About the collinear four-point probe technique´s inability to measure the resistivity of anisotropic electroconductive fabrics

Abstract

The sheet resistance of electrically conducting screen-printed surfaces can be measured with different techniques such as Van Der Pauw or four-point probe. As electroconductive textile materials show anisotropic behaviour we investigate the possibility of using the four-point probe method to measure the anisotropic properties of the electric conductivity. It was proved both theoretically and experimentally that no anisotropy can be detected with the collinear contacts of the four-point probe.

Keywords

Screen printing textiles anisotropy square resistance Van Der Pauw technique four-point probe technique

The four-point probe method is a flexible technique for measuring the sheet conductivity of conducting layers.¹ The most common configuration is to have four collinear contacts. The outer two contacts are used for the electric current supply and the remaining contacts are used to detect the voltage drop. Using four contacts instead of just two eliminates the influence of parasitic contact resistances caused by surface oxidation or moisture, because almost no current flows through the voltage sensing contacts. Contact resistance will not cause any additional voltage drop so the correct sheet resistance will be measured.

Four-point probe equipment is currently commercially available. The probe contains the four contacts and has to be pressed onto the conducting layer to make a measurement. The sheet resistance is then displayed. The measurement can be repeated on several areas of the sheet so that eventual non-homogeneities can be detected. The only condition for a correct measurement is that the conducting layer may not be covered by an insulating protective layer.

A minor drawback is that the four-contact probe should not be too close to the boundaries of the sheet, because an error will be introduced. As a rule of thumb, the distance between the probe and the boundary should be at least three times the distance between the outer contacts. It must be remarked that the error is purely geometrical and one is able to calculate this error exactly by solving the potential equation in the sheet. This problem has been tackled in several papers in the past.^2,3

In this paper we investigate the possibility of using the four-point probe method to measure the anisotropic properties of the electric conductivity of electroconductive textiles. Concerning the conductive layers of electroconductive textiles anisotropy, conduction is normally observed in the direction of the warp and weft. Intuitively, one expects to make a different measurement when the probe is oriented in different directions on an anisotropic conducting sheet. It will be proved both theoretically and experimentally that no anisotropy can be detected with the four-point probe with collinear contacts. It means that one always obtains the same result whatever the orientation on the anisotropic layer.

In order to have a reference value the Van Der Pauw method was used to measure the resistivities on thin conducting sheets.⁴ This method was originally developed for the thin conducting layers frequently used in electronics and microelectronics.⁵ The method was then extended to investigate anisotropic layers.^6,7 Recently, the Van Der Pauw method has been used to investigate electroconductive textile layers.^8–10 It was also proved that the same technique can be used for anisotropic electroconductive layers.¹¹

Only a few papers could be found in the literature dealing with this topic. A mathematical analysis was published in the past.¹² Experimental work was published more recently by Kanagawa et al.¹³ Anisotropic surface-state conductivity was created artificially by depositing In atom chains on a Si substrate. In this way a high anisotropic ratio of about 60 could be obtained.

For the experimental part of this paper, electroconductive layers were screen-printed onto textile substrates. The type of weave gives rise to different structures in the warp and weft directions, leading to anisotropic electric conductivities in the layer deposited on the fabric.¹¹ Anisotropic phenomena in textiles have been reported in several papers but limited to mechanical and optical properties.^14–20 Only a single paper related to anisotropic electric properties was found.²¹

From our investigation it will be clear that the collinear four-point probe should not be used in anisotropic layers; instead, other configurations such as the Van Der Pauw method should be used.

The four-point probe technique for isotropic layers

Consider an infinite conducting sheet with a uniform thickness t_s and an electrical conductivity σ. For practical purposes, it is more convenient to use the so-called square or sheet resistance R_□ defined by

R_{□} = 1 / σ t_{s}

(1)

R_□ is the resistance value measured on a square-shaped sample with two contacts along the opposite sides.

The four contacts of the probe labelled A, B, C and D are collinear and at an intermediate distance a (as shown in Figure 1). Contacts A and D are used to supply the current I whereas the voltage drop V will be sensed between B and C. If a current I is fed to contact A, the potential distribution in the sheet will be given by

Ø (x, y) = - \frac{R_{□} I}{2 π} \ln \sqrt{x^{2} + (y - \frac{3 a}{2})^{2}}

(2)

A similar expression can be written down for the current I extracted from contact D. The potential is then found by superposition

Ø (x, y) = \frac{R_{□} I}{2 π} [- \ln \sqrt{x^{2} + (y - \frac{3 a}{2})^{2}} + \ln \sqrt{x^{2} + (y + \frac{3 a}{2})^{2}}]

(3)

The voltage drop V is then obviously

V = Ø (0, \frac{a}{2}) - Ø (0, - \frac{a}{2}) = \frac{R_{□} I}{2 π} [- ln a + ln 2 a + ln 2 a - ln a] = \frac{R_{□} I}{π} ln 2

(4)

Hence

R_{□} = \frac{V}{I} \frac{π}{\ln 2} = 4.532 \frac{V}{I}

(5)

π/ln 2 = 4.532 is the so-called correction factor. Most equipment takes this correction automatically into account.²²

If the probes are too close to the boundary equation (5) is not valid any more. The potential distribution can still be found numerically for an arbitrary geometry. For a straight boundary an analytical approach is still possible by introducing image sources.⁵ The equation for the potential distribution is then more complicated than equation (3) and different correction factors are obtained.

The four-point probe technique for anisotropic layers

Anisotropy means that different resistances are measured in the x and the y directions, R_xx_,□ and R_yy_,□ respectively. Without loss of generality we can assume that the major axes of the resistivity tensor are along the x and y axes. Hence, the non-diagonal coefficient R_xy_,□ is then zero. For woven textile substrates the main axes usually coincide with the weft (x) and warp (y) directions because the physical properties differ in both directions due to the fabrication process (for example the number of yarns/cm).

Consider the layout shown in Figure 1. For an isotropic layer, the potential ∅ satisfies the Laplace equation

\nabla^{2} Ø = \frac{\partial^{2} Ø}{\partial x^{2}} + \frac{\partial^{2} Ø}{\partial y^{2}} = 0

(6)

Equations (2) and (3) are solutions of equation (6). In those cases where anisotropy exists, the Laplace equation is no longer valid and must be replaced by

\frac{1}{R_{xx, □}} \frac{\partial^{2} Ø}{\partial x^{2}} + \frac{1}{R_{yy, □}} \frac{\partial^{2} Ø}{\partial y^{2}} = 0

(7)

In order to solve equation (7), an affine transformation has to be used to map the (x,y) plane into the (x′, y′) one

x' = x y' = y \sqrt{\frac{R_{yy, □}}{R_{xx, □}}}

(8)

For the particular case that R_yy_,□ = 4R_xx_,□ one gets x′ = x and y′ = 2y. Hence, the four contacts will now be at a distance 2a from each other in the (x′,y′) plane.

Using the affine transformation (8) for equation (7), one gets

\frac{\partial^{2} Ø}{\partial x' 2} + \frac{\partial^{2} Ø}{\partial y' 2} = 0

(9)

which is the Laplace equation, and which is only valid in the (x′,y′) plane. Due to the transformation (8), the layer in the (x′,y′) plane has a uniform resistance R_xx_,□.

A current I is fed through the contacts A and D in the (x, y) plane. This current has to be transformed into a current I′ in the (x′, y′) plane. There is no a priori reason why I′ = I.

In order to find the exact relationshiip between I and I′, one has to start with the potential distribution due to a current I′ in the (x′, y′) plane

Ø (x', y') = - \frac{R_{xx, □} I'}{2 π} ln [\sqrt{x' 2 + y' 2}]

(10)

This solution is only valid for an infinite sheet. In practice it means that the probe should not be too close to the boundary. Without loss of generality, it was assumed in equation (10) that the current I′ was fed into the origin x′ = y′ = 0, whereas the second contact is at infinity. After transforming back to the (x,y) plane, one gets for the potential distribution

Ø (x, y) = - \frac{R_{xx, □} I'}{2 π} ln [\sqrt{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} y^{2}}]

(11)

In order to find the value of the current I′, we need to know the current densities in the x and y direction

J_{x} = - \frac{1}{R_{xx, □}} \frac{\partial Ø}{\partial x} = \frac{I'}{2 π t_{s}} \frac{x}{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} y^{2}}

(12)

J_{y} = - \frac{1}{R_{yy, □}} \frac{\partial Ø}{\partial y} = \frac{I'}{2 π t_{s}} \frac{y}{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} y^{2}}

(13)

We consider now a square-shaped region around the origin (side length = 2ɛ). The total current through the boundary of this square must be obviously equal to I, which can also be found by integrating the current densities (equations (12) and (13)). Due to symmetry it is sufficient to consider only one quarter of the region indicated by EFG in Figure 2. We first integrate the current density J_x along EF in order to obtain the partial current I_x

I_{x} = t_{s} \int_{E}^{F} J_{x} d y = \frac{I'}{2 π} \int_{0}^{ɛ} \frac{ɛ d y}{ɛ^{2} + \frac{R_{yy, □}}{R_{xx, □}} y^{2}} = \frac{I'}{2 π} \sqrt{\frac{R_{xx, □}}{R_{yy, □}}} \int_{0}^{\frac{R_{yy, □}}{R_{xx, □}}} \frac{d v}{1 + v^{2}} = \frac{I'}{2 π} \sqrt{\frac{R_{xx, □}}{R_{yy, □}}} arctan (\sqrt{\frac{R_{yy, □}}{R_{xx, □}}})

(14)

In a similar way the partial current I_y flowing through GF is evaluated as

I_{y} = t_{s} \int_{G}^{F} J_{y} d x = \frac{I'}{2 π} \int_{0}^{s} \frac{ɛ d x}{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} ɛ^{2}} = \frac{I'}{2 π} \sqrt{\frac{R_{xx, □}}{R_{yy, □}}} \arctan (\sqrt{\frac{R_{xx, □}}{R_{yy, □}}})

(15)

Taking into account that

arctan α + arctan 1 / α = π / 2

(16)

The sum of the two partial currents I_x and I_y is given by

I_{x} + I_{y} = \frac{I'}{2 π} \sqrt{\frac{R_{xx, □}}{R_{yy, □}}} [\arctan (\sqrt{\frac{R_{yy, □}}{R_{xx, □}}}) + \arctan (\sqrt{\frac{R_{xx, □}}{R_{yy, □}}})] = \frac{I'}{4} \sqrt{\frac{R_{xx, □}}{R_{yy, □}}}

(17)

Taking into account that I = 4(I_x + I_y), we finally obtain the relation between I and I′

I = I' \sqrt{\frac{R_{xx, □}}{R_{yy, □}}}

(18)

The potential distribution (equation (11)) is then

Ø (x, y) = - \frac{I}{2 π} \sqrt{R_{xx, □} R_{yy, □}} ln [\sqrt{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} y^{2}}]

(19)

Note that equation (19) is the potential distribution due to a current I injected in the origin, x = y = 0.

When a current I is injected in at A and extracted from D, one obtains the following potential distribution

Ø (x, y) = - \frac{I}{2 π} \sqrt{R_{xx, □} R_{yy, □}} [ln \sqrt{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} (y - \frac{3 a}{2})^{2}} - ln \sqrt{x^{2} + \frac{R_{yy, □}}{R_{xx, □}} (y + \frac{3 a}{2})^{2}}]

(20)

The potential drop between the voltage sensing contacts B and C is then obviously

V = Ø (0, \frac{a}{2}) - Ø (0, - \frac{a}{2}) = \frac{I}{π} ln 2 \sqrt{R_{yy, □} R_{xx, □}}

(21)

With the four-point probe method one will measure the value of

\sqrt{R_{yy, □} R_{xx, □}}

, notwithstanding the fact that the four contacts are aligned along the y axis or in any other direction.

If the four contacts are aligned along the x axis or in any other direction, one can easily calculate that the same voltage drop (equation (21)) will be obtained. Hence, the conclusion is that it is not possible to detect electrical anisotropy with the four-point probe method. This statement has only been proved if the four are collinear, which in practice is the most common configuration.

Even when the contacts are not equidistant but still collinear, the voltage drop V remains proportional to $\sqrt{R_{yy, □} R_{xx, □}}$ irrespective of the orientation of the probe.

The conclusion is that the four-point probe method can only be used to measure the geometric mean $\sqrt{R_{yy, □} R_{xx, □}}$ of the two resistance values R_xx_,□ and R_yy_,□.

Experimental results

For the experimental part of this work, electric conducting inks were screen-printed on several textile substrates. These conductive pastes are frequently used on epoxy and ceramic substrates for electronic and microelectronic applications. Three silver inks from SunChemical were applied onto textile materials such as cotton/polyester (CO/PES), cotton (CO), viscose (CV) and polyester (PES) (Table 1).

Table 1.

Properties of applied woven textiles

Material	Yarn density of fabric		Structure	Thickness (mm)	Basic weight (g/m²)
Material	Warp (threads/cm)	Weft (threads/cm)	Structure	Thickness (mm)	Basic weight (g/m²)
CO/PES	42	29	Twill 3/1	0.414	233
CO	47	26	Twill 4/1	0.468	283
CV	20	19	Plain 1/1	0.310	132
PES	57	46	Twill 2/1	0.146	104

The conducting layer deposited on the fabrics were all square-shaped with dimensions 6 × 6 cm² (Figure 3). The mesh used is HM 45-070 T 1/1 with a thickness of 115 µm, a sieve aperture of 148 µm and a sieve opening of 47%. The samples were cured in the oven at the temperature and time as given in the datasheet from the producer (Table 2). The curing process is essential in order to remove the organic solvents from the deposited ink and to guarantee a stable, conducting layer on the substrate.

Table 2.

Properties of applied silver-based conductive inks

Ink type	Ink 1	Ink 2	Ink 3
Solid content (%)*	69–71	70	84–86
Cure conditions (120℃)* (min)	2–10	30	2–15
Surface resistivity (Ω/□)*	0.011	<0.012	≤0.025
Resin†	PES	Vinyl chloride/ ester copolymer	Epoxy

Supplied by the producers; †determined by the authors

Four electroconductive contacts, P, Q, R and S, were provided in the corners for the resistivity measurements according to the Van Der Pauw method.^8,11

Firstly, a current I_PQ was supplied through the contacts P and Q and the voltage drop V_SR was measured. Secondly, a current I_PS was fed through P and S and the resulting voltage drop V_QR between Q and R was recorded. From the relations V_SR/I_PQ and V_QR/I_PS the sheet resistances R_xx_,□ and R_yy_,□ can be found.¹¹ In the case of anisotropic conductivity, V_SR/I_PQ ≠ V_QR/I_PS for a square-shaped sample. For the four-point probe measurements the probe was placed in the middle of the sample, once along the weft and once along the warp direction.

The results from the measurements are listed in Table 3. CO/PES 1 means the cotton/polyester textile with the conductive ink 1 screen-printed onto it.

Table 3.

Measured values of R_xx_,□ and R_yy_,□ according to the Van Der Pauw and four-point probe methods

Textile material	Van Der Pauw method		$\sqrt{R_{xx, □} R_{yy, □}}$ (Ω)	Four-point method
Textile material	R_xx_,□ (Ω)	R_yy_,□ (Ω)	$\sqrt{R_{xx, □} R_{yy, □}}$ (Ω)	x direction (Ω)	y direction (Ω)
CO/PES 1	0.014*	0.018	0.0158	0.01640	0.01691
CO/PES 2	0.007*	0.031	0.0147	0.01381	0.01355
CO/PES 3	0.035*	0.015	0.0229	0.01066	0.01117
CO 1	0.007*	0.023	0.0127	0.01436	0.01326
CO 2	0.035	0.029	0.0318	0.03234	0.03111
CO 3	0.028*	0.036	0.0317	0.02669	0.02718
CV 1	0.019*	0.014	0.0163	0.01649	0.01708
CV 2	0.039	0.042	0.0405	0.01321	0.01360
CV 3	0.007	0.009	0.0079	0.01062	0.01003
PES 1	0.026*	0.015	0.0197	0.01984	0.02010
PES 2	0.015*	0.009	0.0116	0.01236	0.01287
PES 3	0.009	0.007	0.0079	0.00833	0.00858

Electric anisotropy clearly shown.

The second and third columns of Table 3 display the experimental values of R_xx_,□ and R_yy_,□ according to the Van Der Pauw method. The results clearly show electric anisotropy, especially for those samples that are marked with an asterisk.

Four-point probe measurements were carried out on the same samples. The probe was oriented in the x (weft) and the y (warp) directions. These results are also listed in Table 3 and it is clear that there is almost no difference at all for most of the samples. Firstly, all of the four-point probe results give the same value (within 5%) in the x and y directions. Moreover the values are quite close to the value $\sqrt{R_{xx, □} R_{yy, □}}$ obtained from the Van Der Pauw measurements. These experimental results are in full agreement with the theoretical analysis outlined in the foregoing section.

Kanagawa et al.¹³ obtained values of 4.4 kΩ and 5.4 kΩ with the collinear probe arrangement. With the square-probe arrangement, the anisotropy could be detected, giving values of 0.17 kΩ and 10.9 kΩ (anisotropy ratio of 60). The numerical value of $\sqrt{R_{xx, □} R_{yy, □}}$ = $\sqrt{0.17 \times 10.9}$ = 1.32 kΩ, which is quite different from 4.4 kΩ or 5.4 kΩ. A possible explanation might be that the structure of In atom chains on a Si substrate cannot be modeled by a two-dimensional conducting sheet because the Si is conducting as well.

In the theoretical section we have proved that the collinear four-point probe method is not able to detect anisotropy.

Conclusion

In this paper a theoretical and experimental investigation of measuring the square resistance on anisotropic electroconductive textiles is presented. Two techniques are introduced: the collinear four-point probe and, used to give a reference value, the Van Der Pauw technique. It was proved theoretically and verified experimentally that no anisotropy can be detected with the collinear four-point probe technique.

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