surface resistivity and surface resistance measurements

Trek Application Note Number 1005

Surface Resistivity and Surface Resistance MeasurementsUsing a Concentric Ring Probe Technique

William A. Maryniak, Toshio Uehara, Maciej A. Noras

Abstract The relationship between surface resistivity and surface resistance is established andexplained.

1 Introduction

Concepts of surface resistance and surface resis-tivity can be sometimes confusing. Definitions ofboth terms can be found in many books and stan-dards [1–4]. Surface resistance, Rs, is defined inall of the aforementioned literature sources as theratio of a DC voltage U to the current, Is flowingbetween two electrodes of specified configurationthat are in contact with the same side of a materialunder test (Figure 1).

Rs =UIs

(1)

Surface resistivity ρs, on the other hand, is deter-mined by the ratio of DC voltage U drop per unitlength L to the surface current Is per unit width D.

ρs =ULIsD

(2)

L

D

U

electrodes mater ia l

Figure 1: Basic setup for surface resistance andsurface resistivity measurement.

Surface resistivity is a property of a material. The-oretically it should remain constant regardless of

the method and configuration of the electrodesused for the surface resistivity measurement. Aresult of the surface resistance measurement de-pends on both the material and the geometry ofthe electrodes used in the measurement. Thephysical unit of surface resistivity is Ohm (Ω). Thelegitimate unit of the surface resistance is alsoOhm. Because of that surface resistivity and thesurface resistance are often mixed up. In orderto differentiate between the two, surface resistiv-ity is often expressed also in Ohm/square (Ω/ sq.)which is not a valid unit from the dimensional anal-ysis point of view.

2 Surface resistivity andsurface resistance

2.1 Current density and surface cur-rent density

It is possible to establish a relationship betweenthe surface resistance and surface resistivity forany electrode configuration. An idea of the cur-rent density is very helpful in understanding of thatrelationship. Consider two samples of a materialas shown in Figure 2. With a constant voltage Uand both samples made of the same material theamount of current flowing through the material willbe different. The thicker bar (sample #1) conducts"more easily" than the thin bar (sample #2). Onemay use a water pipe analogy - given a constantwater pressure, there will be more water per unittime coming through the pipe with a larger diame-ter. The flow density, be it water or electric current,is the amount of flow passing through a unit areaof the pipe or the sample of the material. The sur-face area is perpendicular to the direction of theflowing current (or water).

TREK, INC. • 190 Walnut Street • Lockport, NY 14094 • Tel: (716) 438-7555

Call: 1 800 FOR TREK • FAX: (716) 201-1804 • E-mail: [email protected] • Web: www.trekinc.com

page 1 of 4Copyright © 2013 TREK, INC. 0623/MAN Rev. 1b



L

D

U

electrodes

mater ia l

D

(a) Sample #1.

L

D/2

U

electrodes

mater ia l

D/2

(b) Sample #2.

Figure 2: Current density.

When the voltage U is kept constant, the currentdensity for the thin and the thick bar is the same.The electric current density is often expressed by:

J =IS

where I is the current and S is the surface area,and is measured in [A/m2]. Surface current den-sity is the next concept helpful in understandingthe relationship between the surface resistanceand surface resistivity. Consider Figure 1, whereboth electrodes are on the same side of the mate-rial. It is assumed that electric current flows on thesurface of the material only. In reality, this is notexactly true. There is always a portion of that cur-rent flowing through the bulk of the material. How-ever, in order to make it possible to compare sur-face properties of various materials, it had beenpresumed that the surface current flows throughinfinitesimally thin surface layer. This layer is sothin, that the thickness of it can be neglected. Sur-face current density Js is therefore defined as:

Js =ID

,

where D is a width of the electrode.

2.2 Concentric ring electrodes con-figuration

The relationship between surface resistivity andthe surface resistance for a concentric ring probegeometry can be found by defining a surface cur-rent density in the area between rings. Knowingthe surface current density, it is possible to find anelectric field intensity between the electrode rings(Figure 3).

R 2

R 1

Figure 3: Surface resistance and surface resistiv-ity measurement configuration for concentric ring

electrodes.

Define:

• R1 outer radius of the center electrode,

• R2 inner radius of the outer ring electrode,

as it is shown in Figure 3 (see also Figure 5). Thesurface current density Js for a concentric ringsconfiguration is determined as:

Js =Is

2π·r, (3)






where the radius r varies from R1 to R2. It isimportant to remember that when testing the sur-face resistivity (or resistance) of any material, it isassumed that all the currents flow between elec-trodes along the surface and do not penetrate intothe bulk of the material. In order to ensure thatthe surface currents are measured properly, somemore advanced techniques for surface resistivitymeasurements have been developed [1,2,4]. TheOhm’s law describes relationship between a cur-rent density J and an electric field intensity E . It isalso valid for the surface currents:

Js =Eρs

, (4)

Therefore, it is possible to find electric field be-tween the concentric rings by solving the followingdependency (using equation 3 and 4):

E =ρsIs2π·r

, (5)

The voltage between electrodes can be found byintegrating the electric field E from R1 to R2:

UR1,R2 =∫ R2

R1

E dr =

=∫ R2

R1

ρsIs2π·r

dr

=ρsIs2π

∫ R2

R1

1r

dr

=ρsIs2π

ln(

R2

R1

)

(6)

Substituting Rs = UIs

:

Rs =ρs

2πln

(

R2

R1

)

(7)

After rearrangements, the surface resistivity is re-lated to the surface resistance by a constant thatdepends on the geometry of the electrodes only:

ρs = Rs2π

ln(

R2R1

) = Rs · k (8)

Where k is frequently called a geometry coeffi-cient.

Figure 4: Surface resistivity measurement setup.

Figure 5: Concentric ring probe.

Figure 4 presents a typical surface resistivity mea-surement setup using a resistivity meter and aconcentric ring probe (Figure 5). The resistiv-ity meter is capable of measuring surface resis-tivities directly, utilizing various configuration of






electrodes. Meter provides a constant voltage Uand measures the current I flowing between elec-trodes. Resistance Rs is then easily calculatedand the value of resistivity is equal to the value ofresistance multiplied by the geometry coefficient.Usually electrodes are especially constructed tosimplify calculations of surface resistivity and thegeometry coefficient is equal to a simple integer.Most of industrial standards use this simplified ap-proach [1,2,4].

Example 1: Consider the following measurementconfiguration:

R1 = 15.3 [mm],

R2 = 28.6 [mm],

U = 10 [V].

The current measured during the test was equalto Is = 1·10-6 [A]. The surface resistivity of the ma-terial under test can be calculated from the equa-tion 8:

ρs = Rs2π

ln(

R2R1

) =

=UIs

2π

ln(

R2R1

) =

=10 [V ]

10-6 [A]2π

ln(

28.6 [mm]15.3 [mm]

) =

= 107 [Ω] ·2π

ln(1.869)=

= 107 [Ω] · 10 =

= 108 [Ω]

3 Additional remarks

While conducting the surface resistivity and thesurface resistance tests, it is important to consider

some additional components affecting the test re-sults. The electric resistivity of any dielectric ma-terial depends on many environmental factors. Itcan change with humidity, temperature, etc. Forthis reason it is recommended to condition the testsample before the measurement. Another impor-tant aspect is to ensure a proper contact betweenelectrodes and the tested material. Electrode sys-tems can be made of various materials and maycome in various shapes and configurations. Theway of electrode contacts the material under testhas a very significant influence on the result ofa measurement. The DC voltage level used fortesting is also an important issue. Usually resis-tivity of the material depends on the value of theapplied voltage and the time span during whichthe sample was energized. All these contributingfactors and criteria are described in appropriateguides and standards [1–4].

References

[1] ASTM Standard D 257-99. Standard testmethods for D-C resistance or conductance ofinsulating materials, 1999.

[2] ESD STM 11.11-2001 Standard. Surfaceresistance measurement of static dissipativeplanar materials, 2001.

[3] Michael B. Heaney. The Measurement, Instru-mentation and Sensors Handbook, chapterElectrical Conductivity and Resistivity. CRCPress, 1999.

[4] IEC 61340-5-1 Standard. Electrostatics - part5-1: Protection of electronic devices fromelectrostatic phenomena - general require-ments, 1998.




surface resistivity and surface resistance measurements

Documents