ohms per square while any kind of resistivity is measured in ohms.cm or ohm.metre. I believe the two expressions are synonymous.

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

A. 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.

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Q. At least one author claims that “surface resistivity is a more precise measurement when dealing with insulating or slightly conductive materials; the problem with evaluating surface resistivity in highly conductive materials is the underlying assumption that electrons move from the negative electrode across the top surface of the conductive layer only to the positive layer.” In other words, he seems to be implying that “you can’t measure surface resistivity on highly conductive materials.” True?

A. True.

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Q. Another author seems to back up #3: “Except in theory, there is no such thing as surface resistivity. Physics handbooks list surface resistivity values for dielectrics (no values below 108W/square), but no surface resistivity values are listed for conductive materials. Volume resistivity values are given for both insulators and conductors. “Insulators have very thin (i.e., several molecular layers thick) conductive surfaces; but the surface conductivity of a conductor’s surface is indistinguishable from its volume conductivity. On an insulative substrate such as sapphire, a thin conductive conductive film has a surface resistivity related to its thickness. Generally, one should not assume that surface and volume resistivities are related.” I find this very interesting. He’s basically saying that an insulator really has a thin conductive layer, and that it is possible to measure it’s resistivity, but it is not possible to measure surface resistivity of a conductor.

A. Surface resistivity – this is a different animal which I am not clear about. As far as I know it can’t be measured with a four point probe. It seems to be done with conductive electrodes pressed onto the specimen and a high voltage applied when the current which flows is measured. I believe ASTM D-991-89 or IEC 93 are applicable. It usually seems related to insulators and I have no experience of this kind of measurement. According to my book of “Tables of Physical and Chemical Constants” (Kaye and Laby) Surface Resistivity is defined as the resistance between opposite edges of a square, and the unit is the ohm (per square). The tables give a figure for various materials after a period of one minute of application of the measurement voltage. This is because the insulator becomes charged.

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Q. You seem to be stating that there is a difference between “sheet” resistance and “surface” resistance. I wasn’t aware of this, which is probably why I’ve been so confused! So is it true there is a distinction between the two?

A. I think that “surface resistance” relates principally to insulators as I remarked previously. I have some more information from Keithley Instruments – see their application note # 314 (volume_surface.pdf) 483K PFD file which refers additionally to ASTM D-257.
Please note: Agilent offers a system as well, the 4339B. We list these here for your convenience, however, Bridge Technology does not offer equipment for measuring near-insulators which are measured as “surface resistivity“. We offer equipment for the well established four point probe measurement which is for measuring semiconductors, metals, oxides, etc., and which provides sheet resistance values (for measuring thin films) expressed in ohm-per-square, and volume (or “bulk”) resistivity values expressed in ohms-cm. Surface resistivity instruments measure resistivities in the range from 10^9 to 10^16 ohms-per-square and are often used to measure materials such as ESD packaging materials, bags, etc. Some of these systems will have contacts on the top and bottom of the substrate while others will have either a contacting ring with one probe in the center, or three probes in a triangle arrangement with a fourth probe in the center. The probes are sometimes made of conductive rubber. More information about surface resistivity measurements is available here:ohms-per-square-what.pdf

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Q. Let’s say I have a sample that’s infinite in 3 dimensions, and I use a 4-point probe measures resistance. Will the resistance be independent of probe spacing? The equations seem to imply this.

A. Such a sample is usually defined as a “semi-infinite volume” if it extends to infinity in all directions below a plane on which four probes are located.
For equidistant probes:
Resistivity = 2 x pi x s x (V/I)
where s is the probe spacing in cm.

Compare this with a “thin” sample when
Resistivity (rho) = pi/(logn2) x V/I x t
where t is the thickness and
pi/(logn2) x V/I
is the sheet resistance.
Hence it can be seen that the formula is independent of spacing.

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Q. As you stated, the equation for volume resistivity of a “semi-infinite volume” is:

Resistivity = 2 x pi x s x (V/I)

where s is the (equidistant) probe spacing in cm.

For a homogenous material, the resistivity is constant, correct? Thus no matter what the probe spacing is, and no matter what the measured resistance value is (V/I), the resistivity value should always be the same, correct?

Rearranging the equation, you have:

resistivity/(2*pi) = s * (V/I)

The left-hand portion should be constant for a homogenous material, regardless of s or resistance value.

This implies the probe spacing and the resistance (V/I) are inversely proportional; if s is decreased, the measured resistance (V/I) must increase, and if s is increased, the measured resistance (V/I) must decrease. Is this correct? But it would seem as if the *opposite* would happen, i.e. as s is increased, the resistance (V/I) would increase.

A. Your argument about s and V/I must be valid. Although one might suppose that a larger spacing on the sample might result in a larger voltage drop, in practice it doesn’t work that way.

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Q. Do you have any papers that explain all of this?

A. The earliest paper is by LB Valdes Proc IRE Feb 1954 “Resistivity Measurements on Germanium for Transistors” – pages 420-427.

There are a host of others, principally concerned with correction factors for special cases.

a) Linear Array Probes

Circular wafers at centre:

  1. 1. D. E. Vaughan, Br.J. Appl. Phys., 12, 414 (1961)
  2. M. A. Logan, Bell Sys. Tech. J., 40, 885 (1961)
    Off centre but on radius:
  3. L. J. Swartzendruber, National Bureau of Standards Technical Note 199
    (1964)
    Perpendicular to radius:
  4. M. P. Albert and J. F. Combs, IEEE Trans. Electron Devices, ED-11, 148
    (1964)
  5. L. J. Swartzendruber, Solid State Electronics, 7, 413 (1964)
    Rectangular sample at centre and off centre:
  6. M. A. Logan, Bell Sys. Tech. J., 46, 2277 (1967)
    Half cylinder:
  7. E. B. Hansen, Appl. Sci. Res., 8B, 93 (1960)
    Circular rod:
  8. H. H. Gegenwarth, Solid State Electronics, 11, 787 (1968)
    Rectangular bar:
  9. A. Marcus and J. J. Oberly, IEEE Trans. Electron. Devices, ED-3, 161
    (1956)

Note: All the foregoing is based on measurement using a four point linear probe, the current being passed between the outer probes and the voltage measured across the inner two probes.

b) Square Array Probes

Small slice at centre:
as 9 above
Small slice along a radius:
as 3 above
Square sample:
10. M. G. Buehler, Solid State Electronics, 10, 801 (1967)
Thick sample near boundary:
11. S. B. Catalano, IEEE Trans. Electron. Devices, ED-10, 185 (1963)
thin infinite sheet:
as 10 above

Note: Square array probes have the current passed between two adjacent probes and the voltage measured across the two opposite when used for resistivity measurement.

written by John Clark

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